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[**3d Abelian Gauge Theories at the Boundary**]{}
[**Lorenzo Di Pietro$^1$, Davide Gaiotto$^1$, Edoardo Lauria$^{2}$ and Jingxiang Wu$^1$\
**]{} ${}^{1}$ Perimeter Institute for Theoretical Physics,\
31 Caroline St N, Waterloo, ON N2L 2Y5, Canada ${}^{2}$ Centre for Particle Theory, Department of Mathematical Sciences\
Durham University, DH1 3LE, UK
`[email protected], [email protected], [email protected], [email protected]`\
A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling $\tau$ in the upper-half plane and by the choice of the CFT in the decoupling limit $\tau \to \infty$. Upon performing an $SL(2,\mathbb{Z})$ transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten’s $SL(2, \mathbb{Z})$ action [@Witten:2003ya]. In particular the cusps on the real $\tau$ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk $S$ transformation, and it also admits a decoupling limit which gives the $O(2)$ model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. We also consider examples with other theories on the boundary, such as large-$N_f$ Dirac fermions –for which the extrapolation to strong coupling can be done exactly order-by-order in $1/N_f$– and a free complex scalar.
Introduction {#sec:intro}
============
The objective of this paper is to study conformal invariant boundary conditions for free Abelian gauge theory in four-dimensions. A striking property of these BCFTs is that they are typically well-defined on some open patch in the space of the four-dimensional gauge coupling.
The simplest way to produce such boundary conditions is to couple the four-dimensional gauge fields to a three-dimensional CFT with a $U(1)$ global symmetry. This is sometimes called a “modified Neumann” boundary condition [@Gaiotto:2008sa]. Assuming that certain mild conditions are satisfied, one obtains a BCFT which is well-defined as long as the four-dimensional gauge coupling is sufficiently small [@Teber:2012de; @Kotikov:2013kcl; @Teber:2014ita; @Teber:2016unz; @Kotikov:2016yrn; @Herzog:2017xha; @Dudal:2018pta]. The conformal data of the BCFT can be computed from the data of the original CFT by perturbation theory in the four-dimensional gauge coupling.
Conversely, there is a general expectation that any BCFT $B$ defined at arbitrarily small 4d gauge coupling will be either a Dirichlet boundary condition or a modified Neumann boundary condition associated to some 3d CFT $T_\infty[B]$ with a $U(1)$ symmetry. Because of electric-magnetic duality, the same statement applies to any other “cusp” $C$ in the space of the complexified gauge coupling, where some dual description of the four-dimensional gauge field becomes arbitrarily weakly coupled. If the BCFT $B$ is defined around the cusp $C$, we can associate to it another 3d CFT $T_C[B]$, which is obtained from $T_\infty[B]$ by applying the $SL(2,\mathbb{Z})$ transformation [@Witten:2003ya] that maps the cusp at infinity to $C$. Therefore, the theories living at the other cusps can be thought of as 3d Abelian gauge theories obtained by gauging the $U(1)$ global symmetry of $T_\infty[B]$.
In the absence of phase transitions, a given BCFT $B$ can be defined on the whole space of 4d gauge couplings and is thus associated to an infinite family $T_*[B]$ of 3d CFTs. The conformal data of the BCFT will admit a similar collection of perturbative expansions in the neighbourhood of each cusp.
In the first part of this paper we study general properties of this family of BCFT’s. A universal feature is the presence in the spectrum of boundary operators of two conserved $U(1)$ currents, the electric and the magnetic currents, that arise as a consequence of the electric and magnetic one-form symmetries in the bulk [@Gaiotto:2014kfa]. The endpoints of bulk line operators carry charge under this $U(1)\times U(1)$ symmetry, while all the local boundary operators are neutral. By matching the bulk and boundary OPE expansions of correlators of the bulk field strength, we show that several BCFT observables –including non-local ones such as the free-energy on a hemisphere background– can be obtained in terms of the coefficients $c_{ij}$ in the two-point correlators of these currents, and of the coefficient $C_{\hat{D}}$ of the two-point function of the displacement operator. The latter relations hold for any $\tau$, provided $B$ exists. We also show that the leading perturbative corrections to $c_{ij}$ and $C_{\hat{D}}$ around a cusp are captured universally in terms of the two-point function of the $U(1)$ current of the 3d CFT living at the cusp, in the decoupling limit.
In the second part of this paper we turn these abstract considerations into a very concrete computational strategy: if some $T_C$ is simple enough for perturbative computations to be feasible, we may study the properties of other $T_*$ theories by re-summing the perturbation theory. If we happen to know, or conjecture, that there are two cusps $C$ and $C'$ such that $T_C$ and $T_{C'}$ are both simple, we may be able to implement an enhanced re-summation which uses both piece of data to predict the properties of the other $T_*$ theories.
This approach gives a new approximation scheme, orthogonal to previously known perturbative approaches to 3d Abelian gauge theories such as the $\epsilon$-expansion [@DiPietro:2015taa; @Giombi:2015haa; @Chester:2015wao; @Janssen:2017eeu; @DiPietro:2017kcd; @DiPietro:2017vsp; @Ji:2018emi; @Zerf:2018csr] or the large-$N$ expansion (see e.g. [@Giombi:2016fct; @Chester:2017vdh; @Gracey:2018fwq; @Benvenuti:2018cwd] for recent results and the review [@Gracey:2018ame]). We will apply this strategy to a very nice boundary condition for a $U(1)$ gauge theory, which is conjecturally associated to a free Dirac fermion at two distinct cusps and to the $O(2)$ model at two other cusps [@Seiberg:2016gmd; @Metlitski:2015eka; @Wang:2015qmt; @Hsiao:2017lch; @Hsiao:2018fsc]. The fact that these theories appear at the cusps can be seen as a consequence of the recently discovered 3d dualities [@Seiberg:2016gmd; @Aharony:2015mjs; @Karch:2016sxi], and it entails the existence of a $\mathbb{Z}_2$ action on $\tau$ that leaves $B(\tau,\bar{\tau})$ invariant. We will do a two-loop calculation at the free-fermion cusp and then extrapolate to the $O(2)$ cusp, finding good agreement with the known data of the $O(2)$ model.
We also consider other applications: Taking the boundary degrees of freedom to be an even number $2N_f$ of free Dirac fermions, setting the gauge coupling to $g^2 = \lambda / N_f$ and taking $N_f$ to infinity with $\lambda$ fixed, we argue that the theory admits a $1/N_f$-expansion, which interpolates between the free theory at $\lambda = 0$ and large-$N_f$ QED$_3$ at $\lambda = \infty$. The exact $\lambda$ dependence can be easily obtained order-by-order in the $1/N_f$ expansion. Applying the general strategy to compute the hemisphere partition function to this case, and taking the limit $\lambda \to \infty$, we obtain the $1/N_f$ correction to the sphere partition function of large-$N_f$ QED$_3$. Another example with a $\mathbb{Z}_2$ duality acting on $\tau$ is conjecturally obtained in the case where the theory on the boundary is a free complex scalar, or equivalently the $U(1)$ Gross-Neveu model [@Rosenstein:1990nm; @ZinnJustin:1991yn]. We consider perturbation theory around the free-scalar cusp, and show the existence of a stable fixed point for the classically marginal sextic coupling on the boundary at large $\tau$. We also discuss an example with two bulk gauge fields coupled to two distinct Dirac fermions on the boundary. We show how to obtain QED$_3$ with 2 fermionic flavors starting with this setup, using the extended electric-magnetic duality group $Sp(4,\mathbb{Z})$ that acts on the two bulk gauge fields.
Structure of the paper
----------------------
We start in section \[sec:mixedDef\] by reviewing the non-interacting boundary conditions for a Maxwell field in four dimensions. We then define the family of interacting boundary conditions $B(\tau,\bar{\tau})$. We derive the general relations that we described above for the bulk two- and three-point functions of the field strength, and obtain the leading corrections in perturbation theory around the cusps in the $\tau$ plane. In section \[sec:FreeEnergy\] we obtain similar results for a different observable, the hemisphere partition function of $B(\tau,\bar{\tau})$. In particular we show how to recover the $S^3$ partition function for the 3d CFTs in the decoupling limit. In section \[sec:MinPhTr\] we put this machinery at work in the example of the boundary condition defined by the $O(2)$ model / a free Dirac fermion. Section \[sec:exa\] contains the other applications that we consider: large-$N_f$ fermions, a complex scalar, and two bulk gauge fields coupled to two Dirac fermions. We conclude in section \[sec:conc\] by discussing some future directions. Several appendices include the details of calculations, and some supplementary material, e.g. a calculation of the anomalous dimension of the boundary stress-tensor using multiplet recombination in appendix \[app:fakestress\], and an explanation of the technique that we used to evaluate the two-loop integrals in appendix \[app:FeynInt\].
Boundary Conditions for 4d Abelian Gauge Field {#sec:mixedDef}
==============================================
Generalities
------------
Boundary Conformal Field Theories for a free $d$-dimensional bulk quantum field theory are interesting theoretical objects. On one hand, the correlation functions of bulk local operators are controlled by the free equations of motion. In particular, they are fully determined by their behaviour near the boundary, which is encoded in some very simple bulk-to-boundary OPE for the bulk free fields.
The free bulk-to-boundary OPE essentially identifies some special boundary local operators as the boundary values of the bulk free fields and their normal derivatives. The correlation functions of these boundary operators determine all correlation functions of bulk operators. These boundary correlation functions, though, can in principle be as complicated as those of any CFT in $(d-1)$ dimensions.
The case of four-dimensional free Abelian gauge theory (with compact gauge group) is particularly interesting because the bulk theory has an exactly marginal gauge coupling.[^1] Furthermore, a BCFT defined for some value of the bulk gauge coupling can typically be deformed to a BCFT defined at a neighbouring value of the bulk gauge coupling by conformal perturbation theory in the gauge coupling. The leading order obstruction is the presence of marginal boundary operators in the bulk-to-boundary OPE of the bulk Lagrangian operators $F^2$ and $F \wedge F$, which can lead to a logarithmic divergence as the bulk perturbation approaches the boundary. Generically, no such operators will be present and the BCFT can be deformed.
In this section, we will discuss the properties of some standard BCFT’s which can be defined in an arbitrarily weakly-coupled gauge theory, starting with free boundary conditions and then including interacting degrees of freedom at the boundary. On general grounds, we expect that any BCFT which can be defined at arbitrarily weak coupling will take this form.
Free Boundary Conditions and $SL(2,\mathbb{Z})$ Action {#sec:Free}
------------------------------------------------------
Consider a $U(1)$ gauge field $A_\mu$ on $\mathbb{R}^3 \times \mathbb{R}_+$. We adopt Euclidean signature, and use coordinates $x = (\vec{x},y)$ where $x^4\equiv y \geq 0$ is the coordinate on $\mathbb{R}_+$, and $\vec{x}$ are the coordinates on $\mathbb{R}^3$. We denote the components of $x$ as $x^\mu$, $\mu = 1,2,3,4$, and those of $\vec{x}$ as $x^a$, $a=1,2,3$. The field strength is $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, its Hodge dual is $\tilde{F}_{\mu\nu} = \tfrac{1}{2}\epsilon_{\mu\nu}^{~~\rho\sigma}F_{\rho\sigma}$ and the self-dual/anti-self-dual components are $F_{\mu\nu}^{\pm} = \tfrac{1}{2}(F_{\mu\nu} \pm \tilde{F}_{\mu\nu})$. They satisfy $\tfrac{1}{2}\epsilon_{\mu\nu}^{~~\rho\sigma} F^{\pm}_{\rho\sigma} = \pm F^{\pm}_{\mu\nu}$.
In the absence of interactions with boundary modes, by varying the action $$\begin{aligned}
S[A, \tau] & = \int_{y\geq 0} dy\,d^3\vec{x} \left(\frac{1}{4 g^2} F_{\mu\nu}F^{\mu\nu} + \frac{i \theta}{32 \pi^2}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\right) \\
& = -\frac{i}{8\pi}\int_{y\geq 0} dy\,d^3\vec{x}\left(\tau F^-_{\mu\nu}F^{-\,\mu\nu} - \bar{\tau} F^+_{\mu\nu}F^{+\,\mu\nu}\right)
~,\label{eq:actau}\end{aligned}$$ we find the bulk equation of motion $\frac{1}{g^2}\partial_\mu F^{\mu\nu} = 0$ and the boundary term $$\begin{aligned}
\delta S_{\partial} & = - \int_{y = 0} d^3\vec{x} \, \delta A^a \left(\frac{1}{g^2} F_{ya} + i \frac{\theta}{4 \pi^2}\tilde{F}_{y a} \right) \\ & = \frac{i}{2\pi} \int_{y = 0} d^3\vec{x} \, \delta A^a(\tau F^-_{ya} - \bar{\tau} F^+_{ya}) ~.\label{eq:bdtau}\end{aligned}$$ Our convention for the orientation is $\epsilon_{abc y} = \epsilon_{abc}$. In equations - we combined $g$ and $\theta$ in the complex coupling $\tau = \frac{\theta}{2\pi} + \frac{2\pi i }{g^2}$. From eq. we see that the possible boundary conditions for the gauge field when no boundary modes are present are
- [[Dirichlet:]{} $\delta A_a\vert_{y=0}=0$, which is equivalent to $$(F^-_{ya} - F^+_{ya})\vert_{y=0} = - \tilde{F}_{ya}\vert_{y=0}=0~;\label{eq:Dbc}$$]{}
- [[Neumann:]{} $$(\tau F^-_{ya} - \bar{\tau} F^+_{ya})\vert_{y=0} = 0~.\label{eq:Nbc}$$ Equivalently, introducing $$\gamma = \frac{\mathrm{Re}\tau}{\mathrm{Im}\tau}= \frac{\theta\,g^2}{4\pi^2}\in \mathbb{R}~,\label{eq:defgamma}$$ we can write this condition as $
(F_{ya} +i \gamma \tilde{F}_{ya})\vert_{y=0}=0
$, in particular for $\gamma = 0$ it simplifies to the standard Neumann condition $
F_{ya}\vert_{y=0}=0
$. ]{}
It is convenient to introduce the boundary currents $$\begin{array}{ll}
2 \pi i\hat{J}_a & = \tau F^-_{ya}(\vec{x},y=0) -\bar{\tau} F^+_{ya}(\vec{x},y=0) ~, \cr 2\pi i \hat{I}_a & = F^-_{ya}(\vec{x},y=0)-F^+_{ya}(\vec{x},y=0)~.
\end{array}\label{eq:IJ}$$ in terms of which the Dirichlet condition is $\hat{I} = 0$, and the Neumann condition is $\hat{J} = 0$.
On $\mathbb{R}^4$ this theory enjoys an $SL(2,\mathbb{Z})$ duality group $$\tau \to \tau' = \frac{a \tau + b}{c \tau + d}\,,\quad a,b,c,d \in \mathbb{Z}\,,~ad-bc = 1~.$$ The duality group acts on the fields as $$\begin{array}{ll}
F^-_{\mu\nu} & \to F^{'-}_{\mu\nu}=( c \tau + d) F^-_{\mu\nu}~,\cr
F^+_{\mu\nu} & \to F^{'+}_{\mu\nu} =(c \bar{\tau} + d) F^+_{\mu\nu}~.
\end{array}\label{eq:ftr}$$ When the boundary is introduced, the group $SL(2,\mathbb{Z})$ also acts on the boundary conditions. From we see that the action on the boundary currents is $$\begin{array}{ll}
\hat{J}_a & \to a \hat{J}_a + b \hat{I}_a~,\cr
\hat{I}_a & \to c \hat{J}_a + d \hat{I}_a~.
\end{array}\label{eq:IJtr}$$
The Dirichlet and Neumann boundary conditions above are exchanged under the $S$ transformation $\tau\to-\frac{1}{\tau}$, i.e. electric-magnetic duality. Indeed, the $S$ transformation exchanges $\hat{J}$ and $\hat{I}$.
However, comparing eq.s - and eq.s - we see that the general $SL(2,\mathbb{Z})$ transformation does not act within the set of boundary conditions that we described above. This is because we assumed that no degrees of freedom are present on the boundary, while the generic $SL(2,\mathbb{Z})$ transformation requires the introduction of topological degrees of freedom on the boundary, namely 3d gauge-fields with Chern-Simons (CS) actions, coupled to the bulk gauge field through a topological $U(1)$ current [@Witten:2003ya; @Gaiotto:2008ak; @Kapustin:2009av]. Note that even in the presence of these topological degrees of freedom the theory is still free, because the action is quadratic. Taking this into account, one finds that the most general free boundary condition for the $U(1)$ gauge field is $$p \hat{J}_a + q \hat{I}_a = 0~, \label{eq:Rbc}$$ where $p,q \in \mathbb{Z}$. This set of boundary conditions is closed under the action of $SL(2,\mathbb{Z})$. We will refer to this more general free boundary condition as “$(p,q)$ boundary condition”. The $(0,1)$ and $(1,0)$ boundary conditions correspond to the Dirichlet and Neumann boundary conditions above, respectively.
When we impose the $(p,q)$ condition, the unconstrained components of the gauge fields give a current operator on the boundary $$p^\prime \hat{J}_a + q^\prime \hat{I}_a$$ with $pq^\prime - p^\prime q =1$, whose correlators are just computed by Wick contraction, i.e. the boundary theory is a mean-field theory for this current. We can always shift $(p',q')$ by a multiple of $(p,q)$, and this gives rise to the same current thanks to the boundary condition.
Since the above boundary conditions preserve conformal symmetry, we can regard this system as a free boundary conformal field theory, and rephrase the boundary conditions in terms of a certain bulk-to-boundary OPE of the field strength $F_{\mu\nu}$. Using the equation of motion and the Bianchi identity one finds that the only primary boundary operators that can appear in the bulk-to-boundary OPE of $F_{\mu\nu}$ are conserved currents, see appendix \[app:detailsBulkDefOPE\] for a derivation. The free boundary conditions described above correspond to having only one conserved current in this OPE, that can be identified with $p^\prime \hat{J}_a + q^\prime \hat{I}_a$. For instance, for the Dirichlet $(0,1)$ boundary condition $$F_{\mu\nu}(\vec{x}, y) \underset{y\to 0}{\sim} - g^2 \hat{J}^a(\vec{x})2\delta_{a[\mu }\delta_{\nu] y}+\dots~,\label{eq:freeOPE}$$ where the dots denote subleading descendant terms, and the square brackets denote antisymmetrization. The general $(p,q)$ case can be obtained from the Dirichlet case by acting with an $SL(2,\mathbb{Z})$ transformation -.
Two-point Function in the Free Theory {#sec:FreeTwoP}
-------------------------------------
In this section we compute the two-point function $\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle$ on $\mathbb{R}^3\times \mathbb{R}_+$ in the free theory. We use that the two-point function is a Green function, i.e. it satisfies the equations of motion $$\frac{1}{g^2}\partial_\mu \langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle = (\delta_{\nu\sigma} \partial_\rho - \delta_{\nu\rho} \partial_\sigma) \delta^4(x_{12})~,$$ and the Bianchi identity $$\epsilon_{\tau\lambda\mu\nu}\partial_\lambda \langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle = 0~.$$ on $y \geq 0$, and it also satisfies the boundary conditions at $y=0$. We are denoting $x_{12} \equiv x_1 - x_2$.
To start with, the Green function on $\mathbb{R}^4$ (i.e. without a boundary) is $$\begin{aligned}
\label{eq:noboundaryF}
\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4} & = \frac{g^2}{\pi^2} G_{\mu\nu,\rho\sigma}(x_{12})~,\\
G_{\mu\nu,\rho\sigma}(x) & \equiv \frac{I_{\mu\rho}(x)I_{\nu\sigma}(x)- I_{\nu\rho}(x)I_{\mu\sigma}(x) }{(x^2)^2}~,\end{aligned}$$ where $I_{\mu\nu}(x)=\delta_{\mu\nu}- \frac{2x_\mu x_\nu}{x^2}$. Starting from and using the method of images we can easily write down the two-point function in the presence of the boundary. The calculation is showed in the appendix \[app:Imag\].
In the case $\gamma = 0$ we find $$\begin{aligned}
\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^3 \times \mathbb{R}_+} & = \frac{g^2}{\pi^2}\left[(1 - s \,v^4 )\, G_{\mu\nu,\rho\sigma}(x_{12}) + s \,v^4 \,H_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)\right]~,\label{eq:NDIm}\\
H_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2) & \equiv 2\frac{1}{(x^2)^2}\left[X_{1\,\mu}X_{2\,\rho}I_{\nu\sigma}(x_{12}) + X_{1\,\nu}X_{2\,\sigma}I_{\mu\rho}(x_{12}) \right. \nonumber \\ & \left. - X_{1\,\mu}X_{2\,\sigma}I_{\nu\rho}(x_{12}) - X_{1\,\nu}X_{2\,\rho}I_{\mu\sigma}(x_{12})\right]~,\end{aligned}$$ for Dirichlet ($s=1$) and Neumann ($s=-1$) conditions. Here $X_{i\,\mu}$ are the conformally covariant vectors [@McAvity:1995zd] $$X_{i\,\mu} \equiv y_i \frac{v}{\xi} \partial_{i\,\mu} \xi = v\left(2 \,\frac{y_i \,s_i \,x_{12\,\mu}}{x_{12}^2}- n_\mu\right)~,\quad i =1,2~,\quad s_1 = - s_2 = 1~,$$ and $\xi$ is the conformally invariant cross-ratio $$\xi \equiv \frac{x_{12}^2}{4y_1y_2}\equiv \frac{v^2}{1-v^2}~.$$
For the more general Neumann boundary condition with $\gamma \neq 0$ we find $$\begin{aligned}
\label{eq:imagesFF}
\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^3 \times \mathbb{R}_+} & = \frac{g^2}{\pi^2} \left[\left(\delta_{[\rho}^{\rho'}\delta_{\sigma]}^{\sigma'} + \,v^4\left( \frac{1-\gamma^2}{1+\gamma^2} \delta_{[\rho}^{\rho'}\delta_{\sigma]}^{\sigma'} - i \frac{\gamma}{1+\gamma^2} \epsilon_{\rho\sigma}^{~~~\rho'\sigma'}\right) \right)\, G_{\mu\nu,\rho'\sigma'}(x_{12})\right.\nonumber\\ & -\,v^4 \,\left.\left( \frac{1-\gamma^2}{1+\gamma^2} \delta_{[\rho}^{\rho'}\delta_{\sigma]}^{\sigma'} - i \frac{\gamma}{1+\gamma^2} \epsilon_{\rho\sigma}^{~~~\rho'\sigma'}\right) H_{\mu\nu,\rho'\sigma'}(\vec{x}_{12},y_1,y_2)\right]~.\end{aligned}$$ Even though not manifest, it can be verified that Bose symmetry is satisfied in this expression. From now on we will drop the subscript $\mathbb{R}^3 \times \mathbb{R}_+$.
It is also useful to rewrite this two point function in terms of the selfdual/antiselfdual components. The selfdual/antiselfdual projectors are $$P_{\mu\nu}^{\pm~\rho\sigma} = \tfrac 12 (\delta_{[\mu}^{\rho}\delta_{\nu]}^{\sigma} \pm \tfrac 12 \epsilon_{\mu\nu}^{~~~\rho\sigma})~.$$ We introduce the following notation $$\begin{aligned}
G^{\pm,\pm}_{\mu\nu,\rho\sigma} & \equiv P_{\mu\nu}^{\pm~\mu'\nu'}P_{\rho\sigma}^{\pm~\rho'\sigma'}G_{\mu'\nu',\rho'\sigma'}~,\\
G^{\pm,\mp}_{\mu\nu,\rho\sigma} & \equiv P_{\mu\nu}^{\pm~\mu'\nu'}P_{\rho\sigma}^{\mp~\rho'\sigma'}G_{\mu'\nu',\rho'\sigma'}~,\end{aligned}$$ and similarly for the structure $H$. The following identities hold $$\begin{aligned}
G^{\pm,\pm} & = 0~, \\
G^{\pm,\mp} - H^{\pm,\mp} & = 0~.\end{aligned}$$ Recalling the definition of $\gamma$, we obtain $$\begin{aligned}
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^+(x_2)\rangle & = \frac{2}{\pi\,\mathrm{Im}\tau}\frac{\tau}{\bar{\tau}} \, v^4 H^{++}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^-(x_2)\rangle & =\frac{2}{\pi\,\mathrm{Im}\tau}\frac{\bar{\tau}}{\tau} \, v^4 H^{--}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^-(x_2)\rangle & = \frac{2}{\pi\,\mathrm{Im}\tau} G^{+-}_{\mu\nu,\rho\sigma}(x_{12})~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^+(x_2)\rangle & = \frac{2}{\pi\,\mathrm{Im}\tau}G^{-+}_{\mu\nu,\rho\sigma}(x_{12})~.\end{aligned}$$ The result above is the field-strength two-point function in the free theory with Neumann boundary conditions. As we argued in section \[sec:Free\], the result for the $(p,q)$ boundary conditions simply follows from an $SL(2,\mathbb{Z})$ transformation -. As an example, for Dirichlet boundary conditions one finds $$\begin{aligned}
\label{FFDirich}
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^+(x_2)\rangle & = \frac{2|\tau|^2}{\pi\,\mathrm{Im}\tau}{}{} \, v^4 H^{++}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^-(x_2)\rangle & =\frac{2|\tau|^2}{\pi\,\mathrm{Im}\tau}{}{} \, v^4 H^{--}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^-(x_2)\rangle & = \frac{2 |\tau|^2}{\pi\,\mathrm{Im}\tau} G^{+-}_{\mu\nu,\rho\sigma}(x_{12})~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^+(x_2)\rangle & = \frac{2|\tau|^2}{\pi\,\mathrm{Im}\tau}G^{-+}_{\mu\nu,\rho\sigma}(x_{12})~.\end{aligned}$$
Coupling to a CFT on the Boundary {#sec:Interaction}
---------------------------------
Consider now a 3d CFT living on the boundary at $y=0$. We assume that the CFT has a $U(1)$ global symmetry, with associated current $\hat{J}_{\text{CFT}\,a}$. We take the Neumann boundary condition for the gauge field, which corresponds to a mean-field current operator $\hat{I}_a$ on the boundary. The two sectors can be coupled in a natural way, simply by gauging the $U(1)$ symmetry via the $y\to 0$ limit of the bulk gauge field. This amounts to adding the boundary coupling $$\int_{y=0} d^3 \vec{x} \, \hat{J}_{\text{CFT}}^a A_a + \text{seagulls}~,$$ and restricting the spectrum of local boundary operators to the $U(1)$ invariant ones. Charged boundary operators can be made gauge-invariant by attaching to them bulk Wilson lines. Therefore, it still makes sense to consider them after the gauging, but as endpoints of line operators rather than as local boundary operators.
The boundary coupling modifies the boundary condition of the gauge field to the “modified Neumann” condition $$\label{eq:modNeu}
\hat{J}_a \equiv \hat{J}_{\text{CFT}\,a} ~.$$ Hence as a consequence of the interactions both $\hat{I}_a$ and $\hat{J}_a$ are nontrivial operators.
As we explained above $\tau$ is an exactly marginal coupling, but we should worry about quantum effects breaking the boundary conformal symmetry by generating beta functions for boundary interactions. If the original 3d CFT has no marginal operators, these boundary beta functions start at linear order in the coupling and can be cancelled order-by-order in perturbation theory by turning on extra boundary interactions of order $\tau^{-1}$.[^2] Barring other non-perturbative phenomena such as the emergence of a condensate, we expect a BCFT to exist for sufficiently large $\tau$, with conformal data perturbatively close to that of the original CFT. We denote this BCFT with $B(\tau, \bar{\tau})$.
If the original 3d CFT has marginal operators the situation is more subtle: turning on boundary couplings $\hat{\lambda}$ will produce a beta function of order $\hat{\lambda}^2$ for the marginal operators. This may or not have the correct sign to cancel the $\tau^{-1}$ contributions. If it does not, we do not expect any unitary BCFT to exist, though one may be able to produce some non-unitary “complex” BCFT with complex couplings.
Conversely, suppose that we are given a BCFT $B(\tau,\bar \tau)$ defined continuously for arbitrarily weak gauge coupling. If $B(\tau,\bar \tau)$ is an interacting boundary condition, we expect that if we take the gauge coupling to $0$ the properties of $B(\tau,\bar \tau)$ will approach those of a 3d CFT with a $U(1)$ global symmetry.
As we will discuss later in this section, the bulk correlation functions are determined by the boundary correlation functions of the two conserved boundary current $\hat{I}_a$ and $\hat{J}_a$ defined in eq. . Due to the boundary condition , at weak coupling, $\hat{J}_a$ is inherited from the boundary degrees of freedom and the corresponding charge is carried by the endpoints of bulk Wilson lines ending at the boundary. On the other hand, $\hat{I}_a$ is analogue to the “topological” charge in three-dimensional $U(1)$ gauge theories and the corresponding charge is carried by the endpoints of bulk ’t Hooft lines ending at the boundary.
When the coupling is turned off, the conformal dimension of endpoints of ’t Hooft lines blows up and the $\langle \hat{I}_a \hat{I}_a \rangle$ correlation functions go to zero. The $\hat{I}_a$ current decouples from the BCFT correlation functions as they collapse to the correlation functions of the underlying 3d CFT $T_{0,1}[B]$ (this is the CFT that we denoted with $T_\infty[B]$ in the introduction).
Boundary Propagator of the Photon
---------------------------------
In order to compute corrections to boundary correlators and beta functions of boundary couplings in perturbation theory at large $\tau$, we need the propagator of the gauge field between two points on the boundary. Since we are perturbing around the decoupling limit, this can be readily obtained from the knowledge of the two-point function in the free theory . Recall from the discussion around eq. that in the free theory $F_{\mu\nu}$ has a non-singular bulk-to-boundary OPE. So the boundary two-point function of the operator $F_{ab}$ is obtained by specifying the indices to be parallel in eq. , and then taking the limit in which both insertion points approach the boundary. When taking this limit, we need to pay attention to possible contact terms that can arise due the following nascent delta-functions $$\frac{y}{(y^2 + \vec{x}^2)^2}\underset{y\to 0}{\longrightarrow} \pi^2 \delta^3(\vec{x})~,$$ and its derivatives. Even though usually we only compute correlators up to contact terms, these kind of contact terms in the two-point functions of 3d currents do actually contain physical information [@Closset:2012vp]. In this context, they encode the $\theta$-dependence of the boundary two-point function of $F_{ab}$. Relatedly, they are also needed to obtain the correct boundary propagator of the photon.
To obtain the $(ab,cd)$ components of the two-point function we need the components $(ab,cd)$ and $(y a^\prime, cd)$ of the structures $G$ and $H$. The structure $G$ gives $$\begin{aligned}
\left(1 + \,v^4\left(\frac{1-\gamma^2}{1+\gamma^2} \right)\right)\, G_{ab,cd}(x_{12})& \underset{y_{1,2}\to 0}{\longrightarrow} \frac{2}{1+\gamma^2}\,G^{\rm 3d}_{ab,cd}(\vec{x}_{12})~,\\
- 2 v^4 \frac{\gamma}{1+\gamma^2}i\epsilon_{ab}^{~~y a'}\, G_{ya',cd}(x_{12}) &\underset{y_{1,2}\to 0}{\longrightarrow} -\frac{2\gamma}{1+\gamma^2} i \,\pi^2\epsilon_{ab [c}(\partial_{\vec{x}_{12}})_{d]}\delta^3(\vec{x}_{12})~.\end{aligned}$$ Here $G^{\rm 3d}_{ab,cd}$ denotes the same structure as in eq. with the replacement of $I_{\mu\nu}$ by the 3d analogue $$\label{eq:I3d}
I^{\rm 3d}_{ab}(\vec{x}) \equiv \delta_{ab} -\frac{2 x^a x^b}{\vec{x}^2}~.$$ On the other hand the only non-zero component of the structure $H$ in the limit $y_{1,2}\to 0$ is $H_{ya,yb}$, hence the $H$ structure completely drops in the calculation of the propagator. The result is $$\langle F_{ab}(\vec{x}_1,0) F_{cd}(\vec{x}_2,0)\rangle = \frac{g^2}{\pi^2}\left[\frac{2}{1+\gamma^2}\,G^{\rm 3d}_{ab,cd}(\vec{x}_{12})-\frac{2\gamma}{1+\gamma^2} i \,\pi^2\epsilon_{ab [c}(\partial_{\vec{x}_{12}})_{d]}\delta^3(\vec{x}_{12})\right]~.\label{eq:twopbpos}$$
It is convenient to go to momentum space, by applying a Fourier transform with respect to the boundary coordinates $$\langle F_{ab}(\vec{x}_1,0) F_{cd}(\vec{x}_2,0)\rangle \equiv \int \frac{d^3 \vec{p}}{(2\pi)^3} \langle F_{ab}(\vec{p},0) F_{cd}(-\vec{p},0)\rangle e^{i \vec{p}\cdot \vec{x}_{12}}~.$$ We obtain $$\langle F_{ab}(\vec{p},0) F_{cd}(-\vec{p},0)\rangle = \frac{2g^2}{1+\gamma^2}\left[|\vec{p}\,|\left( \frac{\delta_{a[c}p_{d]}p_b}{\vec{p\,}^2} -\frac{\delta_{b[c}p_{d]}p_a}{\vec{p\,}^2}\right) + \gamma \epsilon_{ab[c} p_{d]}\right]~.\label{eq:twopbmom}$$ We can finally determine the propagator of the gauge field between two-points in the boundary by imposing that the exterior derivative reproduces the two-point function . The result is $$\langle A_{a}(\vec{p},0) A_{b}(-\vec{p},0)\rangle \equiv \Pi_{ab}(\vec{p}\,) = \frac{g^2}{1+\gamma^2}\left[\frac{\delta_{ab} -(1-\xi)\frac{p_ap_b}{\vec{p\,}^2}}{|\vec{p\,}|} + \gamma \epsilon_{abc} \frac{p^{c}}{\vec{p\,}^2}\right]~.\label{eq:prop}$$ The parameter $\xi$ is not fixed by requiring consistency with eq. , and parametrizes a choice of gauge. From the structure of the propagator we see that the natural perturbative limit is $g^2 \to 0$ with $\gamma$ fixed, which means $\tau \to \infty$ with a fixed ratio $\gamma$ between the real and the imaginary part. Observables are expressed as a power series in $\frac{g^2}{1+\gamma^2}$ with coefficients that are themselves polynomials in $\gamma$, more precisely the coefficient of the order $\mathcal{O}\left(\left(\frac{g^2}{1+\gamma^2}\right)^n\right)$ is a polynomial in $\gamma$ of degree $n$.
### Relations to Large-$k$ and Large-$N_f$ Perturbation Theories {#sec:largeNlargek}
Recall that a 3d Abelian gauge field $a$ with CS action $i \frac{k}{4\pi}\int a \wedge da$ has propagator (up to gauge redundancy) $$\langle a_{a}( \vec{p}) a_{b}( -\vec{p})\rangle = \frac{2\pi}{k}\epsilon_{abc} \frac{p^{c}}{\vec{p\,}^2}~.$$ We see that the contact term in eq. produced a term in the boundary propagator that is identical to the CS one. In particular, from the perturbation theory that we will consider one can immediately recover results for large-$k$ perturbation theory in Abelian 3d gauge theories, simply by setting (recall that $\gamma = \frac{g^2 \theta}{4\pi^2}$) $$\begin{aligned}
\left(\frac{g^2}{1+\gamma^2}\right)^n \gamma^m & \longrightarrow 0~,~~\text{if $m<n$} \\
\left(\frac{g^2}{1+\gamma^2}\right)^n \gamma^n & \longrightarrow \left(\frac{2\pi}{k}\right)^n~.\end{aligned}$$ Indeed, in the limit $g^2 \to \infty$ only the $\theta$-term is left in the bulk action, and the model that we are considering is equivalent to a CS theory on the boundary, with $k = \frac{\theta}{2\pi}$. The only role played by the bulk in this case is to allow generic real values of the CS coupling.
We can also compare to the limit of large number of matter flavors $N_f$, in which observables at the IR fixed point of 3d Abelian gauge theories can be computed perturbatively in $1/N_f$. In this regime, after resumming bubble diagrams, one finds the following “effective” propagator (again, up to gauge redundancy) $$\langle a_{a}(\vec{p}) a_{b}(-\vec{p})\rangle \sim \frac{1}{N_f}\frac{\delta_{ab}}{|\vec{p}\,|}~.$$ The proportionality constant depends on the details of the theory. The resulting “non-local” propagator has precisely the same form of the boundary propagator in the case $\gamma = 0$.[^3] Hence, once again, the two types of perturbation theories inform each other, and results for one case can be applied in the other case as well. Compared to the large-$k$ perturbation theory, here additional care is needed, because the order at which we are computing a certain observable in the $1/N_f$-expansion does not coincide with the number of internal photon lines in the corresponding diagram, owing to the fact that diagrams with a larger number of internal photon lines can get an enhancement by a positive power of $N_f$ from loops of matter fields. Nevertheless, single diagrams computed in one context can be used in the other context, and we will see an application of this observation later. A generalization of the large-$N_f$ limit is obtained by taking both $N_f$ and $k$ large, with a fixed ratio, and was studied recently in [@Chester:2017vdh]. In this case one finds a propagator that contains both terms in eq. , and the same comments about the relation of the two types of perturbation theory apply.
Exploring Strong Coupling {#sec:Strong}
-------------------------
As the coupling is increased, the two currents $\hat{I}_a$ and $\hat{J}_a$ should be treated on an even footing. Indeed, they are rotated into each other by the $SL(2,\mathbb{Z})$ group of electric-magnetic dualities of the bulk theory. Assuming no phase transitions, as we approach cusps $\tau \to -\frac{q}{p}$ where the dual gauge coupling becomes weak in some alternative duality frame, we expect dual statements to be true: the $p \hat{J} + q \hat{I}$ current should decouple from the BCFT correlation functions as they collapse to the correlation functions of a new 3d CFT $T_{p,q}[B]$, which gives the dual weakly coupled description of the original BFCT.
Using the notion of duality walls [@Gaiotto:2008ak; @Kapustin:2009av], one can argue that $T_{p,q}$ should be obtained from $T_{0,1}$ by Witten’s $SL(2,\mathbb{Z})$ action on 3d CFTs equipped with a $U(1)$ global symmetry [@Witten:2003ya]. This involves coupling $T_{0,1}$ to a certain collection of 3d Abelian gauge fields with appropriate Chern-Simons couplings. This statement requires some care and several caveats about the absence of phase transitions as we vary $\tau$.
In an optimal situation where these phase transitions are absent, this picture implies that the data of $B(\tau,\bar \tau)$ will approach the data of an infinite collection of 3d CFTs $T_{p,q}$ as $\tau \to - \frac{q}{p}$, sitting in the same universality classes as certain 3d Abelian gauge theories coupled to $T_{0,1}$. This is depicted in fig. \[fig:Tpq\]. If we “integrate out” the bulk and restrict our attention to the 3d boundary, what we just described can be stated as the existence of a family of non-local 3d conformal theories (i.e. with no stress-tensor in the spectrum) that continuously interpolate between different local 3d CFTs. More precisely, in the decoupling limit the 3d theory is a direct product of a 3d CFT and a non-local sector associated to the boundary condition of the free bulk field. This is reminiscent of the construction of [@Paulos:2015jfa; @Behan:2017emf; @Behan:2017dwr] in the context of the long-range Ising model.
![The family of conformal boundary conditions $B(\tau,\bar{\tau})$ labeled by the variable $\tau$ in the upper-half plane and by a 3d CFT $T_{0,1}$ with $U(1)$ global symmetry. At the cusp at infinity the current $\hat{I}^a$ decouples and we are left with the local 3d theory $T_{0,1}$ on the boundary, with $U(1)$ current $\hat{J}^a$. Approaching this cusp from $T$-translations of the fundamental domain amounts to adding a CS contact term to the 3d theory, or equivalently to redefine the current $\hat{J}^a$ by multiples of the current $\hat{I}^a$ that is decoupling. This is the $T$ operation on $T_{0,1}$ in the sense of [@Witten:2003ya]. In the favorable situation in which no phase transitions occur, the BCFT continuously interpolate to the cusps at the rational points of the real axis $\tau = -q/p$, where again the bulk and the boundary decouple and we find new 3d CFTs $T_{p,q}$. These theories are obtained from $T_{0,1}$ with a more general $SL(2,\mathbb{Z})$ transformation, that involves coupling the original $U(1)$ global symmetry to a 3d dynamical gauge field. \[fig:Tpq\]](fig/Tpq1.pdf){height="7cm"}
Let us mention a possible mechanism for a phase transition. As we change continuously $\tau$ from the neighbourhood of the “ungauged cusp” $T_{0,1}$ towards the “gauged cusps” $T_{p,q}$, the dimension of boundary operators are nontrivial functions of $\tau$. A scalar boundary operator $\hat{O}$ might become marginal at a certain codimension 1 wall in the $\tau$-plane. This possibility is depicted in fig. \[fig:Tpqphasetransition\].
![A cartoon of a possible phase transition at strong coupling. A scalar boundary operator becomes marginal at a certain curve in the $\tau$ plane, i.e. setting $\hat{\Delta}(\tau,\bar{\tau}) = 3$ we find solutions in the upper-half plane. In conformal perturbation theory from a point on the curve, the beta function takes the form . We might be unable to find real fixed points for the marginal coupling. In such a situation, $B(\tau,\bar{\tau})$ can only be defined as a complex BCFT. Assuming that we were able to define $B(\tau,\bar{\tau})$ as a real BCFT in perturbation theory around $\tau \to \infty$ by continuity such a real BCFT is ensured to exist in the full region above the wall, but we might be unable to continue it beyond the wall without introducing complex couplings (or breaking conformality). \[fig:Tpqphasetransition\]](fig/Tpq2.pdf){height="7cm"}
In perturbation theory in the vicinity of the wall, we can repeat the logic that we used in the subsection \[sec:Interaction\] when discussing perturbation theory around $T_{0,1}$ in presence of boundary marginal operators. Namely, the boundary marginal coupling $\hat{\lambda}$ will generically have a non-trivial beta function, which depends both on $\hat{\lambda}$ and $\tau$, and whose leading contributions are [^4] $$\label{eq:betamarg}
\beta_{\hat{\lambda}}(\tau,\bar{\tau},\hat{\lambda}) = b_{(F^-)^2, \hat{O}}\, \delta\tau + b_{(F^+)^2, \hat{O}}\, \delta{\bar{\tau}} + C_{\hat{O}\hat{O}\hat{O}} \hat{\lambda}^2 + \dots ~.$$ Here we are perturbing around a point $\tau_0$ on the wall, the coefficient $b$’s and $C$ are (up to numerical factors) the bulk-to-boundary OPE coefficients [@Karch:2018uft], and the OPE coefficient of the boundary conformal theory, respectively. These OPE coefficients are functions of $\tau_0$. Depending on $\tau_0$ and on the various OPE coefficients, setting $\beta_{\hat{\lambda}} =0$ one might or might not be able to find a real solution for $\hat{\lambda}$. If a real solution can be found perturbing away from the wall in a certain direction, by continuity $B(\tau,\bar{\tau})$ defines a real BCFT in a region of the $\tau$ plane on that side of the wall. Otherwise, on a side of the wall $B(\tau,\bar{\tau})$ exists only as a non-unitary “complex” BCFT.
Two-point Function from the Boundary OPE {#sec:twopointint}
----------------------------------------
In section \[sec:FreeTwoP\] we computed the two-point function of the field strength in free theory using the method of images. We will now compute it in the more general case with interactions on the boundary. We will see that it can be fixed completely in terms of the coefficient of the two-point function of the boundary currents. The method that we will use is an explicit resummation of the bulk-to-boundary OPE.
As a consequence of the interaction, the bulk-to-boundary OPE of the field strength contains two independent primary boundary operators, both of them conserved currents, rather than just one like in the free case. The leading terms in this OPE are $$\label{dOPEF}
F_{\mu\nu}(\vec{x}, y) \underset{y\to 0}{\sim}\hat{V}_1^a(\vec{x})2\delta_{a[\mu }\delta_{\nu] y}- i\epsilon^{abc}\hat{V}_{2\,c}(\vec{x})\delta_{a[\mu}\delta_{\nu] b}+\dots~.$$ The complete form of the above (including all descendants) can be found in . The boundary currents $\hat{V_1}$ and $\hat{V}_2$ can be expressed in terms of $\hat{J}^a$ and $\hat{I}^a$ as follows $$\begin{aligned}
\hat{V}_1^a & = -g^2\left(\hat{J}^a-\frac{\theta}{2\pi}\hat{I}^a\right)~,\\
\hat{V}_2^a & = -2\pi \hat{I}^a ~.\end{aligned}$$ If the 3d CFT that the gauge field couples to has parity symmetry (i.e. symmetry under reflection of one of the coordinates) then the full boundary CFT $B(\tau,\bar{\tau})$ admits such a symmetry when restricted to $\theta = 0$. Under this symmetry $V_1$ transforms like an ordinary vector, while $V_2$ transforms like an axial vector. We can extend this symmetry to the more general case $\theta\neq 0$ by viewing it as a spurionic symmetry that flips the sign of $\theta$.
Plugging the bulk-to-boundary OPE in the two-point function, one obtains the *boundary channel* decomposition. In this case, since only two boundary primaries appear in the OPE, we can explicitly resum the contributions from all the descendants. The result can be written in terms of the structures defined in $$\begin{aligned}
\label{eq:FFfull}
\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle & = \left(\alpha_1\delta_{[\mu}^{\mu'}\delta_{\nu]}^{\nu'} - \,v^4\left(\alpha_2\,\delta_{[\mu}^{\mu'}\delta_{\nu]}^{\nu'} + i \frac{\alpha_3}{2} \,\epsilon_{\mu\nu}^{~~\mu'\nu'}\right) \right)\, G_{\mu'\nu',\rho\sigma}(x_{12})\nonumber\\& +\,v^4 \,\left(\alpha_2\,\delta_{[\mu}^{\mu'}\delta_{\nu]}^{\nu'} + i \frac{\alpha_3}{2} \epsilon_{\mu\nu}^{~~\mu'\nu'}\right) H_{\mu'\nu',\rho\sigma}(\vec{x}_{12},y_1,y_2)~.\end{aligned}$$ with coefficients $$\begin{aligned}
\label{eq:alphadef}
\alpha_1= \tfrac{1}{2}{ \left( c_{11}(\tau,\bar{\tau})+ c_{22}(\tau,\bar{\tau})\right)},\quad\alpha_2= \tfrac{1}{2}\left( c_{11}(\tau,\bar{\tau})- c_{22}(\tau,\bar{\tau})\right),\quad
\alpha_3= - c_{12}(\tau,\bar{\tau})~.\end{aligned}$$ where $$\label{eq:VVco}
\langle \hat{V}_i^a(\vec{x}) \hat{V}_j^b(0) \rangle = c_{ij}(\tau,\bar{\tau})\frac{ I^{\rm 3d\,ab}(\vec{x})}{|\vec{x}|^{4}} + \text{contact term}~.$$ We see that eq. is written explicitly in terms of data of the boundary conformal theory. For the time being we can ignore the contact term in the current two-point function because it cannot contribute to the two-point function of $F_{\mu\nu}$ at separated points.
To make the action of $SL(2,\mathbb{Z})$ more transparent we will also rewrite the above results in the selfdual/antiselfdual components. The bulk-to-boundary OPE takes the following form $$F^{\pm}_{\mu\nu} (\vec{x}, y) \underset{y\to 0}{\sim} \hat{V}_{\pm\,a}(\vec{x}) 4 P_{\mu\nu}^{\pm~ay}+\dots~,$$ where $$\begin{aligned}
\hat{V}_+ & = \frac{1}{2}(\hat{V}_1 -i \hat{V}_2) = -\frac{2\pi}{{\rm Im}\tau}(\hat{J} -\tau \hat{I})~,\\
\hat{V}_- & = \frac{1}{2}(\hat{V}_1 +i \hat{V}_2) = -\frac{2\pi}{{\rm Im}\tau}(\hat{J} -\bar{\tau} \hat{I})~.\end{aligned}$$ An $SL(2,\mathbb{Z})$ transformation acts on $\hat{V}_\pm$ in the same way as it acts on $F^\pm$. In particular under an $S$ transformation $\hat{V}_+ \to \bar{\tau} \,\hat{V}_+$ and $\hat{V}_- \to \tau \,\hat{V}_-$. Using the structures introduced in section \[sec:FreeTwoP\], the result can be rewritten in more compact form $$\begin{aligned}
\label{FFfulldual}
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^+(x_2)\rangle & = (\alpha_2+i \alpha_3)\, v^4 H^{++}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^-(x_2)\rangle & =(\alpha_2-i \alpha_3)\, v^4 H^{--}_{\mu\nu,\rho\sigma}(\vec{x}_{12},y_1,y_2)~,\\
\langle F_{\mu\nu}^+(x_1) F_{\rho\sigma}^-(x_2)\rangle & = \alpha_1 G^{+-}_{\mu\nu,\rho\sigma}(x_{12})~,\\
\langle F_{\mu\nu}^-(x_1) F_{\rho\sigma}^+(x_2)\rangle & = \alpha_1 G^{-+}_{\mu\nu,\rho\sigma}(x_{12})~.\end{aligned}$$ Note that $\alpha_2\pm i \alpha_3 = 2 c_{\pm\pm}$ while $\alpha_1 = 2 c_{+-} = 2 c_{-+}$. In this basis the $SL(2,\mathbb{Z})$ action on the above two-point functions can be immediately read from .
While in this subsection we discussed the two-point function of $F_{\mu\nu}$, clearly a similar computational strategy could be used for an arbitrary $n$-point function, therefore reducing any such bulk correlation functions to correlators of the boundary currents $\hat{J}$, $\hat{I}$. Of course generically for $n>2$ these correlation function are not just captured by the coefficients $c_{ij}$, because they are sensitive to the full spectrum of boundary operators entering in the OPE of the currents.
One-point Functions from the Bulk OPE {#sec:resultsBoot}
-------------------------------------
When $x_{12}^2 \ll y^2$ we can expand the two-point function in the bulk OPE limit, which is controlled by the OPE of free Maxwell theory $$\begin{aligned}
\label{FFope}
F_{\mu\nu}(x) F_{\rho\sigma}(0)\underset{x\to 0}{\sim} \frac{g^2}{\pi^2}G_{\mu\nu,\rho\sigma}(x)+\frac{1}{12}(\delta_{\mu\rho}\delta_{\nu\sigma}-\delta_{\nu\rho}\delta_{\mu\sigma})F^2(0)+\frac{1}{12}\epsilon_{\mu\nu\rho\sigma}F\tilde{F}\,(0)+\dots,\end{aligned}$$ where we neglected spinning bulk primaries (since they do not acquire vev) and descendants, and we used the shorthand notation $F^2 \equiv F_{\mu\nu}F^{\mu\nu}$ and $F\tilde{F} \equiv F_{\mu\nu}\tilde{F}^{\mu\nu}$.
Plugging the bulk OPE into the l.h.s. of one obtains the following [*bulk channel decomposition*]{} of the two-point function $$\begin{aligned}
\label{FFbulk}
\langle F^{\mu\nu}(x_1) F^{\rho\sigma}(x_2) \rangle&\underset{x_1\to x_2}{\sim}{\frac{g^2}{\pi^2} G_{\mu\nu,\rho\sigma}(x_{12})}{}\nonumber\\
&+\frac{1}{12}(\delta^{\mu\rho}\delta^{\nu\sigma}-\delta^{\nu\rho}\delta^{\mu\sigma})\frac{a_{F^2}(\tau,\bar{\tau})}{y_2^4}+\frac{1}{12}\epsilon^{\mu\nu\rho\sigma}\frac{a_{F\tilde{F}}(\tau,\bar{\tau})}{y_2^4}\nonumber\\
&+\dots~,\end{aligned}$$ where $\dots$ denote subleading descendant terms, and we parametrized bulk one-point functions as $$\langle {\mathcal{O}}(\vec{x},y)\rangle=a_{\mathcal{O}}{y^{-\Delta_{\mathcal{O}}}}~.$$ Comparing and (see appendix \[bootapp\] for details) we obtain a constraint from the contribution of the identity $$\label{eq:Idconstr}
c_{11}(\tau,\bar{\tau})+ c_{22}(\tau,\bar{\tau}) = \frac{4}{\pi \,{\rm Im} \tau} ~,$$ and the following expressions for the one-point functions[^5] $$\begin{aligned}
\label{eq:Bootsres}
\quad a_{F^2}(\tau,\bar{\tau})& = \frac38\,\left(c_{22}(\tau,\bar{\tau})- c_{11}(\tau,\bar{\tau})\right) = \frac34\left(c_{22}(\tau,\bar{\tau}) - \frac{2}{\pi \,{\rm Im} \tau}\right) ~,\\
a_{F\tilde{F}}(\tau,\bar{\tau}) & = i\frac34\,c_{12}(\tau,\bar{\tau})~.\end{aligned}$$ This shows that the bulk one-point functions of $F^2$ and $F\tilde{F}$ are determined by the constants $c_{ij}$. Note that these relations are compatible with the (spurionic) parity symmetry, because $a_{F\tilde{F}}$ and $c_{12}$ are odd, while all the other coefficients are even. What we discussed here is a very simple example of the use of the crossing symmetry constraint on bulk two-point functions to determine data of BCFTs [@Liendo:2012hy]. The constraint can be solved exactly in this case because the bulk theory is gaussian.
Equivalently, in selfdual/antiselfdual components $$\quad a_{F_\pm^2}(\tau,\bar{\tau})= \frac{3}{16}\,\left(c_{22}(\tau,\bar{\tau}) \pm 2 i\,c_{12}(\tau,\bar{\tau})- c_{11}(\tau,\bar{\tau})\right) = -\frac{3}{4}c_{\pm\pm}(\tau,\bar{\tau})~.$$
Note that due to the constraint in eq. , the three entries of the matrix $c_{ij}$ actually only contain two independent functions of the coupling. In the appendix \[app:current2ptfunctions\] we show how to express $c_{ij}$ (and also the possible contact terms in ) in terms of two real functions $c_J$ and $\kappa_J$ of $(\tau, \bar{\tau})$, which are the coefficients in the two-point function of $\hat{J}$.
$c_{ij}(\tau,\bar{\tau})$ in Perturbation Theory
------------------------------------------------
Having derived the bulk one-point and two-point functions in terms of the coefficients $c_{ij}(\tau,\bar{\tau})$ in the two-point function of the boundary currents, we will now give the leading order results for these coefficients in perturbation theory in $\tau^{-1}$.
Note that thanks to the modified Neumann condition, at leading order $\hat{J}$ is identified with the U(1) current $\hat{J}_{\text{CFT}}$, whose two-point function can be parametrized as $$\label{eq:JCFT}
\langle\hat{J}^a_{\text{CFT}}(\vec{x}_1)\hat{J}^b_{\text{CFT}}(\vec{x}_2)\rangle = c^{(0)}_J \frac{I^{\rm 3d}_{ab}(\vec{x}_{12})}{|\vec{x}_{12}|^4} - i \frac{\kappa^{(0)}_J}{2\pi} \epsilon_{abc}\partial_{1}^c \delta^3(\vec{x}_{12})~.$$
Using the expression for $c_{ij}(\tau,\bar{\tau})$ in appendix \[app:current2ptfunctions\], and plugging $c_J = c^{(0)}_J +\mathcal{O}(\tau^{-1})$ and $\kappa_J = \kappa^{(0)}_J +\mathcal{O}(\tau^{-1})$, we obtain $$\begin{aligned}
c_{22}(\tau,\bar{\tau}) & =\frac{4}{\pi}\frac{{\rm Im}\tau}{|\tau|^2}-4\frac{({\rm Im}\tau^2 - {\rm Re}\tau^2)\, \pi^2 c^{(0)}_J +4 \,{\rm Im}\tau\,{\rm Re}\tau\,\frac{ \kappa^{(0)}_J}{2\pi}}{|\tau|^4} +\mathcal{O}(|\tau|^{-3})~,\label{eq:c22pert}\\
c_{12}(\tau,\bar{\tau}) & =-\frac{4}{\pi}\frac{{\rm Re}\tau}{|\tau|^2}+\frac{\,{\rm Im}\tau\,{\rm Re}\tau \,\pi^2 c^{(0)}_J -({\rm Im}\tau^2 - {\rm Re}\tau^2)\,\frac{ \kappa^{(0)}_J}{2\pi}}{|\tau|^4} +\mathcal{O}(|\tau|^{-3})~.\label{eq:c12pert}\end{aligned}$$ $c_{11}(\tau,\bar{\tau})$ is obtained by $c_{22}(\tau,\bar{\tau})$ using . Note the compatibility with the (spurionic) parity symmetry, under which both ${\rm Re} \tau$ and $\kappa_J^{(0)}$ flip sign, and $c_{22}$ ($c_{12}$) is even (odd, respectively).
We observe that, to this order, $$\frac{\partial c_{22}}{\partial {\mathrm{Re}}\tau } + \frac{\partial c_{12}}{\partial {\mathrm{Im}}\tau } = 0~.\label{eq:idder}$$ An explanation of this relation, and also a reason why it must hold to all orders in perturbation theory, will be provided in section \[sec:FreeEnergy\].
Going to higher orders in $\tau^{-1}$, the correlators of $\hat{J}$, and in particular the coefficients $c_J$ and $\kappa_J$, will start deviating from those of the CFT. When the CFT is free, these corrections can be computed by ordinary Feynman diagrams on the boundary. We will see examples of this in the following. In the more general case of an interacting CFT, these correction can be computed in conformal perturbation theory, by lowering an insertion of the bulk Lagrangian integrated over the region $y\geq 0$. It would be interesting to characterize the CFT observables that enter the subleading orders of this perturbation theory. We leave this problem for the future.
Displacement Operator
---------------------
In every BCFT with $d$-dimensional bulk there exists a boundary scalar operator of protected scaling dimension $d$, the so-called displacement operator. It can be defined as the only scalar primary boundary operator that appears in the bulk-to-boundary OPE of the bulk stress tensor $$\label{eq:TOPED}
T_{\mu\nu}(\vec{x},y) \underset{y\to 0}{\sim}\frac{d}{d-1} \left( \delta_{\mu y}\delta_{\nu y}-\frac1d\delta_{\mu\nu}\right)\hat{D}(\vec{x}) + \dots~.$$ There is a Ward Identity associated to this operator, namely $$\label{eq:DWI}
\int d^d \vec{x} \langle \hat{D}(\vec{x}) O_1(\vec{x}_1,y_1) \dots O_n(\vec{x}_n,y_n) \rangle = (\partial_{y_1} +\dots +\partial_{y_n}) \langle O_1(\vec{x}_1,y_1) \dots O_n(\vec{x}_n,y_n) \rangle~,$$ that fixes the normalization of the operator. In this normalization its two-point function is $$\langle \hat{D}(\vec{x}_1) \hat{D}(\vec{x}_2)\rangle = \frac{C_{\hat{D}}}{|\vec{x}_{12}|^{2d}}~,$$ and the quantity $C_{\hat{D}}$ is an observable of the BCFT.
It follows from that the displacement operator is the restriction of the component $T_{yy}$ of the stress-tensor to the boundary. In the theory that we are considering the bulk stress-tensor is the usual Maxwell stress-tensor $$T_{\mu\nu} = \frac{{\rm Im}\tau}{2\pi} \left(F_{\mu\rho} F_\nu^{~\rho} -\frac 14 \delta_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma}\right)~.\label{eq:strtensmax}$$ Writing $T_{yy}(y=0)$ in terms of the currents $\hat{I}$ and $\hat{J}$ leads to the following expression for the displacement operator $$\label{eq:dispform}
\hat{D} = \frac{\pi}{\Im \tau}(|\tau|^2 \hat{I}^2+\hat{J}^2 - 2 {\rm Re}\tau\, \hat{I}\hat{J}) = \frac{{\rm Im}\tau}{4 \pi}(\hat{V}_1^2 + \hat{V}_2^2)~.$$ The right-hand side of contains products of two boundary operators at the same point, that are defined through a point-splitting procedure, similarly to the products on the right-hand side of . Such a point splitting makes sense for arbitrary $\tau$ even though generically the boundary currents are not generalized free fields. This is because their dimension and the dimension of $\hat{D}$ are protected, and the contribution of $\hat{D}$ in their OPE is non-singular, so after subtracting the contribution of the identity and possibly of additional operators of scaling dimension $<4$ we can always take the coincident-point limit.
We can use the expression to obtain the first two orders in the perturbative expansion of $C_{\hat{D}}$ universally in terms of the two-point function of the CFT current . The leading order contribution to $C_{\hat{D}}$ at large $\tau$ comes from the contraction of the $\hat{I}$ currents in the $\hat{I}^2$ term, and is therefore proportional to the square of $c_{22}$ at leading order. At next-to-leading order there is a contribution from the correction to $c_{22}$, and a contribution from the $\hat{I}\hat{J}$ term. See fig. \[fig:cd\].
![Diagrams for the two-point function of the displacement operator. The leading order contribution $(a)$ is the square of the two-point function of the topological current $\hat{I}$. At next-to-leading order we have the diagrams $(b.1)$-$(b.2)$-$(b.3)$ that are also sensitive to the electric current $\hat{J}$. The shaded blobs denote insertions/correlators of $\hat{J}$ in the undeformed CFT.\[fig:cd\]](fig/CD){width="0.7\linewidth"}
The result is $$C_{\hat{D}} = \frac{6}{\pi^4} - \frac{12}{\pi} \frac{{\rm Im}\tau}{|\tau|^2}c_J^{(0)} + \mathcal{O}(|\tau|^{-2})~.\label{eq:CDpert}$$ Even though the 3d CFT sector decouples from the bulk in the limit $\tau \to \infty$, and in particular it has a conserved 3d stress tensor, the displacement operator still exists within the sector of boundary operators coming from the free boundary condition of the bulk Maxwell field, and in particular $C_D$ is finite in this limit. Plugging ${\rm Re}\,\tau = 0$ and the value of $c_J^{(0)}$ for a theory of two Dirac fermions, namely $c_J^{(0)} = \frac{1}{4\pi^2}$, we find perfect agreement with [@Herzog:2017xha].
Three-Point Function $\langle\hat{V}_i \hat{V}_j \hat{D}\rangle$
----------------------------------------------------------------
Some of the distinctive features of the conformal theory living on the boundary of $B(\tau,\bar{\tau})$ are
- [the presence of a scalar operator of dimension 4, the displacement operator $\hat{D}$ ; this feature is common to all conformal boundary conditions ;]{}
- [the presence of the two $U(1)$ currents $\hat{V}_1$ and $\hat{V}_2$ .]{}
We will now show that the displacement operator $\hat{D}$ appears in the OPE of the currents, with a matrix of OPE coefficients that can be fixed in terms of the coefficients of the bulk one-point functions $a_{F^2}$ and $a_{F\tilde{F}}$, and the coefficient $C_{\hat{D}}$.
To show this, we consider the three-point correlator between the field strength and the displacement operator $$\begin{aligned}
\langle F_{\mu\nu}(x_1)F_{\rho\sigma}(x_2)\hat{D}(\vec{x}_3)\rangle~.\end{aligned}$$ We compute this three-point function in two OPE channels for $F_{\mu\nu}(x_1)F_{\rho\sigma}(x_2)$. In the boundary channel $y_{1,2} \to 0$, using the OPE this three-point function can be fixed in terms of the OPE coefficients $\langle\hat{V}_i \hat{V}_j \hat{D}\rangle$ that we want to determine. On the other hand, in the bulk OPE channel $x_{12}\to 0$ this three point function can be computed in terms of the bulk-boundary two-point functions $\langle O(x_1) \hat{D}(\vec{x}_3) \rangle$ between the displacement and the operators $O$ in the bulk OPE of two $F$’s. The last step of the argument amounts to relating the latter two-point function to the one-point function of $O$ if $O$ is a scalar operator, or to $C_{\hat{D}}$ if $O$ is the stress-tensor.
The coefficients appearing in the three-point function are [@Costa:2011mg; @Dymarsky:2017xzb] $$\begin{aligned}
\label{eq:VVD3main}
\langle \hat{V}_i^a(\vec{x}_1)\hat{V}_j^b(\vec{x}_2)\hat{D}(\infty)\rangle = \, \lambda_{ij\hat{D}+}^{(1)}\,\, \delta^{ab}+\lambda_{ij\hat{D}-}^{(1)}\,\, {\hat{x}_{12}^c}{}\epsilon^{abc}~.\end{aligned}$$ For simplicity we placed the displacement at infinity. $\lambda_{ij\hat{D}+}^{(1)}$ and $\lambda_{ij\hat{D}-}^{(1)}$ are respectively the parity-even/odd OPE coefficients in the conventions of [@Dymarsky:2017xzb], and $\hat{x}^a=x^a/|\vec{x}|$. Recall that under parity $\hat{V}_1$ is a vector while $\hat{V}_2$ is an axial vector, hence the coefficients $\lambda^{(1)}_{11\hat{D}-},\lambda^{(1)}_{22\hat{D}-},\lambda^{(1)}_{12\hat{D}+}$ are odd under a spurionic parity transformation, while the others are even.
The details of the calculation are showed in the appendix \[app:VVD\], and here we will just give the final result $$\begin{aligned}
\label{eq:3ptfinal}
\lambda_{11\hat{D}+}^{(1)} & =-\frac{8}{3 \pi ^2}a_{F^2}+\frac{g^2}{3}C_{\hat{D}}~,\\
\lambda_{22\hat{D}+}^{(1)} &= \frac{8}{3 \pi ^2}a_{F^2}+\frac{g^2}{3}C_{\hat{D}}~,\\
\lambda_{12\hat{D}+}^{(1)} & = -\frac{8}{3\pi^2}\, i a_{F\tilde{F}}~,\\
\lambda_{ij\hat{D}-}^{(1)} & = 0~.\end{aligned}$$ The parity-odd three-point structures are all set to zero. The spurionic parity symmetry is again satisfied because $\lambda_{12\hat{D}+}^{(1)}$ is proportional to the odd coefficient $a_{F\tilde{F}}$, while the formulas for $\lambda_{11\hat{D}+}^{(1)}$ and $\lambda_{22\hat{D}+}^{(1)}$ are even.
Going to the basis in which the matrix of current-current 2-pt functions is the identity $$U_i^l U_j^k c_{lk} = \delta_{ij}~,$$ the matrix of OPE coefficients becomes $$U\lambda_{D+}^{(1)}U^{T}=\frac{2}{\pi^2}\begin{pmatrix}
\frac{\mathcal{A} - \frac{\pi^2 C_{\hat{D}}}{8}}{\mathcal{A} - \frac{3}{4\pi^2}} & 0 \\
0 & \frac{\mathcal{A} + \frac{\pi^2 C_{\hat{D}}}{8}}{\mathcal{A} + \frac{3}{4\pi^2}}
\end{pmatrix}~,\label{eq:ldiag}$$ where $$\begin{aligned}
\mathcal A \equiv \frac{1}{g^2}\sqrt{a_{F^2}^2 - a_{F\tilde{F}}^2}~.\end{aligned}$$ Recall that $a_{F^2} \in \mathbb{R}$ and $a_{F\tilde{F}} \in i \mathbb{R}$, so $\mathcal{A}$ is real and $\geq 0$. Seemingly the upper entry has a pole at $\mathcal{A} = \frac{3}{4\pi^2}$, which corresponds to the value at the decoupling limit. However recall from that $C_{\hat{D}} \to \frac{6}{\pi^4}$ in the decoupling limit, so that actually the entry is finite in the limit.
The upshot of this analysis is that the OPE coefficients between two currents and the displacement can be completely characterized in terms of the two positive quantities $\mathcal{A}$ and $C_{\hat{D}}$, that can be taken to effectively parametrize the position on the conformal manifold.
Free Energy on a Hemisphere {#sec:FreeEnergy}
===========================
In this section we study the hemisphere free energy for the conformal boundary conditions of the $U(1)$ gauge field.
Following [@Gaiotto:2014gha], to any given conformal boundary condition for a CFT$_4$ we can assign a boundary free energy $F_\partial$, defined as $$\begin{aligned}
\label{eq:defnRatio}
F_\partial=-\frac{1}{2}\log \frac{|Z_{HS^4}|^2}{Z_{S^4}}=-\Re \log Z_{HS^4}+\frac{1}{2}\log Z_{S^4}~.\end{aligned}$$ $Z_{S^4}$ denotes the sphere partition function of the CFT, while $Z_{HS^4}$ denotes the partition function of the theory placed on an hemisphere, with the chosen boundary condition on the boundary $S^3$. In writing we discarded power-law UV divergences, and focused on the universal finite term. Conformal symmetry ensures that the coupling to the curved background can be defined via Weyl rescaling.
In our setup the bulk theory is a $U(1)$ gauge-field with action , so we have $$\begin{aligned}
\label{systemIR}
-8\pi\frac{\partial F_{\partial}}{\partial \Im\tau}=-\Re \int_{HS^4}d^4x \sqrt{g(x)}\langle F^2 (x)\rangle_{HS^4}+\frac{1}{2} \int_{S^4}d^4x\sqrt{g(x)}\langle F^2 (x)\rangle_{S^4},\nonumber\\
-8\pi\frac{\partial F_{\partial}}{\partial \Re\tau}=-\Re \int_{HS^4}d^4x\sqrt{g(x)} \langle i F\tilde{F}(x) \rangle_{HS^4}+\frac{1}{2} \int_{S^4}d^4x \sqrt{g(x)} \langle i F\tilde{F}(x) \rangle_{S^4}.\end{aligned}$$ Using a Weyl transformation the one-point functions can be obtained from those on $\mathbb{R}^3\times \mathbb{R}_+$ as $$\begin{aligned}
\langle F^2 (x) \rangle_{HS^4}=\Omega(x)^{-4}\frac{a_{F^2}}{u(x)^4}+\frac{1}{2}\mathcal{A}, \quad \langle F\tilde{F} (x)\rangle_{HS^4}=\Omega(x)^{-4}\frac{a_{F\tilde{F}}}{u(x)^4}+\frac{1}{2}\widetilde{\mathcal{A}}~.\end{aligned}$$ Here $x$ is a point on the hemisphere, $\Omega(x)$ is the Weyl factor induced by the stereographic projection, and $u(x)$ denotes the chordal distance between the point $x$ and the boundary $S^3$. The shifts $\mathcal{A}$ and $\mathcal{\tilde{A}}$ stand for a scheme-dependent contribution to the one-point function, due to the ambiguity in the definition of the theory on the curved background: we can always add local counterterms given by a scalar density of dimension four built out of the background curvature, multiplied by the real or imaginary part of the marginal coupling $\tau$, and integrated in the interior of the hemisphere. On the other hand, if we compute the partition function on $S^4$ in the same scheme, the one-point functions on $S^4$ receive contribution only from those counterterms, because on $\mathbb{R}^4$ one-point functions must vanish, and there is a relative factor of two because in this case the counterterm is integrated over the whole sphere. Hence $$\begin{aligned}
\langle F^2 \rangle_{S^4}=\mathcal{A}~,\quad \langle F\tilde{F} \rangle_{S^4}=\widetilde{\mathcal{A}}~,\end{aligned}$$ such that the ambiguity precisely cancels in . Here we see the virtue of the choice of normalization in .
The remaining integral on $HS^4$ has a UV divergence when the point $x$ approaches the boundary $S^3$. We introduce a UV regulator $\epsilon \ll 1$ and restrict the integral to the region $u(x) > \epsilon$. The result is $$\int_{u(x)>\epsilon}\sqrt{g(x)}\ \Omega(x)^{-4}\frac{1}{u(x)^4} = \frac{2\pi^2}{3\epsilon^3} -\frac{5\pi^2}{3\epsilon} +\frac{4\pi^2}{3} + \mathcal{O}(\epsilon)~.$$ As implicit in the definition of $F_\partial$, we will neglect the power-law UV divergent term and focus on the universal finite piece. Hence we finally obtain $$\begin{aligned}
\frac{\partial F_{\partial}}{\partial \Im\tau}& =\frac{\pi}{6} \,a_{F^2}=\frac{\pi}{8}c_{22}(\tau,\bar{\tau})-\frac{1}{4\, {\rm Im}\tau}~,\label{eq:onepoint1}\\ \frac{\partial F_{\partial}}{\partial \Re\tau}& =\frac{\pi}{6} \,i\,a_{F\tilde{F}}=-\frac{\pi}{8}c_{12}(\tau,\bar{\tau})~.\label{eq:onepoint2}\end{aligned}$$ We used the relations to rewrite the result in terms of the two-point functions of the conserved currents. A consequence of this equation is that the relation must be valid to all orders in perturbation theory, or more generally whenever $F_{\partial}$ is well-defined.
Plugging - in - and solving the equations we find the following leading behavior of $F_\partial$ at large $\tau$ $$F_\partial \underset{\tau\to\infty}{\sim} -\frac 14 \log\left[\frac{2 \,{\rm Im} \tau}{|\tau|^2}\right]+ C + \pi \frac{\frac{\pi^2}{2} c^{(0)}_J {\rm Im} \tau + \frac{\kappa^{(0)}_J}{2\pi}{\rm Re} \tau}{|\tau|^2}+ \mathcal{O}(|\tau|^{-2})~.\label{eq:Fcuspinf}$$ The first term, which diverges for $\tau \to \infty$, is the value of $F_\partial$ for a free Maxwell field with Neumann boundary conditions [@Gaiotto:2014gha]. Matching eq. with the value of $F_\partial$ for a decoupled system of a Maxwell field with Neumann conditions and a 3d CFT on the boundary, we find that the constant $C$, that remained undetermined by the differential constraint, is in fact the $S^3$ free energy $F_{0,1}$ of the theory $T_{0,1}$.
Using an $SL(2,\mathbb{Z})$ transformation, the same asymptotic behavior holds in the vicinity of any cusp point, upon replacing $\tau$ with the transformed variable $\tau^\prime$ that goes to $\infty$ at the selected cusp, and identifying $C$ with the $S^3$ free energy of the decoupled 3d CFT living at the cusp. Near the cusp where the current $p\hat{J} + q \hat{I}$ decouples from the 3d theory $T_{p,q}$, we have $$F_\partial \underset{\tau^\prime\to\infty}{\sim} -\frac 14 \log\left[\frac{2 \,{\rm Im} \tau^\prime}{|\tau^\prime|^2}\right]+ F_{p,q} + \mathcal{O}(|\tau^{\prime}|^{-1})~.\label{eq:Fcusppq}$$ where $\tau^\prime = \frac{a \tau + b}{p \tau + q}$, with $a q - b p =1$, and $F_{p,q}$ is the $S^3$ free energy of $T_{p,q}$. Note that $$\begin{aligned}
-\frac 14 \log\left[\frac{2 \,{\rm Im} \tau}{|\tau|^2}\right] & \underset{\tau^\prime\to\infty}{\sim} -\frac 14 \log\left[\frac{2 \,{\rm Im} \tau^\prime}{|\tau^\prime|^2}\right] + \frac12 \log |q|~,\label{eq:transF}\\
-\frac 14 \log\left[2 \,{\rm Im} \tau\right] & \underset{\tau^\prime\to\infty}{\sim} -\frac 14 \log\left[\frac{2 \,{\rm Im} \tau^\prime}{|\tau^\prime|^2}\right] + \frac12 \log |p|~.\label{eq:transF2}\end{aligned}$$ Eq. implies that the function $$F_\partial +\frac 14 \log\left[\frac{2 \,{\rm Im} \tau}{|\tau|^2}\right]~,$$ attains the finite value $$\frac12 \log |q| + F_{p,q}~,$$ at all the cusps with $|q| \neq 0$. For the cusp with $q=0$ we can simply use to derive that $$F_\partial +\frac 14 \log\left[ 2 \,{\rm Im} \tau \right]~,$$ approaches $$\frac12 \log |p| + F_{p,q}~.$$ Hence the function $F_\partial(\tau,\bar{\tau})$ contains information about the $S^3$ free energies of an infinite family of 3d Abelian gauge theories, namely all the theories obtained by applying $SL(2,\mathbb{Z})$ transformations to $T_{0,1}$.
We note in passing that the shift by $\frac 12 \log |q|$ in eq. has a nice interpretation in terms of the $S^3$ free energy for a pure CS theory. Indeed, starting with a 4d gauge field with Neumann condition, applying the transformation $S T^{k}$, i.e. $\tau^\prime = -\frac{1}{\tau + k}$, and taking the decoupling limit $\tau^\prime \to \infty$, we are left with a pure CS theory at level $k$ on the boundary. Hence, the free energy $F_\partial$ in this limit should be the sum of the contribution of the decoupled 4d gauge field with Neumann boundary condition, and the contribution from the CS theory at level $k$, which is $\frac12 \log|k|$. This is precisely what eq. gives. Similarly eq. can be interpreted by starting with a 4d gauge field with Dirichlet boundary condition, whose partition function is the left-hand side of , applying $S T^{k} S$, i.e. $\tau^\prime = \frac{\tau}{ - k\tau +1}$, and going to the decoupling limit. Again, we find a decoupled 4d gauge field with Neumann boundary condition, and a CS theory at level $k$ on the boundary. The shift by $\frac 12 \log |p|$ in eq. precisely reproduces the $\frac12 \log|k|$ contribution of the CS theory.
A Minimal Phase Transition {#sec:MinPhTr}
==========================
In this section we will study a non-trivial BCFT which conjecturally describes a second order (boundary) phase transition between two free boundary conditions $(p,q)$ and $(p',q')$ of the 4d gauge field, with $p q' - p' q=1$. In particular, the conjectural BCFT should have a single relevant boundary operator, which can be turned on to flow to either of these free boundary conditions in the IR, depending on the sign of the coupling. We will assume that this BCFT exists for all values of the gauge coupling $\tau$, with no further phase transitions as a function of $\tau$.
Without loss of generality, we can pick two canonical duality frames where the phase transition interpolates between Dirichlet and Neumann boundary conditions or viceversa. We can also pick two duality frames where the phase transition interpolates between Neumann and $(1,1)$ boundary conditions or viceversa.
- If we go to weak coupling in the former duality frames, the boundary degrees of freedom should describe a phase transition between phases with spontaneously broken or unbroken $U(1)$ global symmetry. We expect that to be described by a critical $O(2)$ model.
- If we go to weak coupling in the latter duality frames, the boundary degrees of freedom should describe a phase transition between two gapped phases with unbroken $U(1)$ global symmetry, but background Chern-Simons coupling which differs by one unit. We expect that to be described by a massless Dirac fermion.
Keeping track of the duality transformations between the different frames, we can assemble an overall picture.
- Denote as $\tau_{DN}$ the gauge coupling associated to the description as a phase transition between Dirichlet and Neumann boundary conditions, so that one “$O(2)$ cusp” is at $\tau_{DN} \to \infty$.
- Then $\tau_{ND}= -1/\tau_{DN}$ is the coupling which is weak at the other $O(2)$ cusp, at $\tau_{DN}\to0$.
- Shifting the $\theta$ angle by $2 \pi$ gives an alternative description as a transition between Dirichlet and $(1,-1)$ boundary conditions, with coupling $\tau_{DN''} = \tau_{DN}-1$. Dually, we get a transition between Neumann and $(1,1)$ boundary conditions, with coupling $\tau_{N N'} = -\tau_{DN''}^{-1} = \frac{1}{1-\tau_{DN}}$ which is weak at the “Dirac fermion” cusp, $\tau_{DN} \to 1$.
- If we had shifted the $\theta$ angle in the opposite way, we would arrive to a transition between Neumann and $(1,-1)$ boundary conditions, with coupling $\tau_{N N''} = -\tau_{DN'}^{-1} = -\frac{1}{1+\tau_{DN}}$ which is weak at the second “Dirac fermion” cusp, $\tau_{DN} \to -1$.
In the following we will do most of our calculations in a perturbative expansion around a “Dirac fermion” cusp. The correct boundary theory is actually a Dirac fermion dressed by half a unit of background Chern-Simons coupling [@AlvarezGaume:1984nf; @Witten:2015aba]. It is convenient to absorb that background Chern-Simons coupling into an improperly-quantized shift of the $\theta$ angle, so that the gauge coupling is denoted as $\tau = \tau_{N N'} -\frac12 = \frac12 \frac{1+\tau_{DN}}{1-\tau_{DN}}$. Therefore, denoting with $\psi$ the Dirac fermion, the action that we consider is $$S[A, \tau +\tfrac 12] + \int_{y=0} d^3\vec{x}\, i\bar{\psi}\slashed{D}_A \psi~.
\label{eq:action1fermion}$$ The second Dirac fermion cusp is at $\tau \to 0$ and the $O(2)$ cusps are at $\tau = \pm \frac12$. See fig.s \[fig:tauplaneO2\]-\[fig:tauplane\].
![The upper-half plane of the gauge coupling $\tau_{DN}$, i.e. in the duality frame in which at $\tau_{DN}\to \infty$ we find the $O(2)$ model on the boundary. Thanks to particle-vortex duality, the cusp in the origin $\tau_{DN} = 0$ also gives a decoupled $O(2)$ model on the boundary. Thanks to the boson-fermion duality between $U(1)_1$ coupled to a critical scalar and a free Dirac fermion, the cusps at $\tau_{DN} = \pm 1$ give a free Dirac fermion. \[fig:tauplaneO2\]](fig/O2frame.pdf){height="7cm"}
![The upper-half plane of the gauge coupling $\tau = \tau_{NN'} -\frac 12$, i.e. in the duality frame in which at $\tau\to \infty$ we find a free Dirac fermion on the boundary. Thanks to fermionic particle-vortex duality, the cusp in the origin $\tau = 0$ also gives a free Dirac fermion on the boundary. Thanks to the boson-fermion duality between $U(1)_{\frac 12}$ coupled to a Dirac fermion and the $O(2)$ model, the cusps at $\tau= \pm \frac12$ give the $O(2)$ model. \[fig:tauplane\]](fig/Diracfermionframe.pdf){height="7cm"}
Essentially by construction, the picture is compatible with a well-known duality web of particle-vortex, fermion-boson and fermion-fermion dualities [@Seiberg:2016gmd], which inspired this investigation. In particular, thanks to the particle-vortex duality between the $O(2)$ model and the gauged $O(2)$ model [@Peskin:1977kp; @Dasgupta:1981zz], or equivalently thanks to its fermionic version [@Son:2015xqa], in this case we have a $\mathbb{Z}_2$ subgroup of $SL(2,\mathbb{Z})$ that is a duality of $B(\tau,\bar{\tau})$, i.e. it leaves invariant both the bulk and the boundary condition. This subgroup acts on $\tau = \tau_{NN'} -\frac 12$ as $$\tau \to -\frac{1}{4\tau}~.$$ It is interesting to note that the self-dual point $\tau = \frac{i}{2}$, i.e. $\tau_{DN} = i$, is an extreme of $F_{\partial}$. In our formalism, this is a straightforward consequence of the differential equations -, once we set $c_{11}=c_{22}=\frac{2}{\pi\,{\rm Im}\tau}$ and $c_{12}=0$ – as dictated by self-duality and equation .[^6]
Before proceeding, let us mention some of the previous literature on this theory, and comment on the relation to the results that we will present in the rest of this section. The interplay between the 3d dualities and the 4d electric-magnetic duality in the setup with a 3d Dirac fermion coupled to a bulk gauge field was investigated in [@Seiberg:2016gmd; @Metlitski:2015eka; @Wang:2015qmt; @Hsiao:2017lch; @Hsiao:2018fsc]. In particular [@Hsiao:2017lch; @Hsiao:2018fsc] studied the transport properties of the boundary theory at the self-dual point. For the theory with an even number of Dirac fermions on the boundary, the two-loop two-point function of the boundary current $\hat{J}$ was obtained in [@Teber:2012de] (see also [@Kotikov:2013kcl; @Teber:2014ita; @Teber:2016unz; @Kotikov:2016yrn]) while the Weyl anomalies (or equivalently the two- and three-point functions of the displacement operator) were computed to next-to-leading order in [@Herzog:2017xha; @Herzog:2017kkj] (for the supersymmetric version of the theory see [@Herzog:2018lqz]). The point of view of boundary conformal field theory was first adopted in this theory in [@Herzog:2017xha; @Herzog:2017kkj], but these papers do not consider the action of electric-magnetic duality and the existence of multiple decoupling limits. Besides the transport coefficients and the Weyl anomalies, other boundary observables such as scaling dimensions of operators, or the hemisphere free-energy, were not studied before. Since the duality explained above only exists for the theory with one Dirac fermion, we will first concentrate on this case. Later we will also consider the theory with an even number $2N_f$ of fermions, both at large $N_f$ and in the special case $2 N_f = 2$.
Perturbative Calculation of Scaling Dimensions {#eq:pertscaldim}
----------------------------------------------
We will compute the anomalous dimensions of the first two fermion bilinear operators $O_s$ of spin $s$, namely $$\begin{aligned}
O_0 & = \bar{\psi} \psi~, \\
(O_2)_{ab} & = i\big(\bar{\psi} \gamma_{(a}\overset\leftrightarrow{D}_{b)}\psi -\text{trace}\big)~,\end{aligned}$$ up to two-loop level. Note that in the limit $\tau\to\infty$ of decoupling between bulk and boundary $(O_2)_{ab}$ becomes a conserved current, namely the stress-tensor of the 3d free-fermion CFT.
The anomalous dimension can be obtained from the renormalization of the 1PI correlator of the composite operator with two elementary fields $$\langle O_s(q=0) \psi(-p) \bar{\psi}(p)\rangle_{\rm 1PI}~.\label{eq:3point}$$ We employ dimensional regularization and minimal subtraction, i.e. we set $d = 3-2\epsilon$ and keep the codimension fixed $=1$, expand the dimensionally-continued loop integrals around $\epsilon = 0$, and reabsorb the poles in the renormalization constants $$\begin{aligned}
O_B & = Z_O O~, \\
\psi_B & = Z_{\psi} \psi~, \end{aligned}$$ where the subscript $B$ denotes the bare operators.
Even though the correlator in involves the operator $\psi$ that is not gauge-invariant, it is still sensible to renormalize it. The resulting renormalized correlator, as well as the renormalization constant $Z_{\psi}$, both depend on the choice of gauge-fixing, but the renormalization constant $Z_O$ of the gauge-invariant operator does not, hence we can extract physical information from it.
The renormalization constants admit the loopwise expansion (at small $g^2$ with fixed $\gamma$) $$Z = 1+ \delta Z = 1+\sum_n \left(\frac{g^2}{1+\gamma^2}\right)^n\delta Z^{(n)}~,$$ where $\delta Z^{(n)}$ is a polynomial in $\gamma$ of degree $\leq n$, and furthermore by invariance under space reflections only even powers of $\gamma$ are present. The n-loop term $\delta Z^{(n)}$ contains divergences up to $\epsilon^{-n}$, but the familiar RG argument constrains all the coefficients in terms of the ones at lower loop order, except that of the $\epsilon^{-1}$ divergence.
The anomalous dimension is then given by $$\gamma_O = \frac{d \log Z_O}{d\log \mu}~.$$ The dependence on the renormalization scale $\mu$ is through the $d$-dimensional coupling $$g_B = \mu^{\epsilon} g~.$$ In the latter equation we do not need to include a renormalization of the coupling because, as we explained in section \[sec:Interaction\], $g$ does not run. Therefore we can rewrite $$\gamma_O = -\epsilon\frac{\partial \log Z_O}{\partial \log g}~.$$
To compute we use the Feynman diagrams in fig.\[fig:loopdiagram\], computed in momentum space, and for simplicity we take the composite operator to carry zero momentum. The Feynman rules given in fig.\[fig:Fermionfeynrule1\].
![Feynman rules. $\Pi_{ab}$ is defined in .[]{data-label="fig:Fermionfeynrule1"}](fig/FermionFeynRule.pdf){width="0.8\linewidth"}
![Feynman rules for the zero-momentum insertions of the composite operators. Note that there are two vertices associated to $O_2$.[]{data-label="fig:Fermionfeynrule_ops"}](fig/FermionFeynRule2_ops.pdf){width="0.8\linewidth"}
![One loop and two loops diagrams. We sum over all possible insertions of the composite operators on the internal fermion lines, and also on vertices in the case of $O_2$.[]{data-label="fig:loopdiagram"}](fig/loopdiagram){width="0.9\linewidth"}
We performed the calculation up to two loops. See appendix \[app:FeynInt\] for more details about the computation of the two-loop Feynman diagrams. The resulting renormalization constants are $$\begin{aligned}
\delta Z_\psi& = \frac{g^2}{1+\gamma^2}\frac{2-3\xi}{24\pi^2\epsilon} + \left(\frac{g^2}{1+\gamma^2}\right)^2 \left[\frac{(2-3\xi)^2}{1152 \pi^4 \epsilon^2} -\frac{9(1-2\gamma^2) \pi^2 + 16 }{3456\pi^4 \epsilon} \right] +\mathcal{O}(g^6)~.\label{eq:wf}\\
\delta Z_0& = -\frac{g^2}{1+\gamma^2} \frac{2}{3\pi^2 \epsilon} + \left(\frac{g^2}{1+\gamma^2}\right)^2 \left[\frac{2}{9\pi^4\epsilon^2}+\frac{9\pi^2(1-2\gamma^2)-8}{108\pi^4 \epsilon}\right]+\mathcal{O}(g^6)~.\label{eq:Z0}\\
\delta Z_2& =\frac{g^2}{1+\gamma^2} \frac{2}{5\pi^2 \epsilon} + \left(\frac{g^2}{1+\gamma^2}\right)^2 \left[\frac{2}{25 \pi^4 \epsilon^2} - \frac{75 \pi ^2 + 16}{3000 \pi ^4 \epsilon}\right]+\mathcal{O}(g^6)~,\label{eq:Z2}\end{aligned}$$ where we denoted $\delta Z_s \equiv \delta Z_{O_s}$. Note that indeed $\delta Z_0$ and $\delta Z_2$ do not depend on the gauge-fixing parameter. As a check, we also verified that the operator $O_1 = \bar{\psi}\gamma^a \psi$ does not get renormalized, i.e. we explicitly computed the renormalization up to two-loop order and found $\delta Z_1=0$, as expected for a conserved current. On the other hand, note that $\delta Z_2\neq0$. This is a manifestation of the fact that the boundary degrees of freedom do not define a local 3d theory once we couple them to the bulk: the conservation of the boundary would-be stress-tensor is violated at $g\neq 0$, and the system only admits a stress-tensor in the bulk. This means that the short operator of spin 2 must recombine into a long conformal multiplet. In the appendix \[app:fakestress\], we show that this mechanism can be used to compute the one-loop anomalous dimension, and we use this to check the Feynman diagram calculation.
The resulting anomalous dimensions, expressed as a function of $\tau$ are $$\begin{aligned}
\gamma_0 & = -\frac{8 }{3\pi }\frac{{\rm Im} \tau}{|\tau|^2}+\frac{36\pi^2-32}{27\pi^2} \frac{({\rm Im} \tau)^2}{|\tau|^4}-\frac{8}{3}\frac{({\rm Re} \tau)^2}{|\tau|^4}+\mathcal{O}(|\tau|^{-3})~, \\
\gamma_2 & = \frac{8}{5\pi}\frac{{\rm Im} \tau}{|\tau|^2} -\frac{150 \pi^2+32}{375\pi^2}\frac{({\rm Im} \tau)^2}{|\tau|^4} +\mathcal{O}(|\tau|^{-3})~\label{eq:anomalousO2}.\end{aligned}$$ From these result we can immediately recover the anomalous dimensions for the 3d gauge theory $U(1)_k$ coupled to a Dirac fermion at large $k$ as explained in section \[sec:largeNlargek\]. Since this is a local 3d theory, we expect $\gamma_2 = 0$ and indeed this is what we obtain from . For the anomalous dimension of the scalar bilinear, that starts at two-loop order in this theory, we find $$\gamma_0 = -\frac{8}{3 k^2} + \mathcal{O}(k^{-4})~,$$ in agreement with [@Alves:1998mb].[^7]
Perturbative $F_\partial$ {#sec:perturbativeFdelta}
-------------------------
Thanks to the differential equation -, and to the relations derived in appendix \[app:current2ptfunctions\], the computation of the hemisphere free energy is reduced to the computation of the two-point functions of the boundary current $\hat{J}$. Up to next-to-leading order, we already wrote the universal formula for the hemisphere free energy in terms of the two-point function of the current $\hat{J}_{\text{CFT}}$ of the unperturbed CFT. In this particular setup where the boundary theory at $\tau \to \infty$ is a free Dirac fermion we can do better without much effort, because the correction to the current two-point function, given by the two diagrams in fig. \[fig:currentcurrent\], already exists in the literature. For the parity even part of the two-point function, we can either extract the value of these diagrams from the large-$N_f$ calculation of [@Giombi:2016fct], using the similarities between the two perturbative expansions that we explained in \[sec:largeNlargek\], or alternatively use the computation performed directly in the mixed-dimensional setup in [@Teber:2012de; @Teber:2016unz].[^8] The parity-odd part can be obtained from the large-$k$ calculation in [@Spiridonov:1991ki].
![Leading corrections to the boundary current two-point function for the Dirac fermion.[]{data-label="fig:currentcurrent"}](fig/current_current){width="0.5\linewidth"}
The sum of the diagrams in fig. \[fig:currentcurrent\] is the next-to-leading order correction for the one-photon irreducible two-point function, which we denote by $\Sigma$, see appendix \[app:current2ptfunctions\] for more details. Due to the shift in the real part of $\tau$, i.e. $\tau = \tau_{NN'} -\frac 12$, we have that $\kappa_\Sigma $ vanishes at leading order in perturbation theory, or equivalently $\kappa_J^{(0)} = 0$. The results mentioned above give $$\begin{aligned}
c_\Sigma & = \frac{1}{8\pi^2} + \frac{92-9\pi^2}{144\pi^3}\frac{{\rm Im}\tau}{|\tau|^2} +\mathcal{O}(|\tau|^{-2})~,\\
\kappa_\Sigma & = \frac{4 + \pi^2}{16}\frac{{\rm Re}\tau}{|\tau|^2} +\mathcal{O}(|\tau|^{-2})~.\end{aligned}$$ Using - to obtain the total two-point function of $\hat{J}$, we find $$\begin{aligned}
c_J & = \frac{1}{8\pi^2} +\frac{368-45\pi^2}{576 \pi^3}\frac{{\rm Im}\tau}{|\tau|^2}+\mathcal{O}(|\tau|^{-2})~,\\
\kappa_J & = \frac{16 + 5\pi^2}{64} \frac{{\rm Re}\tau}{|\tau|^2}+\mathcal{O}(|\tau|^{-2})~.\end{aligned}$$ Plugging these values in the formulas - for $c_{22}$ and $c_{12}$, and solving the differential equations - we obtain $$\begin{aligned}
\label{FpartialWeak}
F_\partial & = -\frac{1}{4}\log\left[\frac{2\,{\rm Im}\tau}{|\tau|^2}\right]+ F_{\text{Dirac}}\nonumber \\& ~~~~~+\frac{\pi}{16}\frac{{\rm Im}\tau}{|\tau|^2}+ \frac{(368 -45 \pi ^2 )({\rm Im}\tau)^2 + (144 + 45 \pi^2 ) ({\rm Re}\tau)^2}{2304 |\tau|^4}+\mathcal{O}(|\tau|^{-3})~.\end{aligned}$$ We fixed the integration constant by comparing with the decoupling limit, so that $F_{\text{Dirac}}$ is the $S^3$ free energy for a free Dirac fermion (two complex components) [@Klebanov:2011gs] $$F_{\text{Dirac}} = \frac{\log 2}{4} + \frac{3 \zeta(3)}{8 \pi^2}~.$$
Extrapolations to the $O(2)$ Model
----------------------------------
We can now attempt to extrapolate the perturbative results obtained above around the Dirac fermion cusp to the $O(2)$ cusp (see fig. \[fig:tauplane\]), to obtain the data of the $O(2)$ model. The $O(2)$ model, while being strongly coupled, is a well-studied theory via a variety of techniques, so that we can compare our extrapolations to the known data. Even though so far we only obtained the first two orders in perturbation theory, and one might be wary to already attempt an extrapolation, we will see that the results we obtain are compatible with the known data. We view this as an encouraging indication that the perturbative technique that we are presenting here can indeed be a useful tool to obtain data of 3d Abelian gauge theories, and as a motivation to try to obtain more precise predictions by going to higher orders.
In order to extrapolate, we need to apply a resummation technique. The nice property of our setup is the duality $\tau \leftrightarrow \tau'=-\frac{1}{4\tau}$, which means that the perturbative expansions obtained above also tell us about the behavior of the observables around $\tau' \to \infty$, i.e. the second Dirac fermion cusp. To leverage on this, the idea is to use a “duality-improved” Padé approximant, i.e. a function with a number of free parameters that we can fix by matching to the perturbative result at $\tau\to\infty$, and that is invariant under a duality transformation.
Similar resummations were studied in the context of perturbative string theory [@Sen:2013oza] and $\mathcal{N}=4$ super Yang-Mills (SYM) in [@Beem:2013hha]. In particular [@Beem:2013hha] introduced Padé-like approximants with the property of being invariant under a subgroup of $SL(2,\mathbb{Z})$, and we will borrow their method. Note that the perturbative results of the previous subsections, expressed in terms of $g_s = g^2$ and $\theta$, and expanded for small $g_s$ with $\theta$ fixed take the form $$\begin{aligned}
\gamma_{0} & = -\frac{4 }{3 \pi ^2}g_s - \frac{8- 9 \pi ^2}{27 \pi ^4} g_s^2 +\mathcal{O}\left(g_s^3,g_s^3\theta^2\right)~,\\
\gamma_{2} & = \frac{4 }{5 \pi ^2}g_s -\frac{ 16 + 75 \pi ^2}{750 \pi ^4} g_s^2+\mathcal{O}\left(g_s^3,g_s^3\theta^2\right)~,\\
f_{\partial} &=\frac{1}{32}g_s + \frac{368-45\pi^2}{9216\pi^2}g_s^2 +\mathcal{O}\left(g_s^3,g_s^3\theta^2\right)~,\end{aligned}$$ and $f_\partial$ is the boundary free energy where the contributions from free gauge field as well as the constant term have been subtracted. The expressions above all have the pattern $$a~g_s(1+ b~g_s+ \mathcal{O}(g_s^2,g_s^2\theta^2))~,$$ which can be approximated by the manifestly duality-invariant interpolation functions written in [@Beem:2013hha]. At this order, there are two of their functions that we can use, the integral-power Padé $F_1(g_s,\theta)$ and half-integral-power Padé $F_2(g_s,\theta)$, defined by $$\begin{aligned}
F_1(g_s,\theta) &= \frac{h_1}{g_s^{-1} + (\mathrm{S}\cdot g_s)^{-1} - h_2}~,\label{eq:F1}\\
F_2(g_s,\theta) &= \frac{h_3 \left(g_s^{-1/2}+(\mathrm{S}\cdot g_s)^{-1/2}\right)}{g_s^{-3/2}+(\mathrm{S}\cdot g_s)^{-3/2}+h_4 \left(g_s^{-1/2}+(\mathrm{S}\cdot g_s)^{-1/2}\right)}~.\label{eq:F2}\end{aligned}$$ where $\mathrm{S}\cdot g_s$ is the new gauge coupling under the transformation $\tau \rightarrow -\frac{1}{4 \tau}$, which reads explicitly $$\mathrm{S}\cdot g_s = \frac{g_s^2 \theta ^2+16 \pi ^4}{\pi ^2 g_s}~.$$ The unconventional negative power in the above two Padé approximant was devised in [@Beem:2013hha] to remove the $\theta$ dependence in the Taylor expansion. This is appropriate to match our perturbative expansion up to the order we are considering, because the $\theta$-dependence starts at the subleading order $g_s^3$. On the other hand, while the perturbative expansion of $\mathcal{N}=4$ SYM is independent of $\theta$ to all orders in perturbation theory, and therefore in that context it is desirable to have an ansatz whose Taylor expansion does not contain $\theta$, in our setup observables do depend on $\theta$ even in perturbation theory. Indeed, by taking a different scaling such as $g_s$ small with $\gamma = \frac{\theta g_s}{4 \pi^2}$ fixed, rather than $\theta$ fixed, we would have a non-trivial dependence on $\gamma$ already at the order we are considering, and with this scaling we could not match the observables with the Taylor expansion of the approximants -. The upshot is that in order to use the duality-improved approximants from [@Beem:2013hha] we are forced to consider the expansion at small $g_s$ with $\theta$ fixed, and doing so we throw away some of the information contained in the perturbative calculation, namely the $\frac{ g_s^2\gamma^2}{(1+\gamma^2)^2} = (2\pi)^2\frac{({\rm Re }\tau)^2}{|\tau|^4}$ terms. It would be desirable to find an ansatz that is: (i) duality invariant; (ii) has a final limit to the real $\tau$ axis (or at least to the $O(2)$ cusp); and (iii) can be matched with the perturbative expansion at small $g_s$ and $\gamma$ fixed (at least up to the order $g_s^2$ at which the observables are currently known).
By matching the coefficients in the expansion up to the order $g_s^2$, we find the unknown coefficients $h_i$ to be $$\begin{aligned}
h_1 = a, \quad h_2 = b, \quad h_3 = a, \quad h_4 = \frac{1}{4\pi} - b\end{aligned}$$ In the table \[tab:values\] we show the resulting values of the approximant extrapolated at the $O(2)$ point.
$2+ \gamma_1$ $3 + \gamma_2$ $f_\partial$
---------------------------------- --------------- ---------------- --------------
$\epsilon$ expansion 1.494 — 0.124
Bootstrap $1.5117(25)$ — —
$F_1(g_s = \infty,\theta = \pi)$ 1.406 3.635 1.039
$F_2(g_s = \infty,\theta = \pi)$ 1.560 3.391 0.166
: Comparison of the extrapolations with the known data: for the energy operator we are quoting the value from the conformal bootstrap [@Kos:2015mba], and from the $\epsilon$-expansion [@Kleinert:1991rg]. For the sphere free energy we are comparing to the value from the $\epsilon$-expansion in [@Fei:2015oha].[]{data-label="tab:values"}
The fermion-mass operator is mapped to the energy operator of the $O(2)$ model, whose dimension can be obtained for instance from the conformal bootstrap [@Kos:2015mba], or from the $\epsilon$-expansion [@Kleinert:1991rg]. The spin 2 operator is expected to approach the conserved stress-energy tensor on the boundary in the decoupling limit, hence the dimension should approach the protected value $\Delta_2|_{\text{ cusp}} = 3$. As for the hemisphere free energy, one needs to subtract a finite contribution coming from the decoupled gauge field at the $O(2)$ cusp, and the remaining constant gives the sphere free energy of the $O(2)$ model. To our knowledge this has only been computed using $\epsilon$-expansion [@Fei:2015oha].
We see that both ansatzes give good approximations for the dimension of the energy operator, and in particular $F_2$ is quite close to the values obtained with the other methods. For the other two observables, we see that the ansatz $F_2$ also gives compatible results, while $F_1$ is not as good. In fig.\[fig:Pade\] we show the plots of the approximants at $\theta = \pi$, i.e. the value of the $O(2)$ cusp, as a function of $g_s$ from $0$ to $\infty$.
![Extrapolations of the scaling dimensions from the Dirac fermion point ($\tan^{-1}(g_s) = 0$) to the $O(2)$ point ($\tan^{-1}(g_s) = \pi/2$).[]{data-label="fig:Pade"}](fig/Pade_gamma_0 "fig:"){width="0.46\linewidth"} ![Extrapolations of the scaling dimensions from the Dirac fermion point ($\tan^{-1}(g_s) = 0$) to the $O(2)$ point ($\tan^{-1}(g_s) = \pi/2$).[]{data-label="fig:Pade"}](fig/Pade_gamma_2 "fig:"){width="0.46\linewidth"}
![Extrapolations of the free energy from the Dirac fermion point ($\tan^{-1}(g_s) = 0$) to the $O(2)$ point ($\tan^{-1}(g_s) = \pi/2$).[]{data-label="fig:Pade_freeenergy"}](fig/Pade_free_energy){width="0.46\linewidth"}
Other Examples {#sec:exa}
==============
$2N_f$ Dirac fermions at large $N_f$
------------------------------------
In this section we consider the coupling of $2N_f$ Dirac fermions to the bulk gauge fields, all with the same charge $q=1$, and we take the limit of large $N_f$ with $\lambda = g^2 N_f$ fixed. For simplicity we take $\theta = 0$. We will see that computing observables in $1/N_f$ expansion, and later taking the limit $\lambda \to \infty$, one can recover the $1/N_f$ expansion in QED$_3$. This would be the expected result if we would take $g^2 \to \infty$ first, obtaining the decoupling limit in which on the boundary we have QED$_3$ with $2N_f$ flavors, and later take $N_f$ large. Hence, the observation here is that these two limits commute. This is interesting because order by order in $1/N_f$ we can explicitly follow observables as exact functions of $\lambda$, and see how they interpolate from the “ungauged cusp” at $\lambda = 0$ to the “gauged cusp” at $\lambda =\infty$.
To derive that the limits commute, it is sufficient to observe that in the limit of large $N_f$ with $\lambda = g^2 N_f$ fixed we can obtain an effective propagator for the photon by resumming the fermionic bubbles, see Fig. \[fig:LargeNf\], obtaining (up to gauge redundancy)
![The diagrams that contribute to the boundary propagator of the photon in the limit $N_f \to \infty$ with $\lambda = g^2 N_f$ fixed.[]{data-label="fig:LargeNf"}](fig/LargeNf)
$$\begin{aligned}
\Pi^{(1/N_f)}_{a b}(\vec{p}) = & \frac{1}{N_f |\vec{p}\,|} \lambda \sum_{k=0}^{\infty}\left(- \frac{\lambda}{8}\right)^k\left(\delta_{ab} - \frac{p_a p_b}{\vec{p}^{~2}}\right) \\
= & \frac{8 }{N_f |\vec{p}\,| }\frac{\lambda}{\lambda + 8} \left(\delta_{ab} - \frac{p_a p_b}{\vec{p}^{~2}}\right) ~.\label{eq:effphpr}\end{aligned}$$
In the limit $\lambda \to \infty$ the propagator becomes $$\Pi^{(1/N_f)}_{a b}(\vec{p}) \underset{\lambda\to\infty}{\longrightarrow} \frac{8}{N_f |\vec{p}\,| } \left(\delta_{ab} - \frac{p_a p_b}{\vec{p}^{~2}}\right) ~,$$ which coincides with the effective propagator in QED$_3$ at large $N_f$. It follows that compared to the large-$N_f$ expansion of QED$_3$, in this setup the diagrams that compute $1/N_f$ corrections are simply dressed by a factor $\lambda/(\lambda + 8)$ for each photon propagator. In particular the $1/N_f$-expansion of observables, e.g. boundary scaling dimensions, will approach the corresponding value in large-$N_f$ QED$_3$ upon taking the limit $\lambda \to\infty$. However, recall that in the $1/N_f$-expansion diagrams that contribute at the same order might have different number of internal photon lines, so we cannot just replace $1/N_f$ with $1/N_f\times\lambda/(\lambda + 8)$ everywhere to obtain the exact dependence on $\lambda$ of a certain observable.
Let us now consider the two-point function of the boundary current $\hat{J}$, and obtain from it the hemisphere free energy at large $N_f$. We can obtain the $1/N_f$ correction to the one-photon irreducible two-point function of $\hat{J}$ —computed by the diagrams in Fig. \[fig:currentcurrent\] with the effective photon propagator — by taking the result of the large-$N_f$ calculation in [@Giombi:2016fct] and dressing it by the factor due to the single photon propagator, with the result $$\begin{aligned}
c_\Sigma=\frac{N_f}{4 \pi ^2} \left(1+\frac{1}{N_f}\frac{\lambda}{\lambda+8}\frac{184 - 18 \pi^2}{9 \pi ^2}+{\mathcal{O}}\left({N_f^{-2}}\right)\right)~.\end{aligned}$$ Correspondingly, from equation and we have $c_{12}=0$ and $$\begin{aligned}
c_{22} = \frac{16 }{\pi ^2 N_f}\frac{\lambda}{\lambda + 8}-\frac{32 \left(92 - 9 \pi ^2\right)}{9 \pi ^4 N_f^2}\frac{\lambda^3}{(\lambda + 8)^3}+\mathcal{O}\left(N_f^{-3}\right)~.\end{aligned}$$ We can now plug $c_{22}$ in the differential equation , appropriately rewritten in terms of the variable $\lambda$. Solving for $F_\partial(\lambda)$ up to the order $1/N_f$ we find $$\begin{aligned}
\label{eq:Flambda}
F_\partial(\lambda )= \frac{1}{4}\log \left[\frac{\pi N_f (\lambda + 8)^2}{64 \lambda}\right] + 2N_f
F_{\text{Dirac}}+\frac{\left(92-9 \pi ^2\right) }{18 \pi ^2 N_f }\frac{\lambda^2}{(\lambda + 8)^2} +{\mathcal{O}}\left({N_f^{-2}}\right)~.\end{aligned}$$ Recall that the arbitrary integration constant is fixed by matching with the decoupling limit. In the decoupling limit $F_\partial$ is the sum of a contribution from the free fermions on the boundary, namely $2N_f F_{\text{Dirac}}$, and a contribution from the boundary value of the gauge field with Neumann condition, that we discussed in section \[sec:FreeEnergy\]. The latter contribution is only a function of $g^2$, and when rewritten in terms of $\lambda$ it gives a $\log(N_f)$ constant term. Hence we need to include such a dependence on $N_f$ in the integration constant, and this is how we obtain the $\log(N_f)$ term in . Similarly, we find that a $\lambda$-independent term of order $1/N_f$ needs to be included in the integration constant, to ensure that the $1/N_f$ correction vanishes when $\lambda = 0$. The general lesson here is that when we integrate the equation in the $\lambda$ variable, the integration constant required to reproduce the decoupling limit will be a non-trivial function of the parameter $N_f$.
From the $\lambda \to \infty$ limit of we can extract the sphere free-energy QED$_3$ at large $N_f$. More specifically, the latter is obtained by subtracting to the $\lambda\rightarrow \infty$ limit of the free energy the contribution of the Neumann boundary condition of the bulk gauge field computed at $(g')^2=\frac{4\pi^2}{g^2}$, namely $$\begin{aligned}
F_{\text{QED}_3}& = \lim_{\lambda \to \infty}\left(F_\partial(\lambda) + \left. \frac{1}{4}\log \left[\frac{(g')^2}{\pi}\right]\right\vert_{(g')^2 = \frac{4\pi^2 N_f}{\lambda}}\right) \\
& =2N_f
F_{\text{Dirac}}+\frac{1}{2} \log \left(\frac{\pi N_f}{4}\right)+\frac{92 -9\pi^2}{18 \pi ^2}\frac{1}{N_f}+{\mathcal{O}}\left({N_f^{-2}}\right)~.\label{eq:largeNfF}\end{aligned}$$ Both the logarithmic and the constant terms reproduce perfectly the result of [@Klebanov:2011td]. To our knowledge, the ${\mathcal{O}}\left({N_f^{-1}}\right)$ correction was not computed before, even though its connection with the two-point function of the gauged current is clear from [@Klebanov:2011td].
As we will now briefly review, the free-energy as a function of $N_f$ can be used to diagnose the IR fate of QED$_3$. For $N_f$ smaller than a critical value $N_f^c$ the theory is conjectured to flow to a flavor-symmetry breaking phase rather than to the conformal phase that exists at large $N_f$. A possible scenario for the transition is that the IR scaling dimension of singlet four-fermion operators would cross marginality [@Braun:2014wja; @DiPietro:2015taa; @DiPietro:2017kcd], implying that the IR fixed point that exists at large $N_f$ merges at $N_f = N_f^c$ with a second fixed point in which the quartic operators are turned on, and they both disappear [@Giombi:2015haa; @Gukov:2016tnp]. After the transition they can still be interpreted as complex fixed points [@Kaplan:2009kr; @Gorbenko:2018ncu]. This scenario was recently investigated in [@Benvenuti:2018cwd] using large $N_f$ techniques and in [@Li:2018lyb] using the conformal bootstrap. This merger/annihilation scenario, together with the monotonicity of the sphere free-energy along RG, was used in [@Giombi:2015haa] to constrain $N_f^c$: assuming that $F_{\text{QED}_3}$ can still be interpreted as the free-energy of the nearby complex fixed point when $N_f < N_f^c$, the existence of the RG flow from the vicinity of the complex fixed point towards the symmetry breaking phase requires that $F_{\text{QED}_3} > F_{\text{G.B.}}$ for $N_f < N_f^c$. Here $F_{\text{G.B.}} = (2 N_f^2 +1) F_{\text{scalar}} $ is the free energy of the Goldstone bosons in the symmetry breaking phase. As an application of the calculation above, we can now run this argument using the large-$N_f$ approximation for $F_{\text{QED}_3}$ in eq. . It turns out that the coefficient of the $1/N_f$ term is numerically very small, i.e. $\sim 0.02$, so for the interesting values of $N_f$ of order 1 it does not affect significantly this test, and the resulting estimate is $N_f^c \sim 4.4$. For this value of $N_f$, the $1/N_f^2$ corrections that we are neglecting in are quite small, and assuming that the smallness of the coefficients persists at higher orders this suggests that the estimate might be reliable.
Complex Scalar {#sec:Scalar}
--------------
In section \[sec:MinPhTr\] we studied the case a free fermion on the boundary, and we saw that one of the gauged cusps correspond to the $O(2)$ Wilson-Fisher model. This is a consequence of the boson/fermion dualities that relate a gauged fermion to a critical scalar, or a gauged critical scalar to a free fermion [@Seiberg:2016gmd]. These dualities can be seen as the low-rank analogue of the large-$N$ regular fermion/critical scalar dualities in CS-matter theories [@Giombi:2011kc; @Aharony:2012nh; @GurAri:2012is; @Aharony:2015mjs]. Besides the Wilson-Fisher fixed point, the scalars also admit the Gaussian fixed point consisting of $N$ free complex scalars. Likewise the theory of $N$ Dirac fermions is conjectured to have a second fixed point with quartic interactions turned on, i.e. the UV fixed point of the Gross-Neveu model. The corresponding CS-matter theories are also conjectured to be dual in a level-rank duality fashion, giving the so-called regular boson/critical fermion duality. There is a large amount of evidence for this duality at large $N$, and its extension to finite $N$ was recently studied in [@Aharony:2018pjn; @Dey:2018ykx]. It is not clear whether the duality still holds when $N=1$. Assuming it does, it would have a nice manifestation in our setup: by starting with a free complex scalar on the boundary, one would find that the cusp at $\tau =1$ corresponds to the Gross-Neveu CFT with 1 Dirac fermion.[^9] One crucial new ingredient of the regular boson/critical fermion dualities is the existence of a additional sextic couplings that are classically marginal and potentially lead to multiple fixed points that can be mapped across the duality.
With this motivation in mind, we will now consider the setup with a free complex scalar on the boundary, coupled to the bulk gauge field. The action is $$S[A,\tau] + \int_{y = 0} d^3\vec{x} \ \big(|D_A\phi|^2 + \rho(|\phi|^2)^3\big)~.$$ The couplings $|\phi|^2$ and $|\phi|^4$ are fine-tuned to zero. This fine-tuning might need to be adjusted as a function of the bulk gauge coupling. At least for $\tau$ large enough, these operators are relevant and correspondingly the beta function is linear in the couplings, so this adjustment is possible. On the other hand, the beta function for the classically marginal operator $|\phi|^6$ will start quadratically in $\rho$ and we need to check the existence of (real) fixed points.
We list the Feynman rules in the Fig. \[fig:scalarfeynrule\].
![[]{data-label="fig:scalarfeynrule"}](fig/scalarFeynRule2){width="0.8\linewidth"}
To compute the $\beta$ function of $\rho$ we need to renormalize the six-point vertex. We use the same approach as in the fermion case, i.e. we dimensionally regularize by continuing the dimension of the boundary to $d = 3-2\epsilon$, keeping the codimension fixed $=1$. The boundary action in renormalized variables is $$\int_{y=0} d^d \vec{x}\ |D\phi_B|^2 + \rho_B |\phi_B|^6 = \int_{y=0} d^d \vec{x} \ Z_\phi^2|D\phi|^2 + Z_\rho \rho \mu^{4\epsilon}|\phi|^6 ~,$$ where the subscript $B$ denotes the bare variables. Fig. \[fig:scalarZphioneloop\] shows the diagrams that contribute to the wavefunction renormalization of the field $\phi$, from which we obtain $$\delta Z_\phi = -\frac{ (3 \xi -8)}{24 \pi ^2 \epsilon }\frac{g^2}{1+\gamma^2} + \mathcal{O}(g^4)~.\label{eq:anodimphi}$$
![[]{data-label="fig:scalarZphioneloop"}](fig/Zphi_12){width="0.8\linewidth"}
There are three types of diagram contributing to the six-point vertex counterterm, showed in Fig. \[fig:scalarZrhoa06\] and \[fig:scalarZrhoa12\], from which we can compute $$\rho\delta Z_\rho = \frac{15}{8 \pi ^2 \epsilon } \rho^2 - \frac{3}{4\pi^2\epsilon}\frac{g^2}{1+\gamma ^2}\xi\,\rho -\frac{24(1-3\gamma^2)}{\pi ^2 \epsilon }\left(\frac{g^2}{1+\gamma ^2}\right)^3 + \mathcal{O}(\rho^3, \rho^2 g^2, \rho g^4, g^8)~.$$
![Diagrams contributing to $\mathcal{O}(\rho^2)$ and $\mathcal{O}(\rho g^2)$ in $\beta_\rho$.[]{data-label="fig:scalarZrhoa12"}](fig/Zrho282){width="0.75\linewidth"}
The $\beta$ function is $$\begin{aligned}
\beta_\rho(\rho,g) &=\left.\left(- 4\epsilon\rho - \rho \frac{\partial \log Z_\rho/Z_{\phi}^6}{\partial \log\mu} \right)\right\vert_{\epsilon=0}\\
& = \frac{15}{2\pi^2} \rho^2 -\frac{4}{\pi^2}\rho\frac{g^2}{1+\gamma^2} -\frac{48(1-3\gamma^2)}{\pi^2}\left(\frac{g^2}{1+\gamma^2}\right)^3 + \mathcal{O}(\rho^3, \rho^2 g^2, \rho g^4, g^8)~.\end{aligned}$$ Up to this order we find: a zero at $\rho=\rho_*^{+} > 0$ from the first two terms, and since $\rho_*^{+} = \mathcal{O}(g^2)$ the third term is negligible; and a zero at $\rho=\rho_*^{-}$ from the second and the third therm, and since $\rho_*^{-} = \mathcal{O}(g^4)$ the first term is negligible. The positions of the zeroes are $$\rho_*^{+} = \frac{8}{15}\frac{g^2}{1+\gamma^2} + \mathcal{O}(g^4)~, \qquad \rho_*^{-} = -12(1-3\gamma^2)\left(\frac{g^2}{1+\gamma^2}\right)^2+ \mathcal{O}(g^6)~.$$ The derivative of $\beta_\rho$ is positive at $\rho_*^+$ and negative at $\rho_*^-$. Hence we have found that perturbatively around large $\tau$ there exists a fixed point $\rho=\rho_*^+$ which is IR stable, and gives a scalar potential bounded from below. The fixed point $\rho_*^-$ on the other hand is only physical for $1-3\gamma^2<0$, because otherwise it gives the wrong sign of the scalar potential, and it is unstable in the RG sense.
Having checked the existence of the fixed point in perturbation theory, we proceed to consider the anomalous dimension of boundary operators in this theory, similarly to what we did in section for the fermion case. We consider the mass-squared operator $O = |\phi|^2$. Its anomalous dimension can be obtained from the renormalization of the 1PI correlator of the composite operator with two elementary fields $$\langle O(q=0) \phi(-p) \bar{\phi}(p)\rangle_{\rm 1PI}~. \label{eq:scalar3pointfn}$$
The one-loop (two-loop) diagrams contributing to the three-point function are showed in Fig.\[fig:scalarZphioneloop\] (Fig.\[fig:scalarZmTwoloop\], respectively).
At one loop, using , the renormalization constant of the operator is found to be $$\delta Z_{O} = -\frac{2}{3\pi^2\epsilon} \frac{g^2}{1+\gamma^2} + \mathcal{O}(g^4)~,$$ and correspondingly the anomalous dimension is $$\gamma_O = -\frac{4}{3\pi^2} \frac{g^2}{1+\gamma^2}+ \mathcal{O}(g^4)~.$$
![One loop and two loops diagrams[]{data-label="fig:scalarZmTwoloop"}](fig/scalarTwoLoop2){width="\linewidth"}
Differently from the fermion case, we were not able to evaluate all of the dimensionally-regularized integrals coming from the two-loop diagrams of Fig. \[fig:scalarZmTwoloop\]. See the appendix \[app:FeynInt\] for the details. Knowing the two-loop anomalous dimension would enable an extrapolation to $\tau=1$ that could be compared with the known estimates of the mass operator in the Gross-Neveu CFT. This is therefore an interesting direction left for the future.
QED$_3$ with Two Flavors {#sec:QEDtwoFlavors}
------------------------
In this section we will discuss a realization in our setup of QED$_3$ coupled to two Dirac (complex two-component) fermions of charge $1$. There are several reasons why this is an interesting theory: it is conjectured to describe the easy-plane version of the “deconfined” Néel-VBS quantum phase transition in antiferromagnets [@PhysRevB.70.144407], and enjoy an emergent $O(4)$ symmetry [@Hsin:2016blu; @Cordova:2017kue]; while initially believed to be a second-order transition, recent evidences from simulations of the spin system on the lattice [@Qin:2017cqw] and from the conformal bootstrap [@IliesiuTALK] suggest that this is actually a weakly first-order transition, which can still be compatible with the QED description if the latter has a complex fixed point with $O(4)$ symmetry (see section 5 of [@Gorbenko:2018ncu] and [@Benvenuti:2018cwd]); it is conjectured to enjoy a self-duality [@Hsin:2016blu; @Cordova:2017kue; @Xu:2015lxa; @Karch:2016sxi] and a fermion-boson duality [@Wang:2017txt].
A simple way to realize QED with two flavors in our setup would be to put the CFT of two Dirac fermions on the boundary, and couple a bulk gauge field to the $U(1)$ symmetry that gives charge 1 to both of them. However in this case we only expect a weakly coupled cusp at $\tau \to \infty$. For the purpose of attempting an extrapolation from weak coupling, it would be desirable to have additional weakly coupled cusps, as in the example of section \[sec:MinPhTr\]. With this idea in mind, a more promising approach is to consider a generalization of the former set-up in which we have two Maxwell gauge fields in the bulk and two Dirac fermions on the boundary, namely two decoupled copies of the theory of section \[sec:MinPhTr\]. By performing an $S$-duality for either of the two gauge fields separately we find again two free Dirac fermions on the boundary. On the other hand using the larger electric-magnetic duality group that exists for a theory of two gauge fields, we can also go to a duality frame where in the decoupling limit we have precisely QED with two flavors on the boundary.
In the rest of this section we will first review electric-magnetic duality for multiple Maxwell fields, and then show how to get QED with two flavors starting with two copies of a bulk gauge field coupled to a boundary Dirac fermion. The task of performing perturbative calculations of observables in this theory is left for the future.
### Multiple Maxwell Fields
The action of free bulk $U(1)^n$ gauge theory is determined in terms of $n$ Abelian gauge fields $A^{I}$, such that $F^I=d A^I$ and an $n \times n$ symmetric matrix of complexified gauge couplings $\tau_{IJ}$ $$\begin{aligned}
S[A^I,\tau_{IJ}] &= \int_{y\geq 0} d^4x \left(\frac{1}{4g^2_{IJ}} F^I_{\mu\nu} F^{J,\mu\nu} +\frac{i\theta_{IJ}}{32\pi^2} \epsilon_{\mu\nu\rho\sigma}F^{I,\mu\nu} F^{J,\rho\sigma}\right)\\
&= -\frac{i}{8\pi} \int_{y\geq 0} d^4x (\tau_{IJ} F^{-I}_{\mu\nu}F^{-J,\mu\nu}-\bar{\tau}_{IJ}F^+_{I\mu\nu}F^{+J\mu\nu}),\end{aligned}$$ where $\tau_{IJ}=\frac{\theta{IJ}}{2\pi}+\frac{2\pi i}{g_{IJ}^2}$ and we introduced $F^{\pm,I}_{\mu\nu} = \frac12 (F^I_{\mu\nu} \pm \frac12 \epsilon_{\mu\nu\rho\sigma} F^{I,\rho\sigma})$. This theory enjoys an $Sp(2n,\operatorname{\mathbb{Z}})$ duality group $$\tau'_{IJ}=
({A_I^K\tau_{LM} + B_{IM}})({C^{JN}\tau_{NM} + D^J_M})^{-1},$$ where $$M = \left(\begin{array}{cc}
A & B \\
C & D
\end{array} \right) \in Sp(2n,\operatorname{\mathbb{Z}})~.$$ This duality group is generated by the three types of transformations obtained in [@hua_generators_1949; @Dimofte:2011ju], which we reproduce here [^10]
$$\begin{aligned}
\text{T-type:}\, & \left(\begin{array}{cc}
I & B \\
0 & I
\end{array} \right),\,
&
\begin{array}{ll}
\text{where $I$ the $n\times n$ identity and $B$ is a symmetric}\\
\text{matrix that generates $\tau' = \tau+B$,}
\end{array}\end{aligned}$$
$$\begin{aligned}
\text{S-type:}\, & \left(\begin{array}{cc}
I-J & -J \\
J & I-J
\end{array} \right),\,
&
\begin{array}{ll}
\text{where $J= \text{diag}(j_1,j_2,\dots,j_n)$ and $j_i \in \{0,1\}$.}\\
\text{This gauges those $A_i$'s that have $j_i =1$.}
\end{array}\end{aligned}$$
$$\begin{aligned}
\text{GL-type:}\, & \left(\begin{array}{cc}
U & 0 \\
0 & U^{-1T}
\end{array} \right),\,
&
\begin{array}{ll}
\text{where $U\in SL(n,\operatorname{\mathbb{Z}})$ generate the rotations $A' = U^{-1T} A$.}
\end{array}\end{aligned}$$
In the rest of this section we will be focusing on the case of $n=2$. Following [@hua_generators_1949] we define the generators of $Sp(4,\operatorname{\mathbb{Z}})$ as $$T =
\begin{pmatrix}
\begin{matrix}
1 & \\
& 1
\end{matrix}
& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}& \begin{matrix}
1 & \\
& 0
\end{matrix} \\
\hline
{\mbox{\normalfont\Large\bfseries 0}}& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&
\begin{matrix}
1 & \\
& 1
\end{matrix}
\end{pmatrix}~,
\qquad S =
\begin{pmatrix}
\begin{matrix}
0& \\
&1
\end{matrix}
& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}& \begin{matrix}
-1& \\
&0
\end{matrix} \\
\hline
\begin{matrix}
1& \\
&0
\end{matrix} & {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&
\begin{matrix}
0& \\
&1
\end{matrix}
\end{pmatrix}~,$$ $$R_1 =
\begin{pmatrix}
\begin{matrix}
& 1 \\
1 &
\end{matrix}
& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}& {\mbox{\normalfont\Large\bfseries 0}}\\
\hline
{\mbox{\normalfont\Large\bfseries 0}}& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&
\begin{matrix}
&1 \\
1 &
\end{matrix}
\end{pmatrix}~, \qquad R_2 =
\begin{pmatrix}
\begin{matrix}
1 & 1 \\
0 & 1
\end{matrix}
& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}& {\mbox{\normalfont\Large\bfseries 0}}\\
\hline
{\mbox{\normalfont\Large\bfseries 0}}& {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&
\begin{matrix}
1&0 \\
-1 &1
\end{matrix}
\end{pmatrix}~.$$ Furthermore we use the succinct notation $S[1,0]$, $S[0,1]$ to denote the gauging of $A^1$, $A^2$ (respectively) and $T[m,n]$ for the introduction of the Chern-Simons terms $m A^1dA^1 +n A^2dA^2$.
### Targeting two-flavor QED
We now have all the tools to obtain two-flavour QED$_3$ via an $Sp(4,\operatorname{\mathbb{Z}})$ action from a theory of two free fermions. The action of two-flavour QED$_3$ is [@Cordova:2017kue][^11]
$$S[A'^I,\tau'_{IJ}] + \int_{y=0} \left(i\bar{\psi}_1\cancel{D}_{a} \psi_1 +i\bar{\psi}_2\cancel{D}_{a+A'^1} \psi_2 +\frac{1}{4\pi} ada +\frac{1}{2\pi} adA'^2 -\frac{1}{4\pi} A'^2dA'^2 \right) +2{\rm CS}_g~,
\label{eq:two_flavors}$$
where $A'^{I=1,2}$ are bulk U(1) gauge fields while $a$ is a 3d spin$_c$ connection. The gravitational term ${\rm CS}_g$ is needed because $$\int_{\partial M} \frac{1}{4\pi} ada + 2{\rm CS}_g=2\pi \int_M \left(-\frac{1}{48} \frac{\Tr R\wedge R}{(2\pi)^2} + \frac{1}{8\pi^2}f\wedge f\right)~,$$ which is well-defined for a spin$_c$ connection $a$.[^12]
We want to target this action via an $Sp(4,\operatorname{\mathbb{Z}})$ transformation from $$S[A^I,\tau_{IJ}] +\int_{\partial M} (i\bar{\psi}_1\cancel{D}_{A^1} \psi_1 +i\bar{\psi}_2\cancel{D}_{A^2} \psi_2)~,
\label{eq:free_two_flavors}$$ where $A^{I=1,2}$ are spin$_c$ connections. To this end, we can start from a rotation of the gauge fields by performing a GL-type transformation with $$U= \left(\begin{array}{cc}
1 & -1 \\
0 & 1
\end{array} \right)~,$$ and then act with $T[1,0](-1)S[1,0]T[-1,0]$.[^13] The resulting relation between $\tau$ and $\tau'$ is $$\label{eq:twoflavortau}
\left(\begin{array}{cc}
\tau_{11} & \tau_{12} \\
\tau_{21} & \tau_{22}
\end{array}\right)=\left(
\begin{array}{cc}
-\tau'_{12}+\tau'_{22}-\frac{(\tau'_{12}+1) (-\tau'_{11}+\tau'_{21}+2)}{\tau'_{11}-1} & \frac{-\tau'_{12} (\tau'_{21}+1)+(\tau'_{11}-1) \tau'_{22}}{\tau'_{11}-1} \\
\frac{-(\tau'_{12}+1) \tau'_{21}+(\tau'_{11}-1) \tau'_{22}}{\tau'_{11}-1} & \frac{(\tau'_{11}-1) \tau'_{22}-\tau'_{12} \tau'_{21}}{\tau'_{11}-1} \\
\end{array}
\right)~.$$ The decoupling limit of is $$\begin{aligned}
\left(\begin{array}{cc}
\tau_{11}' & \tau_{12}' \\
\tau_{21}' & \tau_{22}'
\end{array}\right)= \left(\begin{array}{cc}
\infty &0\\
0&\infty
\end{array}\right)~,\end{aligned}$$ which according to corresponds to $$\left(\begin{array}{cc}
\tau_{11} & \tau_{12} \\
\tau_{21} & \tau_{22}
\end{array}\right) = \left(\begin{array}{cc}
1+\infty & \infty \\
\infty & \infty
\end{array} \right)~,\label{eq:QEDtwoflav}$$ by which we mean $\tau_{12} - \tau_{22} = \tau_{21} - \tau_{22}= \tau_{11} -1-\tau_{22} = 0$ is satisfied while taking the limit $\tau_{22} \rightarrow \infty$.
Let us also write down explicitly the self-dualities of the theory .[^14] Recall from section \[sec:MinPhTr\] that $$S[A,\tau]+ \int_{y=0} i \bar{\psi}\cancel{D}_A\psi~,$$ and $$S[A',\tau'] + \int_{y=0} i \bar{\chi} \cancel{D}_{A'} \chi~,$$ are equivalent when $\tau' = ST^{-2}ST^{-1}\circ\tau = (\tau-1)/(2\tau-1)$. Applying this to either $A^1$ or $A^2$ in , we obtain that the decoupling limits in the two following duality frames also correspond to two free Dirac fermions $$\begin{aligned}
\tau''_{IJ} &= S[1,0]T[-2,0]S[1,0]T[-1,0]\circ \tau_{IJ}~, \\
\tau'''_{IJ} &= S[0,1]T[0,-2]S[0,1]T[0,-1]\circ \tau_{IJ}~. \end{aligned}$$ Hence, in the variable $\tau_{IJ}$ the theory has weakly coupled cusps at $$\left(\begin{array}{cc}
\tau_{11} & \tau_{12} \\
\tau_{21} & \tau_{22}
\end{array}\right) =\left(\begin{array}{cc}
\infty & 0 \\
0 & \infty
\end{array} \right)~,\quad\left(\begin{array}{cc}
\pm\frac{1}{2} & 0 \\
0 & \infty
\end{array} \right)~,\quad
\left(\begin{array}{cc}
\infty & 0 \\
0 & \pm\frac{1}{2}
\end{array} \right)~.\label{eq:weaklycoup}$$
To summarize, we showed that the theory of two bulk gauge fields coupled to two Dirac fermions has two additional duality frames in which the boundary theory is still the free theory of two Dirac fermions, and a duality frame in which the boundary theory is QED$_3$ with two flavors. Clearly, additional duality frames corresponding to QED$_3$ with two flavors can be obtained by applying the transformation to either of the additional free-fermions points. This is a promising setup to study QED$_3$ with two flavors via an extrapolation from the weakly-coupled points.
Future Directions {#sec:conc}
=================
We conclude by discussing some directions for future investigation.
- [A universal feature of the setup considered in this paper is the existence of bulk line operators, whose endpoints may be attached to boundary charged operators. It is possible to assign conformal dimensions to the local operators at the location where the line defect ends on the boundary, and these dimensions can be computed perturbatively. Similarly to cusp anomalous dimensions, they are functions of the angle between the defect and the boundary. Starting with the dimensions of the endpoints of ’t Hooft lines (and ’t Hooft-Wilson lines) around $\tau\to\infty$ with a certain CFT on the boundary, it would be interesting to attempt an extrapolation to the cusps on the real axis, where they approach the dimensions of local monopole operators in the gauged version of the initial CFT. Concretely, in the example of section 4, from the dimension of the endpoint of a ’t Hooft line around the Dirac fermion point one can attempt to recover the scaling dimension of the spin operator of the $O(2)$ model.]{}
- [It would be interesting to perform perturbative calculations of anomalous dimensions and of the free energy in the theory with two bulk gauge fields presented in section \[sec:QEDtwoFlavors\], and attempt an extrapolation to QED$_3$ with two flavors. In particular, it is possible to use our setup to test whether this theory exists as a real CFT, by studying the dimension of four-fermion operators and checking whether they cross marginality before we reach the QED cusp, leading to the “phase-transition” described in section \[sec:Strong\].]{}
- [In the model considered in section \[sec:MinPhTr\] we have only used the two-sided extrapolations to give estimates for the $O(2)$ model. However there are infinitely many other cusps on the real axis where strongly-coupled CFTs live, and they are of course amenable to the same extrapolation technique. These theories typically take the form of QED-CS theories, and they also describe interesting phase transitions [@Lee:2018udi]. A direction for the future would be to use our method to give estimates for the observables of these theories.]{}
- [Finally, dualities analogous to the one considered in this paper exist for $\mathcal{N}=2$ gauge theory. One of the simplest examples is the so-called triality [@Intriligator:1996ex; @deBoer:1996ck; @deBoer:1997kr; @Aharony:1997bx] generated by $ST$ transformation [@Dimofte:2011ju; @Dimofte:2011py], with $(ST)^3 = 1$. It would be interesting to see how the triality can improve the extrapolation. Thanks to supersymmetric localization the boundary free energy and dimensions of chiral endpoints of line operators are exactly computable [@Gaiotto:2014gha]. For many other interesting observables, such as the conformal dimensions of operators analogous to $O_0$, which are non-protected, one has to resort to Feynman diagrams.]{}
Acknowledgements {#acknowledgements .unnumbered}
=================
We thank the Simons Collaboration on the Non-perturbative Bootstrap for organizing many stimulating conferences and workshops where part of this work was carried out. We are grateful to M. Baggio, N. Bobev, S. Chester, S. Cremonesi, M. Del Zotto, M. Meineri, G. Tartaglino-Mazzucchelli, E. Stamou, E. Trevisani and B. van Rees for stimulating discussions. EL thanks the Perimeter Institute for Theoretical Physics for hospitality. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. EL is supported by the Simons Foundation grant $\#$488659 (Simons Collaboration on the non-perturbative bootstrap).
Method of Images {#app:Imag}
================
In this appendix we show how to compute the two-point function of $F_{\mu\nu}$ in the free theory using the method of images.
Reflections about the boundary are implemented by the matrix $$R_\mu^{~\nu} = \delta_\mu^{~\nu} - 2n_\mu n^\nu~,$$ where $n^\mu$ is the inward pointing vector normal to the boundary. Note that the reflection of the field strength $$F_{\mu\nu}^R(x) \equiv R_\mu^{~\mu'} R_\nu^{~\nu'}F_{\mu'\nu'}(R\,x)$$ has components $(F^R_{ya}(x),\,\widetilde{F^R_{ya}}(x)) = (-F_{ya}(R\,x),\,\tilde{F}_{ya}(R\,x)) $. Hence, the combination $$\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^3 \times \mathbb{R}_+} \equiv \langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4} - s \langle F_{\mu\nu}(x_1) F^R_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4}~,\label{eq:image}$$ satisfies the equation of motion and Bianchi identity for $y \geq 0$, and also satisfies the Dirichlet (Neumann with $\gamma = 0$) boundary condition upon choosing the sign $s=1$ ($s=-1$, respectively). Even though Bose symmetry is not manifest in , it is satisfied because $\langle F_{\mu\nu}(x_1) F^R_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4} = \langle F^R_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4}$. We can then rewrite the image term using the cross-ratio $\xi$ and the vectors $X_{i\,\mu}$ by means of the following identity $$R_\rho^{~\rho'} I_{\mu\rho'}(x_1 - R x_2) = I_{\mu\rho}(x_{12}) - 2 X_{1\,\mu} X_{2\,\rho} ~.$$ In this way we find .
In the more general case of Neumann boundary condition with $\gamma \neq 0$, consider the combination $$\begin{aligned}
F'_{\mu\nu} & = F_{\mu\nu} + i \gamma \tilde{F}_{\mu\nu} = \mathcal{M}_{\mu\nu}^{~~~\mu'\nu'}F_{\mu'\nu'}~ \\
\mathcal{M}_{\mu\nu}^{~~~\mu'\nu'} & =\delta_{[\mu}^{\mu'}\delta_{\nu]}^{\nu'} + i \frac{\gamma}{2}\epsilon_{\mu\nu}^{~~~\mu'\nu'}~.\end{aligned}$$ For $F'_{\mu\nu}$ the problem is reduced to the Neumann boundary condition with $\gamma =0$, so we have $$\langle F'_{\mu\nu}(x_1) F'_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^3 \times \mathbb{R}_+} \equiv \langle F'_{\mu\nu}(x_1) F'_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4} + \langle F'_{\mu\nu}(x_1) (F')^R_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4}~.\label{eq:fprime}$$ Note that $$\begin{aligned}
(F')^R_{\rho\sigma}(x) & = \overline{\mathcal{M}}_{\mu\nu}^{~~~\mu'\nu'}F^R_{\mu'\nu'}~,\\
\overline{\mathcal{M}}_{\mu\nu}^{~~~\mu'\nu'}& =\delta_{[\mu}^{\mu'}\delta_{\nu]}^{\nu'} - i \frac{\gamma}{2}\epsilon_{\mu\nu}^{~~~\mu'\nu'}~.\end{aligned}$$ Multiplying both sides of by $\mathcal{M}^{-1}\otimes\mathcal{M}^{-1}$ we obtain $$\langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^3 \times \mathbb{R}_+} = \langle F_{\mu\nu}(x_1) F_{\rho\sigma}(x_2) \rangle_{\mathbb{R}^4} + (\mathcal{M}^{-1}\overline{\mathcal{M}})_{\rho\sigma}^{~~~\rho'\sigma'}\langle F_{\mu\nu}(x_1) F^R_{\rho'\sigma'}(x_2) \rangle_{\mathbb{R}^4}~.$$ Finally we use that $$(\mathcal{M}^{-1}\overline{\mathcal{M}})_{\rho\sigma}^{~~~\rho'\sigma'} = \frac{1-\gamma^2}{1+\gamma^2} \delta_{[\rho}^{\rho'}\delta_{\sigma]}^{\sigma'} - i \frac{\gamma}{1+\gamma^2} \epsilon_{\rho\sigma}^{~~~\rho'\sigma'}~,$$ to write the final result for the two-point function in terms of the parameter $\gamma$ and the covariant structures $G$ and $H$, thus obtaining .
Defect OPE of $F_{\mu\nu}$ {#app:detailsBulkDefOPE}
==========================
Let us consider what can appear as a primary inside the bulk-to-boundary OPE of the field strength $F_{\mu\nu}$. By spin selection rules only vectors are admitted, with two possible structures, namely $$\begin{aligned}
\label{dOPEAgen}
F_{\mu\nu}(\vec{x}, y) \underset{y\to 0}{\sim}\frac{1}{y^{2-\widehat{\Delta}_1}}\hat{V}_1^a(\vec{x})2\delta_{a[\mu}\delta_{\nu]y} - \frac{1}{y^{2-\widehat{\Delta}_2}} i \epsilon^{abc}\hat{V}_{2\,c}(\vec{x})\delta_{a [\mu }\delta_{\nu] b}+\dots\end{aligned}$$ and the ellipsis denotes contributions from descendants. Using the bulk eom and Bianchi identity, we have that $$\begin{aligned}
\partial_y F_{y a}\sim \frac{ (\widehat{\Delta}_1-2)}{y^{3-\widehat{\Delta}_1}}\hat{V}_{1\,a}(\vec{x})+\dots ~,\nonumber\\
\partial_y \tilde{F}_{y a}\sim -i \frac{ (\widehat{\Delta}_2-2)}{y^{3-\widehat{\Delta}_1}}\hat{V}_{2\,a}(\vec{x})+\dots ~,\end{aligned}$$ must be boundary descendants. This requires $\widehat{\Delta}_1=\widehat{\Delta}_2=2$. We conclude that the only allowed boundary primaries are conserved currents.
To obtain the complete form of the bulk-to-boundary OPE of $F$ (including all the descendants) we first need the exact $\langle F\hat{V} \rangle $ correlator. This can be easily computed using the techniques of [@Lauria:2018klo] to find $$\begin{aligned}
\label{FtoV}
\langle F_{ya}(x) \hat{V}_{i\,c}(0) \rangle & =\frac{1}{x^{4}}\,\left[\left(\frac{2 y^2 \delta_{ac}}{x^2}-I_{ac}(x)\right)c_{1i}(\tau)- 2i\, c_{2i}(\tau)\frac{y}{x^2} \epsilon_{acd}x^d\right]~,\nonumber\\
\langle F_{ab}(x)\hat{V}_{i\,c}(0) \rangle&=\frac{1}{x^{4}}\,\left[i\left(\frac{2 y^2 \epsilon_{abc}}{x^2}-\epsilon_{abd}I^d_c(x)\right)c_{2i}(\tau)-2c_{1i}(\tau)\frac{y}{x^2} (\delta_{ac}x_b-\delta_{bc}x_a)\right]~,
\end{aligned}$$ where $c_{ij}(\tau)$ are defined in eq. . The bulk-to-boundary OPE of $F$ can now be obtained by expanding both sides of to find $$\begin{aligned}
\label{defectOPE}
F_{ab}(\vec{x},y)=&\sum_{n=0}^\infty(-1)^{n}\left[(\delta_{ac} \delta_{bd}-\delta_{ad} \delta_{bc})y\partial^d\frac{(y^2 \vec{\partial}^{\,2})^n}{(2 n+1)!}\hat{V}_1^c(\vec{x})-i \epsilon_{abc}\frac{(y^2 \vec{\partial}^{\,2})^n}{2n!}\hat{V}_2^c(\vec{x})\right]~,\nonumber\\
F_{ya}(\vec{x},y)=&\sum_{n=0}^\infty(-1)^{n}\left[- \frac{(y^2 \vec{\partial}^{\,2})^n}{2n!}\hat{V}_{1\,a}(\vec{x})+i \epsilon_{abd} \, y\partial^d \frac{(y^2 \vec{\partial}^{\,2})^n}{(2n +1)!}\hat{V}_2^b(\vec{x})\right]~.
\end{aligned}$$ With the bulk-to-boundary OPE above, it is straightforward to obtain the $\langle F F \rangle$ 2-point function in terms of the defect CFT data as in .
Bulk OPE Limit of $\langle F_{\mu\nu} F_{\rho\sigma}\rangle$ {#bootapp}
============================================================
Here we present some details of the bootstrap analysis presented in Section \[sec:resultsBoot\]. To simplify computations, it is convenient to start from a configuration where the two bulk operators lie at the same parallel distance from the defect, i.e. $\vec{x}_{12}=0$. In this case some expressions in simplify considerably, e.g. $$\begin{aligned}
& G_{ay,by}(\vec{x}_{12}=0,y_1-y_2) = -\frac{\delta_{ab}}{(y_1-y_2)^4}~,\\ & H_{ay,by}(\vec{x}_{12}=0,y_1,y_2) = \frac{2X_{1\,y}X_{2\,y}\delta_{ab}}{(y_1-y_2)^4}\underset{y_1\to y_2}{\sim} -\frac{2 \delta_{ab}}{(y_1-y_2)^4}~,\\
& G_{ab,cy}(\vec{x}_{12}=0,y_1-y_2) =0 = H_{ab,c y}(\vec{x}_{12}=0,y_1,y_2)~, \\
& v^4\vert_{\vec{x}_{12}=0} = \frac{(y_1-y_2)^4}{(y_1 + y_2)^4} \underset{y_1\to y_2}{\sim} \frac{(y_1-y_2)^4}{16 y_2^4}~.\end{aligned}$$ It is now a simple exercise to derive the bulk OPE limit of $$\begin{aligned}
\label{FFfull1}
\langle F_{ab}(\vec{x},y_1) F_{cy}(\vec{x},y_2) \rangle &\underset{y_1\to y_2}{\sim} -\frac{i \alpha_3}{16 y_2^4} \epsilon_{abc} +\dots, \nonumber\\
\langle F_{ay}(\vec{x},y_1) F_{by}(\vec{x},y_2) \rangle &\underset{y_1\to y_2}{\sim} -\left(\frac{\alpha_1 }{(y_1-y_2)^4}+\frac{\alpha_2}{16 y_2^4}\right)\delta_{ab}+\dots\end{aligned}$$ where the ellipsis denote contributions from descendants. On the other hand from one finds $$\begin{aligned}
\label{2ptFromBulkOPE}
\langle F_{ab}(\vec{x},y_1) F_{cy}(\vec{x},y_2)\rangle&\underset{y_1\to y_2}{\sim}\frac{1}{12}\frac{a_{F\tilde{F}}(\tau,\bar{\tau})}{y_2^4}\epsilon_{abc}+\dots~, \nonumber\\
\langle F_{ay}(\vec{x},y_1) F_{by}(\vec{x},y_2)\rangle&\underset{y_1\to y_2}{\sim}-\left(\frac{g^2}{\pi^2}\frac{1}{(y_1-y_2)^4}-\frac{1}{12}\frac{a_F^2(\tau,\bar{\tau})}{y_2^4}\right)\delta_{ab}+\dots~.\end{aligned}$$ Crossing symmetry now implies that and must match, therefore $$\begin{aligned}
\alpha_1=\frac{g^2}{\pi^2},\quad a_{F^2}(\tau,\bar{\tau})=-\frac{3}{4}\alpha_2,\quad a_{F\tilde{F}}(\tau,\bar{\tau})=-i\frac{3}{4} \alpha_3. \end{aligned}$$ From the solution above, upon using one obtains $$\begin{aligned}
\label{FperpFperpComp}
c_{11}(\tau,\bar{\tau})+c_{22}(\tau,\bar{\tau})=\frac{2g^2}{\pi^2},\quad a_{F^2}(\tau,\bar{\tau})=\frac{3}{8}(c_{22}(\tau,\bar{\tau})-c_{11}(\tau,\bar{\tau})),\quad a_{F\tilde{F}}(\tau,\bar{\tau})=i\frac{3}{4} c_{12}(\tau,\bar{\tau}). \end{aligned}$$
Current Two-Point Functions {#app:current2ptfunctions}
===========================
In this appendix derive some useful relations between the two-point functions of the conserved boundary currents. The two-point functions of the currents $\hat{V}_i^a$ – see – in momentum space are $$\langle \hat{V}_i^a(p) \hat{V}_j^b(-p)\rangle= - \frac{\pi ^2}{2} c_{ij} p \left(\delta^{ab}-\frac{p^a p^b}{p^2}\right)+\frac{\kappa_{ij}}{2\pi} \epsilon^{abc}p_c~.$$ The main goal is to express the coefficients $c_{ij}$ –that enter directly in the expression of the bulk two-point and one-point functions– in terms of the two-point correlator of the current $\hat{J}^a$, which is more natural to compute in perturbation theory at large $\tau$.
In perturbation theory it is convenient to define a two-point function of $\hat{J}^a$ that cannot be disconnected by cutting a photon line, which we will call one-photon irreducible and denote with the symbol $\Sigma$ $$\langle \hat{J}^a(p) \hat{J}^b(-p)\rangle\vert_{\text{one-photon irr.}}\equiv \Sigma^{ab}(p)= - \frac{\pi^2}{2}c_\Sigma(\tau,\bar{\tau}) p \left(\delta^{ab}-\frac{p^a p^b}{p^2}\right)+\frac{\kappa_\Sigma(\tau,\bar{\tau})}{2\pi} \epsilon^{abc}p_c~.\label{eq:Sigma}$$ Clearly this two-point function reduces to the two-point function of the current of the 3d CFT as $\tau\to \infty$.
![The two-point function of the boundary current $\hat{J}$. The shaded blob represents the one-photon irreducible two-point function $\Sigma(p)$, by which we mean the sum of all the diagrams that cannot be disconnected by cutting a photon line. The full two-point function can be obtained in terms of $\Sigma$, via the geometric sum shown in the figure. \[fig:geometric\]](fig/geometric.pdf){height="2cm"}
By resumming the diagrams in fig. \[fig:geometric\] we obtain $$\begin{aligned}
\langle \hat{J}^a(p) \hat{J}^b(-p)\rangle & = \left(\Sigma(p)\cdot (\mathds{1} - \Pi(p) \cdot \Sigma(p))^{-1}\right)^{ab} \\
& =- \frac{\pi^2}{2}c_J(\tau,\bar{\tau}) p \left(\delta^{ab}-\frac{p^a p^b}{p^2}\right)+\frac{\kappa_J(\tau,\bar{\tau})}{2\pi} \epsilon^{abc}p_c~, \label{eq:JJ}\end{aligned}$$ where $\Pi$ is the boundary propagator of the photon (see eq. ) and $$\begin{aligned}
\frac{\pi^2}{2}c_J & =\frac{\frac{\pi ^2}{2} c_\Sigma \left( \frac{\pi ^2}{2} c_\Sigma g^2+ \gamma ^2+1\right)+ \frac{g^2 \kappa_\Sigma^2}{4\pi^2}}{\left(\frac{\pi^2}{2} c_\Sigma g^2 + 1\right)^2+\left(\gamma +\frac{g^2 \kappa_\Sigma}{2\pi} \right)^2}~, \label{eq:cJfromS}\\
\frac{\kappa_J}{2\pi} & = \frac{\frac{ \gamma}{g^2} \left(\frac{\pi ^2}{2} c_\Sigma g^2\right)^2+ \frac{\kappa_\Sigma}{2\pi} \left(\gamma ^2+\gamma \frac{g^2 \kappa_\Sigma}{2\pi} +1\right)}{\left(\frac{\pi^2}{2} c_\Sigma g^2 + 1\right)^2+\left(\gamma +\frac{g^2 \kappa_\Sigma}{2\pi} \right)^2}~.\label{eq:kJfromS}\end{aligned}$$
We will also need the mixed two-point function $\langle \hat{J} \hat{V}_2\rangle$ which similarly can be parametrized as $$\langle \hat{J}^a(p) \hat{V}_2^b(-p)\rangle = - \frac{\pi ^2}{2} c_{J2} p \left(\delta^{ab}-\frac{p^a p^b}{p^2}\right)+\frac{\kappa_{J2}}{2\pi} \epsilon^{abc}p_c~.$$ Since $\hat{V}_2^a=\frac{i}{2}\, \epsilon^{abc}F_{bc}\vert_{y=0}$, we can readily express the two-point function of $\hat{V}_2$ and the mixed two-point function of $\hat{V}_2$ and $\hat{J}$ in terms of the two-point function of $\hat{J}$ and the boundary propagator of the photon, using the relations depicted in fig. \[fig:V2V2\].
![Relations between the two-point functions involving the current $V_2$ and the two-point function $\langle J J\rangle$. The relation in the second line is only true up to a contact term. \[fig:V2V2\]](fig/V2V2.pdf){height="1.5cm"}
We obtain $$\begin{aligned}
\label{eq:V2V2}
\frac{\pi^2}{2}c_{22} & = \frac{g^2}{1+\gamma ^2}+\left(\frac{g^2}{1+\gamma ^2}\right)^2 \left(\left(\gamma ^2-1\right)\frac{\pi^2}{2} c_J-2 \gamma \frac{\kappa _J}{2\pi}\right)~,\\
\frac{\kappa_{22}}{2\pi} & = -\frac{g^2}{1+\gamma ^2}\gamma+\left(\frac{g^2}{1+\gamma ^2}\right)^2 \left(\gamma \pi^2 c_J + \left(\gamma ^2-1\right) \frac{\kappa _J}{2\pi}\right)~,\\
\frac{\pi^2}{2}c_{J2} & = \frac{g^2}{1+\gamma^2}\left(-\gamma \frac{\pi ^2}{2} c_J+\frac{\kappa_J}{2\pi}\right)~,\\
\frac{\kappa_{J2}}{2\pi} & =1- \frac{g^2}{1+\gamma^2}\left(\frac{\pi ^2}{2} c_J +\gamma \frac{\kappa_J}{2\pi}\right)~.\end{aligned}$$ Finally, using that $\hat{V}_1 = -g^2 \hat{J} - \gamma \hat{V}_2$, we obtain that $$\begin{aligned}
\label{eq:V1V1}
\frac{\pi^2}{2}c_{11} & =\frac{\pi^2}{2}\left( g^4 c_J + 2 g^2 \gamma c_{J2} +\gamma^2 c_{22}\right) \nonumber\\&=\frac{g^2}{1+\gamma ^2} \gamma ^2-\left(\frac{g^2}{1+\gamma ^2}\right)^2 \left( \left(\gamma ^2-1\right) \frac{\pi^2}{2}c_J-2 \gamma \frac{\kappa_J}{2\pi} \right)~,\\
\frac{\kappa_{11}}{2\pi} & = g^4 \frac{\kappa_J}{2\pi} + 2 g^2 \gamma \frac{\kappa_{J2}}{2\pi} +\gamma^2 \frac{\kappa_{22}}{2\pi} \nonumber\\& = \frac{g^2}{1+\gamma ^2} \gamma \left(\gamma ^2+2\right)-\left(\frac{g^2}{1+\gamma ^2}\right)^2 \left(\gamma \pi^2 c_J + \left(\gamma ^2-1\right) \frac{\kappa _J}{2\pi}\right)~,\\
\frac{\pi^2}{2}c_{12} & =- \frac{\pi^2}{2}\left( g^2 c_{J2} + \gamma c_{22} \right)\nonumber\\& =- \frac{g^2}{1+\gamma ^2} \gamma+\left(\frac{g^2}{1+\gamma ^2}\right)^2 \left(\gamma \pi^2 c_J + \left(\gamma ^2-1\right) \frac{\kappa _J}{2\pi}\right)~,\label{eq:V1V2}\\
\frac{\kappa_{12}}{2\pi} & = - g^2 \frac{\kappa_{J2}}{2\pi} - \gamma \frac{\kappa_{22}}{2\pi}\nonumber\\& =-\frac{g^2}{1+\gamma ^2} - \left(\frac{g^2}{1+\gamma ^2}\right)^2 \left(\left(\gamma ^2-1\right)\frac{\pi^2}{2} c_J-2 \gamma \frac{\kappa _J}{2\pi}\right)~.\end{aligned}$$ We see that all the coefficients $c_{ij}$ can be expressed in terms of the functions of the coupling $c_J$ and $\kappa_J$ (or equivalently $c_\Sigma$ and $\kappa_\Sigma$). As a check, note that the first identity in , that was derived from the contribution of the identity in the bulk OPE and relates $c_{11}$ and $c_{22}$, is identically satisfied.
Calculation of $\langle\hat{V}_i \hat{V}_j \hat{D}\rangle$ {#app:VVD}
==========================================================
We start by computing the three-point function $$\begin{aligned}
\langle F_{\mu\nu}(x_1)F_{\rho\sigma}(x_2)\hat{D}(\vec{x}_3)\rangle~.\end{aligned}$$ using the boundary channel. At leading order in the boundary OPE limit the three-point function becomes $$\begin{aligned}
\langle\hat{V}_i^a(\vec{x}_1)\hat{V}_j^b(\vec{x}_2)\hat{D}(\vec{x}_3)\rangle~,\end{aligned}$$ which upon placing the displacement operator at infinity simplifies to [@Costa:2011mg; @Dymarsky:2017xzb] $$\begin{aligned}
\label{eq:VVD3pt}
\langle \hat{V}_i^a(\vec{x}_1)\hat{V}_j^b(\vec{x}_2)\hat{D}(\infty)\rangle\equiv\lim_{\vec{x}_3\rightarrow \infty}|\vec{x}_3|^8\langle \hat{V}_i^a(\vec{x}_1)\hat{V}_j^b(\vec{x}_2)\hat{D}(\vec{x}_3)\rangle= \, \lambda_{ij\hat{D}+}^{(1)}\,\, \delta^{ab}+\lambda_{ij\hat{D}-}^{(1)}\,\, {\hat{x}_{12}^c}{}\epsilon^{abc}~.\end{aligned}$$ Since the result in is a constant, the descendants of the currents in the boundary OPE of $F$ do not contribute. Hence, from the boundary OPE-channel we simply find $$\begin{aligned}
\label{eq:FFDeq}
\langle F_{ay}(x_1)F_{by}(x_2)\hat{D}(\infty)\rangle& = \, \lambda_{11\hat{D}+}^{(1)}\,\, \delta_{ab}+\lambda_{11\hat{D}-}^{(1)}\,\, {\hat{x}_{12}^f}{}\epsilon_{abf}~,\\
\langle F_{ay}(x_1)F_{bc}(x_2)\hat{D}(\infty)\rangle& = -i\,\epsilon_{bc}{}^{e} (\lambda_{12\hat{D}+}^{(1)}\,\, \delta_{ae}+\lambda_{12\hat{D}-}^{(1)}\,\, {\hat{x}_{12}^f}{}\epsilon_{aef})~,\\
\langle F_{ab}(x_1)F_{cd}(x_2)\hat{D}(\infty)\rangle& = -\, \epsilon_{ab}{}^{e}\epsilon_{cd}{}^{g}(\lambda_{22D+}^{(1)}\delta_{eg}+\lambda_{22D-}^{(1)}\,\, \hat{x}_{12}^f\epsilon_{egf})~.\end{aligned}$$
Next, we compute the three-point function using the bulk OPE channel. The Lorentz spin and scaling dimensions of the full set of operators appearing in the OPE of two $F$’s can be found in [@Beccaria:2014zma] – see eq. (2.12) therein – where they are discussed in the context of the so-called minimal type-C higher spin theory on AdS$_5$, the bulk dual to the free Maxwell CFT$_4$. All the operators with scaling dimension $\Delta >4$ in this OPE are higher-spin conserved currents (there is both a family of symmetric traceless tensors and a family of mixed-symmetry ones), and in addition there is the identity operator and a few operators of scaling dimension $\Delta = 4$: the scalar operators $F^2$ and $F\tilde{F}$, the stress tensor $T_{\mu\nu} = (\frac{1}{g^2}F_{\mu \rho}F^{~\rho}_\nu - {\rm trace})$, and a non-conserved operator in the representation $(2,0)\oplus(0,2)$ of rotations, i.e. a tensor with four indices and the same symmetry and trace properties of a Weyl tensor, for this reason we will denote it as $W_{\mu\nu\rho\sigma}$. The three-point function in the bulk OPE channel is written as a sum of the bulk-boundary two-point functions between these operators and the displacement operator. Let us analyze which of these two-point functions can contribute. First of all, it is easy to see that two-point function between the conserved higher-spin currents and the displacement operator must vanish. This is an instance of the more general statement that in boundary CFTs bulk conserved currents $J$ can only have non-zero two-point functions with a scalar boundary operator $\hat{O}$ that has the same scaling dimension. The latter statement can be easily proved by placing the boundary operator at infinity, because in this case invariance under scaling and parallel translations force the two-point function to take the schematic form $$\langle J(y, \vec{x}) \hat{O}(\infty) \rangle = b_{J\hat{O}}\frac{1}{y^{\Delta_J -\Delta_{\hat{O}}}} ~(\text{structure})~,$$ where “structure” denotes an appropriate tensor built out of the $\delta^{\mu\nu}$, the unit normal vector $n^\mu$ and possibly epsilon tensors. Clearly when $\Delta_J \neq \Delta_{\hat{O}}$ this two-point function cannot be compatible with current conservation unless the coefficient $b_{J\hat{O}}$ vanishes. Moreover, rotational invariance implies that also the operator $W_{\mu\nu\rho\sigma}$ has vanishing two-point function with the displacement.[^15] Therefore, the only bulk operators that can contribute to the three-point function are the scalar operators and the stress-tensor. When the displacement is placed at infinity, the corresponding two-point functions are $$\begin{aligned}
\langle F^2(x)\hat{D}(\infty)\rangle & =b_{F^2, \hat{D}}~,\label{eq:twoptD1}\\
\langle F\tilde{F}(x)\hat{D}(\infty)\rangle & =b_{F\tilde{F},\hat{D}}~,\label{eq:twoptD2}\\
\langle T_{\mu\nu}(x)\hat{D}(\infty)\rangle & =b_{T, \hat{D}}\,\left( \delta_{\mu y}\delta_{\nu y}-\frac14\delta_{\mu\nu}\right)~.\label{eq:twoptD3}\end{aligned}$$ Using the OPE and the Ward identity we can express the above two-point function coefficients in terms of the one-point function of the scalar operators, and of the coefficient $C_{\hat{D}}$ in the two-point function of the displacement, namely [@McAvity:1995zd; @Gliozzi:2015qsa; @Billo:2016cpy] $$\begin{aligned}
b_{F^2, \hat{D}} & = -\frac{32a_{F^2}}{\pi^2}~,\label{eq:twoCoeff1}\\
b_{F\tilde{F}, \hat{D}} & = -\frac{32a_{F\tilde{F}}}{\pi^2}~,\label{eq:twoCoeff2}\\
b_{T, \hat{D}} & = \frac{4\,C_{\hat{D}}}{3}~.\label{eq:twoCoeff3}\end{aligned}$$ Since the two-point functions are constant, we can simply plug in the three-point function the leading bulk OPE, ignoring the descendants (and also ignoring the singular contribution from the identity that drops from the three-point function) $$\begin{aligned}
F_{\mu\nu}(x) F^{\rho\sigma}(0)\underset{x\to 0}{\sim} \frac{1}{12}(\delta_\mu^\rho\delta_\nu^\sigma-\delta_\nu^\rho\delta_\mu^\sigma)F^2(0)+\frac{1}{12}\epsilon_{\mu\nu}^{\rho\sigma}F\tilde{F}\,(0) + 2g^2 \delta_{[\mu}^{[\rho}T_{\nu]}^{\sigma]}(0)~.\end{aligned}$$ Using eq.s - in the two-point functions, we find $$\begin{aligned}
\label{eq:FFDbulk}
\langle F_{ay}(x_1)F_{by}(x_2)\hat{D}(\infty)\rangle =&-\left(\frac{8}{3\pi^2}a_{F^2}-\frac{g^2}{3}C_{\hat{D}}\right)\delta_{ab}~,\\
\langle F_{ab}(x_1)F_{cd}(x_2)\hat{D}(\infty)\rangle =& -\left(\frac{8}{3\pi^2}a_{F^2}+\frac{g^2}{3}C_{\hat{D}}\right)\epsilon_{abe}\epsilon_{cde}~,\\
\langle F_{ay}(x_1)F_{bc}(x_2)\hat{D}(\infty)\rangle =&-\frac{8}{3\pi^2}a_{F\tilde{F}}\,\,\epsilon_{abc}~.\end{aligned}$$ Finally, by comparing with we find .
Dimension of the Boundary Pseudo Stress Tensor {#app:fakestress}
==============================================
In section \[sec:MinPhTr\] we mentioned that the conservation of the stress tensor of the 3d CFT is violated at $g\neq 0$ due to multiplet recombination. At $g\neq 0 $ we will call this operator boundary pseudo stress tensor. This is expected from the Ward identities derived in [@Billo:2016cpy]. In this Appendix we exploit this idea, to reproduce the one loop result of . We start from the boundary Lagrangian of a 3d Dirac fermion $\psi$ $$\begin{aligned}
\mathcal{L}= i \,\bar{\psi}\cancel{D}_A \psi,\end{aligned}$$ where $D_a \psi= (\partial_a- i A_a) \psi$ and $D_a \bar\psi= (\partial_a+ i A_a) \bar\psi$. The algebra of gamma matrices is $\{\gamma_a,\gamma_b\}=2\delta_{ab}$. The pseudo boundary stress tensor is $$(O_2)_{ab}=\frac{i}{2}[\bar{\psi}\gamma_{(a}D_{b)}\psi-D_{(a}\bar{\psi}\gamma_{b)}\psi],$$ where the symmetrization includes a factor of $1/2$. Note that the above operator is traceless as a consequence of the equations of motion: $$\gamma^a D_a \psi=0\, \quad D_a\bar{\psi} \gamma^a=0.$$ Using $[D_a,D_b]\psi=-i F_{ab}$ we obtain $$\label{eq:Odef}
\partial_a O_2^{ab}= F^{ab}\bar{\psi}\gamma_a \psi,$$ In the decoupling limit $g\rightarrow 0$ the two-point function of $F_{ab}$ vanishes, hence effectively the right-hand side of is 0 and the operator $O_2^{ab}$ becomes a proper stress tensor for the boundary theory, with conformal dimension $\Delta_{2} = 3$. Upon turning on $g$, this dimension must be lifted from the unitarity bound, i.e. $\Delta_{2} (g)=3+g^2 \Delta^{(2)}_{2}+O(g^4)$. The two-point function of ${O_2}$ is fixed by 3d conformal invariance to be $$\begin{aligned}
\label{eq:tautau}
\langle {O_2}^{ab}(\vec{x}) {O_2}^{cd}(0)\rangle & =\frac{C_{2}(g)}{|\vec{x}|^{2\Delta_{2} (g)}}I^{ab,cd}(\vec{x})~,\nonumber\\
I^{ab,cd}(\vec{x})= \frac{1}{2}[I^{\rm 3d\,ac}(\vec{x})I^{\rm 3d\,bd}(\vec{x}) & +I^{\rm 3d\,ad}(\vec{x})I^{\rm 3d\,bc}(\vec{x})]-\frac{1}{3}\delta_{ab}\delta_{cd}~,\end{aligned}$$ with $I^{\rm 3d\,ac}(\vec{x})$ defined in and $C_{2}(g)=c_{2}^{(0)}+g^2 c_{2}^{(2)}+O(g^4)$, being $c_{2}^{(0)}=\frac{3}{16\pi^2}$ the central charge for a single free 3d Dirac fermion [@Osborn:1993cr]. Furthermore the recombination rule tells us $$\label{Anselmi}
\langle \partial_a {O_2}^{ab}(\vec{x})\, \partial_c O_{2}^{cd}(0)\rangle = \langle (F^{ab}\bar{\psi}\gamma_a \psi) (\vec{x}) (F^{cd}\bar{\psi}\gamma_{c} \psi) (0) \rangle.$$ On one hand, the r.h.s. of can be computed at three level using with the result $$\begin{aligned}
\label{rhs}
\langle (F^{c a}\bar{\psi}\gamma_c \psi) (\vec{x}) (F^{d b}\bar{\psi}\gamma_{d} \psi) (0) \rangle=\frac{{\color{black} 4} g^2 c_J^{(0)}}{\pi^2}\frac{I^{\rm 3d\,ab}(\vec{x})}{|\vec{x}|^8}+O(g^4),\end{aligned}$$ where $c_J^{(0)}= \frac{1}{8\pi^2}$ is the central charge for the $U(1)$ conserved current $\hat{J}_a=\bar{\psi}\gamma_a \psi$ of a free 3d Dirac fermion [@Osborn:1993cr].
On the other hand, taking two derivatives of and expanding to the lowest non trivial order in $g$ gives $$\begin{aligned}
\label{lhs}
\langle \partial_c {O_2}^{ca}(\vec{x})\, \partial_d {O_2}^{db}(0)\rangle=\frac{10}{3}g^2c_{2}^{(0)} \Delta^{(2)}_{2} \frac{I^{\rm 3d\,ab}(\vec{x})}{|\vec{x}|^8}+O(g^4).\end{aligned}$$ Hence the above result, together with and fixes the anomalous dimension of $O_2$ up to $O(g^4)$ terms to be $$\Delta_{2}(g)=3+\frac{\color{black} 6}{5 \pi^2}\frac{c_J^{(0)}}{c_{2}^{(0)}} g^2+O(g^4)=3+\frac{\color{black} 4}{5 \pi^2} g^2+O(g^4),$$ in agreement with .
Two-loop Integrals {#app:FeynInt}
==================
In the perturbative calculations of anomalous dimensions we encountered two-loop diagrams with operator insertions at zero-momentum and two external legs. After performing tensor reduction to get rid of the numerators, the resulting integrals always take the form of a two-loop massless two-point integral, namely $$\begin{aligned}
&G(n_1,n_2,n_3,n_4,n_5) \equiv (4\pi)^d(k^2)^{n_1+n_2+n_3+n_4+n_5 -d} \nonumber \\
&~~~~~~~~~~~~~~~\times\int \frac{d^d p}{(2\pi)^d} \frac{d^d q}{(2\pi)^d}\frac{1}{(p^2)^{n_1}(q^2)^{n_2}((k+p)^2)^{n_3}((k+q)^2)^{n_4}((p-q)^2)^{n_5}}~.\end{aligned}$$ $k$ here is the external momentum associated to the two external legs, and $p$ and $q$ are the loop momenta. The powers $n_i$ depend on the diagram we are considering (and in fact each diagrams will give rise to a linear combination of $G$’s with several different sets of $n_i$’s after reducing the numerators). In order to extract the two-loop renormalization constants we need to find the $1/\epsilon^2$ and $1/\epsilon$ poles in the $\epsilon\to 0$ expansion of the constants $G(n_1,n_2,n_3,n_4,n_5)$, evaluated at $d=3-2\epsilon$. (The coefficient of $1/\epsilon^2$ are fixed by one-loop data, so they do not contain new information.)
The function $G(n_1,n_2,n_3,n_4,n_5)$ enjoys a large group of symmetries [@Barfoot:1987kg] that allows to relate its values at different sets of quintuples of powers. Some of the symmetries are manifest from the definition, e.g. $G(n_1,n_2,n_3,n_4,n_5)=G(n_2,n_1,n_4,n_3,n_5)=G(n_3,n_4,n_1,n_2,n_5)=G(n_4,n_3,n_2,n_1,n_5)$. When one or more of the $n_i$’s vanish, there is a closed expression for $G(n_1,n_2,n_3,n_4,n_5)$ in terms of gamma functions. When all of the $n_i$’s are integer, the strategy to compute $G(n_1,n_2,n_3,n_4,n_5)$ is to use integration-by-parts identities [@Chetyrkin:1981qh; @Tkachov:1981wb] to lower the positive $n_i$’s, until the result is reduced to a linear combination of $G$’s with at least one vanishing entry. However, due to the $1/|p|$ “non-local” propagator of the photon restricted to the boundary, in our setup we encounter diagrams in which two of the $n_i$’s are half-integer, and the remaining three are integer.[^16] In this case it might be impossible to reduce to the case of a vanishing power using integration-by-parts, and a further input is needed. The paper [@Broadhurst:1996ur] derived a closed formula for $G(n_1,n_2,n_3,1,1)$ (and symmetry-related cases), with generic real $n_1,n_2,n_3$, in terms of the generalized hypergeometric function ${}_3F_2$. To recover the $1/\epsilon^2$ and $1/\epsilon$ poles from the result of [@Broadhurst:1996ur], one needs to perform a Taylor expansion of the ${}_3F_2$ in its parameters. This is typically hard to do analytically, but the algorithm of [@Huang:2012qz] can be used to expand numerically to very high precision.
The strategy that we used is then to reduce all of the integrals that we encountered to a small number of “master integrals” using integration-by-parts identities. These master integrals have the property that they can be evaluated with the formula in [@Broadhurst:1996ur], and that using the numerical expansion we can easily recognize the values of the coefficients. To compute anomalous dimensions in the fermion theory of section \[sec:MinPhTr\] we used the following two master integrals $$\begin{aligned}
G(1,\tfrac12,\tfrac12,1,1) & \underset{\epsilon\to0}{\sim} \frac{0}{\epsilon^2} + \frac{0}{\epsilon} + \mathcal{O}(1)~,\\
G(1,\tfrac32,\tfrac12,1,1) & \underset{\epsilon\to0}{\sim} \frac{0}{\epsilon^2} + \frac{4}{\pi \epsilon} + \mathcal{O}(1)~.\end{aligned}$$ We never needed the $1/\epsilon$ coefficient of the master integral in the second line, and the only case in which we needed its $1/\epsilon^2$ coefficient is in the check that the gauged current does not get any anomalous dimension. So all of our non-trivial results only depend on the master integral in the first line. In the scalar theory of section \[sec:Scalar\] we also encountered the integral $G(1,\tfrac12,\tfrac12,1,2)$, which we were not able to compute with this strategy.
We will now give the result that we found for the contribution of each diagram to the renormalization constants. We make reference to the labeling of the diagrams in figure \[fig:loopdiagram\]. In the two-loop calculation we also need to consider the one-loop diagram with the insertions of one-loop counterterms for the vertex or for the internal fermion lines, and we refer to this contribution as “$\text{c.t.}$”. We denote $L \equiv \log (\pi \mu^2) -\gamma_E $ where $\gamma_E$ is the Euler constant and $\mu$ is the scale introduced by dimensional regularization. Locality of counterterms requires that the $L$-dependence must cancel from the coefficient of the $1/\epsilon$ pole when all the diagrams are summed up, but generically it will be present in single diagrams. The cancelation of the $L$-dependence (and also the cancelation of $\xi$ in the gauge-invariant quantities) in the sum of all the diagrams is a check of the calculation.
- [Wavefunction renormalization of the fermion: denoting the external momentum running on the fermion line with $k$, all the diagrams are proportional to $\slashed{k}$, with coefficients $$\begin{aligned}
(a) & =\frac{g^2}{1+\gamma^2} \frac{(2-3 \xi )}{12 \pi ^2 \epsilon }~,\\
(b.1) & =\frac{g^4}{(1+\gamma^2)^2}\left(\frac{(2-3 \xi)^2}{288 \pi ^4 \epsilon ^2}(1+2 \epsilon L)
+\frac{63 \xi ^2-90 \xi +32 }{432 \pi ^4 \epsilon } + \frac{\gamma^2}{96 \pi^2 \epsilon}\right)~,\\
(b.2) & = \frac{g^4}{(1+\gamma^2)^2}\left(-\frac{(2-3 \xi)^2}{144 \pi ^4 \epsilon ^2}(1+2\epsilon L)
-\frac{117 \xi ^2-168 \xi +64}{432 \pi ^4 \epsilon } - \frac{\gamma^2}{192 \pi^2 \epsilon} \right)~,\\
(b.3) & =-\frac{g^4}{(1+\gamma^2)^2} \frac{1-\gamma^2}{192 \pi ^2 \epsilon } ~,\\
\text{c.t.} & = \frac{g^4}{(1+\gamma^2)^2}\left(\frac{ (2-3 \xi )^2}{144 \pi ^4 \epsilon ^2}(1+\epsilon L)+\frac{ 54 \xi ^2-78 \xi +28}{432 \pi ^4 \epsilon }\right)~.
\end{aligned}$$ Requiring the divergence to cancel with $-\delta((Z_\psi)^2) \slashed{k}$, we obtain eq. .]{}
- [Anomalous dimension of $O_0$: summing over all possible insertions in the given topology, the diagrams give $$\begin{aligned}
(a) & =\frac{g^2}{1+\gamma^2}\frac{2+ \xi}{4 \pi ^2 \epsilon }~,\\
(b.1) & =\frac{g^4}{\left(\gamma ^2+1\right)^2} \left(\frac{(2+ \xi ) (10-3 \xi)}{96 \pi ^4 \epsilon ^2}(1+2\epsilon L)
-\!\frac{27 \xi ^2-86 \xi -232}{144 \pi ^4\epsilon }+\! \frac{\gamma^2}{32 \pi^2 \epsilon}\right)~,\\
(b.2) & = \frac{g^4}{\left(\gamma ^2+1\right)^2} \left(\!\!-\frac{(2+\xi) (2-3\xi)}{48 \pi ^4 \epsilon ^2}(1+2\epsilon L)
+\!\frac{63 \xi ^2+40 \xi -112 }{144 \pi ^4\epsilon }+\!\frac{3\gamma^2}{64 \pi^2 \epsilon}\right)~,\\
(b.3) & =-\frac{g^4}{\left(\gamma ^2+1\right)^2}\frac{5-5\gamma^2}{64 \pi ^2 \epsilon }~,\\
\text{c.t.} & = \frac{g^4}{(1+\gamma^2)^2}\left(-\frac{(2+\xi)^2}{16 \pi ^4 \epsilon ^2}(1+\epsilon L)-\frac{2\xi^2 + 7\xi +6}{8 \pi ^4\epsilon}\right)~.
\end{aligned}$$ Requiring the divergence to cancel with $\delta((Z_\psi)^2 Z_0)$, we obtain eq. .]{}
- [Anomalous dimension of $O_2$: we sum over all possible insertions in the given topology. The diagrams are proportional to the tree-level insertion of $O_2$ (see fig. \[fig:Fermionfeynrule\_ops\]) with the following coefficients $$\begin{aligned}
(a) & =-\frac{g^2}{1+\gamma^2} \frac{34-15\xi}{60 \pi ^2 \epsilon }~,\\
(b.1) & =\frac{g^4}{(1+\gamma^2)^2} \left(-\frac{225 \xi ^2-300 \xi +4}{7200 \pi ^4 \epsilon ^2}(1+2\epsilon L)
\right. \nonumber \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.-\frac{5175 \xi ^2-12690 \xi +4096 }{54000 \pi ^4\epsilon }- \frac{\gamma^2}{240\pi^2 \epsilon} \right)~,\\
(b.2) & = \frac{g^4}{(1+\gamma^2)^2}\left(\frac{45 \xi ^2-132 \xi +116}{720 \pi ^4 \epsilon ^2}(1+2\epsilon L)
\right. \nonumber \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.+\frac{1305 \xi ^2-6432 \xi +8416}{10800 \pi ^4\epsilon }- \frac{\gamma^2}{960\pi^2 \epsilon}\right)~,\\
(b.3) & =\frac{g^4}{(1+\gamma^2)^2} \frac{29-5\gamma^2}{960 \pi ^2 \epsilon } ~,\\
\text{c.t.} & = \frac{g^4}{(1+\gamma^2)^2} \left(-\frac{(15 \xi -34)^2}{3600 \pi ^4 \epsilon ^2}(1+\epsilon L)-\frac{675 \xi ^2-9735 \xi +18598}{27000 \pi ^4\epsilon }\right)~.
\end{aligned}$$ Requiring the divergence to cancel with $\delta((Z_\psi)^2 Z_2)$, we obtain eq. .]{}
[^1]: If the gauge group is compact, say $U(1)$, the gauge field has an intrinsic normalization and thus the coefficient in front of the bulk Lagrangian is canonically defined even if the bulk theory is free. Local interactions between the gauge fields and any other degrees of freedom localized in non-zero co-dimension obviously cannot renormalize the bulk gauge coupling. Furthermore, the strength of the interactions between the gauge fields and such other degrees of freedom is controlled by the bulk gauge coupling and by quantized gauge charges and thus cannot get renormalized. The only possible beta functions involve gauge-invariant boundary local operators. This fact is often obfuscated in perturbative treatments and then proven with the help of Ward identities, in a manner analogous to the non-renormalization of gauge charges in QED [@Teber:2012de; @Kotikov:2013kcl; @Teber:2014ita; @Teber:2016unz; @Kotikov:2016yrn; @Herzog:2017xha; @Dudal:2018pta].
[^2]: E.g. if the theory on the boundary is a free scalar field, loop corrections can generate the operator $\phi^2$ on the boundary with coefficient $\sim\tau^{-1}\Lambda^2_{UV}$, where $\Lambda_{UV}$ is the cutoff, but the only implication of this term is that the tuning of $m^2$ needs to be adjusted at order $\tau^{-1}$.
[^3]: The two types of non-locality have different physical origins, in our setup the non-locality on the boundary is due to the existence of the bulk, while in the large-$N_f$ limit it emerges due to the resummation of infinitely-many Feynman diagrams. The fact that the resulting two-point functions of the field strength have the same power of momentum is of course no surprise, because that is just fixed by the scaling dimension of conserved currents in 3d.
[^4]: Note that this expression for the beta function is valid also in the decoupling limit $\tau\to\infty$. Indeed in that limit $b_{(F^-)^2, \hat{O}}\propto \tau^{-2}$ and $b_{(F^+)^2, \hat{O}}\propto\bar{\tau}^{-2}$, from which we recover that the leading contributions from the bulk gauge fields are of order $\tau^{-1}$ and $\bar{\tau}^{-1}$.
[^5]: Note that $a_{F^2}\in\mathbb{R}$ while $a_{F\tilde{F}} \in i \mathbb{R}$. To see this, it is useful to think about these coefficients in radial quantization, as the overlap between the state defined by the local operator $F^2$/$F\tilde{F}$ and the state defined by the conformal boundary condition. Applying an inversion, the overlap gets conjugated. Hence the reality conditions stated above simply follow from the fact that the operator $F^2$/$F\tilde{F}$ is even/odd under inversion.
[^6]: Alternatively, we can implement the reasoning of [@Baggio:2017mas] to show that this property follows from the emergent $\mathbb{Z}_2$ symmetry of the system at the self-dual point.
[^7]: In [@Chen:1992ee] there appears to be a sign mistake in the two-loop diagram that we denoted with (b.2) in fig. \[fig:loopdiagram\]. This mistake leads to the different result for this anomalous dimension given in [@Chen:1993cd]. Upon correcting that sign, we find perfect agreement with our result. We thank E. Stamou for helping us with this check.
[^8]: In comparing with [@Teber:2012de; @Teber:2016unz] one needs to take into account that they consider a 3d interface with the gauge field propagating on both sides, rather than a boundary. The propagator of the photon restricted to an interface has a factor of $\frac 12$ compared to the case of the boundary.
[^9]: The Gross-Neveu CFT is expected to exist also for a small number $N$ of Dirac fermion, the UV completion being provided by a Yukawa theory. See [@Fei:2016sgs] for a recent study in $\epsilon$-expansion.
[^10]: More precisely, these elements generate $Sp(2n,\operatorname{\mathbb{Z}})/\sim$, where we identify $S \sim -S$.
[^11]: Here we are using a different charge normalization compared to [@Cordova:2017kue]. For example, the lowest charged gauge invariant operator is the meson $\bar{\psi}_i\psi_j$, which has charge $1$ under gauge field $A'_1$ in our case but charge $2$ under the gauge field $X$ in [@Cordova:2017kue]. Our choice is necessary if we want to start from , because $Sp(4,\operatorname{\mathbb{Z}})$ respects the charge normalization. The difference between the charge-two theory and charge-one theory is that the former has fewer monopole operators. Starting with the charge-one theory, we can gauge $\operatorname{\mathbb{Z}}_2 \subset U(1)_J$, where $U(1)_J$ is the magnetic $U(1)$ global symmetry. This has the effect of changing the gauge group $G = U(1)$ to $\tilde{G}$ such that $\tilde{G}/\operatorname{\mathbb{Z}}_2 = G$. For example, in this case $G = U(1)$, and we gauge $\operatorname{\mathbb{Z}}_2\subset U(1)_J$, then the new gauge group is $\tilde{G} = U(1)$ but with the replacement of the gauge field $A_\mu \rightarrow 2A_\mu$, namely all the particle charges are multiplied by 2 [@Komargodski:2017keh; @Gaiotto:2014kfa]. In this way we obtain the charge-two theory.
[^12]: In the sense that this combination of boundary CS term is independent of the choices of different extensions of the boundary into bulk mod $2\pi \operatorname{\mathbb{Z}}$
[^13]: We follow the notation in [@Seiberg:2016gmd] that the minus sign in $S^2 = -1$ denotes charge conjugation.
[^14]: Note that here we are not shifting the definition of the bulk coupling $\tau$ by $1/2$ as we did in . So the transformation is the same as the one presented in [@Seiberg:2016gmd] instead of the transformation $\tau' = -1/4\tau$ that we had in the previous section.
[^15]: To see this, consider the projector on the $(2,0)$ representation $$\begin{aligned}
(P^{(2,0)})_{\mu\nu\rho\sigma}^{~~\mu'\nu'\rho'\sigma'} \equiv \frac{1}{2}P_{\mu\nu}^{+~\mu'\nu'}P_{\rho\sigma}^{+~\rho'\sigma'} + \frac{1}{2} P_{\rho\sigma}^{+~\mu'\nu'}P_{\mu\nu}^{+~\rho'\sigma'} - \frac{1}{3} P_{\mu\nu,\rho\sigma}^+ P^{+\,\mu'\nu',\rho'\sigma'}~.\end{aligned}$$ Since the two-point function between $W_{\mu\nu\rho\sigma}(x)$ and $\hat{D}(\infty)$ is a constant, the allowed structures are obtained by acting with this projector on constant four-tensors built out of $\delta$ and $\epsilon$, such as: $\delta_{\mu'\rho'}\delta_{\nu'\sigma'}$, $\delta_{\mu'\rho'}\delta_{\nu' y}\delta_{\sigma' y}$, $\epsilon_{\mu'\nu'\rho'\sigma'}$, $\epsilon_{\mu'\nu'\rho' y}\delta_{\sigma' y}$. Applying the projector to any of these structures we find 0.
[^16]: Specifically, this happens for the diagrams that compute the coefficient of $\frac{({\rm Im}\tau)^2}{|\tau|^2}$ in the two-loop anomalous dimensions. The diagrams that compute the coefficient of $\frac{({\rm Re}\tau)^2}{|\tau|^2}$ do have only integer powers, and in fact they are the same as the diagrams in large-$k$ perturbation theory of CS-matter theories that compute the leading corrections to parity-even observables.
|
---
abstract: 'The collisionally pumped, ground-state 1720MHz maser line of OH is widely recognized as a tracer for shocked regions and observed in star forming regions and supernova remnants. Whereas some lines of excited states of OH have been detected and studied in star forming regions, the subject of excited-state OH in supernova remnants – where high collision rates are to be expected – is only recently being addressed. Modeling of collisional excitation of OH demonstrates that 1720, 4765 and 6049MHz masers can occur under similar conditions in regions of shocked gas. In particular, the 6049 and 4765MHz masers become more significant at increased OH column densities where the 1720MHz masers begin to be quenched. In supernova remnants, the detection of excited-state OH line maser emission could therefore serve as a probe of regions of higher column densities. Using the Very Large Array, we searched for excited-state OH in the 4.7, 7.8, 8.2 and 23.8GHz lines in four well studied supernova remnants with strong 1720MHz maser emission (SgrAEast, W28, W44 and IC443). No detections were made, at typical detection limits of around 10 mJybeam$^{-1}$. The search for the 6GHz lines were done using Effelsberg since the VLA receivers did not cover those frequencies, and are reported on in an accompanying letter [@fish07]. We also cross-correlated the positions of known supernova remnants with the positions of 1612MHz maser emission obtained from blind surveys. No probable associations were found, perhaps except in the SgrAEast region. The lack of detections of excited-state OH indicates that the OH column densities suffice for 1720MHz inversion but not for inversion of excited-state transitions, consistent with the expected results for C-type shocks.'
author:
- 'Ylva M. Pihlström'
- 'Vincent L. Fish'
- 'Loránt O. Sjouwerman'
- 'Laura K. Zschaechner'
- 'Philip B. Lockett'
- Moshe Elitzur
title: 'Excited-state OH Masers and Supernova Remnants'
---
Introduction {#intro}
============
Masers observed in the 1720MHz satellite line of OH are often associated with supernova remnants (SNRs). They originate in the shocked region where the expanding SNR collides with a molecular cloud. During the collision, a non-dissociative C-type shock can produce the temperature and density conditions required for 1720MHz maser emission to occur [@wardle99; @lockett99]. The C-type shock model requires shock speeds of the order of 25 kms$^{-1}$, and shock chemistry predicts the conditions to be right for masers behind the shock wave. Such velocities and spatial positions are in good agreement with previous observations of SNRs [e.g., @claussen97; @yusef-zadeh03b; @frail98]. A natural, next step would be to perform very long baseline interferometric (VLBI) observations to measure the physical sizes of the masing regions, and to track the SNR expansion with time. However, a major hurdle at 1720MHz is the angular broadening due to interstellar scattering. For example, in the direction of the Galactic center, pronounced scattering ($\Theta_{obs}\sim 500$ mas at 1.6GHz) has been measured [@vanlangevelde92]. To overcome this problem, @hoffman03 conducted MERLIN and VLBA observations of the 1720MHz masers in IC443, positioned at a Galactic longitude believed to be little affected by scattering. In another project, @claussen02 used a nearby pulsar to estimate the interstellar scattering effect close to W28.
A more general method that could work at any position on the sky would be using higher frequency transitions tracing the same type of gas (shocked by the SNR/cloud collisions). Higher frequency masers are less susceptible to scattering effects, which scale as $\lambda^{2}$. Finding higher frequency lines would be particularly interesting for the SgrAEast region, since some 1720MHz masers found here have velocities very offset from what is expected in the SNR model [@yusef-zadeh96; @karlsson03]. These offset velocity masers coincide on the sky with the circumnuclear disk and might therefore arise under different conditions. Proper motion studies of masers in the SgrA complex would thus indicate whether the kinematics really differ for the offset velocity masers and the SgrAEast masers. Higher frequency excited-state OH masers thus could be used for astrometric and proper motion VLBI studies along the heavily scattered Galactic plane and in the Galactic center.
Modeling of the three lower rotational states of OH has demonstrated that in star forming regions (SFRs) satellite line 1720 and 4765MHz, and mainline 6035MHz masers can occur under similar conditions in regions of shocked gas [@gray91; @gray92]. The 4765 and 6035MHz masers become more significant at increased column densities where the 1720MHz masers begin to switch off. Detection of excited-state OH maser emission could therefore serve as a probe of regions of higher OH column densities in SNRs ($N_{\rm OH}\sim 3\times
10^{17}$cm$^{-2}$ instead of $N_{\rm OH}\sim 3\times
10^{16}$cm$^{-2}$ for the 1720MHz masers). However, calculations of the $N_{\rm OH}$ resulting from C-type shocks are only able to produce relatively low values, $N_{\rm OH} \simeq 10^{16}$cm$^{-2}$ [@lockett99; @wardle99] which may be a limiting factor in the formation of excited-state OH masers in SNRs. It should be noted that detailed observations of OH and H$_2$O in IC443 require an OH production scenario involving both J- and C-type shocks [@snell05; @hewitt06], indicating that estimating the OH column density could be more complicated than previously thought.
In this work we present the details and results of our search for higher frequency OH transitions in SNRs with the Very Large Array (VLA). We also report on the results of a literature search for collisionally excited 1612MHz masers associated with SNRs.
[llllrccr]{}\[t\]
IC443 & 06 16 43.61 & $+$22 32 39.78 & 4750.7, 4660.2, 4765.6&$-$4.6&0.9 & 5.1$\times$4.3 & 13.2\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & $-$4.6 & 0.6 & 3.0$\times$2.8 & 8.4\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & $-$4.6 & 0.5 & 3.0$\times$2.7 & 6.5\
SgrAEast & 17 45 44.31 & $-$29 01 18.34 & 4750.7 & 64.0 & 0.8 & 7.7$\times$3.4 & 13.8\
& & & 4750.7 & 132.0 & 0.8 & 7.7$\times$3.4 & 14.5\
& & & 4660.2, 4765.6 & 64.0 & 0.8 & 7.7$\times$3.4 & 14.1\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & 64.0 & 0.5 & 7.0$\times$2.5 & 11.4\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 64.0 & 0.5 & 5.7$\times$2.6 & 9.5\
& & & 23817.9, 23826.6, 23838.9, 23805.3 & 64.0 & 0.7 & 1.4$\times$0.8 & 9.4\
SgrAEast & 17 45 40.62 & $-$28 59 43.98 & 4750.7 & 64.0 & 0.8 & 5.8$\times$3.9 & 15.1\
& & & 4750.7 & 132.0 & 0.8 & 5.8$\times$3.9 & 14.1\
& & & 4660.2, 4765.6 & 132.0 & 0.8 & 5.8$\times$3.9 & 14.6\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & 132.0 & 0.5 & 5.4$\times$2.6 & 11.9\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 132.0 & 0.5 & 5.3$\times$2.5 & 9.5\
& & & 23817.9, 23826.6, 23838.9, 23805.3 & 132.0 & 0.7 & 1.2$\times$0.9 & 9.5\
W28A &18 00 45.55&$-$23 17 43.33& 4750.7 & 6.3 & 0.9 & 5.9$\times$4.0 & 8.6\
& & & 4750.7 & 76.3 & 0.9 & 5.9$\times$4.0 & 8.8\
& & & 4660.2, 4765.6 & 6.3 & 0.9 & 6.2$\times$4.2 & 8.7\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & 6.3 & 0.6 & 4.2$\times$2.6 & 9.1\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 6.3 & 0.6 & 4.5$\times$2.6 & 7.4\
W28CD&18 01 39.35&$-$23 25 01.97 & 4750.7 & 14.1 & 0.9 & 5.7$\times$4.0 & 9.2\
& & & 4750.7 & 84.1 & 0.9 & 5.9$\times$4.1 & 8.0\
& & & 4660.2, 4765.6 & 14.1 & 0.9 & 6.2$\times$3.9 & 8.6\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & 14.1 & 0.6 & 4.8$\times$2.6 & 9.1\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 14.1 & 0.5 & 4.1$\times$2.8 & 7.2\
W28EF&18 01 51.64&$-$23 18 21.02 & 4750.7 & 11.7 & 0.9 & 6.7$\times$3.6 & 8.3\
& & & 4750.7 & 81.7 & 0.9 & 6.9$\times$3.5 & 9.2\
& & & 4660.2, 4765.6 & 11.7 & 0.9 & 7.2$\times$4.3 & 8.9\
& & & 7761.7, 7820.1, 7832.0, 7749.9 & 11.7 & 0.6 & 4.6$\times$2.8 & 9.1\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 11.7 & 0.5 & 5.0$\times$2.6 & 7.1\
W44EF & 18 56 33.02 & $+$01 27 54.40 & 4750.7 & 6.3 & 0.9 & 5.3$\times$3.9 & 8.9\
& & & 4750.7 & 76.3 & 0.9 & 5.3$\times$3.9 & 10.3\
& & & 4660.2, 4765.6 & 6.3 & 0.9 & 6.2$\times$3.7 & 8.9\
W44E & 18 56 29.14 & +01 29 14.15 & 7761.7, 7820.1, 7832.0, 7749.9 & 6.3 & 0.6 &3.9$\times$2.5&8.1\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 6.3 & 0.6 & 3.9$\times$2.4 & 6.3\
W44F & 18 56 36.90 & $+$01 26 34.65 & 7761.7, 7820.1, 7832.0, 7749.9 & 6.3 & 0.6 &4.4$\times$2.4&8.0\
& & & 8135.9, 8189,6, 8207.4, 8118.1 & 6.3 & 0.6 & 4.1$\times$2.4 & 6.5\
Data collection
===============
Observations of excited-state OH lines {#observations}
--------------------------------------
Four well studied SNRs with strong 1720MHz masers (ranging from at least 2.5 to 70 Jy) were selected (W44, W28, IC443 and SgrAEast). Under project ID AP490, we have observed all excited-state OH lines with excitation levels less than 500 K above ground in the receiver bands[^1] of the VLA: the $\Lambda$-doubling triplet at 4.7GHz and the quadruplets at 7.8 and 8.2GHz (see Fig.\[energylevels\]). For SgrAEast we have also observed the quadruplet around 23.8GHz (510 K above ground). The 23.8GHz lines were only observed in SgrAEast as this SNR lies in the region with the highest known interstellar scattering, so observations in this region would best benefit from a smaller $\lambda$. In addition, this region is complex and dense, and might therefore provide the best location to search for emission at 23.8GHz. The limited bandwidth available at 23.8GHz ($\sim$36 kms$^{-1}$) may possibly exclude the detections of masers with velocities much offset from the systemic LSR velocity.
W28, W44 and SgrAEast were observed in CnB-configuration 2005 July 5, and IC443 was observed in the C-configuration 2005 September 5. Based on the positions of the strongest 1720MHz masers [$>2.5$Jy; @claussen97], three regions were selected in W28, two regions in W44 and SgrAEast respectively, and one region in IC443. Given that the line widths of most detected 1720MHz masers are relatively narrow, we observed with a channel separation of 12.207kHz ($\lesssim1$ kms$^{-1}$ at 4.7GHz) and a bandwidth of 1.56MHz (which was changed to 48.828kHz separation at 3.1MHz bandwidth for the 23.8GHz observations only). Using 4 IF (Intermediate Frequency) mode at the VLA, we could observe two transitions within each band simultaneously in dual circular polarization centered on the systemic local standard of rest (LSR) velocity of the source. For the 4.7GHz triplet, we used the redundant IF for the 4750MHz line, centered at another velocity (see Table 1).
The data were calibrated using NRAO’s Astronomical Image Processing System (AIPS), and imaged with natural weighting using standard AIPS procedures. If continuum emission existed in the field, this continuum was subtracted in the UV-plane before imaging. In Table 1, we summarize the results of the observations, including the pointing positions. For each pointing position and frequency, we integrated on source for 5 minutes. The typical final rms noise was $\leq10$ mJy per channel, except for SgrAEast where it was $\leq15$ mJy due to image fidelity issues in this complex region. The field of view at 4.7, 7.8, 8.2 and 23.8GHz was 9.6, 5.8, 5.5 and 1.9respectively.
Collisionally excited 1612 MHz OH masers {#coll1612}
----------------------------------------
Models of collisional excitation that predict 1720 MHz maser emission also predict 1612 MHz masers in regions of higher density ($n\sim10^7$cm$^{-3}$) or higher column density [@pavlakis96]. Since 1612 MHz emission traces much denser material than 1720MHz, 1612 MHz masers are not primarily expected in the immediate vicinity of 1720 MHz masers. We therefore cross-correlated the positions of known SNRs with the locations of known 1612 MHz maser emission in the literature. The positions of the SNRs were taken from the SNR catalogue of @green06, and the 1612 MHz masers were selected from blind surveys [@telintel91; @sevenster97a; @sevenster97b; @sevenster01]. Since most 1612 MHz masers with a double-peaked spectrum are associated with evolved stars [e.g., @habing96], we constrained our cross-correlation to sources with single-peaked or irregular spectra, which resulted in a final sample of 184 sources. No probable associations were found within a minimum search radius of 10, suggesting that the higher column densities suitable to produce collisionally excited 1612 MHz SNR maser emission do not occur frequently in SNRs. A special case may be SgrAEast in the Galactic center region where a much more sensitive 1612MHz OH survey is available [@sjouwerman98], and which we will further discuss separately in Sect. \[gc\].
Discussion
==========
Other detections of excited-state OH lines
------------------------------------------
No detection of either absorption or emission was made in any of the lines searched for excited-state OH (but see @fish07 for possible 6030/6035MHz main line absorption in SgrAEast). This may seem somewhat surprising, since modeling of the three lower rotational states of OH in SFRs has demonstrated that 1720, 4765 and 6035MHz masers can occur under similar conditions in regions of shocked gas [@gray91; @gray92]. Observations of SFRs support the coexistence of these lines. Indeed, a search for excited-state OH masers at 4765 and 6035MHz in a sample of SFRs with 1720MHz masers resulted in high detection rates [@macleod97]. About one-third of the 1720MHz masers have associated 4765MHz masers, and as many as about two-thirds display 6035MHz masers, indicating that the excited-state OH masers may form under similar conditions to the 1720MHz masers. This is further supported by detailed mapping of the SFR W3(OH), showing that a third of the 4765MHz spots are spatially coincident with 1720MHz masers [@palmer03]. In several SFRs, rotational lines as high as $\sim500$ K above ground (in the seventh-lowest rotational level, $^2\Pi_{3/2}, J=9/2$) have been detected in absorption and emission, including weak maser emission [e.g. in W3(OH) and SgrB2 @baudry81; @gardner87; @wilson90; @baudry02; @fish07].
Most SFRs show main-line 1665/1667MHz masers, though there are exceptions: @niezurawska04 report on 4765MHz masers associated with SFRs that display 1720MHz emission without main line emission, suggesting that these regions could contain shocks with similar properties to SNR shocks. Despite observations of coexisting 1720, 4765 and 6035MHz masers in SFRs, we have not been able to find higher excitation lines in SNRs. In contrast to SNRs however, where 1720MHz masers are thought to be collisionally pumped, the main-line 1665/1667 and 6035MHz masers are probably radiatively excited. It should be pointed out that the models by @gray91 [@gray92] predicting co-propagating 1720, 4765 and 6035 MHz masers include a strong far-infrared radiation field associated with the parent star, suitable for SFRs. Hence, radiative pumping routes are likely to be critical for producing simultaneous 1720, 4765MHz satellite line emission and 6035MHz main line emission.
Predicted OH maser transitions in SNRs
--------------------------------------
Existing models predict a sequence of inversions as the number density ($n_{\mathrm{H}_2}$) or the column density of OH ($N_{\mathrm{OH}}$) is increased [@pavlakis96; @pavlakis00; @lockett99; @wardle07]. To predict the maser optical depths in all 1.6, 4.7, 6.0, 7.8, 8.2 and 13.4GHz transitions, we have used MOLPOP[^2]. The MOLPOP program solves the molecular level population equations using the escape probability method for a homogeneous slab. We use the @offer94 collision rate coefficients for the lowest 24 energy levels, with an ortho-para ratio of 3:1. Thus, the 23.8GHz transitions are not included in this model. Following the results by @lockett99, we start with post-shock gas properties typically producing 1720MHz masers in SNRs: molecular density $n_{H_2}\sim10^5$cm$^{-3}$, temperature $T\sim75$K, OH fraction $f_{\rm OH}\sim10^{-5}$, and thermal line-widths. Using these values, in Fig. \[offer\] we plot the maser optical depth as a function of OH column density, $N_{\rm
OH}$. These results are consistent with those of @wardle07. With the given parameters, the model does not predict any significant maser optical depths in any of the 7.8, 8.2 and 13.4GHz lines, nor in the individual 1665, 1667, 4660, 4750, 6016, 6030 or 6035MHz lines. Note that the 6035MHz masers modeled by @gray91 [@gray92] thus must be due to IR pumping and therefore will not be discussed further below. At low column densities, 1720MHz maser emission is produced. At progressively higher column densities, first 6049MHz masers turn on, then 1720MHz masers turn off, so that there is an overlap range in which the OH column density is simultaneously low enough to produce a detectable 1720MHz maser and high enough to produce 6049MHz maser emission as well. At still higher column densities, the 4765 and 1612MHz masers turn on and the 6049MHz masers turn off. The transition between 1720MHz masing and 1612/4765MHz masing occurs when the 79$\mu$m transitions between the $^2\Pi_{3/2}, J = 3/2$ states and the $^2\Pi_{1/2}, J = 1/2$ states (Fig. \[energylevels\]) become optically thick [@elitzur76]. It is worth noting that in the low-temperature ($T
< 120$K) regime, the collisionally-excited masers occur only in satellite-line transitions ($\Delta F = \pm 1$, see Fig.\[energylevels\]). At higher temperatures, main lines ($\Delta F =
0$) can become inverted as well via collisions and local line overlap [e.g., @pavlakis96; @pavlakis00].
Inversion of the 1720MHz maser using the Offer collision rates depends on the hydrogen ortho-para ratio [@pavlakis96]. To test the reliability of our results, we also calculated the level populations using hard sphere cross sections and obtained results similar to those found using the Offer rates with an ortho-para ratio of 3.
The only OH transition observed to produce a detectable maser in SNRs is the 1720MHz transition. A targeted search for 6049MHz masers toward 36 SNRs failed to detect a single maser [@mcdonnell07]. Recently, we also used the Effelsberg telescope to search for the four 6GHz lines in W28, W44, IC443 and SgrAEast, with no detections [@fish07]. As noted in Sect.2.1, the 1612MHz masers detected in the @telintel91 [@sevenster97a; @sevenster97b; @sevenster01] blind surveys do not produce any detections associated with SNRs (but see Sect.\[gc\]). Finally, in this work, we fail to obtain positive detections at 4765MHz or any other excited-state transition.
Detectability of predicted masers
---------------------------------
One reason why no excited-state OH lines have been detected may be due to sensitivity. The maser emission will be amplified according to $T_{\rm m}\simeq T_{\rm bg}e^{\tau}$, where $T_{\rm m}$ is the brightness temperature of the maser, $T_{\rm bg}$ is the brightness temperature of the background continuum radiation, and positive $\tau$ is the maser optical depth. By comparing the brightness temperature of a known 1720MHz maser and its background continuum, we can calculate the maser optical depth and compare it to the values predicted in Fig. \[offer\]. If consistent, the results can be scaled to estimate the predicted maser brightness temperatures in the excited states.
As an example, we consider SgrAEast. We select the region where a 0.18 Jy maser occurs at +55 kms$^{-1}$ [@yusef-zadeh96]. At that position, the background continuum at 1720MHz corresponds to a brightness temperature of $T_{\rm bg}\sim 100$K, and the maser brightness temperature is $T_{\rm m}\sim2.3\times10^5$K. Assuming all continuum is in the background, this implies a lower limit to optical depth of $\tau\sim8$, well in agreement with Fig. \[offer\] and other existing models [@lockett99; @wardle07]. To estimate the $T_{\rm m}$ for the transition at 4765MHz, the 1720MHz brightness temperature is scaled using the spectral index $\alpha=-1$ measured in SgrAEast by @pedlar89. The brightness temperature of a synchrotron emitter changes as $T_{\rm b}\propto\lambda^{2-\alpha}$ [@rybicki79], yielding a 4765 $T_{\rm bg}\sim5$K. An optimistic reading of Fig. \[offer\] in the region where 1720MHz emission still occurs will give a $\tau\sim2$, resulting in $T_{\rm m}\sim
40$K. Indeed, this is below the 5-$\sigma$ detection limit in our VLA observations, which is 50mJy or 115K. Thus, it might be difficult to detect 4765MHz masers co-propagating with 1720MHz masers without deeper integrations than those presented in Sect. 2.
A similar estimate for the 6049MHz line, assuming a $\tau\sim6$ and $T_{\rm bg}\sim2.5$K yields $T_{\rm m}\sim1\times10^3$K, which should be easier to detect. With the upgrade of the VLA to EVLA, the 6.0GHz lines will be observable and are thus warranted a deeper search. However, we note that to date, searches for 6049MHz have proved negative. This indicates that the 1720MHz masers occur only in regions of low column densities, insufficient for the formation of 4765 and 6049MHz masers.
Column density effects
----------------------
The most straightforward explanation of the absence of detectable excited-state OH maser emission in SNRs is that highest column density peaks produce the 1720MHz masers but are not high enough to generate maser emission in any other transition. We note that the search presented here is biased toward regions with existing 1720MHz masers, and could thus be biased against regions with the higher column densities needed for the excited-state OH (Fig.\[offer\]). Estimates of the OH column densities are available via thermal absorption in the ground state transitions of OH, when sufficient background continuum emission is present. Absorption of 1667MHz OH in W28 implies $N_{\rm OH}\simeq
2\times10^{16}$cm$^{-2}$, for an adopted excitation temperature $T_{\rm ex}$ of 10K [@yusef-zadeh03a], indicating low column densities with little possible inversion of the excited-state OH (Fig. \[offer\]). Similar observations of IC443 show OH absorption in molecular clumps with $N_{\rm OH}$ ranging between $0.7\times10^{16}-1.4\times10^{17}$cm$^{-2}$ [@hewitt06]. Slightly higher column densities of the order of $2-3\times10^{17}$cm$^{-2}$, assuming $T_{\rm ex}\simeq10$K, are derived for SgrAEast (Sjouwerman, priv. comm). With measured OH column densities of the order of $10^{16}-10^{17}$cm$^{-2}$, models based on the @offer94 collision rates allow the possibility that there might exist a few regions in SNRs with sufficient column densities for the 6049MHz line to be inverted (Fig.\[offer\]). Our non-detections are, however, consistent with regions of lower column densities, where 1720MHz masers are strong, and little or no inversion has occurred in the higher transitions.
Temperature and density
-----------------------
An alternative explanation to the absence of excited-state OH masers could be that the density or temperature is too high. Models of 1720MHz emission imply that the optimal post-shock density in which 1720MHz masers occur is of the order $n_{\rm
H_2}\sim10^5$cm$^{-3}$, with temperatures $T\sim50-125$K [@lockett99; @wardle99]. Millimeter observations of multiple molecular lines in IC443 suggest that there may be regions of much higher temperature and density in SNRs. In particular, in IC443, the 1720MHz masers occur within the IC443 cloud complex G [@denoyer79]. From the millimeter data, the medium in this clump appears to fit a two-component model with both a cool, low-density ($T\sim80$K, $n\sim10^5$cm$^{-3}$) component, and a warmer, higher density ($T\sim200$K, $n\sim3\times10^6$cm$^{-3}$) component [@vandishoeck93]. Similar numbers were derived by @turner92. At densities of the order of $10^6$cm$^{-3}$, but with lower temperature ($T=75$K), previous modeling has already shown that the 1720MHz transition may still be weakly inverted [@lockett99].
Here we calculate the expected maser optical depths in a warmer, higher density region, again using MOLPOP assuming thermal line-widths, $T\simeq200$K, and $n\sim5\times10^6$cm$^{-3}$, we achieve the results shown in Figure \[mpdense\]. As expected, the 1720MHz line is quenched at this higher density. The 4765MHz transition is only modestly inverted with maser optical depths of the order of 0.5–1 at column densities $N_{\rm
OH}\sim10^{16}-10^{17}$cm$^{-3}$. Even weaker inversion is seen in the 6049MHz line. We therefore conclude that if the molecular medium is commonly composed of two components like those in IC443, the 1720MHz masers must be associated with the cooler, lower-density component. A warmer component will not be able to produce excited-state OH masers, and as a consequence, we conclude that it is not the density that primarily constrains the formation of excited-state OH.
We note that at higher densities, the model predicts a strong maser line in the 1612MHz transition (Fig. \[mpdense\]). However, 1612MHz masers in general are not correlated with SNRs (Sect. 2.2), which could indicate that warm, dense regions like the one in IC443G modeled by @vandishoeck93 are rare in SNRs.
A possible exception: SgrAEast {#gc}
------------------------------
Sgr A East is a very complicated region and has been studied in detail in the past. In particular, it has been surveyed for 1612MHz masers to find OH/IR stars [@sjouwerman98 and references therein]. However, 1612MHz emission can also originate from the interaction of a SNR and its surrounding ISM (Sect.\[coll1612\]; Fig. \[mpdense\]). We therefore turned to the list of double peaked 1612MHz masers found by @sjouwerman98, where some maser sources are attributed to the $+$50kms$^{-1}$ molecular cloud (MC). The individual MC spectra shown may not represent the true emission (and absorption) in the cloud, nor be complete for emission of the cloud (e.g., see the “maxmap” in their Fig. 8) as the data reduction and search were focused on finding double peaked OH/IR stars. However, their MC list does partly represent compact 1612MHz emission, either thermal or low-gain masing not attributed to stars, and therefore is relevant in this respect.
In Fig. \[sgraeast\] we plot the locations of the pure MC 1612MHz emission (circles) from @sjouwerman98 on top of the 1720MHz SNR masers (crosses) and 1.7GHz radio continuum (grey scale) taken from @pihlstrom06. The following observations can be made:
- All regions of 1612MHz emission have one or more peaks in the V$_\mathrm{LSR}$ range of 30 to 70 kms$^{-1}$, i.e., closely following the V$_\mathrm{LSR}$ gradient over, and range of the $+$50kms$^{-1}$ molecular cloud.
- The 1612MHz emission avoids regions of 1720MHz emission (the closest possible association is about 0.5pc in projection), which is to be expected as a reflection of column density differences (Figs. \[offer\] & \[mpdense\]).
- The 1612MHz emission is co-located with the radio continuum outlining the SNR, but not on top of the regions of the strongest radio continuum emission.
- No 1612MHz emission is seen near the compact regions in the east and south: i.e., the 1612MHz emission is probably not related to SFRs.
We note that no 4765MHz emission (this work) nor 6049MHz emission [@fish07] is detected toward SgrAEast, so there does not exist a continuous density gradient where the 1720, 6049, 4765 and then 1612MHz transitions, respectively, are purely collisionally pumped. Based on these data, we are not able to determine whether the 1612MHz emission is purely thermal emission from the cloud. It could also be pumped by the interstellar radiation field and perhaps amplifying the background radio continuum, or indeed collisionally pumped stimulated emission resulting from the reverse shock in the SNR. To address these questions in the Galactic center (GC), a more specific observational setup will be required to properly account for details like, for example, missing zero-spacing flux. As the GC is a special case, it is beyond the scope of this paper. Pending a more detailed analysis of the GC region we therefore assume that there generally is no collisionally excited 1612MHz emission due to SNR/MC interactions, with a possible exception in the GC.
Concluding remarks
==================
For OH column densities $N_{\rm OH}\sim 10^{16}-10^{18}$ cm$^{-2}$, models of collisionally-pumped excited-state OH in SNRs predict inversions in the 1720, 4765 and 6049MHz lines, while no significant maser optical depths will be produced in other transitions of the 4.7, 7.8, 8.2 and 23.8GHz lines. We have presented the results from a search of all these excited-state OH transitions in four, well-known SNRs, with no detections. These VLA observations could not tune to the 6049MHz line, which is also predicted to have large optical depths under conditions similar to those of 1720MHz masers. However, a few recent searches for the 6049MHz line in SNRs have yielded no detections [@fish07; @mcdonnell07]. In the future, the EVLA upgrade will allow for a targeted, deeper search for 6049MHz in these SNRs.
Our non-detections are consistent with regions of lower column densities ($N_{\rm OH}\leq5\times10^{16}$cm$^{-2}$), where 1720MHz masers are strong, and little inversion occurs in the higher transitions. Based on VLA and single-dish observations by @yusef-zadeh95 & @hewitt07, such a post-shock medium may have a large filling factor. @hewitt07 found that a large part of the 1720MHz maser flux is undetected using VLA baselines, indicating a widespread distribution of weaker 1720MHz emission. This likely reflects large post-shock regions of low column density, or alternatively, regions of different temperature and density from what is normally expected for 1720MHz maser production. In turn, such a gas component will provide little chance of excited-state OH maser emission. If this is the case, excited-state OH will not be detected either coincident with or offset from detected 1720MHz masers in SNRs. The non-detections imply a low OH column density in SNRs, in agreement with the rather low estimates of the $N_{\rm OH}\simeq 10^{16}$cm$^{-2}$ expected to be produced in C-type shocks [@wardle99; @lockett99].
From these results we draw the conclusion that in the regions where 1720MHz masers are found, the OH column densities are insufficient for excited-state OH masers to exist. This is supported by the absence of 1612MHz masers in SNRs, which require 1–2 orders of magnitude higher column densities than the 1720MHz masers. However, the upper limit to the column density must be even stricter to avoid producing 6049 and 4765MHz masers.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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[^1]: The currently ongoing upgrade to the Expanded VLA (EVLA) will also allow observations of the 6.0GHz lines, as well as the 13.4GHz lines later during the final stages of the upgrade. However, these transitions were not observable during this project. See @fish07 for the 6.0GHz lines in these four SNRs.
[^2]: Available at http://www.pa.uky.edu/ moshe/molpop.zip
|
---
author:
- 'D.I. Bradley'
- 'R. George'
- 'A.M. Guénault'
- 'R.P. Haley'
- 'S. Kafanov$^*$'
- 'M.T. Noble'
- 'Yu.A. Pashkin'
- 'G.R. Pickett'
- 'M. Poole'
- 'J.R. Prance'
- 'M. Sarsby'
- 'R. Schanen'
- 'V. Tsepelin$^\dagger$'
- 'T. Wilcox'
- 'D.E. Zmeev'
title: Operating Nanobeams in a Quantum Fluid
---
Introduction {#sec: Introduction .unnumbered}
============
The relentless drive to reduce the size of electronic components, coupled with the continuous progress in fabrication technology, has given us the capability of creating complex structures on the micron and submicron scale. In consequence MEMS and NEMS are becoming common research tools in the areas of mass and force sensing [@Nature.446.1066.Burg; @NatNanotech.7.301.Chaste], atomic force microscopy [@NatureNano.4.1748.Wilson], nanofluidics [@Nanolett.15.8070.Kara] and quantum behaviour of macroscopic mechanical oscillators [@Nature.464.697.OConnell; @Science.349.952.Wollman].
In this paper we address the efforts to develop new MEMS and NEMS devices, or to adapt those already available, for use in liquids at cryogenic temperatures [@Nanotechnology.11.165.Kraus; @J.Low.Temp.Phys.183.284.Defoort; @PhysRevLett.113.136101.Defoort; @Rev.Sci.Instr.84.025003.Gonzales; @Nat.Phys.3813.Harris]. There are two very major aims here. First, a micro- or nanoscale resonator immersed in a pure superfluid would immediately allow us to probe its properties in the completely novel regimes made accessible when the device dimensions become comparable to the coherence length of the liquid.
Secondly, however, there is a further, wider motivation. Accessing the quantum regime for mechanical systems should lead to the realm of quantum mechanical systems where bulk mechanical objects consisting of billions of atoms are governed by the laws of quantum physics [@PhysToday.58.7.36.Schwab; @PhysRep.511.273.Poot]. At the lowest temperatures and for very small displacements, immersing a NEMS device in a superfluid (which is effectively a “mechanical vacuum” but still has good thermal properties) may enable the cooling of devices[@Nat.Commun.10455.Bradley] down to the mechanical-ground-state level using “brute force” cooling, which so far has been accomplished using much higher frequency systems[@Nature.464.697.OConnell]. As a first step we describe here the use of a 1MHz resonator in superfluid $^4$He down to $\sim$1K. For such a resonator the quantum ground-state regime would be reached at a temperature of $\sim$100$\mu$K. We routinely achieve such temperatures in superfluid $^3$He cooled by the nuclear demagnetization of copper. Thus the devices described here, if immersed in superfluid $^3$He as the thermal contact agent, should readily reach the quantum regime.
Historically, mechanical resonators have been widely used to probe bulk properties of $^3$He and $^4$He superfluids [@PhysicaB.197.390.Pickett; @JPC.4.129.Black], such as their transport properties, to create and detect the presence of topological structures/defects such as quantum vortices [@PhysRevLett.82.4831.Stalp; @Nature.4.46.Bradley; @NatPhys.7.473.Bradley], and to study the pairing configurations of the various $^3$He phases [@Nat.Phys.3813.Bradley]. The most long-standing and commonly used mechanical resonator is the vibrating wire [@JLowTempPhys.62.511.Guenault], typically comprising a few millimetres of superconducting wire, with diameters in the range of $1-100$ micrometres, and with resonance frequencies of up to few kilohertz. A vibrating wire immersed in superfluid can act as both a generator and detector of excitations and as a secondary thermometer down to millikelvin temperatures in $^4$He [@PhysRevLett.100.045301.Yano] and tens of $\mu$K in $^3$He [@Physica.126B.260.Guenault]. During the past decade a range of newer devices have been employed to study bulk quantum fluids: miniature piezoelectric quartz tuning fork oscillators with frequencies up to hundreds of kilohertz [@PhysRevB.85.014501.Bradley; @PhysRevB.89.014515.Ahlstrom], aluminium-coated silicon “goal-post” shaped micro-electro-mechanical systems [@J.Low.Temp.Phys.183.284.Defoort] and comb-drive electrostatic MEMS [@Rev.Sci.Instr.84.025003.Gonzales]. Nevertheless, superconducting wire of a few microns diameter has been the most convenient probe to measure the lowest temperatures inside the superfluid $^3$He via the damping and frequency shift of the resonance curve. It has even been proposed that vibrating wire bolometry might provide ultra-sensitive low temperature dark-matter particle detectors[@Pickett1988; @PhysRevLett.75.1887.Bradley].
Prior to the current work, NEMS devices have only been successfully used at cryogenic temperatures to probe vapour properties [@PhysRevLett.113.136101.Defoort; @Nanotechnology.11.165.Kraus], since bulk cryogenic liquids were perceived to be too difficult to study. Recently, the properties of a superfluid film have been measured by using an opto-mechanical oscillator [@Nat.Phys.3813.Harris]. The work presented here constitutes the first successful measurement of bulk superfluid properties using NEMS resonators and provides a significant step towards both building routine superfluid probes and the ambitious goal of cooling a NEMS beam to its quantum-mechanical ground state using a quantum fluid as the cooling medium.
Results {#results .unnumbered}
=======
Beam Characteristics in Vacuum {#sec: VacuumProp .unnumbered}
------------------------------
The aluminium NEMS beams used in our experiments were formed lithographically on a silicon substrate and after fabrication were measured using a standard microwave magnetomotive technique as shown in figure \[fig: VacuumMeasurements\](a). The beam is driven by an oscillating current which provides the necessary lateral Lorentz force generated in the ambient magnetic field (typically 5T). The details of manufacturing and measurement setup are described in the Methods section.
Initial characterisations of our beams were performed in vacuum at a temperature of 4.2K. An example of the frequency response of a 50$\mu$m beam in vacuum is shown in figure \[fig: VacuumMeasurements\](b). Resonance peaks were detected at frequencies of 1.19MHz, 3.82MHz and 7.11MHz corresponding to the first three odd harmonics of the transverse flexural oscillations. The measured resonance frequencies can be compared to the expected values for a doubly-clamped beam of rectangular cross-section given by [@Bao]: $$\label{eqn: BeamFreq}
f_n=\frac{k_n^2}{\pi\sqrt{48}}\frac{w}{l^2}\sqrt{\frac{E}{\rho_\mathrm{beam}}}\sqrt{1+\gamma_n\left(\frac{l}{w}\right)^2\frac{\Delta l}{l}},$$ where $w$ and $l$ represent the width and the length of the beam, respectively. The coefficients $k_n$ and $\gamma_n$ have different values depending on the eigenmode of the resonance: [@Bao] $k_1=4.7300$, $\gamma_1=0.2949$, $k_2=7.8532$, $\gamma_2=0.1453$, and $k_{n\geqslant 3} = \pi(n + 1/2)$, $\gamma_{n\geqslant 3}=12(k_n-2)/k_n^3$. Despite a weak granularity of our aluminium films, we assume that the Young’s modulus and density of the beams are close to the bulk values of $E = 70$GPa and $\rho_\mathrm{beam}=2700$kgm$^{-3}$ respectively. The ratio $\Delta l/l$ describes the tensile strain of the resonators, which is expected to be substantial at cryogenic temperatures owing to the mismatch in the thermal expansion coefficients of the silicon substrate and aluminium film. Simple calculation shows that the unstressed 50$\mu$m long NEMS beam, with an approximately square cross-section of 0.1$\mu$m$\times$0.1$\mu$m, should have a fundamental resonance at 209kHz. The analysis of measured first, third and fifth eigenmodes shows that the actual average width, $w$, of our beams is about 0.18$\mu$m and the value of the tensile strain, $\Delta l/l\approx4.3\cdot10^{-4}$, is an order of magnitude lower than that measured for shorter aluminium resonators [@Appl.Phys.Lett.92.043112.Li; @Nano.Lett.10.4884.Sulkko]. We note that, due to an extremely high aspect ratio ($l/w\gtrsim10^2$), our resonators have a significant compression stress at room temperature, which is clearly observable as bending on the micrograph of our typical beam (see figure \[fig: VacuumMeasurements\](a)), arising from the forced match of the substrate and deposited lattices at the time of deposition. Such obvious compression stress is not seen in shorter samples in other experiments [@Appl.Phys.Lett.92.043112.Li; @Nano.Lett.10.4884.Sulkko].
The power dependence of the frequency response of the 50$\mu$m aluminium beam near the first harmonic is presented in figure \[fig: VacuumMeasurements\](c). At low excitations the beam exhibits a linear response and has a quality factor ($Q$-factor) on the order of 10$^3$. Similar $Q$-factors are observed for higher harmonics and other beams in vacuum at 4.2K. At high excitations the nonlinearities stiffen the beam and the resonance peak position shifts to a higher frequency as expected for the Duffing oscillator [@MicrosystemTechnologies.1432.1.Tajaddodianfar]. In addition to Duffing-like non-linearity, above intermediate applied power (-88dBm) a second resonance becomes apparent which we believe is the lower-level excitation of the second of the two near-degenerate flexural modes of the almost square cross-section beam.
Since our ultimate goal is to probe liquid helium, it is instructive to deduce the force-velocity characteristics of the beam from the frequency response as a function of power to confirm the velocity below which the beam response remains linear. We have used the definition of quality factor $Q = f_1/\Delta f = \pi f_1 \, m_\mathrm{beam} v^2/P_{\max}$ to deduce the force $F_\mathrm{L}$ and velocity $v$ values, and applied them even when the resonance curve became non-Lorentzian: $$v\approx\sqrt{\dfrac{P_{\max}}{\pi m_\mathrm{beam} \Delta f}};
\qquad
F_\mathrm{L}\approx\sqrt{\pi m_\mathrm{beam} P_{\max} \Delta f}.
\label{eqn: ForceVelocity}$$ Here $m_\mathrm{beam}=\rho_{\mathrm{beam}}V$ is the mass of the beam, $\Delta f$ is the width of the resonance and $P_{\max} = F_\mathrm{L}v$ is the maximum observable power amplitude. We conclude that the beam behaves linearly up to a velocity amplitude of 0.1ms$^{-1}$ corresponding to a displacement of 13nm, above which it behaved non-linearly. Therefore any non-linear behaviour observed in a liquid medium below that velocity should be attributed to the interaction of the beam with the fluid.
Beam response in liquid helium {#sec: ProbingLiquid .unnumbered}
------------------------------
Figure \[fig: LiquidHe4\](a) presents the temperature dependence of the frequency response of the beam around the fundamental resonance. The resonance curves for all temperatures are very broad with widths of order 100kHz, *i.e.* with quality factors of order 10. This guarantees that in-liquid measurements are completely dominated by liquid-induced dissipation since the corresponding intrinsic beam dissipation is negligible in comparison. The resonance frequency of the beam is also significantly lower ($\sim$10%) than the vacuum value owing to the increase in effective mass of the beam arising from the induced flow of surrounding liquid helium. Helium flow can be thought of as having two components, the “pure potential” backflow of liquid needed to allow the passage of the beam, and the additional liquid that moves because it is viscously “clamped” to the beam.
The frequency sweeps in normal helium, between 4.2K and 2.178K, are almost indistinguishable and for clarity only representative data sets at 4.2K and 2.6K are plotted in the figure. As the temperature falls below the superfluid transition at 2.178K, the damping experienced by the beam decreases and the resonance peak shifts towards higher frequencies (since below the transition decreasing masses of liquid are dragged along by the wire). The observed increase of the beam velocity with falling temperature reflects the decreasing damping with falling normal fluid fraction. At the lowest measured temperature of 1.3K, where the helium has the highest superfluid fraction $\sim$95%, the beam has the smallest observed resonance width with a corresponding maximum velocity of 2.8mms$^{-1}$ (which we note is much lower than the velocity for the onset of any intrinsic nonlinear behaviour of the beam in vacuum).
For a quantitative description of the beam behaviour in liquid helium, we compare our results with the phenomenological two-fluid model for the superfluid. In the linear regime, at low beam velocities, we can treat our “beam + liquid” system as an externally-driven damped harmonic oscillator with a variable effective mass $\tilde{m}$ and a constant effective elastic constant $k$ [@J.Low.Temp.Phys.146.537.Blaauwgeers]. The ratio of resonance frequencies of the “bare” oscillator in vacuum, $f_1=\sqrt{k/4\pi^2m_\mathrm{beam}}$, and our “beam + liquid” system, $f=\sqrt{k/4\pi^2\tilde{m}}$, can be written as: $$\label{eqn: FreqRatio}
\left(\dfrac{f_1}{f}\right)^2=\dfrac{\tilde{m}}{m_\mathrm{beam}}=1 + \beta \dfrac{\rho}{\rho_\mathrm{beam}} + B\dfrac{\rho_\mathrm{n}}{\rho_\mathrm{beam}}\dfrac{S}{V}\sqrt{\frac{\eta}{\pi\rho_\mathrm{n} f}}.$$ The effective mass $\tilde{m}$ of NEMS beam with volume $V$ and surface area $S$ immersed in a fluid with the total density $\rho$, the normal component density $\rho_\mathrm{n}$ and viscosity $\eta$ consists of a self mass of the resonator $m_\mathrm{beam}$ and two additional contributions associated with: (*i*) a mass of fluid proportional to the volume of the oscillating body $m_\rho\propto\rho V$; (*ii*) a mass of fluid in a layer of thickness the viscous penetration depth $\delta=\sqrt{\eta/(\pi \rho_\mathrm{n} f)}$ dragged along by the oscillating motion $m_\eta\propto \rho_\mathrm{n} S\delta$. Coefficients $B$ and $\beta$ are geometry-dependent parameters. The theoretical predicted values in the case of an infinitely long rectangular cross-section beam with height $h$ and width $w$ moving perpendicular to the width are $\beta = (\pi/4)h/w$ and $B=1$ [@JAppPhys.84.64.Sader].
The damping experienced by our system includes any intrinsic damping such as inter-crystalline friction within the material of the resonator itself, and external forces such as Stokes’ drag which describes the interactions with the surrounding fluid. At a temperature of 4.2K this latter contribution dominates even for a beam oscillating in a rarefied helium gas at pressures as low as $\sim$10Pa. The Stokes’ drag force is proportional to the velocity of the oscillating body. The exact calculation of the proportionality coefficient requires the full solution of the flow field around the oscillating body and in the high-frequency limit is given by $CS\sqrt{\pi\rho_\mathrm{n}\eta f}$, where $C$ is a geometrical constant independent of the fluid. For an infinitely long cylinder oscillating perpendicular to its axis, the value of this constant is [@J.Low.Temp.Phys.146.537.Blaauwgeers] $C=2$. For our 50$\mu$m beam with a resonance frequency of 1.2MHz the viscous penetration depth is 80nm at 4.2K (150nm at 1.5K) and is comparable to the beam thickness, which means that the high-frequency approximation used here is marginal. That said, treating the cross-over from high-frequency to steady-flow behaviour even for a cylinder is a formidable (non-trivial) task even in classical hydrodynamics[@LandauLifshitz:FluidMechanics] and a full treatment would not significantly benefit the description and core understanding of our main results. The distance of the beam to the substrate at 2$\mu$m is far enough away not to influence the dynamics in our temperature range. Based on these assumptions, the resonance width for an oscillator immersed in a fluid can be expressed as [@J.Low.Temp.Phys.146.537.Blaauwgeers]: $$\Delta f = C\frac{S}{2 V\rho_\mathrm{beam}} \sqrt{\frac{\eta\rho_\mathrm{n} f}{\pi}} \left(\frac{f}{f_1}\right)^2.
\label{eqn: DeltaF}$$
Figure \[fig: LiquidHe4\](b) shows as a function of temperature the measured change in the resonance frequency, presented as the square of the ratio of the resonance frequency in vacuum, $f_{1}$ to that measured in the liquid, $f$ (as expressed in equation (\[eqn: FreqRatio\])). The rapid increase in the frequency $f$ below the superfluid transition temperature arises almost entirely from the reduction in the liquid viscously “clamped” to the beam as a consequence of the falling normal fluid fraction. The red solid line is the least-square fit of equation (\[eqn: FreqRatio\]) using the parameters for liquid $^4$He tabulated by Donnelly and Barenghi [@JourPhysChemRefData.27.1217.Donnelly] and yields the following values for the geometric parameters: $\beta=1.18 \pm 0.02$ and $B=1.19 \pm 0.01$. The measured data are in excellent agreement with the model over the whole temperature range. Similar results have been observed for our beams with shorter lengths and higher resonant frequencies.
Figure \[fig: LiquidHe4\](c) shows the measured resonance width of the beam $\Delta f$ in liquid helium as function of temperature. We found that the Stokes’ drag model fits the measured data excellently until temperature of 1.7K below which the model starts to deviate from our measurements. The solid red curve corresponds to the least-square fit from equation (\[eqn: DeltaF\]) for the data between 4.2K and 1.7K with the relevant liquid $^4$He parameters from Donnelly and Barenghi [@JourPhysChemRefData.27.1217.Donnelly]. The least-square fit yields the geometric parameter $C=2.62\pm0.06$. The dashed red line shows the expected behaviour for the Stokes‘ model and highlights the presence of excess damping detected by the beam below 1.7K. It is clear that the intrinsic damping of this beam plays no role in the observed discrepancy since it was nearly *an order of magnitude smaller* in vacuum measurements. Several effects might contribute to this difference: (*i*) we could be generating quantum vorticity or exciting Kelvin waves on existing vortices attached to the beam; (*ii*) it is possible that we start to observe the onset of transition from the hydrodynamic to ballistic regime in the liquid phase; (*iii*) finally, we may be seeing the effect of acoustic emission.
Regarding the effect of turbulence, we would expect the nucleation of quantum turbulence to be triggered only at velocities an order of magnitude higher [@PNAS.111.4699.Vinen]. However, this has not hitherto been studied with such small objects and high frequencies. As to the possibility that we are seeing transition to the ballistic regime, unfortunately temperatures below 0.8K needed to observe the $T^4$ dependence expected for the phonon damping in the ballistic regime, are not accessible with the current experimental apparatus. It is also unlikely that acoustic damping is solely responsible for the discrepancy observed since some high frequency beams exhibit lower measured damping. However, a standing acoustic resonance for a particular beam and cell cannot be ruled out as the reason for the higher-than-expected dissipation at a certain temperature. Since various NEMS samples have slightly different discrepancies, it is clear that systematic measurements of several beams of various length, and consequently frequencies, down to much lower temperatures are needed to determine the actual nature of these discrepancies.
Discussion {#sec: Discussion .unnumbered}
==========
It is instructive to compare the temperature dependence of the resonance frequency of our NEMS with other devices which have been successfully used to probe the properties of superfluid helium. Using the two-fluid model we can make a reasonable prediction of which beam-like devices with a characteristic size of cross-section $d^2$, density $\rho_\mathrm{beam}$ and the resonance frequency $f$ will have the highest frequency sensitivity to the temperature-dependent changes in superfluid. First, to ease the analysis we rewrite equation (\[eqn: FreqRatio\]) as: $$\label{eqn: FreqSens}
\left(\dfrac{f_1}{f}\right)^2 - 1 = \frac{1}{\rho_\mathrm{beam}} \left( \beta\rho + B \sqrt{\frac{\eta\rho_\mathrm{n}}{\pi}} \frac{4}{d\sqrt{f}}\right).$$ Then, setting the geometrical coefficients $B$ and $\beta$ to be on the order of unity for all resonators, and taking the right hand side (RHS) terms one by one, it is immediately clear from the leading coefficient that beams with lower material densities would be expected to be more sensitive. Quartz tuning forks ($\rho_\mathrm{Q}=2690$kgm$^{-3}$) and aluminium beams ($\rho_\mathrm{Al}=2700$kgm$^{-3}$) are clearly favoured over denser NbTi wires ($\rho_\mathrm{NbTi}=6500$kgm$^{-3}$). The first term on the RHS corresponds to the backflow around the resonator, has only weak temperature dependence and can be neglected in the discussion. The second term, describing viscous clamping, predicts that an object with the smallest cross-section and the lowest frequency will yield the highest frequency sensitivity. Putting this all together we find that of the devices we have tested, the most sensitive should be the smallest diameter vibrating wire resonator manufactured from NbTi ($d = 0.9$$\mu$m, $f\approx10^3$) closely followed by the aluminium 50$\mu$m long beam ($d = 0.18$$\mu$m, $f\approx10^6$). Despite being manufactured from a denser material, the vibrating wire resonator should show a sensitivity approximately two times better than our 50$\mu$m long beam owing to its submicron size and very low frequency. Finally the thinnest available tuning fork ($d = 25$$\mu$m, $f\approx10^4$) comes in at a sensitivity $\approx2.5$ times worse than our 50$\mu$m aluminium beam.
Figure \[fig: Comparison\](a) shows the fractional frequency change for these NEMS, as a function of the normal fluid fraction $\rho_n/\rho$ from the superfluid transition down to 1.1K. It is clear that our 50$\mu$m and 15$\mu$m NEMS show a significant fractional frequency change over the whole temperature range, and are comparable to the smallest vibrating wire resonator and much more sensitive than a tuning fork. It is also worth noting that our NEMS do not show a significant saturation down to the lowest measured temperatures.
While a good frequency sensitivity is important for identifying the best resonator, a high $Q$-factor is also essential for a high signal-to-noise required for measurements. Using similar two-fluid-model arguments to those used above we can deduce from equation (\[eqn: DeltaF\]) that the $Q$-factor can be written as: $$Q = 2C^* \sqrt{\frac{\pi}{\eta\rho_\mathrm{n}}} \left(\frac{f_1}{f}\right)^2 \sqrt{f} d \rho_\mathrm{beam},
\label{eqn: Qfactor}$$ where we can see that the thickest, densest and highest frequency beams will have the best quality factors. This of course is a necessary trade off since these factors tend to reduce the frequency sensitivity.
The insert of figure \[fig: Comparison\](b) presents the $Q$-factors of various resonators in liquid helium as a function of temperature. $Q$-factors at the lowest temperatures vary by almost three orders of magnitude; the superconducting NbTi wire with a 0.9$\mu$m diameter and 25$\mu$m thick quartz tuning fork show the lowest and highest $Q$-factors, respectively. The figure insert supports our conclusions, with the NEMS devices showing much better $Q$-factors than the NbTi 0.9$\mu$m diameter wire, owing to their higher resonance frequencies.
Interestingly, the equation also suggests that the temperature dependence of the reduced $Q$-factor, $Q/(\sqrt{f_1}d\rho_\mathrm{beam})$, should be almost identical for all resonators, provided the in-liquid and vacuum resonance frequencies are similar, which is the case to within 10% for all the data presented here. We have used the $C^*$ coefficient as a scaling parameter to normalise all of the available data and reproduce the reduced quality factor in figure \[fig: Comparison\](b). Despite the very different geometries of the resonators studied, the value of $C^*$ only varies by a factor of 4 (from 0.35 to 1.4), for all the resonators. In the figure it is clear that the reduced $Q$-factors of the 15$\mu$m beam with resonance frequency of 8MHz and 25$\mu$m tuning fork below $\sim$1.6K increases more rapidly with decreasing temperature than those of the others. This would indicate that acoustic emission, which should increase very rapidly with the resonant frequency [@PhysRevB.85.014501.Bradley], may not become the dominant damping for NEMS even at these low temperatures. Overall our results demonstrate that the use of NEMS beams as local superfluid temperature probes has a very exciting prospect since tuning forks and vibrating wires already work extremely well at low temperatures.
The absence of any saturation of the $Q$-factor of the high frequency beam down to $1.2\,\mathrm{K}$ warrants extending measurements down to dilution fridge temperatures. The device should enter the ballistic regime below 0.8K with the $Q$-factor showing a $T^4$ temperature dependence, which would then reveal whether other mechanisms such as intrinsic damping in the beam or acoustic emission become significant. Furthermore, low temperatures and correspondingly lower damping should allow us to investigate the nucleation of quantum turbulence as inferred from the force-velocity dependence of “beam + quantum fluid” system. Possibly single-vortex sensitivity may eventually become achievable using NEMS owing to the well-defined geometry, high sensitivity and small size.
Armed with the knowledge that we can run these devices in superfluid $^4$He, we can now presume that they will also run in superfluid $^3$He. We know $^3\mathrm{He}$ is the best medium to cool electronic devices to the lowest temperatures [@Nat.Commun.10455.Bradley]. Here the liquid temperatures can be taken routinely to temperatures below 100$\mu$K. This would have the immediate advantages that, first, the intrinsic $Q$-factor (inverse damping) of vibrating wire resonators immersed in superfluid $^3$He usually exceeds the $Q$-factor measured in vacuum at low resonator velocities (because the liquid $^3$He cools the material of the resonator to lower temperatures than can usually be achieved by cooling through the leads in a vacuum). Secondly, the small dimensions of the beam should make it an extremely sensitive detector of ambient quasiparticles and could lead to high resolution visualization of quantum turbulence or other topological defects present in $^3$He-B. Thirdly, the coherence length of superfluid $^3$He-B at zero pressure is 80nm and is of the same order as the beam dimensions, thus opening another regime to probe superfluid $^3$He-B.
However, the wider advantage would be that at these temperatures, as pointed out in the introduction, we are at the point when the low frequency nanobeams are on the brink of reaching their quantum ground state, especially if we choose a beam with somewhat higher frequency, say $\sim$10MHz. The challenge of performing such experiments in $^3$He is that they would require both small magnetic fields and minute power dissipation. To achieve the ultimate intrinsic $Q$-factors for the resonators, the highest sensitivity to the liquid and the necessary low dissipation would require that the aluminium is in the superconducting state. This requires an ambient field below 10mT, which makes the excitation and detection of the beam motion rather demanding (bearing in mind that the results reported here were made in a 5T field). However, this is within the capabilities of a SQUID-based voltmeter [@JLowTempPhys.119.703.Bradley]. Additionally, recently developed superconducting diamond beams with a critical field of several Tesla [@Carbon.72.100.Bautze] could be employed, or measurements utilising a microwave reflection technique that drive and detect beam motion electrostatically[@Nano.Lett.10.4884.Sulkko] could be adapted. We would also have to take precautions to reduce the ambient noise, but we are confident that this is possible in the ultra-quiet environments of our microkelvin cryostats [@JLowTempPhys.114.547.Cousins].
Methods {#methods .unnumbered}
=======
Fabrication and Experimental Setup {#sec: Fabrication .unnumbered}
----------------------------------
The NEMS resonators used in our experiments are formed on a silicon substrate by standard nanofabrication methods: electron-beam lithography, metal deposition and reactive ion etching. Our manufacturing technique allows the creation of doubly-clamped aluminium resonators over a broad range of lengths ($l$) from 0.5$\mu$m up to 500$\mu$m. All beams have a lithographically-defined width ($w$) and thickness ($h$) with dimensions of $\approx$0.1$\mu$m and are clamped at both ends to a wider film of the same thickness. The beam is suspended above the substrate by 2$\mu$m. A scanning-electron microscope image of a typical aluminium NEMS beam is shown in figure \[fig: VacuumMeasurements\](a) together with the principle measurement schematics.
The NEMS are mounted on a chip within a small container inside the cryogenic system and connected by two coaxial cables to a network analyser. This setup allows the measurement of the transmission through the beam, which is characterised by a magnetomotive measurement scheme [@ApplPhysLett.81.2253.Ekinci]. The 5T magnetic field $B$ necessary for the measurements is provided by a superconducting solenoid surrounding the sample. The beam orientation is such that the Lorentz force $F_L\sim I B l$, a result of an AC current $I$ flowing through the beam, induces in–plane vibrations of the beam, with the displacement parallel to the chip surface. The transmitted signal is amplified by a low-noise high-frequency amplifier mounted at room temperature and then directed to the corresponding port of the network analyser.
All measurements on the beams were conducted in a $^4$He evaporation cryostat operating over the temperature range from 4.2K down to a base temperature of $\sim$1.2K. After the initial characterization of a beam in vacuum at 4.2K, the cell was slowly filled with helium gas while the beam resonance was monitored and followed. At pressures greater than the saturated vapour pressure at 4.2K, $^4$He condenses and the cell fills with a liquid. We used the damping of a vibrating wire resonator located above the beam to monitor the helium level and to confirm that the beam was indeed fully submerged. The temperature of the cryostat was monitored and controlled both by a calibrated RuO$_2$ resistor and from a measurement of the $^4$He saturated vapour pressure [@JourPhysChemRefData.27.1217.Donnelly]. The $^4$He saturated vapour pressure was measured at room temperature just outside the cryostat on the pumping line. Both temperature measurement methods agreed to within 20mK.
Data and software availability {#data-and-software-availability .unnumbered}
==============================
All data used in this paper are available at http://dx.doi.org/10.17635/lancaster/researchdata/139, including descriptions of the data sets.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank S.M. Holt, A. Stokes, and M.G. Ward for excellent technical support. This research was supported by the UK EPSRC Grants No. EP/L000016/1, No. EP/I028285/1 and No. EP/K01675X/1. Yu.A.P also acknowledges support from the the Royal Society Grant No. WM110105. J.R.P. acknowledges support of the People Programme (Marie Curie Actions) of the European FP7 Programme under REA grant agreement 618450.
Contributions {#contributions .unnumbered}
=============
R.G., S.K., M.S. and Yu.A.P. designed, fabricated and packaged the beams. D.I.B., A.M.G., S.K., M.T.N., Yu.A.P., J.R.P., M.P., M.S., R.S., V.T. and D.E.Z. developed custom measurement instrumentation and methods. S.K., M.P., M.S. and T.W. performed measurements. R.P.H., S.K., Yu.A.P., G.R.P, M.S. and V.T. performed calculations and analysis. S.K. and V.T. drafted the manuscript. All authors discussed the results and implications, and commented on the manuscript at all stages.
Competing financial interests {#competing-financial-interests .unnumbered}
=============================
The author(s) declare no competing financial interests.
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![The behaviour of the aluminium beams in vacuum. **(a)** A micrograph of a typical doubly-clamped aluminium beam as used in experiments. Despite an obvious compression of the resonator at room temperature, our beams have a tensile stress at cryogenic temperatures due to the significant differential thermal contraction of the silicon substrate and the deposited aluminium film. The resonator shown here has a nominal length of 15$\mu$m with a cross-sectional dimensions of 0.1$\mu$m$\times$0.1$\mu$m, with an expected resonance frequency at cryogenic temperatures of 8.5MHz. The beam is placed in a perpendicular magnetic field and connected in the microwave circuit, shown schematically in the figure. Transmission measurements are performed with a network analyser. The microwave drive from the network analyser is attenuated by -40dB with an attenuator located at 4.2K; the microwave signal transmitted through the sample is amplified by a 40dB amplifier at room-temperature. **(b)** A typical microwave transmission measurements of our $50\,\mathrm{\mu m}$ beam in vacuum at 4.2K in a 5T magnetic field. A wide frequency sweep at -90dBm of applied power clearly shows the first three odd harmonics of the beam. The quality factors of all three harmonics are approximately 10$^3$. **(c)** The frequency characteristics of electromotive power generated by a 50$\mu$m aluminium beam measured over a range of applied powers. The beam drive power in dBm is shown in the figure. The typical frequency response shows a Lorentzian shape at low driving power. However, all measured resonators demonstrate a Duffing response at drive powers above -86dBm. This particular beam has an additional parasitic resonance becoming visible above -88dBm, probably the second “near-degenerate’’ perpendicular beam mode.[]{data-label="fig: VacuumMeasurements"}](fig1final_f-eps-converted-to.pdf){width="\linewidth"}
![The response of the beam in liquid $^4$He. **(a)** The velocity resonance curves of the 50$\mu$m beam oscillating in liquid $^4$He. The output power of the network analyser was maintained at a constant -50dBm with an additional -40dB external attenuation. The shape of the resonance is independent of temperature in the normal fluid above the transition temperature $T_\lambda=2.178$K. The decrease in density of the normal component of the liquid below $T_\lambda$ results in a decrease of the Stoke’s drag, leading to a significant increase in the NEMS quality factor along with an increase in the resonant frequency as less fluid is being dragged with the beam. **(b, c)** The temperature dependence of the resonance frequency and the resonance width of the 50$\mu$m nanomechanical resonator. Each data point of figures **(b)** and **(c)** was obtained from a complete resonance curve for the beam. The solid lines are the theoretical models given by equations (\[eqn: FreqRatio\]) and (\[eqn: DeltaF\]) for the **(b)** and **(c)** respectively. We treated all three geometrical factors as fitting parameters and have found the following values: $\beta=1.18 \pm 0.02$, $B=1.19 \pm 0.01$ and $C=2.62\pm0.06$. The values obtained are close to the theoretical expectations and broadly agree with the values observed for vibrating wire resonators and tuning forks. The resonance width was fitted at temperature range between 4.2K and 1.7K, where the hydrodynamic model agrees well with the experimental data. The red dashed line highlights the difference between Stokes’ model and experimental data below 1.7K.[]{data-label="fig: LiquidHe4"}](fig2final_g-eps-converted-to.pdf){width="0.9\linewidth"}
![A comparison of the frequency response of the NEMS oscillators with superconducting NbTi vibrating wires [@LeMiere2013; @marktheodorenoble2015] and quartz tuning-fork [@JLowTempPhys.184.1080.Bradley] operated in the superfluid. **(a)** The frequency shift as a function of temperature. The lightness and small geometrical sizes of the NEMS resonators give a higher sensitivity to the normal fluid component of the $^4$He than the other commonly used beam-like devices. **(b)** Temperature dependence of the reduced Q-factor for various beam-like resonators. The response of the NEMS devices in the liquid is similar to that of the other devices and promises excellent prospects for their use as detectors in quantum fluids. **Insert:** The temperature dependencies of the $Q$-factors for all resonators around the superfluid transition. Since the $Q$-factors are determined by the conventional Stokes drag, they are almost independent of temperature above the lambda-point transition $T_{\lambda}=2.178$K, but begin to increase significantly with the reduction of the temperature below $T_{\lambda}$ from the reduction in the density of the normal component.[]{data-label="fig: Comparison"}](fig3final_c-eps-converted-to.pdf){width="0.9\linewidth"}
|
---
abstract: 'In the context of texture 4 - zero and texture 5 - zero hierarchical quark mass matrices, CP violating asymmetry in $B^o_d({\bar B}^o_d) \rightarrow \psi K_S$ (sin2$\beta$) has been evaluated by considering quark masses at $m_Z$ scale. For a particular viable texture 4 - zero mass matrix the range of 2 is: 0.27 - 0.60 and for the corresponding texture 5 - zero case it is 0.31 - 0.59. Further our calculations reveal a crucial dependence of sin2$\beta$ on light quark masses as well as the phase in this sector.'
author:
- |
Monika Randhawa and Manmohan Gupta\
[*Department of Physics,*]{}\
[*Centre of Advanced Study in Physics,*]{}\
[*Panjab University, Chandigarh- 160 014, India.*]{}
title: 'Texture Specific Mass Matrices and CP Violating Asymmetry in $B^o_d({\bar B}^o_d) \rightarrow \psi K_S$ '
---
21truecm 14truecm psfig 2[sin$2\beta$]{}
The recent first measurements of time dependent CP asymmetry $a_{\psi}K_S$ in $B^o_d({\bar B}^o_d) \rightarrow \psi K_S$ decay by BABAR and BELLE collaborations suggest that these values could be smaller than the expectations from Standard Model analysis of the CKM unitarity triangle. For example, the reported asymmetry by BABAR and BELLE are as follows, a\_K\_S& =&0.12 0.37 0.09 [BABAR]{} [@babar], \[babar\]\
a\_K\_S&=&0.45\^[+0.43 +0.07]{}\_[-0.44 -0.09]{} [BELLE]{} [@belle], \[belle\] whereas the earlier CDF measurements gave [@cdf], a\_K\_S\^[[CDF]{}]{}=0.79\^[+0.41]{}\_[-0.44]{}, \[cdf\] and a recent global analysis of the CKM unitarity triangle [@parodi] gives the value a\_K\_S\^[[SM]{}]{} = 0.75 0.06 . \[parodi\] Recently, several authors [@kagan] - [@buras] have studied the implications of the possibility of low value of in comparison to the CDF measurements as well as the standard model expectations. In particular, Silva and Wolfenstein [@silva] have examined the possibilities of physics beyond the standard model in case $\leq~0.2$.
In the context of texture specific mass matrices, has been evaluated in the leading approximations [@sin2; @hall], however without going into the detailed implications of on the texture as well as the mass scale at which the quark masses are evaluated. Recently, it has been demonstrated [@monica]-[@frz] that texture 4 - zero quark mass matrices not only accommodate the CKM phenomenology but are also able to reproduce a neutrino mixing matrix which can accommodate Solar Neutrino Problem, Atmospheric Neutrino Problem and the oscillations observed at LSND. In particular, Randhawa [*et al.*]{} [@monica] have shown that there is a unique set of viable texture 4 - zero mass matrices in the quark sector as well as in the lepton sector. The purpose of the present communication is to investigate in detail and beyond the leading order the implications of measurements for particular viable case of texture 4 - zero mass matrices as well as for texture 5 - zero mass matrices. It would also be interesting to examine the implications of low values of , in particular of $\leq$ 0.2, a benchmark for Physics beyond the standard model as advocated by Silva and Wolfenstein [@silva].
We begin with the unique set of texture 4 - zero quark mass matrices considered by Randhawa [*et al.*]{} [@monica], for example, 0.2cm $ M_u=\left( \ba {ccc} 0 & A_u & 0\\ A_u^* & D_u & B_u\\
0 & B_u^* & C_u \ea \right)$ & $M_d = \left( \ba {ccc} 0 & A_d & 0\\ A_d^* & D_d & B_d\\
0 & B_d^* & C_d \ea \right)$\
\[mat\] 0.2cm where $A_u=|A_u|e^{i\alpha_u}$, $A_d=|A_d|e^{i\alpha_d}$, $B_u=|B_u|e^{i\beta_u}$ and $B_d=|B_d|e^{i\beta_d}.$ The elements of the mass matrices follow the following mass hierarchy [@monica; @gill; @hierarchy] A\_i D\_i \~B\_i C\_i, i=u,d. \[hier\]
Within the Standard Model, is related to the angle $\beta$ of the unitarity triangle, expressed as a\_K\_S\^[[SM]{}]{} = [sin]{}2, , \[beta\] 2 can be calculated by evaluating the elements $V_{cd}$, $V_{cb}$, $V_{td}$ and $V_{tb}$ from the above mass matrices.
The above matrices can be diagonalized exactly and the corresponding CKM matrix elements can easily be found, for details we refer the reader to reference [@gill]. However for the sake of readability of manuscript as well as for facilitating the discussion to evaluate 2, we reproduce below the exact expressions of $V_{cd}$, $V_{cb}$, $V_{td}$ and $V_{tb}$. V\_[cd]{}&=&-ae\^[-i\_1]{} + c + ce\^[i\_2]{}, \[vcd\]\
V\_[cb]{}&=& -acd\^2e\^[-i\_1]{} +\
& & - e\^[i\_2]{}, \[vcb\]\
V\_[td]{}&=&ab \^2e\^[-i\_1]{} + c\
& & - ce\^[i\_2]{}, \[vtd\]\
V\_[tb]{}&=&acb\^2d\^2 e\^[-i\_1]{} +\
& & + e\^[i\_2]{}, \[vtb\] where $a=\sqrt{m_u/m_c}$, $b=\sqrt{m_c/m_t}$, $c=\sqrt{m_d/m_s}$, $d=\sqrt{m_s/m_b}$, $\phi_1=\alpha_u-\alpha_d.$, $\phi_2=\beta_u-\beta_d.$ $R_u=D_u/m_t$ and $R_d=D_d/m_b$. In principle, 2 can be calculated using equations \[beta\]-\[vtb\], however before doing that we first ensure that by varying the various input parameters, the CKM matrix elements are within their respective range given by PDG [@pdg]. In carrying out these calculations, we have taken the quark masses at $m_Z$ scale as recently advocated by Fusaoka and Koide [@koide] as well as by Fritzsch and Xing [@frz]. For the sake of completion, however, we have also repeated the whole analysis with masses at 1 GeV scale, the scale conventionally used.
To facilitate the analysis, without loss of generality we first consider $\phi_2=0$ as advocated by several authors [@monica; @frz]. As mentioned earlier we have carried out our calculations at two different mass scales i.e. at $m_Z$ scale and at 1 GeV, the corresponding input masses are summarized in Table \[tabinp\]. For calculating the limits on 2, we scanned the full ranges of all the input masses at different CLs as well as at both the scales, varying $\phi_1$ from $0^o$ to $180^o$. It may be of interest to point out that while carrying out the variations in $R_u$ and $R_d$, we have restricted their variation upto 0.2 only as the values higher than that are not able to reproduce the CKM elements within their range given by PDG. This is in accordance with our earlier calculations [@monica] as well as the hierarchical structure of mass matrices described by equation \[hier\]. Having taken care of the CKM matrix elements being within the limits mentioned by PDG, we proceed to find a range for 2 using expressions \[beta\] - \[vtd\]. A similar exercise is carried out for (i) $D_u$ = 0, $D_d~ {\not=}~0 $, (ii) $D_u~ {\not=}~0$, $D_d$ = 0, the two cases corresponding to texture 5 - zero matrices.
In Table \[tab1\], we have summarized the results of our calculations at the different mass scales for texture 4 - zero and texture 5 - zero mass matrices. From the table one can immediately find that the range of 2 in the case of texture 4 - zero mass matrices, with input masses at $m_Z$ scale and at 1$\sigma$ CL, is given by sin2= 0.27 - 0.60 . \[sinmzb1\] This range looks to be narrow in comparison with the BELLE and BABAR results and is ruled out by the SM analysis. The corresponding range for 2 narrows further when quark masses are considered at 1 GeV scale, for example, sin2= 0.39 - 0.54. \[sinb1\] This can be easily understood from the fact that the light quark masses at $m_Z$ scale show much more scatter compared to masses at 1 GeV scale. In view of the sensitive dependence of 2 on the quark masses, in figures \[figa\] - \[figd\], we have plotted the variation of 2 with mass ratios $m_u/m_c$, $m_d/m_s$, $m_c/m_t$ and $m_s/m_b$. From these figures, it is easy to conclude that 2 is very sensitively dependent on the ratios of the light quark masses, $m_u/m_c$ and $m_d/m_s$, while variations in $m_c/m_t$ and $m_s/m_b$ do not affect 2 much. This gets further emphasized when one closely examines the figures, for example, 2 varies from 0.40 to 0.52 when $m_u/m_c$ varies from 0.0026 to 0.0045, while it varies only from 0.464 to 0.457 when $m_c/m_t$ varies from 0.0032 to 0.0044. Similarly 2 varies from 0.52 to 0.41 when $m_d/m_s$ varies from 0.0383 to 0.0658, while it varies only from 0.458 to 0.461 when $m_s/m_b$ varies from 0.0258 to 0.0364. It is perhaps desirable to mention that while considering the above mentioned variation of 2 on a given mass ratio all other masses have been kept at their mean values at $m_Z$ scale, whereas $\phi_1 = 90^o$ and $R_u=R_d=0.1$.
In view of the scale sensitivity of 2, it is perhaps desirable to study the affect of quark masses on 2 at higher confidence levels of quark masses in comparison to the $1 \sigma$ CL corresponding to equations \[sinmzb1\] - \[sinb2\]. In the table \[tab1\], we have also listed the results for 2 with the input quark masses being at their 2$\sigma$ and 3$\sigma$ confidence levels. A look at the table reveals that when the quark masses are considered at $2 \sigma$ CL, we obtain the following ranges for 2 for the set of texture 4 - zero matrices given in equation \[mat\]: 0.057 - 0.68. These ranges get further broadened when masses are considered at their $3 \sigma$ CL, for example, 0.04 - 0.75. Thus we see that with input masses at their $2 \sigma$ and $3 \sigma$ CL, the entire range of BABAR and BELLE is covered, once again emphasizing the sensitivity of 2 on the quark masses. This brings into focus the better evaluation of light quark masses.
Further scrutiny of the Table \[tab1\] reveals interesting results for texture 5 - zero case. For example, we obtain the following range for sin2$\beta$ with $D_u~ {\not=}~0$, $D_d$ = 0 and with quark masses at $m_Z$ scale and at 1$\sigma$ CL, sin2= 0.31 - 0.59. \[sinmzb2\] and correspondingly with the masses at 1 GeV we get, sin2= 0.45 - 0.54 \[sinb2\] Thus, in comparison to the corresponding ranges for texture 4 - zero matrices, the lower bound on 2 goes up somewhat while there is not much change in the upper bound. This can be understood by an examination of equations \[beta\] and \[vtd\] where $D_d=0$ results in lowering the upper bound on , thus pushing up the lower bound on 2. In the other texture 5 - zero case, for example $D_d~ {\not=}~0$ and $D_u$ = 0, we find that it is not meaningful to talk of the range of 2 as in this case the CKM matrix elements do not show overlap with the PDG CKM matrix even after the full variation of all the parameters.
A few comments are in order. In view of the dependence of 2 on $\phi_1$ and $\phi_2$ through $V_{cd}$, $V_{cb}$, $V_{td}$ and $V_{tb}$, we have also studied the case when both $\phi_1$ and $\phi_2$ are taken non zero. The results in this case do not show much deviation from the case when $\phi_2 = 0$ and $\phi_1$ is varied. However when $\phi_1 = 0$ and $\phi_2$ is given full variation, interestingly we find that we are not able to reproduce the CKM matrix elements and hence finding a range for 2 in this case is meaningless. Thus, it seems that the CP violating phase resides only in the light quark sector, in agreement with the conclusions of Fritzsch and Xing [@sin2] based on leading order calculations only.
Interestingly, from equations \[sinmzb1\] - \[sinb2\], we find that a value of 2 lower than 0.2 would rule out, with good deal of confidence, texture 4 - zero and texture 5 - zero matrices. This would force one to consider texture 3 - zero and texture 2 - zero mass matrices, which would be discussed elsewhere. Therefore it seems that a sharper measurement of 2 will have strong bearings on the specific textures of mass matrices.
To conclude, we have found a range for 2 using texture 4 - zero and texture 5 - zero hierarchical quark mass matrices with input quark masses at $m_Z$ scale. In the texture 4 - zero case with masses at 1$\sigma$ CL, we get 2=0.27 - 0.60 and in the texture 5 - zero case we get 2 =0.31 - 0.59. Both texture 5 - zero and texture 4 - zero matrices are ruled out if 2 is found to be $\leq$ 0.2 and one may have to go to texture 3 - zero matrices. Our analysis indicates a sensitive dependence of 2 on the light quark masses as well as the phase in this sector.\
M.G. would like to thank S.D. Sharma for useful discussions. M.R. would like to thank CSIR, Govt. of India, for financial support and also the Chairman, Department of Physics, for providing facilities to work in the department.
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& At $\mu$=1 GeV & At $\mu = m_Z$\
& &\
$m_u$ & 0.00488 $\pm$ 0.00057 & $0.00233^{+0.00042}_{-0.00045}$\
& &\
$m_d$ & 0.00981 $\pm$ 0.00065 & $0.00469^{+0.00060}_{-0.00066}$\
& &\
$m_s$ & 0.1954 $\pm$ 0.0125 & $0.0934^{+0.0118}_{-0.0130}$\
& &\
$m_c$ & $1.506^{+0.048}_{-0.037}$ & $0.677^{+0.056}_{-0.061}$\
& &\
$m_b$ & $7.18^{+0.59}_{-0.44}$ & 3.0 $\pm$ 0.11\
& &\
$m_t$ & $475^{+86}_{-71}$ & 181 $\pm$ 13\
& &\
[@koide]. \[tabinp\]
& &\
& texture 4 zeros & texture 5 zeros\
($D_d=0$, $D_u{\not =0}$)\
& texture 4 zeros & texture 5 zeros\
($D_d=0$, $D_u{\not =0}$)\
\
Sin2$\beta$\
(with quark masses\
at 1$\sigma$ CL)\
& 0.27 - 0.60 & 0.31 - 0.59 & 0.39 - 0.54 & 0.45 - 0.54\
Sin2$\beta$\
(with quark masses\
at 2$\sigma$ CL)\
& 0.057 - 0.68 & 0.08 - 0.65 & 0.27 - 0.57 & 0.38 - 0.56\
Sin2$\beta$\
(with quark masses\
at 3$\sigma$ CL)\
& 0.04 - 0.75 & 0.05 - 0.73 & 0.06 - 0.61 & 0.07 - 0.58\
|
---
abstract: 'We discuss the smoothness and strict convexity of the solution of the $L_p$-Minkowski problem when $p<1$ and the given measure has a positive density function.'
address:
- 'Dipartimento di Matematica e Informatica “U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, Italy I-50134'
- 'Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reltanoda u. 13-15, H-1053 Budapest, Hungary, and Department of Mathematics, Central European University, Nador u 9, H-1051, Budapest, Hungary'
- 'Dipartimento di Matematica e Informatica “U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, Italy I-50134'
author:
- 'Gabriele Bianchi, Károly J. Böröczky, and Andrea Colesanti'
title: |
Smoothness in the $L_p$ Minkowski problem\
for $p<1$
---
[^1]
Introduction
============
Given a convex body $K$ in the class $\mathcal {K}_0^n$ of convex convex sets with non-empty interior in ${{\mathbb{R}}}^n$ containing the origin $o$, we write $h_K$ and $S_K$ to denote its support function and its surface area measure, respectively, and for $p\in{{\mathbb{R}}}$, $S_{K,p}$ to denote its $L_p$-area measure, where $d S_{K,p}=h_K^{1-p}dS_K$. The $L_p$-area meaure defined by Lutwak [@Lut] is a central notion in convexity, see say Barthe, Guédon, Mendelson and Naor [@BG], Böröczky, Lutwak, Yang and Zhang [@BLYZ2], Campi and Gronchi [@CG], Chou [@KSC], Cianchi, Lutwak, Yang and Zhang [@CLYZ], Gage and Hamilton [@GH], Haberl and Parapatits [@HP], Haberl and Schuster [@HS1; @HS2], Haberl, Schuster and Xiao [@HSX], He, Leng and Li [@HLK], Henk and Linke [@HENK], Ludwig [@LU2], Lutwak, Yang and Zhang [@LYZ1; @LYZ4], Naor [@NAO], Naor and Romik [@NR], Paouris [@PAO], Paouris and Werner [@PW] and Stancu [@ST3].
The $L_p$ Minkowski problem asks for the existence of a convex body $K\in\mathcal{K}_{0}^n$ whose $L_p$ area measure is a given finite Borel measure $\nu$ on $S^{n-1}$. When $p=1$ this is the classical Minkowski problem solved by Minkowski [@MIN] for polytopes, and by Alexandrov [@Ale38] and Fenchel and Jessen [@FeJ38] in general. The smoothness of the solution was clarified in a series of papers by Nirenberg [@NIR], Cheng and Yau [@CY], Pogorelov [@POG] and Caffarelli [@Caf90a; @Caf90b]. For $p>1$ and $p\neq n$, the $L_p$ Minkowski problem has a unique solution according to Chou and Wang [@CW], Guan and Lin [@GL] and Hug, Lutwak, Yang and Zhang [@HLYZ2]. The smoothness of the solution is discussed in Chou and Wang [@CW], Huang and Lu [@HL1] and Lutwak and Oliker [@LO]. In addition, the case $p<1$ has been intensively investigated by Böröczky, Lutwak, Yang and Zhang [@BLYZ], Böröczky and Hai T. Trinh [@BoT], Chen [@WC], Chen, Li and Zhu [@CSL0; @CSL01], Ivaki [@Iva13], Jiang [@JI], Lu and Wang [@LWA], Lutwak, Yang and Zhang [@LYZ5], Stancu [@ST1; @ST2] and Zhu [@Z2; @Z3; @Zhu15; @Zhu16].
The solution of the $L_p$-Minkowski problem may not be unique for $p<1$ according to Chen, Li and Zhu [@CSL01] if $0<p<1$, according to Stancu [@ST2] if $p=0$, and according to Chou and Wang [@CW] if $p<0$ small.
In this paper we are interested in this problem when $p<1$ and $\nu$ is a measure with density with respect to the Hausdorff measure $\mathcal{H}^{n-1}$ on $S^{n-1}$, i.e. in the problem $$\label{Kdensityfunction}
dS_{K,p}=f\,d{\mathcal H}^{n-1}\quad\text{ on $S^{n-1}$,}$$ where $f$ is a non-negative function in $S^{n-1}$.
According to Chou and Wang [@CW], if $-n<p<1$ and $f$ is bounded from above and below by positive constants, then has a solution. More general existence results are provided by the recent works Chen, Li and Zhu [@CSL0] if $p=0$, Chen, Li and Zhu [@CSL01] if $0< p<1$, and Bianchi, Böröczky and Colesanti [@BiBoCoY] if $-n<p<0$ and $0<p<1$.
We observe that $h$ is a non-negative positively $1$-homogeneous convex function in ${{\mathbb{R}}}^n$ which solves the Monge-Ampère equation $$\label{MongeAmpere_Sn}
h^{1-p}\det(\nabla^2h+h I)=nf \quad \text{ on $S^{n-1}$}$$ in the sense of measure if and only if $h$ is the support function of a convex body $K\in{\mathcal K}_{0}^n$ which is the solution of (see Section \[secPreliminaries\]). Naturally, if $h$ is $C^2$, then (\[MongeAmpere\_Sn\]) is a proper Monge-Ampère equation. The function $h$ may vanish somewhere for certain functions $f$, and when this happen and $p<1$ the equation is singular at the zero set of $h$.
In this paper we study the smoothness and strict convexity of a solution $K\in{\mathcal K}_{0}^n$ of assuming $\tau_2>f>\tau_1$ for some constants $\tau_2>\tau_1>0$. Concerning these aspects for $p<1$, we summarize the known results in Theorem \[th\_regularity\], and the new results in Theorem \[th\_regularity-new\].
We say that $x\in\partial K$ is a smooth point if there is a unique tangent hyperplane to $K$ at $x$ and that $K$ is smooth if each $x\in\partial K$ is smooth (see Section \[secPreliminaries\] for all definitions). For $z\in\partial K$, the exterior normal cone at $z$ is denoted by $N(K,z)$, and for $z\in{\rm int}\,K$, we set $N(K,z)=\{o\}$. Theorem \[th\_regularity\] (i) and (ii) are essentially due to Caffarelli in [@Caf90a] (see Theorem \[Caffarelli-smooth\]), and Theorem \[th\_regularity\] (iii) is due to Chou, Wang [@CW]. If the function $f$ in is $C^\alpha$ for $\alpha>0$, then Caffarelli [@Caf90b] proves (iv).
\[th\_regularity\] If $K\in{\mathcal K}_{0}^n$ is a solution of for $n\geq 2$ and $p<1$, and $f$ is bounded from above and below by positive constants, then the following assertions hold:
(i) \[th\_regularity\_a\_\] The set $X_0$ of the points $x\in \partial K$ with $N(K,x)\subset N(K,o)$ is closed, any point of $X=\partial K\backslash X_0$ is smooth and $X$ contains no segment.
(ii) \[th\_regularity\_b\_\] If $o\in\partial K$ is a smooth point, then $K$ is smooth.
(iii) \[th\_regularity\_c\_\] If $p\leq 2-n$, then $o\in{\rm int}\,K$, and hence $K$ is smooth and strictly convex.
(iv) \[th\_regularity\_x\_\] If $o\in{\rm int}\,K$ and the function $f$ in is positive and $C^\alpha$, for some $\alpha>0$, then $\partial K$ is $C^{2,\alpha}$.
Concerning strict convexity Claim (iii) here is optimal because Example \[non-strictly-convex\] shows that if $2-n<p<1$, then it is possible that $o$ belongs to the relative interior of an $(n-1)$-dimensional face of a solution $K$ of where $f$ is a positive continuous function. Therefore the only question left open is the smoothness of the solution if $2-n<p<1$.
We note that if $p<1$ and $K$ is a solution of with $f$ positive and $o\in\partial K$, then $$\label{NKosmall}
{\rm dim}\,N(K,o)\leq n-1.$$ Therefore Theorem \[th\_regularity\] (ii) yields that the solution $K$ is smooth if $n=2$. In general, we have the following partial results.
\[th\_regularity-new\] If $K\in{\mathcal K}_{0}^n$ is a solution of for $n\geq 2$ and $p<1$, and $f$ is bounded from above and below by positive constants, then the following assertions hold:
(i) \[th\_regularity\_e\_\] If $n=2$, $n=3$ or $n>3$ and $p<4-n$, then $K$ is smooth.
(ii) \[th\_regularity\_f\_\] If $\mathcal{H}^{n-1}(X_0)=0$ for the $X_0$ in Theorem \[th\_regularity\] (i), then $K$ is smooth.
Our results differ in some cases from the ones in Chou and Wang [@CW], possibly because [@CW] considers the equation $$\label{MongeAmpere_CW}
\det(\nabla^2h+h I)=nfh^{p-1}\quad \text{ on $S^{n-1}$}$$ instead of . In the context of non-negative convex functions being a solution of this last equation is a priori more restrictive than being a solution of , even if obviously the two notions coincide where $h$ is positive (see Section \[secPreliminaries\] for more on this point). Chou and Wang [@CW] proves, under our same assumptions on $f$, the strict convexity of the solution $h$ of , and uses this to prove that the body $K$ is smooth. In our opinion is the right equation to consider and using it we obtain weaker results.
To give an example of the differences of the two equations, the support function $h$ of the body $K$ in Example \[non-strictly-convex\] (where $o$ belongs to the relative interior of an $(n-1)$-dimensional face) is a solution of but it is not a solution of .
According to Chou and Wang [@CW] (see also Lemma \[MongeAmpereRn-lemma\] below), the Monge Ampère equation (\[MongeAmpere\_Sn\]) can be transferred to a Monge-Ampère equation $$\label{MongeAmpereRn0}
v^{1-p}\det( \nabla^2 v )=g$$ for a convex function $v$ on ${{\mathbb{R}}}^{n-1}$ where $g$ is a given non-negative function.
The proofs of Claims and in Theorem \[th\_regularity\] use as an essential tool a result proved by Caffarelli in [@Caf90a] regarding smoothness and strict convexity of convex solutions of certain Monge-Ampére equation of type (see Theorem \[Caffarelli-smooth\]). Proving that ${{\partial}}K$ is $C^1$ is equivalent to prove that $h_K$ is strictly convex, and [@Caf90a] is the key to prove this property in $\{y\in S^{n-1} : h(y)>0\}$.
The proof of Claim in Theorem \[th\_regularity-new\] is based on the following result for the singular inequality $v^{1-p}\det\nabla^2 v\geq g$.
\[vanishingonsegment\] Let $\Omega\subset{{\mathbb{R}}}^n$ be an open convex set, and let $v$ be a non-negative convex function in $\Omega$ with $S=\{x\in\Omega:\,v(x)=0\}$. If for $p<1$ and $\tau>0$, $v$ is the solution of $$\label{MongeAmpereOmega}
v^{1-p}\det \nabla^2 v \geq \tau \quad \text{ in $\Omega\setminus S$}$$ in the sense of measure, and $S$ is $r$-dimensional, for $r\geq 1$, then $p\geq-n+1+2r$.
The underlying idea behind the proof of this result is that on the one hand, the graph of $v$ near $S$ is close to be ruled, hence the total variation of the derivative is “small", and on the other hand, the total variation of the derivative is “large" because of the Monge-Ampere inequality.
The inequality $p\geq-n+1+2r$ in this result is optimal, at least when $r=1$. Indeed Example \[example\_vanishingonsegment\] shows that for any $p>-n+3$ there exists a non-negative convex solution of in $\Omega$ which vanish on the intersection of $\Omega$ with a line.
Proposition \[vanishingonsegment\] yields actually somewhat more than Claim in Theorem \[th\_regularity-new\]; namely, if $r\geq 2$ is an integer, $p<\min\{1,2r-n\}$ and $K\in{\mathcal K}_{0}^n$ is a solution of with $o\in\partial K$, then ${\rm dim}\,N(K,o)<r$. As a consequence, we have the following technical statements about $K$, where we also use Theorem \[th\_regularity-new\] for Claim (ii).
\[cor\_regularity\_new\] If $p<1$ and $K\in{\mathcal K}_{0}^n$, $n\geq 4$, is a solution of with $o\in\partial K$, then
(i) \[th\_regularity\_g\_\] ${\rm dim}\,N(K,o)<\frac{n+1}2$;
(ii) \[th\_regularity\_h\_\] if in addition $n=4,5$ and $K$ is not smooth, then ${\rm dim}\,N(K,o) =2$ and ${\rm dim}\,F(K,u)=n-1$ for some $u\in N(K,o)$.
We review the notation used in this paper in Section \[secPreliminaries\]. Section \[sec-Monge-Ampere\] contains results and examples regarding Monge-Ampère equations in ${{\mathbb{R}}}^n$, namely Proposition \[vanishingonsegment\], Example \[example\_vanishingonsegment\] and Proposition \[ustrictconvex\]. This last result is the key to prove Theorem \[th\_regularity-new\] . In Section \[sec-th\_regularity\] we show, for the sake of completeness, how to prove Theorem \[th\_regularity\] using ideas due to Caffarelli [@Caf90a; @Caf90b] and Chou and Wang [@CW]. Theorem \[th\_regularity-new\] and Corollary \[cor\_regularity\_new\] are proved in Section \[sec-th\_regularity-new\].
Notation and preliminaries {#secPreliminaries}
==========================
As usual, $S^{n-1}$ denotes the unit sphere and $o$ the origin in the Euclidean $n$-space ${{\mathbb{R}}}^n$. If $x,y\in{{\mathbb{R}}}^n$, then $\left<x,y\right>$ is the scalar product of $x$ and $y$, while $\|x\|$ is the euclidean norm of $x$. By $[x,y]$ we denote the segment with endpoint $x$ and $y$. We write $\mathcal{H}^k$ for $k$-dimensional Hausdorff measure in ${{\mathbb{R}}}^n$.
We denote by $\partial E$, ${{\operatorname{int }}}E$, ${{\operatorname{cl}}}E$, and $1_E$ the [*boundary*]{}, [*interior*]{}, *closure*, and [*characteristic function*]{} of a set $E$ in ${{\mathbb{R}}}^n$, respectively. The symbols ${{\operatorname{aff }}}E$ and ${{\operatorname{lin }}}E$ denote respectively the *affine hull* and the *linear hull* of $E$. The *dimension* $\dim E$ is the dimension of ${{\operatorname{aff }}}E$. With the symbol $E\mathbin|L$ we denote the orthogonal projection of $E$ on the linear space $L$.
For notions and facts about Monge-Ampère equations, see the survey Trudinger and Wang [@TrWa]. Given a function $v$ defined on a subset of ${{\mathbb{R}}}^n$, $\nabla v$ and $\nabla^2 v$ denote its gradient and its Hessian, respectively. When $v$ is a convex function defined in an open convex set $\Omega$, the subgradient ${{\partial}}v(x)$ of $v$ at $x\in\Omega$ is defined as $${{\partial}}v(x) =\{z\in{{\mathbb{R}}}^n : v(y)\geq v(x)+\langle z,y-x\rangle \text{ for each $y\in\Omega$}\},$$ which is a compact convex set. If $\omega\subset\Omega$ is a Borel set, then we denote by $N_v(\omega)$ the image of $\omega$ through the gradient map of $v$, i.e. $$N_v(\omega)=\bigcup_{x\in\omega}{{\partial}}v (x).$$ The associated Monge-Ampère measure is defined by $$\label{Monge-Ampere-measure}
\mu_v(\omega)=\mathcal{H}^{n}\Big(N_v\big(\omega\big)\Big).$$ For $p<1$ and non-negative $g$ on ${{\mathbb{R}}}^n$, we say that the non-negative convex function $v$ satisfies the Monge-Ampère equation $$v^{1-p}\det( \nabla^2 v )=g$$ in the sense of measure (or in the Alexandrov sense) if $$v^{1-p}\,d\mu_v=g\,d\mathcal{H}^n.$$
A [*convex body*]{} in ${{\mathbb{R}}}^n$ is a compact convex set with nonempty interior. The treatise Gardner [@GAR1], Gruber [@PM], Schneider [@SCH] are excellent general references for convex geometry. The function $$h_K(u)=\max\{\left<u,y\right>: y\in K\},$$ for $u\in{{\mathbb{R}}}^n$, is the [*support function*]{} of $K$. When it is clear the convex body to which we refer we will drop the subscript $K$ from $h_K$ and write simply $h$. Any convex body $K$ is uniquely determined by its support function.
If $S$ is a convex set in ${{\mathbb{R}}}^n$, then a $z\in S$ is an extremal point if $z=\alpha x_1+(1-\alpha)x_2$ for $x_1,x_2\in S$ and $\alpha\in(0,1)$ imply $x_1=x_2=z$. We note that if $S$ is compact convex, then it is the convex hull of its extremal points. Next let $C$ be a convex cone; namely, $\alpha_1u_1+\alpha_2u_2\in C$ if $u_1,u_2\in C$ and $\alpha_1,\alpha_2\geq 0$. For $u\in C\backslash\{o\}$, we say that $\sigma=\{\lambda u:\,\lambda\geq 0\}$ is an extremal ray if $\alpha_1 x_1+\alpha_2x_2\in\sigma$ for $x_1,x_2\in C$ and $\alpha_1,\alpha_2>0$ imply $x_1,x_2\in\sigma$. Now if $C\neq\{o\}$ is a closed convex cone such that the origin is an extremal point of $C$, then $C$ is the convex hull of its extremal rays.
The *normal cone* of a convex body $K$ at $z\in K$ is defined as $$N(K,z)=\{u\in{{\mathbb{R}}}^n : \langle u, y\rangle\leq \langle u, z\rangle\text{ for all $y\in K$}\}$$ where $N(K,z)=\{o\}$ if $z\in {\rm int} K$ and ${\rm dim}\,N(K,z)\geq 1$ if $z\in{{\partial}}K$. This definition can be written also as $$\label{duality_body_support1}
N(K,z)=\{u\in{{\mathbb{R}}}^n : h_K(u)=\langle z,u\rangle\}.$$ In particular, $N(K,z)$ is a closed convex cone such that the origin is an extremal point, and $$\label{notstrictlyconvex}
h_K(\alpha_1u_1+\alpha_2u_2)=\alpha_1h_K(u_1)+\alpha_2h_K(u_2)
\text{\ for $u_1,u_2\in N(K,z)$ and $\alpha_1,\alpha_2>0$.}$$ A convex body $K$ is smooth at $p\in{{\partial}}K$ if $N(K,p)$ is a ray, and $K$ is a smooth convex body if each $p\in{{\partial}}K$ is a smooth point. In the latter case, ${{\partial}}K$ is $C^1$, which is equivalent to saying that the restriction of $h_K$ to any hyperplane not containing $o$ is strictly convex, by .
We say that $K$ is strictly convex if ${{\partial}}K$ contains no segment, or equivalently, $h_K$ is $C^1$ on ${{\mathbb{R}}}^n\backslash\{o\}$ (see ).
The *face* of $K$ with outer normal $u\in{{\mathbb{R}}}^n$ is defined as $$F(K,u)=\{z\in K : h_K(u)=\langle z,u\rangle\},$$ which lies in ${{\partial}}K$ if $u\neq o$. Schneider [@SCH Thm. 1.7.4] proves that $$\label{duality_body_support2}
{{\partial}}h_K(u)=F(K,u).$$ In particular, for any Borel $\omega\subset S^{n-1}$, the surface area measure $S_K$ satisfies $$S_K(\omega)=\mathcal{H}^{n-1}\big(\cup_{u\in\omega} F(K,u)\big)
=\mathcal{H}^{n-1}\big(\cup_{u\in\omega} {{\partial}}h_K(u)\big),$$ and hence $S_K$ is the analogue of the Monge-Ampère measure for the restriction of $h_K$ to $S^{n-1}$.
Given a convex body $K$ containing $o$ and $p<1$, let $S_{K,p}$ denote the [$L_p$ area measure]{} of $K$; namely, $$\label{p_area_measure}
d S_{K,p}=h_K^{1-p}d S_K.$$ In particular, for a positive measurable $f:S^{n-1}\to{{\mathbb{R}}}$, $h_K$ is a solution of in the sense of measure if and only if the following conditions (a) and (b) hold:
(a) $\dim N(K,o)<n$; or equivalently, $$\label{sol_alex_a}
\mathcal{H}^{n-1}\big(\{y\in S^{n-1} : h_K(y)=0\}\big)=0,$$
(b) for each Borel set $\omega\subset \{y\in S^{n-1}:\,h_K(y)>0\}$, we have $$\label{sol_alex_b}
S_K(\omega)
=\int_{\omega} nf(y)h_K(y)^{p-1}\, d\mathcal{H}^{n-1}(y).$$
Let us compare these two conditions to the conditions for $h_K$, $K\in{\mathcal K}_{0}^n$, being a solution of for $p<1$ and positive $f$. On the one hand, we have and . However, since the exponent $p-1<0$, we have to add the condition $$\label{sol_alex_c}
S_K\big(N(K,o)\cap S^{n-1}\big)=\mathcal{H}^{n-1}\left(\bigcup\{F(K,u):\,u\in N(K,o)\cap S^{n-1}\}\right)=0.$$ In particular, if $K\in{\mathcal K}_{0}^n$ is a solution of for $p<1$ and $f$ is bounded from below and above by positive constants, then combining , Theorem \[th\_regularity\] (i) and Theorem \[th\_regularity-new\] (ii) shows that $K$ is smooth, as it was verified by Chou and Wang [@CW].
Some results on Monge-Ampère equations in Euclidean space {#sec-Monge-Ampere}
=========================================================
Lemma \[MongeAmpereRn-lemma\] is the tool to transfer the Monge-Ampère equation (\[MongeAmpere\_Sn\]) on $S^{n-1}$ to a Euclidean Monge-Ampère equation on ${{\mathbb{R}}}^{n-1}$. For $e\in S^{n-1}$, we consider the restriction of a solution $h$ of to the hyperplane tangent to $S^{n-1}$ at $e$.
\[MongeAmpereRn-lemma\] If $e\in S^{n-1}$, $h$ is a convex positively $1$-homogeneous non-negative function on ${{\mathbb{R}}}^n$ that is a solution of for $p<1$ and positive $f$, and $v(y)=h(y+e)$ holds for $v:\,e^\bot\to {{\mathbb{R}}}$, then $v$ satisfies $$\label{MongeAmpereRn}
v^{1-p}\det( \nabla^2 v )=g \quad \text{ on $e^\bot$}$$ where, for $y\in e^\bot$, we have $$g(y)=\left(1+\|y\|^2\right)^{-\frac{n+p}2} f\left(\frac{e+y}{\sqrt{1+\|y\|^2}}\right).$$
Let $h=h_K$ for $K\in\mathcal{K}_0^n$, and let $$\widetilde{S}=\{u\in S^{n-1}:\,h_K(u)=0\},$$ which is a possibly empty spherically convex compact set whose spherical dimension is at most $n-2$, by (\[sol\_alex\_a\]). According to (\[sol\_alex\_b\]), the Monge-Ampère equation for $h_K$ can be written in the form $$\label{hKnozero}
dS_K=h_K^{p-1}f\,d \mathcal{H}^{n-1}\mbox{ \ \ on $S^{n-1}\backslash \widetilde{S}$}.$$
We consider $\pi:e^\bot\to S^{n-1}$ defined by $$\pi(x)=(1+\|x\|^2)^{\frac{-1}2}(x+e),$$ which is induced by the radial projection from the tangent hyperplane $e+e^\bot$ to $S^{n-1}$. Since $\langle \pi(x),e\rangle=(1+\|x\|^2)^{\frac{-1}2}$, the Jacobian of $\pi$ is $$\label{Dpi}
\det D\pi(x)=(1+\|x\|^2)^{\frac{-n}2}.$$
For $x\in e^\bot$, (\[duality\_body\_support2\]) and writing $h_K$ in terms of an orthonormal basis of ${{\mathbb{R}}}^n$ containing $e$, yield that $v$ satisfies $${{\partial}}v(x)={{\partial}}h_K(x+e)|e^\bot=F(K,x+e)|e^\bot=F(K,\pi(x))|e^\bot.$$ Let $S=\pi^{-1}(\widetilde{S})$. For a Borel set $\omega\subset e^\bot\backslash S$, we have $$\begin{aligned}
\mathcal{H}^{n-1}(N_v(\omega))&=&
\mathcal{H}^{n-1}\left(\cup_{x\in\omega}{{\partial}}v(x)\right)\\
&=&\mathcal{H}^{n-1}\left(\cup_{u\in\pi(\omega)}\left(F(K,u)|e^\bot\right)\right)
=\int_{\pi(\omega)}\langle u,e\rangle\,dS_K(u)\\
&=&\int_{\pi(\omega)}\langle u,e\rangle h_K^{p-1}(u)f(u)\,d \mathcal{H}^{n-1}(u)\\
&=&\int_\omega(1+\|x\|^2)^{\frac{-n-p}2} f(\pi(x))v(x)^{p-1}\,d \mathcal{H}^{n-1}(x)\end{aligned}$$ where we used at the last step that $$v(x)=h_K(x+e)=(1+\|x\|^2)^{\frac{1}2}h_K(\pi(x)).$$ In particular, $v$ satisfies the Monge-Ampère type differential equation $$\det D^2v(x)=(1+\|x\|^2)^{\frac{-n-p}2} f(\pi(x))v(x)^{p-1}
\mbox{ \ \ on $e^\bot\backslash S$}.$$ Since, ${\rm dim}\,S\leq n-2$ by , $v$ satisfies (\[MongeAmpereRn\]) on $e^\bot$.
Having Lemma \[MongeAmpereRn-lemma\] at hand showing the need to understand related Monge-Ampère equations in Euclidean spaces, we prove Propositions \[vanishingonsegment\] and \[ustrictconvex\], and quote Caffarelli’s Theorem \[Caffarelli-smooth\].
Up to restricting $\Omega$ and changing coordinate system, we may assume, without loss of generality, that $\Omega=\{(x_1,x_2)\in{{\mathbb{R}}}^r\times{{\mathbb{R}}}^{n-r}: \|x_1\|< s_1, \|x_2\|< s_2\}$ and that $S=\{(x_1,x_2) : x_2=0\}$, and $v$ is continuous on ${\rm cl}\,\Omega$.
Let $\alpha=\max_{{\rm cl}\,\Omega} v$ and let us consider the convex body $$M=\{(x_1,x_2,y)\in {{\mathbb{R}}}^r\times{{\mathbb{R}}}^{n-r}\times{{\mathbb{R}}}: \|x_1\|\leq s_1, \|x_2\|\leq s_2, v(x_1,x_2)\leq y\leq \alpha\}.$$ For $t\in(0,s_2/2]$, let $$\Omega_t=\{(x_1,x_2)\in {{\mathbb{R}}}^r\times{{\mathbb{R}}}^{n-r} : \|x_1\|\leq s_1/2, \|x_2\|\leq t\}.$$
We estimate $\mathcal{H}^{n}\big(N_v(\Omega_t\setminus S)\big)$. Let $(x_1,x_2)\in \Omega_t\setminus S$ and let $(z_1,z_2)\in {{\mathbb{R}}}^r\times{{\mathbb{R}}}^{n-r}$ belong to ${{\partial}}v (x_1,x_2)$. We prove that $$\label{bound_subgradient}
\|z_2\|\leq \frac{2\alpha}{s_2}\quad\text{and}\quad \|z_1\|\leq\frac{4\alpha}{s_1 s_2} t.$$ If $z_2=0$ the first inequality in holds true. Assume $z_2\neq0$. The vector $(z_1,z_2,-1)$ is an exterior normal to $M$ at $p=(x_1,x_2,u(x_1,x_2))$. Since $$q_1=(x_1,x_2+\frac{s_2 z_2}{2\|z_2\|},\alpha)\in M$$ (because $\big\|x_2+s_2 z_2/(2\|z_2\|)\big\|\leq \|x_2\|+s_2/2\leq s_2$) then $\langle q_1-p, (z_1,z_2,-1)\rangle\leq0$. This implies $$\|z_2\|\leq\frac{2}{s_2}(\alpha-v(x_1,x_2))$$ and the first inequality in . Again, if $z_1=0$ then the second inequality holds true. Assume $z_1\neq0$. We have $$q_2=(x_1+\frac{s_1 z_1}{2 \|z_1\|}, 0, u(x_1,x_2))\in M,$$ because $\big\|x_1+s_1 z_1/(2\|z_1\|)\big\|\leq s_1$, $(x_1+s_1 z_1/(2\|z_1\|),0)\in S$ and therefore $v(x_1,x_2)\geq 0=v(x_1+s_1 z_1/(2\|z_1\|),0)$. The inequality $\langle q_2-p, (z_1,z_2,-1)\rangle\leq0$ implies the second inequality .
The inequalities in imply $$\label{th_regularity_c_estimate_Omegat}
\mathcal{H}^{n}\big(N_v(\Omega_t\setminus S)\big)\leq c\ t^r,$$ for a suitable constant $c$ independent on $t$.
Now we estimate $\int_{\Omega_t\backslash S} v(x)^{p-1}\ dx$. The inclusion of the convex hull of $S\times\{0\}$ and $\{\|x_1\|\leq s_1, \|x_2\|\leq s_2, y=\alpha\}$ in $M$ implies that $v(x_1,x_2)\leq\frac{\alpha}{s_2}\, \|x_2\|$ for each $(x_1,x_2)\in\Omega_t$ by the convexity of $v$. Using this estimate it is straightforward to compute that $$\label{th_regularity_c_estimate_integral}
\int_{\Omega_t\backslash S} v(x)^{p-1}\ dx\geq d\ t^{n+p-r-1},$$ for a suitable constant $d$ independent on $t$. The inequalities and and the differential inequality satisfied by $v$ imply, as $t\to0^+$, $$c t^r\geq \mathcal{H}^{n}\big(N_v(\Omega_t\setminus S)\big)\geq \int_{\Omega_t\backslash S} \tau v(x)^{p-1}\ dx\geq \tau d\ t^{n+p-r-1}.$$ This inequality implies $p\geq -n+1+2r$.
\[example\_vanishingonsegment\]Let us show that for any $p>-n+3$ there exists a non-negative convex solution of in $\Omega=\{(x_1,x_2)\in{{\mathbb{R}}}\times{{\mathbb{R}}}^{n-1}: x_1\in[-1,1], \|x_2\|\leq 1\}$ which vanish on the $1$-dimensional space $S=\{(x_1,x_2)\in{{\mathbb{R}}}\times{{\mathbb{R}}}^{n-1} : x_2=0\}$.
To prove this let $$v(x_1,x_2) = \|x_2\|+f(\|x_2\|)g(x_1)$$ where $f(r)=r^\alpha$, with $\alpha=(p+n-1)/2$, and $g(x_1)=(1+\beta x_1^2)$, with $\beta>0$ sufficiently small. Note that $\alpha>1$ exactly when $p>-n+3$.
The function $v$ is invariant with respect to rotations around the line containing $S$. To compute $\det \nabla^2 v$ at an arbitrary point, it suffices to compute it at $(x_1,0,\dots,0,r)$, $r\ge0$. We get $$\begin{aligned}
&v_{x_1x_1}=f(r) g''(x_1), & &\\
&v_{x_1x_i}=0 & &\text{when $1<i<n$,}\\
&v_{x_1x_n}=f'(r) g'(x_1), & &\\
&v_{x_ix_i}=\frac1{r}+\frac{f'(r)}{r} g(x_1) & &\text{when $1<i<n$,}\\
&v_{x_i x_j}=0 & &\text{when $i\neq j$, $(i,j)\neq(1,n)$, $(i,j)\neq(n,1)$,}\\
&v_{x_nx_n}=f''(r)g(x_1). & &\end{aligned}$$ The function $v$ is convex if $\beta$ is sufficiently small. Indeed, the eigenvalues of $\nabla^2 v$ are $\frac1{r}+\frac{f'(r)}{r} g(x_1)$, with multiplicity $n-2$, and those of the matrix $$\left(
\begin{array}{ll}
f g'' & f'g'\\
f'g' & f''g
\end{array}
\right).$$ The determinant of the latter matrix is $$2\alpha\beta r^{2(\alpha-1)}\Big(\alpha-1-(1+\alpha)\beta x_1^2\Big),$$ which is positive if $\beta>0$ is sufficiently small. Thus all eigenvalues of $\nabla^2 v$ are positive.
We get $$\det \nabla^2 v= \Big( f'' g f g'' -( f' g')^2 \Big) \Big(\frac1{r}+\frac{f'}{r} g\Big)^{n-2}$$ which has the same order as $r^{2\alpha -n}$ as $r\to 0^+$. Clearly $v$ has order $r$, and $v^{1-p}\det \nabla^2 v$ has order $r^{2\alpha-n+1-p}$, which is uniformly bounded from above and below for our choice of $\alpha$.
The next statement is a slight revision of Lemmas 3.2 and 3.3 from Trudinger and Wang [@TrWa]. We remark that Lemma 3.2 in [@TrWa] proves with $\sup_\Omega|v|$ instead of $|v(o)|$. The inequality follows from that and the observation that if $u$ is any convex function in $\Omega$, which vanishes on $\partial\Omega$, and $tE\subset \Omega\subset E$ then $|u(o)|\geq t/(t+1) \sup_\Omega |u|$.
\[MongeAmperepointinside\] Let $v$ be a convex function defined on the closure of an open bounded convex set $\Omega\subset {{\mathbb{R}}}^n$ satisfying the Monge-Ampere equation $$\det \nabla^2 v=\nu$$ for a finite non-negative measure $\nu$ on $\Omega$, let $v\equiv 0$ on $\partial \Omega$ and let $tE\subset\Omega\subset E$ for $t>0$ and an origin centered ellipsoid $E$.
(i) \[MApi\_i\] If $z\in\Omega$ satisfies $(z+s\, E)\cap\partial\Omega\neq \emptyset$ for $s>0$, then $$|v(z)|\leq s^{1/n}c_0\mathcal{H}^n(\Omega)^{1/n}\nu(\Omega)^{1/n}$$ for some $c_0>0$ depending on $n,t$.
(ii) \[MApi\_ii\] If $\nu(t\Omega)\geq b\,\nu(\Omega)$ for $b>0$, then $$\label{formula_lemmaTW_2}
|v(0)|\geq c_1\mathcal{H}^n(\Omega)^{1/n}\nu(\Omega)^{1/n}$$ for some $c_1>0$ depending on $n$, $t$ and $b$.
The proof of Claim in Theorem \[th\_regularity-new\] is based on the following proposition. This proposition is related to a step in the proof of Theorem E (a) in [@CW], however the argument that we use to prove it is substantially different from that in [@CW].
\[ustrictconvex\] Let $v$ be a non-negative convex function defined on the closure of an open convex set $\Omega\subset {{\mathbb{R}}}^n$, $n\geq 2$, such that $S=\{x\in \Omega:v(x)=0\}$ is non-empty and compact, and $v$ is locally strictly convex on $\Omega\backslash S$. Let $\psi:(0,\infty)\to[0,\infty)$ be monotone decreasing and not identically zero; assume that $\tau_2>\tau_1>0$ and $v$ satisfy $$\label{utau1tau2}
\tau_1\psi(v)\leq \det \nabla^2 v\leq \tau_2\psi(v)$$ in the sense of measure on $\Omega\backslash S$. If ${\rm dim}\,S\leq n-1$ and $\mu_v(S)=0$ for the associated Monge-Ampère measure $\mu_v$, then $S$ is a point.
Note that means that for each Borel set $\omega\subset \Omega\setminus S$ we have $$\tau_1\int_\omega\psi(v(x))\,dx\leq \mu_v(\omega)\leq \tau_2\int_\omega\psi(v(x))\,dx,$$ where $\mu_v$ has been defined in .
We may assume that $\Omega$ is bounded. We suppose that $\dim({{\operatorname{aff }}}\,S)\geq 1$, and seek a contradiction. We may assume that $o$ is the center of mass of $S$, let $L={{\operatorname{aff }}}\, S={{\operatorname{lin }}}\, S$ and let $e=(o,1)\in{{\mathbb{R}}}^n\times{{\mathbb{R}}}$. Let $\varepsilon_0>0$ be the minimum of $v$ on $\partial \Omega$ and let us consider the convex body $$M=\{(x,y)\in{{\mathbb{R}}}^{n}\times{{\mathbb{R}}}:v(x)\leq y\leq \varepsilon_0\}.$$ Since $\Omega$ is bounded and $v$ is locally strictly convex on $\Omega\setminus S$, no supporting hyperplane to $M$ intersects both $S$ and the top facet $F(M,e)$. Therefore there exists a constant $\varrho_0$ such that if a hyperplane $H$ intersects both $\{(x,y)\in {{\mathbb{R}}}^n\times{{\mathbb{R}}}: x=o, 0\leq y\leq\varepsilon_0/2\}$ and the top face $\{(x,y)\in M : y=\varepsilon_0\}$, then both components of $M\backslash H$ are of volume at least $\varrho_0$. We choose $\varepsilon_1\in(0,\varepsilon_0/2)$ such that the volume of the cap $\{(x,y)\in M:\,y\leq \varepsilon_1\}$ is less than $\varrho_0$.
For $\varepsilon\in(0,\varepsilon_1)$, let $H_\varepsilon$ be the hyperplane
(i) containing $L+\varepsilon e$ and
(ii) cutting off the minimal volume from $M$ (on the side containing the origin) under condition (i).
We write $l_\varepsilon$ to denote the linear function on ${{\mathbb{R}}}^n$ whose graph is $H_\varepsilon$, and define $$\Omega_\varepsilon=\{x\in{{\mathbb{R}}}^n:v(x)<l_\varepsilon(x)\}.$$ Since $\Omega$ is bounded, we have ${{\operatorname{cl}}}\ \Omega_\varepsilon\subset \Omega$ by the choice of $\varepsilon_1$.
Let us prove that for each $w\in L^\perp\cap{{\mathbb{R}}}^n$ we have $$\label{baricenter_onL}
\int_{\Omega_\varepsilon}\langle x,w\rangle\,dx=0.$$ Indeed, for $t\in{{\mathbb{R}}}$ with $|t|$ small, let $$F(t)=\int_{\{x\in\Omega: l_\varepsilon(x)+t\langle x,w\rangle-v(x)>0\}} l_\varepsilon(x)+t\langle x,w\rangle -v(x)\ dx.$$ By definition of $l_\varepsilon$, $F$ has a local minimum at $t=0$. We have $$\begin{gathered}
\frac{F(t)-F(0)}{t}=\int_{\{x\in\Omega: l_\varepsilon(x)-v(x)>0\}} \langle x,w\rangle\ dx\\
+\int_\Omega\Big( \frac{l_\varepsilon(x)-v(x)}{t}+\langle x,w\rangle \Big)\Big(1_{\{x: l_\varepsilon(x)+t\langle x,w\rangle-v(x)>0\}}-1_{\{x: l_\varepsilon(x)-v(x)>0\}}\Big)\ dx.\end{gathered}$$ The set where $1_{\{x: l_\varepsilon(x)+t\langle x,w\rangle-v(x)>0\}}-1_{\{x: l_\varepsilon(x)-v(x)>0\}}$ differs from $0$ is contained in $$A_t=\{x\in\Omega :\left|l_\varepsilon(x)-v(x)\right|<|t \langle x,w\rangle|\}$$ and there exists $c$ independent on $t$ such that $\mathcal{H}^n(A_t)< c t$ and $\sup_{A_t}|l_\varepsilon(x)-v(x)|< ct$. Therefore we have $$\frac{d F}{dt}(0)=\int_{\Omega_\varepsilon} \langle x,w\rangle\ dx,$$ which proves .
Note that implies that the center of mass of $\Omega_\varepsilon$ is contained in $L$. Therefore (see [@SCH Lemma 2.3.3]) $\Omega_\varepsilon$ contains the reflection of $\Omega_\varepsilon$ with respect to this center of mass, scaled, with respect to the same center of mass, by a factor $1/n$. We deduce from this that $$\label{Omegaepsnproj}
-(\Omega_\varepsilon|L^\bot)\subset n (\Omega_\varepsilon|L^\bot).$$
It follows from the definition of $\Omega_\varepsilon$ that $$\label{SOmegaep}
S\subset\Omega_\varepsilon\mbox{ \ and \ }\lim_{\varepsilon\to 0^+}(L\cap \Omega_\varepsilon)=S,$$ where the last limit is in the sense of the Hausdorff distance. In particular, there exists $\varepsilon_2\in(0,\varepsilon_1)$ such that if $\varepsilon\in(0,\varepsilon_2)$, then $$\label{SOmegaep2}
L\cap \Omega_\varepsilon\subset 2S.$$ We observe that $$\label{uminusl}
v(x)-l_\varepsilon(x)=
\left\{\begin{array}{rl}
0&\mbox{ if $x\in\partial \Omega_\varepsilon$}\\
-\varepsilon& \mbox{ if $x\in S$}.
\end{array} \right.$$
We claim that for any $\varepsilon\in(0,\varepsilon_2)$, there exists an ellipsoid $E_\varepsilon$ centered at the origin such that $$\label{Omegaepsellipsoid}
\frac1{8 n^3}E_\varepsilon\subset \Omega_\varepsilon\subset E_\varepsilon.$$ According to Loewner’s or John’s theorems, there exists an ellipsoid $\widetilde{E}$ centered at the origin and $z_1\in\Omega_\varepsilon$ such that $$z_1+\frac1n\,\widetilde{E}\subset \Omega_\varepsilon\subset z_1+\widetilde{E}.$$ It follows from (\[Omegaepsnproj\]) that there exists $z_2\in\Omega_\varepsilon$ such that $z_2|L^\bot=\frac{-1}n\,z_1|L^\bot$. In particular, $y_1=\frac1{n+1}z_1+\frac{n}{n+1}z_2\in\Omega_\varepsilon$ satisfies that $y_1|L^\bot=o$, or in other words, $y_1\in L\cap \Omega_\varepsilon$. In addition, $$y_1+\frac1{2n^2}\,\widetilde{E}\subset \frac1{n+1}
\left(z_1+\frac1n\,\widetilde{E}\right)+\frac{n}{n+1}z_2\subset \Omega_\varepsilon.$$ Let $m={\rm dim}\,L\leq n-1$. Since $y_1\in L\cap \Omega_\varepsilon$ and (\[SOmegaep2\]) imply $\frac12\,y_1\in S$, and the origin is the centroid of $S$, we deduce that $y_2=\frac{-1}{2m}\,y_1\in S$. As $2m+1<2n$, we have $$\frac1{4n^3}\,\widetilde{E}\subset \frac1{2m+1}
\left(y_1+\frac1{2n^2}\,\widetilde{E}\right)+\frac{2m}{2m+1}y_2\subset \Omega_\varepsilon.$$ As $\Omega_\varepsilon\subset 2\widetilde{E}$ follows from $o\in z_1+\widetilde{E}$, we may choose $E_\varepsilon=2\widetilde{E}$, proving (\[Omegaepsellipsoid\]).
Let us apply Lemma \[MongeAmperepointinside\] to $\Omega_{{\varepsilon}}$ and to the function $v-l_\varepsilon$. Let $\nu$ denote the Monge-Ampère measure $\mu_{(v-l_\varepsilon)}$ restricted to $\Omega_\varepsilon$. If $\Omega_0$ is an open set such that $\Omega_\varepsilon\subset \Omega_0\subset{\rm cl}\,\Omega_0\subset \Omega$ then the set $N_v(\Omega_0)$ is bounded and this implies $$\nu(\Omega_{{\varepsilon}})=\mathcal{H}^{n}(N_{(v-l_\varepsilon)}(\Omega_\varepsilon))\leq \mathcal{H}^{n}(N_v(\Omega_0))<\infty.$$
Let $t=1/(8 n^3)$. Formula yields that $tE_\varepsilon\subset\Omega_{{\varepsilon}}\subset E_\varepsilon$.
Let us prove that $\nu(t\Omega_{{\varepsilon}})\geq b \nu(\Omega_{{\varepsilon}})$ if $b=\tau_1t^n/\tau_2$. The function $v$ is convex and attains its minimum at $o$, thus $v(x)\geq v(tx)$ for any $x\in \Omega_{{\varepsilon}}$. By this, the monotonicity of $\psi$, and the assumptions on $S$, we deduce that $$\begin{aligned}
\nu(t\Omega_{{\varepsilon}})=\nu(t(\Omega_{{\varepsilon}}\setminus S))&\geq\tau_1\int_{t(\Omega_\varepsilon\setminus S)}\psi(v(x))\,dx\\
&=\tau_1t^n \int_{\Omega_\varepsilon\setminus S}\psi(v(tz))\,dz\\
&\geq\tau_1t^n \int_{\Omega_\varepsilon\setminus S}\psi(v(z))\,dz\\
&\geq \frac{\tau_1t^n}{\tau_2}\,\nu(\Omega_\varepsilon\setminus S)=\frac{\tau_1t^n}{\tau_2}\,\nu(\Omega_\varepsilon).\end{aligned}$$
Let $c_0$ and $c_1$ be the constants appearing in Lemma \[MongeAmperepointinside\]. It follows from and Lemma \[MongeAmperepointinside\] that if $\varepsilon\in(0,\varepsilon_2)$, then $$\label{epsilonb}
\varepsilon=|v(o)-l_\varepsilon(o)|\geq c_1\mathcal{H}^n(\Omega_{{\varepsilon}})^{1/n}\nu(\Omega_{{\varepsilon}})^{1/n}.$$ On the other hand, let $s=c_1^n/(2c_0)^n$. It follows from , $\dim\, S\geq 1$ and from the fact that the origin is the centroid of $S$ that there exists $\varepsilon\in(0,{{\varepsilon}}_1)$ small enough, such that $S\subset L\cap \Omega_\varepsilon\subset(1+s)S$. In particular, there exists $z_\varepsilon\in S$ such that $(z_\varepsilon+sE_\varepsilon)\cap \partial \Omega_\varepsilon\neq\emptyset$. It follows from Lemma \[MongeAmperepointinside\] that $$\varepsilon=|v(z_\varepsilon)-l_\varepsilon(z_\varepsilon)|\leq c_0s\,\mathcal{H}^n(\Omega_{{\varepsilon}})^{1/n}\nu(\Omega_{{\varepsilon}})^{1/n}
= \frac{c_1}2\,\mathcal{H}^n(\Omega_\varepsilon)^{1/n}\nu(\Omega_\varepsilon)^{1/n}.$$ This contradicts (\[epsilonb\]), and in turn proves Proposition \[ustrictconvex\].
We will actually use the following consequence of Proposition \[ustrictconvex\].
\[ChouWang\_ustrictconvex\] Let $\tau_2>\tau_1>0$, and let $g$ be a function defined on an open convex set $\Omega\subset{{\mathbb{R}}}^n$, $n\geq 2$, such that $\tau_2>g(x)>\tau_1$ for $x\in\Omega$. For $p<1$, let $v$ be a non-negative convex solution of $$v^{1-p}\det \nabla^2 v= g \quad\text{ in $\Omega$}.$$ If $S=\{x\in \Omega:v(x)=0\}$ is non-empty, compact and $\mu_v(S)=0$, and $v$ is locally strictly convex on $\Omega\backslash S$, then $S$ is a point.
All we have to check that ${\rm dim}\, S\leq n-1$. It follows from the fact that the left hand side of the differential equation is zero on $S$, while the right hand side is positive.
The following result by Caffarelli (see Theorem 1 and Corollary 1 in [@Caf90a]), handles the part of the boundary of a convex body $K$ where the support function at some normal vector is positive.
\[Caffarelli-smooth\] Let $\lambda_2>\lambda_1>0$, and let $v$ be a convex function on an open convex set $\Omega\subset {{\mathbb{R}}}^n$ such that $$\label{aggiunta}
\lambda_1\leq \det \nabla^2v\leq \lambda_2$$ in the sense of measure.
(i) If $v$ is non-negative and $S=\{x\in\Omega:\,v(x)=0\}$ is not a point, then $S$ has no extremal point in $\Omega$.
(ii) If $v$ is strictly convex, then $v$ is $C^1$.
We recall that is equivalent to saying that for each Borel set $\omega\subset \Omega$ we have $$\lambda_1\mathcal{H}^n(\omega)\leq \mu_v(\omega)\leq \lambda_2\mathcal{H}^n(\omega),$$ where $\mu_v$ has been defined in .
Proof of Theorem \[th\_regularity\] {#sec-th_regularity}
===================================
The next lemma provides a tool for the proof of Theorem \[th\_regularity\] (iii). The same result is also proved in Chou and Wang [@CW]; we present a short argument for the sake of completeness.
\[oinside\] For $n\geq 2$ and $p\leq 2-n$, if $K\in{\mathcal K}_{0}^n$ and there exists $c>0$ such that $S_{K,p}(\omega)\geq c\,{\mathcal H}^{n-1}(\omega)$ for any Borel set $\omega\subset S^{n-1}$, then $o\in{\rm int}\,K$.
We suppose that $o\in\partial K$ and seek a contradiction. We choose $e\in N(K,o)\cap S^{n-1}$ such that $\{ \lambda e :\lambda\geq 0\}$ is an extremal ray of $N(K,o)$. Let $H^+$ be a closed half space containing ${{\mathbb{R}}}e$ on the boundary such that $N(K,o)\cap {\rm int} H^+=\emptyset$. Let $B^n$ be the unit ball centered at the origin $o$, and let $$V_0=S^{n-1}\cap (e+B^n)\cap {\rm int} H^+.$$ It follows by the condition on $S_{K,p}$ that $$\label{honV0}
c\int_{V_0}h_K(u)^{p-1}\,d{\mathcal H}^{n-1}\leq
\int_{V_0}h_K(u)^{p-1}\,dS_{K,p}= S_K(V_0)<\infty.$$
However, since $h_K$ is convex and $h_K(e)=0$, there exists $c_0>0$ such that $$h_K(x)\leq c_0\|x-e\|\mbox{ \ for $x\in e+B^n$}.$$ We observe that the radial projection of $V_0$ onto the tangent hyperplane $e+e^\bot$ to $S^{n-1}$ at $e$ is $e+V'_0$ for $$V'_0=e^\bot\cap (\sqrt{3}\,B^n)\cap {\rm int} H^+.$$ If $y\in V'_0$, then $u=(e+y)/\|e+y\|$ verifies $\|u-e\|\geq \|y\|/2$. It follows that $$\begin{aligned}
\int_{V_0}h_K(u)^{p-1}\,d{\mathcal H}^{n-1}&\geq&
c_0^{p-1}\int_{V_0}\|u-e\|^{p-1}\,d{\mathcal H}^{n-1}(u)\\
&\geq&
\frac{c_0^{p-1}}2\int_{V'_0}\frac{\|y\|^{p-1}}{(1+\|y\|^2)^{n/2}}\,d{\mathcal H}^{n-1}(y)\\
&\geq &
\frac{c_0^{p-1}}{2^{n+1}}\int_{V'_0}\|y\|^{p-1}\,d{\mathcal H}^{n-1}(y)=\infty\end{aligned}$$ as $p\leq 2-n$. This contradicts , and hence verifies the lemma.
*Claim .* For $u_0\in S^{n-1}\backslash N(K,o)$, we choose a spherically convex open neighbourhood $\Omega_0$ of $u_0$ on $S^{n-1}$ such that for any $u\in{\rm cl}\,\Omega_0$, we have $\langle u,u_0\rangle>0$ and $u\not\in N(K,o)$. Let $\Omega\subset u_0^\bot$ be defined in a way such that $u_0+\Omega$ is the radial image of $\Omega_0$ into $u_0+u_0^\bot$, and let $v$ be the function on $\Omega$ defined as in Lemma \[MongeAmpereRn-lemma\] with $h=h_K$. Since $h_K$ is positive and continuous on ${\rm cl}\,\Omega$, we deduce from Lemma \[MongeAmpereRn-lemma\] that there exist $\lambda_2>\lambda_1>0$ depending on $K$, $u_0$ and $\Omega_0$ such that $$\label{vzinXsmooth}
\lambda_1\leq \det \nabla^2v\leq \lambda_2$$ on $\Omega$.
First we claim that $$\label{zinXsmooth}
\mbox{ if $z\in\partial K$ and $N(K,z)\not\subset N(K,o)$, then
$z$ is a smooth point.}$$ We suppose that ${\rm dim}\,N(K,z)\geq 2$, and seek a contradiction. Since $N(K,z)$ is a closed convex cone such that $o$ is an extremal point, the property $N(K,z)\not\subset N(K,o)$ yields an $e\in (N(K,z)\cap S^{n-1})\backslash N(K,0)$ generating an extremal ray of $N(K,z)$. We apply the construction above for $u_0=e$. The convexity of $h_K$ and imply $h_K(x)\geq \langle z,x\rangle$ for $x\in{{\mathbb{R}}}^n$, with equality if and only if $x\in N(K,z)$. We define $S\subset \Omega$ by $S+e=N(K,z)\cap(\Omega+e)$ and hence $o$ is an extremal point of $S$. It follows that the function $\tilde{v}$ defined by $\tilde{v}(y)=v(y)-\langle z,y+e\rangle$ is non-negative on $\Omega$, satisfies , and $$S=\{y\in\Omega:\, \tilde{v}(y)=0\}.$$ These properties contradict Caffarelli’s Theorem \[Caffarelli-smooth\] (i) as $o$ is an extremal point of $S$, and in turn we conclude .
Next we show that $$\label{u0notinNKo}
\mbox{ $h_K$ is differentiable at any $u_0\in S^{n-1}\backslash N(K,o)$.}$$ We apply again the construction above for $u_0$. If $u\in\Omega_0$ and $z\in F(K,u)$ clearly $K$ is smooth at $z$ (i.e. $N(K,z)$ is a ray) by . Therefore, by , $v$ is strictly convex on $\Omega$ and Caffarelli’s Theorem \[Caffarelli-smooth\] (ii) yields that $v$ is $C^1$ on $\Omega$. In turn, we conclude .
In addition, $F(K,u)$ is a unique smooth point for $u\in\Omega_0$ (see ), yielding that $\Omega_*=\cup\{F(K,u):\,u\in \Omega_0\}$ is an open subset of ${{\partial}}K$. Therefore $\Omega_*\subset X$, any point of $\Omega_*$ is smooth (by ) and $\Omega_*$ contains no segment (by ), completing the proof of Claim .
*Claim .* We suppose that $o\in{{\partial}}K$ is smooth, and there exists $z\in {{\partial}}K$ such that $K$ is not smooth at $z$. Claim yields that $z\in X_0$, and hence $N(K,z)\subset N(K,o)$, which is a contradiction, verifying Claim .
*Claim .* This is a consequence of Lemma \[oinside\] and Claim .
*Claim .* This is a consequence of Lemma \[MongeAmpereRn-lemma\], Claim and Caffarelli [@Caf90b].
\[non-strictly-convex\] If $n\geq2$ and $p\in(-n+2,1)$, then there exists $K\in{\mathcal K}_{0}^n$ with smooth boundary such that $o$ lies in the relative interior of a facet of $\partial K$ and $dS_{K,p}=f\,d{\mathcal H}^{n-1}$ for a strictly positive continuous $f:\,S^{n-1}\to{{\mathbb{R}}}$.
Let $q=(p+n-1)/(p+n-2)$. We have $q>1$. Let $$g(r)=\begin{cases}
(r-1)^q&\text{when $r\geq1$;}\\
0&\text{when $r\in[0,1)$;}
\end{cases}$$ and $\bar g(x_1,\dots,x_{n-1})=g(\|(x_1,\dots,x_{n-1})\|)$. Let $K\in{\mathcal K}_{0}^n $ be such that $K\cap \{x : x_n\leq1\}=\{x : 1\geq x_n\geq \bar g(x_1,\dots,x_{n-1})\}$ and ${\partial}K\cap \{x : x_n>0\}$ is a $C^2$ surface with Gauss curvature positive at every point. Clearly $K\cap \{x : x_n=0\}$ is a $(n-1)$-dimensional face of $K$ which contains $o$ in its relative interior and has unit outer normal $(0,\dots,0,-1)$.
To prove that $dS_{K,p}=f\,d{\mathcal H}^{n-1}$ for a positive continuous $f:\,S^{n-1}\to{{\mathbb{R}}}$, it suffices to prove that there is a neighborhood of the South pole where $dS_{K,p}/d{\mathcal H}^{n-1}$ is continuous and bounded from above and below by positive constants. Let $h$ be the support function of $K$ and, for $y\in{{\mathbb{R}}}^{n-1}$, let $v(y)=h(y,-1)$ be the restriction of $h$ to the hyperplane tangent to $S^{n-1}$ at the South pole. It suffices to prove that in a neighborhood $U$ of $o$, $v$ satisfies the equation $v^{1-p}\det \nabla^2 v=G$ with a function $G$ which is bounded from above and below by positive constants.
If $y\in U\setminus\{o\}$ we have $$\label{non-strictly_convex_f1}
v(y)=h(y,-1)=\langle (x',\bar g(x')),(y,-1)\rangle\quad\text{ where }\quad \nabla \bar g(x')=y.$$ If $U$ is sufficiently small then $v(y)$ depends only on $\|y\|$. Let $y=(z,0,\dots,0)$, with $z>0$ small and let $r=1+(z/q)^{1/(q-1)}$. We have $$\nabla \bar g(r,0,\dots,0))=(z,0,\dots,0)$$ and gives $$\begin{aligned}
\label{non-strictly_convex_f2}
v(z,0,\dots,0)=&r q(r-1)^{q-1}-(r-1)^q\\
=&z+\frac{q-1}{q^{n-1+p}}z^{n-1+p}.\end{aligned}$$ (Note that $n-1+p>1$.) Clearly $v(0,\dots,0)=h(0,\dots,0,-1)=0$. When $z>0$ we have $$\begin{aligned}
&v_{y_1y_1}=\frac{q-1}{q^{n-1+p}}(n-1-p)(n-2-p)z^{n-3+p}\\
&v_{y_iy_i}=\frac1{z}+\frac{q-1}{q^{n-1+p}}(n-1-p)z^{n-3+p} \quad\quad\quad\text{when $i\neq1$}\\
&v_{y_iy_j}=0\quad\quad\quad\text{when $i\neq j$,}\end{aligned}$$ and, as $z\to0^+$ $$v(z,0,\dots,0)^{1-p}\det \nabla^2 v(z,0,\dots,0)=c+o(1),$$ for a suitable constant $c>0$. This implies the existence of a function $G$ positive and continuous on $U$ such that $$\mathcal{H}^{n-1}\big(N_v(\omega\cap\{v>0\})\big)=\int_{\omega\cap\{v>0\}} nG(y)v(y)^{p-1}\ dy.$$ for any Borel set $\omega\subset U$. To conclude the proof that $v$ is a solution in the sense of Alexandrov of $v^{1-p} \det \nabla^2 v=G$ in $U$ it remains to prove that $\mathcal{H}^{n-1}\big(\{y\in U : v(y)=0\}\big)=0$, but this is obvious since $\{y\in U : v(y)=0\}=\{o\}$.
We remark that $h$ is not a solution of because fails.
Proofs of Theorem \[th\_regularity-new\] and Corollary \[cor\_regularity\_new\] {#sec-th_regularity-new}
================================================================================
We may assume that $o\in {{\partial}}K$ since otherwise $K$ is smooth, by Theorem \[th\_regularity\]. Let $e\in N(K,o)\cap S^{n-1}$ be such that $\langle u,e\rangle>0$ for any $u\in N(K,o)\cap S^{n-1}$. Let $v$ be defined on $\Omega=e^\bot$ as in Lemma \[MongeAmpereRn-lemma\] with $h=h_K$ and let $S=\{x\in e^\bot:\,v(x)=0\}$. We have $$\label{th_reg_new_a}
S+e=N(K,o)\cap(e^\bot +e),$$ by . If $K$ is not smooth at $o$ then $\dim S\geq 1$ and, by Proposition \[vanishingonsegment\], $p\geq n-4$ (note that here the dimension of the ambient space is $n-1$). This proves Theorem \[th\_regularity-new\] (i).
To prove Theorem \[th\_regularity-new\] (ii) we observe that $$N_{h_K}(e+S)=\bigcup_{u\in N(K,o)}F(K,u)=X_0,$$ where $X_0$ is defined as in Theorem \[th\_regularity\] (i). The equality on the left in this formula follows by and the equality on the right follows by Theorem \[th\_regularity\] (i). Thus $$N_v(S)=X_0|e^\bot,$$ and if $\mathcal{H}^{n-1}(X_0)=0$ then $\mu_v(S)=0$. We observe that $S$ is compact, by , that $v$ is locally strictly convex, by Theorem \[th\_regularity\] (i), and that ${\rm dim}\,S\leq n-2$, by . Hence Theorem \[th\_regularity-new\] (ii) follows by Corollary \[ChouWang\_ustrictconvex\] and .
Claim is an immediate consequence of , Proposition \[vanishingonsegment\] and Lemma \[MongeAmpereRn-lemma\]. This claim implies that when $n=4$ or $n=5$ and $K$ is not smooth then $\dim N(K,o)=2$. In this case $N(K,o)\cap S^{n-1}$ is a closed arc: let $e_1$ and $e_2$ be its endpoints. If $u\in N(K,o)\cap S^{n-1}$, $u\neq e_1$, $u\neq e_2$, then $F(K,u)$ is contained in the intersection of the two supporting hyperplanes $\{x\in{{\mathbb{R}}}^n: \langle x,e_i\rangle=h_K(e_i)\}$, $i=1,2$. Thus $$\mathcal{H}^{n-1}\Big(\bigcup\{F(K,u) : u\in N(K,o)\cap S^{n-1}, u\neq e_1, u\neq e_2\}\Big)=0.$$ Therefore $\dim F(K,e_1)=n-1$ or $\dim F(K,e_2)=n-1$, because otherwise $$\bigcup\{F(K,u) : u\in N(K,o)\cap S^{n-1}\},$$ which coincides with $X_0$ by Theorem \[th\_regularity\] (i), has $(n-1)$-dimensional Hausdorff measure equal to zero and, by Theorem \[th\_regularity-new\] (ii), $K$ is smooth.
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[^1]: First and third authors are supported in part by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Second author is supported in part by NKFIH grants 116451 and 109789.
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---
abstract: 'We obtain the Schwarzschild solution based on teleparallel gravity (TG) theory formulated in a space-time with torsion only. The starting point is the Poincaré gauge theory (PGT).The general structure of TG and its connection with general relativity (GR) are presented and the Schwarzschild solution is obtained by solving the field equations of TG. Most of calculations are performed using the GRTensorII package, running on the MapleV platform.'
address: 'Technical University ”Gh.Asachi”, Department of Physics, Iasi, Romania'
author:
- Gheorghe Zet
title: 'Schwarzschild solution on a space-time with torsion'
---
Introduction
=============
In general relativity (GR), the curvature of the space-time is used to describe the gravitation.The geometry of the space-time replaces the concept of force. On the other hand, teleparallel gravity (TG) attributes gravitation to torsion \[1,2\] and it is a gauge theory for the group of space-time translations. The gravitational interactions are described in TG by forces, similar to the Lorentz forces in electrodynamics. Therefore, the gravitational interactions can be described alternatively in terms of curvature as is usually done in GR, or in terms of torsion as in TG. It is believed that requesting a curved or a torsioned space-time to describe gravity is a matter of convention \[3\].
In this paper we obtain the Schwarzschild solution in the case of TG. The general structure of TG and its connection with GR are presented and the Schwarzschild solution is obtained by solving the field equations. Section 2 contains a review of Poincaré gauge theory (PGT) on a Minkowski space-time as base manifold. In Section 3 a non-symmetric connection is constructed starting with gauge fields of PGT. The field equations of TG are then obtained in the Section 4 under a general form. The Schwarzschild solution of these equations and the analytical program are presented in Section 5. A comparison with GR case is also presented. The conclusions and some remarks are presented in Section 6.
Poincaré gauge theory
=====================
We will denote the generators of Poincaré group $P$ by $\{P_{a},M_{ab}\}$, where $a,b=0,1,2,3$. Here, $P_{a\text{ }}$ are the generators of space-time translations and $M_{ab}=-M_{ba}$ are the generators of the Lorentz rotations. Suppose now that $P$ is a gauge group for gravitation \[4\]. Correspondingly, we introduce the 1-form potential $A$ with values in Lie algebra of the Poincaré group, defined by formula:$\qquad $
$$A=e^{a}P_{a}+\frac{1}{2}\omega ^{ab}M_{ab} \tag{1}$$
where $e^{a}=$ $e_{\mu }^{a}$ $dx^{\mu }$ and $\omega ^{ab}=\omega _{\mu
}^{ab}$ $dx^{\mu }$ are ordinary 1-forms. The 1-form defines a connection on the space-time $M_{4}$ of our gauge model with $e_{\mu }^{a}$ and $\omega
_{\mu }^{ab}$ as gauge fields. The 2-form of curvature $F$ is given by the expression $$F=dA+\frac{1}{2}\left[ A,A\right] \tag{2}$$ Inserting (1) in (2) and identifying the result with the definition $$F=T^{a}P_{a}+\frac{1}{2}R^{ab}M_{ab} \tag{3}$$
we obtain the 2-forms of torsion $T^{a}$ and of curvature $R^{ab}$ in the form $$T^{a}=de^{a}+\omega ^{a}\text{ }_{b}\wedge e^{b}, \tag{4}$$ and respectively $$R^{a}\text{ }_{b}=d\omega ^{a}\text{ }_{b}+\omega ^{a}\text{ }_{c}\wedge
\omega ^{c}\text{ }_{b}. \tag{5}$$ We use the Minkowski metric $\eta _{ab}=diag\left( 1,-1,-1,-1\right) $ on the Poincaré group manifold to rise and lower the latin indices $a,$ $b,$ $c$. Written on the components, the equations (4) and (5) give: $$T_{\mu \nu }^{a}=\partial _{\mu }e_{\nu }^{a}-\partial _{\nu }e_{\mu
}^{a}+\left( \omega _{\mu }^{ab}e_{\nu }^{c}-\omega _{\nu }^{ab}e_{\mu
}^{c}\right) \eta _{bc}, \tag{6}$$ and respectively $$R^{ab}\text{ }_{\mu \nu }=\partial _{\mu }\omega _{\nu }^{ab}-\partial _{\nu
}\omega _{\mu }^{ab}+\left( \omega _{\mu }^{ac}\omega _{\nu }^{db}-\omega
_{\nu }^{ac}\omega _{\mu }^{db}\right) \eta _{cd}. \tag{7}$$ The quantity $T_{\mu \nu }^{a}$ is the torsion tensor and the quantity $R^{ab}$ $_{\mu \nu }$ is the curvature tensor of the connection $A$ defined by the equation (1). The connection $A$ defines a structure Einstein-Cartan (EC) on the space-time and we will denote the corresponding space by $U_{4}$. This space have both torsion and curvature. In the next Section, we will consider the case of a space-time with non-null torsion and vanishing curvature to construct the teleparallel theory of gravity ($TG$).
Teleparallel gravity
=====================
We interpret $e_{\nu }^{a}$ as tetrad fields and $\omega _{\nu }^{ab}$ as spin connection. A model of gauge theory based on Poincaré group and implying only torsion can be obtained choosing $\omega _{\nu }^{ab}=0$. Then the curvature tensor $R^{ab}$ $_{\mu \nu }$ vanishes and the torsion in equation (6) becomes: $$T^{a}\text{ }_{\mu \nu }=\partial _{\mu }e_{\nu }^{a}-\partial _{\nu }e_{\mu
}^{a} \tag{8}$$ Expressed in a coordinate basis, this tensor has the components: $$T^{\rho }\text{ }_{\mu \nu }=e_{a}\text{ }^{\rho }\partial _{\mu }e_{\nu
}^{a}-e_{a}\text{ }^{\rho }\partial _{\nu }e_{\mu }^{a}, \tag{9}$$ where $e_{a}$ $^{\rho }$ is the inverse of $e^{a}$ $_{\rho }$.
Now, we define the Cartan connection $\Gamma $ on the space-time $M_{4}$ with nonsymmetric coefficients $$\Gamma ^{\rho }\text{ }_{\mu \nu }=e_{a}\text{ }^{\rho }\partial _{\nu
}e_{\mu }^{a}. \tag{10}$$ This definition is suggested by the expression (9) of the torsion components. Therefore, the connection $\Gamma $ has the torsion given by the usually formula $$T^{\rho }\text{ }_{\mu \nu }=\Gamma ^{\rho }\text{ }_{\nu \mu }-\Gamma
^{\rho }\text{ }_{\mu \nu }. \tag{11}$$ With respect to the connection $\Gamma $, the tetrad field is parallel, that is: $$\nabla _{\mu }e^{a}\text{ }_{\nu }=\partial _{\mu }e_{\nu }^{a}-\Gamma
^{\rho }\text{ }_{\nu \mu }\text{ }e^{a}\text{ }_{\rho }=0. \tag{12}$$
Curvature and torsion have to be considered as properties of the connections and therefore many different connections are allowed on the same space-time \[5\]. For example, starting with the tetrad field $e^{a}$ $_{\mu }$ we can define the Riemannian metric: $$g_{\mu \nu }=\eta _{ab}\text{ }e^{a}\text{ }_{\mu }\text{ }e^{a}\text{ }_{\nu }. \tag{13}$$ Then, we can introduce the Levi-Civita connection $$\overset{\circ }{\Gamma }_{\mu \nu }^{\sigma }=\frac{1}{2}g^{\sigma \rho
}\left( \partial _{\mu }g_{\rho \nu }+\partial _{\nu }g_{\rho \mu }-\partial
_{\rho }g_{\mu \nu }\right) . \tag{14}$$ This connection is metric preserving: $$\overset{\circ }{\nabla _{\rho }}g^{\mu \nu }=\partial _{\rho }g+\overset{\circ }{\Gamma }_{\sigma \rho }^{\mu }g^{\sigma \nu }+\overset{\circ }{\Gamma }_{\sigma \rho }^{\nu }g^{\sigma \mu }=0. \tag{15}$$ The relation between the two connections $\Gamma $ and $\overset{\circ }{\Gamma }$ is $$\Gamma ^{\sigma }\text{ }_{\mu \nu }=\overset{\circ }{\Gamma }_{\mu \nu
}^{\sigma }+K^{\sigma }\text{ }_{\mu \nu }, \tag{16}$$ where $$K^{\sigma }\text{ }_{\mu \nu }=\frac{1}{2}\left( T_{\mu }\text{ }^{\sigma }\text{ }_{\nu }+T_{\nu }\text{ }^{\sigma }\text{ }_{\mu }-T^{\sigma }\text{ }_{\mu \nu }\right) \tag{17}$$ is the contortion tensor.
The curvature tensor of the Levi-Civita Connection $\overset{\circ }{\Gamma }
$ is: $$\overset{\circ }{R}^{\sigma }\text{ }_{\rho \mu \nu }=\partial _{\mu }\overset{\circ }{\Gamma }_{\rho \nu }^{\sigma }+\overset{\circ }{\Gamma }_{\tau \mu }^{\sigma }\overset{\circ }{\Gamma }_{\rho \nu }^{\tau }-\left(
\mu \leftrightarrow \nu \right) \tag{18}$$ Because the connection coefficients $\overset{\circ }{\Gamma }_{\mu \nu
}^{\sigma }$are symmetric in the indices $\mu $ and $\nu $, its torsion is vanishing.Therefore, the Levi-Civita connection $\overset{\circ }{\Gamma }$ have non-null curvature, but no torsion. Contrarily, the Cartan connection $\Gamma $ presents torsion, but no curvature. Indeed, using the definition (10), we can verify that the curvature of the connection $\Gamma $ vanishes identically: $$\overset{}{R}^{\sigma }\text{ }_{\rho \mu \nu }=\partial _{\mu }\overset{}{\Gamma }_{\rho \nu }^{\sigma }+\overset{}{\Gamma }_{\tau \mu }^{\sigma }\overset{}{\Gamma }_{\rho \nu }^{\tau }-\left( \mu \leftrightarrow \nu
\right) \equiv 0. \tag{19}$$ Then, substituting (16) into the expression (19), we obtain: $$R^{\sigma }\text{ }_{\rho \mu \nu }=\overset{\circ }{R}^{\sigma }\text{ }_{\rho \mu \nu }+Q^{\sigma }\text{ }_{\rho \mu \nu }\equiv 0, \tag{20}$$ where $$Q^{\sigma }\text{ }_{\rho \mu \nu }=D_{\mu }K^{\sigma }\text{ }_{\rho \nu
}+\Gamma ^{\sigma }\text{ }_{\tau \nu }K^{\tau }\text{ }_{\rho \mu }-\left(
\mu \leftrightarrow \nu \right) \tag{21}$$ is the non-metricity tensor. Here $$D_{\mu }K^{\sigma }\text{ }_{\rho \nu }=\partial _{\mu }K^{\sigma }\text{ }_{\rho \nu }+\Gamma ^{\sigma }\text{ }_{\tau \mu }K^{\tau }\text{ }_{\rho
\nu }-\Gamma ^{\sigma }\text{ }_{\tau \nu }K^{\tau }\text{ }_{\rho \mu }
\tag{22}$$ is the teleparallel covariant derivative.
The equation (20) has an interesting interpretation \[6\]: the contribution $\overset{\circ }{R}^{\sigma }$ $_{\rho \mu \nu }$ coming from the Levi-Civita connection $\overset{\circ }{\Gamma }$ compensates exactly the contribution $Q^{\sigma }$ $_{\rho \mu \nu }$ coming from the Cartan connection $\Gamma $, yielding an identically zero Cartan curvature tensor $R^{\sigma }$ $_{\rho \mu \nu }$.
Now, according to GR theory, the dynamics of the gravitational field is determined by the Lagrangian \[6\]: $$L_{GR}=\frac{\sqrt{-g}c^{4}}{16\pi G}\overset{\circ }{R}, \tag{23}$$ where $\overset{\circ }{R}$ $=$ $g^{\mu \nu }\overset{\circ }{R}^{\rho }$ $_{\mu \rho \nu }$ is the scalar curvature of the Levi-Civita connection $\overset{\circ }{\Gamma }$, $G$ is the gravitational constant and $g=\det
\left( g_{\mu \nu }\right) $. Then, substituting $\overset{\circ }{R}$ as obtained from (20), one obtains up to divergences \[6\]: $$L_{TG}=\frac{ec^{4}}{16\pi G}S^{\rho \mu \nu }T_{\rho \mu \nu }\text{ },
\tag{24}$$ where $e=\det (e^{a}$ $_{\mu })=\sqrt{-g}$, and $$S^{\rho \mu \nu }=-S^{\rho \nu \mu }=\frac{1}{2}\left( K^{\mu \nu \rho
}-g^{\rho \nu }T^{\sigma \mu }\text{ }_{\sigma }+g^{\rho \mu }T^{\sigma \nu }\text{ }_{\sigma }\right) \tag{25}$$ is a tensor written in terms of the Cartan connection only. The equation (24) gives the Lagrangian of the TG as a gauge theory of gravitation for the translation group.
It is proven \[7\] that the translational gauge theory of gravitation TG with the Lagrangian $L_{TG}$ quadratic in torsion is completely equivalent to general relativity GR with usual Lagrangian $L_{GR}$ linear in the scalar curvature. Therefore, the gravitation presents two equivalent descriptions: one GR in terms of a metric geometry and another one TG in which the underlying geometry is provided by a teleparallel structure.
In the next Section we will obtain the field equations of gravitation within TG theory.
Field equations
===============
Taking the variation of the Lagrangian $L_{TG}$ in Eq. (24) with respect to the gauge field $e^{a}$ $_{\mu }$, one obtains the teleparallel version of the gravitational field equations: $$\partial _{\nu }\left( eS_{a}\text{ }^{\nu \rho }\right) -\frac{4\pi G}{c^{4}}\left( ej_{a}\text{ }^{\rho }\right) =0, \tag{26}$$ where $S_{a}$ $^{\nu \rho }=e_{a}$ $^{\mu }S_{\mu }$ $^{\nu \rho }$ and $$S_{\mu }\text{ }^{\nu \rho }=g_{\mu \tau }S^{\tau \nu \rho }=\frac{1}{4}\left( T_{\mu }\text{ }^{\nu \rho }+T^{\nu }\text{ }_{\mu }\text{ }^{\rho
}-T^{\rho }\text{ }_{\mu }\text{ }^{\nu }\right) -\frac{1}{2}\left( \delta
_{\mu }\text{ }^{\rho }T_{\sigma }\text{ }^{\nu \sigma }-\delta _{\mu }\text{
}^{\nu }T_{\sigma }\text{ }^{\rho \sigma }\right) .$$
The quantity $j_{a}$ $^{\rho }$ in Eq.(26) is the gauge gravitational current, defined analogous to the Yang-Mills theory: $$j_{a}\text{ }^{\rho }=\frac{1}{e}\frac{\partial L_{TG}}{\partial e_{a}\text{
}^{\rho }}=-\frac{c^{4}}{4\pi G}e_{a}\text{ }^{\sigma }S_{\mu }\text{ }^{\nu
\rho }T^{\mu }\text{ }_{\nu \sigma }+\frac{1}{e}e_{a}\text{ }^{\rho }L_{TG.}
\tag{27}$$ The current $j_{a}$ $^{\rho }$ represents the energy-momentum of the gravitational field. The term $eS_{a}$ $^{\sigma \rho }$ is called superpotential in the sense that its derivative yields the gauge current $ej_{a}$ $^{\rho }$ . Due to the anti-symmetry of $S_{a}$ $^{\sigma \rho }$ in the indices $\sigma $ and $\rho $ the quantity $ej_{a}$ $^{\rho }$ is conserved as a consequence of the field equations, i.e. $$\partial _{\rho }\left( ej_{a}^{\text{ \ }\rho }\right) =0. \tag{28}$$ Making use of Eq. (10) to express $\partial _{\rho }e_{a}$ $^{\sigma }$, the field equations (26) can be written in a purely space-time form: $$\frac{1}{e}\partial _{\sigma }\left( eS_{\mu }\text{ }^{\sigma \rho }\right)
-\frac{4\pi G}{c^{4}}\left( t_{\mu }\text{ }^{\rho }\right) =0, \tag{29}$$ where $t_{\mu }$ $^{\rho }$ is the canonical energy-momentum pseudo-tensor of the gravitational field \[5\], defined by the expression: $$t_{\sigma }^{\text{ \ \ }\rho }=\frac{c^{4}}{4\pi G}\Gamma ^{\mu }\text{ }_{\nu \sigma }S_{\mu }\text{ }^{\nu \rho }+\frac{1}{e}\delta _{\sigma }^{\text{ \ }\rho }L_{TG}. \tag{30}$$ It is important to notice that the canonical energy-momentum pseudo-tensor $t_{\mu }$ $^{\rho }$ is not simply the gauge current $j_{a}^{\text{ \ }\rho
} $ with the Lorentz index ”$a$” changed to the space-time index ”$\mu $”. It incorporates also an extra term coming from the derivative term of Eq. (26) $$t_{\sigma }^{\text{ \ \ }\rho }=e_{\text{ \ }\sigma }^{a}j_{a}^{\text{ \ \ }\rho }+\frac{c^{4}}{4\pi G}\Gamma ^{\mu }\text{ }_{\sigma \nu }S_{\mu }\text{
}^{\nu \rho }. \tag{31}$$ Like the gauge current $ej_{a}^{\text{ \ }\rho }$, the pseudo-tensor $et_{\mu }$ $^{\rho }$ is conserved as a consequence of the field equation: $$\partial _{\rho }\left( et_{\mu }^{\text{ \ \ }\rho }\right) =0. \tag{32}$$ But, due to the pseudo-tensor character of $t_{\mu }$ $^{\rho }$, this conservation law can not be expressed with a covariant derivative, in contrast with $j_{a}^{\text{ \ }\rho }$ case.
Using the previous results, we will prove in the next Section that the Schwarzschild solution can be obtained from the field equations (29) of the teleparallel theory of gravity.
Schwarzschild solution and analytical program
=============================================
Because we are looking for a spherically symmetric solution of the field equations, we will choose the Minkowski metric $$ds^{2}=dt^{2}-dr^{2}-r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\varphi
^{2}\right) \tag{33}$$ on the space-time manifold. The coordinates $x^{0},x^{1},x^{2},x^{3}$ correspond to $ct,r,\theta ,\varphi $ respectively. Then we will consider the gauge theory based on Poincaré group with $\omega _{\nu }^{ab}=0$ described in Section 3. The tetrad field $e_{\text{ \ }\mu }^{a}$ will be chosen under the form: $$(e_{\text{ \ }\mu }^{a})=\left(
\begin{array}{cccc}
e^{A/2} & 0 & 0 & 0 \\
0 & e^{B/2} & 0 & 0 \\
0 & 0 & r & 0 \\
0 & 0 & 0 & r\sin \theta
\end{array}
\right) , \tag{34}$$ where $A=A(r)$ and $B=B(r)$ are functions only of the 3D radius $r$. The inverse of $e_{\text{ \ }\mu }^{a}$ is therefore: $$(e_{a}^{\text{ \ \ }\mu })=\left(
\begin{array}{cccc}
e^{-A/2} & 0 & 0 & 0 \\
0 & e^{-B/2} & 0 & 0 \\
0 & 0 & \frac{1}{r} & 0 \\
0 & 0 & 0 & \frac{1}{r\sin \theta }
\end{array}
\right) \tag{35}$$ The metric $g_{\mu \nu }$ $=\eta _{ab}$ $e^{a}$ $_{\mu }$ $e^{a}$ $_{\nu }$ will have then the form: $$\left( g_{\mu \nu }\right) =\left(
\begin{array}{cccc}
e^{A} & 0 & 0 & 0 \\
0 & -e^{B} & 0 & 0 \\
0 & 0 & -r^{2} & 0 \\
0 & 0 & 0 & -r^{2}\sin ^{2}\theta
\end{array}
\right) \tag{36}$$ where the Eq. (33) have been used. The inverse of $g_{\mu \nu }$ is evidently: $$\left( g^{\mu \nu }\right) =\left(
\begin{array}{cccc}
e^{-A} & 0 & 0 & 0 \\
0 & -e^{-B} & 0 & 0 \\
0 & 0 & -\frac{1}{r^{2}} & 0 \\
0 & 0 & 0 & -\frac{1}{r^{2}\sin ^{2}\theta }
\end{array}
\right) . \tag{37}$$
We use the above expressions to compute the coefficients $\Gamma ^{\rho }$ $_{\mu \nu }$ of the Cartan connection, the components $T_{\mu }^{\text{ \ \ }\nu \rho }$ of the torsion tensor, of the tensor $S_{\mu }^{\text{ \ \ }\nu
\rho }$ and of the canonical energy-momentum pseudo-tensor $t_{\mu }$ $^{\rho }$. From this point at end we performed all the calculations using an analytical program conceived by us and which is given in Section 6. For example, the non-null components of the tensor $T_{\mu }^{\text{ \ \ }\nu
\rho }$ are: $$T_{0}^{\text{ \ \ }01}=\frac{A^{\prime }e^{-B}}{2},\text{ \ \ \ \ }T_{2}^{\text{ \ \ }21}=T_{3}^{\text{ \ \ }31}=\frac{e^{-B}}{r},\text{ \ \ }T_{3}^{\text{ \ \ }32}=\frac{\cot \theta }{r^{2}}, \tag{38}$$ where $A^{\prime }=\frac{dA}{dr}$ denote de derivative of the function $A(r)$ with respect to the variable $r$. We will use the same notation for the derivative of $B(r)$, that is $B^{\prime }=\frac{dB}{dr}.$
We list also the non-null components of the tensor $S_{\mu }$ $^{\sigma \rho
}$ and of the canonical energy-momentum pseudo-tensor $t_{\mu }$ $^{\rho }$ of the gravitational field. Thus, for $S_{\mu }$ $^{\sigma \rho }$ we have: $$S_{0}\text{ }^{10}=\frac{e^{-B}}{r},\text{ \ }S_{0}\text{ }^{20}=S_{1}\text{
}^{21}=\frac{\cot \theta }{2r^{2}},\text{ \ }S_{2}\text{ }^{12}=S_{3}\text{ }^{13}=\frac{e^{-B}\left( rA^{\prime }+2\right) }{4r},$$ and for $t_{\mu }$ $^{\rho }$: $$\begin{aligned}
t_{0}\text{ }^{0} &=&t_{1}\text{ }^{1}=t_{2}\text{ }^{2}=t_{3}\text{ }^{3}=\frac{c^{4}}{4\pi G}\frac{e^{-B}\left( rA^{\prime }+1\right) }{2r^{2}},\text{
} \\
\text{\ }t_{1}\text{ }^{2} &=&-\frac{c^{4}}{4\pi G}\frac{\left( A^{\prime
}+B^{\prime }\right) \cot \theta }{4r^{2}},\text{ }t_{2}\text{ }^{1}=-\frac{c^{4}}{4\pi G}\frac{e^{-B}\left( rA^{\prime }+2\right) \cot \theta }{4r}.\end{aligned}$$
Now, using these components we obtain from (29) the following equations of gravitational field in $TG$ theory: $$e^{-B}\left( \frac{B^{\prime }}{r}-\frac{1}{r^{2}}\right) +\frac{1}{r^{2}}=0,
\tag{39}$$ $$e^{-B}\left( \frac{A^{\prime }}{r}+\frac{1}{r^{2}}\right) -\frac{1}{r^{2}}=0,
\tag{40}$$ $$2A^{\prime \prime }+(\frac{2}{r}+A^{\prime })(A^{\prime }-B^{\prime })=0,
\tag{41}$$ where $A^{\prime \prime }=\frac{d^{2}A}{dr^{2}}$ is the second derivative of $A(r)$ with respect to $r$. It is easy to verify that the third field equation (41) is a combination of the first two (39) and (40). Therefore, the equation (39) and (40) are the only independent field equations and they determine the two unknown functions $A(r)$ and $B(r)$.
The solution of the equations (39) and (40) are \[8\]: $$e^{-B}=e^{A}=1+\frac{\alpha }{r}, \tag{42}$$ where $\alpha $ is a constant of integration. It is known that the constant $\alpha $ can be expressed by the mass $m$ of the body which is the source of the gravitational field with spherically symmetry \[8\]: $\alpha =-\frac{2Gm}{c^{2}}$. Therefore, we obtain the Scwarzschild solution $$ds^{2}=(c^{2}-\frac{2Gm}{r})dt^{2}-\frac{dr^{2}}{\left( 1-\frac{2Gm}{c^{2}r}\right) }-r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}\right) ,
\tag{43}$$ within the frame-work of $TG$ theory of the gravitational field.
Finally, we emphasis again on the conclusion given in Ref. \[2\]: the gravitation presents two equivalent descriptions, one in terms of a metric geometry, and another one in which the underlying geometry is that provided by a teleparallel structure.
Program ”TELEPARALLEL GRAVITY.MWS”
restart: grtw( ):
grload (minkowski, ‘spheric.mpl’);
grdef(‘ev{a miu}’); grcalc(ev(up,dn));
grdef(‘evinv{ a miu}’); grcalc(evinv(dn,up));
grdef(‘eta{a b}‘); grcalc(eta(dn,dn));
grdef(‘etainv{a b}‘); grcalc(etainv(up,up));
grdef(‘ge{miu niu}:=ev{a miu}ev{betha niu}eta{a b}‘); grcalc(ge(dn,dn));
grdef(‘geinv{miu niu}:=etainv{a b}evinv{a miu}evinv{b niu}‘); grcalc(geinv(up,up));
grdef(‘gama{sigma miu niu} :=evinv{a sigma}ev{a miu,niu}‘);
grcalc(gama(up,dn,dn));
grdef(‘TORS{sigma miu niu}:=gama{sigma niu miu}-gama{sigma miu niu}‘);
grcalc(TORS(up,dn,dn));
grdef(‘TS1{miu niu rho}:=ge{miu sigma}geinv{niu lambda}geinv{rho tau}
TORS{sigma lambda tau}‘); grcalc(TS1(dn,up,up));
grdef(‘TS2{niu miu rho}:=geinv{rho sigma}TORS{niu miu sigma}‘); grcalc(TS2(up,dn,up));
grdef(‘S{miu niu rho}:=(1/4)\*(TS1{miu niu rho}+TS2{niu miu rho}-TS2{rho miu
niu})-(1/2)\*(kdelta{miu rho}TS1{sigma niu sigma}-kdelta{miu niu}TS1{sigma rho sigma})‘);
grcalc(S(dn,up,up));
grdef(‘ed:=r2\*sin(theta)\*exp((A(r)+B(r))/2); grcalc(e);
grdef(‘eS{lambda sigma rho}:=ed\*S{lambda sigma rho}‘);
grcalc(eS(dn,up,up));
grdef(‘t{lambda rho}:=(c4/(4\*pi\*G))\*(gama{miu niu lambda}S{miu niu rho}+(1/4)\*kdelta{lambda
rho}S{tau miu niu}TORS{tau miu niu})‘); grcalc(t(dn,up));
grdef(‘EQ{lambda rho}:=(1/ed)\*eS{lambda sigma rho,sigma}-(4\*pi\*G/c4)\*t{lambda rho}‘);
grcalc(EQ(dn,up)); grdisplay(\_);
Concluding remarks
==================
We obtained the Schwarzschild solution within the teleparallel theory (TG) of gravity which is formulated in a space-time with torsion only. This can be interpreted as an indication that source of the torsion can be also the mass of the bodies that create the gravitational field, not only the spin. Therefore the torsion and curvature of the space-time is determined by the mater distribution in the considered region.
Most of the calculations have been performed using an analytical program conceived by us. The program allows to calculate the components of all quantities appearing in the model and to obtain also the equations of the gravitational field.
The TG theory can be used also to unify the gravitational field with other fundamental interactions (electromagnetic, weak and strong). Some results about this problem are given in our paper \[9\].
References
==========
1\. Calcada, M., Pereira, J.G.: Int.J.Theor.Phys. Vol. 41, p.729, 2002
2\. Andrade, V.C., Pereira, J.G.: Torsion and electromagnetic field, arXiv:qr-qc/9708051, v2-11 Jan. 1999
3\. Capozziello, S., Lambiase, G., Stornaiolo: Ann. Phys. (Leipzig) Vol. 10, p.8, 2001
4\. Zet, G., Manta, V.: Int.J.Modern Phys. C, Vol.13, p. 509, 2002
5\. Blagojevic, M.: Three lectures on Poincaré gauge theory, arXiv:qr-qc/0302040, v1-11 Feb. 2003
6\. Andrade, V.C., Guillen, L.C.T., Pereira, J.G.: Teleparallel gravity:an overview, arXiv:qr-qc/0011087, 2000
7\. Andrade, V.C., Pereira, J.G.: Phys.Rev. D56, p. 4689, 1997
8\. Landau, L., Lifchitz, E.: Théorie du champ, Ed. Mir, Moscou, 1966
9\. Zet, G.: Unified theory of fundamental interactions in a space-time with torsion (in preparation)
|
---
abstract: 'Studies of low-frequency resistance noise show that the glassy freezing of the two-dimensional (2D) electron system in the vicinity of the metal-insulator transition occurs in all Si inversion layers. The size of the metallic glass phase, which separates the 2D metal and the (glassy) insulator, depends strongly on disorder, becoming extremely small in high-mobility samples. The behavior of the second spectrum, an important fourth-order noise statistic, indicates the presence of long-range correlations between fluctuators in the glassy phase, consistent with the hierarchical picture of glassy dynamics.'
author:
- 'J. Jaroszyński'
- Dragana Popović
- 'T. M. Klapwijk'
title: 'Universal Behavior of the Resistance Noise Across the Metal-Insulator Transition in Silicon Inversion Layers'
---
Despite many theoretical and experimental efforts, the metal-insulator transition [@SAK2000] (MIT) in two-dimensional (2D) systems remains controversial. Since the apparent MIT occurs in the regime where both electron-electron interactions and disorder are strong, it has been suggested that the 2D system undergoes glassy ordering in the vicinity of the MIT. The proposals include freezing into a Coulomb [@thakur; @PastorD99], Wigner [@sudip], or spin glass [@Sachdev]. Indeed, a recent study of low-frequency resistance noise in an extremely low-mobility (high disorder) 2D electron system in Si demonstrated [@Bogd2002] glassy freezing, which occurred in the metallic phase as a precursor to the MIT.
Here we report a detailed study of low-frequency resistance noise in a 2D electron system (2DES) in Si in the opposite limit of very low disorder, where the metallic drop of resistivity $\rho$ with decreasing temperature $T$ is most pronounced. Such samples have been studied extensively [@Heemskerk98; @SAK2000] in the context of a 2D MIT using magnetotransport measurements. We find that, similar to the case of low-mobility samples, the behavior of several spectral characteristics of noise in these high-mobility devices indicates a sudden and dramatic slowing down of the electron dynamics at a well-defined electron density $n_s=n_g$, corresponding to the transition to a glassy phase. Since the two sets of devices, Si metal-oxide-semiconductor field-effect transistors (MOSFETs), differ considerably by their peak mobility, which is a rough measure of the disorder, and span essentially the entire range of Si technology, we conclude that the observed glass transition is a universal phenomenon in Si inversion layers. The experiments, however, have also revealed an important difference between low- and high-mobility samples. In low-mobility devices, $n_g\approx 1.5\,
n_c$ [@Bogd2002], where $n_c$ is the critical density for the MIT determined from the vanishing of activation energy [@activated], and the temperature coefficient of $\rho$ changes sign at $n_{s}^{\ast}>n_g>n_c$. In high-mobility structures, on the other hand, the onset of glassy dynamics seems almost to coincide with the MIT, [*i*. e.]{} $n_g\approx n_c\approx
n_{s}^{\ast}$. It is interesting that the observed strong dependence on disorder of the size of the metallic glass phase ($n_c<n_s<n_g$), which separates the 2D metal and the (glassy) insulator, is consistent with recent predictions of the model of interacting electrons near a disorder-driven MIT [@PastorD02]. Furthermore, by analyzing the second spectrum [@Weissman88; @Weissman93], an important fourth-order noise statistic, we have established the presence of long-range correlations between fluctuators in the glassy phase, which provides an unambiguous evidence for the onset of glassy dynamics at $n_g$. The results are consistent with the picture in which noise, in the glassy phase, results from transitions between many metastable states with a hierarchical structure [@Ogielski] with each transition being a reconfiguration of a large number of electrons.
Measurements were carried out on n-channel Si MOSFETs with the peak mobility $\mu\approx 2.5$ m$^2$/Vs at 4.2 K, fabricated in a Hall bar geometry with Al gates, and oxide thickness $d_{ox}=147$ nm [@Heemskerk98]. The resistance $R$ was measured down to $T=0.24$ K using a standard four-probe ac technique (typically $2.7$ Hz) in the Ohmic regime. A precision DC voltage standard (EDC MV116J) was used to apply the gate voltage, which controls $n_s$. Contact resistances and their influence on noise measurements were minimized by using a split-gate geometry, which allows one to maintain high $n_{s}$ ($\approx10^{12}$ cm$^{-2}$) in the contact region while allowing an independent control of $n_s$ of the 2D system under investigation in the central part of the sample ($120\times 50~\mu$m$^2$) (Fig. \[average\] inset).
Nevertheless, care was taken to ensure that the observed noise did not come from either the current contacts or the regions of gaps in the gate. For example, since the noise measured across a resistor connected in series with the sample and having a similar resistance was at least three times lower than the noise from the central part of the sample, the effect of the contact noise on the excitation current $I_{exc}$ could be easily ruled out. Similarly, the resistance and the noise measured between the voltage contact in the region of high $n_s$ ([*e*. g.]{} \#5 in Fig. \[average\] inset) and the one in the central part (\#6) were much smaller than those measured between contacts in the central part ([*e*. g.]{} \#6 and \#7). In fact, they were in agreement with what is expected based on the geometry of the sample, which proves that the submicron gap regions did not contribute to either the measured resistance or noise. In order to minimize the influence of fluctuations of both $I_{exc}$ and $T$, some of the noise measurements were carried out with a bridge configuration [@Scol87]. The difference voltage was detected using two PAR124A lock-in amplifiers, and a cross-spectrum measurement was performed with an HP35665A spectrum analyzer in order to reduce the background noise even further [@Verbruggen89]. The output filters of the lock-in amplifiers and/or spectrum analyzer served as an antialiasing device. Most of the noise spectra were obtained in the $f=(10^{-4}-10^{-1})$ Hz bandwidth, where the upper bound was set by the low frequency of $I_{exc}$, limited by the low cut-off frequency of RC filters used to reduce external electromagnetic noise as well as by high $R$ of the sample.
Fig. \[average\] shows the time-averaged resistivity $\langle\rho\rangle$ as a function of $T$ for different $n_s$; $d\langle\rho\rangle/dT=0$ at $n_{s}^{\ast}\approx 9.7\times 10^{10}$cm$^{-2}$. For the lowest $n_s$ and $T$, the data are described by the simply activated form $\langle\rho\rangle\propto\exp (E_{A}/k_{B}T)$. The vanishing of $E_A$ is often used as a criterion to determine $n_c$ [@activated]. This method (Fig. \[average\] inset) yields $n_c\approx n_{s}^{\ast}$, in agreement with other studies [@activated].
Figures \[spectra1\](a) and (b) show the time series of the relative changes in resistance $\Delta R(t)/\langle R\rangle$,
and the corresponding power spectra $S_R(f)\propto 1/f^{\alpha}$, respectively. In order to compare the noise magnitudes under different conditions, the spectra were averaged over two octaves \[$(0.5-2)\times 10^{-3}$ Hz\] around $f=10^{-3}$ Hz. The resulting fraction of power $S_R(\mbox{$f=1$~mHz})$ is taken as the measure of noise, and its dependence on $n_s$ is shown in Fig. \[expalpha\](a) for
$T=0.24$ K. The exponent $\alpha$ is displayed in Fig. \[expalpha\](b), while Fig. \[expalpha\](c) shows the dependence of $S_R(f=1$ mHz) on $T$ for several $n_s$. Below $T\approx 3$ K, the noise increases with decreasing $T$, but at high $n_s$ where $d\langle\rho\rangle/dT>0$, $S_R$ depends rather weakly on both $T$ and $n_s$. In the vicinity of $n_c$, however, a dramatic change in the behavior of $S_R$ is observed. The noise amplitude starts to increase strongly with decreasing $n_s$, and $\alpha$ rises rapidly from $\approx1$ to $\approx1.8$ [@alphaT]. This shift of the spectral weight towards lower $f$ indicates a sudden and dramatic slowing down of the electron dynamics at $n_g\approx 10\times
10^{10}$cm$^{-2}$, which is attributed to the freezing of the electron glass. The same qualitative behavior, together with other manifestations of glassiness (slow relaxations, history dependence), was observed [@Bogd2002] in Si MOSFETs with a much higher amount of disorder: $\mu$ was a factor of 40 lower than in the samples studied here, and not surprisingly, $n_g$ was almost an order of magnitude higher. We note that the two sets of devices also differ substantially by their geometry, size, and many fabrication details (see Refs. ), all of which leads us to conclude that the observed glass transition is a universal phenomenon in Si inversion layers, at least in those with conventional ($d\langle\rho\rangle/dT>0$) metallic behavior [@novel].
In addition to affecting the values of $n_c$, $n_g$, and $n_{s}^{\ast}$, the disorder clearly plays another, nontrivial role. In particular, $n_c$ and $n_g$ were found to differ from each other considerably in low-mobility devices ($n_c$, $n_g$, and $n_{s}^{\ast}$ were 5.0, 7.5, and 12.9, respectively, in units of $10^{11}$cm$^{-2}$) [@Bogd2002], whereas in high-mobility devices $n_g$ is at most a few percent higher than $n_c$ \[see Figs. \[expalpha\](a), (b)\]. Therefore, the emergence of glassy dynamics here seems almost to coincide with the MIT. Obviously, the size of the intermediate ($n_c<n_s<n_g$) glass phase depends strongly on disorder, in agreement with theoretical predictions [@PastorD02].
Earlier studies of noise in $R$ in Si MOSFETs were performed mostly at $T>4.2$ K and $n_s>10^{12}$cm$^{-2}$ [@Ralls; @Kirton]. The observed random telegraph and $1/f$ noise were attributed to charging and discharging of traps in the oxide close to the Si/SiO$_2$ interface, leading to $dS_R/dT>0$ [@Rogers84]. On the other hand, studies carried out at lower $n_s$ and $T$ demonstrated [@Voss; @Koch] clearly that the observed $1/f$ noise was an intrinsic property of the conduction in a 2D channel and not due to charge trapping. Moreover, $dS_R/dT<0$ was found [@Koch] for $T=1.5$, 4.2 K. Our measurements at much lower $T$ reveal a dramatic [*increase*]{} of $S_R$ with decreasing $T$. This rules out models of thermally activated charge trapping [@Weissman88; @DH81; @Rogers84], noise generated by fluctuations of $T$ [@VC], and a model of noise near the Anderson transition [@Ovadyahu92], as possible explanations. Likewise, the models of noise in the Mott and Efros-Shklovskii variable-range hopping regimes [@Shklovskii80] do not describe the data because they predict either $dS_R/dT>0$ or a saturation of $S_R$ below 10-100 Hz, both in clear disagreement with the experiment. Therefore, the observed noise cannot be a result of single electron hops even when Coulomb interactions are included through the Coulomb gap. We note that no such low-frequency saturation of $S_R$ was found in computer simulations of a Coulomb glass, where $1/f$ noise was a result of transitions between many metastable states, with each transition being a reconfiguration of a large number of electrons [@Kogan98].
We have established that the exponent $\alpha\approx 1$ in the 2D metallic phase (above $n_g$) in both low- and high-mobility samples. On the other hand, $\alpha\approx 1.8$ in the glassy phase, similar to $\alpha$ found in some spin glasses [@jjprl98; @Strunk2000], and submicron wires in the quantum Hall regime [@wrobel]. In general, such noise with spectra closer to $1/f^2$ than to $1/f$ is typical of a system far from equilibrium, in which a step does not lead to a probable return step. Such high values of $\alpha$ may be also obtained if noise results from a superposition of a small number of independent two-state systems (TSS) [@DH81; @Weissman88]. However, even though some distinguishable discrete events can be seen at low $n_s$ \[Fig. \[spectra1\](a)\], they do not show the characteristic repetitive form of stable TSS. On the contrary, both the shape and the magnitude of noise exhibit random, nonmonotonic (which exclude aging) changes with time. A quantitative measure of such spectral wandering is the so-called second spectrum $S_2(f_2,f)$, which is the power spectrum of the fluctuations of $S_{R}(f)$ with time [@Weissman93], [*i*. e.]{} the Fourier transform of the autocorrelation function of the time series of $S_{R}(f)$. If the fluctuators ([*e*. g.]{} TSS) are not correlated, $S_2(f_2,f)$ is white (independent of $f_2$) [@Weissman88; @Weissman93] and equal to the square of the first spectrum. Such noise is called Gaussian. On the other hand, $S_2$ has a nonwhite character, $S_2\propto
1/f_{2}^{1-\beta}$, for interacting fluctuators [@Weissman88; @Weissman93]. Therefore, the deviations from Gaussianity provide a direct probe of correlations between fluctuators.
We investigate $S_2$ using digital filtering [@Seid96a] in a given frequency range $f=(f_L,f_H)$ (usually $f_H=2f_L$). The normalized second spectra, with the Gaussian background subtracted, are shown in Fig. \[second\](a) for two $n_s$, just above and just below $n_g$.
It is clear that there is a striking difference in the character of the two spectra. Similar differences are observed between various spin glasses (Fig. \[second\](a) inset), where $S_2$ is white [@jjprl98] in the absence of long range interactions, and nonwhite [@Weissman93] when long range RKKY interaction leads to hierarchical glassy dynamics [@Ogielski]. A detailed dependence of the exponent $(1-\beta)$ on $n_s$ has been determined for both high- and low-mobility samples (Figs. \[second\](b) and (c), respectively). In both cases, $S_2$ is white for $n_s>n_g$, indicating that the observed $1/f$ noise results from uncorrelated fluctuators. It is quite remarkable that $S_2$ changes its character in a dramatic way exactly at $n_g$ in both types of samples. For $n_s<n_g$, $S_2$ is strongly non-Gaussian, which demonstrates that the fluctuators are strongly correlated. This, of course, rules out independent TSS (such as traps) as possible sources of noise when $n_s<n_g$. In fact, a sudden change in the nature of the fluctuators ([*i*. e.]{} correlated [*v*s.]{} uncorrelated) as a function of $n_s$ rules out [*any*]{} traps, defects, or a highly unlikely scenario that the observed glassiness may be due to some other time dependent changes of the disorder potential itself. Instead, it provides an unambiguous evidence for the onset of glassy dynamics in a 2D electron system at $n_g$.
In the studies of spin glasses, the scaling of $S_2$ with respect to $f$ and $f_2$ has been used [@Weissman93] to unravel the glassy dynamics and, in particular, to distinguish generalized models of interacting droplets or clusters ([*i*. e.]{} TSS) from hierarchical pictures. In the former case, the low-$f$ noise comes from a smaller number of large elements because they are slower, while the higher-$f$ noise comes from a larger number of smaller elements, which are faster. In this picture, which assumes compact droplets and short-range interactions between them, big elements are more likely to interact than small ones and, hence, non-Gaussian effects and $S_2$ will be stronger for lower $f$. $S_2(f_2,f)$, however, need to be compared for fixed $f_2/f$, [*i*. e.]{} on time scales determined by the time scales of the fluctuations being measured, since spectra taken over a fixed time interval average the high-frequency data more than the low-frequency data. Therefore, in the interacting “droplet” model, $S_2(f_2,f)$ should be a decreasing function of $f$ at constant $f_2/f$. In the hierarchical picture, on the other hand, $S_2(f_2,f)$ should be scale invariant: it should depend only on $f_2/f$, not on the scale $f$ [@Weissman93]. Fig. \[scaling\] shows that no systematic dependence of $S_2$ on $f$ is seen in our samples, which
signals that the system wanders collectively between many metastable states related by a kinetic hierarchy. Metastable states correspond to the local minima or “valleys” in the free energy landscape, separated by barriers with a wide, hierarchical distribution of heights and, thus, relaxation times. Intervalley transitions, which are reconfigurations of a large number of electrons, thus lead to the observed strong, correlated, $1/f$-type noise, remarkably similar to what was observed in spin glasses with a long-range correlation of spin configuration [@Weissman93]. We note that, unlike droplet models [@Fisher], hierarchical pictures of glassy dynamics [@Binder] do allow for the existence of a finite $T$ (or finite Fermi energy) glass transition in presence of a symmetry-breaking field, such as the random potential in an electron glass.
In summary, by studying the statistics of low-$f$ resistance noise, we have established that the glassy ordering of a 2DES near the MIT occurs in all Si inversion layers. The size of the metallic glass phase, which separates the 2D metal and the glassy insulator, depends strongly on disorder, becoming extremely small in high-mobility samples. The properties of the entire glass phase are consistent with the hierarchical picture of glassy dynamics, similar to spin glasses with long-range correlations.
We are grateful to the Silicon Facility at IBM, Yorktown Heights for fabricating low-mobility samples, and to S. Bogdanovich and V. Dobrosavljević for useful discussions. This work was supported by NSF grant DMR-0071668 and NHMFL through NSF Cooperative Agreement DMR-0084173.
[99]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
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The rise in $\alpha$ becomes sharper with decreasing $T$, and it will be described in detail elsewhere; see also Ref. .
For “unconventional” metallic behavior ($d\langle\rho\rangle/dT<0$), see X. G. Feng [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 2625 (2001); low-$f$ noise has not been studied in such samples yet.
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abstract: 'The structure and electronic density of states in layered LnFeAsO$_{1-x}$F$_x$ (Ln=La,Sm; $x$=0.0, 0.125, 0.25) are investigated using density functional theory. For the $x$=0.0 system we predict a complex potential energy surface, formed by close-lying single-well and double-well potentials, which gives rise to the tetragonal-to-orthorhombic structural transition, appearance of the magnetic order, and an anomaly in the specific heat capacity observed experimentally at temperatures below $\sim$140–160 K. We propose a mechanism for these transitions and suggest that these phenomena are generic to all compounds containing FeAs layers. For $x>$0.0 we demonstrate that transition temperatures to the superconducting state and their dependence on $x$ correlate well with the calculated magnitude of the electronic density of states at the Fermi energy.'
author:
- 'Peter V. Sushko,$^{1,2,*}$ Alexander L. Shluger,$^{2}$ Masahiro Hirano,$^{3}$ Hideo Hosono$^{3}$'
title: ' Mechanism of phase transitions and the electronic density of states in (La,Sm)FeAsO$_{1-x}$F$_x$ from [*ab initio*]{} calculations'
---
The discovery of a new superconductor LaFeAsO$_{1-x}$F$_{x}$ with a high transition temperature ($T_c$=26 K) \[\] has triggered a global search for other Fe-based alternatives to Cu-based superconductors, which have dominated the field since their discovery in 1986 \[\]. Substituting As, Fe, and La for other pnicogens \[\], transition metals \[\] and lanthanides \[\], respectively, applying external pressure \[\], and optimizing the doping level have pushed the $T_c$ to 54.5 K \[\]. However, since then its value seems to have saturated. As doubts have been expressed that $T_c$ can be raised any further \[\], it became apparent that generic guiding principles for the $T_c$ optimization need to be developed.
LaFeAsO is a member of the layered Fe-pnicogens, in which FeAs sheets are separated from each other by spacers such as layers of ionic oxide, e.g. LaO in LaFeAsO, \[Fig. \[fig\_struct\](a)\] or metal atoms, e.g. Ba in BaFe$_2$As$_2$ \[\]. In spite of the difference in the nature of the spacers, FeAs-based materials show surprising similarities in the temperature dependence of their structural parameters, anomalies in the electric resistance and specific heat capacity, and in their magnetic properties (e.g. \[\]).
A series of theoretical and computational reports appeared recently describing the electronic properties, magnetic interactions, phonon structure, and the origin of the superconductivity in LaFeAsO and related compounds (e.g. \[\]). The aim of this work is twofold: (i) to develop a model for the phase transitions observed in FeAs-based materials and (ii) to investigate a correlation between electronic density of states at the Fermi energy and the experimentally observed values of the $T_c$ and its dependence on the doping level.
![(Color online) (a) Structure of 1$\times$1 LnFeAsO (Ln = La, Sm) unit cell. (b) Schematics of several spin configurations within Fe layers shown for a $\sqrt{2}\times\sqrt{2}$ supercell. The circles show the positions of Fe atoms within the Fe layer. Here and below, up and down arrows indicate “up” and “down” spins, respectively. See text for details.[]{data-label="fig_struct"}](fig1_stuct.pdf){width="8.5cm"}
The calculations were carried out using density functional theory (DFT), the generalized gradient approximation functional PW91 \[\] and the projected augmented waves method \[\] implemented in the VASP code \[\]. The plane-wave basis set cutoff was set to 600 eV. The supercells containing eight (1$\times$1, Fig. \[fig\_struct\]), 16 ($\sqrt{2}$$\times$$\sqrt{2}$), and 32 (2$\times$2) atoms and Monkhorts-Pack grids of 252, 132, and 36 $k$-points, respectively, were used. For the analysis of the electronic structure, the charge-density was decomposed over atom-centered spherical harmonics.
In the first part of the paper we consider the relation between configurations of the spins associated with Fe 3$d$ electrons and the lattice structure. Several ordered antiferromagnetic configurations in a $\sqrt{2}$$\times$$\sqrt{2}$ supercell are shown in Fig. \[fig\_struct\](b). In AF1, the spins on the neighboring Fe atoms are antiparallel. In configurations AF2$'$ and AF2$''$ spins are parallel along $y$- and $x$-axes respectively; AF2$'$ and AF2$''$ are equivalent in the case of the high-temperature tetragonal ([**T**]{}) phase.
![Potential energy surfaces for the AF1 and AF2 configurations. Dots correspond to calculated energy values. Open circles indicate the spin pairs, which are different in [**O$_1$**]{} and [**O$_2$**]{} configurations.[]{data-label="fig_eng"}](fig2_pes.pdf){width="8.5cm"}
After minimization of the total energies with respect to both the atomic positions and the lattice parameters, the AF1 configuration maintains the [**T**]{} structure (Table \[tbl\_struct\]). Configurations AF2$'$ and AF2$''$ relax to two equivalent orthorhombic ([**O**]{}) structures [**O$_1$**]{} and [**O$_2$**]{}, in which the Fe 3$d$ spins along the short Fe–Fe bonds are parallel and those along the long Fe–Fe bond are antiparallel \[see Fig. \[fig\_struct\](c)\]. The lattice parameters $a$, $b$, and $c$ for [**O$_1$**]{} and [**O$_2$**]{} relate as $a_1$=$b_2$, $b_1$=$a_2$, $c_1$=$c_2$, and $a_1$$>$$b_1$ (see also Fig. \[fig\_eng\]). The calculated values for the lattice parameters for the low-temperature [**O**]{}-phase agree with the experimental data to within 0.4 %. (Table \[tbl\_struct\]). The ferromagnetic configuration is considerably less stable than antiferromagnetic ones and is not considered here.
Integration of the AF2$''$ charge-density within LaO and FeAs layers shows that the layers are charged: (LaO)$^{+\delta}$(FeAs)$^{-\delta}$ with $\delta$ = 0.15 $|e|$. Thus, one can consider LnFeAsO as a super-ionic compound, in which ionic and ion-covalent bonding within the LnO and FeAs layers, respectively, is accompanied by the weak ionic bonding of these layers. The magnetic moments on Fe atoms calculated for AF2 are 1.56 $\mu_{B}$ (Ln=La) and 1.33 $\mu_{B}$ (Ln=Sm). These differ significantly from the values suggested by Mössbauer measurements ($\sim$0.35 $\mu_B$) \[\].
To find the energy barrier separating the fully relaxed AF2$'$ and AF2$''$ configurations, we calculated the total energies $E_{AF2'}$ and $E_{AF2''}$ along the path $\ell_1$ connecting [**O$_1$**]{} and [**O$_2$**]{} (inset in Fig. \[fig\_eng\]). Path $\ell_1$ is parallel to the vector $\mathbf{n}$=(1,–1) in the $a$-$b$ plane. The $E_{AF2'}(Q)$ and $E_{AF2''}(Q)$, where $Q$=$a$–$b$, are plotted in Fig. \[fig\_eng\]. For comparison, we also calculated $E_{AF1}(Q)$ along the path $\ell_2 || \mathbf{n}$.
The calculated values of $E_1$, $E_2$, and $E_3$ are 0.005 eV, 0.025 eV, and 0.15 eV, respectively, for LaFeAsO and 0.006 eV, 0.026 eV and 0.11 eV for SmFeAsO. We note that approximate exchange-correlation functionals, such as PW91, can underestimate the values of energetic characteristics by as much as 100%. More reliable values of $E_1$, $E_2$, and $E_3$, as well as those of Fe magnetic moments, can be obtained by applying methods, which include exact exchange interaction and allow for coupling of different many-electron states \[\].
At $Q$=0, AF2$'$ and AF2$''$ have the same atomic structures and $E_{AF2'}$=$E_{AF2''}$, yet, their electronic states are different. This situation leads to Jahn-Teller (JT) instability \[\] and formation of a conical intersection at the crossover of the potential energy surfaces (PESs) $E_{AF2'}$ and $E_{AF2''}$. Correcting for non-adiabatic behavior near the intersection, together with taking into account the coupling of many-electron states, introduces an effective interaction $V$, which splits the $E_{AF2'}$ and $E_{AF2''}$ into a higher-energy single-well potential ($E_S$) and a lower-energy double-well potential ($E_D$) \[\] as shown in Fig. \[fig\_eng\]. We can conservatively estimate that $0<V<E_1$.
The lattice dynamics, described by $E_S$ and $E_D$, has three regimes depending on the temperature ($T$):
1\. For $T < E_1 - V$, the atoms vibrate near the their positions defined by one of the orthorhombic energy minima of $E_D$ (e.g. [**O$_1$**]{}). In this case the magnetic structure is dominated by AF2$'$ configuration (see Fig. \[fig\_eng\]).
2\. For $E_1 - V < T < E_1 + V$, motion of atoms is determined by parabolic branches of $E_D$, although the effect of the barrier separating its energy minima can not be neglected. The difference between the average distribution of short and long Fe–Fe bonds decreases with increasing temperature, which corresponds to a gradual transition from [**O**]{} to [**T**]{} symmetry. Magnetic order is lost because the Fe spins adjust themselves to the momentary local atomic structure, so as the spins are parallel for Fe atoms forming short Fe–Fe bonds and anti-parallel otherwise. In other words, thermal fluctuations of Fe–Fe bond lengths cause reorientation of Fe spins (Fig. \[fig\_eng\]).
3\. For $T>E_1+V$, the lattice dynamics is determined by parabolic branches of $E_S$ and $E_D$ and the effect of the barrier in $E_D$ can be neglected. The lattice has the [**T**]{}-symmetry. There is no magnetic order because the orientation of the spins changes according to the local atomic structure, as described in 2, and also due to coupling of electronic states of $E_S$ and $E_D$.
Experimental observations of the structural and magnetic phase transitions in LaFeAsO (e.g. \[\]) suggest that the [**T**]{}$\rightarrow$[**O**]{} transition takes place gradually, with the $Q$=$a$–$b$ order parameter exhibiting two kinks at $T_{max}$ ($\sim$160 K) and $T_{min}$ ($\sim$140 K), and that the magnetic phase transition occurs at $T_{min}$ or slightly below it. In addition, specific heat capacity displays two peaks, which also seem to coincide with $T_{max}$ and $T_{min}$ \[\]. Similar data have been reported for other FeAs-based materials \[\]. These results are consistent with the model for the three regimes of the lattice dynamics outlined above, in which two phase transition temperatures $T_{max}$ and $T_{min}$ correspond to $E_1 + V$ and $E_1 - V$, respectively. We can also speculate, that the decrease in the amplitude of atomic vibrations during [**T**]{}$\rightarrow$[**O**]{} transition \[\] can contribute to the abrupt drop in the electrical resistivity observed, for example, in \[\].
![Density of states for (La,Sm)FeAsO$_{x}$F$_{1-x}$. Letters [**O**]{} and [**T**]{} refer to the orthorhombic and tetragonal phases, respectively. The Fermi energy is at 0.0 eV.[]{data-label="fig_doped"}](fig3_dos_all.pdf){width="8.5cm"}
We now consider the effect of F-doping on the atomic and electronic structures of LnFeAsO. The doping provides additional electrons to the FeAs layer so as the charge distribution becomes (LnO)$^{+\delta+x}$(FeAs)$^{-\delta-x}$ and the lattice parameter $c$ decreases due to the increased inter-layer ionic bonding (Table \[tbl\_struct\]). This leads to opening up of a narrow gap in the $N(\varepsilon)$ at $\sim$2.5 eV below the $\varepsilon_F$ (not shown).
We find that the spin-density distribution in the FeAs layers is not independent on the arrangement of the F impurities. For example, for $x$=0.25 ($\sqrt{2}\times\sqrt{2}$ cell), the spin-down density is localized on a single Fe atom nearest to the F$^-$ impurity, while the remaining three Fe atoms share the spin-up density. At the doping level of $x$=0.125 (2$\times$2 cell), the effect is more subtle. The lowest energy state is similar to that of the undoped LnFeAsO: the lattice structure corresponds to [**O**]{}-symmetry of the $\sqrt{2}\times\sqrt{2}$ cell and the spin-arrangement is the same as in AF2, although the values of $\mu_{Fe}$ are reduced to 1.32 (Ln=La) and 0.75 $\mu_{B}$ (Ln=Sm). We also found a spin-disordered state, which has the [**T**]{}-symmetry and is $\sim$7 (Ln=La) and $\sim$5 (Ln=Sm) meV per Fe atom higher than the ground state. Taking into account the generally random distribution of the F impurities over O lattice sites in realistic samples, we suggest that such spin-disordered state realizes in practice.
For $x$=0.125 (2$\times$2 cell) we distinguish two sets of non-equivalent Ln and As atoms with different values of their $z$-coordinates. The effect of such structure on the lattice phonons and on the charge- and spin-density distributions needs to be considered separately.
Finally, we investigate the correlation between the structure and doping level and the electronic density of states \[$N(\varepsilon)$\] calculated for the fully relaxed AF1, AF2, and doped LnFeAsO$_{1-x}$F$_{x}$ (Fig. \[fig\_doped\]). In all cases the $N(\varepsilon)$ near the Fermi energy ($\varepsilon_F$) is dominated by the Fe 3$d$ states and the polarization of spin-up and spin-down states is negligible.
In stoichiometric LnFeAsO, $N_{AF2}(\varepsilon)$ has a pronounced depression near $\varepsilon_F$, while the $N_{AF1}(\varepsilon)$ has a narrow deep minimum separating a steep rise at $\varepsilon<\varepsilon_F$ and a peak at $\varepsilon$$>$$\varepsilon_F$ \[\]. Projecting $N_{AF1}(\varepsilon)$ on the $d$-states shows that this peak is dominated by $d_{xz}$ and $d_{yz}$ states. The same $d_{xz}$+$d_{yz}$ peaks near $\varepsilon_F$ are evident for the doped LnFeAsO (Fig. \[fig\_doped\]).
[l|cc|ccc|cc]{} $x$ & & $a$, Å& $b$, Å& $c$, Å& $z$(Ln) & $z$(As)\
\
0.0 & AF1 & $T$ & 5.6873 & 5.6899 & 8.6185 & 0.1448 & 0.6383\
0.0 & AF2 & $T$ & 5.7305 & 5.6672 & 8.6948 & 0.1433 & 0.6438\
0.0 & 300 K & $E$ \[\] & 5.7031 & 5.7031 & 8.74111 & 0.1413 & 0.6517\
0.0 & 120 K & $E$ \[\] & 5.6826 & 5.7104 & 8.71964 & 0.1417 & 0.6513\
0.125 & & $T$ & 5.6829 & 5.6829 & 8.5630 & 0.1560 & 0.6405\
& & & & & & 0.1452 & 0.6394\
0.25 & & $T$ & 5.6873 & 5.6831 & 8.4859 & 0.1562 & 0.6410\
0.14 &120 K & $E$ \[\] & 5.6844& 5.6844 & 8.6653 & 0.1477 & 0.6527\
\
0.0 & AF1 & $T$ & 5.5955 & 5.5918 & 8.3435 & 0.1406 & 0.6472\
0.0 & AF2 & $T$ & 5.6232 & 5.5623 & 8.4142 & 0.1396 & 0.6515\
0.125 & & $T$ & 5.5834 & 5.5834 & 8.2884 & 0.1523 & 0.6496\
& & & & & & 0.1413 & 0.6479\
0.25 & & $T$ & 5.5888 & 5.5902 & 8.2046 & 0.1529 & 0.6493\
\[tbl\_struct\]
According to the standard BCS theory of superconductivity, the transition temperature $T_c$ is proportional to $\left< \omega \right> exp[-1/\lambda N(\varepsilon_F)]$, where $\left< \omega \right> $ is a typical phonon frequency and $\lambda$ is the electron-phonon coupling constant. As shown in Ref. \[\], the limitation of $T_c <$40 K, suggested by Migdal’s theorem for BCS superconductors, is not justified and, therefore, much higher values of the $T_c$ can be achieved by optimizing $\left< \omega \right>$, $\lambda$, and $N(\varepsilon_F)$. We can tentatively suggest that $\left< \omega \right>$ and $\lambda$ do not vary strongly for FeAs-based compounds, since the conductivity is confined to the FeAs layers. Then $T_c$ can be considered as a function of a single parameter $N(\varepsilon_F)$.
Thus, we consider the correlation between the behavior of $N(\varepsilon)$ for $\varepsilon$ close to $\varepsilon_F$ (Fig. \[fig\_doped\]) and experimentally observed properties of LnFeAsO$_{1-x}$F$_{x}$ superconductors. First, we notice that as $x$ increases and $\varepsilon_F$ shifts across the $d_{xz}$+$d_{yz}$ peak, the value of $N(\varepsilon_F)$ increases as well, then reaches its maximum and then decreases. The details of the peak structure depend of the value of $x$ but its general shape is reminiscent of the experimentally observed dependence of the $T_c$ on $x$ (e.g. \[\]).
Furthermore, the maximum of the $d_{xz}$+$d_{yz}$ peak ($x$=0.0) in SmFeAsO is higher and further away from $\varepsilon_F$ than that in LaFeAsO. This correlates with the observations that the optimal $T_c$ is higher in SmFeAsO$_{1-x}$F$_{x}$ (46 K, $x$=0.15 \[\]) than in LaFeAsO$_{1-x}$F$_{x}$ (26 K, $x$=0.05–0.12 \[\]) and that it is achieved at larger values of $x$. The slope of $N(\varepsilon_F)$ calculated for $x$=0.125 is negative for Ln=La and positive for Ln=Sm, which indicates that maximum of $N(\varepsilon_F)$ can be found at $x<$0.125 for Ln=La and $x>0.125$ for Ln=Sm. This is consistent with the optimal values of $x$ found for these compounds as $\sim$0.11 (Ln=La) \[\] and $\sim$0.20 (Ln=Sm) \[\].
Finally, we notice that the $d_{xz}$+$d_{yz}$ peak in LaFeAsO is wider than that in SmFeAsO (this is clearly seen for $x$=0.0 and 0.125), which suggests that $T_c$ has a stronger dependence on $x$ in SmFeAsO as observed in \[\]. While these observations say little about the mechanism of the superconductivity in FeAs-based materials, they suggest that the highest $T_c$ can be achieved in those, which have the largest magnitude of the $d_{xz}$+$d_{yz}$ peak close to $\varepsilon_F$.
To summarize, we investigated the PESs for different magnetic states of stoichiometric LnFeAsO (Ln=La,Sm) and found that the properties of this system are determined by two close-lying PESs: a lower-energy double-well potential, where each well corresponds to the orthorhombic symmetry, and a higher-energy single-well potential of the tetragonal symmetry. This complex potential energy surface gives rise to three temperature ranges, and, therefore, two transition temperatures, and can explain the experimentally observed structural phase transition, the appearance of the magnetic order, and the anomaly in the temperature dependence of the specific heat capacity.
We noticed a correlation between the calculated profile of $N(\varepsilon)$ near $\varepsilon_F$ and experimentally observed dependence of the $T_c$ on the dopant concentration $x$ and on the type of Ln atom. This correlation can be used for computational prescreening of the promising LnFeAsO derivatives as well as for predicting optimal dopant concentrations via relatively inexpensive electronic structure calculations.
The authors thank C. Rüegg and A. M. Stoneham for their comments on the manuscript and S. W. Kim, Y. Kamihara, T. Nomura, and T. Kamyia for valuable discussions. P. V. S. is grateful to Japan Science Foundation and WPI-AIMR at Tohoku University. The access to HPCx is provided via the Materials Chemistry Consortium.
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00[$\pi^{0}\pi^{0}$]{} 3[$K_{l3}$]{} 4[$K_{l4}$]{} 00[K$^{0}_{S} \rightarrow \pi^{0}\pi^{0}$]{} Ł3p[K$^{0}_{L} \rightarrow \pi^{+}\pi^{-}\pi^{0}$]{} 0[P$_{S0}$]{} 1[P$_{L1}$]{}
Contributed paper to [*Lepton Photon 2001*]{}\
Rome, July 23-28 2001
[Studies of decays with the KLOE\
detector at\
]{}
[**The KLOE Collaboration:**]{}
A. Aloisio$^g$, F. Ambrosino$^g$, A. Antonelli$^c$, M. Antonelli$^c$, C. Bacci$^l$, G. Barbiellini$^n$, F. Bellini$^l$ G. Bencivenni$^c$, S. Bertolucci$^c$, C. Bini$^j$, C. Bloise$^c$, V. Bocci$^j$, F. Bossi$^c$, P. Branchini$^l$, S. A. Bulychjov$^f$, G. Cabibbo$^j$, R. Caloi$^j$, P. Campana$^c$, G. Capon$^c$, G. Carboni$^k$, M. Casarsa$^n$, V. Casavola$^e$, G. Cataldi$^e$, F. Ceradini$^l$, F. Cervelli$^h$, F. Cevenini$^g$, G. Chiefari$^g$, P. Ciambrone$^c$, S. Conetti$^o$, E. De Lucia$^j$, G. De Robertis$^a$, P. De Simone$^c$, G. De Zorzi$^j$, S. Dell’Agnello$^c$, A. Denig$^c$, A. Di Domenico$^j$, C. Di Donato$^g$, S. Di Falco$^d$, A. Doria$^g$, M. Dreucci$^c$, O. Erriquez$^a$, A. Farilla$^l$, G. Felici$^c$, A. Ferrari$^l$, M. L. Ferrer$^c$, G. Finocchiaro$^c$, C. Forti$^c$, A. Franceschi$^c$, P. Franzini$^{c,j}$, C. Gatti$^h$, P. Gauzzi$^j$, A. Giannasi$^h$, S. Giovannella$^c$, E. Gorini$^e$, F. Grancagnolo$^e$, E. Graziani$^l$, S. W. Han$^{b,c}$, M. Incagli$^h$, L. Ingrosso$^c$, W. Kluge$^d$, C. Kuo$^d$, V. Kulikov$^f$, F. Lacava$^j$, G. Lanfranchi $^c$, J. Lee-Franzini$^{c,m}$, D. Leone$^j$, F. Lu$^{b,c}$, M. Martemianov$^{c,f}$, M. Matsyuk$^{c,f}$, W. Mei$^c$, A. Menicucci$^k$, L. Merola$^g$, R. Messi$^k$, S. Miscetti$^c$, M. Moulson$^c$, S. Müller$^d$, F. Murtas$^c$, M. Napolitano$^g$, A. Nedosekin$^c$, M. Palutan$^l$, L. Paoluzi$^k$, E. Pasqualucci$^j$, L. Passalacqua$^c$, A. Passeri$^l$, V. Patera$^{c,j}$, E. Petrolo$^j$, D. Picca$^j$, G. Pirozzi$^g$, L. Pontecorvo$^j$, M. Primavera$^e$, F. Ruggieri$^a$, P. Santangelo$^c$, E. Santovetti$^k$, G. Saracino$^g$, R. D. Schamberger$^m$, B. Sciascia$^j$, A. Sciubba$^{c,j}$, F. Scuri$^n$, I. Sfiligoi$^c$, J. Shan$^c$, P. Silano$^j$, T. Spadaro$^j$, E. Spiriti$^l$, G. L. Tong$^{b,c}$, L. Tortora$^l$, E. Valente$^j$, P. Valente$^c$, B. Valeriani$^d$, G. Venanzoni$^h$, S. Veneziano$^j$, A. Ventura$^e$, Y. Wu$^{b,c}$, G. Xu$^{b,c}$, G. W. Yu$^{b,c}$, P. F. Zema$^h$, Y. Zhou$^c$
\
[$^b$Institute of High Energy Physics of Academica Sinica, Beijing, China]{}\
[$^c$Laboratori Nazionali di Frascati dell’INFN, Frascati, Italy]{}\
[$^d$Institut für Experimentelle Kernphysik, Universität Karlsruhe, Germany]{}\
[$^e$Dipartimento di Fisica dell’Università e Sezione INFN, Lecce, Italy]{}\
[$^f$Institute for Theoretical and Experimental Physics, Moscow, Russia]{}\
[$^g$Dipartimento di Scienze Fisiche dell’Università e Sezione INFN, Napoli, Italy]{}\
[$^h$Dipartimento di Fisica dell’Università e Sezione INFN, Pisa, Italy]{}\
[$^j$Dipartimento di Fisica dell’Università “La Sapienza” e Sezione INFN, Roma, Italy]{}\
[$^k$Dipartimento di Fisica dell’Università “Tor Vergata” e Sezione INFN, Roma, Italy]{}\
[$^l$Dipartimento di Fisica dell’Università “Roma Tre” e Sezione INFN, Roma, Italy]{}\
[$^m$Physics Department, State University of New York at Stony Brook, USA]{}\
[$^n$Dipartimento di Fisica dell’Università e Sezione INFN, Trieste, Italy]{}\
[$^o$Physics Department, University of Virginia, USA]{}
1.cm
INTRODUCTION
============
The KLOE detector at , the Frascati $\phi$-factory, has started physics data taking in April 1999. It has collected about 30 pb$^{-1}$ by the end of year 2000.
The $\phi$(1020) meson decays $\sim$ 34$\%$ of the times into a - pair; at peak energy, about 1 million of such decays occur every delivered pb$^{-1}$. is therefore an exceptional source of almost monochromatic, tagged particles, allowing for detailed studies of their more rare decays.
In the present paper, the status of the analysis about two different decay channels is presented.
Firstly, a measurement of the ratio among the branching ratios into two charged and neutral pions is presented. This is relevant for CP violation studies, since it enters the double ratio from which is derived. Moreover it is of interest for chiral perturbation theory studies, especially if the radiation of soft photons in the charged decay is properly taken into account.
Secondly, a measurement of the branching ratio of the decay is presented. Up to now, only one measurement of this branching ratio exists, based on a data sample of 75 events [@semnov]. In the present analysis the measurement is performed using a sample of about 600 event candidates with a background contamination of less than 10$\%$.
THE KLOE DETECTOR
=================
The KLOE (KLOngExperiment) detector [@kp], designed with the primary goal of measuring with a sensitivity of the order of one part in ten thousand, is also particularly well suited to perform studies on all charged and neutral decays of the meson.
It consists of a large tracking chamber, a hermetic electromagnetic calorimeter and a large magnet surrounding the whole detector, consisting of a superconducting coil and an iron yoke (see figure \[k1\]).
The tracking chamber [@dcp; @dpap] (DC) is a cylindrical, 2 m radius, 3.3 m long drift chamber. The total number of wires is 52140, out of which 12582 are the sense ones. It operates with a low-Z, He gas mixture, to minimize multiple scattering of charged particles and regeneration of ’s. The 58 concentric layers of wires are strung in an all-stereo geometry, with constant inward radial displacement at the chamber center. A spatial resolution better than 200 $\mu$m is obtained. The momentum resolution for 510 MeV/c electrons and positrons is 1.3 MeV/c, in the angular range 130$^{\circ} > \theta >$ 50$^{\circ}$.
The electromagnetic calorimeter[@tp; @epap] (EmC) is a lead scintillating fibers sampling calorimeter, divided into a barrel section and two endcaps. The modules of both sections are read out at the two ends by a total of 4880 photomultipliers. In order to minimize dead zones in the overlap region between barrel and endcaps, the modules of the latter are bent outwards with respect to the decay region.
The calorimeter was designed to detect with very high efficiency photons with energy as low as 20 MeV, and to accurately measure their energy and time of flight. Absolute calibrations of energy and time scales are performed using collision data. An energy resolution of 5.7$\%$/$\sqrt{E(GeV)}$ is achieved throughout the whole calorimeter together with a linearity in energy response better than 1$\%$ above 80 MeV and 4$\%$ between 20 to 80 MeV.
Moreover, $\gamma$ samples from different processes are selected to measure the time resolution at various energies; it scales according to the law $\sigma_{t}$=(54/$\sqrt(E(GeV) \oplus$ 147) ps, where the first term is in agreement with test beam data, while the second, to be added in quadrature, is dominated by the intrinsic time spread due to the bunch length.
STUDIES ON PHYSICS CHANNELS
===========================
Tagging of decays
------------------
When a $\phi$ meson decays into two neutral kaons $C$-parity invariance forces the two kaons to be in a - state. The observation of a , therefore, [*tags*]{} the presence of the in the opposite hemisphere. Similarly, decays can be selected by observing the on the other side.
As tagging strategy, one can either look for a charged vertex well inside the DC volume, or identify a EmC cluster compatible with being due to a slowly moving ($\beta \approx$ 0.22) neutral particle (so called ’KCRASH’ event). Actually, more than one half of the ’s reach the calorimeter before they decay. For the above reason the ’KCRASH’ tag provides a particularly clean, high statistics sample.
More specifically, events are selected on the basis of the two following requests:
1. [The presence of a EmC cluster with energy larger than 50 MeV, and transverse radius larger than 60 cm, due to the decay; it is needed to determine the t$_{0}$ of the event, i.e. the time at which the $\phi$ production and decay occurred.]{}
2. [The presence of a EmC cluster in the barrel region with energy larger than 100 MeV and time compatible with being due to a particle moving at a velocity in the $\phi$ rest frame 0.195 $< \beta^{\ast} <$ 0.2475 (the KCRASH).]{}
The tag efficiency is slightly dependent on the decay type, since the time zero estimate (first point above) is determined by particles with different velocities (prompt photons in the case of 00 events, pions in ones, pions or electrons for semileptonic decays). For instance, the distributions for the reconstructed $\beta^{\ast}$ for charged and neutral two pions decays are shown in figure \[fbeta\]; it turns out that the ratio of the efficiencies for having a KCRASH in the above mentioned velocity interval is $\epsilon^{+-}$/$\epsilon^{00}$ = (95.030 $\pm$ 0.005)$\%$, where the error is statistical only.
In the following, all events are tagged making use of the KCRASH prescription.
and 00 decays
--------------
The decays into two neutral pions are selected requiring the presence of four EmC clusters with a timing compatible with the hypothesis of being due to prompt photons (within 5 $\sigma$’s), and energy larger than 20 MeV. The prompt clusters distribution for the data taken during summer 2000 is shown in figure \[cprom\] together with the Monte Carlo expectation. The distribution agree well between each other. The energy spectrum and the angular distribution for the photons of the events with four prompt clusters are shown in figure \[pdist\]. Again, good agreement between data and Monte Carlo is observed.
Photon detection efficiency is estimated by real data using $\gamma$’s in the decays $\phi \rightarrow \pi^{+}\pi^{-}\pi^{0}$ as a control sample. The final selection efficiency for the 00 decay channel is $\epsilon_{00}$=(56.7$\pm$0.1)$\%$, dominated by acceptance.
The selection of events proceeds through the request of two oppositely charged tracks with polar angle in the interval 30$^{\circ} < \theta <$ 150$^{\circ}$, originating in a cylinder of 4 cm radius and 10 cm length around the interaction point. A further request is applied on the measured momenta to remove the residual background due to charged kaon decays: 120 $<$ p(MeV/c) $<$ 300 (see figure \[trch\]). Both tracks are also required to impinge to the calorimeter, in order to enhance the probability for having a good t$_{0}$ determination.
The track reconstruction efficiency is measured in momentum and polar angle bins from data subsamples. The final selection efficiency is $\epsilon_{+-}$=(58.5$\pm$0.1)$\%$, again dominated by acceptance.
The trigger efficiency is determined with real data for both decay types. It is (99.69 $\pm$ 0.03)$\%$ for the neutral decay and (96.5 $\pm$ 0.1)$\%$ for the charged one. The above figure includes also the probability for having at least one good cluster to determine the t$_{0}$ of the event, as explained in the previous paragraph.
Background levels are kept well below 1$\%$ for both decay types.
Using part of the data acquired in year 2000, corresponding to $\sim$10 pb$^{-1}$, 872748 and 414118 00 decays have been selected, providing: [**B()/B(00)**]{} = 2.23 $\times$ (1 $\pm$ 0.35$\times$ 10$^{-2}$ (stat) $\pm$ 1.5$\times$ 10$^{-2}$ (syst) )
to be compared with the present PDG value [@pdg] 2.197 $\times$ (1 $\pm$ 1.2$\times$ 10$^{-2}$ (stat) $\pm$ 1.5$\times$ 10$^{-2}$ (syst) ).
Systematics are dominated by residual uncertainties in photon counting and in the understanding of the difference between the tagging efficiencies for the two channels. More precise studies are presently under way.
decays
-------
In order to search for decay candidates, events with a KCRASH and two oppositely charged tracks from the interaction region are initially selected. Events are then rejected if the two tracks invariant mass (in the pion hypothesis) and the resulting momentum in the $\phi$ rest frame are compatible with those expected for a decay. According to Monte Carlo, this preselection has an efficiency, after the tag, of $\sim$ 62.4$\%$ on the signal.
In order to perform a time of flight identification of the charged particles, both tracks are required to be associated with a EmC cluster. The acceptance for such request, estimated by Monte Carlo, is (51.1 $\pm$ 0.2) $\%$. The time of flight difference $\Delta\delta$t for the two charged particles in both e-$\pi$ and $\pi$-$\pi$ hypotheses is then computed; events are accepted if $| \Delta\delta$t($\pi$-$\pi $)$| >$ 1.5 ns and $ | \Delta\delta$t($\pi$-e)$| <$ 1 ns and $ | \Delta\delta$t(e-$\pi$)$| >$ 3 ns. The efficiency on the signal, estimated by means of ’s decaying into $\pi$e$\nu$ before the DC internal wall, is (82.0$\pm$0.7)$\%$ .
Also the trigger efficiency as well as the one for correctly associating a track to a cluster and for having a good t$_{0}$ determination is measured directly on data, making use of , $\phi \rightarrow \pi^{+}\pi^{-}\pi^{0}$ and subsamples. The product of these efficiencies turns out to be (81.7 $\pm$ 0.5) $\%$.
The event is finally kinematically closed. The momentum is estimated making use of the measured direction of the and of the $\phi$ 4-momentum. The missing energy and momentum of the -$\pi$-e system, corresponding to the neutrino’s ones, are then computed. Their difference is distributed as in figure \[fitse\]; it must be equal zero for the signal. Data are fit using MC spectra for both signal and the residual background, due mostly to events with an early decay of one of the two pions.
Using data corresponding to a luminosity of $\sim$ 17 pb$^{-1}$, the measured yield is N() = 627 $\pm$ 30 events, for a total efficieny of (21.8 $\pm$ 0.3) $\%$. The total number of events is then normalised to the amount of observed events, to give [**B()**]{} = (6.8 $\pm$ 0.3 (stat))$\times$10$^{-4}$. In the ratio, the tagging efficiency, which is the largest cause of systematic uncertainty, cancels out identically. Other systematic effects, presently under study, are preliminarly estimated to be at a few percent level.
This result can be compared with the one obtained by the CMD2 Collaboration[@semnov]: B() = (7.2 $\pm$ 1.2)$\times$10$^{-4}$, and with the prediction obtained assuming $\Gamma_{S}$=$\Gamma_{L}$ : B() = (6.70 $\pm$ 0.07)$\times$10$^{-4}$.
[99]{} R. R. Akhmetshin et al. (CMD2 Collaboration) , Phys.Lett [**B456**]{} (1999), 90-94. A. Aloisio et al. (The KLOE Collaboration), A general purpose detector for , LNF-92/019 (1992). A. Aloisio et al. (The KLOE Collaboration), The KLOE detector, Technical Proposal, LNF-93/002 (1993). A. Aloisio et al. (The KLOE Collaboration), The KLOE Central Drift Chamber, Addendum to the Technical Proposal, LNF-94/028 (1994). M. Adinolfi et al., The Tracking detector of the KLOE experiment, LNF-01/016 (P) (2001), Submitted to Nucl. Inst. Meth. A. M. Adinolfi et al., The KLOE electromagnetic calorimeter, LNF-01/017 (P) (2001), Submitted to Nucl. Inst. Meth. A) A. Aloisio et al. (The KLOE Collaboration), The KLOE Trigger System, Addendum to the Technical Proposal, LNF-96/043 (1996). A. Aloisio et al. (The KLOE Collaboration), The KLOE Data Acquisition System, Addendum to the Technical Proposal, LNF-95/014 (1995). G. Cabibbo, PhD Thesis (2000), Universita’ La Sapienza Roma. D.E. Groom et al., The European Physical Journal [**C15**]{} (2000).
|
---
abstract: 'Resistively-detected NMR measurements on 2D electron systems near the $\nu = 1$ quantum Hall state are reported. In contrast to recent results of Gervais *et al.* \[Phys. Rev. Lett. $\bf 94$, 196803 (2005)\], a dispersive lineshape is found at all RF powers studied and Korringa-like nuclear spin-lattice relaxation is observed. The shape of the unexplained dispersive lineshape is found to invert when the temperature derivative of the longitudinal resistance changes sign. This suggests that both Zeeman and thermal effects are important to resistively-detected NMR in this regime.'
author:
- 'L. A. Tracy$^1$, J. P. Eisenstein$^1$, L. N. Pfeiffer$^2$, and K. W. West$^2$'
title: 'Resistively-Detected NMR in a Two-Dimensional Electron System near $\nu = 1$: Clues to the Origin of the Dispersive Lineshape'
---
Two dimensional electron systems (2DES) in semiconductors are weakly coupled to the nuclear magnetic moments of the host material via the hyperfine interaction. This coupling allows for studies of the spin degree of freedom in the electronic system via nuclear magnetic resonance (NMR) techniques. Among the several important findings that have stemmed from this connection, the discovery of multi-spin “skyrmion” excitations in the quantum Hall effect (QHE) regime is particularly significant[@barrett1]. Resistively-detected NMR (RDNMR), in which the resistance of the 2DES is modified by NMR excitation, has also led to intriguing observations, including signatures of competition between collective electronic phases with different electronic spin configurations[@kronmuller1; @smet1; @stern; @spielman1].
Typically, the hyperfine coupling between nuclear moments and 2D electron spins is expressed in terms of an effective magnetic field $B_N$. This field, which does not influence the orbital motion of the 2D electrons, contributes to their spin Zeeman energy: $E_Z = g\mu_B(B+B_N)$, with $g$ the electron $g$-factor, $\mu_B$ the Bohr magneton, and $B$ the externally applied magnetic field. $B_N$ depends upon the nuclear spin polarization and can reach several tesla in GaAs. Consequently, it is often assumed that, at least for electronic phases in which the Zeeman energy contributes to the energetics and transport in the 2DES, modification of $B_N$ via radio-frequency (RF) excitation is what enables RDNMR. Interestingly, $B_N$ and $B$ have opposite signs in GaAs, owing to the negative $g$-factor ($g \sim -0.4$) of electrons in the conduction band. As a result, destruction of nuclear polarization via NMR excitation usually $increases$ the electronic Zeeman energy.
NMR-induced modification of $E_Z$ in 2D systems is almost certainly the origin of some of the RDNMR responses which have been reported[@kronmuller1; @smet1; @stern; @spielman1]. However, more complex RDNMR effects have also been found. Desrat, [*et al.*]{}[@desrat] reported a curious “dispersive” RDNMR lineshape (vs. frequency) near the integer QHE state with one fully occupied spin-resolved Landau level (i.e. near filling factor $\nu = 1$). They found that in slow frequency sweeps through the NMR line the longitudinal resistance $R_{xx}$ displays roughly equal positive and negative excursions from its equilibrium value. Desrat, [*et al.*]{} suggested that this unusual resonance shape might be due to the formation of a skyrmion lattice. Interestingly, dispersive lineshapes have also been reported at certain other filling factors[@stern; @gervais2].
Here we report RDNMR studies of 2D electron systems in the QHE regime, focussing on the regions surrounding $\nu = 1$. In agreement with Desrat, [*et al.*]{}[@desrat], we find that the equilibrium RDNMR response has a dispersive frequency dependence near $\nu = 1$. While the amplitude of this response evolves continuously with RF power, its dispersive shape remains the same down to the lowest powers studied. Using a frequency-jumping technique we determine the nuclear spin-lattice relaxation rate $T_1^{-1}$ near $\nu = 1$. We find that $T_1^{-1}$ exhibits a Korringa-like temperature dependence near $\nu = 1$. Most interestingly, we find that the shape of the dispersive RDNMR line inverts at a well-defined filling factor on the flank of the $\nu = 1$ QHE. Remarkably, this shape inversion coincides with a change in sign of $dR_{xx}/dT$, the temperature derivative of the longitudinal resistance. This suggests that Zeeman effects are not solely responsible for RDNMR near $\nu = 1$ and that thermal effects also play a role.
The samples used in the present experiments are modulation-doped GaAs/AlGaAs heterostructures containing high mobility 2DESs. For the data presented here, the 2DES is confined in GaAs at a single interface with AlGaAs and is laterally patterned into a wide (500 $\mu$m) Hall bar geometry. The density of the 2DES is $N_s \approx 1.6 \times 10^{11} \rm cm^{-2}$ and its low temperature mobility is $\mu \approx 8 \times 10^6 \rm cm^2/Vs$. Diffused In ohmic contacts enable low frequency (13 Hz, typically) magneto-transport measurements. Cooling of the 2DES occurs primarily through the well-heat-sunk (and filtered) Au wires attached to these contacts. Importantly, the thermal relaxation time of the 2DES (measured via pulsed ohmic heating experiments) is quite short, less than 0.1 sec. throughout the regime of these experiments.
An approximately rectangular 8-turn NMR coil is wound around the sample for applying a RF magnetic field $H_1$ parallel to the 2DES plane and perpendicular to the applied dc magnetic field. We estimate $H_1$ to be in the 0.1 –- 0.5 $\mu$T range[@H1estimate]. These RF fields are far smaller than the typical nuclear dipolar field, $H_d \sim 100 ~\mu$T. Heating induced by the RF was readily calibrated out using the measured temperature dependence of the 2DES resistivity; all temperatures quoted here have been corrected for this effect.
Figure 1 displays the longitudinal resistance $R_{xx}$ in the present sample as a function of perpendicular magnetic field at $T = 70$ mK. The broad minimum centered around $B = 6.4$ T reflects the $\nu = 1$ QHE state in which the 2DES density $N_s$ matches the degeneracy $eB/h$ of the lowest spin-resolved Landau level. The inset to Fig. 1 shows the response of $R_{xx}$ to a slow sweep of the frequency of RF excitation of the NMR coil surrounding the sample. For these data the magnetic field is set to $B = 7.1$ T, i.e. on the high field flank of the $\nu = 1$ QHE minimum. At this field $\nu \approx 0.90$ and $R_{xx}$ is just becoming significant as quasi-holes in the lowest Landau level delocalize and begin to conduct. A strong resonant response is apparent in the figure and corresponds to NMR of the $^{75}$As nuclei in the sample. As first observed by Desrat, [*et al.*]{}[@desrat] the resonance has a dispersive, or “derivative” shape: $R_{xx}$ falls below its equilibrium value on the low frequency side of the resonance and then rises above it on the high frequency side. The approximately 15 kHz width of the resonance is about one order of magnitude larger than that expected from simple nuclear dipolar broadening. The distribution of Knight shifts resulting from the shape of the 2D electron subband wavefunction is almost certainly a significant contributor to the observed width[@barrett1]. Varying the excitation current $I$ used for the magneto-transport measurement from 2 nA to 100 nA produced no qualitative effect on the shape of these resonances. We emphasize that while dispersive lineshapes like that in Fig. 1 are also found on the low field flank of the $\nu = 1$ QHE, they are by no means ubiquitous in the QHE regime. Non-dispersive, or “conventional” RDNMR lineshapes are more commonly seen[@smet1; @stern; @desrat; @hashimoto]. For example, at $\nu = 1/2$ we and others have observed simple, unipolar RDNMR lineshapes[@stern; @spielman1; @tracy].
The RDNMR spectrum shown in Fig. 1 was obtained by sweeping the RF frequency slowly (0.13 kHz/sec) through the line. At this rate the difference between sweeping the frequency up vs. down is small. RDNMR spectra can also be obtained by abruptly adjusting the frequency to a value $f$ within the resonance region from a value $f_0$ several linewidths away[@desrat]. The resistance $R_{xx}$ responds by relaxing to a new value. After equilibrium is reached, the frequency is then reset to $f_0$ and $R_{xx}$ returns to its off-resonance value. Collecting a series of such transients, with different on-resonance frequencies $f$, allows construction of the entire RDNMR spectrum. Good agreement with slowly swept spectra like that in Fig. 1 is obtained. This frequency-jumping technique also allows for examination of the dependence of the RDNMR signal $\Delta R_{xx}/R_{xx}$ on the RF power delivered to the NMR coil. Figure 2 shows this power dependence in a typical dispersive RDNMR line at the two frequencies where $\Delta R_{xx}/R_{xx}$ is maximally positive and negative. At the maximum power shown we estimate $H_1 \approx 0.5~\mu$T. In addition to demonstrating that the nuclear spin system is not saturated, these data prove that the dispersive character of the RDNMR lineshape in our sample persists down to the lowest powers used ($H_1 \approx 0.1~\mu$T).
The transient relaxation of $R_{xx}$ following the return of the RF frequency $f$ to its off-resonance value $f_0$ offers a simple way to assess the nuclear spin lattice relaxation rate $T_1^{-1}$. We find that $T_1^{-1}$ varies monotonically with frequency across the RDNMR line, faster rates being observed on the low frequency side of the line. This variation is not surprising given the distribution of Knight shifts arising from the shape of the 2D electron subband wavefunction. The low frequency side of the RDNMR resonance corresponds to the largest Knight shifts and thus derives from those nuclei lying near the maximum of the subband wavefunction. These same nuclei will relax faster than those in locations where the subband wavefunction is small, consistent with our observations. While the transient responses of $R_{xx}$ when jumping off the low frequency side of the RDNMR line are well-fit by a simple exponential, this is not the case on the high frequency side. Here the transients exhibit a long tail which we speculate arises from nuclei outside the quantum well whose influence is felt via slow nuclear spin diffusion. In the following we report $T_1$ values obtained from the low frequency side of the line, at the frequency where the RDNMR response is maximally negative[@T1].
In agreement with previous work[@barrettT1; @smet2; @hashimoto], we find short $T_1$ times ($\sim$5 sec. at $T = 50$ mK being typical) near $\nu = 1$, with local maxima in the rate $T_1^{-1}$ both below and above $\nu = 1$. $\rm C\hat{o} t \acute{e}$, [*et al.*]{}[@cote] have attributed this fast relaxation to low-lying spin-wave modes of a skyrmion solid and used it to explain the enhanced heat capacity reported by Bayot, [*et al.*]{}[@bayot]. The maximum in $T_1^{-1}$ on the low $\nu$ side of $\nu = 1$ at $T = 70$ mK is shown in Fig. 3a. At each point in Fig. 3a a dispersive RDNMR lineshape is observed. Figure 3b shows the temperature dependence of $T_1^{-1}$ at $\nu = 0.88$, i.e. near the maximum in $T_1^{-1}$ vs. $\nu$. The data reveal that $T_1^{-1}$ is linear in temperature from $T = 120$ mK down to about 45 mK. The solid line in the figure corresponds to the best-fit Korringa law: $T_1T$ = 0.28 sec-K. This temperature dependence is consistent with the theory of $\rm C\hat{o} t \acute{e}$, [*et al.*]{}[@cote].
Recently, Gervais, [*et al.*]{}[@horst] have reported RDNMR measurements near $\nu = 1$ on a 2DES with mobility $17 \times 10^6 \rm ~cm^2/Vs$ and density $1.6 \times 10^{11} ~\rm cm^{-2}$, confined to a 40 nm GaAs quantum well. They find a dispersive lineshape only at high RF powers. For RF powers corresponding to $H_1 \approx 1 ~\mu$T they instead find that $\Delta R_{xx}$ shows only negative excursions as the RF frequency is swept through resonance. In contrast, we find a dispersive lineshape down to $H_1 \approx 0.1 ~\mu$T. Gervais, [*et al.*]{} also report long $T_1$ times (from $\sim 20$ to 600 sec.) and that $T_1^{-1}$ *rises* with falling temperature, in the same regime of temperature and filling factor where we find the opposite behavior. Although these discrepancies might be related to sample and/or measurement differences, we have reproduced our basic findings on the lineshape and the $T_1$ temperature dependence using a 2DES with mobility $14 \times 10^6 \rm ~cm^2/Vs$, confined to a 30 nm GaAs quantum well.
A dispersive lineshape is inconsistent with the usual model in which RDNMR is due solely to a hyperfine-induced increase of the electronic Zeeman energy. The low temperature transport properties of the $\nu = 1$ QHE are believed to reflect the existence of skyrmionic quasiparticles, the energy gap for which increases monotonically with $E_Z$. As a result, the $\nu = 1$ QHE should be strengthened by an NMR-induced increase in $E_Z$ and the resistance $R_{xx}$ exhibit a simple minimum vs. frequency.
Alternatively, the dispersive RDNMR lineshape might reflect the temperature of the electron gas, driven out of equilibrium via interaction with the nuclear spin system and the RF electromagnetic field. Since $dR_{xx}/dT > 0$ near well-developed QHE states, the data in Fig. 1 could imply that the electron gas is colder than the background thermal reservoir (crystal lattice, metallic ohmic contacts, etc.) on the low frequency side of the NMR line, but hotter than it on the high frequency side.
To investigate this possibility, we have examined the relationship between the dispersive RDNMR resonance and the magnitude and sign of $dR_{xx}/dT$. While $dR_{xx}/dT > 0$ in the immediate magnetic field vicinity of $\nu = 1$, it crosses zero and changes sign on moving further away. Sign changes such as these are commonplace in the QHE regime, although a complete understanding of them does not exist. If the RDNMR lineshape reflects the electron temperature, its shape ought to invert when $dR_{xx}/dT$ changes sign. Figure 4 demonstrates that this is precisely what occurs. In Fig. 4a the $R_{xx}$ peak on the high magnetic field flank of the $\nu = 1$ QHE is shown at three closely-spaced temperatures: 62, 65, and 70 mK. A clear sign change in $dR_{xx}/dT$ is observed at $B \approx 7.65$ T. Figs. 4b and 4c display RDNMR spectra at $B =$ 7.6 and 7.7 T, fields which straddle the sign change in $dR_{xx}/dT$. The two spectra have clearly inverted shapes. In both cases, the electron gas appears to cool on the low frequency side of the resonance and to warm on the high frequency side. Not surprisingly, RDNMR signals are very weak closer to the sign change where $dR_{xx}/dT \approx 0$.
At higher temperatures the QHE minimum in $R_{xx}$ narrows and the field where $dR_{xx}/dT$ changes sign shifts. Figure 4d shows that the RDNMR shape inversion faithfully tracks these shifts[@cooldown]; the solid dots give the magnetic field of the sign change while the open circles and squares are fields where RDNMR spectra with the same basic inverted shapes shown in Fig. 4b and 4c, respectively, are observed. These fields closely straddle the location of the sign change and thus strongly suggest that the RDNMR shape inversion and the $dR_{xx}/dT$ sign change are coincident. We emphasize that while the $T \approx 65$ mK data in Fig. 4a suggest that the $dR_{xx}/dT$ sign change coincides with the maximum in $R_{xx}$, this is not true in general. At higher temperatures the sign change occurs at a significantly lower field than the maximum in $R_{xx}$. From these several observations we conclude that the RDNMR spectra are essentially proportional to $dR_{xx}/dT$, at least in the vicinity of $\nu =1$. This supports the idea that the RDNMR lineshape reflects small ($\sim 1$ mK) temperature changes of the 2DES.
A model which might account for these observations is built around the idea that RF photons, detuned slightly from resonance with the nuclear Larmor frequency, can nonetheless induce nuclear spin flips if the energy mismatch is accomodated by the 2D electron gas[@girvin]. For RF frequencies $\omega_{RF}$ less than the nuclear Larmor frequency $\omega_N$ (positive detuning) the electron gas must supply energy to the nuclear system. As a result, cooling of the electrons occurs. For negative detuning, $\omega_{RF} > \omega_N$, the electron gas absorbs the excess energy and heats up. However, it is difficult to reconcile this model with the usual descriptions of non-saturated NMR under low power continuous RF illumination. Since the dispersive RDNMR lineshape is independent of the frequency sweep rate (so long as it is sufficiently slow), a steady-state description ought to be possible. In this case the Bloch equations reveal that the nuclear spins are heated out of equilibrium and energy is flowing into the lattice degrees of freedom, including the 2D electron system. Cooling of the electron gas thus seems unlikely.
An alternative model of the dispersive lineshape recognizes that NMR-induced heating of the nuclear spin system has two distinct effects on the electron gas. One is the increase in the electronic Zeeman energy described in the introduction. In addition, however, the hot nuclei will naturally heat the electron gas to some degree. Near the $\nu = 1$ QHE the former effect reduces $R_{xx}$ while the latter increases it. Importantly, the two effects have different dependences on RF frequency. The Zeeman effect, just like the Knight shift, is largest on the low frequency side of the NMR line. In contrast, the NMR-induced heating of the electron gas is roughly independent of frequency across the NMR line. These different frequency dependences could thus explain the dispersive shape of the RDNMR line. But this model also has flaws: While the heating effect on $R_{xx}$ will change sign with $dR_{xx}/dT$ it not obvious why the Zeeman effect would. We can only comment that the Zeeman energy dependence of $R_{xx}$ is well understood only close to $\nu = 1$; how it behaves at larger $|\nu –- 1|$ is unknown.
In conclusion, we have examined resistively-detected NMR near the $\nu = 1$ quantized Hall effect. We find a dispersive lineshape at all RF power levels. Accurate measurements of the nuclear spin-lattice relaxation time are in good agreement with theoretical expectations. Surprisingly, we find the shape of the RDNMR line inverts when the temperature derivative of the longitudinal resistance, $dR_{xx}/dT$, changes sign. While this is not yet understood, it strongly suggests that both Zeeman and thermal effects are important in RDNMR near $\nu = 1$.
It is a pleasure to acknowledge several helpful conversations with S.M. Girvin. This work was supported by the DOE under Grant No. DE-FG03-99ER45766 and the NSF under Grant No. DMR-0242946.
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These estimates were derived from a careful circuit analysis of the 50-ohm coax + NMR coil system.
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The observed RDNMR transient response times exceed the electronic thermal relaxation time by at least a factor of 25. This justifies our association of the transient response time with the nuclear $T_1$.
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|
---
author:
- 'Gerardo Iñiguez, Zhongyuan Ruan, Kimmo Kaski, János Kertész and Márton Karsai'
title: Service adoption spreading in online social networks
---
Introduction {#sec:intro}
============
A human society abounds with examples of collective patterns of behaviour that arise due to the correlated decisions of a large number of individuals. This is evidenced in the spread of religious beliefs and political movements, in the behavioural, cultural, and opinion shifts in a population, in the adoption of technological and medical innovations, in the rise of popularity of political and media figures, in the growth of bubbles in financial markets, and in the use of products and online services. All of these phenomena tend to evolve similarly over time, as they start with individuals that independently from their peers and due to external influence such as mass media, take the risk by adopting a certain behaviour [@valente-thresholds-1996; @Toole2012Modeling]. Then, these processes continue as friends, colleagues, and acquaintances observe such individuals and engage with the same behaviour, therefore participating in a spreading process throughout society [@Kleinberg2007Cascading].
The way ideas, products, and behaviour spread throughout a population over time, commonly know as [*innovation diffusion*]{}, was first observed empirically in the mid 20th century by the likes of Rogers [@Rogers2003Diffusion] and Bass [@Bass1969]. In the following decades, many mathematical models were introduced with the goal of identifying mechanisms by which behaviour diffuses through society [@Granovetter1978Threshold; @schelling1969models; @Axelrod1997Dissemination]. One of the first (and arguably simplest) is the Bass model for forecasting sales of new consumer durables [@Bass1969], which characterises the diffusion of innovation as a process of contagion initiated by some external influence [@Toole2012Modeling] (e.g. mass communication, news media) and promoted by internal, social influence [@Centola2010Spread] (via word-of-mouth, viral marketing, etc.). The model assumes a homogeneous population of adopters and it predicts that aggregated sales data has an s-shaped pattern as a function of time [@Granovetter1978Threshold; @Zhang2016Empirically].
Despite the success of the Bass model and similar diffusion-like models to capture qualitatively the temporal behaviour of adoption processes, macroscopic models only provide empirical generalisations based on the behaviour of society as a whole (by means of aggregated data on adoption rates, for example). Hence, these models do not take into account individual heterogeneities and the complex structure and dynamics of social processes [@Kiesling2012Agent]. In other words, since the same macro-level behaviour may arise from several individual-level mechanisms (like learning, externalities, or contagion), it is difficult for these models to assess what mechanisms are actually responsible for large-scale spreading phenomena [@Goel2012Structure; @BorgeHolthoefer2013Cascading; @Goel2015Structural]. In order to overcome this issue, agent-based diffusion models consider behavioural heterogeneities, networked social interactions [@RevModPhys.81.591; @Bakshy2012The] and decision-making processes based on the cognitive capacities of individuals [@Holt06; @bikhchandani-hirshleifer-welch-92; @Karsai2014Complex]. Then, behaviour at the level of society emerges dynamically from the interplay between network structure and the actions of people. This microscopic approach allows for the modelling of varying behaviour across individuals, while recognising that social interactions and interpersonal communication are essential in determining adoption [@valente-thresholds-1996; @Romero2011Differences].
Under the network approach, the Bass model is an archetypal example of [*simple contagion*]{} [@Barrat2008Dynamical] where, akin to the transmission of a disease, information and individuals’ willingness to adopt may propagate with exposure to a single person engaging in some particular behaviour. However, when adoption turns out costly, risky or controversial, the spread of ideas and products often requires social reinforcement and exposure to several sources, a phenomenon usually called [*complex contagion*]{} [@Centola2007Complex; @porter2016]. The requirement of multiple interactions for adoption was first implemented theoretically by Granovetter via behavioural thresholds, namely [*‘the number or proportion of others who must make one decision before a given actor does so’*]{} [@Granovetter1978Threshold]. Following this idea various agent-based network models have been introduced and analysed by Watts and others [@valente-thresholds-1996; @Watts2002Simple; @Handjani1997Survival; @Neill2005Cascade; @Watts2007Influentials; @Melnik2013Multistage; @Gomez2010Modeling; @Karampourniotis2015The; @Miller2015Complex] in order to understand the properties of threshold-driven social contagion.
Despite the allure of social influence as the reason behind innovation diffusion, it is more challenging to identify causal mechanisms in adoption spreading than in biological contagion, since the same empirical, large-scale observations may be obtained as effects of social influence [@Onnela2010Spontaneous], homophily [@McPherson2001], or the environment. For example, collective adoption patterns may appear as a consequence of homophilic structural correlations, where interacting individuals adopt due to their similar interests and not due to actual social influence [@porter2016]. Hence distinguishing between the effects of social influence and homophily at the individual level remains a challenge [@Aral2009; @Shalizi2011]. Moreover, regarding the particular role social influence may have in adoption spreading, several assumptions have been proposed about its functional dependency on the number of adopters necessary to influence an individual. While Granovetter and others [@Granovetter1978Threshold; @Watts2002Simple] suggest a simple linear dependency, as observed in some large techno-social systems [@Karsai2014Complex], Latané [@Latane1981The] argues for non-linear effects that have been demonstrated empirically by online experiments at different scales [@Centola2010Spread; @Centola2011An; @Suri2011Cooperation].
Perhaps one of the most intriguing features of threshold-driven social contagion is its ability to capture what Watts calls the [*robust yet fragile*]{} nature of complex systems [@Watts2002Simple]. This means that a population may be robust and disregard many ideas and products, but suddenly exhibit fast system-wide adoption patterns known as [*behavioural cascades*]{}. While homophily suggests that adoption behaviour is only seemingly correlated, and simple contagion implies that external influence always induces global adoption in a connected population, complex contagion captures the additional feature that large cascades of behavioural patterns tend to happen only rarely, and may be triggered by actions at the individual level that are indistinguishable from the rest. Indeed, behavioural cascades are rare but potentially disrupting social spreading phenomena, where collective patterns of exposure arise through reinforcement as a consequence of small initial perturbations [@Motter2017Unfolding]. Examples include the rapid emergence of political and grass-root movements [@GonzalezBailon2011Dynamics; @BorgeHolthoefer2011Structural; @EllisInformation], or the fast spreading of information [@Goel2012Structure; @Watts2007Influentials; @Dow2013Anatomy; @Gruhl2004Information; @Banos2013Role; @Hale2013Regime; @Leskovec2005Patterns; @Leskovec2007Dynamics] and behavioural patterns [@Fowler2009Cooperative]. Moreover, cascades may appear in both online [@Leskovec2007Patterns; @Duan2009Informational; @Bond2012Million; @Hui2012Information; @Hodas2014Simple] and offline [@Green2017Modeling] social environments.
The characterisation [@Goel2012Structure; @BorgeHolthoefer2013Cascading; @Hackett2013Cascades; @Gleeson2008Cascades; @Brummitt2011; @GhoshCascadesArxiv2010] and modelling [@Watts2002Simple; @Hurd2013Watts; @Singh2013Thresholdlimited; @Gleeson2007Seed; @Gleeson2014Simple] of behavioural cascades have received a lot of attention in the past and provide some understanding of the causal mechanisms and structure of empirical and synthetic cascades on various types of networks [@Yagan2012Analysis; @Brummitt2015Cascades; @Karimi2013Threshold; @Backlund2014Effects]. However, these studies fail in addressing the temporal dynamics of the emerging cascades, which may vary among empirical examples of social contagion. In other words, previous works do not answer why real-world cascades may evolve either slowly or rapidly over time. In contrast to the cases of rapid cascading mentioned above, the propagation of products in social networks is typically slower, with adoption spreading gradually, even if it is driven by threshold mechanisms and may eventually cover a large fraction of the total population [@Karsai2014Complex]. This slow behaviour characterises the adoption of online services such as Facebook, Twitter, LinkedIn and Skype (Fig.\[fig:0\]a), since their yearly maximum relative growth rate of cumulative adoption [@SocialMedia] is lower than in the case of rapid cascades, as suggested in standard models of threshold-driven social contagion like the Watts threshold (WT) model [@Watts2002Simple].
![**The speed and layers of online service adoption.** **(a)** Yearly maximum relative growth rate (RGR) of cumulative adoptions obtained by taking the maximum of the yearly adoption rate (yearly count of adoptions) normalised by the final observed number of adoptions of a given service. We show it for several online social-communication services [@SocialMedia] (black bars), including three paid Skype services (s1 - “subscription”, s2 - “voicemail”, and s3 - “buy credit”). The dark grey bar corresponds to a rapid cascade of adoption as suggested by the Watts threshold model, while the light grey bar is the prediction of our model for Skype s3. **(b)** Schematic layer structure of online service adoption systems. The lowest layer represents a real, offline social network; the middle layer corresponds to any online social network; and the top layer is the adoption of a service within the social network. As an advantage in this study we have full knowledge about the Skype online social network in this multi-layer structure, while we follow a paid service spreading on the online network.[]{data-label="fig:0"}](Fig0.pdf){width="1.\textwidth"}
In this chapter we review recent works [@Ruan2015; @Karsai2016Local] focusing on the empirical characterisation and mathematical modelling of the slow, threshold-driven spreading of service adoption in online social networks, particularly in the case of Skype. We first provide empirical evidence of the distribution of individual adoption thresholds and other structural and dynamical features of the worldwide Skype adoption cluster. We then show how to incorporate the observed structural and threshold heterogeneities into a dynamical threshold model where multiple individuals may adopt spontaneously (i.e. firstly among their acquaintances). We find that if the fraction of users who reject to adopt a product or idea in the model is large, the system enters a quenched state where the evolution and structure of the global adoption cluster is very similar to our observations of services within Skype. Model calculations and the analysis of the real social contagion process suggest that the evolving structure of an adoption cluster differs radically from previous expectations [@Watts2002Simple], since it is triggered by several spontaneous adoptions arriving at a constant rate. Furthermore, the stable adopters (who initially resist exposure) are actually responsible for the emergence of global social adoption.
Empirical observations {#sec:emp}
======================
In order to observe service adoption dynamics we analyse an example of an online diffusion process, where we have access to individual service adoption events as well as the underlying social network. Our aim is to identify the crucial mechanisms necessary to consider in models of complex contagion to match them better with reality, and define a model that incorporates these mechanisms and captures the possible dynamics leading to the emergence of real-world global cascades.
To fully understand service adoption processes on online social structures, we need to keep in mind some of their proxy characteristics. People of a society constitute a social network by being connected with ties of several kinds that are maintained in various ways. However, and despite their recent popularity, online social systems are not capable of mapping the entire social network as offline, occasionally maintained, temporary, or ill-favored social ties may remain invisible in such systems. Therefore, these networks provide only a proxy sample of the real social structure (Fig.\[fig:0\]b), with important but also insignificant social ties present. Moreover, data available for social network studies commonly arrives as a sample of a larger online social system, which unavoidably leads to observational biases. In addition, connections in an online social structure cannot precisely assign the flow of direct social influence among the connected individuals, only the possibility of it. Finally, just like real social networks, online social systems evolve over time via the creation and dissolution of social ties or by nodes entering or leaving the system. Due to all these limitations it is rather challenging to make unbiased observations about any unfolding dynamical processes, without making some assumption about the underlying online social systems.
In our study we use the social network of one of the largest voice-over-internet providers in the world, the network of Skype, which actually copes well with the limitations listed above. It maps all connections in the Skype network without sampling, thus it provides us with a complete, unbiased map of the underlying social network, maintaining the diffusion of services available only for registered users in the network. This network evolves as a function of time via adoption, churning, and link creation dynamics. We have shown in an earlier study [@Karsai2014Complex] that while rates of these actions increase considerably with time, the adoption processes can be well characterised by the net rate of the actual number of users. We also found that while spontaneous adoptions and churning evolve with a constant rate, the probability of peer-pressured adoptions corresponds linearly to the strength of social influence, giving rise to a non-linear dynamics at the system level, which enables its modelling as a complex contagion process.
In our study we concentrate on the adoption dynamics of a paid service that unfolds over the Skype social network (Fig.\[fig:0\]b). Since this adoption process evolves in a considerably faster time-scale than the underpinning social network, we can validly assume a time-scale separation. Thus, from here on we consider the network structure to be static, which may give us a good first approximation while concentrating on the adoption dynamics unfolding on its fabric. To identify the effects of social influence in our empirical system we also present a null model study (Section \[sec:sinf\]).
Data description
----------------
In our social network nodes represent users and edges between pairs of users exist if they are in each other’s contact lists. A user’s contact list is composed of *friends*. If user $u$ wants to add another user $v$ to his/her contact list, $u$ sends $v$ a contact request, and the edge is established at the moment $v$ approves the request (or not, if the contact request is rejected). For the purpose of our study we use the largest connected component of the aggregated free Skype service network, which was recorded from September 2003 to November 2011 (i.e. over $99$ months) and contains roughly 4.4 billion links and 510 million registered users worldwide [@SkypeIPO]. The data is fully anonymised and considers only confirmed connections between users after the removal of spammers and blocked nodes.
To study an example of service adoption dynamics we follow the purchases of the “buy credit” paid service for $89$ months starting from 2004. Data includes the time of first payment of each adopting user, an individual and conscious action that tracks adoption behaviour. Note that other examples about the adoption dynamics of similar services are presented in [@Karsai2016Local].
Degree and threshold heterogeneities
------------------------------------
In his seminal work on modelling adoption cascades [@Watts2002Simple], Watts identified two structural characteristics that control the emergence of collective adoption cascades. One is the distribution $P(k)$ of degrees (i.e. number of neighbours of a node), with average $z = \langle k \rangle$, and the other is the distribution $P(\varphi)$ of adoption thresholds (with average $w = \langle \varphi \rangle$), defined as the necessary fraction of exposed neighbours that triggers the adoption of an individual under study, or central ego.
Degree heterogeneities have been in the focus of network science for a while now, and a broad degree distribution $P(k)$ is one of the main characteristics of complex networks [@Newman2010Networks; @Barabasi2016NetSci]. This distribution has been usually described as a power-law, but a log-normal fit has often turned out to work better [@Mitzenmacher2004]. The latter is the case with our data: $$P(k) \propto k^{-1} \exp[-(\ln k - \mu_D)^2/(2\sigma_D^2)],
\label{eq:Pk}$$ where the best fit is obtained with $k \geq \kmin$ and parameters $\mu_D=1.2$, $\sigma_D=1.39$ and $\kmin=1$ (Fig. \[fig:1\]a), giving an average degree $z = 8.56$.
{width="\textwidth"}
It is a challenging task to quantify individual adoption thresholds, as their observation simultaneously requires information about the underlying network structure and the dynamical adoption process evolving on top. Therefore, beside measuring the number $k$ of friends of an ego in the Skype social network (already needed for the degree distribution), for $k$-degree users at the time of their adoption we measure the number $\Phi_k$ of their neighbours who have adopted the service earlier, i.e. the integer threshold [@Gleeson2008Cascades]. To our knowledge, this is the first detailed study measuring the number of adopting neighbors of adopters in an empirical setting. The obtained distribution $P(\Phi_k)$ for varying $k$ is shown in the inset of Fig. \[fig:1\]b. The importance of our empirical findings is amplified by the observation that these distributions can be scaled together when using the fractional threshold variable $\varphi=\Phi_k/k$, i.e. the fraction of adopting neighbors at the time of adoption (Fig. \[fig:1\]b main panel). Thus, in a discussion of whether the number or the ratio of adopting neighbours matters in behavioural adoption [@Watts2002Simple; @Centola2007Complex], our results give strong support to the latter.
Using fractional thresholds and the relationship $P(\Phi_k, k) = k P(\Phi_k/k)$, all distributions collapse to a master curve, which is once again well-approximated by a log-normal function of the following form, $$P(\Phi_k/k)=P(\varphi) \propto \varphi^{-1} \exp[-(\ln\varphi - \mu_T)^2/(2\sigma_T^2],
\label{eq:Pphi}$$ with parameters $\mu_T=-2$ and $\sigma_T=1$ as constrained by the average threshold $w = 0.19$ [@Karsai2016Local]. These empirical observations, in addition to the broad degree distribution, provide quantitative description of the heterogeneous nature of adoption thresholds.
Dynamics and structure of adoption cascades
-------------------------------------------
Since we know the complete structure of the online social network, as well as the first time of service usage for all adopters, we can follow the temporal evolution of the adoption dynamics. By counting the number of adopting neighbours of an ego, we identify innovators ($\Phi_k=0$), and vulnerable ($\Phi_k=1$) or stable ($\Phi_k>1$) nodes, in accordance with the categorisation of Watts [@Watts2002Simple]. As we show in Fig. \[fig:2\]a, the adoption rates for these categories behave rather differently from previous suggestions [@Watts2002Simple]. First, there is not only one seed but an increasing fraction of innovators in the system who, after an initial period, adopt approximately at a constant rate (denoted by the grey shaded area in Fig. \[fig:2\]a). Second, vulnerable nodes adopt approximately with the same rate as innovators, which suggests a strong correlation between these types of adoption. This stationary behaviour is rather surprising as environmental effects, like competition or marketing campaigns, could potentially influence the adoption dynamics. On the other hand, the overall adoption process accelerates due to the increasing rate of stable adoptions induced by social influence.
To better understand how innovation spreads throughout the social network, we take a closer look at the internal structure of the service adoption process. To do so, we consider individual adoption times and construct an evolving adoption network, where links exist between users who have adopted the service before time $t$ and are connected in the social network underneath. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between connected nodes whose adoption did not happen at the same time. This way links in the adoption graph indicate ties where social influence among individuals could have existed. By observing the evolution of the adoption network, we are interested in its connectedness and its composition of sub-components of adopters of different kinds.
The size distribution $P(s_a)$ of connected components in the adoption network shows the emergence of a giant percolating component over time (Fig. \[fig:2\]c main panel), along with several other small clusters. Moreover, after decomposition we observe that the giant cluster builds up from several innovator seeds that induce small vulnerable trees locally (Fig. \[fig:2\]d), each with small depth [@Karsai2016Local; @Bakshy11; @Goel2012Structure]. At the same time the stable adoption network (considering connections between all stable adopters at the time) has a giant connected component, indicating the emergence of a percolating stable cluster with size comparable to the largest adoption cluster (Fig. \[fig:2\]c, inset). These observations suggest a scenario for the evolution of the global adoption component where multiple innovators adopt at different times and trigger local vulnerable trees, which in turn induce a percolating component of the connected stable nodes holding the global adoption cluster together. Consequently, in the structure of the adoption network primary triggering effects are important only locally, while external and secondary triggering mechanisms seem to be responsible for the emergence of global-scale adoption.
![ **(a)** Adoption rate of innovators \[$R_i(t)$\], vulnerable nodes \[$R_v(t)$\], and stable nodes \[$R_s(t)$\], as well as the net service adoption rate \[$R(t)$\], where the rates are measured with a 1-month time window, and $q$ and $\tau$ are arbitrary constants. The shaded area indicates the regime where innovators adopt approximately with constant rate. [**(b)**]{} Null model rates where times of adoption are randomly shuffled. **(c)** Empirical connected-component size distribution at different times for the adoption \[$P(s_a)$, main panel\] and stable adoption \[$P(s_s)$, inset\] networks, with $s_a$ and $s_s$ relative to system size. **(d)** Empirical connected-component size distribution $P(s_v)$ for the relative size of innovator-induced vulnerable trees at different times.[]{data-label="fig:2"}](Fig2.pdf){width="\textwidth"}
Despite of this expansion dynamics and connected structure of the service adoption network, we need to take a closer look at spurious effects, which could potentially induce the observed behaviour. First, during our analysis we assume that the adoption process is exclusively driven by social influence, without any direct information about the presence of the influence itself. One can argue that the observed phenomena is simply explained by homophily, i.e. by frequent links between people who are both interested in the given service and who would adopt independently from each other. Second, the service reaches less than $6\%$ of the total number of active Skype users over a period of $7$ years [@SkypeIPO]. Since this adopting minority is connected within a giant adopting cluster, it may indicate local effects of social influence but also raises the question about the role of non-adopting users. Finally, we observe that the giant adoption cluster evolves over several years, which could simply be the consequence of individual decisions of users to wait to adopt the service even after their threshold has been reached. In the following we further investigate these questions to better understand the adoption process. First we present a null model study to underline the overall effects of social influence as compared to homophily; we also perform a time re-scaling experiment to explore the role of waiting times on the global adoption dynamics; and finally we propose a dynamical threshold model [@Ruan2015; @Karsai2016Local], which helps us understand the role of multiple innovators and non-adopters in the unfolding of the service adoption processes.
Social influence vs. homophily {#sec:sinf}
------------------------------
Studies of social contagion phenomena assume that social influence is responsible for the correlated adoption of connected people. However, an alternative explanation for the observed correlated adoption patterns is homophily: a link creation mechanism by which similar egos get connected in a social structure. In the latter case, the correlated adoption of a connected group of people would be explained by their similarity and not necessarily due to social influence. Homophily and influence are two processes that may simultaneously play a role during the adoption process. Nevertheless, distinguishing between them on the individual level is very challenging using our or any similar datasets [@Shalizi2011; @Aral2009]. Fortunately, at the system level one may identify which process is dominant in the empirical data. To do that we first need to elaborate on the differences between these two processes.
Influence-driven adoption of an ego may take place once one or more of its neighbours have adopted, since then their actions may influence the decision of the central ego. Consequently, the time ordering of adoptions of the ego and its neighbours matters. Homophily-driven adoption is, however, different. Homophily drives social tie formation such that similar people tend to be connected in the social structure. In this case connected people may adopt because they have similar interests, but the time ordering of their adoptions would not matter. Therefore, it is valid to assume that adoption could evolve in clusters due to homophily, but these adoptions would appear in a more-or-less random order.
To test this hypothesis we define a null model where we take the adoption times of users and shuffle them randomly among all adopting egos. This way a randomly selected time is assigned to each adopter, while the adoption rate and the final set of adopters remain the same. Moreover, this procedure only destroys correlations between adoption events induced by social influence, but keeps the social network structure and node degrees unchanged. In this way, during the null model process the same egos appear as adopters, but the rates of adoption may in principle change (or not), corresponding to social influence (or homophily) as a dominant factor during the adoption process. If adoption is mostly driven by homophily, the rates of adoption would not change considerably beyond statistical fluctuations. On the other hand, if social influence plays a role in the process, rates of adoption in the null model should be very different from the empirical curves, implying that the time ordering of events matters in the adoption process. In this case, the rate of innovators should be higher than in the empirical data, since nodes that are in the adoption cluster originally without being directly connected, would have a greater chance to appear as innovators, due to a random adoption time that is not conditional to the time ordering of the adopting neighbours.
After calculating the adoption rates of different user groups in the shuffled null model, we observe the latter situation (Fig. \[fig:2\]b): the rate of innovators becomes dominant, while the rates of stable and vulnerable adoptions drop considerably as they appear only by chance. This suggests that the temporal ordering of adoption events matters a lot in the evolution of the observed adoption patterns, and thus social influence may play a strong role here. Of course one cannot decide whether influence is solely driving the process or homophily has some impact on it; in reality it probably does to some extent. However, we can use this null model measure to demonstrate the presence and importance of the mechanism of influence during the adoption process.
Waiting time of adoption {#sec:tw}
------------------------
![ Distribution $P(\tau_w)$ of times between the last adoption in the egocentric network of an individual and his/her own adoption. [**(b)**]{} Cumulative adoption rates before and after the removal of waiting times \[$CR(t)$ and $CR_{\tau_w}(t)$, respectively\]. $n$ and $\tau$ are arbitrary constant values.[]{data-label="fig:3"}](Fig3.pdf){width="\textwidth"}
As we mentioned earlier, one reason behind the slow evolution of the adoption process could be due to the time users wait after their personal adoption threshold is reached and before adopting the service. This lag in adoption can be due to individual characteristics, or can come from the fact that social influence does not spread instantaneously (as commonly assumed in threshold models). This waiting time $\tau_w$ can be estimated by measuring the time difference between the last adoption in a user’s egocentric network and the time of his/her adoption. This time is $\tau_w=0$ by definition for innovators, but $\tau_w$ can take any positive value for vulnerable and stable adopters up to the length of the observation period.
We find that waiting times are broadly distributed for adopters in our dataset (Fig. \[fig:3\]a), meaning that many users adopt the service shortly after their personal threshold is reached, but a considerable fraction waits long before adopting the service. This heterogeneous nature of waiting times may be a key element behind the observed adoption dynamics. One way to figure out its effect on the speed of cascade evolution is by removing them. We can extract the waiting time from the adoption time of adopters and assign a rescaled adoption time for each of them. The rescaled adoption time of a user is the last time when his/her fraction of adopting neighbours changed and the adoption threshold was (hypothetically) reached. After this procedure, we can calculate a new adoption rate function by using the rescaled adoption times and compare this rate to the original. From Fig. \[fig:3\]b we conclude that although adoption becomes faster, the rescaled adoption dynamics is still not rapid. On the contrary, it suggests that the rescaled adoption dynamics is still very slow and quite similar to the original. Consequently, waiting times cannot explain the observed slow dynamics of adoption.
Note that long waiting times can have a further effect on the measured dynamics. After the ‘real’ threshold of a user is reached and he/she waits to adopt, some neighbours may adopt the product. Hence all observed measures are in this sense ‘effective’: observed thresholds are larger or equal than real thresholds; the innovator rate is smaller or equal; the vulnerable and stable rates will be larger or equal; and waiting times will be shorter or equal than the real values. Consequently the process may actually be faster than that we observe in Fig. \[fig:3\]b after removing the effective waiting times. However, this bias becomes important only after the majority of individuals in the social network has adopted the service and the spontaneous emergence of adopting neighbours becomes more frequent. As the fraction of adopters in our dataset is always less than $6\%$ [@SkypeIPO], we expect minor effects of this observational bias on our measurements.
Modelling social contagion {#sec:model}
==========================
In order to understand better the possible microscopic mechanisms behind the empirical observations of online service adoption described previously, we introduce and analyse two agent-based network models of threshold-driven social contagion. First we discuss the WT model as originally proposed by Watts [@Watts2002Simple], and secondly an extended, dynamical threshold model devised by us [@Ruan2015; @Karsai2016Local], where both multiple innovators and non-adopters have a role in social contagion.
The Watts model {#sec:wmodel}
---------------
Under the complex contagion hypothesis by Granovetter, Centola and others [@Granovetter1978Threshold; @Centola2007Complex], social contagion may be modelled as a binary-state process evolving in a network and driven by a threshold mechanism. In this framework individuals are represented by agents or network nodes, each in either a susceptible (0) or adopter (1) state, while the influence by an agent is achieved by transferring information via social ties. Nodes are connected in a network with degree distribution $P(k)$ and average degree $z = \langle k \rangle$. Moreover, each node has an individual threshold $\varphi \in [0, 1]$ drawn from a distribution $P(\varphi)$ with average $w = \langle \varphi \rangle$. The threshold $\varphi$ determines the minimum fraction of exposed neighbours that triggers adoption, capturing the resistance of an individual against engaging in a given behaviour. Hence, in case a node has $m$ adopting neighbours and $m \geq k \varphi$ (the so-called [*threshold rule*]{}), it switches state from $0$ to $1$ and remains so until the end of the dynamics. In his seminal paper about threshold dynamics [@Watts2002Simple], Watts classified nodes into three categories based on their threshold and degree: He first identified [*innovator*]{} nodes that spontaneously change state to $1$ and therefore start the spreading process. Such nodes have a trivial threshold $\varphi=0$. Then there are nodes with threshold $0 < \varphi \leq 1/k$, called [*vulnerable*]{}, which need one adopting neighbour before their own adoption. Finally, there are more resilient nodes with threshold $\varphi>1/k$, known as [*stable*]{}, representing individuals in need of strong social influence to follow the actions of their acquaintances.
In the WT model [@Watts2002Simple], small perturbations (like the spontaneous adoption of a single seed node) can trigger network-wide cascading patterns. However, their emergence is subject to the following [*cascade condition*]{}: the innovator seed has to be linked to a percolating vulnerable cluster, which adopts immediately afterwards and further triggers a [*global*]{} cascade (i.e. a set of adopters larger than a fixed fraction of a finite network, or a nonzero fraction of adopters in an infinite network). The cascade condition is satisfied if the network is inside a bounded regime in $(w, z)$-space [@Watts2002Simple]. When considering a vanishingly small innovator seed and a configuration-model network [@Newman2010Networks] \[i.e., by ignoring structural correlations in the social network and characterising it solely by its degree distribution $P(k)$\], a generating function approach allows us to write the cascade condition as $$\label{eq:wattsCond}
\sum_k \frac{k}{z} (k - 1) P(k) f(k, 1) > 1,$$ where $f(k, 1) = C(1/k)$ is the probability that a randomly-selected node with degree $k$ is vulnerable, and $C$ is the cumulative distribution function of $P(\varphi)$. More generally, $f(k, m)$ (for $m = 0,\ldots,k$) is also known as a response function of the monotone binary dynamics defining the WT model [@porter2016; @Ruan2015].
As Eq.(\[eq:wattsCond\]) shows, the cascade regime depends on degree and threshold heterogeneities [@Watts2002Simple] and may change its shape if several innovators start the process [@Singh2013Thresholdlimited]. In addition, while models with more sophisticated functional forms of social influence may be introduced [@Latane1981The; @Dodds2004Universal], the original assumption proposed by Watts and Granovetter seems to be sufficient to interpret our observations.
Dynamical threshold model with immune nodes {#sec:dmodel}
-------------------------------------------
Our modelling framework is an extension to the WT model and similar threshold dynamics on networks, studied by Watts, Gleeson, Singh, and others, where all the nodes are initially susceptible and innovators are only introduced as an initial seed of arbitrary size [@Watts2002Simple; @Karampourniotis2015The; @Gleeson2007Seed; @Singh2013Thresholdlimited]. Apart from the above discussed threshold rule and motivated by the empirical observations in the spread of online services within Skype, our model considers two additional features, namely that (i) a fraction $r$ of ‘immune’ nodes never adopts, indicating a lack of interest in the online service, and that (ii) due to external influence, susceptible nodes adopt the service spontaneously (i.e. become innovators) throughout the time with constant rate $\pn$, rather than only at the beginning of the dynamics. In this way, the dynamical evolution of the system is completely determined by the online social network, the distribution $P(\varphi)$ of thresholds, and the parameters $r$ and $\pn$ (Fig. \[Fig:4\]). For the sake of simplicity, we consider a configuration-model network and statistical independence between degrees and thresholds [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high]. We remark that the somewhat similar concepts of ‘stubborn nodes’, mimicking individuals’ resistance against adoption [@Brummitt2012Multiplexity; @Lee2014Threshold], and ‘global nodes’, capturing adoption driven by external effects [@Kobayashi2015Trend], have also been considered in threshold models and show a rich variety of effects on cascading behaviour.
![[**Immune individuals in social contagion.**]{} Numerical simulation of our dynamical threshold model in an empirical network, with a single adoption threshold $\varphi = 0.2$ for all the nodes, rate of spontaneous adopters $\pn = 0.0005$, and fraction of immune nodes $r = 0.1$. The network is an ego sample of Facebook friendships with size $N = 96$ and average degree $z = 10.63$ [@Leskovec2012Learning]. The network shows how susceptible nodes adopt spontaneously with rate $\pn$, or after a fraction $\varphi$ of their neighbours has adopted, while immune nodes never adopt.[]{data-label="Fig:4"}](Fig4.pdf){width="\linewidth"}
Our threshold model [@Ruan2015; @Karsai2016Local] can be studied analytically by extending the framework of approximate master equations (AMEs) for monotone binary-state dynamics, as recently developed by Gleeson [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high], where the transition rate between susceptible and adoption states only depends on the number $m$ of network neighbours that have already adopted. We describe a node by the property vector $\kvec = (k, c)$, where $k = k_0, k_1, \ldots k_{M-1}$ is its degree and $c = 0, 1, \ldots, M$ its type, i.e. $c = 0$ is the type of the fraction $r$ of immune nodes, while $c \neq 0$ is the type of all non-immune nodes that have threshold $\varphi_c$. In this way, $P(\varphi)$ is substituted by the discrete distribution of types $P(c)$ (for $c > 0$). The integer $M$ is the maximum number of degrees (or non-zero types) considered in the AME framework, which can be increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical computation.
We characterise the static social network by the extended distribution $\Pk$, where $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$. Non-immune and susceptible nodes with property vector $\kvec$ adopt spontaneously with a constant rate $\pn$, otherwise they adopt only if a fraction $\varphi_c$ of their $k$ neighbours has adopted before. These rules are condensed into the probability $\Fkm dt$ that a node will adopt within a small time interval $dt$, given that $m$ of its neighbours are already adopters, $$\label{eq:thresRule}
\Fkm =
\begin{cases}
\pr & \text{if} \quad m < k \varphi_c \\
1 & \text{if} \quad m \geq k \varphi_c
\end{cases}, \quad \forall m \; \text{and} \; k, c \neq 0,$$ with $F_{(k,0),m} = 0$ $\forall k, m$ and $F_{(0,c),0} = \pr$ $\forall c \neq 0$ (for immune and isolated nodes, respectively). The rescaled rate $\pr = \pn / (1 - r)$ (with $\pr = 1$ for $\pn > 1 - r$) is necessary if we wish to obtain a rate $\pn$ of innovators for early times of the dynamics, regardless of the value of $r$.
The dynamics of adoption is well described by an AME for the fraction $\skm(t)$ of $\kvec$-nodes that are susceptible at time $t$ and have $m=0,\ldots,k$ adopting neighbours [@porter2016; @gleeson2013binary; @gleeson2011high], $$\label{eq:AMEsThres}
\dskm = -\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo,$$ where $$\label{eq:rateBs}
\bs(t) = \frac{\sumk \Pk \summ (k - m) \Fkm \skm(t)}{\sumk \Pk \summ (k - m) \skm(t)},$$ and the sum is over all the degrees and types, i.e. $\sumk \bullet = \sum_k \sum_c \bullet$. To reduce the dimensionality of Eq. (\[eq:AMEsThres\]), we consider the ansatz $$\label{eq:AMEansatz}
\skm(t) = \Bkm [\nu(t)] e^{-\pr t}
\quad \text{for} \; m < k\varphi_c \; \text{and} \; c \neq 0,$$ with $\nu(t)$ the probability that a randomly-chosen neighbour of a susceptible node is an adopter.
Introducing the ansatz of Eq. (\[eq:AMEansatz\]) into the AME system of Eq. (\[eq:AMEsThres\]) leads to the condition $\dot{\nu} = \bs (1 - \nu)$. With some algebra, the AMEs for our dynamical threshold model are reduced to the pair of ordinary differential equations
\[eq:reducedAMEs\] $$\begin{aligned}
\dot{\rho} &= h(\nu, t) - \rho, \\
\dot{\nu} &= g(\nu, t) - \nu,\end{aligned}$$
where $\rho(t) = 1 - \sumk \Pk \summ \skm(t)$ is the fraction of adopters in the network, and the initial conditions are $\rho(0) = \nu(0) = 0$. Here, $$\label{eq:hTerm}
h = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} P(k) P(c) \sum_{m \geq k\varphi_c} \Bkm(\nu) \Big],$$ and $$\label{eq:gTerm}
g = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} \frac{k}{z} P(k) P(c) \sum_{m \geq k\varphi_c} \Bkom(\nu) \Big],$$ where $\ft = 1 - (1 - \pr) e^{-\pr t}$, and $\Bkm(\nu) = \binom{k}{m} \nu^m (1 - \nu)^{k - m}$ is the binomial distribution. The fraction of adopters $\rho$ is then obtained by solving Eq. (\[eq:reducedAMEs\]) numerically. Since the susceptible nodes adopt spontaneously with rate $p_n$, the fraction of innovators $\rho_0(t)$ in the network is given by $$\label{eq:innovFrac}
\rho_0(t) = \pr \int_0^t [1 - r - \rho(\tau)] d\tau.$$
![[**A dynamical threshold model for the adoption of online services.**]{} [**(a-b)**]{} Surface plot of the normalised fraction of adopters $\rho / (1 - r)$ in $(w, z)$-space, for $r = 0.73$ and $t = 89$. Contour lines signal the parameter values for which $20\%$ of non-immune nodes have adopted, for fixed $r$ and varying time (a), and for fixed time and varying $r$ (b). The continuous contour line and dot indicate parameter values of the last observation of Skype s3. A regime of maximal adoption ($\rho \approx 1 - r$) grows as time goes by, and shrinks for larger $r$. [**(c)**]{} Time series of the fraction of adopters $\rho$ for fixed $p_n = 0.00019$ and varying $r$ (main), and for fixed $r = 0$ and varying $p_n$ (inset). These curves are well approximated by the solution of Eq. (\[eq:reducedAMEs\]) for $k_0 = 3$, $k_{M-1} = 150$ and $M = 25$ (dashed lines). The dynamics is clearly faster for larger $p_n$ values. As $r$ increases, the system enters a regime where the dynamics is slowed down and adopters are mostly innovators. [**(d)**]{} Final fraction of innovators $\rho_{0,\infty}$ and the time $t_c$ when $50\%$ of non-immune nodes have adopted as a function of $r$, both simulated and theoretical. The crossover to a regime of slow adoption is characterised by a maximal fraction of innovators and time $t_c$. Unless otherwise stated, $p_n=0.00019$ and we use $N=10^4$, $\mu_D=1.09$, $\sigma_D=1.39$, $\kmin=1$, $\mu_T=-2$, and $\sigma_T=1$ to obtain $z = 8.56$ and $w = 0.19$ as in Skype s3. The difference in $\mu_D$ between data and model is due to finite-size effects. Numerical results are averaged over $10^2$ (a-b) and $10^3$ (c-d) realisations. \[fig:5\]](Fig5.pdf){width="\textwidth"}
We may also implement our dynamical threshold model numerically via a Monte Carlo simulation in a network of size $N$, with a log-normal degree distribution and a log-normal threshold distribution as observed empirically in the case of Skype. Hence we can explore the behaviour of $\rho$ and $\rho_0$ as a function of $z$, $w$, $p_n$ and $r$, both in the numerical simulation and in the theoretical approximation given by Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]). For $p_n > 0$ some nodes adopt spontaneously as time passes by, leading to a frozen state characterised by the final fraction adopters $\rho(\infty) = 1 - r$. However, the time needed to reach such a state depends heavily on the distribution of degrees and thresholds, as indicated by a region of large adoption ($\rho \approx 1 - r$) that grows in $(w, z)$-space with time (contour lines in Fig. \[fig:5\]a). If we fix the time in the dynamics and vary the fraction of immune nodes instead, this region shrinks as $r$ increases (contour lines in Fig. \[fig:5\]b). In other words, the set of networks (defined by their average degree and threshold) that allow the spread of adoption is larger at later times in the dynamics, or when the fraction of immune nodes is small. When both $t$ and $r$ are fixed, the normalised fraction of adopters $\rho / (1 - r)$ gradually decreases for less connected networks with larger thresholds (surface plot in Fig. \[fig:5\]a and b).
Both numerical simulations and analytical approximations show how the dynamics of spreading changes by introducing immune individuals in the social network. For $r \approx 0$, the adoption cascade appears sooner for larger $\pn$, since this parameter regulates how quickly we reach the critical fraction of innovators necessary to trigger a cascade of fast adoption throughout all susceptible nodes (Fig. \[fig:5\]c, inset). Yet as we increase $r$ above a critical value $r_c$ (and thus introduce random quenching), the system enters a regime where rapid cascades disappear and adoption is slowed down, since stable nodes have more immune neighbours and it is difficult to fulfil their threshold condition. The crossover between these fast and slow regimes is gradual, as seen in the shape of $\rho$ for increasing $r$ (Fig. \[fig:5\]c, main panel). We may identify $r_c$ in various ways: by the maximum in both the final fraction of innovators $\rho_{0,\infty} = \rho_0(\infty)$ and the critical time $t_c$ when $\rho = (1-r)/2$ (Fig. \[fig:5\]d), or as the $r$ value where the inflection point in $\rho$ disappears. These measures indicate $r_c \approx 0.8$ for parameter values calibrated with Skype data. All global properties of the dynamics (like the functional dependence of $\rho$ and $\rho_0$) are very well approximated by the solution of Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]) (dashed lines in Fig. \[fig:5\]c and d). Indeed, the AME framework is able to capture the shape of the $\rho$ time series, the crossover between regimes of fast and slow adoption, as well as the maximum in $\rho_{0,\infty}$ and $t_c$.
In the simplified case of an Erdős-Rényi random graph as the underlying social network, the crossover between fast and slow regimes of spreading may also be characterised by a percolation-type transition in the asymptotic limit ($t \to \infty$) of the size distribution $P(s)$ of [*induced*]{} adoption clusters, i.e. connected components of adopters disregarding innovators [@Ruan2015]. For early times $P(s)$ includes small induced clusters only, which in turn indicates that a larger fraction of spontaneous adopters is crucial for global spreading in the absence of a percolating vulnerable component. However, for late times the behaviour of $P(s)$ differs between regimes: in the regime of fast spreading the distribution becomes bimodal due to the appearance of a global cluster of induced adopters, while in the slow regime it remains unimodal until the end of dynamics.
Finally, in the extreme case of $\pn = 0$ (corresponding to the WT model with immune nodes), the reduced AME system of Eq. (\[eq:reducedAMEs\]) can be used to derive a cascade condition and thus give insight into the dynamics of spreading in the presence of immune individuals [@porter2016; @Ruan2015]. Eq. (\[eq:reducedAMEs\]) has an equilibrium point for the initial condition $(\rho(0), \nu(0)) = (0, 0)$. If this equilibrium point is linearly unstable, the perturbation of a single innovator seed may move the dynamical system away from equilibrium and create a global cascade. A linear stability analysis shows that this condition is equivalent to $$\label{eq:AMEsCond}
(1 - r) \sum_k \frac{k}{z} (k - 1) P(k) f(k, 1) > 1,$$ where $f(k, 1) = C(1/k)$ implements the response of a non-immune node of degree $k$ to one adopting neighbour, and $C$ is the cumulative distribution function of $P(\varphi)$ (for non-immune nodes with $c > 0$). When $r = 0$, Eq. (\[eq:AMEsCond\]) reduces trivially to the cascade condition of the original WT model in Eq. (\[eq:wattsCond\]). This shows that the shape of the cascade regime can be obtained either by using generating functions in percolation theory, or by performing a stability analysis of the AMEs.
Validation {#sec:val}
==========
As demonstrated above, our model provides insight on the role of innovators and immune nodes in controlling the speed of the adoption process. However, in empirical datasets information about the fraction of non-adopters is usually not available, which makes it difficult to predict the future dynamics of service adoption. Here we use our modelling framework to perform data-driven simulations with parameters determined from Skype for two reasons: (a) to estimate the fraction $r$ of immune nodes in the real system; and (b) to validate our modelling as compared to real data.
To set up our data-driven simulations we use the Skype data to directly determine all model parameters, apart from the fraction $r$ of immune nodes. As we already discussed, the best approximation of the degree distribution of the real network is a log-normal function (Eq. \[eq:Pk\]) with parameters $\mu_D=1.2$, $\sigma_D=1.39$, minimum degree $\kmin = 1$ and average degree $z = 8.56$. To account for finite-size effects in the model results for low $N$, we decrease $\mu_D$ slightly to obtain the same value of $z$ as in the real network. We also observe in Fig. \[fig:1\]b that the threshold distribution of each degree group collapses into a master curve after normalisation by using the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$. This master curve can be well-approximated by the log-normal distribution shown in Eq. \[eq:Pphi\], with parameters $\mu_T=-2$ and $\sigma_T=1$ as determined by the empirical average threshold $w = 0.19$ and standard deviation $0.233$. We estimate a rate of innovators $p_n = 0.00019$ by fitting a constant function to $R_i(t)$ for $t > 2\tau$ (Fig. \[fig:2\]a). The fit to $\pn$ also matches the time-scale of a Monte Carlo iteration in the model to 1 month. To model the observed dynamics and explore the effect of immune nodes, we use a configuration-model network [@Newman2010Networks] with log-normal degree and threshold distributions and $p_n$ as the constant rate of innovators, all determined from the empirical data. Model results in Fig. \[fig:6\] (and Fig. \[fig:7\]) are averaged over $100$ networks of size $N=10^5$ ($10^6$) after $T=89$ iterations, matching the length of the observation period in Skype.
![**Empirical cluster statistics and simulation results.** Average size of the largest ($LC$, upper panel) and the 2nd largest ($LC^{2nd}$, lower panel) components of the model network (‘Net’, squares), adoption network (‘Casc’, circles), stable network (‘Stab’, diamonds), and induced vulnerable trees (‘Vuln’, triangles) as a function of the fraction $r$ of immune nodes. Dashed lines show the observed relative size of the real $LC$ of the adopter network in $2011$ (Fig.\[fig:2\]c) and the predicted $r^{emp}$ value. Dotted lines on the lower panel indicate the critical percolation points for the full ($r_c^{casc}$) and stable ($r_c^{stab}$) adoption networks. \[fig:6\]](Fig6.pdf){width="80.00000%"}
As a function of $r$, the underlying and adoption networks pass through three percolation-type phase transitions. First, the appearance of immune nodes (for increasing $r$) can be considered as a removal process of nodes available for adoption from the underlying network structure. After the appearance of a critical fraction of immune nodes, $r_c^{net}$, the effective network structure available for adoption will be fragmented and will consist of small components only, limiting the size of the largest adoption cluster possible. Second, $r$ also controls the size of the emergent adoption cascades evolving on top of the network structure. While for small $r$ the adoption network is connected into a large component, for larger $r$ cascades cannot evolve since there are not enough nodes to fulfill the threshold condition of susceptible stable nodes, even if the underlying network is still connected. The transition point between these two phases of the adoption network is located at $r_c^{casc}\leq r_c^{net}$, limited from above by the critical point $r_c^{net}$. Finally, we observe from the empirical data and model results that the adoption network is held together by a large connected component of stable nodes. Consequently, for increasing $r$ the stable adoption network goes through a percolation transition as well, with a critical point $r_c^{stab}\leq r_c^{casc}\leq r_c^{net}$.
To characterise these percolation phase transitions we compute the average size of the largest ($LC$) and second largest ($LC^{2nd}$) connected components (Fig. \[fig:6\]). We measure these quantities for the underlying network, and for the stable, vulnerable and global adoption networks, as a function of the fraction of immune nodes $r$. After $T=89$ iterations (matching the length of the real observation period), we identify three regimes of the dynamics: if $0<r<0.6$ (dark-shaded area) the spreading process is very rapid and evolves as a global cascade, which reaches most of the nodes of the shrinking susceptible network in a few iteration steps. About $10\%$ of adopters are connected in a percolating stable cluster, while vulnerable components remain very small in accordance with empirical observations. In the crossover regime $0.6<r<0.8$ (light-shaded area), the adoption process slows down considerably (Fig. \[fig:6\], upper panel), as stable adoptions become less likely due to the quenching effect of immune nodes. The adoption process becomes the slowest at $r_c^{stab}=0.8$ when the percolating stable cluster falls apart, as demonstrated by a peak in the corresponding $LC^{2nd}$ curve in Fig. \[fig:6\] (diamonds in lower panel). Finally, around $r_c^{casc}=0.9$ the adoption network becomes fragmented and no global cascade takes place. Since the underlying network has a broad degree distribution, it is robust against random node removal processes¬[@Newman2010Networks]. That is why its critical percolation point $r_c^{net}$ appears after $95\%$ or more nodes are immune. Note that similar calculations for another service have been presented before [@Karsai2016Local] with qualitatively the same results, but with the crossover regime shifted towards larger $r$ due to different parameter values of the model process.
![**Additional empirical cluster statistics and simulation results.** **(a)** Distribution $P(d)$ of depths of induced vulnerable trees in both data and model for several $r$ values, showing a good fit with the data for $r=0.73$. The difference in the tail is due to finite-size effects. **(b)** Correlation $\langle s_v \rangle (k)$ between innovator degree and average size of vulnerable trees in both data and model with the same $r$ values as in (a). Model calculations correspond to networks of size $N=10^6$ and are averaged over $10^2$ realisations. \[fig:7\]](Fig7.pdf){width="\textwidth"}
We can use these calculations to estimate the only unknown parameter, namely the fraction $r$ of immune nodes in Skype, by matching the relative size of the largest component ($LC_{Net}$) between real and model adoption networks at time $T$. Empirically, this value is the relative size $s_a^{LC}\simeq 0.043$ corresponding to the last point on the right-hand side of the distribution for $2011$ in Fig. \[fig:2\]c (main panel). Matching this relative size with the simulation results (see the observation line in Fig. \[fig:6\] upper panel), we find that it corresponds to $r^{emp} = 0.73$ (prediction line in Fig. \[fig:6\]), suggesting that the real adoption process lies in the crossover regime. In other words, large adoption cascades could potentially evolve in Skype but with reduced speed, as $73\%$ of users might not be interested in adopting a service within the network.
To test the validity of the predicted $r^{emp}$ value we perform three different calculations. First we measure the maximum relative growth rate of cumulative adoptions and find a good match between model and data (see Skype s3 and Model Skype s3 in Fig. \[fig:0\]). In other words, the model correctly estimates the speed of the adoption process. Second, we measure the distribution $P(d)$ of the depths of induced vulnerable trees (Fig. \[fig:7\]a). Vulnerable trees evolve with a shallow structure in the empirical and model processes. After measuring the distribution $P(d)$ for various $r$ values below, above and at $r^{emp}$, we find that the distribution corresponding to the predicted $r^{emp}$ value fits the best with the empirical data. Finally, in order to verify earlier theoretical suggestions [@Singh2013Thresholdlimited], we look at the correlation $\langle s_v \rangle (k)$ between the degree of innovators and the average size of vulnerable trees induced by them (Fig. \[fig:7\]b). Similar to the distribution $P(d)$, we perform this measurement on the real data and in the model for $r=0.6$ and $0.9$, as well as for the predicted value $r_{emp}=0.73$. We find a strong positive correlation in the data, explained partially by degree heterogeneities in the underlying social network, but surprisingly well emulated by the model as well. More importantly, although this quantity appears to scale with $r$, the estimated $r$ value fits the empirical data remarkably well, thus validating our estimation method for $r$ based on a matching of relative component sizes.
Conclusion and future directions {#sec:concl}
================================
The analysis and modelling of the diffusion of services and innovations is a long-standing scientific challenge, with recent developments built on large digital datasets registering adoption processes in a society with a large population. Due to these advancements we are currently at the position to simultaneously observe various types of adoption processes and the underlying social structure. Individual-level observations of social and adoption behaviour are crucial in identifying the mechanisms that fuel collective patterns of rapid or slow adoption cascades. In this chapter, using one of the first datasets of this kind, we observe the worldwide spread of an online service in the techno-social communication network of Skype. First we provide novel empirical evidence about heterogeneous adoption thresholds and non-linear dynamics of the adoption process. We have also identified two additional components necessary to introduce into the modelling of product adoption, namely (a) a constant flow of innovators, which may induce rapid adoption cascades even if the system is initially out of the cascading regime, and (b) a fraction of immune nodes that forces the system into a quenched state where adoption slows down. These features are responsible for a critical structure of empirical adoption components that radically differs from previous theoretical expectations. We incorporate these mechanisms into a threshold model that, despite containing several simplifying assumptions, successfully recovers and predicts real-world adoption scenarios such as the spreading of Skype services.
Our aim in this chapter has been to provide empirical observations as well as methods and tools to model the dynamics of social contagion phenomena, with the hope that it will foster thoughts for future research. One possible direction is the observation of the reported structure and evolution of the global adoption cluster in other systems similar to the ones studied in [@BorgeHolthoefer2011Structural; @Dow2013Anatomy; @Gruhl2004Information; @Goel2012Structure; @BorgeHolthoefer2013Cascading; @Bakshy11]. Other promising directions are the consideration of structural homophilic or assortative correlations, the evolving nature of the underpinning social network with timely created and dissolved social ties (as studied in [@Karsai2014Complex]), and the effects of interpersonal influence or leader-follower mechanisms on the social contagion process. We hope that our results provide a direction for data-driven modelling of these phenomena, and serve as a scholarly example in future studies of the dynamics of service adoption processes.
The results presented in this chapter are adapted from [@Ruan2015; @Karsai2016Local] and were obtained in collaboration with Riivo Kikas. The authors gratefully acknowledge the support of M. Dumas, A. Saabas, and A. Dumitras from STACC and Microsoft/Skype Labs. GI acknowledges a Visiting Fellowship from the Aalto Science Institute. JK and ZR were supported by FP7 317532 Multiplex and JK by H2020 FETPROACT-GSS CIMPLEX 641191. KK is supported by the Academy of Finland’s project COSDYN project, No. 276439 and EU HORIZON 2020 FET Open RIA IBSEN project No. 662725.
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abstract: 'Music genre classification is a widely researched topic in music information retrieval (MIR). Being able to automatically tag genres will benefit music streaming service providers such as JOOX, Apple Music, and Spotify for their content-based recommendation. However, most studies on music classification have been done on western songs which differ from Thai songs. Lukthung, a distinctive and long-established type of Thai music, is one of the most popular music genres in Thailand and has a specific group of listeners. In this paper, we develop neural networks to classify such Lukthung genre from others using both lyrics and audios. Words used in Lukthung songs are particularly poetical, and their musical styles are uniquely composed of traditional Thai instruments. We leverage these two main characteristics by building a lyrics model based on bag-of-words (BoW), and an audio model using a convolutional neural network (CNN) architecture. We then aggregate the intermediate features learned from both models to build a final classifier. Our results show that the proposed three models outperform all of the standard classifiers where the combined model yields the best $F_1$ score of 0.86, allowing Lukthung classification to be applicable to personalized recommendation for Thai audience.'
author:
-
bibliography:
- 'lukthungbib.bib'
title: Lukthung Classification Using Neural Networks on Lyrics and Audios
---
at (current page.south) ;
Introduction
============
Lukthung, a unique type of music genre, originated from rural communities in Thailand. It is one of the most prominent genres and has a large listener base from farmers and urban working-class people [@c13]. Lyrically, the songs contain a wide range of themes, yet often based on Thai country life: rural poverty, romantic love, aesthetic appeal of pastoral scenery, religious belief, traditional culture, and political crisis [@c11]. The vocal parts are usually sung with unique country accents and ubiquitous use of vibrato and are harmonized with western instruments (e.g. brass and electronic devices), as well as traditional Thai instruments such as Khene (mouth organ) and Phin (lute).
It is normal to see on many public music streaming platforms such as Youtube that many non-Lukthung playlists contain very few or even none of Lukthung tracks, compared to other genres e.g. Pop, Rock, R&B which are usually mixed together in the same playlists. This implies that only a small proportion of users listen to both Lukthung and other genres, and non-Lukthung listeners rarely listen to Lukthung songs at all. Therefore, for the purpose of personalized music recommendation in the Thai music industry, identifying Lukthung songs in hundreds of thousands of songs can reduce the chance of mistakenly recommending them to non-Lukthung listeners.
Many musical genre classification methods rely heavily on audio files, either raw waveforms or frequency spectrograms, as predictors. Previously, traditional approaches focused on hand-crafted feature extraction to be input to classifiers [@c3; @c18], while more recent network-based approaches view spectrograms as 2-dimensional images for musical genre classification [@c10]. Lyrics-based music classification is less studied, as it has generally been less successful [@c12]. Early approaches designed lyrics features that were comprised of vocabulary, styles, semantics, orientations and, structures for SVM classifiers [@c14]. Since modern natural language processing techniques have switched to recurrent neural networks (RNNs), lyrics-based genre classification can also employ similar architectures. On the other hand, as Lukthung songs often contain dialects and unconventional vocabulary, we will show later that a simpler bag-of-words (BoW) model on our relatively much smaller dataset can also achieve satisfying results.
Recently, both lyrics and audios have been incorporated into classifiers together [@c16; @c17]. The assumption is that lyrics contain inherent semantics that cannot be captured by audios. Therefore, both features provide information that complements each other.
Although both audio and lyrics features have been used in musical genre classification, to the best of our knowledge, none of the previous works have been done on Lukthung classification. In principle, many existing methods can be adopted directly. However, classifying Lukthung from other genres is challenging in practice. This is because machine learning models, especially deep neural networks, are known to be specific to datasets on which they are trained. Moreover, Lukthung songs, while uniquely special, are sometimes close to several other Thai traditional genres. Proper architecture design and optimal parameter tuning are still required.
In this paper, we build a system that harnesses audio characteristics and lyrics to automatically and effectively classify Lukthung songs from other genres. Since audio and lyrics attributes are intrinsically different, we train two separate models solely from lyrics and audios. Nevertheless, these two features are complementary to a certain degree. Hence, we build a final classifier by aggregating the intermediate features learned from both individual models. Our results show that the proposed models outperform the traditional methods where the combined model that make use of both lyrics and audios gives the best classification accuracy.
Related work {#Sec:literature}
============
For automatic music tagging tasks including genre classification, the conventional machine learning approaches involve creating hand-crafted features and using them as the input of a classifier. Rather than the raw waveforms, early classification models were frequently trained on Mel-frequency cepstrum coefficients (MFCCs) extracted from audio waves for computational efficiency [@c1; @c2]. Other hand-designed features were introduced in addition to MFCCs for music tagging [@c3]. Like MFCCs, these audio features such as MFCC derivatives and spectral features typically represent the physical or perceived aspects of sounds. Since these derived features are frame-dependent attributes, their statistics such as means and variances are computed across multiple time frames to generate a feature vector per an audio clip. However, the main difficulty with hand-crafted features is that task-relevant features are challenging to be designed. Many feature selection techniques have been introduced to tackle this problem [@c5], but the right solution is yet to emerge.
Recently, deep learning approaches have been widely explored to combine feature extraction and modeling, allowing relevant features to be learned automatically. Following the success in speech recognition [@c6], deep neural networks (DNNs) have been currently used for audio data analysis [@c7; @c8; @c9]. The hidden layers in DNNs can be interpreted as representative features underlying the audio. Without requiring hand-crafted features from the audio spectrograms, a neural network can automatically learn high-level representations while classification being trained. Instead, the only requirement is to determine the network architecture e.g. the number of nodes in hidden layers, the number of filters (kernels) and the filter size in convolution layers such that meaningful features can be captured.
In [@c10], several filters different in sizes were explicitly designed to capture important patterns in two-dimensional spectrograms derived from audio files. Unlike the filters used in standard image classification, they introduced vertical filters lying along the frequency axis to capture pitch-invariant timbral features simultaneously with long horizontal filters to learn such time-invariant attributes as tempos and rhythms. The underlying intuition is that the musical features of interest residing in a 2D spectrogram are not spatially invariant along the frequency and time axes. Accordingly, the common symmetric filters are not effective for feature extraction on spectrograms.
Lyrics-based models for music genre classification are similar to those used in text classification. [@c14] categorized lyrics features into five main classes: vocabulary, style, semantics, orientation and song structure, and used them for music genre classification. With the recent success in recurrent neural networks (RNNs) in natural language processing, a modern approach used Hierarchical attention networks (HAN) for music genre classification [@c15]. This advance in deep learning methods allows us to encapsulate both meanings and the structure of lyrics. However, this model relied on a large lyrics corpus to train word embedding, which is not practical for small datasets.
Dataset
=======
We collected 10,547 Thai songs dated from the year 1985 to 2019 where both lyrics and audios were available. The genre of each song, along with other tags related to mood, tempo, musical instruments, to name a few, was tagged by music experts. Genre annotations are Pop, Rock, Lukthung (Thai rural style), Lukkrung (Thai urban style in the 30s to 60s eras), Hip Hop, and Phuea Chiwit (translated as songs for life) which we will refer as Life for the rest of this paper. Here, we are interested in distinguishing Lukthung songs from the others. Fig. \[fig:genre\_year\_plot\] shows the number of songs labeled in each genre and era. We found that Lukthung songs occupy approximately 20% of the entire dataset. There are few songs before 2000s. The genre Others in Fig. \[fig:genre\_year\_plot\] refers to less popular genres such as Reggae and Ska. Since the era also affects the song style, the model performance may vary. We will also discuss the results based on different eras in Section \[Sec:experiments\].
![Number of instances in each genre and era classes[]{data-label="fig:genre_year_plot"}](images/genre_year.png){width="\linewidth"}
Data preprocessing {#Sec:preprocessing}
==================
Our goal is to build machine learning models from both lyrics and audio inputs. We describe the preprocessing steps carried out in our data preparation for each data type below.
{width="\linewidth"}
Lyrics input {#Sec:lyricfeature}
------------
We constructed word-based features using the entire lyrics from the beginning to the end of the songs. The lyrics were firstly tokenized. We tried several Thai word tokenizers and chose the `deepcut` library [^1] due to its best performance on this task. It is noted that the artists’ names were not included, allowing the model to perform classification on songs from unknown or never-seen-before artists. An example of tokenized lyrics is shown in Fig. \[fig:overallarch\] (top-left), which is passed through the lyrics model described in Section \[Sec:lyricmodel\].
Audio input {#Sec:audiofeature}
-----------
For each song, we excerpted a 10-second clip from an audio file in its chorus part. We used the chorus part solely not only for computational reasons but also because it usually contains all musical instruments present in the song, hence containing the richest information. We approximated that the chorus part came after 30% of the total song duration from the start. Finally, we extracted the Mel spectrograms from the excerpted audio clips with the following specifications:
- Sampling rate: 22050 Hz i.e. frame size = 4.53e-5 s
- Frequency range: 300-8000 Hz
- Number of Mel bins: 128
- Window of length (n\_fft): 2048
- Time advance between frames (hop size): 512
![Mel spectrograms (shown only half of the input time length) along with vertical and horizontal filters[]{data-label="fig:lukthung_pop_filters"}](images/lukthung_pop_filters.jpg){width="\linewidth"}
Examples of extracted spectrograms are demonstrated in Fig. \[fig:overallarch\] (center-left) and Fig. \[fig:lukthung\_pop\_filters\]. We may see from Fig. \[fig:lukthung\_pop\_filters\] that audio clips from different genres have different characteristics. We will discuss the audio model that captures such features in Section \[Sec:audiomodel\].
Proposed models {#Sec:model}
===============
We propose separate models for the lyrics and audio features. We also propose the combined model that aggregates the intermediate features from both models.
Lyrics Model: BoW-MLP {#Sec:lyricmodel}
---------------------
Since most Lukthung songs have a different set of vocabulary compared to other genres, we propose a simple bag-of-words (BoW) model. The vocabulary $\mathcal{V}$ is constructed using a larger set of unlabeled and labeled lyrics, from roughly 85k songs. We filter out words that are longer than 20 characters and words that appear less than 10 times in the entire lyrics corpus. The lyrics of each song is represented by a normalized bag-of-words vector $\bm{a}_i$ using the vocabulary $\mathcal{V}$. Let $c_{i,j}$ denote the number of occurrences of word $j$ in the lyrics of song $i$. The normalized count of this word, $a_{i,j}$, is computed as in .
$$a_{i,j} = \frac{\log c_{i,j}}{max \log \bm{c}_{i}}
\label{eq:normalized_count}$$
The logarithm transformation is applied to smooth the discrete count values. We normalize by the maximum word count within the lyrics of each song because it preserves the sparsity of the BoW features while scaling each feature value to between 0 and 1, inclusive. The BoW is then fed into a two-layer fully connected multi-layer perceptron (MLP). The input layer is comprised of $|\mathcal{V}|$ nodes where $|\mathcal{V}|$ is the vocabulary size, followed by 100 hidden nodes in each intermediate layer before connecting to a single neuron in the output layer. We put rectified linear unit (ReLU) activation functions on the hidden nodes to allow the model to learn non-linear mapping, and place a sigmoid activation function on the output node to obtain the probability whether a given song is Lukthung. The graphical model architecture is depicted in the Fig.\[fig:overallarch\] (top).
Audio Model: Spectro-CNN {#Sec:audiomodel}
------------------------
We develop the same CNN-based model as stated in [@c10] with some architectural modifications to be suited in our data set as elaborated below. The overview of the model is illustrated in Fig. \[fig:overallarch\] (bottom).
The model is aimed to automatically learn timbral and temporal properties for the genre classification task. Overall, the model starts with the input layer. These inputs are passed through an inception network to extract features, followed by a residual network. Then, the output from the residual network is fed to a fully connected layers to predict a binary output.
For each audio file, we construct a 2-dimensional spectrogram spanning across 431 time frames with 128 Mel frequency bins, as described earlier in Section \[Sec:audiofeature\]. The spectrogram inputs are then put forward through the feature extraction layers where both timbral and temporal features are extracted in parallel using a set of convolutional filters in different dimensions.
Along the frequency axis on the spectrograms, the timbral elements are represented by a stack of bright lines with different degrees of intensity, which involve perceived vocal and tone-related sound. In addition to fundamental frequencies, Lukthung songs often contain Thai traditional musical instruments and vocals that produce overtone partials (either multiples of the fundamental frequencies or any higher constituents of frequencies). To detect such timbral features located along the frequency axis of the spectrograms, we apply vertical filters with variable heights of mel bins across short time intervals, where the taller vertical filters aim to capture the higher overtones. Following the parallel convolution layer, a max pooling is performed across the frequency axis. Examples of vertical filters are highlighted on the left part of Fig \[fig:lukthung\_pop\_filters\].
In parallel with the timbral feature extraction, we place horizontal blocks of filters to learn the temporal elements including tempos and rhythms which can be detected by a drastic change in energy along the time axis. We handle songs with different tempos and rhythms using variable lengths of filters along the time axis. We refer to such filters as horizontal filters as depicted on the right part of Fig. \[fig:lukthung\_pop\_filters\]. Unlike the architecture in [@c10] where a 1D convolution was performed across the time axis after the spectrograms were mean-pooled along the frequency axis, we expand the horizontal filters to cover a small frequency bins and apply them on the inputs before a max pooling layer. This is because we aim to preserve vibrato or coloratura, a remarkable trait presented in Lukthung singing style. Vibrato is a characteristic of sound oscillating within a small range of frequencies over time, considered a hybrid timbral-temporal features. An example of vibrato is the wavy bright lines in the spectrogram illustrated in the left part of Fig. \[fig:lukthung\_pop\_filters\]. In contrast, spectrograms of Pop songs, as depicted in the right part of Fig. \[fig:lukthung\_pop\_filters\], have only straight lines. We reason that this type of feature might vanish if we perform the average pooling process directly on the spectrogram inputs as done in [@c10]. Table \[tab:filters\] summarized all filter sizes and the number of filters used in our feature extraction module.
--------- ------------ --------- -----
Feature Frequency Time
type (Mel bins) (units)
115 7 32
115 3 64
115 1 128
51 7 32
51 3 64
51 1 128
7 32 32
7 64 32
7 128 32
7 165 32
--------- ------------ --------- -----
: Summary of filters used in our feature extraction module[]{data-label="tab:filters"}
After obtaining the representations of both timbral and temporal features, we concatenate them along the frequency axis and pass them into the binary classification module to predict whether a given audio clip is considered a Lukthung genre. Within the classification module, a residual network with three convolution layers followed by a fully connected network is implemented. Details of the residual network architecture can be found in [@c19]. Briefly, a residual network is stacked of convolution layers with alternative connections that skip from one layer to the next. Such bypass connections mitigate the effect of gradient vanishing and importantly allow the model to learn to identity functions, ensuring that the higher layers perform at least as well as the lower layers.
Combined model {#Sec:combinedmodel}
--------------
Since both lyrics and audios carry rich information about the Lukthung genre, we decide to combine both data to perform the classification task. Instead of using the single predicted probabilities from each model, we extract the learned representations from the last layer of both models and concatenate them to form a new feature vector. We reason that some features extracted from one type of inputs are complementary to the other and should be learned simultaneously. Based on these pre-trained features, we construct an additional feed-forward neural network comprising 800 nodes (100 from the lyrics model and 700 from the audio model) in the input layer, fully connected with 2 layers with ReLU activation and a single output node for the binary classification. The process is illustrated in the right part of Fig. \[fig:overallarch\].
Results {#Sec:experiments}
=======
We randomly split the dataset into a training set, validation set, and test set using the ratio of 0.55:0.2:0.25. We trained our models, BoW-MLP, Spectro-CNN, Combined, and all baselines, on INTEL Xeon CPU E5-1650 v3 3.50 GHz with 12 cores, 80GB RAM, and a GeForce GTX 1080 GPU. For comparison, we built the following baseline classifiers on the audio inputs using default parameters implemented in the Scikit-learn library[^2]. Using the MFCCs derived from the audio files as inputs, a simple logistic regression (LR), a tree-based random forest (RF), and support vector machines with a linear kernel (SVM-Lin) and a polynomial kernel (SVM-Poly) were trained.
Input Model Precision Recall $F_1$
------------- ----------------- ------------ -------- --------
Lyrics **BoW-MLP** 0.8581 0.7631 0.7905
RF **0.9500** 0.1313 0.2308
LR 0.5263 0.5760 0.5501
SVM-Lin 0.6939 0.0783 0.1408
SVM-Poly 0.5574 0.5369 0.5469
**Spectro-CNN** 0.8494 0.7397 0.7730
Lyrics $\&$
Audios
: $Precision$, $recall$, and $F_1$ score on test dataset[]{data-label="tab:result_test"}
We evaluated the models based on precision, recall, and $F_1$ scores for handling the imbalanced data. The results are shown in Table \[tab:result\_test\] where our proposed models and the best performing scores for each measure are highlighted in bold. Despite the best precision, RF performed unacceptably poor to recall Lukthung songs, yielding an extremely low $F_1$ score. In contrast, our models trained solely on either lyrics or audios already had significantly higher performance in both recall and precision than other traditional models. The best classifier based on $F_1$ scores lies on the combined model, supporting our hypothesis that lyrics and audio features are complementary determinants for Lukthung classification.
![Feature values of selected songs, taken from the input layer of the combined model[]{data-label="fig:featureHeatmap"}](images/heatmap.jpg){width="\linewidth"}
We selected the top-20 songs with high predictive confidence to perform feature analysis in these categories: true positive (TP), false positive (FP), false negative (FN), and true negative classes (TN). As an input in the combined model, the learned representations in the last intermediate layer prior to the output node from the lyrics and audio models were extracted and examined below.
Fig. \[fig:featureHeatmap\] shows such feature values of the selected songs. Each column represents a feature, while rows are songs grouped by the categories mentioned above, respectively. We also clustered the features values (columns) so that similar values were put together. The columns in light blue correspond to the lyrics (word) features, whereas those in dark blue represent the audio features. Note that the combined model carried 800 nodes, 100 word features and 700 audio features, in the input layer.
The heatmap clearly divides the lyrics features into two groups which were exploited differently by the model. Songs predicted as Lukthung had positive values on the first group of lyrics features and negative values on the other, and vice versa for songs predicted as non-Lukthung. We can see that lyrics features are similar within the predicted classes, separating the top (TP and FP) half from the bottom (FN and TN) half.
We inspected the false positive songs and found that some of them were Life and Lukkrung as well as Lukthung songs sung with Pop/Rock artists. The extracted lyrics and audio features from these genres are typically similar to Lukthung by nature.
Having scrutinized the list of false negative songs, we categorized them into two sets. The first type is recent Lukthung songs whose lyrics are composed of the standard Thai dialect but may be sung using a non-standard accent with no or only few vibratos. Our audio features hardly capture such accent. Moreover, the musical instruments played in this class of songs are more similar to Pop/Rock songs. This phenomemon is common in songs from new eras. The other type is non-Lukthung songs incorrectly labelled as Lukthung in the dataset, or non-Lukthung songs sung by Lukthung artists.
On the other hand, the values of audio features in the true negative class are much different from other classes. These songs were mainly labelled as Pop and Rock. This indicates that, the audio features of non-Lukthung songs, in general, considerably differ from Lukthung songs.
![Feature embeddings from the last hidden layer in the combined model[]{data-label="fig:tsne_embeddings"}](images/tsne_last_layer.png){width="\linewidth"}
To visualize the effectiveness of the learned features, we extracted the last hidden layer of the combined model and plotted them on a lower dimensional space using t-SNE as shown in Fig. \[fig:tsne\_embeddings\]. We can clearly see Lukthung and non-Lukthung songs are substantially separated, implying that the features were well extracted by the lyrics and audio models. Most of the falsely predicted songs are on the boundary between Lukthung and non-Lukthung songs. While the false positive songs form a single cloud at the boundary, the Lukthung songs classifed as non-Lukthung (false negatives) are scattered over the non-Lukthung space. This dispersion supports our previous explanation that some Lukthung songs are similar to Pop/Rock songs and moreover not limited to just one group of them.
Conclusion {#Sec:conclusion}
==========
In this paper, we have presented novel classification models to identify Lukthung music from other genres using lyrics and audio inputs. Due to a unique set of vocabulary usually used in Lukthung songs, a bag of words representation together with a simple neural network with a few hidden layers is sufficient to distinguish Lukthung from non-Lukthung songs. The audio inputs, on the other hand, require a more sophisticated model to find patterns across frequency bins and time intervals. Our approach applies multiple filters on the raw audio spectrograms to automatically learn different types of features such as overtones, tempos, and vibratos. These abstract features are used later for classification using a residual network with skip connections in deep networks. Using each input type individually yield satisfying results, outperforming all of the standard classifiers. Moreover, we show that extracting the pre-trained features from both models and combining them substantially improve the overall performance for Lukthung classification.
Country songs, which includes Lukthung, Lukkrung, Life and Mor-lam, bear some resemblance to each other in the distributions of words used in lyrics. This problem may be tackled with document-level, instead of word-level, representation such as semantic word vectors together with such sequence models as recurrent neural network. With more exposure to contemporary culture, some modern Lukthung songs are now adopting musical instruments and several sound techniques in close proximity to non-Lukthung songs in the old days. However, vocals might serve as the main remaining determinant that makes Lukthung differentiable from other genres. Thus, isolating singing voice from instrumental and designing vocal-specific filters may beneficially improve the classification outcomes. One example of voice-specific features is to capture the accent of the singer.
Our approach for Lukthung classification can effectively accommodate personalized music recommendation. Using our model, the system can classify streaming songs and automatically generate a comprehensive list of Lukthung songs in preparation for further music suggestion. Additionally, further analysis on the features extracted from the models can advance our understanding on how Lukthung songs evolve over eras.
[^1]: <https://github.com/rkcosmos/deepcut>
[^2]: <https://scikit-learn.org/stable/>
|
---
author:
- 'Matteo Colangeli, Adrian Muntean, Omar Richardson and Thoa Thieu'
bibliography:
- 'literature.bib'
title: Modelling interactions between active and passive agents moving through heterogeneous environments
---
Key words: Crowd dynamics; lattice gas model; fire and smoke dynamics; particle methods; heterogeneous domains.
PACS: 02.70.Uu, 07.05.Tp, 05.06.-k.
MSC 2010: 65Z05, 82C80, 91E30.
Introduction
============
Unlike fluid flows, pedestrian flows are rarely uniform. Hence, their motion is difficult to predict accurately. The main source of non-uniformity stems from the fact that pedestrian flows are thinking flows, i.e., both agent-agent interactions and agent-structure interactions are always active and are much more complex than the standard Van der Waals-like (attraction-repulsion) interactions which govern to a large extent the molecular description of fluids and gases. In this framework, we consider a particular type of non-uniformity. Looking at a heterogeneous environment (e.g. a complex office building), we consider our target pedestrian flow to contain the dynamics of interacting agents from two distinct populations:
- *active agents*, knowing where to go (they are aware of a predetermined optimal velocity field leading towards the exits),
- *passive agents*, randomly exploring the environment (they have no information about the exit routes, but base their motion on interaction with other agents).
We are particularly interested in investigating what mechanisms can be responsible for the minimization of the residence time of the pedestrians when an emergency evacuation situation has occurred, for instance, due to the unexpected occurrence of a fire that produces a significant amount of smoke. Our standing assumption is that the use of a purely macroscopic crowd model, which encodes the motion of a uniform flow, is prone to underestimate the residence time and does not properly capture crowd interaction.
In this chapter, we present conceptually different crowd dynamics models that describe the joint evolution of such passive and active agents. One of the models employs systems of nonlinear stochastic differential equations of motion one-way coupled with the diffusive-convective dynamics of the smoke, while an other model is a lattice-gas-type approach based on a Monte Carlo stochastic dynamics. Both models give estimates of the residence time of the particles as well as of the local occupancy (local pedestrian densities). When treating such scenarios, the complexity of the work is high. One of the difficulties is the handling of agent-structure interaction. It is worth noting that even if all agents were active and their wanted path is known [*a priori*]{}, if their number is sufficiently high, given a certain prescribed internal geometry of the facility, the agent-agent and agent-structure interactions normally lead to clogging or to the faster-is-slower effect; see e.g. [@Zuriguel2014] and [@Garcimartin2015]. Another difficulty is to handle the presence of the fire, and consequently, of the smoke and of the increased discomfort the agents feel. We refer the reader to [@OmarMSC] for one possible way of treating the presence of obstacles and to [@Omar2017] for hints on how to introduce the fire physics in the evolution equations describing the dynamics of the crowd. In this framework, we focus exclusively on the effect of knowledge of the geometry on the actual dynamics of the agents.
After reviewing a number of relevant related contributions, we proceed with the description of two closely-related modeling scenarios where the type of models previously mentioned apply. Then we solve the models numerically and illustrate the typical behavior of the output: positions, residence times, discomfort values, etc. We also discuss a few basic aspects concerning the mathematical well-posedness of one crowd model related to Model 1. We close the chapter with a discussion section where we also include hints towards further potential contributions in this context. The results reported here should be seen as preliminary. More efforts are currently invested to develop these research directions.
Related contributions {#sec:related_contributions}
=====================
Escape evacuation and social human behaviour are closely connected. In an emergency situation, building occupants require information about the surrounding environment and social interactions in order to evacuate successfully. The experiments in [@Horiuchi1986] can serve as a typical example for the relevance of distinguishing between two groups of occupants: regular users of the building and those less familiar with it.
In the model presented in [@Chu2014], the building occupants are modelled as agents who decide their evacuation actions on the basis of their infrastructural knowledge and their interactions with the social groups and the neighboring crowd therein. The authors showed that both familiar agents with the geometry building and social influence can dramatically impact on egress performance. As a multi-agent evacuation simulation tool, the ESCAPES system (presented in [@Tsai2011]) describes a realistic spread of knowledge to model two types of different knowledge: exit knowledge together with event knowledge. The conclusions made based on these models are supported by experimental findings such as those reported in [@ronchi17], where an evacuation was performed and the exit choice of participants was investigated, as well as the effect of the evacuation geometry.
Commonly, agent-based crowd models are based on developing individual trajectories. Yet for dense crowds, additional dynamics come into play. This has been observed in, for instance, [@corbetta14], where the interaction in dense crowds has been measured and analyzed, obtaining statistics for aggregate dynamics. These *macroscopic* properties have been observed from a theoretical perspective as well in e.g. [@luding07]. One way to bridge the gap between models for regular and dense crowds is to use models defined on different spatial scales, giving rise to a so-called multiscale model. In [@cristiani11], a multiscale model is proposed in terms of a granular flow formulation to display both microscopic as well as macroscopic crowd behaviour. For an investigation of handling contacts in such flows of granular matters applied to crowds, we refer the reader to the work of Maury and co-authors, compare [@Faure2015]. All these papers assume that the exits are visible. For study cases when the walking environment is not visible due to the lack of light, we refer the reader to [@Cialela]. There the main question is whether the grouping of the agents (involving higher coordination costs and information overload) has a chance to favorize an eventually quicker evacuation. From a different perspective, interesting connections to crisis management issues are made in [@Bellomo2016_1] and references cited therein.
We refer the reader to further related contributions on modeling crowd dynamics as reported, for instance, in [@Nguyen2013], [@Nguyen2015], [@Anh2012], as well as in [@Bellomo2015] and [@Colombo2015_2].
Agent-based dynamics (Model 1) {#sec:model_1}
==============================
In this section we introduce an agent-based model in a continuous two-dimensional multiple connected region $\Omega$, containing obstacles with a fixed location, a fire that produces smoke, and an exit. $\Omega$ represents the environment in which the crowd is present and tries to find the fastest way to the exit, avoiding any obstacles and the fire. The crowd is represented by the two aforementioned groups, active and passive agents. At time $t=0$, the crowd starts to evacuate from $\Omega$. In the rest of this section, $\Omega$ refers to the geometry displayed in Figure \[fig:example\].
![Basic geometry for our case study cf. Model 1. Agents are initialized in a random location within the geometry and have to reach the exit (green) while avoiding the obstacles (black).[]{data-label="fig:example"}](geometry.png){width="50.00000%"}
Active agents have a perfect knowledge of the environment and the locations of the obstacles, but are not aware of the location of the fire prior to experiencing sensory cues. Passive agents have no information on the environment and follow their neighbours to reach the exit. A similar model as the one described below has been presented in [@Omar2017].
Active and passive agents are seen as members from the sets $X_A = \{a_1,...,a_{N_A}\}$ and $X_B = \{b_1,...,b_{N_B}\}$, respectively. The dynamics governing their motion are described in the following sections.
Active agents
-------------
The motion of the active agents is governed by a potential field model proposed by Hughes in [@hughes02] and adapted in [@treuille06]. It functions similarly to a floor field function, its counterpart in lattice models presented in e.g. [@tan15] and [@cao2014]. We modify the potential field model to account for the presence of obstacles and the effects of fire and smoke.
The potential field agrees with the principle of *minimization of effort*, serving as a dynamic generalized distance transform. Let $\vec{x}$ be an arbitrary point selected in $\Omega$. We introduce a *marginal cost field* $u(\vec{x})>0$, defined as $$u(\vec{x}) = \alpha + u_{\mathrm{obs}}(\vec{x}) + wH(\vec{x}).$$ The marginal cost field represents the effort of moving through a certain location and consists of a base level of constant walking effort $\alpha>0$, information on the geometry and the obstacles $u_{\mathrm{obs}}$, and information on the fire source $wH$. Here, $w$ takes value 1 if the agent is aware of the location of the fire and 0 otherwise.
Let $S$ be a path going from point $\vec{x}_p$ to point $\vec{x}_q$. Then the effort of walking on the path $S$ can be expressed as $$\int_S u(\vec{\xi})d\vec{\xi}=\int_S\alpha + u_{\mathrm{obs}}(\vec{\xi}) + wH(\vec{\xi})d\vec{\xi}.$$ At the beginning of the simulation, $w$ is 0 for all agents. When an active agent experiences a significant increase in temperature because of his proximity to the location of the fire, $w$ is set to 1 and $S$ changes, and as a result, the fire is avoided. Let $G\subset \Omega$ be the set of all inaccessible locations in the geometry (i.e. those parts of $\Omega$ covered by obstacles). Then for all $\vec{x}\in \Omega$, the geometry information (i.e. the obstacle cost field) can be expressed as $$u_{\mathrm{obs}}(\vec{x}) =
\begin{cases}
\infty &\mbox{if }\vec{x} \in G,\\
\frac{1}{|d(\vec{x},G)|} & \mbox{if }\vec{x} \notin G\mbox{ and }d(\vec{x},G) \leq r_G,\\
0 &\mbox{if } d(\vec{x},G) > r_G,
\end{cases}
\label{eq:pot_obs}$$ where $r_G$ is a parameter of the order of the size of the agents. The obstacle cost makes sure that obstacle locations are inaccessible, and $r_G$ adds a tiny layer of repulsion around each obstacle to ensure the basic fact that agents do not run into walls.
The preferred path $S^*$ for an agent with location $\vec{x}_p$ and motion target $\vec{x}_q$ is determined as $$S^* = \operatorname*{arg\,min}_S \int_S u(\vec{\xi})d\vec{\xi},$$ where we minimize over the set of all possible motion paths $S$ from $\vec{x}_p$ to $\vec{x}_q$. In this framework, the active agents are aware of all exits, and the optimal path $S^*$ is made available by means of the potential function $\Phi$, a solution to the equation $$\left|\left|\nabla \Phi(\vec{x})\right|\right| = u(\vec{x}),
\label{eq:eikonal}$$ where $||\cdot||$ denotes the standard Euclidean norm. Passive agents do not have access to the optimal paths.
Figure \[fig:potential\_standard\] and Figure \[fig:evac\_path\_standard\] display the potential field and the corresponding paths for our case study. Figure \[fig:potential\_fire\] and Figure \[fig:evac\_path\_fire\] display the adaption active agents make as soon as they become aware of the fire locations and take an alternative route out.
![Paths generated from the potential field in Figure \[fig:potential\_standard\].[]{data-label="fig:evac_path_standard"}](potential_without_fire.pdf){width="\textwidth"}
![Paths generated from the potential field in Figure \[fig:potential\_standard\].[]{data-label="fig:evac_path_standard"}](paths_without_fire.pdf){width="\textwidth"}
![Paths generated from the potential field in Figure \[fig:potential\_fire\], avoiding the fire.[]{data-label="fig:evac_path_fire"}](potential_with_fire.pdf){width="\textwidth"}
![Paths generated from the potential field in Figure \[fig:potential\_fire\], avoiding the fire.[]{data-label="fig:evac_path_fire"}](paths_with_fire.pdf){width="\textwidth"}
Let $\vec{x}_{a_i}(t)$ denote the position of active agent $i$ at time $t$. We express their motion within the geometry $\Omega$ by $$\begin{cases}
\displaystyle
\frac{d\vec{x}_{a_i}}{dt} &= -v_s(\vec{x}_{a_i},t)\frac{\nabla\Phi(\vec{x}_{a_i}) - \nabla p(\vec{x}_{a_i},t)}{||\nabla\Phi(\vec{x}_{a_i}) - \nabla p(\vec{x}_{a_i},t)||},\\
\vec{x}_{a_i}(0) &= \vec{x}_{a_i,0},
\end{cases}
\label{eq:x_a}$$ where $\vec{x}_{a_i,0}$ represents the initial configuration of the active agents and $v_s$ represents a predefined walking speed. In , $p$ represents a given discomfort term that influences agent interactions at the macroscopic scale. The discomfort measures how much agents locally have to deviate from their ideal velocity. We are on purpose vague concerning this macroscopic discomfort. In a follow-up publication, we will tackle a multiscale non-uniform crowd model where $p$ will be part of the solution to a macro-micro flow problem.
Passive agents
--------------
Since we assume in this context that passive agents are unfamiliar with their environment, it is reasonable to postulate that to obtain information, they rely solely on neighbouring agents. This is a modelling assumption which has been confirmed for e.g. primates in [@meunier06]. This idea has already been applied in other crowd dynamics models (cf. e.g. [@helbing00]).
To model this strategy, we choose to apply a Cucker-Smale-like model which averages the velocity of nearby agents (an idea introduced originally in [@cucker07]). A Brownian term $\mathbf{B}_i$ is added to this swarming-like model to represent disorienting and chaotic effects which inherently appear while moving through an unknown environment. We denote the positions and velocities of agent $i$ from population $X_B$ as $\vec{x}_{b_i}$ and $\vec{v}_{b_i}$, and positions and velocities from member $j$ of the complete set $X_A \cup X_B$ as $\vec{x}_j$ and $\vec{v}_j$, respectively.
We express the motion of passive agents in the following way $$\begin{cases}
\frac{d\vec{v}_{b_i}}{dt} &= \sum_{j\in X} (\vec{v}_{j} - \vec{v}_{b_i})w_{ij}- \nabla {H}(\vec{x}_{b_i},t)\\
&+ \frac{\vec{v}_{b_i} - \nabla p}{||\vec{v}_{b_i} - \nabla p||}\Upsilon\left(s(\vec{x}_{b_i},t)\right)+ \mathbf{B}_i(t),\\
\frac{d\vec{x}_{b_i}}{dt}&=\vec{v}_{b_i},\\
\vec{v}_{b_i}(0) &= \vec{v}_{b_i,0},\\
\vec{x}_{b_i}(0) &= \vec{x}_{b_i,0}.
\end{cases}
\label{eq:x_b}$$ In , $w_{ij}$ are weight factors, decreasing as a function of distance, defined as $$w_{ij} \sim \frac{1}{r_s^2}\exp\left(-\frac{|\vec{x_{b_i}} - \vec{x_{j}}|^2}{r_s^2}\right).
\label{eq:weight_factor}$$ In , $r_s$ is the sight radius in the agents’ location, affected by smoke level $s(\vec{x},t)$ (see Section \[sec:smoke\]). It should also be noted that we do not take into account those walls that block the transfer of information between agents, since they are ignored in . However, in the simulations described in the next section, the size of the walls generally exceeds the size of the interaction radius. The term $\Upsilon\left(s(\vec{x}_{b_i},t)\right)$ is simply an [*a priori*]{} known normalization factor depending of the smoke level; one can take $\Upsilon\left(s(\vec{x}_{b_i},t)\right)=1$ just for simplicity. In the context of , the gradient in the discomfort level[^1] $\nabla p$ bounds asymptotically the speed of the passive agents.
Note that, based on , passive agents follow a set of coupled second-order differential equations (a social force-like model), while following , the active agents are expected to respect a set of coupled first-order differential equations (a social velocity-like model). We believe that the ’social inertia’ is much higher in the case of passive agents, so we keep the classical Langevin structure of the balance of forces, while for the active agents we choose an overdamped version.
Another important observation is that in this model, passive agents do not know which of the other agents are active, and which are passive themselves; they follow others indiscriminately.
Smoke effects {#sec:smoke}
-------------
In addition to repelling the agents, the fire produces smoke which propagates in $\Omega$ and reduces the visual acuity of the agents. The creation and propagation of the smoke is modelled as a diffusion-dominated reaction-advection-diffusion process.
The smoke density $s(\vec{x},t)$, is assumed to respect the following equation $$\displaystyle
\begin{cases}
\partial_ts = \operatorname{div}(D\nabla s) - \operatorname{div}(\vec{v}s) + y_s H(\vec{x})&\mbox{ in } \Omega \setminus G,\\
(-D\nabla s+ \vec{v}s)\cdot \vec{n} = 0 &\mbox{ on } \partial \Omega \cup \partial G,\\
s(\vec{x},0) = 0 &\mbox{ in } \Omega,
\end{cases}
\label{eq:smoke}$$ where $D>0$ represents the smoke diffusivity, determined by the environment, $\vec{n}$ is the outer normal vector to $\partial \Omega \cup \partial G$, $\mathbf{v}$ is a given drift corresponding to, for instance, ventilation systems or indoor airflow, while $H(\vec{x})$ encodes the shape and intensity of the fire, viz. $$H(\vec{x}) = \begin{cases}
R &\mbox{ if }|\vec{x} - \vec{x}_0| < r_0\\
0 &\mbox{ otherwise }
\end{cases}.
\label{eq:fire}$$ In our context, $D>0$ is the molecular diffusion coefficient for the smoke and a slight space dependence in $D$ is allowed. At a later stage, maybe eventually also an $s$-dependence of $D$ can be foreseen, if one would replace by an averaged version where the free motion paths and the geometry are perceived as some sort of homogenized porous medium.
Figure \[fig:smoke\_prop\] illustrates a snapshot of the smoke density for our case study.
![Smoke density in the environment at $t=60$.[]{data-label="fig:smoke_prop"}](smoke.pdf){width="60.00000%"}
Results Model 1: Agent-based dynamics {#sec:results_1}
=====================================
This section contains our numerical results obtained using the agent-based dynamics described in the previous section.
The results are run in crowd simulation prototyping application *Mercurial* ([@mercurial]). This is an open-source framework developed in Python and Fortran to simulate hybrid crowd representations as the one described in Section \[sec:model\_1\]. It provides both agent-based- and continuum-level visualizations and supports the design of arbitrary two-dimensional geometries. More details on the structure and implementation of *Mercurial* are found in [@OmarMSC].
Figure \[fig:example\] shows the geometry of our case study. It has a fairly simple structure to ensure the exit can be reached even without environment knowledge. However, the placement of the obstacles is such that zones of congestion easily occur and paths to the exit will necessarily have to be curved.
The simulation was run twice with 1000 agents, varying the ratio between active and passive agents. The first run (Case 1), of which a snapshot is presented in Figure \[fig:active\_configuration\], contains a total of 800 active agents and 200 passive agents. The second run (Case 2), illustrated with a snapshot in Figure \[fig:passive\_configuration\], contains a total of 200 active agents and 800 passive agents.
![Discomfort observed in the simulation snapshot of Figure \[fig:active\_configuration\].[]{data-label="fig:active_discomfort"}](active_configuration.pdf){width="80.00000%"}
![Discomfort observed in the simulation snapshot of Figure \[fig:active\_configuration\].[]{data-label="fig:active_discomfort"}](active_discomfort.pdf){width="\textwidth"}
![Discomfort observed in the simulation snapshot of Figure \[fig:passive\_configuration\].[]{data-label="fig:passive_discomfort"}](passive_configuration.pdf){width="80.00000%"}
![Discomfort observed in the simulation snapshot of Figure \[fig:passive\_configuration\].[]{data-label="fig:passive_discomfort"}](passive_discomfort.pdf){width="\textwidth"}
Figure \[fig:active\_discomfort\] and Figure \[fig:passive\_discomfort\] illustrate a coherent discomfort field in an ongoing simulation due to a large number of agents with conflicting directions. The corresponding agents configuration (i.e. their spatial distribution) is displayed in Figure \[fig:active\_configuration\] and Figure \[fig:passive\_configuration\]. Visible is that close to the fire a lot of discomfort is generated. The main cause for this congestion is the conflict between active agents who have identified the location of the fire and want to move in different directions and active agents which are still unaware and want to exit the geometry through that particular corridor. This reminisces of the panic zone that occurs in crowd disasters close to the origin of the panic. Notice how this zone is much more present in Case 1 than in Case 2, due to the lack of active agents in Case 2. While the passive agents take a lot longer to reach the exit, their following-dominated behaviour amounts to less discomfort in doing so.
It would be interesting to have a partial differential equation describing at least approximately the macroscopic space-time evolution of such discomfort field available. Also, such an object would be very useful from a practical point of view – it would allow a fast detection of zones of high discomfort, which could be helpful in taking management decisions to reduce the potential of risks and accidents.
In Case 1 (Figure \[fig:active\_configuration\]), we observe that all agents belong to a collective moving towards the exit, regardless of population. As one would expect, Case 2 (Figure \[fig:passive\_configuration\]) displays less order than Case 1. Most agents move in smaller groups, either guided by active agents or randomly moving throughout the geometry.\
Figure \[fig:active\_evac\_times\] and Figure \[fig:passive\_evac\_times\] depict the agents leaving the environment as a function of time. In Figure \[fig:active\_evac\_times\] we observe three stages: the first stage (from $t=0$ to $t\approx 100$) corresponds to the group of active agents that exit the geometry without any obstructions, guiding most of the passive agents while doing so. The second stage (from $t\approx 100$ to $t\approx 400$) has virtually no agents that reach the exit; all the remaining agents are trapped in the high discomfort panic-like zone close to the fire. The third stage shows the final active agents have escaped the panic zone, reaching the exit.
Figure \[fig:passive\_evac\_times\] displays a similar first stage, but because the discomfort zones are a lot less intensive, there is no pronounced second and third stage. Notice how after the bulk of the active agents have left the geometry, the egress of the passive agents has reduced to a random walk.\
![Agent exit times in Case 2.[]{data-label="fig:passive_evac_times"}](active_exit_times.pdf){width="\textwidth"}
![Agent exit times in Case 2.[]{data-label="fig:passive_evac_times"}](passive_exit_times.pdf){width="\textwidth"}
These observations are supported by Figure \[fig:active\_cum\_discomfort\] and Figure \[fig:passive\_cum\_discomfort\], where the cumulative discomfort for each location $\vec{x}$ in $\Omega$ is displayed. Case 1 (Figure \[fig:active\_cum\_discomfort\]) shows significantly higher discomfort both near the fire and where the geometry narrows itself. Case 2 (Figure \[fig:passive\_cum\_discomfort\]) shows a much higher usage of the space in the geometry, i.e. agents walking in locations that do not belong to any shortest path. However, the high discomfort zones are an order of magnitude lower than in Case 1, due to the flexibility of passive agents.
![Logarithmic heat-map of discomfort zones that developed in the scenario with 20% active agents. White regions indicate no experienced discomfort.[]{data-label="fig:passive_cum_discomfort"}](active_cum_discomfort.pdf){width="\textwidth"}
![Logarithmic heat-map of discomfort zones that developed in the scenario with 20% active agents. White regions indicate no experienced discomfort.[]{data-label="fig:passive_cum_discomfort"}](passive_cum_discomfort.pdf){width="\textwidth"}
Concluding, simulations support the following observations. Differences in environment knowledge can have a significant impact on several aspects on the dynamics of crowds in e.g. evacuations. While it is true that additional knowledge decreases evacuation time, the autonomy of active agents can cause problems when their information turns out to be incorrect. When steering passive agents it is significantly more difficult to maintain order in the evacuation, but the fact that they can be guided can relieve discomfort and reduce congestion.
Lattice gas dynamics (Model 2) {#sec:model_2}
==============================
The second model we shall tackle here is a lattice gas model. Namely, we consider a Simple Exclusion Process (SEP) [@kipnis1998] on a two-dimensional lattice $\Lambda$: $$\Lambda=\{(i,j)\in \mathbb{Z}^2 :i=1,...,L_x \; \text{and} \; j=1,...,L_y\} \;.$$ According to the basic tenets of the SEP dynamics, there can be only one particle per site, and particles jump independently towards one of the nearest neighbor sites on the lattice, provided that the arrival site is empty. We shall hereafter assume that the system is *closed*, namely particles may not hop outwards from any of the boundary sites of $\Lambda$, except from a subset of lattice sites $\mathcal{D}=\{(i,j) \in \Lambda : j=L_y \; \text{and} \; i\in[i_{ex},i_{ex}+w_{ex}]\}$, called the “exit door”: any particle located in $\mathcal{D}$ and hopping upwards is annihilated. Note that particles may just leave the system through the exit door: no inward flux of particles is considered in this model. The numerical investigation of the lattice gas model aims, indeed, to shed light on the characteristic time scales characterizing the particle evacuation from the system.\
As in the case of Model 1, we distinguish between two species of particles, namely *aware* or active particles, and, respectively, *unaware* or passive particles. For simplicity of the notation, we shall refer to them as particles “$A$” and “$U$” in this section. While the species $U$ performs a symmetric simple exclusion dynamics on the lattice, particles of the species $A$ experience both a horizontal and a vertical drift, denoted below as $\epsilon_x$ and $\epsilon_y$, that enhance the rates at which particles of such species hop towards the exit door. The microscopic dynamics is defined as follows. Call $\eta^{(U)}(x)$ and $\eta^{(A)}(x)$ the occupation number on the site $x$ (which is either $0$ or $1$) of the species, respectively, $U$ and $A$. Given two nearest neighbor sites $x,y \in \Lambda$, $|x-y|=1$, such that the *bond* joining $x$ to $y$ is entirely contained in $\Lambda$, we define the hopping *rate* from $x$ to $y$ of a particle of the species $U$ (no matter if the jump occurs along a horizontal or a vertical bond) as: $$c^{(U)}(x,y)=\eta^{(U)}(x)\left[1-\eta^{(U)}(y)-\eta^{(A)}(y)\right] \;.
\label{cU}$$ To define the corresponding hopping *rate* for particles of the species $A$ we shall distinguish between vertical and horizontal bonds. For the vertical bonds, i.e. when $i_y=i_x$, we set: $$c^{(A)}(x,y)=\left\{\begin{array}{cc} (1+\epsilon_y)\eta^{(A)}(x)\left[1-\eta^{(A)}(y)-\eta^{(U)}(y)\right] & \quad \hbox{if $j_y>j_x$}\; ,\\ \\
0 & \quad \hbox{if $j_y<j_x$}\;.
\label{cA1}
\end{array}\right.$$ for bonds directed upwards and downwards, respectively. The microscopic dynamics of the species $A$, ruled by Eq. , highlights the intrinsic bias of the species $A$ to move upwards, i.e. towards the exit, and prevents any redundant vertical motion in the opposite direction. Moreover, also includes a drift term $\epsilon_y$ that marks the tendency of the species $A$ to reach the exit door with a rate that is higher than the unitary rate defining the unbiased dynamics of the species $U$, see Eq. . For the horizontal bonds, i.e. when $j_y=j_x$, we shall first consider the case $i_x \notin [i_{ex},i_{ex}+w_{ex}]$, for which we set: $$c^{(A)}(x,y)=\left\{\begin{array}{cc}
(1+\epsilon_x)\eta^{(A)}(x)\left[1-\eta^{(A)}(y)-\eta^{(U)}(y)\right] & \quad \hbox{if $(i_y-i_x) (i_{ex}-i_y)\geq0$}\; ,\\ \\
\eta^{(A)}(x)\left[1-\eta^{(A)}(y)-\eta^{(U)}(y)\right] & \quad \hbox{if $(i_y-i_x) (i_{ex}-i_y)<0$}\;.
\end{array}\right.
\label{cA2}$$ The presence of a horizontal drift term $\epsilon_x$, in Eq. reminds us that particles of the species $A$, unlike particles of the species $U$, move preferably towards the right (resp. the left), when the horizontal coordinate of the departure site is $i_x<i_{ex}$ (resp. $i_x>i_{ex}+w_{ex}$). Instead, when $j_y=j_x$ and $i_x\in[i_{ex},i_{ex}+w_{ex}]$, we set: $$c^{(A)}(x,y)=0 \;.
\label{cA3}$$ Equation says that, when $i_x\in [i_{ex},i_{ex}+w_{ex}]$, particles of the species $A$ may only hop upwards, namely they point directly towards the exit door without wandering along the horizontal direction.\
The rates associated to those bonds joining any boundary site of $\Lambda$, that is not part of the exit door, to any external site, are all set equal to 0. Finally, the rates $c_{ex}$ associated to the vertical bonds joining a site $x\in \mathcal{D}$ with a site $y\notin \Lambda$, are defined as follows:
$$c_{ex}(x)=\eta^{(U)}(x)(1-\eta^{(A)}(x))+(1+\epsilon_y)\eta^{(A)}(x)(1-\eta^{(U)}(x))\;.$$
The proposed lattice gas model also accounts for the presence of fixed obstacles inside the domain, that correspond to a subset $\Lambda_{ob}\subset \Lambda$ of lattice sites that are inaccessible, represented by the black spots shown in Figure \[fig:config\]. To complete the description of the microscopic dynamics we shall, hence, also set equal to zero all the rates associated to those bonds joining two sites, one (or even both) of them belonging to $ \Lambda_{ob}$.
The study of the evacuation of particles shall be pursued by considering, for each species, the behavior of the number of particles and the particle current (through the exit door) as a function of time. The particle current $J$ is defined as $$J(t)=\frac{N(0)-N(t)}{t}\;,
\label{curr}$$ where $N(0)$ and $N(t)$ denote, respectively, the number of particles of a given species at the initial time and at the time $t$.
Results Model 2: Lattice gas dynamics {#sec:results_2}
=====================================
This section contains our preliminary numerical results obtained using Model 2.\
The dynamics was implemented by running a set of Kinetic Monte Carlo (KMC) simulations ([@Landau:2005]). KMC methods are notoriously suited to describe transient phenomena, in which physical time plays a crucial role in the microscopic evolution ([@Bortz; @Voter]). Note that, denoting by $T\in\mathbb{N}$ the number of time steps considered in the KMC simulation, the physical time $t\in\mathbb{R}$, considered in , is obtained as $t=\sum_{k=1}^{T} t_k$, where each $t_k$ (corresponding to the time elapsed between two consecutive particle jumps on the lattice) is an exponentially distributed random variable with a parameter given by the sum of all the rates associated to the lattice bonds, defined in Section \[sec:model\_2\], see refs. [@CM; @CCM] for details.
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The results of the KMC simulations for this model are portrayed in Figure \[fig:config\] and Figure \[fig:part\]. In Figure \[fig:config\] we show the microscopic configurations of the particles $A$ and $U$ at different times, namely from the initial configuration (top left panel) until the time when the evacuation of particle $A$ is essentially completed (cf. the bottom right panel, corresponding to some $3\times 10^6$ time steps of the dynamics).
In the left panel of Figure \[fig:part\], we show the effect of varying the width $w_{ex}$ of the exit door as well as the drifts $\epsilon_x$ and $\epsilon_y$ on the total number of particles $A$ and $U$ as a function of time. Clearly, by increasing the width of the exit door, particles of both species evacuate with a higher pace. The left panel of Figure \[fig:part\] also highlights an interesting effect that is obtained by varying $\epsilon_x$ and $\epsilon_y$: an increase of the drift terms induces a higher evacuation rate for particles of the species $A$, and also, consequently, for particles of the species $U$, which have access to a larger number of empty sites on the lattice. In the right panel of Figure \[fig:part\] we show the behavior of the particle current, defined in , for the two species $A$ and $U$ as a function of time, for fixed values of the parameters $w_{ex}$, $\epsilon_x$ and $\epsilon_y$. The higher evacuation rate observed for the aware particles stems directly from the definition of the rates for the two species $A$ and $U$, given in section \[sec:model\_2\].
Mathematical aspects of social dynamics in mixed populations {#sec:analysis}
============================================================
In this section, we discuss the solvability of a social dynamics model of mixed populations, resembling an overdamped version of Model 1. Note that Model 2 is well-posed by construction. Here, interesting questions would be pointing towards the rigorous derivation of the corresponding hydrodynamic limit equations [@DMP], and/or the numerical evaluation of non-equilibrium collective effects (e.g. the inclusion of a reaction mechanism within the microscopic dynamics, allowing particles to switch from one species to the other, or the presence of long-range interactions between particles), but these aspects are not in our focus for the moment.
This section contains a couple of technical preliminaries needed to state the evolution problem in a functional analytic framework. We use standard methods to handle the well-posedness of a coupled set of SDEs for the agents dynamics, also linked to a linear parabolic equation describing the motion of the smoke.
Technical preliminaries, notation and assumptions {#sectio_pre}
-------------------------------------------------
### Geometry
We consider a two dimensional domain, which we refer to as $\tilde{\Omega}$. This domain presents the geometry of the evacuation scenario. In addition, as a building geometry, parts of the domain are filled with obstacles ($G_1$ and $G_2$) denoted by $G := G_1 \cup G_2$ and the fire denoted by $\tilde{F}$. Moreover, the domain has exits denoted by $E$. Let $\Omega:=\tilde{\Omega}\backslash(G \cup E \cup \tilde{F}) \subset \mathbb{R}^d$ for $d=2$ and $\partial \Omega$ be $C^2$, or at least satisfying the exterior sphere condition. A typical example of such $\Omega$ is depicted in Figure \[fig:example\_again\].
![Basic geometry for our case study cf. Model 1. Obstacles are colored black, while the exit is colored green.[]{data-label="fig:example_again"}](geometry.png){width="50.00000%"}
### Function spaces
In this section, we employ a number of Sobolev spaces; see e.g. [@Adams03], [@Evans1997] for details on their definition and properties.
The space $H^m(\Omega)$, $m\in \mathbb{N}$, is endowed with the norm $$\begin{aligned}
\|v\|_{H^m(\Omega)}^2 :=\left(\sum_{|\alpha|\leq m}\int_{\Omega}|D^{\alpha}v|^2dx\right)^{1/2} \textrm{ for all }v \in H^m(\Omega),\nonumber\end{aligned}$$ while for the space $W^{m,\infty}(\Omega)$ we consider the norm $$\begin{aligned}
\|v\|_{m,\infty}(\Omega) := \sum_{|\alpha|\leq k}\textrm{ess} \sup_{\Omega}|D^{\alpha}v| \textrm{ for all } v \in W^{m,\infty}(\Omega), m \in \mathbb{N}; k = 0,\ldots,m.\nonumber\end{aligned}$$
Our analysis of the stochastic differential equations (SDEs) describing the evolution of our populations follows the line of reasoning from [@Flandoli95] and [@Prato14] (the compactness method of SPDEs and martingale solutions). We refer to [@Evans2013] and [@Pavliotis2014] for the basic concepts and usual notations.
Let $\textbf{x}_t$ be a continuous-time stochastic process. We define the family of laws $$\begin{aligned}
\label{laws}
\left\{Q(\textbf{x}_t^n); t \geq 0, n\geq 1\right\}.\end{aligned}$$ This is a family of probability distribution of $\textbf{x}_t^n$.
Recall the classical Ascoli-Arzelà theorem:\
A family of functions $F \subset C([0,T]; \mathbb{R}^d)$ is relatively compact (in uniformly topology) if
1. for every $t \in [0,T]$, the set $\{f(t);f \in F\}$ is bounded.
2. for every $\varepsilon>0$ and $t,s \in [0,T]$ there is $\delta > 0$ such that $$\begin{aligned}
|f(t)-f(s)| \leq \varepsilon,
\end{aligned}$$ whenever $|t - s| \leq \delta$ for all $f \subset F$.
We introduce the definition of Hölder seminorms, for $f: [0,T] \to \mathbb{R}^d$ as $$\begin{aligned}
\label{holder_norm}
[f]_\alpha = \sup_{t\neq s}\frac{|f(t)-f(s)|}{|t-s|}, \end{aligned}$$ and the supremum norm as $$\begin{aligned}
\label{infty_norm}
\|f\|_{\infty} = \sup_{t \in [0,T]}|f(t)|.\end{aligned}$$ Using Ascoli-Arzelà theorem, starting from the facts:
1. there is $M_1>0$ such that $\|f\|_{\infty}\leq M_1$ for all $f \in F$.
2. for some $\alpha \in (0,1)$, there is an $M_2>0$ such that $[f]_{C^\alpha} \leq M_2$ for all $f \in F$,
we infer that the set $$\begin{aligned}
\label{relativelycompactKMP}
K_{M_1M_2} = \left\{f\in C([0,T];\mathbb{R}^d); \|f\|_{\infty}\leq M_1, [f]_{C^\alpha} \leq M_2\right\}\end{aligned}$$ is relatively compact in $C([0,T]; \mathbb{R}^d)$.
For $\alpha \in (0,1)$, $T>0$ and $p>1$, the space $W^{\alpha,p}(0,T;\mathbb{R}^d)$ is defined as the set of all $f \in L^p(0,T;\mathbb{R}^d)$ such that $$\begin{aligned}
[f]_{W^{\alpha,p}}:= \int_{0}^{T}\int_{0}^{T}\frac{|f(t) - f(s)|^p}{|t-s|^{1 + \alpha p}}dtds < \infty. \nonumber\end{aligned}$$ This space is endowed with the norm $$\begin{aligned}
\|f\|_{W^{\alpha,p}} = \|f\|_{L^p} + [f]_{W^{\alpha,p}}.\nonumber\end{aligned}$$ Moreover, we know that if $\alpha p> 1$, then $$\begin{aligned}
W^{\alpha,p}(0,T; \mathbb{R}^d) \subset C^\gamma([0,T];\mathbb{R}^d) \quad \textrm{ for } (\alpha p - \gamma) >1\nonumber\end{aligned}$$ and $[f]_{C^\gamma} \leq C_{\gamma,\alpha,p}\|f\|_{W^{\alpha,p}}$. Relying on the Ascoli-Arzelà theorem, we have the following situation:
1. for some $\alpha \in (0,1)$ and $p>1$ with $\alpha p>1$, there is $M_2>0$ such that $[f]_{W^{\alpha,p}} \leq M_2$ for all $f \in F$.
If $\textrm{i'}$ and $\textrm{ii''}$ hold, then the set $$\begin{aligned}
\label{relativelycompactKMP_2}
K'_{M_1M_2} = \left\{f \in C([0,T]; \mathbb{R}^d); \|f\|_{\infty} \leq M_1, [f]_{W^{\alpha,p}} \leq M_2\right\}\end{aligned}$$ is relatively compact in $C([0,T]; \mathbb{R}^d)$, if $\alpha p>1$.
### Hypotheses
In this framework, we require the following assumptions:
1. $\phi\in C^2(\Omega)$ (see also section \[eiko\]).
2. $p_{\max}=N|\Omega| <\infty$ (bounded maximal discomfort).
3. The smoke matrix diffusion coefficient $D = \textbf{D}(x) \in W^{m,\infty}(\Omega)$ satisfies the uniform ellipticity condition, i.e. there exists positive constants $\underline{\theta}, \overline{\theta}$ such that $$\underline{\theta}|\xi|^2 \leq \textbf{D}(x)\xi_i\xi_j \leq \overline{\theta}|\xi|^2 \textrm{ for any } \xi \in \Omega.$$
4. The smoke interface exchange coefficient on the boundary of our domain $\lambda:=\Lambda(x) \in W^{m,\infty}(\partial \Omega)$ is such that there exist positive constants $\underline{\gamma}, \overline{\gamma}$ satisfying $$-\underline{\gamma}|\xi|^2 < \lambda(x)\xi_i\xi_j \leq \overline{\gamma}|\xi|^2 \textrm{ for any } \xi \in \partial \Omega.$$
Changing the functional framework will naturally lead to a reconsideration of these assumptions.
### First-order social agents dynamics
We focus on the interaction between two groups of pedestrians, one familiar (active agents) and one unfamiliar (passive agents, visitors) with the geometry. To keep the presentation simple, we decide to tackle here the case when both active and passive agents follow a first-order dynamics (overdamped Langevin equations). To this end, we modify the dynamics of the passive agents, deviating this way from Model 1.
Let $\textbf{x}_{a_i}$ denote the position of the agent $i$ from group $A$ at time $t$. The crowd dynamics in group A is expressed by the first-order differential equation encoding optimal environment knowledge within the domain $\Omega$, viz. $$\begin{aligned}
\label{micro_eqn1}
\begin{cases}
\frac{d\textbf{x}_{a_i}(t)}{dt} = -\Upsilon(s(\textbf{x}_{a_i},t))\left(\frac{\nabla \phi(\textbf{x}_{a_i})}{\|\nabla \phi(\textbf{x}_{a_i})\|}\right)(p_{\max} - p(\textbf{x}_{a_i},t)),\\
\textbf{x}_{a_i}(0) = \textbf{x}_{a_i,0},
\end{cases}\end{aligned}$$ where $p_{\max}$ is a discomfort threshold proportional to the overall population size, say $p_{\max} = N|\Omega|$, with $N := N_A + N_B$ and $p(\textbf x,t)$ is the local discomfort (realization of social pressure) so that $$\begin{aligned}
\label{pressure}
p(\textbf x,t)=\int_{\Omega\cap B(\textbf x,\tilde{\delta})}\sum_{j=1}^{N}\delta(z - \textbf{x}_{c_j}(t))dz.\end{aligned}$$ In , $\delta$ is the Dirac (point) measure and $B(\textbf{x},\tilde{\delta})$ is a ball center $\textbf{x}$ with small enough radius $\tilde{\delta}$ such that $\tilde{\delta} >0$. Hence, the discomfort $p(\textbf{x},t)$ represents a finite measure on the bounded set $\Omega \cap B(\textbf{x},\tilde{\delta})$. In addition, we assume the following structural relation between the smoke extinction and the walking speed: $$\begin{aligned}
\Upsilon(s(\textbf x,t)) = - a s(\textbf x,t) + b,\nonumber\end{aligned}$$ where $a, b$ are given positive numbers. Note that every member of this group wants to follow the motion path explicitly given by $\nabla\phi$ (with $\phi$ the potential function solving the Eikonal equation), which minimizes the distance between particle positions $\textbf{x}_{a_i}$ and the exit location $E$.
As mentioned before, concerning the second population, since the agents do not know the geometry, they must rely on the information from their neighbour. The unfamiliarity with the local environment is expressed here by means of a Brownian motion term $B_i$. Moreover, the passive agents like to be stay away the fire – for this to happen we use a repulsion term pinpointing to the location of the fire source $\nabla H_\epsilon$. Hence, the dynamics is described as a stochastic differential equation as follows $$\begin{aligned}
\label{micro_eqn2}
\begin{cases}
\frac{d\textbf{x}_{b_k}(t)}{dt} = \sum_{j = 1}^N\frac{(\textbf{x}_{c_j} - \textbf{x}_{b_k})}{\|\textbf{x}_{c_j} - \textbf{x}_{b_k}\|}w(\hat{\delta},s(\textbf{x}_{b_k},t)) - \nabla H_\epsilon(\textbf{x}_{b_k},t) + \tilde{D}B_k(t),\\
\textbf{x}_{b_k}(0) = \textbf{x}_{b_k,0}.
\end{cases}\end{aligned}$$ Here $\tilde{D}$ is the constant diffusion coefficient matrix, while $\hat{\delta} \sim \|\textbf{x}_{c_j} - \textbf{x}_{b_k}\|$, and $w$ is a weight factor decreasing as a function of distance. They are defined as $$\begin{aligned}
\label{weight_factor}
w(x,y) \sim \frac{1}{r_s^2}\exp\left(-\frac{(x-y)^2}{r_s^2}\right).\end{aligned}$$ In , $r_s$ is the sight radius in the evacuees location. Since $H$ is in general not differentiable everywhere (cf. e.g. ), in order to be able to take the gradient of $H$, we consider from the start a mollified $H$, say $H_\epsilon$. Furthermore, note that $\tilde{D}$ can in principle also depend on the space position. This way the random effects can be skipped in the regions where the geometry is not available for walking. It is worth noting that we have many ways to express how the active agents sense the fire. We choose here to introduce the fire location as a region to be avoided and impose it in the definition domain of the Eikonal equation. It is worth comparing this model for the evolution of the passive agents and the one prescribed in Model 1. Notice here the following important aspects: not only the dynamics is over-damped, but also the expression of the social velocity is slightly adapted to avoid an implicit definition.
Well-posedness
--------------
Our evolution system consists of an ODE coupled to an SDE . Therefore, due to the randomness incorporated in the SDE , the ODE becomes an SDE after coupling. So, we can consider and as a SDE system. Note that this system is one-way coupled with the reaction-diffusion-drift equation describing the smoke evolution.
For convenience, we rephrase the solution to the system and in terms of the vector $\textbf{x}_t^n$ such that $$\begin{aligned}
\textbf x_t^n &= ({\textbf x}_{a_i}^n(t),{\textbf x}_{b_k}^n(t)),\\ F_1({\textbf x}_t^n,t) &:= -\Upsilon(s({\textbf x}_{a_i}^n,t))\frac{\nabla \phi({\textbf x}_{a_i}^n)}{\|\nabla \phi({\textbf x}_{a_i}^n)\|}(p_{\max} - p({\textbf x}_{a_i}^n,t)),\\
F_2({\textbf x}_t^n,t) &:= \sum_{j = 1}^N\frac{({\textbf x}_{c_j}^n - {\textbf x}_{b_k}^n)}{\|{\textbf x}_{c_j}^n - {\textbf x}_{b_k}^n\|}w(\hat{\delta},s({\textbf x}_{b_k}^n,t)) - \nabla H_\epsilon({\textbf x}_{b_k}^n,t).\end{aligned}$$ Furthermore, we set $$\begin{aligned}
b_n(\textbf{x}_t^n,t):= \begin{bmatrix}
F_1(\textbf{x}_t^n,t)\\F_2(\textbf{x}_t^n,t)
\end{bmatrix} \textrm{ and }
\tilde{\sigma}:=\begin{bmatrix}
\tilde{0}\\\tilde{D}
\end{bmatrix},\end{aligned}$$ with $$\begin{aligned}
\tilde{0}:= \begin{bmatrix}
0 &0\\
0 &0
\end{bmatrix} \quad \text{ and } \tilde{D}:=\begin{bmatrix}
D_{11} & D_{12}\\D_{21} & D_{22}
\end{bmatrix}\end{aligned}$$ and initial datum $$\begin{aligned}
\textbf{x}_0^n := \begin{bmatrix}
\textbf{x}_{a_i,0}^n\\ \textbf{x}_{b_k,0}^n
\end{bmatrix}.\end{aligned}$$ In this section, we use the compactness method for proving the existence of solutions; we follow the arguments by G. Da Prato and J. Zabczyk ($2014$) (cf. [@Prato14], Section $8.3$) and a result of F. Flandoli (1995) (cf. [@Flandoli95]) for martingale solutions. The starting point of this argument is based on considering a sequence $\{\textbf{x}_t^n\}$ of solutions of the following stochastic differential equation $$\begin{aligned}
\label{approxi_formsde}
\begin{cases}
d\textbf{x}_t^n = b_n(\textbf{x}_t^n,t) dt + \tilde{\sigma} dB_t\\
\textbf{x}_0^n = \textbf{x}_0^n
\end{cases}\end{aligned}$$ To ensure the applicability of the compactness argument, we need the following structural assumptions:
1. $b_n$ be a consequence of continuous functions and uniformly Lipschitz in $x$.
2. $b_n$ be equi-bounded $\|b_n\|_{\infty} \leq C$.
It is not difficult to see that in our case, Assumptions ($\textrm{A}_4$) and ($\textrm{A}_5$) are fulfilled. By $s\in C([0,T];C^1(\Omega))$ from Remark \[rm\_C1\], we have $\tilde{v}_s$ Lipschitz in $x$. Moreover, by the Assumption $(\text{A}_1)$, we obtain $\nabla\phi$ is Lipschitz for $x\in \Omega$. On the other hand, the term $p_{\max} - p(\textbf{x}_{a_i},t)$ is a finite measure on bounded sets – it is automatically Lipschitz. These considerations lead to the fact that $F_1$ is Lipschitz in $x\in \Omega$. In addition, by $(\text{A}_2)$ together with taking $H_\epsilon$ (as a mollified $H$) implies that $\nabla H_\epsilon$ is uniformly Lipschitz in $x\in \Omega$. By the formula , the weight factors are Lipschitz in $x \in \Omega$. Thus, $F_2$ inherites the Lipschitz property. Clearly, from these arguments, we obtain not only that $F_1$ and $F_2$ are Lipschitz, but also that these functions are equibounded $\|F_1\|_{\infty} \leq C$ and $\|F_2\|_{\infty} \leq C$. Hence, we have $b_n$ satisfying both assumptions ($\textrm{A}_4$) and ($\textrm{A}_5$).
The compactness argument proceeds as follows. We begin with solutions $\textbf{x}_t^n, n \in \mathbb{N}$ of the system and , describing in . The construction of these solutions can be investigated on a probability space $(\Omega,\mathcal{F}, P)$ with a filtration $\{\mathcal{F}_t\}$ and a Brownian motion $B(t)$. Next, let $Q^n$ be the laws of $\textbf{x}_t^n$ which is defined cf. . Then, by using Prokhorov’s theorem, we show that the sequence of laws $\{Q^n(\textbf{x}_t^n)\}$ is weakly convergent to $Q(\textbf{x}_t)$ in $C([0,T]; \mathbb{R}^d)$. Then, by using the Skorohod representation Theorem, the weak convergence is in a new probability space with a new stochastic process, for a new filtration. This leads to some arguments for weak convergence results of two stochastic processes in two different probability spaces that we need to use to obtain the existence of our SDE system. Finally, we prove the uniqueness of solutions to our system.
Let us start with handling the tightness of the laws $\{Q^n\}$ through the following lemma.
\[tightness\] Assume ($\textrm{A}_4$) and ($\textrm{A}_5$) hold. The family of $\{Q^n\}$ is tight in $C([0,T];\mathbb{R}^d)$
In order to prove the tightness, let us recall the following compact sets $K_{M,P}$ (as in the preliminaries section \[sectio\_pre\]): $$\begin{aligned}
K_{M_1M_2}=\left\{f \in C([0,T];\mathbb{R}^d); \|f\|_{\infty} \leq M_1, [f]_{C^{\alpha}} \leq M_2\right\}\nonumber
\end{aligned}$$ Now, we will show that for a given $\varepsilon >0$, there are $M_1, M_2 > 0$ such that $$\begin{aligned}
P(\textbf{x}_{\cdot}^n \in K_{M_1M_2}^c) < \varepsilon, \text{ for all } n\in \mathbb{N}.\nonumber
\end{aligned}$$ This means that $$\begin{aligned}
P(\|\textbf{x}_{\cdot}^n\|_{\infty} > M_1 \text{ or } [\textbf{x}_{\cdot}^n]_{C^{\alpha}} > M_2) < \varepsilon.\nonumber
\end{aligned}$$ A sufficient condition is $$\begin{aligned}
\label{pair1}
P(\|\textbf{x}_{\cdot}^n\|_{\infty} > M_1) < \frac{\varepsilon}{2} \text{ and } P([\textbf{x}_{\cdot}^n]_{C^{\alpha}} > M_2) < \frac{\varepsilon}{2}.
\end{aligned}$$ Now, we consider the first one $P(\|\textbf{x}_{\cdot}^n\|_{\infty} > M_1) < \frac{\varepsilon}{2}$. Using Markov’s inequality (cf. [@Jacod2004], Corollary 5.1), we get $$\begin{aligned}
P(\|\textbf{x}_{\cdot}^n\|_{\infty} > M_1) \leq \frac{1}{M_1}E\left[\sup_{t \in [0,T]}\left|\textbf{x}_t^n\right|\right],\nonumber
\end{aligned}$$ but $$\begin{aligned}
\sup_{t \in [0,T]}\left|\textbf{x}_t^n\right| = \sup_{t \in [0,T]}\left\{\left|\textbf{x}_{a_i,0}^n + \int_{0}^{t}F_1(\textbf{x}_y^n,y)dy\right|,\left|\textbf{x}_{b_k,0}^n + \int_{0}^{t}F_2(\textbf{x}_y^n,y)dy+\int_{0}^{t}\tilde{\sigma} dB_y\right|\right\}.\nonumber
\end{aligned}$$ We estimate $$\begin{aligned}
\sup_{t \in [0,T]}\left|\textbf{x}_t^n\right| \leq \sup_{t \in [0,T]}\left\{|\textbf{x}_{a_i,0}^n| + \left|\int_{0}^{t}F_1(\textbf{x}_y^n,y)dy\right|,|\textbf{x}_{b_k,0}^n|+ \left|\int_{0}^{t}F_2(\textbf{x}_y^n,y)dy\right|+\left|\int_{0}^{t}\tilde{\sigma} dB_y\right|\right\}\nonumber
\end{aligned}$$ Since $F_1,F_2$ bounded, then we have $$\begin{aligned}
\int_{0}^T\left|F_1(\textbf{x}_y^n,y)\right|dy &= \int_{0}^{T}\left|-\Upsilon(s({\textbf x}_{a_i}^n,y))\left(\frac{\nabla \phi({\textbf x}_{a_i}^n)}{\|\nabla \phi({\textbf x}_{a_i}^n)\|}\right)(p_{\max} - p({\textbf x}_{a_i}^n,y))\right|dy \leq C,\\
\int_{0}^T\left|F_2(\textbf{x}_y^n,y)\right|dy &= \int_{0}^T\left|\sum_{j = 1}^N\frac{({\textbf x}_{c_j}^n - {\textbf x}_{b_k}^n)}{\|{\textbf x}_{c_j}^n - {\textbf x}_{b_k}^n\|}w(\hat{\delta},s({\textbf x}_{b_k}^n,y)) - \nabla H_\epsilon({\textbf x}_{b_k}^n,y)\right|dy \leq C.
\end{aligned}$$ Taking the expectation, we have the following estimate $$\begin{aligned}
E\left[\sup_{t \in [0,T]}|\textbf{x}_t^n|\right]\leq C.\nonumber
\end{aligned}$$ Hence, for $\varepsilon > 0$, we can choose $M_1>0$ such that $P(\|\textbf{x}_\cdot^n\|_{\infty} > M_1) < \frac{\varepsilon}{2}$.
From now on, we consider the second inequality in . This reads $$\begin{aligned}
P([\textbf{x}_{\cdot}^n]_{C^{\alpha}} > M_2)=P\left(\sup_{t \neq r} \frac{|\textbf{x}_t^n - \textbf{x}_t^r|}{|t - r|} > M_2\right) \leq \frac{\varepsilon}{2}. \nonumber
\end{aligned}$$ Let us introduce another class of compact sets now in the Sobolev space $W^{\alpha,p}(0,T; \mathbb{R}^d)$ (which for suitable exponents lies in $C^{\gamma}([0,T], \mathbb{R}^d)$). Additionally, we recall the relatively compact sets $K'_{M_1M_2}$ in such that $$\begin{aligned}
K'_{M_1M_2} = \left\{f \in C([0,T];\mathbb{R}^d); \|f\|_{\infty} \leq M_1, [f]_{W^{\alpha,p}} \leq M_2 \right\}.\nonumber
\end{aligned}$$ A sufficient condition for $K'_{M_1M_2}$ to be relatively compact in the underlying space is $\alpha p > 1$. Having this in mind, we wish to prove that there exist $\alpha \in (0,1)$ and $p > 1$ with $\alpha p > 1$ together with the property: given $\varepsilon > 0$, there is $M>0$ such that $$\begin{aligned}
P([\textbf{x}_{\cdot}^n]_{W^{\alpha,p}}> M_2) < \frac{\varepsilon}{2},\nonumber
\end{aligned}$$ for every $n \in \mathbb{N}$.
Using Markov’s inequality, we obtain $$\begin{aligned}
P([\textbf{x}_{\cdot}^n]_{W^{\alpha,p}} > M_2) &\leq \frac{1}{M}E\left[\int_0^T\int_0^T \frac{|\textbf{x}_t^n - \textbf{x}_r^n|^p}{|t - r|^{1 + \alpha p}}dtdr\right] \nonumber\\
&= \frac{C}{M}\int_0^T\int_0^T\frac{E\left[|\textbf{x}_t^n - \textbf{x}_r^n|^p\right]}{|t - r|^{1+\alpha p}}dtdr.\nonumber
\end{aligned}$$ For $t \geq r$, we have $$\begin{aligned}
\textbf{x}_t^n - \textbf{x}_r^n = \begin{pmatrix}
\int_{r}^{t}F_1(\textbf{x}_y^n,y)dy \\
\int_{r}^{t}F_2(\textbf{x}_y^n,y)dy
\end{pmatrix} + \begin{pmatrix}
0\\ \int_{r}^{t}\tilde{\sigma} dB_y
\end{pmatrix}.\nonumber
\end{aligned}$$ Let us introduce some further notation. For a vector $u = (u_1,u_2)$, we set $|u| := |u_1|+|u_2|$. At this moment, we consider the following expression: $$\begin{aligned}
\label{modulus_def}
|\textbf{x}_t^n - \textbf{x}_r^n| = \left|\int_{r}^{t}F_1(\textbf{x}_y^n,y)dy\right| + \left|\int_{r}^{t}F_2(\textbf{x}_y^n,y)dy+\int_{r}^{t}\tilde{\sigma} dB_y\right|.
\end{aligned}$$ Taking the modulus up to the power $p>1$, reads $$\begin{aligned}
\label{modulus_p}
|\textbf{x}_t^n - \textbf{x}_r^n|^p &= \left( \left|\int_{r}^{t}F_1(\textbf{x}_y^n,y)dy\right| + \left|\int_{r}^{t}F_2(\textbf{x}_y^n,y)dy+\int_{r}^{t}\tilde{\sigma} dB_y\right|\right)^p\nonumber\\
&\leq \left|\int_{r}^{t}F_1(\textbf{x}_y^n,y)dy\right|^p + \left|\int_{r}^{t}F_2(\textbf{x}_y^n,y)dy\right|^p + \left|\int_{r}^{t}\tilde{\sigma} dB_y\right|^p\nonumber\\
&\leq \int_{r}^{t}\left|F_1(\textbf{x}_y^n,y)\right|^pdy+\int_{r}^{t}\left|F_2(\textbf{x}_y^n,y)\right|^pdy+\left|\int_{r}^{t}\tilde{\sigma} dB_y\right|^p\nonumber\\
&\leq C(t-r)^p +\left|\int_{r}^{t}\tilde{\sigma} dB_y\right|^p.
\end{aligned}$$ Taking the expectation on , we obtain the following estimate $$\begin{aligned}
\label{expectation_modu}
E[|\textbf{x}_t^n - \textbf{x}_r^n|^p] \leq C(t-r)^p +E\left[\left|\int_{r}^{t}\tilde{\sigma} dB_y\right|^p\right].
\end{aligned}$$ Now, we consider the second term of the right hand side of . By using the Burkholder-Davis-Gundy inequality (cf. [@Prato14], Hypothesis 6.4), we obtain $$\begin{aligned}
\label{Bur_ine}
E\left[\left|\int_{r}^{t} \tilde{\sigma} dB_y\right|^p\right] \leq CE\left[\left(\int_{r}^{t} dy\right)^{p/2}\right] \leq C(t - r)^{p/2}.
\end{aligned}$$ Combining and , we have the upper bound $$\begin{aligned}
E[|\textbf{x}_t^n - \textbf{x}_r^n|^p] \leq C(t - r)^{p/2}.\nonumber
\end{aligned}$$ On the other hand, the integral $$\int_{0}^{T}\int_{0}^{T} \frac{1}{|t - r|^{1 + (\alpha - \frac{1}{2})p}}dtdr$$ is finite if $\alpha < \frac{1}{2}$. Consequently, we can pick $\alpha < \frac{1}{2}$. Taking now $p>2$ together with the constraint $\alpha p > 1$, we can find $M_2 > 0$ such that $$\begin{aligned}
P\left([\textbf{x}_{\cdot}^n]_{W^{\alpha,p}} > M_2\right) < \frac{\varepsilon}{2}.\nonumber
\end{aligned}$$ This complete the proof of the Lemma.
Assume $(\text{A}_1)$ and $(\text{A}_2)$ hold. There exits a solution of the microscopic dynamics SDE system and .
From Lemma \[tightness\], we have obtained that the sequence $\{Q^n\}$ is tight in $C([0,T];\mathbb{R}^d)$. Applying the Prokhorov’s Theorem (cf. [@Billingsley1999], Theorem $5.1$), there are subsequences $\{Q^{n_k}\}$ which converge weakly. For simplicity of the notation, we denote these subsequences by $\{Q^{n}\}$. This means that we have $\{Q^n\}$ converges weakly to some probability measure $Q$ on Borel sets in $C([0,T];\mathbb{R}^d)$.
Since we have that $Q(\textbf{x}_t^n)$ converges weakly to $Q(\textbf{x}_t)$, by using the Skorohod Representation Theorem (cf. [@Prato14], Theorem $2.4$) , there exists a probability space $(\widetilde{\Omega}, \tilde{\mathcal{F}},\tilde{P})$ with the filtration $\{\tilde{\mathcal{F}}_t\}$ and $\tilde{\textbf{x}}_t^n$, $\tilde{\textbf{x}}_t$ belong to $C([0,T]; \mathbb{R}^d)$ with $n\in \mathbb{N}$, such that $Q(\tilde{\textbf{x}}) = Q(\textbf{x})$, $Q(\tilde{\textbf{x}}_t^n) = Q(\textbf{x}_t^n)$ with $n \in \mathbb{N}$, and $\tilde{\textbf{x}}_t^n \to \tilde{\textbf{x}}_t$ as $n \to \infty$, $\tilde{P}-$a.s.
By using this argument, we get that $\tilde{\textbf{x}}_t^n$ converges to $\tilde{\textbf{x}}_t$ a.s. in the uniform topology on compacts sets, and then $\tilde{\textbf{x}}_t^n$ converges in probability towards $\tilde{\textbf{x}}_t$. It leads to $$\begin{aligned}
\int_{r}^{t}\tilde{b}_n(\tilde{\textbf{x}}_y^n,y)dy &\rightarrow \int_{r}^{t}\tilde{b}(\tilde{\textbf{x}}_y,y)dy \end{aligned}$$ in probability. To prove that these new processes satisfy the SDEs, we rely on an argument of Bensoussan cf. [@Bensoussan1995]. Essentially, we need to check that the pair $(\tilde{\textbf{x}}_{\cdot}^n, \tilde{B}_{\cdot})$ satisfies the following equation $$\begin{aligned}
\label{equalweak1}
\tilde{\textbf{x}}_t^n = \tilde{\textbf{x}}_0^n + \int_{0}^tb_n(\tilde{\textbf{x}}_y^n,y)dy + \int_0^t\tilde{\sigma}d\tilde{B}_y.
\end{aligned}$$ Let us call $$\begin{aligned}
\tilde{\mathcal{M}}_t^n:=\tilde{\textbf{x}}_t^n - \tilde{\textbf{x}}_0^n - \int_{0}^tb_n(\tilde{\textbf{x}}_y^n,y)dy - \int_0^t\tilde{\sigma}d\tilde{B}_y.\nonumber
\end{aligned}$$ To prove , we define the following equation $$\begin{aligned}
\mathcal{M}_t^n:=\textbf{x}_t^n - \textbf{x}_0^n - \int_{0}^tb_n(\textbf{x}_y^n,y)dy - \int_0^t\tilde{\sigma}dB_y.\nonumber
\end{aligned}$$ Clearly, this definition implies $\mathcal{M}_t^n=0 \quad P \text{ a.s.}$. Hence, we have $
E\left[\frac{\mathcal{M}_t^n}{\mathcal{M}_t^n + 1}\right] = 0$. Now, we want to check that $$\begin{aligned}
\tilde{E}\left[\frac{\tilde{\mathcal{M}}_t^n}{\tilde{\mathcal{M}}_t^n + 1}\right] = 0.\label{MM}
\end{aligned}$$ Consider the fact that $$\begin{aligned}
\frac{\mathcal{M}_t^n}{\mathcal{M}_t^n + 1} = \phi^n(\textbf{x}_{\cdot}^n,B_{\cdot}) \quad \text{ and } \frac{\tilde{\mathcal{M}}_t^n}{\tilde{\mathcal{M}}_t^n + 1} = \phi^n(\tilde{\textbf{x}}_{\cdot}^n,\tilde{B}_{\cdot}),\nonumber
\end{aligned}$$ where $\phi^n$ belong to $\mathcal{B}(E)$ which is the Borel sets of $E$ with $E := C([0,T]; \mathbb{R}^d)$.\
We note that $$\begin{aligned}
\tilde{E}\left[\frac{\tilde{\mathcal{M}}_t^n}{\tilde{\mathcal{M}}_t^n + 1}\right] = \tilde{E}[\phi^n(\tilde{\textbf{x}}_{\cdot}^n,\tilde{B}_{\cdot})] = \int_{\mathcal{B}(E)}\phi^n dQ^n = E[\phi^n(\textbf{x}_{\cdot}^n,B_{\cdot})] = E\left[\frac{M_t^n}{\mathcal{M}_t^n + 1}\right].\nonumber
\end{aligned}$$ Thus, holds. This implies $\tilde{\mathcal{M}}_t^n = 0 \quad \tilde{P} \text{ a.s.}$ Therefore, the new process, posed in the new probability space, satisfies the SDE.
The solution of SDE system and is unique.
Assume that we have two distinct solutions $\textbf{x}_1$ and $\textbf{x}_2$ belonging to $C([0,T];\mathbb{R}^d)$ with continuous sample paths almost surely. Then it also holds $$\begin{aligned}
\textbf{x}_1(t) - \textbf{x}_2(t) = \int_{0}^{t}(b(\textbf{x}_1,y) - b(\textbf{x}_2,y))dy,\nonumber
\end{aligned}$$ and hence, $$\begin{aligned}
\label{2solminus}
E(|\textbf{x}_1(t) - \textbf{x}_2(t)|) \leq E\left(\left|\int_{0}^{t}b(\textbf{x}_1(y),y) - b(\textbf{x}_2(y),y)dy\right|\right).
\end{aligned}$$ For a detailed check, we consider $$\begin{aligned}
E\begin{pmatrix}\label{expect_1}
|\int_{0}^{t}F_1(\textbf{x}_1(y),y) - F_1(\textbf{x}_2(y),y)dy|\\
|\int_{0}^{t}F_2(\textbf{x}_1(y),y) - F_2(\textbf{x}_2(y),y)dy|
\end{pmatrix}.
\end{aligned}$$ Since the terms of $F_1$ is Lipschitz, the first line of reads $$\begin{aligned}
\label{1st_line}
&\left|\int_{0}^{t}F_1(\textbf{x}_1(y),y) - F_1(\textbf{x}_2(y),y)dy\right| \nonumber\\&=\Bigg|\int_{0}^{t}\Bigg(-\tilde{v}_{s}({\textbf x}_{a_i}^1(y),y)\left(\frac{\nabla \phi({\textbf x}_{a_i}^1(y))}{\|\nabla \phi({\textbf x}_{a_i}^1(y))\|}\right)(p_{\max} - p({\textbf x}_{a_i}^1(y),y)) \nonumber\\&+ \tilde{v}_{s}({\textbf x}_{a_i}^2(y),y)\left(\frac{\nabla \phi({\textbf x}_{a_i}^2(y))}{\|\nabla \phi({\textbf x}_{a_i}^2(y))\|}\right)(p_{\max} - p({\textbf x}_{a_i}^2(y),y))\Bigg)dy\Bigg|\nonumber\\
&\leq C\int_{0}^{t}|\textbf{x}_{a_i}^2(y) - \textbf{x}_{a_i}^1(y)|dy.
\end{aligned}$$ By the same argument, the second line of becomes $$\begin{aligned}
\label{2nd_line}
&\left|\int_{0}^{t}F_2(\textbf{x}_1(y),y) - F_2(\textbf{x}_2(y),y)dy\right|=\nonumber\\
&\Bigg|\int_{0}^{t}\Bigg(\sum_{j = 1}^N\frac{({\textbf x}_{c_j}^1 - {\textbf x}_{b_k}^1)}{\|{\textbf x}_{c_j}^1 - {\textbf x}_{b_k}^1\|}w(\hat{\delta},s({\textbf x}_{b_k}^1,y)) - \nabla H_\epsilon({\textbf x}_{b_k}^1,y) \nonumber\\&- \sum_{j = 1}^N\frac{({\textbf x}_{c_j}^2 - {\textbf x}_{b_k}^2)}{\|{\textbf x}_{c_j}^2 - {\textbf x}_{b_k}^2\|}w(\hat{\delta},s({\textbf x}_{b_k}^2,y)) + \nabla H_\epsilon({\textbf x}_{b_k}^2,y)\Bigg)dy\Bigg|= \nonumber\\
\nonumber\\
&\Bigg|\int_{0}^t\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^1,y))-\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))\nonumber\\
&+\sum_{j=1}^{N_B}\frac{\textbf{x}_{b_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{b_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^1,y))-\sum_{j=1}^{N_B}\frac{\textbf{x}_{b_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{b_j}^2 - \textbf{x}_{b_k}^2\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y)) \nonumber\\
&\nabla H_\epsilon({\textbf x}_{b_k}^2,y) - \nabla H_\epsilon({\textbf x}_{b_k}^1,y)dy
\Bigg|\nonumber\\
&= \left|\int_{0}^{t}(A_1 + A_2 + A_3)dy\right|,
\end{aligned}$$ where $$\begin{aligned}
A_1&:= \sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^1,y))-\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y)),\\
A_2&:= \sum_{j=1}^{N_B}\frac{\textbf{x}_{b_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{b_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^1,y))-\sum_{j=1}^{N_B}\frac{\textbf{x}_{b_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{b_j}^2 - \textbf{x}_{b_k}^2\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y)),\\
A_3&:= \nabla H_\epsilon({\textbf x}_{b_k}^2,y) - \nabla H_\epsilon({\textbf x}_{b_k}^1,y).
\end{aligned}$$ By the Lipschitz property of the weight factors, the term $\left|\int_{0}^{t}A_1dy\right|$ reads $$\begin{aligned}
\label{A_1}
\left|\int_{0}^{t}A_1dy\right| &= \Bigg|\int_{0}^t \sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^1,y))-\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))\nonumber\\ &+ \sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))-\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2\|}w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))dy\Bigg|\nonumber\\
&=\Bigg|\int_{0}^t\sum_{j=1}^{N_A}\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|}\left(w(\hat{\delta},s(\textbf{x}_{b_k}^1,y)) - w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))\right) \nonumber\\
&+\sum_{j=1}^{N_A}\left(\frac{\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|} -\frac{\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2}{\|\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2\|} \right)w(\hat{\delta},s(\textbf{x}_{b_k}^2,y))dy \Bigg|\nonumber\\
&\leq C_1\int_{0}^{t}\left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y)\right|dy + C_2\int_{0}^{t}\Bigg(\left|\textbf{x}_{a_j}^1(y) - \textbf{x}_{a_j}^2(y)\right| \nonumber\\&+ \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y) \right|\Bigg)dy.
\end{aligned}$$ In , $C_1, C_2$ are constants with $C_2$ defined as $$\begin{aligned}
C_2 := \max\left\{\frac{1}{\|\textbf{x}_{a_j}^1 - \textbf{x}_{b_k}^1\|},\frac{1}{\|\textbf{x}_{a_j}^2 - \textbf{x}_{b_k}^2\|}\right\}. \nonumber
\end{aligned}$$ Thus, can be written as $$\begin{aligned}
\label{ineA1}
\left|\int_{0}^{t}A_1dy\right|\leq C\int_{0}^t\left|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)\right| + \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y) \right|dy.
\end{aligned}$$ By the same argument, we consider $\left|\int_{0}^{t}A_2dy\right|$, then we obtain $$\begin{aligned}
\label{ineA2}
\left|\int_{0}^{t}A_2dy\right| &\leq C\int_{0}^{t}\left|\textbf{x}_{b_j}^1(y) - \textbf{x}_{b_j}^2(y)\right| + \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y)\right|dy\nonumber\\
&\leq C\int_{0}^{t}\left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y)\right|dy.
\end{aligned}$$ Now, using the Lipschitz property of $\nabla H_\varepsilon$, we estimate the last term $\left|\int_{0}^{t}A_3dy\right|$ by $$\begin{aligned}
\label{ineA3}
\left|\int_{0}^{t}A_3dy\right|\leq C\int_{0}^{t}\left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y)\right|dy.
\end{aligned}$$ Combining , and , gives $$\begin{aligned}
\left|\int_{0}^{t}(A_1 + A_2 + A_3)dy\right| \leq C\int_{0}^t\left|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)\right| + \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y) \right|dy.\nonumber
\end{aligned}$$ Therefore, from , , together with taking expectation, we obtain $$\begin{aligned}
&E\begin{pmatrix}
|\int_{0}^{t}F_1(\textbf{x}_1(y),y) - F_1(\textbf{x}_2(y),y)dy|\\ |\int_{0}^{t}F_2(\textbf{x}_1(y),y) - F_2(\textbf{x}_2(y),y)dy|
\end{pmatrix} \nonumber\\&\leq C \begin{pmatrix}\int_{0}^t|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)|dy\\
\int_{0}^t\left|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)\right| + \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y) \right|dy
\end{pmatrix}\nonumber
\end{aligned}$$ From , we get the following estimate $$\begin{aligned}
\begin{pmatrix}
E\left(\left|\textbf{x}_{a_i}^1 - \textbf{x}_{a_i}^2\right|\right)\\
E\left(\left|\textbf{x}_{b_k}^1 - \textbf{x}_{b_k}^2\right|\right)
\end{pmatrix} \leq C \begin{pmatrix}\int_{0}^t|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)|dy\\
\int_{0}^t\left|\textbf{x}_{a_i}^1(y) - \textbf{x}_{a_i}^2(y)\right| + \left|\textbf{x}_{b_k}^1(y) - \textbf{x}_{b_k}^2(y) \right|dy
\end{pmatrix}\nonumber
\end{aligned}$$ Thanks to the Grönwall lemma, we obtain $$\begin{aligned}
\begin{pmatrix}
E\left(\left|\textbf{x}_{a_i}^1 - \textbf{x}_{a_i}^2\right|\right)\\
E\left(\left|\textbf{x}_{b_k}^1 - \textbf{x}_{b_k}^2\right|\right)
\end{pmatrix} = \begin{pmatrix}
0\\0
\end{pmatrix}\nonumber
\end{aligned}$$ This implies that $\textbf{x}_1(t) = \textbf{x}_2(t)$ almost surely that $$\begin{aligned}
P\left(\sup_{t \in [0,T]}\left|\textbf{x}_1(t) - \textbf{x}_2(t) \right|=0\right)=1.\nonumber
\end{aligned}$$
Background results
------------------
This section contains a few remarks about the regularity of the agents’s paths as well as of the concentration of smoke. These results are fairly standard; we add them here for the sake of completeness of our arguments.
### A regularized Eikonal equation {#eiko}
In this section, we regularize the Eikonal equation introduced for Model 1; see . This is often referred to as a viscous Eikonal equation.
For $\varepsilon>0$, we introduce the following semilinear viscous problem approximating as $\varepsilon\to 0$ our Eikonal equation:
Find $\phi_\varepsilon\in C(\overline{\Omega})\cap C^2(\Omega)$ satisfying $$\begin{aligned}
\begin{cases}
-\varepsilon\Delta \phi_\varepsilon + |\nabla\phi_\varepsilon|^2 = f^2 \quad &\text{ in } \Omega,\\
\phi_\varepsilon(x) = 0 \quad &\text{ at } \partial \Omega \cup \partial G,\\
\nabla\phi_\varepsilon\cdot \textbf{n} = g \quad &\text{ at } E,
\end{cases}\end{aligned}$$ With suitable assumptions on $f,g,\Omega$, this problem with mixed Dirichlet-Neumann boundary conditions can be shown to be well-posed; see e.g. Theorem 2.1 , p.10, in [@Schieborn:2006] for the case of the Dirichlet problem. Note also that it is sometimes convenient to transform this semilinear PDE via $$\begin{aligned}
w_a := \exp(-\varepsilon^{-1}\phi_\varepsilon) - 1,\end{aligned}$$ where $a = \frac{1}{\varepsilon}$. Then $w_a$ becomes a solution of the following linear PDE with mixed Dirichlet-Robin boundary conditions: $$\begin{aligned}
\begin{cases}\label{transform:eqn}
-\Delta w_a + f^2 a^2 w_a + a^2 = 0 \quad &\text{ in } \Omega,\\
w_a = 0 \quad &\text{ at } \partial \Omega \cup \partial G,\\
\nabla w_a\cdot \textbf{n} = \tilde{g}(w_a) \quad &\text{ at } E,
\end{cases}\end{aligned}$$ where $\tilde{g}(w_a)= -\varepsilon^{-1}(w_a + 1)g.$
### Higher regularity estimates for the smoke concentration
We introduce the evolution of fire throughout a diffusion-dominated convection process. The production and spreading of smoke, with the smoke density $s(\textbf{x},t)$, are described as the following diffusion-drift-reaction equation: $$\begin{aligned}
\begin{cases}\label{paraboliceqn1}
\partial_t s + \textrm{div} (-D\nabla s + \textbf{v}_ds) = y_{s}H(\textbf{x},t) &\textrm{ in } \Omega \times (0,T],\\
\left(-D\nabla s + \textbf{v}_ds\right)\cdot\textbf{n} = 0 &\textrm{ on } \partial \Omega \cup \partial G\times (0,T], \\
\left(-D\nabla s + \textbf{v}_ds\right)\cdot\textbf{n} = \lambda s &\textrm{ at } \partial E \times (0,T],\\
s(x,0) = s_0 &\textrm{ in } \Omega \times \{t = 0\},
\end{cases}\end{aligned}$$ where $D$ is the smoke diffusive coefficient, $\textbf{v}_d$ is a given drift corresponding (e.g. wind’s velocity,…), $y_{s}$ is a smoke production coefficient, while $H$ represents the shape and intensity of the fire. The center of the fire location is denoted by $\textbf{x}_0$ with radius $r_0$. $H$ reads $$\begin{aligned}
\label{H_def}
H(\textbf x,t) = \begin{cases}
R(\textbf x,t) \quad \text{ if } |\textbf x-{\textbf x}_0| <r_0,\\
0 \quad \text{ otherwise },
\end{cases}\end{aligned}$$ where $R(\textbf x,t)$ is defined by $$\begin{aligned}
R(\textbf{x},t) = c(t)\exp\left(-\kappa\frac{|\textbf x-{\textbf x}_0|}{L}\right).\nonumber\end{aligned}$$ Here, $\kappa$ is the convection heat transfer constant coefficient, $c(t)$ is a constant function depending on $t$, $L$ is the typical length of a stationary temperature distribution within the geometry and $\lambda$ is an interface exchange smoke coefficient. For convenience, in order to take the gradient of $H$, we consider $H_\epsilon$ a suitable mollification of $H$. In our case, from now on, we consider the coefficient $y_s$ as a constant $c_y$ and put $f(x,t) := c_yH_\epsilon(x,t)$, then becomes $$\begin{aligned}
\begin{cases}\label{paraboliceqn}
\partial_t s + \text{div} (-D\nabla s + \textbf{v}_ds) = f(\textbf{x},t) &\text{ in } \Omega \times (0,T],\\
\left(-D\nabla s + \textbf{v}_ds\right)\cdot\textbf{n} = 0 &\text{ on } \partial \Omega \cup \partial G\times (0,T], \\
\left(-D\nabla s + \textbf{v}_ds\right)\cdot\textbf{n} = \lambda s &\text{ at } \partial E \times (0,T],\\
s(\textbf{x},0) = s_0 &\text{ in } \Omega \times \{t = 0\},
\end{cases}\end{aligned}$$ In order to have a well-posed dynamics of pedestrians model, we need the solution of to belong to $C([0,T];C^1(\Omega))$. Since the pedestrian dynamics system couple one way with the smoke equation, the solution $s$ of should be Lipschitz to guarantee the well-posedness of the system. In the next part, we adapt the approach in [@Pankavich15] to get a short proof of increased parabolic regularity for a bounded domain $\Omega$ in $\mathbb{R}^d$. Moreover, from now on, we assume the boundaries $\partial \Omega \cup \partial G$ and $\partial E$ are $C^2$ (or, at least, they satisfy the exterior sphere condition).
\[lower\_regularity\]\[Lower-order regularity\] Assume Assumptions $(\text{A}_3)$, $(\text{A}_4)$ to hold. Suppose $f\in H^1(\Omega)$ and $\textbf{v}_d \in W^{1,\infty}(\Omega)$. Then, for any $T>0$, $t \in (0,T]$, there exists a unique $$\begin{aligned}
s \in C([0,T];H^1(\Omega)) \text{ and } s' \in L^2(0,T;H^{-1}(\Omega))\nonumber
\end{aligned}$$ that solves . Furthermore, the following a priori estimates hold $$\begin{aligned}
\sup_{t \in [0,T]}\|s\|_{L^2(\Omega)}^2 \leq C_T\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{L^2(\Omega)}^2\right) \text{ and } \|\nabla s\|_{L^2(\Omega)}^2 \nonumber\\\leq \frac{C_T}{t}\left(\|s_0\|_{L^2(\Omega)}^2+\|f\|_{H^1(\Omega)}^2\right).\nonumber
\end{aligned}$$
We adapt the arguments from [@Pankavich15] to our setting and split the proof into fourth steps:
- Step 1: [*Galerkin approximation*]{}
Firstly, we assume that the functions $w_k = w_k(x) (k \in \mathbb{N})$ are smooth and that $$\begin{aligned}
\{w_k\}_{k=1}^{\infty} \text{ is an orthonormal basis of } H^1(\Omega).
\end{aligned}$$ We are looking for an approximation of in the form $$\begin{aligned}
\label{approxima_galerkin1}
s_m(t) := \sum_{k=1}^{m}d_n^k(t)w_k,
\end{aligned}$$ where the coefficients $d_m^k$ satisfy the following system $$\begin{aligned}
\label{weakformgalerkin1}
\begin{cases}
\langle s'_m, w_k\rangle_{L^2(\Omega)} + \langle D\nabla s_m,\nabla w_k\rangle_{L^2(\Omega)} - \langle\textbf{v}_ds_m,\nabla w_k\rangle_{L^2(\Omega)} \\+ \langle \lambda s_m,w_k\rangle_{L^2(\partial E)} = \langle f,w_k \rangle_{L^2(\Omega)}, \\
s_m(0) = s_{0m} \text{ with } k = 1\ldots m,
\end{cases}
\end{aligned}$$ where $$\begin{aligned}
\label{s0approximateform1}
s_{0m} = \sum_{k = 1}^{m}c_{m}^{k}w_k \to s_0
\end{aligned}$$ strongly in $L^2(\Omega)$.
- Step 2: [*A priori estimates*]{}
The goal of this step is to obtain some useful a priori estimates. Multiplying by $d_m^k(t)$, taking the summation for $k\in \{1,\ldots,m\}$. Then recalling , using Green’s formula together with the mixed boundary condition, we arrive at $$\begin{aligned}
\label{weakform_test1}
\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2 + \int_{\Omega}\nabla s_m\cdot D\nabla s_mdx + \int_{\partial E}\lambda s_m^2d\sigma(E) \nonumber\\= \int_{\Omega}s_m\textbf{v}_d\cdot\nabla s_mdx + \int_{\Omega}fs_mdx.
\end{aligned}$$ Thanks to Cauchy-Schwarz’s inequality $\varepsilon$ for an $\varepsilon >0$, we have $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2 + \int_{\Omega}\nabla s_m\cdot D\nabla s_mdx + \int_{\partial E}\lambda s_m^2d\sigma(E) \leq + \|\textbf{v}_d\|_{1,\infty}\Big(\varepsilon\|s_m\|_{L^2(\Omega)}^2 \nonumber\\+\frac{1}{\varepsilon}\|\nabla s_m\|_{L^2(\Omega)}^2 \Big) + \frac{1}{2}\left(\|f\|_{L^2(\Omega)}^2 + \|s_m\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ Next, by using the ellipticity property of the diffusion coefficient $D$ and the assumption on the interface exchange coefficient, we obtain $$\begin{aligned}
\label{prior_1}
\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2 \leq -\underline{\theta}\|\nabla s_m\|_{L^2(\Omega)}^2 + \underline{\gamma}\|s_m\|_{L^2(\partial E)}^2 + \|\textbf{v}_d\|_{1,\infty}\Bigg(\frac{1}{\varepsilon}\|s_m\|_{L^2(\Omega)}^2 \nonumber\\+\varepsilon\|\nabla s_m\|_{L^2(\Omega)}^2 \Bigg) + \frac{1}{2}\left(\|f\|_{L^2(\Omega)}^2 + \|s_m\|_{L^2(\Omega)}^2\right).
\end{aligned}$$ By the trace inequality applied to $\|s_m\|_{L^2(\partial E)}^2$, reads $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2 \leq -\underline{\theta}\|\nabla s_m\|_{L^2(\Omega)}^2 + C(\underline{\gamma})\|s_m\|_{H^1(\Omega)}^2 + \|\textbf{v}_d\|_{1,\infty}\Bigg(\frac{1}{\varepsilon}\|s_m\|_{L^2(\Omega)}^2 \nonumber\\+\varepsilon\|\nabla s_m\|_{L^2(\Omega)}^2 \Bigg) + \frac{1}{2}\left(\|f\|_{L^2(\Omega)}^2 + \|s_m\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ By choosing $\varepsilon = \underline{\theta}(2\|\textbf{v}_d\|_{1,\infty})^{-1}$, we get the following estimate $$\begin{aligned}
\label{priori_1}
\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2 \leq C\left(\|f\|_{L^2(\Omega)}^2 + \|s_m\|_{H^1(\Omega)}^2\right) + \left(C(\underline{\gamma}) - \frac{\underline{\theta}}{2}\right)\|\nabla s_m\|_{L^2(\Omega)}^2.
\end{aligned}$$ Multiplying with $\varphi = \partial_x s_m$ differentiated with respect to $x$, we obtain after integrating by part that $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\partial_x s_m\|_{L^2(\Omega)}^2 + \int_{\Omega}\nabla \partial_x s_m\cdot D \nabla \partial_x s_mdx + \int_{\Omega}\nabla\partial_x s_m \cdot \partial_x D\nabla s_mdx \nonumber\\- \int_{\Omega}\nabla\partial_x s_m\cdot\textbf{v}_d\partial_xs_mdx -\int_{\Omega}\nabla\partial_xs_m\cdot \partial_x \textbf{v}_ds_mdx+ \int_{\partial E}\lambda|\partial_x s_m|^2d\sigma(E)\nonumber\\ + \int_{\partial E}\partial_x(\lambda s_m)\partial_xs_md\sigma (E) = \int_{\Omega}\partial_x f \partial_x s_mdx.\nonumber
\end{aligned}$$ This leads to $$\begin{aligned}
\label{priori_test2}
\frac{1}{2}\frac{d}{dt}\|\partial_x s_m\|_{L^2(\Omega)}^2=-\int_{\Omega}\nabla \partial_x s_m\cdot D \nabla \partial_x s_mdx - \int_{\Omega}\nabla\partial_x s_m \cdot \partial_x D\nabla s_mdx\nonumber\\ + \int_{\Omega}\nabla\partial_x s_m\cdot\textbf{v}_d\partial_xs_mdx +\int_{\Omega}\nabla\partial_xs_n\cdot \partial_x \textbf{v}_ds_mdx - \int_{\partial E}\lambda|\partial_x s_m|^2d\sigma(E)\nonumber\\ - \int_{\partial E}\partial_x(\lambda s_m)\partial_xs_md\sigma(E) + \int_{\Omega}\partial_x f \partial_x s_mdx.
\end{aligned}$$ Using the assumptions on $D$ and $\lambda$ as well as Cauchy-Schwarz’s inequality for the RHS of , we obtain the following estimate $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\partial_x s_m\|_{L^2(\Omega)}^2 \leq -\underline{\theta} \|\nabla \partial_x s_m\|_{L^2(\Omega)}^2 + \|D\|_{W^{1,\infty}}\Bigg(\varepsilon_1\|\nabla\partial_xs_m\|_{L^2(\Omega)}^2 \nonumber\\+ \frac{1}{\varepsilon_1}\|\nabla s_m\|_{L^2(\Omega)}^2\Bigg)
+ |\textbf{v}_d|\left(\varepsilon_{2'}\|\nabla \partial_xs\|_{L^2(\Omega)}^2+\frac{1}{\varepsilon_{2'}}\|\partial_xs\|_{L^2(\Omega)}^2\right) \nonumber\\+ \|\textbf{v}_d\|_{1,\infty}\left(\varepsilon_2\|\nabla \partial_xs\|_{L^2(\Omega)}^2 + \frac{1}{\varepsilon_2}\|s\|_{L^2(\Omega)}^2\right) +\underline{\gamma}\|\partial_xs_m\|_{L^2(\partial E)}^2 \nonumber\\+ \|\lambda\|_{1,\infty}\|\partial_xs_m\|_{L^2(\partial E)}^2 + \frac{1}{2}\left(\|\partial_xf\|_{L^2(\Omega)}^2 + \|\partial_xs_m\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ By choosing $\varepsilon_1= \underline{\theta}(4\|D\|_{1,\infty})^{-1}$, $\varepsilon_{2'}=\underline{\theta}(8C)^{-1}$, $\varepsilon_2 = \underline{\theta}(8\|\textbf{v}_d\|_{1,\infty})^{-1}$ together with the use of the trace inequality to handle the boundary terms, we arrive at $$\begin{aligned}
\label{priori_2}
\frac{1}{2}\frac{d}{dt}\|\partial_x s_m\|_{L^2(\Omega)}^2 &\leq -\frac{\underline{\theta}}{2}\|\nabla\partial_xs_m\|_{L^2(\Omega)}^2 + C\|\nabla s_m\|_{L^2(\Omega)}^2 + C\|\partial_xs_m\|_{L^2(\Omega)}^2\nonumber\\&+ C\|s_m\|_{L^2(\Omega)}^2 +C(\underline{\gamma})\left(\|\partial_x s_m\|_{L^2(\Omega)}^2 + \|\nabla \partial_x s_m\|_{L^2(\Omega)}^2\right)
\nonumber\\&+\frac{1}{2}\left(\|\partial_x f\|_{L^2(\Omega)}^2 + \|\partial_x s_m\|_{L^2(\Omega)}^2\right)\nonumber\\
&\leq \left(-\frac{\underline{\theta}}{2} + C(\underline{\gamma})\right)\|\nabla\partial_xs_m\|_{L^2(\Omega)}^2 \nonumber\\&+ C\left(\|\nabla s_m\|_{L^2(\Omega)}^2 + \|\partial_xs_m\|_{L^2(\Omega)}^2 + \|s_m\|_{L^2(\Omega)}^2\right).
\end{aligned}$$ Taking the summation over all first order derivatives, we have $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla s_m\|_{L^2(\Omega)}^2 \leq \left(C(\underline{\gamma})- \frac{\underline{\theta}}{2}\right)\|\nabla^2s_m\|_{L^2(\Omega)}^2 + C\left(\| s_m\|_{H^1(\Omega)}^2 + \|f\|_{H^1(\Omega)}^2\right).\nonumber
\end{aligned}$$ Let us introduce a linear expansion in $t$ as follow $$\begin{aligned}
\label{expansion}
\zeta_1(t) = \|s_m\|_{L^2(\Omega)}^2 + \frac{C(\underline{\theta},\underline{\gamma})t}{2}\|\nabla s_m\|_{L^2(\Omega)}^2.
\end{aligned}$$ Taking the derivative of with respect to $t$, we obtain $$\begin{aligned}
\zeta_1'(t) = \frac{d}{dt}\|s_m\|_{L^2(\Omega)}^2 + \frac{C(\underline{\theta},\underline{\gamma})}{2}\|\nabla s_m\|_{L^2(\Omega)}^2 + \frac{C(\underline{\theta},\underline{\gamma})t}{2}\frac{d}{dt}\|\nabla s_m\|_{L^2(\Omega)}^2.\nonumber
\end{aligned}$$ Combining and , we are led to the following estimate $$\begin{aligned}
\zeta'(t) \leq 2C\left(\|f\|_{L^2(\Omega)}^2 + \|s_m\|_{H^1(\Omega)}^2\right) - 2\left(C(\underline{\gamma})- \frac{\underline{\theta}}{2}\right)\|\nabla s_m\|_{L^2(\Omega)}^2 \nonumber\\+ \frac{C(\underline{\theta},\underline{\gamma})}{2}\|\nabla s_m\|_{L^2(\Omega)}^2 + \frac{C(\underline{\theta},\underline{\gamma})t}{2}\Bigg[\left(C(\underline{\gamma})- \frac{\underline{\theta}}{2}\right)\|\nabla^2s_m\|_{L^2(\Omega)}^2 \nonumber\\+ C\left(\| s_m\|_{H^1(\Omega)}^2 + \|f\|_{H^1(\Omega)}^2\right)\Bigg].\nonumber
\end{aligned}$$ Choosing $\underline{\theta},\underline{\gamma}$ such that $- \frac{\underline{\theta}}{2} + C(\underline{\gamma})< 0$ and put $- \frac{\underline{\theta}}{2} + C(\underline{\gamma}) =: -C(\underline{\theta},\underline{\gamma})$, we obtain $$\begin{aligned}
\label{bef_gronwall}
\zeta'(t) \leq C_T\left(\|f\|_{H^1(\Omega)}^2 + \zeta(t)\right) \textrm{ for a.e. } t \in (0,T).
\end{aligned}$$ Applying Grönwall’s inequality to , we have the following estimate $$\begin{aligned}
\label{Gronwall}
\zeta(t) \leq C_{T}\left(\zeta(0) + \|f\|_{H^1(\Omega)}^2\right) = C_T\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{H^1(\Omega)}^2\right).
\end{aligned}$$ Combining and gives $$\begin{aligned}
\label{pri_resl1}
\|s_m(t)\|_{L^2(\Omega)}^2 \leq C_T\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{H^1(\Omega)}^2\right)
\end{aligned}$$ and $$\begin{aligned}
\label{pri_resl2}
\|\nabla s_m\|_{L^2(\Omega)}^2 \leq \frac{C_T}{Ct}\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{H^1(\Omega)}^2\right).
\end{aligned}$$ The estimates and imply that $s_m$ is a bounded sequence in $H^1(\Omega)$ and a.e $t \in (0,T)$.
- Step 3: [ *Passage to the limit $m \to \infty$*]{}
Using the [*a priori*]{} estimates and , we obtain the following inequality $$\begin{aligned}
\int_{0}^{T}\frac{1}{2}\frac{d}{dt}\|s_m(t)\|_{L^2(\Omega)}^2dt + \int_{0}^{T}\|\nabla s_m\|_{L^2(\Omega)}^2dt \leq C_T\int_{0}^{T}\left(\|f\|_{H^1(\Omega)}^2+\|s_0\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ This implies that $(s_m)$ is a bounded sequence in $L^2(0,T;H^1(\Omega))$.
On the other hand, in order to use Aubin-Lions’s lemma, we additionally need to prove $s_m' \in L^2(0,T;H^{-1}(\Omega))$. Take an arbitrary $v \in H^1(\Omega)$, with $\|v\|_{H^1(\Omega)} \leq 1$. We can deduce for a.e. $0 < t < T$ that $$\begin{aligned}
\langle s_m', v\rangle_{L^2(\Omega)} = \langle f,v\rangle_{L^2(\Omega)} + \langle \textbf{v}_ds_m,\nabla v\rangle_{L^2(\Omega)} - \langle D\nabla s_m,\nabla v\rangle_{L^2(\Omega)} - \langle\lambda s_m,v\rangle_{L^2(\partial E)}.\nonumber
\end{aligned}$$ Then, we get $$\begin{aligned}
\label{est_st11}
|\langle s_m', v\rangle| \leq C\|s_m\|_{H^1(\Omega)} + C\|f\|_{L^2(\Omega)}.
\end{aligned}$$ for $\|v\|_{W^{1,2}(\Omega)} \leq 1$. Moreover, implies that $$\begin{aligned}
\label{est_st12}
\|s_m'\|_{H^{-1}(\Omega)} \leq C\left(\|s_m\|_{H^1(\Omega)} + \|f\|_{L^2(\Omega)}\right) .
\end{aligned}$$ Integrating on $(0,T)$, we obtain the following estimate $$\begin{aligned}
\label{est_st13}
\int_{0}^{T}\|s_m'\|_{H^{-1}(\Omega)}^2dt &\leq C\int_{0}^{T} \|s_m\|_{H^1(\Omega)}+\|f\|_{L^2(\Omega)}dt
\nonumber\\&\leq C\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{L^2(0,T;L^2(\Omega))}\right).
\end{aligned}$$ Thus, $s_m' \in L^2(0,T;H^{-1}(\Omega))$. Therefore, we conclude that $$\begin{aligned}
\begin{cases}
s_m \rightharpoonup s \text{ weakly in } L^2(0,T;H^1(\Omega)),\\
{s'}_m \rightharpoonup s' \text{ weakly in } L^2(0,T;H^{-1}(\Omega)).
\end{cases}\nonumber
\end{aligned}$$ Relying on Aubin-Lions lemma in [@Boyer2013] with $p,q = 2$, $$E_0 = H^1(\Omega), \quad E = L^2(\Omega), \quad E_1 = H^{-1}(\Omega)$$ together with Rellich theorem (cf. [@Evans1997], Section 5.7, Theorem 1) for the compactness embedding $H^1(\Omega) \subset L^2(\Omega)$, we have the sequence $\{s_m\}$ is relatively compact in $L^2(0,T;L^2(\Omega))$ in the strong topology. This sequence also weakly relatively compact in $L^2(0,T;H^1(\Omega))$ and weakly star relatively compact in $C([0,T];L^2(\Omega))$. Hence, there exists a subsequence $s_{m_k}$ (just for simplicity of notation, let us denote it by $s_m$) which converges to a function $s$ belonging to $L^2(0,T;H^1(\Omega))$ and $C([0,T];L^2(\Omega))$. Therefore, we can conclude that there exists a solution $s \in L^2(0,T;H^1(\Omega)) \cup C([0,T];L^2(\Omega))$ satisfying equation .
- Step 4: [*Uniqueness of solutions*]{}
Assume that equation admits $2$ solutions $s_1$ and $s_2$ belonging to\
$L^2(0,T;H^1(\Omega)) \cup C([0,T];L^2(\Omega))$. Denote $w = s_1 - s_2$. Then equation becomes $$\begin{aligned}
\begin{cases}
\partial_t w + \text{div}(-D\nabla w + \textbf{v}_dw) = 0\quad \text{ in } \Omega \times (0,T],\\
(-D\nabla w + \textbf{v}_dw)\cdot \textbf{n} = 0\quad \text{ on } \partial \Omega \cup \partial G\times (0,T],\\
(-D\nabla w + \textbf{v}_dw)\cdot \textbf{n} = \lambda w \quad\text{ at } \partial E \times (0,T],\\
w(t = 0) = 0 \quad \text{ in } \Omega \times \{t=0\},
\end{cases}\nonumber
\end{aligned}$$ Recalling , we note that $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}w^2dx + \int_{\Omega}D|\nabla w|^2dx + \int_{\partial E}\lambda w^2d\sigma(E) = \int_{\Omega}w\textbf{v}_d\cdot\nabla wdx ,\nonumber
\end{aligned}$$ which leads to $$\begin{aligned}
\frac{d}{dt}\left(\|w\|_{L^2(\Omega)}^2\right) + \overline{\theta}\|\nabla w\|_{L^2(\Omega)}^2 + \overline{\gamma}\|w\|_{L^2(\partial E)}^2 \leq C\|w\|_{L^2(\Omega)}^2.\nonumber
\end{aligned}$$ This also implies $$\begin{aligned}
\label{gronwall_12}
\frac{d}{dt}\left(\|w\|_{L^2(\Omega)}^2\right) \leq C\|w\|_{L^2(\Omega)}^2.
\end{aligned}$$ Integrating on $(0,T)$, gives $$\begin{aligned}
\|w\|_{L^2(\Omega)}^2 \leq \|w(0)\|_{L^2(\Omega)}^2 + C\int_{0}^t\|w\|_{L^2(\Omega)}^2.\nonumber
\end{aligned}$$ Grönwall’s lemma ensure $$\begin{aligned}
\|w\|_{L^2(\Omega)}^2 \leq \|w(0)\|_{L^2(\Omega)}^2(1 + Cte^{Ct}),\nonumber
\end{aligned}$$ which for $w(0) = 0$, gives $\|w\|_{L^2(\Omega)} = 0$. So, $w = 0$ a.e. in $\Omega$ and everywhere in $[0,T]$, which ensures the desired uniqueness.
Now, let us show that $s \in C([0,T];H^1(\Omega))$. We consider $w_r(t) = s(t+r) - s(t)$, then $w_r(t)$ satisfies the equation with $f = 0$, $w(0) = s_0 - s(r)$ and $\lambda s(t+r) - \lambda s(t) = \lambda w_r(t)$. By using similar argument, we obtain $$\begin{aligned}
\|w_r(t)\|_{L^2(\Omega)}^2 + \frac{C(\underline{\theta},\underline{\gamma}) t}{2}\|\nabla w_r(t)\|_{L^2(\Omega)}^2 \leq C_T\left(\|s_0 - s(r)\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ Since we have $s \in C([0,T];L^2(\Omega))$, then $\lim_{r \to \infty}\|s(t+r)-s(t)\| = 0$ and\
$\lim_{r \to \infty}\|\nabla s(t+r)-\nabla s(t)\| = 0$ for $t>0$. Therefore, we obtain $s \in C([0,T];H^1(\Omega))$.
\[Higer\_order\]\[High-order regularity\] Assume $(A_3)$, $(A_4)$ to hold. Suppose $f\in H^m(\Omega)$ and $\textbf{v}_d \in W^{m,\infty}(\Omega)$ for every $m \in \mathbb{N}$, and $s_0 \in L^2(\Omega)$. Then, for any $T>0$, $t \in [0,T]$, the solution of satisfies the following estimate $$\begin{aligned}
\|\nabla^k s\|_{L^2(\Omega)}^2 \leq \frac{C_T}{t^k}\left(\|s_0\|_{L^2(\Omega)}^2 + \|f\|_{H^k(\Omega)}^2\right) \text{ for } k=0,1,\ldots,m.\nonumber
\end{aligned}$$
We use the method of induction on $m \in \mathbb{N}$, using the fact that we have done the first case $m=1$ of induction mathematically in Theorem \[lower\_regularity\]. As a notation for derivatives, we define $$\begin{aligned}
\|\nabla^k s\|_{L^2(\Omega)}^2 := \sum_{|\alpha|\leq k}\|\partial_x^{\alpha}s\|_{L^2(\Omega)}^2.\nonumber
\end{aligned}$$
Now, taking the $k-$order derivative with respect to $x$ for $k\in \mathbb{N}$ which is denoted by $\partial_x^{\alpha}$ of the equation , multiplying by $\partial_x^\alpha s$ and integrating the results by parts together with using Green’s theorem for the equation, we obtain $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\partial_x^\alpha s(t)\|_{L^2(\Omega)}^2 + \int_{\Omega}\nabla\partial_x^{\alpha}s\cdot \sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}D\nabla\partial_x^{\gamma}sdx \nonumber\\-\int_{\Omega}\nabla\partial_x^{\alpha}s\cdot \sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}\textbf{v}_d\partial_x^{\gamma}sdx + \int_{\partial E}\partial_x^{\alpha}s\lambda\partial_x^{\alpha}sd\sigma(E)\nonumber\\+\int_{\partial E}\partial_x^{\alpha}(\lambda s)\cdot \partial_x^{\alpha}sd\sigma(E)
=\int_{\Omega}\partial_x^{\alpha}f\partial_x^{\alpha}sdx,\nonumber
\end{aligned}$$ and thus $$\begin{aligned}
\label{high_derivative}
\frac{1}{2}\frac{d}{dt}\|\partial_x^\alpha s(t)\|_{L^2(\Omega)}^2 =- \int_{\Omega}\nabla\partial_x^{\alpha}s\cdot \sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}D\nabla\partial_x^{\gamma}sdx\nonumber\\+\int_{\Omega}\nabla\partial_x^{\alpha}s\cdot\sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}\textbf{v}_d\partial_x^{\gamma}sdx -\int_{\partial E}\partial_x^{\alpha}s\lambda\partial_x^{\alpha}sd\sigma(E)\nonumber\\ - \int_{\partial E}\partial_x^{\alpha}(\lambda s)\cdot \partial_x^{\alpha}sd\sigma(E)
+\int_{\Omega}\partial_x^{\alpha}f\partial_x^{\alpha}sdx.
\end{aligned}$$ Denote $$\begin{aligned}
A&:=- \int_{\Omega}\nabla\partial_x^{\alpha}s\cdot \sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}D\nabla\partial_x^{\gamma}sdx\nonumber\\
&= -\int_{\Omega}\nabla\partial_x^{\alpha}s\cdot D \nabla\partial_x^{\alpha}sdx - \int_{\Omega}\nabla\partial_x^{\alpha}s\cdot \sum_{j=1}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}D\nabla\partial_x^{\gamma}sdx.\nonumber
\end{aligned}$$ We can estimate $|A|$ from above, it follows $$\begin{aligned}
\label{est_A}
|A| \leq -\underline{\theta}\|\nabla\partial_x^{\alpha}s\|_{L^2(\Omega)}^2 + C\|D\|_{1,\infty}\left(\varepsilon_1\|\nabla \partial_x^{\alpha}s\|_{L^2(\Omega)}^2 + \frac{1}{\varepsilon_1}\|s\|_{H^{k-1}(\Omega)}^2\right)\nonumber\\
\leq \frac{-\underline{\theta}}{2}\|\nabla\partial_x^{\alpha}s \|_{L^2(\Omega)}^2 + C\|s\|_{H^{k-1}(\Omega)}^2,
\end{aligned}$$ where we choose $\varepsilon_1 = \underline{\theta}(4C\|D\|_{m,\infty})^{-1}$. Set $$\begin{aligned}
B:=\int_{\Omega}\nabla\partial_x^{\alpha}s\cdot\sum_{j=0}^{k}\sum_{|\beta| = j,\beta+\gamma = \alpha}{\scalebox{0.8}{$\dbinom{\alpha}{\beta}$}}\partial_x^{\beta}\textbf{v}_d\partial_x^{\gamma}sdx,\nonumber
\end{aligned}$$ and obtain the upper bound $$\begin{aligned}
\label{est_B}
|B|\leq C\|\textbf{v}_d\|_{m,\infty}\left(\varepsilon_2 \|\nabla \partial_x^{\alpha}s\|_{L^2(\Omega)}^2 + \frac{1}{\varepsilon_2}\|\partial_x^{\alpha}s\|_{H^{k-1}(\Omega)}^2\right).
\end{aligned}$$ Now, let us label the third and fourth terms in the right hand side of as follow $$\begin{aligned}
\tilde{C}:=-\int_{\partial E}\partial_x^{\alpha}s\lambda\partial_x^{\alpha}sd\sigma(E) - \int_{\partial E}\partial_x^{\alpha}(\lambda s)\cdot \partial_x^{\alpha}sd\sigma(E).\nonumber
\end{aligned}$$ Using the assumptions on $\lambda$ together with applying Cauchy’s inequality, trace inequality for $\tilde{C}$, we have the following estimate: $$\begin{aligned}
\label{est_C}
\tilde{C} \leq \underline{\gamma}\|\partial_x^{\alpha}s\|_{L^2(\partial E)}^2 + \|\lambda\|_{m,\infty}\|\partial_x^\alpha s\|_{L^2(\partial E)}^2
\leq C(\underline{\gamma})\left(\|\nabla\partial_x^{\alpha}s\|_{L^2(\Omega)}^2 + \|\partial_x^{\alpha}s\|_{L^2(\Omega)}^2\right).
\end{aligned}$$ Finally, we estimate the last term of , by using Cauchy’s inequality, we obtain $$\begin{aligned}
\label{est_D}
\int_{\Omega}\partial_x^{\alpha}f\partial_x^{\alpha}sdx
\leq \frac{1}{2}\left(\|\partial_x^{\alpha}f\|_{L^2(\Omega)}^2 + \|\partial_x^{\alpha}s\|_{L^2(\Omega)}^2\right).
\end{aligned}$$ Combining - and choosing $\varepsilon_2 = \underline{\theta}(4C\|\textbf{v}_d\|_{1,\infty})^{-1}$, we have the following estimate $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\partial_x^\alpha s(t)\|_{L^2(\Omega)}^2 \leq \frac{-\underline{\theta}}{2}\|\nabla\partial_x^{\alpha}s \|_{L^2(\Omega)}^2 + C\|s\|_{H^{k-1}(\Omega)}^2 + C\|\partial_x^\alpha s\|_{H^{k-1}(\Omega)}^2 \nonumber\\+ C(\underline{\gamma})\|\nabla \partial_x^{\alpha}s\|_{L^2(\Omega)}^2 + \|f\|_{H^k(\Omega)}^2.\nonumber
\end{aligned}$$ Now, summing all of first-order derivatives, we obtain $$\begin{aligned}
\label{nablak_est}
\frac{1}{2}\frac{d}{dt}\|\nabla^k s(t)\|_{L^2(\Omega)}^2 \leq \left(C(\underline{\gamma}) - \frac{\underline{\theta}}{2} \right)\|\nabla^{k+1}s\|_{L^2(\Omega)}^2 + C\left(\|f\|_{H^k(\Omega)}^2 + \|s\|_{H^k(\Omega)}^2\right).
\end{aligned}$$ Now, we aim to find $s \in C([0,T],H^{m-1}(\Omega))$ using the induction hypothesis under the assumptions $f \in H^{m-1}(\Omega)$ and $D,\lambda,\textbf{v}_d \in W^{m,\infty}(0,T;\Omega)$. Using the same argument as in the case $m=1$, we define $$\begin{aligned}
\label{expansion_2}
\zeta_2(t) := \sum_{k=1}^{n}\frac{(C(\underline{\theta},\underline{\gamma})t)^k}{2^kk!}\|\nabla^k s\|_{L^2(\Omega)}^2.
\end{aligned}$$ Taking the derivative of with respect to $t$, we obtain $$\begin{aligned}
\zeta'_2(t) &= \sum_{k=1}^m\frac{(C(\underline{\theta},\underline{\gamma}))^kt^{k-1}}{2^k(k-1)!}\|\nabla^ks\|_{L^2(\Omega)}^2 + \sum_{k=0}^{m}\frac{(C(\underline{\theta},\underline{\gamma}))^k}{2^kk!}\frac{d}{dt}\|\nabla^ks\|_{L^2(\Omega)}^2\nonumber\\
&:= G_1+G_2.\nonumber
\end{aligned}$$ $$\begin{aligned}
\label{G_2}
G_2 &\leq \sum_{k=0}^m\frac{(C(\underline{\theta},\underline{\gamma})t)^k}{2^kk!}\left(-C(\underline{\theta},\underline{\gamma})\|\nabla^{k+1}s\|_{L^2(\Omega)}^2 + C\left(\|f\|_{H^k(\Omega)}^2 + \|s\|_{H^k(\Omega)}^2\right)\right)\nonumber\\
&=-2\sum_{k=0}^m\frac{(C(\underline{\theta},\underline{\gamma}))^{k+1}t^k}{2^{k+1}k!}\|\nabla^{k+1}s\|_{L^2(\Omega)}^2 \nonumber\\&+ C\sum_{k=0}^{m}\frac{(C(\underline{\theta},\underline{\gamma}))^k}{2^kk!}\left(\|f\|_{H^k(\Omega)}^2 + \|s\|_{H^k(\Omega)}^2\right)\nonumber\\
&\leq -2G_1 - \frac{2(C(\underline{\theta},\underline{\gamma}))^{m+1}t^m}{2^{m+1}m!}\|\nabla^{m+1}s\|_{L^2(\Omega)}^2 \nonumber\\&+ C\sum_{k=0}^{m}\frac{(C(\underline{\theta},\underline{\gamma}))^k}{2^kk!}\left(\|f\|_{H^k(\Omega)}^2 + \|s\|_{H^k(\Omega)}^2\right).
\end{aligned}$$ On the other hand, the induction hypothesis gives the following inequality $$\begin{aligned}
\label{induction_hypo}
\|s\|_{H^{k-1}(\Omega)}^2 \leq \frac{C_T}{t^{k-1}}\left(\|f\|_{H^{k-1}(\Omega)}^2+\|s_0\|_{L^2(\Omega)}^2\right).
\end{aligned}$$ Combining and , we obtain $$\begin{aligned}
\zeta'_2(t) &\leq C\sum_{k=0}^{m}\frac{(C(\underline{\theta},\underline{\gamma}))^k}{2^kk!}\left(\|f\|_{H^k(\Omega)}^2 + \|s\|_{H^k(\Omega)}^2\right)\nonumber\\
&\leq C_T\left(\|f\|_{H^m(\Omega)}^2 + \sum_{k=0}^m\frac{(C(\underline{\theta},\underline{\gamma})t)^k}{2^kk!}\left[\|\nabla^k s\|_{L^2(\Omega)}^2 + \|s\|_{H^{k-1}(\Omega)}^2\right]\right)\nonumber\\
&\leq C_T\left(\|f\|_{H^m(\Omega)}^2 + \zeta_2(t) +\sum_{k=0}^m\frac{(C(\underline{\theta},\underline{\gamma})t)^k}{2^kk!}\frac{C_T}{t^{k-1}}\left[\|f\|_{H^{k-1}(\Omega)}^2 + \|s_0\|_{L^2(\Omega)}^2\right] \right)\nonumber\\
&\leq C_T\left(\|f\|_{H^m(\Omega)}^2 + \|s_0\|_{L^2(\Omega)}^2 + \zeta_2(t)\right).\nonumber
\end{aligned}$$ Grönwall’s inequality yields $$\begin{aligned}
\zeta_2(t) \leq C_T\left(\|f\|_{H^m(\Omega)}^2 + \|s_0\|_{L^2(\Omega)}^2 + \zeta_2(0)\right) \leq C_T\left(\|f\|_{H^m(\Omega)}^2 + \|s_0\|_{L^2(\Omega)}^2\right).\nonumber
\end{aligned}$$ The bound on $\zeta_2(t)$ gives the following estimate $$\begin{aligned}
\|\nabla^m s\|_{L^2(\Omega)}^2 \leq \frac{C_T}{(C(\underline{\theta},\underline{\gamma})t)^m},\nonumber
\end{aligned}$$ which completes the induction proof.
\[rm\_C1\] From Theorem \[Higer\_order\], for $m = 3$, $\Omega \subset \mathbb{R}^d$ with $d=2$, there exists a unique solution $s \in C([0,T]; C^1(\Omega))$ and $s' \in L^2(0,T;H^{-1}(\Omega))$ that solves the equation .
By the same arguments as in Theorem \[lower\_regularity\], this also implies that $s \in C([0,T];H^m(\Omega))$. On the other hand, in our model, we consider our domain in $\Omega \subset \mathbb{R}^d$ with $d = 2$. Moreover, assume $\Omega$ satisfies the strong locally Lipschitz condition (cf. [@Adams03], Theorem 4.12), taking $m=3$, hence $H^3(\Omega)$ compact embedding into $C^1(\Omega)$, i.e. $H^3(\Omega) \subset C^1(\Omega)$. As a conclusion, we obtain $s \in C([0,T]; C^1(\Omega))$. This property ensures that the smoke concentration $s$ is Lipschitz with respect to the space variable – a fact needed to handle the well-posedness of our SDEs.
Discussion {#sec:discussion}
==========
In this chapter, we presented various models aimed at modelling crowds of mixed populations (active and passive) moving inside heterogeneous environments.
Based on our numerical experiments, we observed the impact of passive agents on the residence times of the population and conclude that the lack of environment knowledge can have a substantial impact on the evacuation. Additionally, we notice that the size of the obstacles and doors have a significant influence on the overall dynamics.
While the presence of passive agents increases the evacuation time, we speculate that by manipulating the spacial distribution of active particles, it is possible to optimize the residence time of the passive agents. We plan to investigate these aspects in a forthcoming publication.
From the mathematical point of view, the situation becomes a lot more challenging when there is a feedback mechanism between the agent-based dynamics and the environment (fire, smoke, geometry). Formulating this relationship mathematically would allow for an optimization approach, eventually in a multiscale setting. The main advantage of such a mathematical framework would be to contribute to an intelligent design of building interiors and to provide a basis for smart evacuation signaling systems.
[^1]: Here, we assume that the discomfort is perceptible, known.
|
---
author:
- 'Maxim Yu. Kagan[^1]'
- 'Vitaly A. Mitskan'
- 'Maxim M. Korovushkin'
date: 'Received: date / Revised version: date'
title: 'Phase diagram of the Kohn-Luttinger superconducting state for bilayer graphene'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
In recent years, there has been an increased interest in the possibility of the development of the Cooper instability in graphene under appropriate experimental conditions. Although so far this possibility has not been confirmed, it was experimentally shown [@Heersche07; @Shailos07; @Du08; @Ojeda09; @Kanda10; @Han14] that graphene becomes superconducting when it is in a contact with ordinary superconductors. This fact stimulated theoretical studies on possible implementation of the superconducting phase in an idealized monolayer and bilayer graphene where the authors did not take into account the effect of nonmagnetic impurities and van der Waals potential of the substrate.
Along with the numerous studies of this problem using the electron-phonon mechanism [@Kopnin08; @Basko08; @Lozovik10; @Einenkel11; @Classen14], pairing mechanisms caused by electron correlations [@Black07; @Honerkamp08; @Vucicevic12; @Milovanovic12], and other exotic superconductivity mechanisms [@Hosseini12a; @Hosseini12b], some authors widely discuss the possibility of the development of Cooper instability in the above-mentioned systems using the Kohn-Luttinger mechanism [@Kohn65], which suggests the emergence of superconducting pairing in the systems with the purely repulsive interaction [@Fay68; @Kagan88; @Kagan89; @Baranov92; @Kagan14a].
As it was shown in [@Gonzalez08], the Cooper instability can occur in an idealized graphene single layer due to the strong anisotropy of the Fermi contour for Van Hove filling $n_{VH}$, which, in fact, originates from the Kohn-Luttinger mechanism. According to the results obtained in [@Gonzalez08], this Cooper instability in graphene evolves predominantly in the $d-$wave channel and can be responsible for the critical superconducting transition temperatures up to $T_c\sim10\,K$, depending on the proximity of the chemical potential level to the Van Hove singularity. The theoretical analysis of the competition between the ferromagnetic and superconducting instabilities showed [@McChesney10] that the tendency to superconductivity due to strong modulation of the effective interaction along the Fermi contour, i.e., due to electron-electron interactions alone, prevails. In this case, the superconducting instability evolves predominantly in the $f$-wave channel.
The competition between the Kohn-Luttinger superconducting phase and the spin density wave phase at the Van Hove filling and near it in the graphene single layer was analyzed in [@Nandkishore12; @Kiesel12] using the functional renormalization group method. It was found that superconductivity with the $d+id-$wave symmetry of the order parameter prevails in a large domain near the Van Hove singularity, and a change in the calculated parameters may lead to a transition to the phase of the spin density wave. According to [@Kiesel12], far away from the Van Hove singularity, the long-range Coulomb interactions change the form of the $d+id-$wave function of a Cooper pair and can facilitate superconductivity with the $f-$wave symmetry of the order parameter. The competition between the superconducting phases with different symmetry types in the wide electron density range $1<n\leq n_{VH}$ in the graphene single layer was studied in [@Kagan14; @Nandkishore14]. It was demonstrated that at intermediate electron densities the long-range Coulomb interactions facilitate implementation of superconductivity with the $f-$wave symmetry of the order parameter, while at approaching the Van Hove singularity, the superconducting pairing with the $d+id-$symmetry type evolves [@Kagan14; @Nandkishore14].
The conditions for the Kohn-Luttinger superconducting pairing was analyzed also in graphene bilayer [@Vafek10a; @Vafek10b; @Guinea12; @Vafek14]. According to the results of [@Gonzalez13], the ferromagnetic instability near the Van Hove singularities dominates over the Kohn-Luttinger pairing in graphene bilayer. It should be noted, however, that in these calculations only the Coulomb repulsion of electrons on one site was taken into account. Authors of [@Hwang08] calculated the screening function of Coulomb interaction in the doped and undoped bilayer graphene in the random phase approximation (RPA). They established that the static polarization operator in the doped regime contains the singular part (the Kohn anomaly) that significantly exceeds one calculated for monolayer or 2D electron gas. As it is known, the Kohn anomaly [@Migdal58; @Kohn59] facilitates the effective attraction between two particles, inducing a contribution that always exceeds the repulsive contribution connected with the regular part of the polarization operator for the angular momenta $l\neq0$ of two particles [@Kohn65]. Therefore, one can expect that the critical superconducting temperature $T_c$ in an idealized bilayer can exceed the corresponding value for graphene monolayer.
Additionally, it was shown in papers [@Kagan91; @KaganValkov11a] that the value of $T_c$ can be increased in the framework of the Kohn-Luttinger mechanism even for low carrier densities if the spin-polarized two-band situation or a multilayer system is considered. In this situation, the role of the pairing spins “up” is played by electrons of one band (layer), while the role of the screening spins “down” is played by electrons of another band (layer). Coupling between the electrons from the two bands occurs owing to the interband (interlayer) Coulomb interaction. In this case, the following mechanism is possible: electrons of one sort form a Cooper pair by polarizing the electrons of another sort [@Kagan91; @KaganValkov11a]. This mechanism can be realized also in quasi-2D systems.
In this paper, in the Born weak-coupling approximation, we consider the Kohn-Luttinger superconducting pairing in an idealized graphene bilayer. We calculate the phase diagram, which reflects the competition between the superconducting phases with different types of the symmetry of the order parameter, taking into account the second-order contributions in the Coulomb interaction to the effective interaction of electrons in the Cooper channel. We analyze modification of the phase diagram with allowance for the Coulomb repulsion between electrons of the same, of the nearest, and of the next-to-nearest carbon atoms in a single layer, as well as the interlayer Coulomb interactions. We demonstrate the importance of taking into account the Coulomb repulsion of electrons on different crystal lattice sites and in different layers of bilayer graphene. The account of Coulomb repulsion changes the phase diagram of the superconducting state and, under certain conditions, increases the critical temperature.
Theoretical model
=================
We consider an idealized graphene bilayer, assuming that two layers are arranged in accordance with the $AB$ type, i.e., one layer is rotated on 60$^o$ relative to the other one [@McCann06; @McCann13]. Let us choose the arrangement of the sublattices in the layers in such a way that the sites from different layers located one above another belong to the sublattices $A_1$ and $A_2$ respectively, while the other sites belong to the sublattices $B_1$ and $B_2$ (Fig. \[bilayer\_structure\]). In the Shubin-Vonsovsky (extended Hubbard) model [@Shubin34], the Hamiltonian for the graphene bilayer which takes into account electron hoppings between the nearest and next-to-nearest atoms, as well as the Coulomb repulsion between electrons of the same and of the adjacent atoms and the interlayer Coulomb interaction of electrons, in the Wannier representation has the form: $$\begin{aligned}
\label{HamiltonianBilayer}
\hat{H}&=&\hat{H}_0+\hat{H}_{int},\\
\hat{H}_0&=&(\varepsilon-\mu)\Biggl(\sum_{if\sigma}\hat{n}^{A}_{if\sigma}+
\sum_{ig\sigma}\hat{n}^{B}_{ig\sigma}\Biggr)\nonumber\\
&-&t_1\sum_{f\delta\sigma}(a^{\dag}_{1f\sigma}b_{1,f+\delta,\sigma}+
a^{\dag}_{2f\sigma}b_{2,f-\delta,\sigma}+\textrm{h.c.})\nonumber\\
&-&t_2\sum_{i\sigma}\Biggl(\sum_{\langle\langle
fm\rangle\rangle}a^{\dag}_{if\sigma}a_{im\sigma}+\sum_{\langle\langle
gn\rangle\rangle}b^{\dag}_{ig\sigma}b_{in\sigma}+
\textrm{h.c.}\Biggr)\nonumber\\
&-&\gamma_1\sum_{f\sigma}(a^{\dag}_{1f\sigma}a_{2f\sigma}+\textrm{h.c.})\nonumber\\
&-&\gamma_3\sum_{g\delta\sigma}(b^{\dag}_{1g\sigma}b_{2,g+\delta,\sigma}+\textrm{h.c.})\nonumber\\
&-&\gamma_4\sum_{f\delta\sigma}(a^{\dag}_{1f\sigma}b_{2,f-\delta,\sigma}+
a^{\dag}_{2f\sigma}b_{1,f+\delta,\sigma}+\textrm{h.c.}),\label{H0Bilayer}\\
\hat{H}_{int}&=&U\biggl(\sum_{if}
\hat{n}^{A}_{if\uparrow}\hat{n}^{A}_{if\downarrow}+ \sum_{ig}
\hat{n}^{B}_{ig\uparrow}\hat{n}^{B}_{ig\downarrow}\biggr)+\nonumber\\
&+&V_1\sum_{f\delta\sigma\sigma'}
\Bigl(\hat{n}^{A}_{1f\sigma}\hat{n}^{B}_{1,f+\delta,\sigma'}+
\hat{n}^{A}_{2f\sigma}\hat{n}^{B}_{2,f-\delta,\sigma'}\Bigr)+\nonumber\\
&+&\frac{V_2}{2}\sum_{i\sigma\sigma'}\Biggl(\sum_{\langle\langle
fm\rangle\rangle}\hat{n}^{A}_{if\sigma}\hat{n}^{A}_{im\sigma'}+\sum_{\langle\langle
gn\rangle\rangle}\hat{n}^{B}_{ig\sigma}\hat{n}^{B}_{in\sigma'}\Biggr)+\nonumber\\
&+&G_1\sum_{f\sigma\sigma'}
\hat{n}^{A}_{1f\sigma}\hat{n}^{A}_{2f\sigma'} +
G_3\sum_{g\delta\sigma\sigma'}
\hat{n}^{B}_{1g\sigma}\hat{n}^{B}_{2,g+\delta,\sigma'}+\nonumber\\
&+&G_4\sum_{f\delta\sigma\sigma'}
\Bigl(\hat{n}^{A}_{1f\sigma}\hat{n}^{B}_{2,f-\delta,\sigma'}+
\hat{n}^{A}_{2f\sigma}\hat{n}^{B}_{1,f+\delta,\sigma'}\Bigr).\label{HintBilayer}\end{aligned}$$ In (\[HamiltonianBilayer\])–(\[HintBilayer\]), the operators $a^{\dag}_{1f\sigma}(a_{1f\sigma})$ create (annihilate) an electron with the spin projection $\sigma=\pm1/2$ at site $f$ of the sublattice $A_1$; $\hat{n}^{A}_{1f\sigma}=
a^{\dag}_{1f\sigma}a_{1f\sigma}$ denotes the operators of the numbers of fermions at the $f$ site of the sublattice $A_1$ (analogous notations are used for the sublattices $A_2$, $B_1$, and $B_2$). Vector $\delta (-\delta)$ connects the nearest atoms of the hexagonal lattice of the lower (upper) layer. Index $i=1,2$ in Hamiltonian (\[HamiltonianBilayer\]) denotes the number of layer. We assume that the one-site energies are identical ($\varepsilon_{Ai}=\varepsilon_{Bi}=\varepsilon$) and the position of the chemical potential $\mu$ and number of carriers $n$ in graphene bilayer can be controlled by a gate electric field. In the Hamiltonian, $t_1$ is the hopping integral between the neighboring atoms (hoppings between different sublattices), $t_2$ is the hopping integral between the next-to-nearest neighboring atoms (hoppings in the same sublattice), $U$ is the parameter of Coulomb repulsion between electrons of the same atom with the opposite spin projections (Hubbard repulsion), and $V_1$ and $V_2$ are the Coulomb interactions between electrons of the nearest and the next-to-nearest carbon atoms in a single layer. The symbol $\langle\langle~\rangle\rangle$ indicates that summation is made only over next-to-nearest neighbors; the symbols $\gamma_1,\,\gamma_3,\,\gamma_4$ denote the parameters of the interlayer electron hoppings (Fig. \[bilayer\_structure\]), and $G_1$, $G_3$ and $G_4$ are the interlayer Coulomb interactions between electrons.
We diagonalize the Hamiltonian $\hat{H}_0$ using the Bogolyubov transformation $$\begin{aligned}
\label{uv2}
\alpha_{i\vec{k}\sigma}&=& w_{i1}(\vec{k}){a_{1 \vec{k}\sigma }} +
w_{i2}(\vec{k}){a_{2\vec{k}\sigma
}}\\
&+&w_{i3}(\vec{k}){b_{1\vec{k}\sigma }} +
w_{i4}(\vec{k}){b_{2\vec{k}\sigma }},\quad i=1,2,3,4.\nonumber\end{aligned}$$ As a result, $\hat{H}_0$ acquires the form $$\begin{aligned}
\hat H_0 =\sum\limits_{i=1}^4 \sum\limits_{ \vec{k}\sigma }
E_{i\vec{k}}
{\alpha_{i\vec{k}\sigma}^{\dag}\alpha_{i\vec{k}\sigma}}.\end{aligned}$$ Since the results of ab initio calculations for graphite [@Dresselhaus02; @Brandt88] showed a very small value of the interlayer hopping parameter $\gamma_4$, hereinafter we assume that $\gamma_4=0$. Then, the four-band energy spectrum of the graphene bilayer is described by the expressions $$\begin{aligned}
\label{spectra}
&&E_{i\vec{k}}=\varepsilon\pm\sqrt{A_{\vec{k}}\pm\sqrt{B_{\vec{k}}}}-t_2f_{\vec{k}},\\
&&A_{\vec{k}}=\frac14\Bigl(2a^2+4|b_{\vec{k}}|^2+2|d_{\vec{k}}|^2\Bigr),\nonumber\\
&&B_{\vec{k}}=\frac14\Bigl(|d_{\vec{k}}|^2(|d_{\vec{k}}|^2-2a^2+4|b_{\vec{k}}|^2)+a^4+4a^2|b_{\vec{k}}|^2\nonumber\\
&&\qquad+4ab^2_{\vec{k}}d_{\vec{k}}+4ab_{\vec{k}}^{*2}d^*_{\vec{k}}\Bigr),\nonumber\\
&&a=\gamma_1,\quad b_{\vec{k}}=t_1u_{\vec{k}},\quad
d_{\vec{k}}=\gamma_3u_{\vec{k}}\nonumber,\end{aligned}$$ where the following notation has been introduced: $$\begin{aligned}
\label{f_k}
&&f_{\vec{k}}=2\cos(\sqrt{3}k_y)+
4\cos\biggl(\frac{\sqrt{3}}{2}k_y\biggr)\cos\biggl(\frac{3}{2}k_x\biggr),\\
&&u_{\vec{k}}=\displaystyle\sum_{\delta}e^{i
\vec{k}\delta}=e^{-ik_x}+
2e^{\frac{i}{2}k_x}\cos\biggl(\frac{\sqrt{3}}{2}k_y\biggr),\label{u_k}\\
&&|u_{\vec{k}}|=\sqrt{3+f_{\vec{k}}}.\end{aligned}$$
In this paper, the conditions for the implementation of the Kohn-Luttinger superconductivity are analyzed by considering the situation when upon doping of the graphene bilayer the chemical potential falls into the two upper energy bands $E_{1\vec{k}}$ and $E_{2\vec{k}}$ (Fig. \[two\_contours\]a). Then, if $\gamma_1\neq0$ and the inequality $\mu>\gamma_1$ is valid, the Fermi contour will consist of two lines (Fig. \[two\_contours\]b) in the vicinity of each Dirac point for the electron densities $1<n<n_{VH}$, where $n$ is the electron density calculated per atoms of one layer.
The coefficients of the Bogolyubov transformation can be found from the system of homogeneous equations $$\begin{aligned}
\label{wz}
\left(%
\begin{array}{cccc}
x_i & a & b^*_{\vec{k}} & 0 \\
a & x_i & 0 & b_{\vec{k}} \\
b_{\vec{k}} & 0 & x_i & d^*_{\vec{k}} \\
0 & b^*_{\vec{k}} & d_{\vec{k}} & x_i \\
\end{array}%
\right)\left(%
\begin{array}{c}
w_{i1} \\
w_{i2} \\
w_{i3} \\
w_{i4} \\
\end{array}%
\right)=0,\end{aligned}$$ where $x_i=E_{i\vec{k}}-\varepsilon+t_2f_{\vec{k}}$.
In the Bogolyubov representation, the Hamiltonian $\hat H_{int}$ (\[HintBilayer\]) in terms of the operators $\alpha_{1\vec{k}\sigma},\,\alpha_{2\vec{k}\sigma},\,\alpha_{3\vec{k}\sigma}$ and $\alpha_{4\vec{k}\sigma}$ reads as follows: $$\begin{aligned}
\label{Hint_ab}
\hat H_{int} &=& \frac{1}{N}\sum\limits_{ijlm\sigma\atop
\vec{k}\vec{p}\vec{q}\vec{s}}
\Gamma_{ij;lm}^{||}(\vec{k},\vec{p}|\vec{q},\vec{s})
\alpha_{i\vec{k}\sigma}^\dag \alpha_{j\vec{p}\sigma}^\dag
\alpha_{l\vec{q}\sigma}\alpha_{m\vec{s}\sigma}
\nonumber\\
&\times&\delta (\vec{k}+\vec{p}-\vec{q}-\vec{s})\\
&+& \frac{1}{N}\sum\limits_{ijlm\atop
\vec{k}\vec{p}\vec{q}\vec{s}}\Gamma_{ij;lm}^{\bot}(\vec{k},\vec{p}|\vec{q},\vec{s})
\alpha_{i\vec{k}\uparrow}^\dag \alpha_{j\vec{p}\downarrow}^\dag
\alpha_{l\vec{q}\downarrow}
\alpha_{m\vec{s}\uparrow}\nonumber\\
&\times&\delta (\vec{k}+\vec{p}-\vec{q}-\vec{s}),\nonumber\end{aligned}$$ where $\delta(x)$ is the Dirac delta-function and $\Gamma_{ij;lm}^{||}(\vec{k},\vec{p}|\vec{q},\vec{s})$ and $\Gamma_{ij;lm}^{\bot}(\vec{k},\vec{p}|\vec{q},\vec{s})$ are the initial amplitudes. The quantity $$\begin{aligned}
\Gamma_{ij;lm}^{||}&&(\vec{k},\vec{p}|\vec{q},\vec{s})\nonumber\\
&&= \frac12\Bigl(V_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+V_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(1)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(1)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(3)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(3)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(4)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(4)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\Bigr),\\
V_{ij;lm}&&(\vec{k},\vec{p}|\vec{q},\vec{s})=V_1\Bigl(
u_{\vec{q}-\vec{p}} w_{i1}(\vec{k}) w_{j3}(\vec{p})
w^*_{l3}(\vec{q}) w^*_{m1}(\vec{s})\nonumber\\
&&+u_{\vec{q}-\vec{p}}^* w_{i2}(\vec{k}) w_{j4}(\vec{p})
w^*_{l4}(\vec{q}) w^*_{m2}(\vec{s})\Bigr)\nonumber\\
&&+\frac{V_2}{2}\sum_{r=1}^4 f_{\vec{q}-\vec{p}}w_{ir}(\vec{k})
w_{jr}(\vec{p}) w^*_{lr}(\vec{q}) w^*_{mr}(\vec{s}),\end{aligned}$$ $$\begin{aligned}
G^{(1)}_{ij;lm}&&(\vec{k},\vec{p}|\vec{q},\vec{s})=G_1
w_{i1}(\vec{k}) w_{j2}(\vec{p}) w^*_{l2}(\vec{q})
w^*_{m1}(\vec{s}),\\
G^{(3)}_{ij;lm}&&(\vec{k},\vec{p}|\vec{q},\vec{s})\nonumber\\
&&=G_3 u_{\vec{q}-\vec{p}} w_{i3}(\vec{k}) w_{j4}(\vec{p})
w^*_{l4}(\vec{q})w^*_{m3}(\vec{s}),\\
G^{(4)}_{ij;lm}&&(\vec{k},\vec{p}|\vec{q},\vec{s})=G_4\Bigl(
u_{\vec{q}-\vec{p}}^* w_{i1}(\vec{k}) w_{j4}(\vec{p})
w^*_{l4}(\vec{q}) w^*_{m1}(\vec{s})\nonumber\\
&&+u_{\vec{q}-\vec{p}} w_{i2}(\vec{k}) w_{j3}(\vec{p})
w^*_{l3}(\vec{q}) w^*_{m2}(\vec{s})\Bigr)\end{aligned}$$ corresponds to the intensity of the interaction of fermions with parallel spin projections, while the quantity $$\begin{aligned}
\Gamma_{ij;lm}^{\bot}&&(\vec{k},\vec{p}|\vec{q},\vec{s})
=U_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
\nonumber\\
&&+V_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})+
V_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(1)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(1)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(3)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(3)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q})\nonumber\\
&&+G^{(4)}_{ij;lm}(\vec{k},\vec{p}|\vec{q},\vec{s})
+G^{(4)}_{ji;ml}(\vec{p},\vec{k}|\vec{s},\vec{q}),\\
U_{ij;lm}&&(\vec{k},\vec{p}|\vec{q},\vec{s})
\nonumber\\
&&=U\sum_{r=1}^4 w_{ir}(\vec{k}) w_{jr}(\vec{p}) w^*_{lr}(\vec{q})
w^*_{mr}(\vec{s})\end{aligned}$$ describes the interaction of fermions with antiparallel spin projections. Indices ${i,j,l,m}$ correspond to the number of the energy band and acquire the values 1, 2, 3, or 4.
Effective interaction and equation for the superconducting order parameter
==========================================================================
In this paper, we use the Born weak-coupling approximation, in which the hierarchy of model parameters has the form $$\label{hierarchy}
W>U>V_1>V_2>G_1>G_3,\,G_4,$$ where $W$ is the bandwidth in graphene bilayer (\[spectra\]). In the calculation of the scattering amplitude in the Cooper channel, the condition (\[hierarchy\]) allows us to limit the consideration to only the second-order diagrams in the effective interaction of two electrons with opposite values of the momentum and spin and use the quantity $\widetilde{\Gamma}(\vec{p},
\vec{k})$ for it. Figure \[diagrams\] depicts the sum of diagrams which determines $\widetilde{\Gamma}(\vec{p}, \vec{k})$. Here, solid lines correspond to Green’s functions for the electrons with opposite spin projections $+\frac12$ (light arrows) and $-\frac12$ (black arrows). The first diagram describes the initial interaction of two electrons in the Cooper channel. Here, the wavy lines correspond to the initial interaction. The next four diagrams in Fig. \[diagrams\] correspond to the second-order scattering processes $\delta\widetilde{\Gamma}(\vec{p}, \vec{k})$ and describe the polarization effects of the filled Fermi sphere. In the diagrams, the presence of solid lines without arrows means the summation over the spin projections values.
The possibility of the Cooper pairing is determined by the features of the energy structure and the effective interaction of electrons near the Fermi level [@Gor'kov61]. If we assume that the chemical potential in doped graphene bilayer is located in the two upper bands $E_{1\vec{k}}$ and $E_{2\vec{k}}$ (Fig. \[two\_contours\]a), we can consider the situation in which the initial and final momenta of electrons in the Cooper channel also belong to the two upper bands and analyze the conditions for the Kohn-Luttinger superconducting pairing. At that, indices $i$ and $j$ in the diagrams (Fig. \[diagrams\]) will acquire the values of 1 or 2.
Introducing the analytical expressions for the diagrams, we get the effective interaction in the form $$\begin{aligned}
\label{Gamma_wave}
&&\widetilde{\Gamma}(\vec{p},\vec{k})=
\widetilde{\Gamma}_0(\vec{p},\vec{k})
+\delta\widetilde{\Gamma}(\vec{p},\vec{k}),\\
&&\widetilde{\Gamma}_0(\vec{p},\vec{k})=\Gamma^{\bot}_{ii;jj}(\vec{p},
-\vec{p}| -\vec{k},
\vec{k}),\\
&&\delta \tilde \Gamma (\vec{p},\vec{k})=
\frac{1}{N}\sum\limits_{l,m, \vec{p}_1}
\Gamma^{\bot}_{il;jm}(\vec{p}, \vec{q}_2| -\vec{k},
\vec{p}_1)\\
&&\times\Gamma^{\bot}_{mi;lj}(\vec{p}_1,
-\vec{p}|\vec{q}_2,\vec{k})
\chi_{l,m}(\vec{q}_2, \vec{p}_1)\nonumber\\
&&+\frac{2}{N}\sum\limits_{l,m, \vec{p}_1} \Bigl\{
\Gamma^{\bot}_{im;lj}( \vec{p}, \vec{p}_1| \vec{q}_1,
\vec{k})\nonumber\\
&&\times\left[\Gamma^{||}_{li;mj}( \vec{q}_1, -\vec{p}| \vec{p}_1,
-\vec{k}) -
\Gamma^{||}_{li;jm}( \vec{q}_1, -\vec{p}| -\vec{k}, \vec{p}_1) \right] \Bigr.\nonumber\\
&&+\Bigl.\Gamma^{\bot}_{li;jm}(\vec{q}_1, -\vec{p}| -\vec{k}, \vec{p}_1)\nonumber\\
&&\times\left[\Gamma^{||}_{im;jl}( \vec{p}, \vec{p}_1| \vec{k},
\vec{q}_1) - \Gamma^{||}_{im;lj}( \vec{p}, \vec{p}_1| \vec{q}_1,
\vec{k})\right] \Bigr\}\chi_{l,m}(\vec{q}_1,\vec{p}_1).\nonumber\end{aligned}$$ Here, we use the notations for the generalized susceptibilities $$\chi_{l,m}(\vec{k},\vec{p}) = \frac{f(E_{l\vec{k}}) -
f(E_{m\vec{p}})} {E_{m\vec{p}} - E_{l\vec{k}}},$$ where $f(x)=(\exp(\frac{x-\mu}{T})+1)^{-1}$ is the Fermi-Dirac function and the energies $E_{i\vec{k}}$ are defined by the expressions (\[spectra\]). Additionally, we have introduced the following notations for the combinations of the momenta $$\label{q1q2}
\vec{q}_1 = \vec{p}_1 + \vec{p} - \vec{k},\qquad \vec{q}_2 =
\vec{p}_1-\vec{p}-\vec{k}.$$
The renormalized expression for the effective interaction allows us to analyze the conditions for the occurrence of superconductivity in the system. It is known [@Gor'kov61] that the development of the Cooper instability can be established from the consideration of the homogeneous part of the Bethe-Salpeter equation. At that, the dependence of the scattering amplitude $\Gamma(\vec{p},\vec{k})$ on momentum $\vec{k}$ is factorized and we get the integral equation for the superconducting order parameter $\Delta(\vec{p})$. After the integration over the isoenergetic contours, the problem of the Cooper instability can be reduced to the eigenvalue problem [@Scalapino86; @Baranov92; @Hlubina99; @Raghu10; @Alexandrov11] $$\label{IntegralEqPhi}
\frac{3\sqrt{3}}{8\pi^2}\oint\limits_{\varepsilon_{\vec{q}}=\mu}
\frac{d\hat{\vec{q}}} {v_F(\hat{\vec{q}})}
\widetilde{\Gamma}(\hat{\vec{\vec{p}}},\hat{\vec{q}})
\Delta(\hat{\vec{q}})=\lambda\Delta(\hat{\vec{p}}),$$ where the eigenvector is the superconducting order parameter $\Delta(\hat{\vec{q}})$ and the eigenvalues $\lambda$ satisfy the relation $\lambda^{-1}\simeq \ln(T_c/W)$. Here, the momenta $\hat{\vec{p}}$ and $\hat{\vec{q}}$ belong to the Fermi surface and $v_F(\hat{\vec{q}})$ is the Fermi velocity. Equation (\[IntegralEqPhi\]) is solved in accordance with the common scheme described in [@Kagan14; @Kagan14a]. The integration is fulfilled with the allowance for the fact that the Fermi contour near each Dirac point consists of two lines (Fig. \[two\_contours\]b).
Results and discussion
======================
Let us consider the phase diagram of the superconducting state of the graphene bilayer and the modifications of this diagram in the different regimes obtained by solving Eq. ($\ref{IntegralEqPhi}$). When building the phase diagram, we divided the multisheet Fermi contour into 180 intervals and the Brillouin zone of the graphene bilayer, into $5\cdot10^4$ cells. It was established that the chosen method of division is sufficient for the correct description of the dependence of the effective coupling constant $\lambda$ on the electron density $n$ [@Kagan14]. Based on the obtained dependences $\lambda(n)$ for different values of the intersite $V_1$ and interplane $G_1,\,G_3$ and $G_4$ Coulomb interactions, we built the phase diagrams of the Shubin-Vonsovsky model for bilayer graphene, which reflect the competition between the superconducting phases with different types of symmetry of the order parameter.
So far, there has been no agreement regarding the values of parameters of the intra- and interplanar Coulomb interactions in the graphene bilayer. The ab initio calculations for graphite [@Wehling11] showed that the value of Hubbard repulsion is $U=8.0\,\textrm{eV}$, which is consistent with the estimation made in [@Levin74] and contradicts the intuitively expected small value of $U$ and weak-coupling limit $U<W$ (it is known [@Reich02] that $t_1\approx2.8\,\textrm{eV}$). The authors of [@Wehling11] calculated the parameters of Coulomb repulsion between electrons of the nearest and the next-to-nearest carbon atoms: $V_1=3.9\,\textrm{eV}$ and $V_2=2.4\,\textrm{eV}$, respectively. At the same time, the other authors (see, for example, [@Perfetto07]) consider these parameters to be much smaller. The authors of [@Milovanovic12] mentioned that the estimation of the parameters of Coulomb interaction, including the Hubbard repulsion, in the graphene bilayer strongly depends on the calculation scheme which is used. In our calculation, we apply the parameter hierarchy (\[hierarchy\]), which allows us to use the Born weak-coupling approximation. For interlayer hopping parameters $\gamma_1$ and $\gamma_3$, we use the values similar to those determined in [@Dresselhaus02; @Brandt88] for graphite.
First, let us consider the limiting case when the bilayer energy spectrum is described by the only one hopping parameter ($t_1\neq0,\,t_2=\gamma_1=\gamma_3=0$). The Hubbard repulsion is also taken into account $U=2$ (hereinafter, all the parameters are given in units of $|t_1|$). The Coulomb repulsion between electrons ($V_1\neq0$) of the neighboring carbon atoms in the same layer is taken into account as well. At the same time, the interlayer Coulomb interactions are not taken into account ($G_1=G_3=G_4=0$). Thus, in the chosen regime, the graphene bilayer consists of two isolated single layers. The phase diagram of the superconducting state shown as a function of the variables “$n-V_1$” for this case is presented in Fig. \[PD\_limiting\]a. It can be seen that the phase diagram comprises three regions. At low electron densities $n$, the ground state of the system corresponds to the superconductivity with the $d+id-$wave symmetry of the order parameter, which is described by the 2D representation $E_2$, the contribution to which is determined by the harmonics $$\begin{aligned}
\label{E2}
g_{m}^{(d+id)}(\phi)=\frac{1}{\sqrt{\pi}}\,(A\,\textrm{sin}\,
(2m+2)\phi+B\,\textrm{cos}\,(2m+2)\phi)\nonumber,\end{aligned}$$ where subscripts $m$ run over the values for which the coefficients $(2m+2)$ are not multiples of 3. At the intermediate electron densities, the superconducting $f-$wave pairing is implemented, the contribution to which is determined by the harmonics $g_{m}^{(f_1)}(\phi)=\displaystyle\frac{1}{\sqrt{\pi}}\,
\textrm{sin}\,(6m+3)\phi$ (here $m\in[\,0,\infty)$), while the contribution of the harmonics $g_{m}^{(f_2)}(\phi)=\displaystyle\frac{1}{\sqrt{\pi}}\,
\textrm{cos}\,(6m+3)\phi$ is absent. At the large values of $n$, the domain of the superconducting $d+id-$wave pairing occurs [@Nandkishore12]. With the increase of the parameter $V_1$ of the intersite Coulomb interaction, in the region of small values of $n$, the $d+id-$wave pairing is suppressed and the pairing with the $f-$wave symmetry of the order parameter is implemented. Thin blue lines in Fig. \[PD\_limiting\] are the lines of the equal values of the effective coupling constant $|\lambda|$. It can be seen that in this case in the proximity of the Van Hove filling $n_{VH}$ (solid curve in Fig. \[DOS\_bilayer\]) the effective coupling constant attains the values $|\lambda|=0.1$.
It should be noted that to avoid the summation of the parquet diagrams [@Dzyaloshinskii88a; @Dzyaloshinskii88b; @Zheleznyak97], we do not analyze here the electron density regions that are very close to the Van Hove singularity in the density of electron states of bilayer graphene (Fig. \[DOS\_bilayer\]). For this reason, the boundaries between different domains of the implementation of the Kohn-Luttinger superconducting pairing, as well as the lines of the equal value of $|\lambda|$ that are very close to the Van Hove singularity are indicated in the phase diagram by the dashed lines.
Thus, in the numerical calculation for the graphene bilayer for the chosen parameters, we made the limiting transition to the results obtained by us previously for the graphene monolayer [@Kagan14; @Kagan14a].
Let us consider the modification of the phase diagram for the isolated graphene single layers with regard to the long-range intraplane Coulomb interactions between electrons $V_2$. It can be seen in Fig. \[PD\_limiting\]b for the fixed ratio between the parameters of the long-range Coulomb interactions $V_2=0.6V_1$ that when $V_2$ is taken into account, the phase diagram changes qualitatively. This change involves the suppression of a large domain of the superconducting state with the $f-$wave symmetry at the intermediate electron densities and the implementation of the superconducting pairing with the $p+ip-$wave symmetry of the order parameter. In addition, when $V_2$ is taken into account, the effective coupling constant increases to the value $|\lambda|=0.3$.
Now, let us consider the modification of the phase diagram of the superconducting state with respect to the interplanar interactions. When the interlayer electron hoppings $\gamma_1=0.12$ and $\gamma_3=0.1$ are taken into account while the other parameters being the same as in Fig. \[PD\_limiting\], the phase diagram of the graphene bilayer remains nearly unchanged.
Inclusion of the Coulomb interaction $G_1$ in the consideration weakly shifts the boundaries of the $f_1-$wave and $d+id-$wave pairing in the phase diagram in Fig. \[PD\_limiting\] and does not affect the absolute values of $\lambda$. Figure \[PD\_G3G4\] shows the effect of taking into account the interlayer Coulomb interactions $G_3$ and $G_4$. Figure \[PD\_G3G4\]a shows the phase diagram of the Shubin-Vonsovsky model for the graphene bilayer for the set of parameters $t_2=0,\,\gamma_1=0.12,\,\gamma_3=0.1,\,U=2$ and $V_2=0$ for the chosen ratios between the interlayer and intersite Coulomb interactions $G_1=0.5V_1,\,G_3=G_4=0.4V_1$, according to the hierarchy of the parameters (\[hierarchy\]). The calculation shows that the separate increase of the parameters $G_3$ and $G_4$ suppresses the $d+id-$wave pairing and, at the same time, broadens the $f-$wave pairing region at small electron densities. The superconducting $d+id-$phase is suppressed the most effectively by enhancing the parameter $G_4$ of the interlayer Coulomb interaction. When the interactions $G_3$ and $G_4$ are simultaneously taken into account (Fig. \[PD\_G3G4\]a), then along with the intensive suppression of the superconducting $d+id-$wave pairing at small electron densities and the implementation of the superconductivity with the $f-$wave symmetry of the order parameter, the growth of the absolute values of effective coupling constant $\lambda$ is also observed.
Figure \[PD\_G3G4\]b depicts the phase diagram of the graphene bilayer calculated for the same parameters as in Fig. \[PD\_G3G4\]a but with respect to the long-range intraplane Coulomb repulsion between electrons $V_2$. Comparison of Figs. \[PD\_G3G4\]b and \[PD\_limiting\]b shows that the account for $G_3\neq0$ and $G_4\neq0$ leads to the strong competition between the $d+id-$wave and $p+ip-$wave pairings with the significant suppression of the $p+ip-$wave pairing in the region of the intermediate electron densities. In this case, in the remained region of the $p+ip-$wave pairing, $|\lambda_{p+ip}|$ slightly exceeds $|\lambda_f|$.
The account for electron hoppings to the next-to-nearest carbon atoms $t_2$ does not qualitatively affects the competition between the superconducting phases (Fig. \[PD\_G3G4\]). Figure \[PD\_t2\] depicts the phase diagram of the graphene bilayer obtained for the parameters $t_2=0.1,\,\gamma_1=0.12,\,\gamma_3=0.1,\,U=2,\,G_1=0.5V_1$ and $G_3=G_4=0.4V_1$. Such a behavior of the system is explained by the fact that switching on of the hoppings $t_2>0$ or $t_2<0$ for the graphene bilayer, similarly to the case of the monolayer investigated by us in [@Kagan14; @Kagan14a], does not significantly modify the density of electron states in the carrier concentration regions between the Dirac point and both points $n_{VH}$ (Fig. \[DOS\_bilayer\]). However, it can be seen in Fig. \[PD\_t2\] that the account for the hoppings $t_2$ leads to an increase of the effective interaction in the absolute values and, consequently, to the higher superconducting transition temperatures in an idealized graphene bilayer.
It should be noted that the Kohn–Luttinger superconductivity in the graphene single layer and bilayer never develops near the Dirac points. The calculations show that in the vicinity of these points, where the linear approximation for the energy spectrum of the graphene single layer and the parabolic approximation for the spectrum of the graphene bilayer work pretty well, the density of states is very low and the effective coupling constant $|\lambda|<10^{-2}$. The higher values of $|\lambda|$, which are indicative of the development of the Cooper instability, arise at the electron densities $n>1.15$. However, at such densities, the energy spectrum of the bilayer along the direction $KM$ of the Brillouin zone (Fig. \[two\_contours\]b) already significantly differs from the Dirac approximation.
Conclusions
===========
In the work, we have analyzed the conditions for the Kohn-Luttinger superconductivity in a semimetal with the Dirac spectrum using as an example an idealized graphene bilayer, disregarding the van der Waals potential of the substrate and both magnetic and non-magnetic impurities. The electronic structure of graphene bilayer is described in the Shubin-Vonsovsky model taking into account not only the Coulomb repulsion of electrons of the same carbon atom, but also the intersite and interlayer Coulomb interactions. It was shown that in such a system, the Kohn-Luttinger polarization contributions lead to the effective attraction between electrons in the Cooper channel. The constructed superconducting phase diagram of the system determines the Cooper pairing domains with the different types of the symmetry of the order parameter, depending on the intersite Coulomb interactions and the electron densities. The analysis of the phase diagram showed that the inclusion of the Kohn-Luttinger renormalizations up to the second order of perturbation theory inclusively and the allowance for the long-range Coulomb interactions $V_1$ and $V_2$ determine, to a considerable extent, the competition between the superconducting phases with the $f-$wave, $p+ip-$wave, and $d+id$-wave types of the symmetry of the order parameter. They also lead to a significant increase in the absolute values of the effective interaction. It was shown that the allowance for the interlayer Coulomb interactions $G_3$ and $G_4$, as well as for the distant electron hoppings $t_2$, leads to an additional increase in the effective interaction and, hence, to the higher superconducting transition temperatures in an idealized graphene bilayer.
Our calculation showed that the Kohn-Luttinger mechanism can lead to the superconducting transition temperatures $T_c\sim
20\div40~K$ in an idealized graphene bilayer. Contrary to these rather optimistic estimations, in real graphene, as it was mentioned in Introduction, superconductivity has not been found yet. This material is only close to superconductivity.
For a few reasons, the results of the theoretical calculations reported here can differ from the experimental situation. First, we did not take into account the effect of the van der Waals potential of the substrate [@Gomez09; @Bostrom12; @Klimchitskaya13]. It seems that the effect of this potential should be weakened with the increase of number of layers. However, even in the multilayer systems the van der Waals forces can degrade the conditions for the development of the Cooper instability.
Second, as we mentioned in Section 4, there has been no agreement regarding the values of the parameters of the intraplane and interplanar Coulomb interactions in the graphene bilayer in the literature. In this work, we used the values of the intraplane Coulomb interactions that are close to those obtained from the ab initio calculation in [@Wehling11] for graphite. The values of the interplanar Coulomb interactions were chosen to satisfy the hierarchy of the parameters of the Born weak-coupling approximation.
Third, in our calculations, we considered a pure graphene bilayer with the ideal structure, whereas the real material contains numerous impurities and structural defects. It is well known that, in contrast to the traditional $s-$wave pairing, for the anomalous pairing with the $f$-wave, $p+ip$-wave, and $d+id$-wave symmetries of the order parameter, nonmagnetic impurities and structural defects can destroy the superconducting order [@Black14b].
In addition, we should mention one more possible reason for the discrepancy between the theoretical calculations on superconductivity in graphene and the experimentally observed situation. In recent paper [@Kats14], the effect of quantum fluctuations ($T=0$) on the graphene layers was investigated. It was shown that these fluctuations initiate the logarithmic corrections to the moduli of elasticity and bending of the layers. In other words, according to [@Kats14], the quantum fluctuations connected with the bending vibrations of the graphene layers can lead to the situation when the electrons do not move along the atomically smooth layers but along the strongly curved string-like trajectories, as in quantum chromodynamics. This situation requires further investigations, although in this case the superconductivity is not at all excluded and even can be enhanced by the exchange of bending vibration quanta between the pairing electrons.
We thank V.V. Val’kov for useful discussions. This work is supported by the Russian Foundation for Basic Research (projects nos. 14-02-00058 and 14-02-31237). One of the authors (M.Yu.K.) gratefully acknowledges support from the Basic Research Program of the National Research University Higher School of Economics. Another one (M.M.K.) thanks the scholarship SP-1361.2015.1 of the President of the Russian Federation and the Dynasty foundation.
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[^1]: *e*-mail: [email protected]
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abstract: 'The phase behavior of a cross-linked polymer blend made of two incompatible species, $A$ and $B$, of different chemical nature is analyzed. Besides a homogeneous phase, this system also exhibits two microphases and a phase of total segregation. The transition between the homogeneous and the microphase is continuous along a $\lambda$-line; a first-order phase boundaries separate the microphase and the disordered phase from the phase of complete segregation. The critical line meets the first-order phase boundaries at an end point. Scaling arguments indicate that, close to [*any*]{} end point, the equations for the first-order phase boundaries exhibit nonanaliticities associated with the singularities present at the thermodynamic functions near the critical line. Explicit expressions for the phase boundaries near the end point for a cross-linked polymer mixture are obtained and checked for singularities.'
author:
- |
Edilson Vargas and Marcia C. Barbosa\
Instituto de Física, Universidade Federal do Rio Grande do Sul\
Caixa Postal 15051, CEP 91501-970 , Porto Alegre, RS, Brazil
title: 'Phase-Boundaries near Critical End points: Applications to Cross-linked Copolymers'
---
**Introduction**
================
Polymer mixtures were usually phase separated [@De77]. When the chains are long, the translational entropy of the chain is small and any chemical difference between the two species leads to a repulsion and consequent phase separation at a temperature $T=T_0$. One way to prevent phase-separation is to cross-link the system at high temperatures and then brought it to the coexistence region. A competition between the natural tendency of the system to phase separate and the elastic forces due to the presence of the cross-links is established. As a result of this competition, there is a microphase separation [@De79]. For strongly cross-linked system, the distance between two cross-links is fixed and the cross-links are permanent. In this case, the microphase separation happens at a temperature lower than the temperature for which complete segregation would had occurs $T_m<T_0$ [@De79].
When the number of cross-links is not too large, the distance between cross-links it is not fixed and the position of each $A$-$B$ link fluctuates in space. Then, as these inhomogeneities in the links are to be taken taken into account, the phase-diagram of the system changes quite drastically [@Sc94][@St94][@Va97]. Besides the homogeneous phase and the microphase, one also finds a phase of complete segregation. This phase is separated from the homogeneous and microphase by first-order phase boundaries. The critical line between the disordered and the microphase meets these phase boundaries at an end point where the three phases coexist.
It is well known that in the vicinity of continuous transitions, singularities in the thermodynamic functions such as compressibility, specific heat and density should be expected. These nonanalyticities are usually expressed through universal critical exponents and universal amplitude ratios that one can measure. However, some time ago [@Ba1][@Ba2], it was shown, through a phenomenological argument, that in the vicinity of an end point, singularities in the first order phase boundaries should also be expected and that these singularities could also be expressed in terms of universal exponents and universal amplitude ratios. Nevertheless, this assumption is based on scaling arguments and needs to be checked. This was already done in the spherical model [@Ba3]. Unfortunately the spherical model has no physical realization and, consequently, we indeed need to check our theory in an more realistic model.
In this sense, our aim in this paper is to show that these singularities are actually present in the first-order phase boundaries of the polymer mixture discribed above. In order to do so, on basis of an effective mean-field theory that takes into account anisotropy between the two species and inhomogeneities in the cross-links, we calculate the phase boundaries near the end point. The remainder of this paper is organized as follows. In sec. $2$, we briefly review the scaling arguments that led us to the conclusion that near the end points singularities in the phase boundaries might exist. Accepting the plausibility but the uncertainty of this conclusion, in sec. $3$, we test them in a model for polymer mixture. Our results are summarized in sec. $4$.
**Phase Boundaries near Critical End points**
=============================================
Before analyzing the phase-diagram of the copolymers, it would be helpful to review the theory for singularities in the phase boundaries near end points [@Ba1]-[@Ba4]. In the space $(T,g,h)$ ( here $g$ is a intensive parameter that can be a pressure, external field, chemical potential, etc), an end point is the locus where two phases, $II^+$ and $II^-$, become critical in the presence of a third noncritical phase, $III$ (see Fig.$1$). Now, what is the form of the phase boundary $g_{\rho}(T,h)$ that separates the critical phases from the spectator phase when $(T,h)$ pass through $(T_e,h=0)$? At the end point $g$ has a definite value, $g_{e}$, and so do the slopes $g_{1}\equiv dg_{\rho}/dT\mid_e$ and $g_{2}\equiv dg_{\rho}/dh\mid_e$. Thus, it seems likely that most people would assume that $g_{\rho}=g_{e}+g_{1}t+g_{2}h+...$ and further [*analytic terms*]{} in the Taylor series expansion in powers of of $ t \equiv (T-T_{e})/T_{e})$ and $h$. However, this is wrong. Fisher and Barbosa [*(FB)*]{} [@Ba1] have shown that the full expression for $g_{\rho}$ given by $$\begin{aligned}
\label{pb1}
g_{\rho}&=& g_{e}+g_{1} t + g_{2} h - X_{\pm}\mid t \mid^{2-\alpha}
-Y\mid t \mid^{\beta}\mid h \mid - Z_{\pm} \mid t \mid^{-\gamma} h^2+ ...\end{aligned}$$ contains nonanaliticities related with the singularities at the critical line. Here $+$ means the disordered phase ($I$) and the $-$ refers to the ordered phases ($II^{\pm}$), the exponents $\alpha,\beta$ and $\gamma$ are the specific heat, the order parameter and the susceptibility critical exponents associated with $\lambda$ line. Similarly the amplitudes $X_{\pm}$, $Y$ and $Z_{\pm}$ are universally related to the critical amplitudes by $$\begin{aligned}
\label{r}
\frac{X_{+}}{X_{-}}&=&\frac{A_{+}}{A_{-}}\\ \nonumber
\frac{Z_{+}}{Z_{-}}&=&\frac{C_{+}}{C_{-}}\\ \nonumber
\frac{X_{+}Z_{+}}{Y^2}&=&\frac{A_{+}C_{+}}{(2-\alpha)(1-\alpha)B^2}\end{aligned}$$ where $A_{\pm}$, $B$ and $C_{\pm}$ are the critical amplitudes for the specific heat, for the order parameter and for the susceptibility near the critical line respectively.
But this is not the only nonanaliticities present in this phase boundary. Indeed the intersection of the surface $\rho$ with the plane $t=0$ is a curve that near the end point can be written as $$\begin{aligned}
\label{pb2}
g_{c}(h)&=&g_{e}-Y_{c}\mid h \mid^{1+1/\delta}\end{aligned}$$ where $\delta=\beta\Delta$ and where $\Delta$ is the gap exponent of the critical region. Here, as before, the coefficient $Y_{c}$ is also related to the critical amplitudes by $$\begin{aligned}
\label{ryc}
\frac{Y^{\delta-1}Z_{+}}{Y^{\delta}_{c}}&=&(\frac{\delta+1}{\delta})^{\delta}
\frac{B^{\delta-1}A_{+}}{B^{\delta}_{c}} .\end{aligned}$$ Therefore, if the $\lambda$-line would be classical, [*FB’s*]{} theory predicts that the phase boundary $g_{\rho}$ given by Eq. (\[pb1\]) would have two nonanalytic terms with exponents $\beta=1/2$ and $\gamma=1$, while the phase boundary $g_{c}$ given by Eq. (\[pb2\]) would have one singular term with exponent $\delta=3$. These are the usual mean-field critical exponents. Besides, [*FB*]{} approach also predicts that the coefficients $Z_{\pm}$, $Y$ and $Y_{c}$ would be universally related to the critical amplitudes $C_{\pm}$, $B$ and $B_{c}$ by $$\begin{aligned}
\label{rmf}
\frac{Z_{+}}{Z_{-}}&=&\frac{C_{+}}{C_{-}}=2 \\
\frac{Y^{\delta-1}Z_{+}}{Y^{\delta}_{c}}&=&(\frac{\delta+1}{\delta})^{\delta}
\frac{B^{\delta-1}A_{+}}{B^{\delta}_{c}}=(\frac{4}{3})^{3}. \nonumber\end{aligned}$$
**Phase-Diagram of Crossliked Copolymers**
===========================================
In this section, we investigate the melt of a non compatible and cross-linked mixture of polymers $A$ and $B$ using the Landau-Ginzburg-Wilson-de Gennes’ Hamiltonian [@De77][@Va97] $$\begin{aligned}
\label{H}
\beta H&=&\int_{}^{} d^{3}r
\{ \frac{(a\nabla \phi(r))^2}{48}+\frac{\tau}{2}\phi(r)^2+
u\phi(r)^4-h(r)\phi(r)+\frac{C(r) P(r)^2}{2}\}\end{aligned}$$ where $a$ is the size of one monomer and where the order parameter $\phi(r)$ is given in terms of the local fluctuations of the density of each specie, $\phi_A(r)$ and $\phi_B(r)$ by $$\begin{aligned}
\label{phi}
\phi_A(r)&=&\frac{1}{2}(1+l(r)+\phi(r))\nonumber \\
\phi_B(r)&=&\frac{1}{2}(1-l(r)-\phi(r)) .\end{aligned}$$ Here $\langle \phi_A(r) \rangle=1/2(1+\langle l(r) \rangle)$ and $\langle \phi_B(r)\rangle=1/2(1-\langle l(r) \rangle)$ are the volume fractions of each type of polymer. A nonzero value of $\langle l(r) \rangle$ allows for different volume fractions of each specie. The term linear in $\phi (r)$ in Eq. (\[Heff\]) contains the difference in the chemical potential between the two types of polymer. In each cross-link, two monomers one belonging to the specie $A$ and another to the specie $B$ are tied together. However they can be displaced slightly, leading to an elastic “polarization” given by $$\begin{aligned}
\label{P}
\vec{P}&=&\frac{1}{V}(\sum_{i\in A}^{}\vec{r}_i - \sum_{j\in B}^{}\vec{r}_j)\end{aligned}$$ where $\vec{r}_i$ is the position of the $i$ monomer at a polymer of type $A$ while $\vec{r}_j$ is the position of the $j$ monomer of type $B$ and where $V$ is the total volume of the system. In the same way that for the electrostatic case, polarization and charge are not independent quantities , here the elasticity and the volume fraction of each specie are also related by $\nabla \cdot \vec{P}=\phi(r)+l(r)$.
Consequently, the last term at the Hamiltonian Eq. (\[H\]) contains the elastic contribution associated with the cross-links. For simplicity, we assume that this term has a quadratic form that resembles the energy of a spring system. $C(r)$ is the internal rigidity is given by given by $C(r)=C_0\sum_{\vec{r}_i}^{}\delta(\vec{r}-\vec{r}_i)$ where here $\{\vec{r}_i\}$ correspond to coordinates of $N_c$ cross-links distributed in the volume $V$ according a Poisson distribution characterized by $\langle C(r_1)C(r_2) \rangle = C_{0} \langle C(r_{1}) \rangle\delta(r_1-r_2)$.
Since the cross-links are not permanent, they can open and close. Consequently the disorder is assumed to be annealed. Then, the resulting effective hamiltonian obtained after averaging over this distribution is given by $$\begin{aligned}
\label{Heff}
\beta H_{eff}&=&\int_{}^{} d^{3}r
\{ a^2\frac{(\nabla \phi(r))^2}{48}+\frac{\tau}{2}\phi(r)^2+
u\phi(r)^4-h(r)\phi(r) \nonumber \\
&+&\frac{1}{n}\int_{}^{} d^3 r [1- e^{C_{0} P^{2}/2}]\}.\end{aligned}$$ where $n=V/N_c$ ( actually $1/n$ is the density of cross-links). If the gel is very dense, $n$ will be related to the average number of monomers between two cross-links.
Now, one has to eliminate $\vec{P}$ in favor of $\phi)(r)$, using the constraint $\nabla \cdot \vec{P}=\phi(r)+l(r)$. Then, the expression for the free energy $\beta F_{eff}=-\ln e^{-\beta H_{eff}}$ can be evaluated at the mean-field level by taking the saddle point approximation, what leads to $$\begin{aligned}
\label{F}
\beta F_{eff}&=&\frac{1}{2}[\tau+\frac{(q_ca)^2}{24}]\psi_{q_c}\psi_{-q_c}
+u\psi_{q_c}^2\psi_{-q_c}^2
-h_{-q_c}\psi_{q_c}+\frac{1}{n}[1-e^{- cn\psi_{q_c}\psi_{-q_c}/(2q_c^2)}]\nonumber
\\\end{aligned}$$ where the expressions for $\psi_{q_c}$ and $q_c$ are given by the equations $$\begin{aligned}
\label{dF1}
\frac{\partial \beta H_{eff}}{\partial{\phi(q)}}\mid_{\phi_q=\psi_{q_c},q=q_c}
&=&[\tau+\frac{(q_c a)^2}{24}]\psi_{-q_c}+4u\psi_{-q_c}^2\psi_{q_c}
-h_{-q_c}\nonumber\\
&+&\frac{c}{q_c^2}\psi_{-q_c}e^{- cn\psi_{q_c}\psi_{-q_c}/(2q_c^2)}\end{aligned}$$ and $$\begin{aligned}
\label{dF2}
\frac{\partial \beta H_{eff}}{\partial{q}}\mid_{\phi_q=\psi_{q_c},q=q_c}&=&
a^2\frac{q_c }{24}\psi_{q_c}\psi_{-q_c}-\frac{c}{q_c^3}
e^{-cn\psi_{q_c}\psi_{-q_c}/(2q_c^2)}\end{aligned}$$ and where $c=C_0/n$. From the above equations we can see that the system exhibits four possible phases :
$(a)$ , a [*homogeneous phase*]{} where $\psi_I\rightarrow 0$ as $h_{q_{I}}\rightarrow 0$ and where $q_{c}=q_I\neq 0$;;
$(b)$, two microphases where [*partial segregation*]{} occurs, where $\psi_{II}\not\rightarrow 0$ as $h_{q_{II}}\rightarrow 0$ and where $q_{c}=q_{II}\neq 0$.
$(c)$ , a [*complete segregated phase*]{}, where $\psi_{III}\not\rightarrow 0$ as $h_{q_c}\rightarrow 0$ and where $q_c=q_{III}=0$;
The free energy associated with each one of these phases is given by $$\begin{aligned}
\label{I}
\beta F_I&=&\frac{1}{2}[\tau+\frac{(a q_I)^2}{24}]
\psi_I^2+u\psi_I^4+\frac{1}{n}[1-e^{-[cn\psi_I^2/(2q_I)]}]
-h_{-q_I} \psi_I\end{aligned}$$ for the phase $I$, $$\begin{aligned}
\label{II}
\beta F_{II}&=&\frac{1}{2}[\tau+\frac{(aq_{II})^2}{24}]\psi_{II}^2
+u\psi_{II}^4+\frac{1}{n}[1-e^{-[cn\psi_{II}^2/(2q_{II})]}]-h_{-q_{II}} \psi_{II}
\nonumber \\ \end{aligned}$$ for phase $II$ and $$\begin{aligned}
\label{III}
\beta F_{III}&=&\frac{\tau}{2}\psi_{III}^2+\frac{u}{4}\psi_{III}^4+\frac{1}{n}\end{aligned}$$ for the phase $III$. Here the values of $\psi_I,\psi_{II},\psi_{III},q_{I}$ and $q_{II}$ are given by the saddle point solutions of Eq. (\[dF1\]) and Eq. (\[dF2\]).
By comparing the above free energies we find the phase-diagram illustrated in fig. $2$ that goes as follows [@Va97]. At high values of $\tau$ that here plays the hole of temperature, the two species are mixed at phase $I$. For strongly cross-linked system, low values of $n$, as the temperature is decreased, one finds at $\tau=\tau_e$ a continuous phase transition to the microphase ( phase $II$). If the temperature is decreased even further, one meets a first-order phase boundary between the microphase, phase $II$, and a state where the two species are completely segregated, phase $III$. When the system is weakly cross-linked, the number of monomers between two cross-links, $n$, is large and consequently, as the temperature is decreased, one finds a first-order phase transition between the homogeneous state, phase $I$, and the state where the segregation is total, phase $III$. The three phases, $I$, $II$ and $III$ meet at the end point $e$ at $(\tau=\tau_e,n=n_e,h=0)$.
Then, in order to check if our predictions summarized at the Eq. (\[pb1\]), Eq. (\[pb2\]), Eq. (\[r\]) and Eq. (\[ryc\]), are correct, we will obtain the expression for the phase boundary $\rho$ near the end point. First, by equating the free energy of the homogeneous phase given by Eq. (\[I\]) to the free energy of the phase $III$ given by Eq. (\[III\]) we find the expression $$\begin{aligned}
\label{rho+}
\tau_{\rho +}&=& \tau_{e}+g_{1} (n-n_e) - X_{+}\mid n-n_e \mid^{2-\alpha}
- \frac{1}{2}Z_{+} \mid n-n_e \mid^{-\gamma} h^2\end{aligned}$$ for the phase boundary $\rho_{+}$ near the end point when $n>n_e$. Here the exponents are $\alpha=0$ and $\gamma=1$ while the non-universal coefficients are $$\begin{aligned}
\label{c+}
g_{1}&=& \frac{c^{3/2}}{u} \\ \nonumber
X_{+}&=& \frac{c^{5/2}}{4u^2} \\ \nonumber
Z_{+}&=& \frac{u^2}{c^2} .\end{aligned}$$ Similarly, equating the free energy of the microphase, phase $II$, given by Eq. (\[II\]) to the free energy of the phase $III$ given by Eq. (\[III\]), we obtain the expression for the phase boundary $\rho_{-}$ near the end point when $n<n_e$ namely $$\begin{aligned}
\label{rho-}
\tau_{\rho -}&=& \tau_{e}+g_{1} (n-n_e) - X_{-}\mid n-n_e \mid^{2-\alpha}
-Y\mid n-n_e \mid^{\beta}\mid h \mid \nonumber \\
&- &Z_{-} \mid n-n_e \mid^{-\gamma} h^2 \end{aligned}$$ where $\alpha=0$, $\beta=1/2$ and $\gamma=1$, where the coefficient $g_{1}$ is the same as the one obtained in Eq. (\[c+\]) as we have predicted ( see Eq.(\[pb1\])) and where $$\begin{aligned}
\label{c-}
X_{-}&=&\frac{5c^{5/2}}{4u^2} \\ \nonumber
Y&=& \sqrt{2}c^{1/4} \\ \nonumber
Z_{-}&=&\frac{u^2}{2c^2} .\end{aligned}$$ Now, one can easily verify that the ratios Eq. (\[r\]) built on basis of Eq. (\[c+\]) and Eq. (\[c-\]) are universal and given by the values predicted at Eq. (\[rmf\]).
The intersection of the plane $n=n_e$ with the surface $\rho$ is a curve given by $$\begin{aligned}
\label{rhoc}
\tau_{c}(h)&=&\tau_{e}-Y_{c}\mid h \mid^{1+1/\delta}\end{aligned}$$ where $\delta=3$ as we have predicted and where $$\begin{aligned}
\label{cc}
Y_{c}&=& \frac{4 (2u^{2})^{1/3}}{3 c^{1/2}} .\end{aligned}$$ Using the value above for the coefficient $Y_c$ together with Eq. (\[c+\]) and Eq. (\[c-\]) we obtain the universal values predicted by Eq. (\[rmf\]).
**Results and Conclusions**
===========================
Scaling arguments indicate that near any critical end point the phase boundaries must exhibit a nonanalyticities related to the singularities of the critical line.
In order to verify if this prediction based in a phenomenological approach is actually correct, we have looked for a realistic system that exhibits an end point. Therefore, we have studied the phase behavior of a mixture of two incompatible polymer species, $A$ and $B$, that at high temperatures are cross-linked. The phase-diagram for an asymmetric polymer blend with inhomogeneities in the cross-links, illustrated in figure $2$, exhibits a critical end point where the four phases present in this system, a homogeneous phase, two microphases and a phase of complete segregation, meet. Then, using a mean-field theory, expressions for the phase boundaries separating the critical phases from the non-critical phase were derived. Finally we checked that these equations exhibit nonanalyticities associated with the mean-field critical exponents of the specific heat, $\alpha=0$, of the order parameter , $\beta=1/2$, of the isothermal compressibility, $\gamma=1$ and of the order parameter at the critical temperature $\delta=3$. We also verified that the coefficients $X_{\pm}, Z_{\pm}, Y$ and $Y_c$ are universally related to the critical amplitudes $A_{\pm}, C_{\pm}, B$ and $B_{c}$ as was predicted in Eq. (\[r\]) and in Eq.(\[ryc\]). The universal ratios assume the classical values given by Eq. (\[rmf\]).
*ACKNOWLEDGMENTS*
This work was supported in part by CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico and FINEP - Financiadora de Estudos e Projetos, Brazil.
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**FIGURE CAPTION**
Figure $1$. Schematic phase-diagram $t \times g \times h$. The phase $I$ is the disordered phase, phase $II_{\pm}$ are the two ordered phases and phase $III$ is the noncritical phase. The line $\lambda$, dashed line, is a continuous transition, the planes $\rho$ and $\eta$ are first-order phase boundaries and $e$ locates the end point. The first-order lines $\sigma_{\pm}$ are the intersection of the surface $\rho$ with the plane $h=0$.
Figure $2$ . Phase-diagram $n \times \tau \times h$ for a $A$-$B$ polymer bend. The phase $I$ is the homogeneous phase, phases $II_{\pm}$ are the microphases and phase $III$ is the phase where the segregation is total. The line $\lambda$, dashed line, is a continuous transition, the planes $\rho$ and $\eta$ are first-order phase boundaries and $e$ locates the end point. The first-order lines $\sigma_{\pm}$ are the intersection of the surface $\rho$ with the plane $h=0$.
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---
abstract: 'We focus on the problem of teaching a robot to solve tasks presented sequentially, i.e., in a continual learning scenario. The robot should be able to solve all tasks it has encountered, without forgetting past tasks. We provide preliminary work on applying Reinforcement Learning to such setting, on 2D navigation tasks for a 3 wheel omni-directional robot. Our approach takes advantage of state representation learning and policy distillation. Policies are trained using learned features as input, rather than raw observations, allowing better sample efficiency. Policy distillation is used to combine multiple policies into a single one that solves all encountered tasks.'
bibliography:
- 'references.bib'
---
Introduction
============
In realistic real-life reinforcement learning scenarios, involving for instance service robots, tasks evolve over time either because the context of one task changes or because new tasks appear [@Doncieux18]. Our end goal is therefore to have an embodied agent in real-life that learns incrementally as time passes. One example would be a robot tasked with wrapping gifts. Most gifts are rectangular packages (cuboids), so the robot would first learn to wrap cuboids. Then if a soccer ball appears, the robot would have to learn how to wrap a sphere while still being able to wrap cuboids. Indeed, the robot should add additional knowledge to his repertoire. Moreover, even if it would be easier to learn how to wrap spheres and cuboids before test time, there are potentially many other shapes that have to be considered, and thus, learning continually seems more natural and convenient than trying to learn all skills at once.
Continual Learning (CL) and State Representation Learning (SRL) are essential to build agents that face such challenge. SRL allows to build strong representations of the world since agents should be able to understand their surroundings, and extract general concepts from sensory inputs of complex scenes. An agent better sees a chair as an object, not as a bunch of pixels together in an image. CL allows to learn such representation without forgetting in settings where the distribution of data change through time and is needed for agents that learn in the real-world and are required to adapt to changes. Combining CL and SRL would then allow to create strong representation robust to catastrophic forgetting.
Reinforcement learning (RL) is a popular approach to learn robot controllers that also has to face the CL challenges, and can take advantage of SRL to learn faster and to produce more robust policies. Therefore we perform our experiments (Fig. \[fig:real\_life\_tasks\]) in a setup where tasks are encountered sequentially and not all at once. Note that it differs from a setting where we can pick and shuffle experiences, often encountered in the multi-task RL literature (cf Section \[subsec:multi\]).
![Having access to task 1 only first, and then task 2 only, we learn a single policy that solves two real-life navigation tasks using policy distillation and sim2real transfer.[]{data-label="fig:real_life_tasks"}](images/photo_real_life_two_tasks.png)
In our approach we aim to take advantage of simulations to create this scenario. We demonstrate that deploying a policy in real-life which has continually learned two tasks in simulation is successful with our approach.
Our contribution consists on applying two major paradigms for robotics in real life: a state representation learning approach for compact and efficient representation that facilitates learning a policy, and a policy that learns continually in a sequential manner. The approach is deployed in a real robot thanks to policy distillation and sim2real transfer. Furthermore, in opposition to most methods in reinforcement learning, the solution we propose does not need a task indicator at test time. Indeed, the information about the task to be solved can be found from a different color tag in the image.
The rest of the article is structured as follows. Section \[ref:relatedwork\] introduces state representation learning, multi-task RL and continual learning paradigms in an RL setting, Sec.\[ref:experiments\] details the robotics settings and tasks performed; Sec.\[ref:methods\] details the methods utilized and Sec.\[ref:discussion\] concludes with future insights from our experiments.
Related work {#ref:relatedwork}
============
State representation learning (SRL)
-----------------------------------
Scaling end-to-end reinforcement learning to control real robots from vision presents a series of challenges, in particular in terms of sample efficiency. Against end-to-end learning, SRL [@Lesort18] can help learn a compact, efficient and relevant representation of states. Previous works such as [@Finn15; @Watter15; @Hoof16; @Lesort17; @Sermanet17; @Thomas17; @raffin2019decoupling] have shown that SRL can speed up policy learning, reducing the number of samples needed while additionally being easier to interpret.
Multi-task RL {#subsec:multi}
-------------
Multi-task RL aims at constructing one single policy module that can solve a number of different tasks. The CURIOUS algorithm [@colas2018curious] selects through exploration the tasks to be learned that improve an absolute learning progress metric the most. Policy distillation [@rusu2015policy] can also be used to merge different policies into one module/network. The Distral algorithm [@teh2017distral] is one successful example of such approach: a shared policy distills common behaviours from task-specific policies. Then, the distilled policy is used to guide task-specific policies via regularization using a Kullback-Leibler (KL) divergence. Other approaches like SAC-X [@riedmiller2018learning] or HER [@andrychowicz2017hindsight] take advantage of Multi-task RL by learning auxiliary tasks in order to help the learning of an objective task.
Continual Learning
------------------
Continual learning (CL) is the ability of a model to learn new skills without forgetting previous knowledge. In our context, it means learning several tasks sequentially and being able to solve any task at the end of the sequence. This differs from the easier multi-task scenario, where tasks can be experienced all at once. Most CL approaches can be classified into four main methods that differ in the way they handle the memory from past tasks.
The first method, referred to as *rehearsal*, keeps samples from previous tasks [@rebuffi2017icarl; @nguyen2017variational]. The second approach consists in applying *regularization*, either by constraining weight updates in order to maintain knowledge from previous tasks [@kirkpatrick2017overcoming; @Zenke17; @Maltoni18], or by keeping an old model in memory and distilling knowledge [@hinton2015distilling] into it later to remember [@li2018learning; @schwarz2018progress; @rusu2015policy]. The third category of strategies, *dynamic network architectures*, maintains past knowledge thanks to architectural modifications while learning [@rusu2016progressive; @fernando2017pathnet; @li2018learning; @fernando2017pathnet]. The fourth and more recent method is *generative replay* [@shin2017continual; @lesort2018generative; @wu2018memory; @lesort2018marginal], where a generative model is used as a memory to produce samples from previous tasks. This approach has also been referred to as pseudo-rehearsal. Note that all four type of methods can used for classification as well as for generation.
RL in real-life
---------------
Applying RL to real-life scenarios is a major challenge that has been studied widely. Most attempts fall into two categories: games and robotics.
For games, AlphaGo Zero [@silver2018general] has mastered the game of Go from scratch without any human supervision by combining RL, self-play and Monte Carlo Tree Search [@chaslot2008monte]. AlphaStar [@alphastar] and OpenAI Five [@OpenAI_dota] were both able to get competitive results against professional human players on the game Starcraft and DOTA2, respectively. Both solutions are based on RL, and current research is still investigating how to master the game with the same constraints as humans (e.g. same FPS).
In robotics, there is a plethora of successful attempts at deploying RL on real robots. One common approach is training policies in simulation and then deploying them in real-life hoping that they will successfully transfer, considering the gap in complexity between simulation and the real world. Such approaches are termed *Sim2Real* [@Golemo19], and have been successfully applied [@christiano2016transfer; @matas2018sim] in many scenarios. In order to cope with the unpredictable nature of the real world, one can use Domain Randomization [@tobin2017domain], which we use in our approach. This technique trains policies in numerous simulations that are randomly different from each other (different background, colors, etc.). Using this technique, the transfer to real life is easier.
Others have tried to train a policy directly on real robots, facing the hurdle of the lack of sample efficiency that RL suffers from. SAC-X [@riedmiller2018learning] is one example where a successful policy is learned directly on the real robot.
One can also find applications of reinforcement learning in other domains: TacTex’13 [@AAAI14-urieli], relying on online RL, is an autonomous broker agent that maximizes profit through energy trading; and [@DBLP:journals/corr/LiMRGGJ16] propose using policy gradients for dialogue generation using a set of reward functions designed to increase the diversity and length of generated responses. In education, a faster teaching policy by [[POMDP]{}]{} (partially-observable Markov decision process) planning [@Rafferty2016FasterTV] leverages a probabilistic learner model in order to achieve a long-term teaching objective.
In the literature, most approaches focus on the single-task or simultaneous multi-task scenario. In this paper, we attempt to train a policy on several tasks sequentially and deploy it in real life. Hence, we attempt to apply RL in real life in a continual learning setting.
Methods {#ref:methods}
=======
In this section we present the method proposed to combine state representation learning (SRL) and continual learning (CL) in a real life reinforcement learning setting. First we present how a single task is learned and how the SRL part works, secondly we explain how to learn continually and thirdly, we explain how we evaluate learning in the different phases of the learning sequence.
Learning on one task {#subsec:oneTask}
--------------------
Each task $i$ is learned following to procedure we describe here. First, as we use an SRL approach, we need to learn a state representation encoder. We sample data from the environment $Env_i$ (where $i$ refers to the task) with an agent guided by a random policy. We call this dataset $D_{Random,i}$. $D_{Random,i}$ is then used to train an SRL model composed of an inverse model and an auto-encoder. This architecture is inspired from [@raffin2019decoupling], and illustrated in Fig. \[fig:split-model\].
{width=".9\textwidth"}
Once the SRL model is trained, we use it’s encoder $E_i$ to provide with features for learning a reinforcement learning policy $\pi_i$ with the model $M(\theta)$ ($\theta$ represents the model parameters). Once $\pi_i$ is learned, we use it to generate sequences of on-policy observations with associated actions, which will eventually be used for distillation (Fig. \[fig:overview\], right). We call this the distillation dataset $D_{\pi i}$. We generate $D_{\pi i}$ the following way: we randomly sample a starting position and then let the agent generate a trajectory. At each step we save both the observation and associated action. We stop the sequence when enough reward is gathered (see Section \[ref:experiments\]). From each task is only kept the dataset $D_{\pi_i}$. As soon as we change task, $D_{Random,i}$ and $Env_i$ are not available anymore. In our setting, in order to decrease training time, we generate $D_{Random,i}$ in simulation and learn $\pi_i$ also in simulation. However, at the end of $T$ tasks, $\pi_{D_0,...,D_{T-1}}$ are tested in a real robot. In order to pass the reality gap, the datasets generated are augmented with different luminosity variations.
Learning continually {#subsec:continual_learning}
--------------------
To learn continually we use a distillation algorithm [@rusu2015policy]. Once we learned several tasks, we can aggregate the distillation datasets $D_{\pi_i}$ and distill the knowledge into a new model $M_Di(\theta')$ to produce a single flexible policy (Fig. \[fig:overview\], right). $\theta'$ are the parameters of the distillation model.
The distillation consists in learning in a supervised fashion the action probability associated to an observation at a timestep $t$. Each dataset $D_{\pi_i}$ allows to distill the policy $\pi_i$ into a new network. We name the distilled policy $\pi_{D_i}$. With the aggregation of several distillation datasets, we can distill several policies into the same network. By extension of the previous nomenclature we call a model where policy 1 and policy 2 have been distilled in, $\pi_{D_{1,2}}$. At test time, we do not need a task indicator; however, we assume that the observations and state space visually allow to recognize the current task. In the context of continual RL, the task signal is mandatory if the observation does not give any clue about the policy to be run. In our setting, as the policy can be inferred from a different color target tag, we do not need it.
The method presented allows to learn continually several policies without forgetting. On the other hand, $M(\theta)$ also learn on the sequence of task but without any memorization mechanism, its leads to catastrophic forgetting. The dataset $D_{\pi_i}$ contains 10k samples per task, which allows to learn the distillation very quickly (a few minutes are needed to learn $\pi_{D_i}$ while several hours are needed to learn $\pi_i$).
Evaluation {#subsec:evaluation}
----------
The main evaluation is the performance of the single and final policy, which can supposedly achieve all previous tasks, as well as being deployed in real life. For that, we report the mean and standard error on 5 runs of the policy on each task in simulation \[fig:final\_perf\], and provide videos to show the behaviour of the final policy.
On the other hand we also would like to analyze the learning process. In order to have an insight on the evolution of the distilled model, we save distillation datasets at different checkpoints in the sequence of tasks. Those checkpoints are saved regularly during the RL training. By distilling and evaluating at several time steps, we are able to evaluate the evolution of learning and forgetting on all environments, both separately and jointly. At each checkpoint, we evaluate the actual policy $\pi_i$ on past tasks to assess forgetting and compare it to $\pi_{D_{0,..,t}}$. It is important to note that, even if we consider $Env_i$ as not available anymore at task $i+1$, we did use it for evaluation purposes at any time.
Experimental setup {#ref:experiments}
==================
We apply our approach to learn continually two 2D navigation tasks on a real mobile robot.
Robotic setup
-------------
The experiments consists of 2D navigation tasks using a 3 wheel omni-directional robot. It is similar to the 2D random target mobile navigation ([@raffin2018s], identical reward setting and possibility of movement). The robot is identified by a black QR code and the scene is recorded from above.
We are able to simulate the experiment, since the robot’s input is a fixed RGB image of the scene recorded from above. The robot uses 4 high level discrete actions (move left/right, move up/down in a cartesian plane relative to the robot) rather than motor commands. The room where the real-life robotic experiments are to be performed is subject to illumination changes. The input image is a top-down view of the floor, which is lighted by surroundings windows and artificial illumination of the room. Hence, the illumination changes depending on the weather and time of the day. We use domain randomization [@tobin2017domain] to improve the chances of the policies learned in simulation to better transfer to the real world, by being robust to weather and time conditions. During RL training, at each timestep, the color of the background is randomly changed.
Continual learning setup
------------------------
Our continual learning scenario is composed of two similar environments, where the robot is rewarded according to the associated task. In both environments, the robot is free to navigate for up to 250 steps, performing only discrete actions within the boundaries identified by a red square.
In environment 1, the robot gets at each timestep $t$ a positive reward +1 for reaching the target identified by a red square marker (task 1), a negative reward $R_{t, bump}=-1$ for bumping into the boundaries, and no reward otherwise.
In environment 2, the robot gets at each timestep $t$ a reward $R_t$ (where $z_t$ is the 2D coordinate position with respect to the center of the circle, see eq. \[eq:reward-circular-task\]), which is highest when the agent is both keeping a distance to the target equal to a radius $r_{circle}$ (see eq. \[eq:reward-cicle\]), and has been moving for the previous $k$ steps (see eq. \[eq:reward-moving\]). An additional penalty term $R_{t, bump}=-1$ is added to the reward function in case of bump with the boundaries, and a coefficient $\lambda=10$ is introduced to balance the behaviour. $R_t$ is designed for agents to learn the task of circling around a central blue tag (task 2).
$$R_{t, circle} = 1 - \lambda (\|z_t\| - r_{circle}) ^2
\label{eq:reward-cicle}$$
$$R_{t, movement} = \|z_t -z_{t-k} \|_{2}^2
\label{eq:reward-moving}$$
$$R_t = \lambda R_{t, circle} * R_{t, movement} + \lambda ^ 2 R_{t, bump}
\label{eq:reward-circular-task}$$
It is important to note that as the tags associated to each scenario’s target are of different color, the algorithm can automatically infer which policy it needs to run and thus, does not need task labels at test time.
Moreover, while generating on-policy datasets $D_\pi 1$ (see Section \[subsec:oneTask\]) for task 1, we allow the robot a limited number of contacts with the target to reach ($N_{contacts}=10$) in order to mainly preserve the frames associated with the correct reaching behaviour. There are no such additional constraints when recording for task 2, the limit is the standard episode size, i.e. 250 time-steps.
The main software related to our experimental setting can be found at the url: <https://github.com/kalifou/robotics-rl-srl/tree/circular_movement_omnibot>\
Results
=======
Main result
-----------
Our main result is the continual learning of a single policy that solves both tasks in simulation, as presented in Fig. \[fig:overview\][^1]. The two teacher policies are learnt separately (i.e. independently) on each environment. Then, distillation is used to combine the two teacher policies into a single policy that can solve the two tasks.
Fig. \[fig:final\_perf\] demonstrates the efficiency of our approach. We can see that the single student distilled policy achieves close to maximum reward in both tasks.
Evaluation of distillation
--------------------------
We performed a more explicit evaluation of distillation in the task 2 (target circling (TC) around). While we train a policy using RL, we save the policy every 200 episodes (50K timesteps), and distill it into a new student policy which we test. This is illustrated in Fig. \[fig:distillation\_cc\]. Both curves are very close, which indicates distillation works as intended. It is able to transfer a policy using only a limited distillation dataset, with limited loss in the policy performance.
![Demonstration of the effectiveness of distillation. Blue: RL training curve of PPO2 on target circling task. Green: Mean and std performance on 8 seeds of distilled student policy. The blue policy is distilled into a student policy at regular time-step (1 episode = 250 timesteps). Both curves are very close, which indicates distillation works as intended. []{data-label="fig:distillation_cc"}](images/single_distillation_TC.png)
Discussion and future work {#ref:discussion}
==========================
Our work is preliminary and offers many possibilities for improvement. Our roadmap include having not only a policy learned in a continual way, but also the SRL model associated. We would need to update the SRL model as new tasks are presented sequentially. One possible approach would be to use Continual SRL methods like S-TRIGGER [@caselles2019s] or VASE [@achille2018life].
We also expect to encounter issues when scaling continual learning approaches to more tasks or environments. Indeed, the agent should not accumulate knowledge blindly, but rather make connections between different types of information (i.e. generalize) and/or selectively forget non-useful knowledge.\
Moreover, we intend to soon provide with supplementary quantitative results and videos of these tasks deployed in the real-life setup. We would like to train policies directly on the real robot, as it is the end goal scenario for this research. One promising approach would be to use model-based RL while learning the SRL modelto improve sample efficiency. The final goal would be to learn the policy on a real robot in a reasonable amount of time.
Conclusion
==========
In this paper we provide preliminary results towards a proper real life continual learning setup, where a real robot would encounter tasks presented in a sequence and be asked to accumulate knowledge in a scalable manner. The building blocks for achieving a single policy that solves all presented tasks consists of RL that uses state representation learning models, and distillation into a single policy. This model shows to be a good candidate for transfer to real life and future work should evaluate it in more and more complex tasks.
Acknowledgement
===============
This work is supported by the EU H2020 DREAM project (Grant agreement No 640891).
[^1]: The deployment and evaluation in real life is part of future work
|
---
abstract: |
The derivation by Alan Hodgkin and Andrew Huxley of their famous neuronal conductance model relied on experimental data gathered using neurons of the giant squid. It becomes clear that determining experimentally the conductances of neurons is hard, in particular under the presence of spatial and temporal heterogeneities. Moreover it is reasonable to expect variations between species or even between types of neurons of a same species. Determining conductances from one type of neuron is no guarantee that it works across the board.
We tackle the inverse problem of determining, given voltage data, conductances with non-uniform distribution computationally. In the simpler setting of a cable equation, we consider the Landweber iteration, a computational technique used to identify non-uniform spatial and temporal ionic distributions, both in a single branch or in a tree. Here, we propose and (numerically) investigate an iterative scheme that consists in numerically solving two partial differential equations in each step. We provide several numerical results showing that the method is able to capture the correct conductances given information on the voltages, even for noisy data.
address: 'Departamento de Modelagem Computacional, Laboratório Nacional de Computação Científica, Av. Getúlio Vargas 333, 25651-070 Petrópolis, RJ, Brazil'
author:
- 'Jemy A. Mandujano Valle, Alexandre L. Madureira, Antonio Leitão'
bibliography:
- 'mybibfile.bib'
date: 'October 05, 2018'
title: A Computational Approach for the inverse problem of neuronal conductances determination
---
Introduction.
=============
The seminal model of Hodgkin and Huxley [@H-H1952] of neuronal voltage conductance describes how action potential occurs and propagate. It is a landmark model, and present an outstanding combination of modeling based on physical arguments and experimental data, needed to determine the behavior of ion channels. Our Holy Grail is to determine such behavior as well, but directly from voltage measurements, not relying on excruciating data fitting.
Finding the conductances is crucial of one want to emulate the neuronal voltage propagation using computational models, since the conductances *are part of the data requires by the models*. Mimicking the work of Hodgkin and Huxley for every single neuron and or experimental condition is unfeasible. What we offer is a computational way to determine the conductances based on experimental data that is readily available to the researcher. Our method can also be extended to several computational models, such as the FitzHugh-Nagumo, Morris-Lecar, Hodgkin-Huxley, etc.
We use a simplified neuronal model, the cable equation [@bell1990; @ermentrout2010; @schutter2009], given by a parabolic partial differential equation. We consider first the case of a single branch of length $L$, represented by the interval $[0,L]$. The more general case of a branched tree is described in the Section \[subs2.3\]. In the cable model the membrane electrical potential $V:[0,T]\times[0,L]\to{\mathbb R}$ solves $$\label{equ1}
C_M{\frac{\partialV}{\partial t}}=\frac1{R_I+R_E}{\frac{\partial^2V}{\partial x^2}}+I_\ion\quad\text{in }(0,T]\times(0,L),$$ where $R_I$, $R_E$ are the internal and external neuronal resistance; $C_M$ represents the membrane capacitance per unit area. For the passive cable models, the ionic current is given by $$I_{\ion}(t,x)=\sum_{i\in\Ion}G_i(t,x)\bigl(V(t,x)-E_i\bigr),$$ where $\Ion$ is the set of ions being considered in the modeling, for example, $\Ion=\{\K,\Na,\Leak\}$. Also, $G_i(t,x)$ is the conductance for each ion $i\in\Ion$, and it might depend on spatial and temporal variables, as indicated in the notation. In this paper, these functions are not known. Finally, $E_i$ is the Nernst potential for each ion $i\in\Ion$.
To equation we add boundary and initial conditions given by $$\label{equ2}
{\frac{\partialV}{\partial x}}(t,0)=p(t), \qquad{\frac{\partialV}{\partial x}}(t,L)=q(t),
\qquad
V(0,x)=r(x),$$ We assume that the constants $C_M$, $R_I$, $R_E$ and $E_i$, and the functions $p$, $q$ and $r$ are given data.
Let $c=C_M(R_I+R_E)$ be positive, and $g_i(t,x)=G_i(t,x)(R_I+R_E)$. We gather then from and that $$\label{equ3}
\left \{\begin{array}{l}
V_{xx}(t,x)=c V_t(t,x)+\sum_{i \in Ion}g_i(t,x)[V(t,x)-E_i],
\\
V(0,x)=r(x),\;\hspace*{2.9cm} x \in [0,L],
\\
V_x(t,0) =p(t),\;\; V_x(t,L)=q(t) \;\;\;t \in [0,T],
\end{array} \right.$$
The inverse problem of finding “correct” conductances $g_i$ given some measurements of the voltage is “hard”, in the sense that it leads to ill-posed problems [@white1992], and that might explains why the vast majority of related research avoids the problem of finding *spatially dependent parameters*. There are several approaches to deal with the problem in hand, but certainly no panacea.
Hodgkin and Huxley [@H-H1952] tackled such problem by data fitting. Wilfrid Rall and co-authors considered several related questions for the cable equation [@rall1959; @rall1960; @rall1962; @rall1977; @rall1992-1; @rall1992-2]. See also [@SS98; @jack1971; @brown1981; @durand1983; @Aguanno1986; @schierwagen1990; @kawato1984]. In [@willms1999] there is an interesting attempt to introduce heterogeneity into the Hodgkin and Huxley model.
We consider next references with a stronger mathematical flavor; see however [@alain2017] for a biologically inclined work where the authors consider the branched cable equation with the chemical synapses, and convert somatic conductances to dendritic conductances.
Uniqueness of solutions for finding constant parameters in the cable equation, and related methods, are consider in [@cox1998; @cox2000; @cox2004], and [@cox2001-1] for a nonlinear model; see also [@avdonin2013; @MR534419] for further considerations related to existence and uniqueness. In [@cox2001-2] a more involved problem was tackled based on the FitzHugh–Nagumo and Morris–Lecar models, and where nonlinear functions modeling the conductances are sought. The method is based on fixed point arguments, and despite its ingenuity, it is not clear how to extend it to more involved models or to accommodate for spatially distributed ions channels.
In [@bell2005; @tadi2002; @cox2006; @avdonin2013; @avdonin2015], the question of determining spatially distributed conductances is investigated through different techniques and algorithms. They differ considerably from our method, and seem harder to generalize for other situations, as, for instance, when the domain is given by trees (with the obvious exception of [@avdonin2013; @avdonin2015]), for time dependent conductances, and for general nonlinear equations, our ultimate goal.
Inverse problems are ill-posed, and, under certain conditions, the *Landweber method* [@landweber1951] provides convergent iterative scheme. The main goal of the present paper is to develop the *Nonlinear Landweber method* [@hanke1991; @hanke1995; @binder1996; @neubauer2000; @chapko2004; @kaltenbacher2009] to solve the inverse problem of recovering the conductances in the cable equation. We also test the scheme under different scenarios.
We next outline the contents of the paper. In Section \[s:LM\] we present the Landweber method, detailing how it should be applied in the cases of a non-branched and branched cable, where the geometry is given by a tree. In Section \[s:numerics\] presents the related numerical results. In Section \[s:conc\] we present some concluding remarks, and in the Appendix we describe an abstract formulation of the Landweber method.
The Landweber Method applied to the conductance determination {#s:LM}
=============================================================
We describe the abstract form of the Landweber method in the Appendix. Here we consider its application to the problem at hand, that is, knowing the voltage $V$ at the space-time domain $\Gamma$, we want to determine $g_i$, assuming that holds. We consider three different cases, depending on where the voltage is known. In the first case, we assume that $V$ is known at all points, i.e., $\Gamma=[0,T]\times[0,L]$. In the remaining cases, we assume that the voltage is known at both or only one end points, and all times. Thus $\Gamma=[0,T]\times\{0,L\}$ or $\Gamma=[0,T]\times\{L\}$. This is summarized in Table \[tab0\], along with other definitions.
[CASE I]{} CASE II CASE III
------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------- --
$\Gamma=[0,T]\times[0,L]$ $\Gamma=[0,T]\times\{0,L\}$ $\Gamma=[0,T]\times\{L\}$
$R(F)=L^2\left( [0,T]\times{[0,L]}\right)$ $R(F)=\left[ L^2 [0,T]\right]^2$ $R(F)=L^2 [0,T]$
$V|_{\Gamma}=V(\cdot,\cdot)$ $V|_{\Gamma}=\left(- V(\cdot,0),V(\cdot,L)\right)$ $V|_{\Gamma}=V(\cdot,L)$
$W|_{\Gamma}=W(\cdot,\cdot)$ $W|_{\Gamma}=\left(- W(\cdot,0),W(\cdot,L)\right)$ $W|_{\Gamma}=W(\cdot,L)$
$\alpha_1=1;\hspace*{0.6cm}\alpha_2=0;\hspace*{0.6cm}\alpha_3=0$ $\alpha_1=0;\hspace*{0.6cm}\alpha_2=-1;\hspace*{0.6cm}\alpha_3=1$ $\alpha_1=0;\hspace*{0.6cm}\alpha_2=0;\hspace*{0.6cm}\alpha_3=1 $
: Summary of the three different cases considered in this paper. We seek the conductances $g_i$ assuming that holds and that the voltage $V$ is known at the space-time domain $\Gamma$ defined above along with other definitions.[]{data-label="tab0"}
Let $\Omega=[0,T]\times[0,L]$, $N_\ion$ the number of ions of the set , and the vector $\bg=(g_1,\dots,g_{N_\ion})$. In this article we work with the functional Hilbert space of square integrable functions $L^2(\Omega)$, and the Banach space of “essentially” bounded functions $L^\infty(\Omega)$ (see [@kreyszig1978] for precise definitions). Given $f\in L^2(\Omega)$ and $g$ continuous and bounded in $\Omega$, we define the norms $$\|f\|_{L^2(\Omega)}^2=\int_\Omega |f(\xi)|^2\,d\xi, \qquad
\|g\|_{L^\infty(\Omega)}^2=\sup_{\xi\in\Omega} |g(\xi)|,$$ where $\sup$ stands for *supremum*.
Consider the nonlinear operator $$F:D(F)\rightarrow R(F)$$ defined by $F(\bg)=V|_\Gamma$, where $V$ solves and $D(F)=\bigl(L^\infty(\Omega)\bigr)^{N_\ion}$. Also $R(F)$ and $V|_{\Gamma}$ are defined as in Table \[tab0\].
We consider the inverse problem of finding an approximation for $\bg$, given the noisy data $V^\delta|_\Gamma$, where $$\label{in4}
\|V-V^{\delta}\|_{L^\infty(\Gamma)}\le\delta,$$ for some known noise threshold $\delta>0$. That makes sense since in practice, the data $V|_\Gamma$ are never known exactly, and it is why we work with the data actually obtained $V^\delta|_\Gamma$, within a certain given precision $\delta>0$. In section 3 we detail the type of noise introduced.
Define in $R(F)$ the inner product $$\begin{gathered}
\label{equ5}
\la V^\delta|_\Gamma-V^{k,\delta}|_\Gamma,W^k|_\Gamma\ra_{R(F)}
=\alpha_1\int_0^L\int_0^T\bigl(V^\delta(t,x)-V^{k,\delta}(t,x)\bigr)W^k(t,x)\,dt\,dx
\\
-\alpha_2\int_0^T\left(V^\delta(t,0)-V^{k,\delta}(t,0)\right)W^k(t,0)\,dt+\alpha_3\int_0^T\left(V^\delta(t,L)-V^{k,\delta}(t,L)\right)W^k(t,L)\,dt, \end{gathered}$$ where $\alpha_1$, $\alpha_2$ and $\alpha_3$ are as in Table \[tab0\].
Given the initial guess $\bg^{1,\delta}\in\bigl(L^{\infty}(\Omega)\bigr)^{N_{ion}}$, the Landweber approximation for $\bg$ is defined by the sequence $$\label{equ6}
\bg^{k+1,\delta}=\bg^{k,\delta}+F'(\bg^{k,\delta})^*(V^\delta|_\Gamma-F(\bg^{k,\delta}))\quad\text{for }k\in{\mathbb N}.$$ As stopping criteria we use the *discrepancy principle* with $\tau > 2$ ( See [@kaltenbacher2008]), i.e., $$\label{equ7}
\|V^\delta|_\Gamma-F(\bg^{k_*,\delta})\|_{R(F)}
\le \tau\delta
\le\|V^\delta|_\Gamma-F(\bg^{k,\delta})\|_{R(F)},$$ for all $0\le k<k_*$. We next compute the Gâteux derivative $F'$ and its adjoint $F'(\cdot)^*$.
The adjoint operator $F'(\cdot)^*$ {#subs2.1}
----------------------------------
Given $\bg^{k,\delta}\in D(F)$ and the vector $\btheta=(\theta_1,\dots,\theta_{N_\ion})$, the Gâteux derivative of $F$ at $\bg^{k,\delta}$ in the direction $\btheta\in\bigl(L^{\infty}(\Omega)\bigr)^{N_{ion}}$ is given by $$\label{equ8}
F'(\bg^{k,\delta})(\btheta)
=\lim_{\lambda\to0}\frac{F(\bg^{k,\delta}+\lambda\btheta)-F(\bg^{k,\delta})}{\lambda}
=W^k|_\Gamma,$$ where $W^k$ solves $$\label{equ9}
\begin{gathered}
W_{xx}^k(t,x)-cW_t^k(t,x)-\sum_{i\in\Ion}g_i^{k,\delta}(t,x)W^k(t,x)
=\sum_{i\in\Ion}\theta_i(V^{k,\delta}(t,x)-E_i)\quad\text{in }\Omega,
\\
W^k(0,x)=0\quad\text{for }x \in [0,L],
\qquad
W^k_x(t,0)=W^k_x(t,L)=0\quad\text{for }t \in [0,T],
\end{gathered}$$ and $V^{k,\delta}$ solves with $g_i$ replaced by $g_i^{k,\delta}$. To obtain from , it is enough to consider the difference between problem with coefficients $\bg^{k,\delta}+\lambda\btheta$ and $\bg^{k,\delta}$, divide by $\lambda$ and take the limit $\lambda\to0$.
Consider now the following PDE with *final condition*: $$\label{equ10}
\left \{\begin{array}{l}
-U_{xx}^k(t,x)-c U_t^k(t,x)+\sum_{i\in\Ion}g_i^{k,\delta}(t,x)U^k(t,x)
=\alpha_1\left(V^\delta(t,x)-V^{k,\delta}(t,x)\right),
\\
U^k(T,x)=0,\hspace*{4.5cm} x \in [0,L],
\\
U^k_x(t,0)=\alpha_2\left(V^\delta(t,0)-V^{k,\delta}(t,0)\right), \;\;\;\;\;t \in [0,T],\\
U^k_x(t,L)=\alpha_3\left(V^\delta(t,L)-V^{k,\delta}(t,L)\right), \;\;\;t \in [0,T].
\end{array} \right.$$ The variables $\alpha_1$, $\alpha_2$ and $\alpha_3$ are defined in Table \[tab0\]. Let $V^{k,\delta}|_\Gamma=F(\bg^{k,\delta})$. From the Landweber iteration , we gather that $$\begin{gathered}
\label{equ11}
\la\bg^{k+1,\delta}-\bg^{k,\delta},\btheta\ra_{\left(L^2(\Omega)\right)^{N_{ion}}}
=\la F'(\bg^{k,\delta})^*(V^\delta|_\Gamma-F(\bg^{K,\delta})),\btheta\ra_{\left(L^2(\Omega)\right)^{N_{ion}}}
\\
=\la F'(\bg^{k,\delta})^*(V^\delta|_\Gamma-V^{k,\delta}|_\Gamma),\btheta\ra_{\left(L^2(\Omega)\right)^{N_{ion}}}
=\la V^\delta|_\Gamma-V^{k,\delta}|_\Gamma,F'(\bg^{k,\delta})\cdot\btheta\rangle_{R(F)}
\\
=\la V^\delta|_\Gamma-V^{k,\delta}|_\Gamma,W^k|_\Gamma \ra_{R(F)}, \end{gathered}$$ Although yields an interesting relation, it carries an impeding dependence on $\btheta$ through $W^k$. It is possible to avoid that by performing some “trick” manipulations. Multiplying the first equation of by $-W^k$, and integrating in the intervals $[0,T]$ and $[0,L]$ we gather that $$\begin{gathered}
\label{equ12}
\int_0^L\int_0^T U^k_{xx}(t,x)W^k(t,x)\,dt\,dx+\int_0^L\int_0^Tc\;U^k_t(t,x)W^k(t,x)\,dt\,dx
\\
-\int_0^L\int_0^T\sum_{i \in ion}g_i^{k,\delta}(t,x)\;U^k(t,x)W^k(t,x)\,dt\,dx=
\\
-\alpha_1\int_0^L\int_0^T \left(V^\delta(t,x)-V^{k,\delta}(t,x)\right) W^k(t,x)\,dt\,dx. \end{gathered}$$ Integrating for parts the first term of with respect to the space variable twice, and using the boundary conditions for $W^k$ we have $$\label{equ13}
\int_0^L\int_0^TU^k_{xx}(t,x)W^k(t,x)\,dt\,dx=\int_0^L\int_0^TU^k(t,x)W_{xx}^k(t,x)\,dt\,dx+\int_0^TU^k_x(t,x)W^k(t,x)|_0^L\,dt,$$ where we denote $U^k_x(t,x)W^k(t,x)|_0^L=U^k(t,L)W^k(t,L)-U^k(t,0)W^k(t,0)$. Similarly, integrating for parts the second term of with respect to time and using the initial condition of $W^k$ and the final condition of $U^k$, we gather that $$\label{equ14}
\int_0^L\int_0^Tc\;U^k_{t}(t,x)W^k(t,x)\,dt\,dx=-\int_0^L\int_0^TcU^k(t,x)W_t^k(t,x)\,dt\,dx.$$ Substituting and in , it follows that $$\begin{gathered}
\int_0^L\int_0^T\bigl(W_{xx}^k(t,x)-cW_t^k(t,x)-\sum_{i \in ion}g_i^{k,\delta}(t,x) \bigr)U^k(t,x)\,dt dx=
\\
-\alpha_1\int_0^L\int_0^T \bigl(V^\delta(t,x)-V^{k,\delta}(t,x)\bigr)W^k(t,x)\,dt\,dx-\int_0^TU^k_x(t,x)W^k(t,x)|_0^L\,dt. \end{gathered}$$ Substituting the first equation of in the previous equation, we obtain $$\begin{gathered}
\int_0^L\int_0^T\sum_{i \in ion}\theta_i(V^{k,\delta}(t,x)-E_i) U^k(t,x) \,dt \,dx =-\alpha_1\int_0^L\int_0^T \left(V^\delta(t,x)-V^{k,\delta}(t,x)\right) W^k(t,x)\,dt dx\\
-\int_0^TU^k_x(t,x)W^k(t,x)|_0^L\,dt. \end{gathered}$$ From the boundary conditions of , the following expression holds $$\begin{gathered}
\int_0^L\int_0^T\sum_{i \in ion}\theta_i(V^{k,\delta}(t,x)-E_i) U^k(t,x) \,dt dx =
-\alpha_1\int_0^L\int_0^T \left(V^\delta(t,x)-V^{k,\delta}(t,x)\right) W^k(t,x)\,dt dx\\
+\alpha_2\int_0^T\left(V^\delta(t,0)-V^{k,\delta}(t,0)\right)W^k(t,0)-\alpha_3\int_0^T\left(V^\delta(t,L)-V^{k,\delta}(t,L)\right)W^k(t,L)\,dt.\end{gathered}$$ From the previous equation and the definition of the inner product , we have $$\label{equ15}
\int_0^L\int_0^T\sum_{i \in ion}\theta_i(V^{k,\delta}(t,x)-E_i) U^k(t,x) \,dt dx =-\la V^\delta|_\Gamma-V^{k,\delta}|_\Gamma,W^k|_\Gamma \ra _{R(F)}.$$ From and we have $$\int_0^L\int_0^T\sum_{i\in\Ion}\theta_i\left( g_i^{k+1,\delta}(t,x)-g_i^{k,\delta}(t,x)\right)\,dt dx
=-\int_0^L\int_0^T\sum_{i\in\Ion}\theta_i(V^{k,\delta}(t,x)-E_i) U^k(t,x) \,dt dx.$$ Since $\btheta\in\bigl(L^2(\Omega)\bigr)^{N_{ion}}$ is arbitrary, we gather that the following iteration holds: $$\label{equ16}
g_i^{k+1,\delta}(t,x)=g_i^{k,\delta}(t,x)-(V^{k,\delta}(t,x)-E_i)U^k(t,x)\quad\text{for all }i\in\Ion$$
\[remark1\] Note from that $g_i^{k+1,\delta}(T,x)=g_i^{k,\delta}(T,x)$ for all $x\in[0,L]$ and every $k\in{\mathbb N}$, since, from , $U^k(T,x)=0$. Thus, $g_i^{k,\delta}$ is *never corrected at final time $T$*. To recover $g_i$ at time $T$, we consider multiple experiments $($Landweber-Kaczmarcz method [@kaltenbacher2008]$)$, one forward and another *backward in time*. The derivations for the backward in time case are the same as above, except that we change the signal of the derivatives with respect to time in the PDEs and . We also change the following conditions: $V(0,x)=r(x)$ by $V(T,x) = r(x)$ and $U^k(T,x)=0$ by $U^k(0,x)=0$. We detail such changes in Section \[subs2.2\]
In the case of a single experiment, the numerical scheme would be as follows. Check Table \[tab0\] for notation.
\[a:Land\] Choose $\bg^{1,\delta}$ as an initial approximation for $\bg$ Compute $V^{1,\delta}|_{\Gamma}$ from by replacing $\bg$ by $\bg^{1,\delta}$ k=1
\[remark2\] Whenever $\bg$ is time independent, and in this case we write $\bg(t,x)=\bg(x)$, the interaction is defined by $$\label{equ17}
g_i^{k+1,\delta}=g_i^{k,\delta}-\frac1T\int_0^T(V^{k,\delta}-E_i)U^k\,dt
\quad\text{for } i\in\Ion.$$
\[remark3\] Note that the numerical solutions of two PDEs are needed for each iteration. Of course, the solutions are obtained numerically, and for that we use finite difference scheme in space coupled with backward Euler in time. To compute the integral in we use the trapezoidal rule. In what follows we assume that the numerical approximations are accurate enough. All the experiments performed using Matlab^^.
Multiple experiments {#subs2.2}
--------------------
It might be convenient to have data from multiple experiments to guarantee a better approximation for the conductivities. In this case, a simple modification of the Algorithm \[a:Land\] is necessary [@kaltenbacher2008]. In our case, such multiple experimental approach is also necessary, as noted in Remark \[remark1\]. We detail here the necessary changes.
So assume we have two experiments yielding the data $V_F^\delta|_{\Gamma}$ and $V_B^\delta|_{\Gamma}$ (the letter “F” stands for *forward* and “B” for *backward*). Assume that the first experiment yields $V_F^\delta|_{\Gamma}$, and obeys with $r=r_F$, $p=p_F$ and $q=q_F$ given. Assume that the second experiment yields $V_B^\delta|_{\Gamma}$ and follows a similar equation, but *backwards in time*. By performing the change of variables $t\to T-t$, we gather that the same equation holds, but now with $$\label{equ18}
\left \{\begin{array}{l}
V_{xx}(t,x)=cV_t(t,x)+\sum_{i \in Ion} g_i(T-t ,x) [V(t,x)-E_i],
\\
V(0,x)=r_B(x),\;\hspace*{4.8cm} x \in [0,L],
\\
V_x(t,0)=p_B(T-t),\;\; V_x(t,L)=q_B(T-t) \;\;\;t \in [0,T].
\end{array} \right.$$ The actual solution is $V_B(t,x)=V(T-t,x)$.
We also define the backward in time equation equivalent of (with final condition), and again perform the change of variables $T-t$, yielding $$\label{equ19}
\left \{\begin{array}{l}
-U_{xx}^k(t,x)+c U_t^k(t,x)+\sum_{i\in\Ion}g_i^{k,\delta}(T-t,x)U^k(t,x)
=\alpha_1\left(V_B^\delta(T-t,x)-V_B^{k,\delta}(T-t,x)\right),
\\
U^k(0,x)=0,\hspace*{6.5cm} x \in [0,L],
\\
U^k_x(t,0)=\alpha_2\left(V_B^\delta(T-t,0)-V_B^{k,\delta}(T-t,0)\right), \;\;\;\;\;t \in [0,T],\\
U^k_x(t,L)=\alpha_3\left(V_B^\delta(T-t,L)-V_B^{k,\delta}(T-t,L)\right), \;\;\;t \in [0,T].
\end{array} \right.$$ The actual solution is $U^k_B=U^k$. Again, the variables $\alpha_1$, $\alpha_2$ and $\alpha_3$ are as in Table \[tab0\].
In terms of the algorithm, we gather the following.
\[a:Land2\] Choose $\bg^{1,\delta}$ as an initial approximation for $\bg$ Compute $V_F^{1,\delta}|_{\Gamma}$ from by replacing $\bg$ by $\bg^{1,\delta}$ and $r$, $p$, $q$ by $r_F$, $p_F$, $q_F$ Compute $V_B^{1,\delta}|_{\Gamma}$ from by replacing $\bg$ by $\bg^{1,\delta}$ k=1
A parallel version of the above algorithm is obtained by updating $\bg$ *simultaneously*. Also, the modification to accommodate several experiments is trivial.
The Landweber Method applied to the conductance determination defined on a tree {#subs2.3}
-------------------------------------------------------------------------------
Following the notation of [@avdonin2013; @avdonin2015], we let $\Theta =\mathcal{E}\cup\mathcal{V} $ be a tree, where $\mathcal{E}=\{e_1,e_2,\cdots,e_N\}$ is a set of edges, $\mathcal{V}=\{\nu_1,\nu_2,\cdots,\nu_M\}$ is a set of vertices, and the edges are connected at the vertices $\nu_j$. Let $\{\gamma_1,\gamma_2,\cdots, \gamma_m\}=\partial \Theta \subset \mathcal{V}$, i.e. if the index of a vertex, $\id(\nu)$, is the number of edges incident to it, then $\partial \Theta=\{\nu \in \mathcal{V}:\,\id(\nu)=1\}$. Hence $\mathcal{V} \setminus\partial \Theta=\{\nu\in \mathcal{V}:\,\id(\nu)> 2\}$. In Figure 1 we depict a simple example of a tree with one bifurcation point.
\[f:tree\]
Our cable equation model defined on a tree is given by $$\label{equ20}
\left \{\begin{array}{l}
V_{xx}(t,x)=cV_t(t,x) +\sum_{i \in Ion} g_i(t,x)\left[V(t,x)-E_i\right], \hspace*{1.15cm}\text{in}\;\;\;\;(0,T)\times\mathcal{E},
\vspace*{0.2cm}
\\
V(0,x)=r(x),\hspace*{2.8cm}\text{ in }x \in \Theta,
\vspace*{0.1cm}
\\
V_x(t,\gamma_k)=f_k(t),\hspace*{2.51cm}\text{ at each vertex }\gamma_k\in\partial \Theta\text{ and }t \in [0,T],
\vspace*{0.2cm}
\\
\sum\limits_{e_j\sim \nu}V'_j(t,\nu)=0,\hspace*{2.61cm}\text{ at each vertex } \nu \in\mathcal{V} \setminus\partial \Theta\text{ and }t \in [0,T],
\end{array} \right.$$ where $c$, $r$, $f_k$ and $\bg=(g_1,\dots,g_{N_\ion})$ are the given data; cf .
The last equation, of the EDP $(\ref{equ20})$, $V_j'(\nu)$ denotes the derivative of $V$ at the vertex $\nu$ taken along the edge $e_j$ in the direction towards the vertex. Also, $e_j\sim \nu $ means edge $e_j$ is incident to vertex $\nu$, and the sum is taken over all edges incident to $\nu$. Since $\partial \Theta $ consists of $m$ vertices, $f_k$ can be naturally identified with a function acting from $[0,T]$ to $\mathbb{R}^m$.
Let $\Omega=(0,T)\times\Theta$ and define the operator $$F:L^2(\Omega)\rightarrow L^2(\Omega)$$ such that $F(\bg)=V(\cdot,\cdot)$, where $V$ solves . The objective of this section is to, given $V^\delta$, obtain an approximation to $\bg$, using the method . To compute the adjoint operator $F^{'}(\cdot)^*$, we define the following PDE: $$\label{equ21}
\left \{\begin{array}{l}
-U_{xx}^k(t,x)-cU_t^k(t,x) +\sum_{i \in Ion} g_i(t,x)U^k(t,x)=V^\delta(t,x)-V^{k,\delta}(t,x),\hspace*{0.5cm}\text{in }(0,T)\times\mathcal{E},
\vspace*{0.2cm}
\\
U^k(T,x)=0,\hspace*{2.8cm}\text{ in }x \in \Theta,
\vspace*{0.2cm}
\\
U^k_x(t,\gamma_k)=0,\hspace*{2.8cm}\text{ at each vertex }\gamma_k\in\partial \Theta\text{ and }t \in [0,T],
\vspace*{0.2cm}
\\
\sum\limits_{e_j\sim \nu}U'_j(t,\nu)=0,\hspace*{2.25cm}\text{ at each vertex } \nu \in\mathcal{V} \setminus\partial \Theta\text{ and }t \in [0,T].
\end{array} \right.$$ By doing the same procedure as subsection \[subs2.1\], we obtain . Remarks \[remark1\], \[remark2\] and \[remark3\] also hold in this problem.
Numerical Simulation {#s:numerics}
====================
To design our numerical experiments, we first choose $\bg$ and compute $V$ from , obtaining then $V|_\Gamma$. Of course, in practice, the values of $V|_\Gamma$ are given by some experimental measures, and thus subject to experimental/measurement errors. In our examples, $V^\delta|_{\Gamma}$ is obtained by $$\label{equ22}
V^\delta(t,x)=V(t,x)+\rand_\delta, \;\;\;\text{for all } (t,x) \in \Gamma$$ where $\rand_\delta$ is a uniformly distributed random variable taking values in the range $[-\delta,\delta]$.
Next, given the initial guess $\bg^{1,\delta}$ and the data $V^\delta|_{\Gamma}$ and $\delta$, we start to recover $\bg$ using the Algorithm \[a:Land\]. Note that unlike in PDE problems where the exact solution usually has to be computed by numerical over-kill, here we have the exact $\bg$, and we use that to gauge the algorithm performance. We denote the following terms $$\label{equ23}
\Res_{k_*}= \| V^{\delta}|_{\Gamma}-F(\bg^{ {k_*},\delta})\|_{R(F)},
\qquad
\Error_{k_*}=\frac1{N_{ion}}{ \sum_{i\in Ion}\frac{\|g_i-g_i^{k_*,\delta}\|_{L^{\infty}(\Omega)}}{\|g_i\|_{L^\infty(\Omega)}}}\times 100 \%.$$
In this section we will present three numerical simulations. The first example considers only an ion $(\Ion=\{\K\})$, with the conductance $(\bg=g_{\K})$ dependent only the spatial variable. In the second example, still with one ion $(\Ion=\{\K\})$, the conductance depends on the spatial and temporal variable. Finally, in the third example, we consider two ions $(\Ion = \{\K,\Na\})$, where the conductance $(\bg=(g_\K, g_\Na))$ depends only on the spatial variable. In each example we consider two different values of $\tau$, based on experimental and theoretical considerations. \[Exa3.1\] We first consider a test problem as in [@bell2005]\*[Example 1]{}, which is a particular case of where $N_\ion=1$, $E_{\K}=0$, $L=1$, $c=1$, $T=1$, $g(t,x)=g_\K(x)$ and $$r(x)=\cos(x)\sin\bigl(0.5\tan(x)\bigr),
\qquad
p(t)=0.5\exp(-t),
\qquad q(t)=0.068\exp(-t).$$ The goal in this numerical test is to find $g_{\K}(x)=-0.25\sec^4(x)$ given the boundary condition $V^\delta|_{\Gamma}=(-V^\delta(\cdot,0),V^\delta(\cdot,L))$.
Here we introduce perturbation on the boundary data, according to . Thus, given $g_{\Na}$, we compute $V^\delta(\cdot,0)$ and $V^\delta(\cdot,1)$ according to and . In this tests we consider the initial guess $g_{\K}^{1,\delta}(x)=2x$. We partition the time variable in 30 points and the spatial variable in 20 points. In our experiments we test two different values of $\tau$ ($2.01$ and $4$), showing its influence.
----------- --------- ---------------- -------------------- -------- ---------------- --------------------
$k_* $ $\Error_{k_*}$ $\Res_{k_*}$ $k_* $ $\Error_{k_*}$ $\Res_{k_*}$
$0.1$ 58 135 % $2.0\times10^{-1}$ 0 168% $2.6\times10^{-1}$
$0.1/5$ 167 50 % $4.0\times10^{-2}$ 144 62% $7.9\times10^{-2}$
$0.1/5^2$ 203 42 % $7.8\times10^{-3}$ 187 44% $1.6\times10^{-2}$
$0.1/5^3$ 17421 39 % $1.6\times10^{-3}$ 241 40% $3.2\times10^{-3}$
$0.1/5^4$ 267460 24 % $3.2\times10^{-4}$ 88535 34% $6.4\times10^{-4}$
$0.1/5^5$ 1128345 4 % $6.4\times10^{-5}$ 725625 10% $1.3\times10^{-4}$
----------- --------- ---------------- -------------------- -------- ---------------- --------------------
: Numerical results for Example \[Exa3.1\]. The first column describes the noise level $\delta$, as in . From the second to the fourth column we present results for $\tau=2.02$, where the second column contains the number of iterations according to , and the third column contains the error according to . The fourth column presents the residual as in . We repeat the same information from the fifth to the seventh column, this time with $\tau=4$.[]{data-label="t:ex1"}
Table \[t:ex1\] presents the results for various levels of noise. When $\delta$ decreases, the number of iterations grow resulting in a better approximation for $g_{\K}$ and smaller residuals. As expected, the results of the fourth and seventh columns are close to $\tau\delta$, related to the stopping criteria .
In Figure \[f:figura1a\], we plot some results for $\delta=0.02$. On the left, we plot the initial guess $g^{1,\delta}$ and the corresponding approximate solutions for $\tau=2.01$ and $\tau=4$. On the right we present the actual error and the residual as a function of the iteration (log plot). Figure \[f:figuraVVdelta\] displays $V$ and $V^\delta$.
![For Example \[Exa3.1\], the plot on the left is for $\delta=0.02$ and shows the conductances as functions of the spatial variable. The red line is the exact solution, and the green and blue light line are the approximations obtained for $\tau=2.01$ and $\tau=4$. The blue curve is the initial guess of our iterative procedure. The plot on the right is for $\delta=0.02$ and displays the percentile error and the residual as functions of the iteration number. Furthermore, it shows the “stopping time” for both values of $\tau$. []{data-label="f:figura1a"}](Exemplo1a.pdf "fig:"){height="6cm" width="7.5cm"} ![For Example \[Exa3.1\], the plot on the left is for $\delta=0.02$ and shows the conductances as functions of the spatial variable. The red line is the exact solution, and the green and blue light line are the approximations obtained for $\tau=2.01$ and $\tau=4$. The blue curve is the initial guess of our iterative procedure. The plot on the right is for $\delta=0.02$ and displays the percentile error and the residual as functions of the iteration number. Furthermore, it shows the “stopping time” for both values of $\tau$. []{data-label="f:figura1a"}](Exemplo1b.pdf "fig:"){height="6cm" width="7.5cm"}
\[Exa3.2\] In this example we used multiple experiments and . For the first experiment the initial condition is $r_F(x)=\sin(x)$ we assume that we know $V_F^\delta$. For the second experiment the final condition is $r_B(x)=\cos(x)$ we assume that we know $V_B^\delta$. For the two experiments $N_\ion=1$, $E_{\K}=2$, $L=4$, $c=1$, $T=1$, $\bg(t,x)=g_\K(t,x)$ and $$p(t)=p_F(t)=p_B(t)=\exp(t),\qquad q(t)=q_F(t)=q_B(t)=\exp(t).$$ The goal of this example is to find $g_{\K}(t,x)=(x-2)\exp(-(x-2)^2-(4t-2)^2)$ given $V_F^\delta|_\Gamma=V_F^\delta(\cdot,\cdot)$ and $V_B^\delta|_\Gamma=V_B^\delta(\cdot,\cdot)$.
Given $g_{\K}$, we compute $V_F^\delta(\cdot,\cdot)$ according to and , and we compute $V_B^\delta(\cdot,\cdot)$ according to and . We consider the initial guess $g^{1,\delta}(t,x)=0$, and set both the time step and the mesh size as $1/64$.
We present in Table \[t:ex2\] the results for various levels of noise.
------------ -------- ---------------- --------------------- -------- ---------------- ---------------------
$k_* $ $\Error_{k_*}$ $\Res_{k_*}$ $k_* $ $\Error_{k_*}$ $\Res_{k_*}$
$0.05$ 2 81% $8.1\times 10^{-2}$ 1 100% $8.1\times 10^{-2}$
$0.05/5$ 8 50 % $3.6\times 10^{-3}$ 4 67% $6.5\times 10^{-3}$
$0.05/5^2$ 30 18 % $1.3\times 10^{-4}$ 23 23% $2.6\times 10^{-4}$
$0.05/5^3$ 84 5% $5.9\times 10^{-6}$ 66 7.2% $1.1\times 10^{-5}$
$0.05/5^4$ 186 2.6% $2.0\times 10^{-7}$ 152 2.7% $4.6\times 10^{-7}$
$0.05/5^5$ 575 2.2% $8.2\times 10^{-9}$ 360 2.3% $1.8\times 10^{-8}$
------------ -------- ---------------- --------------------- -------- ---------------- ---------------------
: Results related to Example \[Exa3.2\]. See Table \[t:ex1\] for a description of the contents.[]{data-label="t:ex2"}
\[Exa3.3\] In this example we consider two different ions, $\Na$ and $K$, where $E_\K=1$, $E_\Na=2$, $L=2$, $c=1$, $T=1$, $\bg(x)=\bigl(g_\K(x),g_\Na(x)\bigr)$ and $$r(x)=\cos(x+\pi/2),\qquad p(t)=\exp(-t),\qquad q(t)=0.$$ The goal is to find $\bg(x)=(\sin(x),\cos(x))$ given $V^\delta|_\Gamma=V^\delta(\cdot,\cdot)$. We compute $V^\delta(\cdot,\cdot)$ according to and , and as initial guess we set $\bg^{1,\delta}(x)=(3x,\exp(x))$. The time step is $1/64$ and the spatial mesh size is $1/128$.
In Table \[t:ex4\] we present the results for various levels of noise.
----------- -------- ---------------- --------------------- -------- ---------------- ---------------------
$k_* $ $\Error_{k_*}$ $\Res_{k_*}$ $k_* $ $\Error_{k_*}$ $\Res_{k_*}$
$10^{0}$ 1 645 % $9.4\times 10^{-1}$ 1 645% $1.8\times 10^{0}$
$10^{-1}$ 39 310 % $2.0\times 10^{-1}$ 27 373% $3.9\times 10^{-1}$
$10^{-2}$ 1366 70% $2.0\times 10^{-2}$ 601 142% $4.0\times 10^{-2}$
$10^{-3}$ 3182 25% $2.0\times 10^{-3}$ 2411 35% $4.0\times 10^{-3}$
$10^{-4}$ 17138 17% $2.0\times 10^{-4}$ 9021 19% $4.0\times 10^{-4}$
$10^{-5}$ 241044 8% $2.0\times 10^{-5}$ 100764 10% $4.0\times 10^{-5}$
----------- -------- ---------------- --------------------- -------- ---------------- ---------------------
: Results for Example \[Exa3.3\]. See Table \[t:ex1\] for a description of the contents.[]{data-label="t:ex4"}
\[Exa3.4\] As a particular case of , the set of edges is $\mathcal{E}=\{e_1,e_2,e_3\}$, the set of vertices $\mathcal{V}=\{\nu_1,\nu_2,\nu_3,\nu_4\}$, the border points $\partial \Theta=\{\gamma_1,\gamma_2,\gamma_3\}$, with one bifurcation point, as in Figure 1. The edge $e_1$ has vertices $\nu_1$ and $\nu_2$, the edge $e_2$ has vertices $\nu_2$ and $\nu_3$, finally the edge $e_3$ has vertices $\nu_2$ and $\nu_4$. The length of the edges are: $|e_1|=1$, $|e_2|=1$ e $|e_3|=2$. The numerical value of the vertices are: $\nu_1=0$, $\nu_2=1$, $\nu_3=2$ e $\nu_4=3$. The numerical value of the border points are: $\gamma_1=0$, $\gamma_2=2$ e $\gamma_3=3$. We denote by $V^i=V|e_i$ the restriction of $V$ to the edge $e_i$. In this example $N_{ion}=1$, $Ion=\{\K\}$, $E_{\K}=2$, $c=2$, $T=1$. For a point $p \in \Theta $ we define the initial condition, $$V(0,p)=\left \{\begin{array}{l}
\dist(\nu_1,p)+2\;\;\;if \;\;\;\;p\in e_1,
\\
\dist(\nu_2,p)+3\;\;\;if \;\;\;\;p\in e_1\cup e_2,
\end{array} \right.$$ where $\dist (a,b)$ is the distance between the points $a$ and $b$, the function $V(0,\cdot)$is continuous.
The boundary conditions are: $V_x(t,\gamma_1)=2t$, $V_x(t,\gamma_2)=\cos(t)$ and $V_x(t,\gamma_3)=0$. The condition at the bifurcation point is ${V_x}_{e_1}(t,\nu_2)-{V_x}_{e_2}(t,\nu_2)-{V_x}_{e_3}(t,\nu_2)=0$
The goal of this example is to find $g_{\K}(x)=\exp (x) $ given $V(\cdot,\cdot)$.
Thus, given $g_{\K}$, we compute $V^\delta(\cdot,\cdot)$ according to and . We consider the initial guess $g^{1,\delta}(t,x)=\sin (x)$. the time step is $1/300$ and the spatial mesh size is $1/61$. In Table \[t:ex5\] we present the results for various levels of noise.
The x-axis of figure \[f:figura8a\]-a corresponds to the edge $ e_1 $, the x-axis of figure \[f:figura8a\]-b corresponds to the edge $ e_2 $ and the x-axis of figure \[f:figura8a\]-c corresponds to the edge $e_3$.
\(a) (b)\
(c) (d)
----------- --------- ---------------- ---------------------- -------- ---------------- ---------------------
$k_* $ $\Error_{k_*}$ $\Res_{k_*}$ $k_* $ $\Error_{k_*}$ $\Res_{k_*}$
$10^{0} $ 3 962 % $1.9 \times 10^{0} $ 1 1995% $2.3\times 10^{0}$
$10^{-1}$ 250 76 % $2.0 \times 10^{-1}$ 68 157% $4.0\times 10^{-1}$
$10^{-2}$ 2631 17% $2.0 \times 10^{-2}$ 1288 28% $4.0\times 10^{-2}$
$10^{-3}$ 17608 5% $2.0 \times 10^{-3}$ 8407 8% $4.0\times 10^{-3}$
$10^{-4}$ 163865 1% $2.0 \times 10^{-4}$ 70548 2% $4.0\times 10^{-4}$
$10^{-5}$ 1174605 0.2% $2.0 \times 10^{-5}$ 621842 0.5% $4.0\times 10^{-5}$
----------- --------- ---------------- ---------------------- -------- ---------------- ---------------------
: Results for Example \[Exa3.4\]. See Table \[t:ex1\] for a description of the contents.[]{data-label="t:ex5"}
Conclusions {#s:conc}
===========
In this paper we develop and test a numerical scheme to find conductances of a passive cable. This has important applications and neuroscience, and is a hard problem. The method showed promising results, and the even harder problem of determining the conductances of “real” (i.e., nonlinear) neurons is currently under investigated.
The Landweber method has a somewhat straightforward description, but is not practical in the original formulation. Indeed, computing the adjoint of the Gâteux derivative seems impossible in general. The development of auxiliary equations to overcome such hurdle is more art than science, and is done in a case-by-case basis.
However, when the method can be implemented, it yields good results even in the presence of noise, as shown here. It is also general enough to accommodate for different geometries (straight cables and trees), and different measured data (end point, whole cable).
Abstract Formulation
====================
In practice, $V(\cdot,0)$ and $V(\cdot,L)$ are part of the data. To account for the possibility of measurement noise we denote the actual *measured data* by $V^\delta(\cdot,0)$ and $V^\delta\cdot,L)$. Given $V^\delta$ and under the assumption that holds, the inverse problem under consideration is to recover or approximate the conductances $g_i$.
The lack of stability characteristic of ill-posed problems can by tamed by regularization methods [@engl1996; @kaltenbacher2008; @kirsch2011], in particular by the non-linear Landweber method, that we describe next.
Consider the Hilbert spaces $\H_1$ and $\H_2$, with inner-products $\la\cdot,\cdot\ra_{\H_i}$ for $i=1,2$, and the operator $F:\D(F)\to\H_2$, where $\D(F)\subset\H_1$ is the domain of $F$, not necessary a Hilbert space. Assume that $F(\X)=\Y$, and that $\Y^\delta$ is known and represents a “noisy approximation” of the data $\Y$, where for a given $\delta>0$, $$\|\Y-\Y^\delta\|_{\H_2}\le\delta.$$ Our goal is to find an approximation for $\X$.
The Landweber iteration defines $\X^{k,\delta}$ by $$\label{in}
\X^{ {k+1},\delta}=\X^{ k,\delta}+F'(\X^{ k,\delta})^*(\Y-F(\X^{ k,\delta})).$$ For each fixed $\X^{ k,\delta}$, the Gâteaux derivative $F'(\X^{ k,\delta}):\D(F)\to\H_2$ defines a linear operator such that for each $\widetilde\X\in\D(F)$, $$F'(\X^{ k,\delta})(\widetilde\X)
=\lim_{t\to0}\frac{F(\X^{ k,\delta}+t\widetilde\X)-F(\X^{ k,\delta})}t.$$ Note in particular that it is possible to extend the domain of $F'(\X^{ k,\delta})$ to $\H_1$ when $F'(\X^{ k,\delta})$ is bounded and $\D(F)$ is dense in $\H_1$. We assume that.
The adjoint operator $F'(\X^{ k,\delta})^*:\H_2\to\H_1$ is such that $$\la F'(\X^{ k,\delta})^*\widetilde\Y,\widetilde\X\ra_{\H_1}
=\la\widetilde\Y,F'(\X^{ k,\delta})\widetilde\X\ra_{\H_2}
\quad\text{for all }\widetilde\X\in\H_1,\widetilde\Y\in\H_2.$$
One possible stopping criteria for the iterative scheme is given by the *discrepancy principle*, i.e., the iteration stops at the minimum $k_*=k(\delta,\Y^\delta)$, such that, for a given $\tau>2$, $$\label{in100}
\|\Y^\delta-F(\X^{ {k_*},\delta})\|_{\H_2}\le\tau\delta.$$ It is possible to show that, under certain conditions, $\X^{ {k_*},\delta}$ converges to a solution of $F(\X)=\Y$ as $\delta\to0$ [@kaltenbacher2008]\*[Theorem 2.6]{}.
|
---
abstract: 'We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given.'
author:
- 'B.Sh. Kulpeshov, S.V. Sudoplatov'
title: 'On structures in hypergraphs of models of a theory[^1]'
---
[**Keywords:**]{} hypergraph of models, elementary theory, elementarily substructural set, lattice structure.
Hypergraphs of models of a theory are derived objects allowing to obtain an essential structural information about both given theories and related semantic objects including graph ones [@CCMCT14; @Su013; @Su08; @Baik; @SudKar16; @KulSud17; @Sud17; @KulSud171; @KS12018]. Studying of hypergraphs of models of a theory is closely related with a series of papers on description of lattices of substructures [@Paris1; @Gaifman; @Paris2; @Wilkie; @Schmerl1; @Mills; @Schmerl11; @Schmerl2; @Schmerl3; @KosSch; @Schmerl4; @Schmerl5; @AQ].
In the presented paper we define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory as well as for structures on sets of isomorphism types of models of a theory are given.
Preliminaries
=============
Recall that a [*hypergraph*]{} is a pair of sets $(X,Y)$, where $Y$ is some subset of the Boolean $\mathcal{P}(X)$ of the set $X$.
Let $\mathcal{M}$ be some model of a complete theory $T$. Following [@SudKar16], we denote by $H(\mathcal{M})$ a family of all subsets $N$ of the universe $M$ of $\mathcal{M}$ that are universes of elementary submodels $\mathcal{N}$ of the model $\mathcal{M}$: $H(\mathcal{M})=\{N\mid
\mathcal{N}\preccurlyeq\mathcal{M}\}$. The pair $(M,H(\mathcal{M}))$ is called the [*hypergraph of elementary submodels*]{} of the model $\mathcal{M}$ and denoted by $\mathcal{H}(\mathcal{M})$.
[[@KulSud171]. Let $\mathcal{M}$ be a model of a theory $T$ with a hypergraph $\mathcal{H}=(M,H(\mathcal{M}))$ of elementary submodels, $A$ be an infinite definable set in $\mathcal{M}$, of arity $n$: $A\subseteq M^n$. The set $A$ is called [*$\mathcal{H}$-free*]{} if for any infinite set $A'\subseteq A$, $A'=A\cap Z^n$ for some $Z\in H(\mathcal{M})$ containing parameters for $A$. Two $\mathcal{H}$-free sets $A$ and $B$ of arities $m$ and $n$ respectively are called [*$\mathcal{H}$-independent*]{} if for any infinite $A'\subseteq A$ and $B'\subseteq B$ there is $Z\in H(\mathcal{M})$ containing parameters for $A$ and $B$ and such that $A'=A\cap Z^m$ and $B'=B\cap Z^n$.]{}
Note the following properties [@KulSud171].
1\. Any two tuples of a $\mathcal{H}$-free set $A$, whose distinct tuples do not have common coordinates, have same type.
Indeed, if there are tuples $\bar{a},\bar{b}\in A$ with ${\rm
tp}(\bar{a})\ne{\rm tp}(\bar{b})$ then for some formula $\varphi(\bar{x})$ the sets of solutions of that formula and of the formula $\neg\varphi(\bar{x})$ divide the set $A$ into two nonempty parts $A_1$ and $A_2$, where at least one part, say $A_1$, is infinite. Taking $A_1$ for $A'$ we have $A_1=A\cap Z^n$ for appropriate $Z\in H(\mathcal{M})$ and $n$. Then by the condition for tuples in $A$ we have $A_2\cap Z^n=\emptyset$ that is impossible since $Z$ is the universe of an elementary submodel of $\mathcal{M}$.
Thus the formula $\varphi(\bar{x})$, defining $A$, implies some complete type in $S^n(\emptyset)$, and if $A$ is $\emptyset$-definable then $\varphi(\bar{x})$ is a principal formula.
In particular, if the set $A$ is $\mathcal{H}$-free and $A\subseteq M$, then the formula, defining $A$, implies some complete type in $S^1(\emptyset)$.
2\. If $A\subseteq M$ is a $\mathcal{H}$-free set, then $A$ does not have nontrivial definable subsets, with parameters in $A$, i.e., subsets distinct to subsets defined by equalities and inequalities with elements in $A$.
Indeed, if $B\subset A$ is a nontrivial definable subset then $B$ is defined by a tuple $\bar{a}$ of parameters in $A$, forming a [*finite*]{} set $A_0\subset A$, and $B$ is distinct to subsets of $A_0$ and to $A\setminus C$, where $C\subseteq A_0$. Then removing from $A$ a set $B\setminus A_0$ or $(A\setminus B)\setminus A_0$, we obtain some $Z\in H(\mathcal{M})$ violating the satisfiability for $B$ or its complement. It contradicts the condition that $Z$ is the universe of an elementary submode of $\mathcal{M}$.
3\. If $A$ and $B$ are two $\mathcal{H}$-independent sets, where $A\cup B$ does not have distinct tuples with common coordinates, then $A\cap B=\emptyset$.
Indeed, if $A\cap B$ contains a tuple $\bar{a}$, then, choosing infinite sets $A'\subseteq A$ and $B'\subseteq B$ with $\bar{a}\in
A'$ and $\bar{a}\notin B'$, we obtain $\bar{a}\in A'=A\cap Z^n$ for appropriate $Z\in H(\mathcal{M})$ and $n$, as so $\bar{a}\in
B\cap Z^n=B'$. This contradiction means that $A\cap B=\emptyset$.
[[@KulSud17]. The [*complete union*]{} of hypergraphs $(X_i,Y_i)$, $i\in I$, is the hypergraph $\left(\bigcup\limits_{i\in I}X_i,Y\right)$, where $Y=\left\{\bigcup\limits_{i\in I}Z_i\mid Z_i\in Y_i\right\}$. If the sets $X_i$ are disjoint, the complete union is called [*disjoint*]{} too. If the set $X_i$ form a $\subseteq$-chain, then the complete union is called [*chain*]{}.]{}
By Property 3 we have the following theorem on decomposition of restrictions of hypergraphs $\mathcal{H}$, representable by unions of families of $\mathcal{H}$-independent sets.
\[dcu\] [[@KulSud171].]{} A restriction of hypergraph $\mathcal{H}=(M,H(\mathcal{M}))$ to a union of a family of $\mathcal{H}$-free $\mathcal{H}$-independent sets $A_i\subseteq M$ is represented as a disjoint complete union of restrictions $\mathcal{H}_i$ of the hypergraph $\mathcal{H}$ to the sets $A_i$.
Proof. Consider a family of $\mathcal{H}$-independent sets $A_i\subseteq M$. By Property 3 these sets are disjoint, and using the definition of $\mathcal{H}$-independence we immediately obtain that the union of restrictions $\mathcal{H}_i$ of $\mathcal{H}$ to the sets $A_i$ is complete.
Recall that a subset $A$ of a linearly ordered structure $M$ is called [*convex*]{} if for any $a, b\in A$ and $c\in M$ whenever $a<c<b$ we have $c\in A$. A [*weakly o-minimal structure*]{} is a linearly ordered structure $M=\langle M,=,<,\ldots \rangle$ such that any definable (with parameters) subset of the structure $M$ is a union of finitely many convex sets in $M$.
In the following definitions $M$ is a weakly o-minimal structure, $A, B\subseteq M$, $M$ be $|A|^+$-saturated, $p,q\in S_1(A)$ be non-algebraic types.
[@bbs1] We say that $p$ is not [*weakly orthogonal*]{} to $q$ ($p\not\perp^w q$) if there exist an $A$-definable formula $H(x,y)$, $\alpha \in p(M)$ and $\beta_1, \beta_2 \in q(M)$ such that $\beta_1 \in H(M,\alpha)$ and $\beta_2 \not\in H(M,\alpha)$.
[@k2003] We say that $p$ is not [*quite orthogonal*]{} to $q$ ($p\not\perp^q q$) if there exists an $A$-definable bijection $f: p(M)\to q(M)$. We say that a weakly o-minimal theory is [*quite o-minimal*]{} if the notions of weak and quite orthogonality of 1-types coincide.
In the work [@KS] the countable spectrum for quite o-minimal theories with non-maximal number of countable models has been described:
\[KS\_apal\] Let $T$ be a quite o-minimal theory with non-maximal number of countable models. Then $T$ has exactly $3^k\cdot 6^s$ countable models, where $k$ and $s$ are natural numbers. Moreover, for any $k,s\in\omega$ there exists a quite o-minimal theory $T$ having exactly $3^k\cdot 6^s$ countable models.
Realizations of these theories with a finite number of countable models are natural generalizations of Ehrenfeucht examples obtained by expansions of dense linear orderings by a countable set of constants, and they are called theories of [*Ehrenfeucht type*]{}. Moreover, these realizations are representative examples for hypergraphs of prime models [@CCMCT14; @Su08; @SudKar16]. We consider operators for hypergraphs allowing on one hand to describe the decomposition of hypergraphs of prime models for quite o-minimal theories with few countable models, and on the other hand pointing out constructions leading to the building of required hypergraphs by some simplest ones.
Having nontrivial structures like structures with some orders it is assumed that “complete” decompositions are considered modulo additional conditions guaranteing the elementarity for substructures with considered universes. So we use the [*conditional*]{} completeness taking unions with the properties of density, linearity etc.
Below we illustrate this conditional completeness for structures with dense linear orders.
Denote by $(M,H_{\rm dlo}(\mathcal{M}))$ the hypergraph of (prime) elementary submodels of a countable model $\mathcal{M}$ of the theory of dense linear order without endpoints.
\[no441\] [The class of hypergraphs $(M,H_{\rm dlo}(\mathcal{M}))$ is closed under countable chain complete unions, modulo density and having an encompassing dense linear order without endpoints. Thus, any hypergraph $(M,H_{\rm
dlo}(\mathcal{M}))$ is represented as a countable chain complete, modulo density, union of some its proper subhypergraphs.]{}
Any countable model of a theory of Ehrenfeucht type is a disjoint union of some intervals, which are ordered both themselves and between them, and of some singletons. Dense subsets of the intervals form universes of elementary substructures. So, in view of Remark \[no441\], we have:
\[KS\_apr\] [[@KulSud17].]{} A hypergraph of prime models of a countable model of a theory of Ehrenfeucht type is represented as a disjoint complete, modulo density, union of some hypergraphs in the form $(M,H_{\rm
dlo}(\mathcal{M}))$ as well as singleton hypergraphs of the form $(\{c\},\{\{c\}\})$.
\[no442\] [Taking into consideration links between sets of realizations of $1$-types, which are not weakly orthogonal, as well as definable equivalence relations, the construction for the proof of Theorem \[KS\_apr\] admits a natural generalization for an arbitrary quite o-minimal theory with few countable models. Here conditional complete unions should be additionally [*coordinated*]{}, i.e., considering definable bijections between sets of realizations of $1$-types, which are not quite orthogonal.]{}
Elementarily substructural sets
===============================
Let $\mathcal{M}$ be a model of theory $T$, $(M,H(\mathcal{M}))$ be a hypergraph of elementary submodels of $\mathcal{M}$. The sets $N\in H(\mathcal{M})$ are called [*elementarily submodel*]{} or [*elementarily substructural*]{} in $\mathcal{M}$.
Elementarily substructural sets in $\mathcal{M}$ are characterized by the following well-known Tarski–Vaught Theorem, which is called the Tarski–Vaught test.
\[thTV\] Let $\mathcal{A}$ and $\mathcal{B}$ be structures in a language $\Sigma$, $\mathcal{A}\subseteq\mathcal{B}$. The following are equivalent:
[(1)]{} $\mathcal{A}\preccurlyeq\mathcal{B}$[;]{}
[(2)]{} for any formula $\varphi(x_0,x_1,\ldots,x_n)$ in the language $\Sigma$ and for any elements $a_1,\ldots,a_n\in A$, if $\mathcal{B}\models\exists x_0\,\varphi(x_0,a_1,\ldots,a_n)$ then there is an element $a_0\in A$ such that $\mathcal{B}\models\varphi(a_0,a_1,\ldots,a_n)$.
\[coess1\] A set $N\subseteq M$ is elementarily substructural in $\mathcal{M}$ if and only if for any formula $\varphi(x_0,x_1,\ldots,x_n)$ in the language $\Sigma(\mathcal{M})$ and for any elements $a_1,\ldots,a_n\in N$, if $\mathcal{M}\models\exists x_0\,\varphi(x_0,a_1$, $\ldots$, $a_n)$ then there is an element $a_0\in N$ such that $\mathcal{M}\models\varphi(a_0,a_1,\ldots,a_n)$.
\[press1\] Let $A$ be a definable set in an $\omega_1$-saturated model $\mathcal{M}$ of a countable complete theory $T$. Then exactly one of the following conditions is satisfied:
$(1)$ $A$ is finite and contained in any elementarily substructural set in $\mathcal{M}$;
$(2)$ $A$ is infinite and has infinitely many distinct intersections with elementarily substructural sets in $\mathcal{M}$, and all these intersections are infinite; these intersections can be chosen forming an infinite chain/antichain by inclusion.
Proof. If $|A|<\omega$ then $A$ is contained in ${\rm
acl}(\emptyset)$, and so it is contained in any elementary submodel of $\mathcal{M}$.
If $A=\varphi(\mathcal{M},\bar{a})$ is infinite, we construct a countable submodel $\mathcal{N}_0\prec\mathcal{M}$ containing parameters in $\bar{a}$. Since $A$ is infinite, the set $A\cap
N_0$ is countable. By compactness, since $\mathcal{M}$ is $\omega_1$-saturated, the set $A\setminus N_0$ is infinite. Adding to $N_0$ new elements of $A$ we construct a countable model $\mathcal{N}_1$ such that $\mathcal{N}_0\prec\mathcal{N}_1\prec\mathcal{M}$. Continuing the process we build an elementary chain of models $\mathcal{N}_k$, $k\in\omega$, such that $\mathcal{N}_k\prec\mathcal{M}$ and $A\cap
N_k\subset A\cap N_{k+1}$, $k\in\omega$.
Constructing the required antichain of intersections $A\cap N$ with elementarily substructural sets $N$, it suffices to use [@KS12018 Theorem 2.10] allowing to separate disjoint finite sets, whose elements do not belong to ${\rm acl}(\emptyset)$.
The arguments for the proof of Proposition \[press1\] stay valid for a countable saturated model $\mathcal{M}$. Thus, we have the following
\[press2\] Let $A$ be a definable set in a countable saturated model $\mathcal{M}$ of a small theory $T$. Then exactly one of the following conditions is satisfied:
$(1)$ $A$ is finite and contained in any elementarily substructural set in $\mathcal{M}$;
$(2)$ $A$ is infinite and has infinitely many distinct intersections with elementarily substructural sets in $\mathcal{M}$, and all these intersections are infinite; these intersections can be chosen forming an infinite chain/antichain by inclusion.
The following example illustrates that if $\mathcal{M}$ is not saturated then the conclusions of assertions \[press1\] and \[press2\] can fail.
\[exess1\] [Let a set $A$ is defined by a unary predicate $P$ and includes infinitely many language constants $c_i$, $i\in I$. Then there is, in the language $\{P\}\cup\{c_i\mid i\in I\}$, a structure $\mathcal{M}$ having only finite set $A_0$ of elements in $A$, which are not interpreted by constants. Since elementarily substructural sets $N$ take all constants, there are only finitely many possibilities for intersections $A\cap N$.]{}
In view of aforesaid arguments it is interesting to describe possible cardinalities both for sets $H(\mathcal{M})$ and their restrictions $H(\mathcal{M})\upharpoonright
A\rightleftharpoons\{A\cap N\mid N\in H(\mathcal{M})\}$ on definable sets $A\subseteq M$.
Since in Example \[exess1\] intersections $A\cap N$, taking all constants $c_i$, can include an arbitrary subset of $A_0$, then for this example we have $|H(\mathcal{M})\upharpoonright
A|=2^{|A_0|}$. The same formula holds for infinite sets $A_0$, but in such a case the set $H(\mathcal{M})\upharpoonright A$ is transformed from finite one directly to a set with continuum many elements.
Note that for $\mathcal{H}$-free sets $A\subseteq M$, [*modulo*]{} ${\rm acl}(\emptyset)$ (i.e., for sets $A$, whose each subset $B\subseteq A\setminus {\rm acl}(\emptyset)$ has a representation $B\cup ({\rm acl}(\emptyset)\cap A)=A\cap N$ for some $N\in
H(\mathcal{M})$), the equality $|H(\mathcal{M})\upharpoonright
A|=2^{|A\setminus{\rm acl}(\emptyset)|}$ holds. Thus, we have the following [*dichotomy theorem*]{}.
\[thess2\] For any $\mathcal{H}$-free, modulo ${\rm acl}(\emptyset)$, set $A\subseteq M$ its restriction to any elementary submodel $\mathcal{M}_0\prec\mathcal{M}$ satisfies either $|H(\mathcal{M}_0)\upharpoonright A|=2^n$ for some $n\in\omega$, or $|H(\mathcal{M}_0)\upharpoonright A|=2^\lambda$ form some $\lambda\geq\omega$.
Similar to Example \[exess1\], the following example illustrates the dichotomy for hypergraphs of elementary submodels.
\[exess2\] [Consider the structure $\mathcal{M}$ of rational numbers, $\langle{\bf Q},<,c_q\rangle_{q\in{\bf Q}}$, in which every element is interpreted by a constant. This structure does not have proper elementary substructures, therefore $|H(\mathcal{M})|=1=2^0$. Extending $\mathcal{M}$ to a structure $\mathcal{M}_1$ by addition of $n$ elements for pairwise distinct $1$-types, defined by cuts, we have $|H(\mathcal{M}_1)|=2^n$. If $\mathcal{M}$ is extended till a structure $\mathcal{M}_2$ by addition of at least two elements of fixed cut or of infinitely many elements for distinct cuts, then by density the summarized number of added elements occurs infinite and $|H(\mathcal{M}_2)|=2^\lambda$ holds for some $\lambda\geq\omega$.]{}
At the same time there are examples of hypergraphs of elementary submodels, for which the conclusion of Theorem \[thess2\] fails. For instance, as shown in [@Wilkie], there are hypergraphs for the theory of arithmetic of natural numbers such that $|H(\mathcal{M})|=5$ and the lattice of elementary submodels is isomorphic to the lattice $P_5$.
Lattice structures associated with hypergraphs of models of a theory
====================================================================
For given structure $\mathcal{M}$ we define the structure $L(\mathcal{M})=\langle H(\mathcal{M});\wedge,\vee\rangle$ by the following relations for $\mathcal{M}_1,\mathcal{M}_2\prec\mathcal{M}$: $\mathcal{M}_1\wedge\mathcal{M}_2=\mathcal{M}_1\cap\mathcal{M}_2$ and $\mathcal{M}_1\vee\mathcal{M}_2=\mathcal{M}(M_1\cup M_2)$.
Consider the following question: when the structure $L(\mathcal{M})$ is a lattice?
Clearly, answering this question we have to characterize the conditions $\mathcal{M}_1\cap\mathcal{M}_2\prec\mathcal{M}$ and $\mathcal{M}(M_1\cup M_2)\prec\mathcal{M}$. Assuming that $\mathcal{M}$ is infinite, the structures $\mathcal{M}_1\cap\mathcal{M}_2$ should be infinite too, in particular, $M_1\cap M_2\ne\emptyset$. By [@SudKar16 Theorem 3.2], assuming that $\mathcal{M}$ is $\lambda$-saturated, it can not contain separated sets $A$ and $B$ of cardinalities $<\lambda$, such that ${\rm acl}(A)\cap{\rm
acl}(B)=\emptyset$.
By Theorem \[thTV\] we have the following theorems characterizing the elementarity of substructures.
\[thTV2\] Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be elementary substructures of structure $\mathcal{M}$ in a language $\Sigma$, $M_1\cap M_2\ne\emptyset$. The following are equivalent:
[(1)]{} $(\mathcal{M}_1\cap\mathcal{M}_2)\prec\mathcal{M}$[;]{}
[(2)]{} for any formula $\varphi(x_0,x_1,\ldots,x_n)$ of the language $\Sigma$ and for any elements $a_1,\ldots,a_n\in M_1\cap
M_2$ if $\mathcal{M}\models\exists x_0\,\varphi(x_0,a_1, \ldots,
a_n)$ then there is an element $a_0\in M_1\cap M_2$ such that $\mathcal{M}_i\models\varphi(a_0,a_1,\ldots,a_n)$, $i=1,2$.
\[thTV3\] Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be elementary substructures of structure $\mathcal{M}$ in a language $\Sigma$. The following are equivalent:
[(1)]{} $\mathcal{M}(M_1\cup M_2)\prec\mathcal{M}$[;]{}
[(2)]{} for any formula $\varphi(x_0,x_1,\ldots,x_n)$ of the language $\Sigma$ and for any elements $a_1,\ldots,a_n\in M_1\cap
M_2$ if $\mathcal{M}\models\exists x_0\,\varphi(x_0,a_1, \ldots,
a_n)$ then there is an element $a_0\in M(M_1\cup M_2)$ such that $\mathcal{M}(M_1\cup M_2)\models\varphi(a_0,a_1,\ldots,a_n)$.
The following examples illustrate valuations of the conditions (2) in Theorems \[thTV2\] and \[thTV3\].
\[exess3\] [Consider a structure $\mathcal{M}$ in a graph language $\{R^{(2)}\}$ with a symmetric irreflexive relation $R$ and elements $a_1,a_2,a_3,a_4$ such that $$R=\{[a_1,a_3],[a_1,a_4],[a_2,a_3],[a_2,a_4]\}.$$ The substructures $\mathcal{M}_1\subset\mathcal{M}$ and $\mathcal{M}_2\subset\mathcal{M}$ with the universes $\{a_1,a_2,a_3\}$ and $\{a_1,a_2,a_4\}$ respectively satisfy the formula $\varphi(a_1,a_2)\rightleftharpoons\exists
x(R(a_1,x)\wedge R(a_2,x))$ whereas $\mathcal{M}_1\cap\mathcal{M}_2$ does not satisfy that formula since appropriate elements for $x$ belong to $M_1\oplus M_2$.]{}
\[exess4\] [Consider a structure $\mathcal{M}$ of graph language $\{R^{(2)}\}$ with symmetric irreflexive relation $R$ and with elements $a_1,a_2,a_3$ such that $R=\{[a_1,a_3]$, $[a_2,a_3]\}$. The substructures $\mathcal{M}_1\subset\mathcal{M}$ and $\mathcal{M}_2\subset\mathcal{M}$ with the universes $\{a_1\}$ and $\{a_2\}$ form the substructure $\mathcal{M}(M_1\cup M_2)$ with the universe $\{a_1,a_2\}$ and it does not satisfy the formula $\varphi(a_1,a_2)$ in Example \[exess3\]. At the same time the structure $\mathcal{M}$ satisfies this formula.]{}
Since in some cases elementary substructures of given structure $\mathcal{M}$ form the lattice with respect to the operations $\mathcal{M}_1\wedge\mathcal{M}_2=\mathcal{M}_1\cap\mathcal{M}_2$ and $\mathcal{M}_1\vee\mathcal{M}_2=\mathcal{M}(M_1\cup M_2)$, the study of hypergraphs $\mathcal{H}(\mathcal{M})$, for these cases, is reduced to study of the lattices $L(\mathcal{M})$. As Example in [@Wilkie] shows, the lattices $L(\mathcal{M})$ can be non-distributive unlike the description in Theorem \[thess2\], where correspondent lattices are distributive, and for finite $H(\mathcal{M}_0)$ even form Boolean algebras.
In the given context hypergraphs/lattices with minimal, i.e. least structures play an important role. These structures can be obtained from an arbitrary structure by addition of constants interpreted by all elements of the structure. Besides, these minimal structures exist for finite sets $H(\mathcal{M})$.
In [@Tan], the following theorem on dichotomy for minimal structures is proved.
\[thess3\] Let $\mathcal{M}_0$ be a minimal structure, $\mathcal{M}$ be its saturated elementary extension and $p\in S_1(\mathcal{M}_0)$ be unique non-algebraic $1$-type. Then exactly one of the following conditions holds:
[(I)]{} the structure $(p(M), {\rm Sem}_p)$ is a pregeometry, where ${\rm Sem}_p$ is the relation of semi-isolation on the set of realizations of the type $p$, i.e. the following conditions are satisfied:
[(S1)]{} Monotony: if $A\subseteq B$ then $A\subseteq{\rm
Sem}_p(A)\subseteq{\rm Sem}_p(B)$;
[(S2)]{} Finite character: ${\rm Sem}_p(A)=\bigcup\{{\rm
Sem}_p(A_0)\mid A_0$ is a finite subset of $A\}$;
[(S3)]{} Transitivity: ${\rm Sem}_p(A)={\rm Sem}_p({\rm
Sem}_p(A))$;
[(S4)]{} Exchange property [(]{}Symmetry[)]{}: if $a\in{\rm
Sem}_p(A\cup\{b\})\setminus{\rm Sem}_p(A)$ then $b\in {\rm
Sem}_p(A\cup\{a\})$;
[(II)]{} for some finite $A\subset M$ there exists an infinite set $C_0\subseteq{\rm dcl}(A\cup M_0)$ and a definable quasi-order $\leq$ on $\mathcal{M}$ such that $C_0$ orders a type over $A$:
[(D1)]{} for any $c\in C_0$ the set $\{x\in C_0\mid c\leq x\}$ is a cofinite subset of $C_0$;
[(D2)]{} $C_0$ is an initial segment of $\mathcal{M}$: if $c\in
C_0$ and $m\leq c$, then $m\in C_0$.
Basic examples illustrating Theorem \[thess3\] are represented by ordered structures $\langle\omega,<\rangle$ and $\langle\omega+\omega^\ast,<\rangle$. The conclusion of Theorem \[thess2\] holds for both structures. Moreover, for $\mathcal{M}_1\equiv\langle\omega,<\rangle$ and $\mathcal{M}_2\equiv\langle\omega+\omega^\ast,<\rangle$ the structures $L(\mathcal{M}_1)$ and $L(\mathcal{M}_2)$ form atomic Boolean algebras, whose atoms are defined by equivalence classes, being closures of singletons, not in $\omega+\omega^\ast$, taking all predecessors and successors.
Return to Example \[exess2\]. It is known that the intersection of convex sets is convex, whereas the intersection of dense orders can be not dense. For instance, any interval $[a,b]$ contains countable dense subsets $X,Y$ such that $X\cap Y=\{a,b\}$. It means that for the structure $\mathcal{M}'\equiv\langle{\bf
Q},<,c_q\rangle_{q\in{\bf Q}}$ the structure $L(\mathcal{M}')$ forms a lattice, moreover, a Boolean algebra, if and only if each type in $S_1({\rm Th}(\mathcal{M}'))$ has at most one realization in $\mathcal{M}'$. If $\mathcal{M}'$, with the lattice $L(\mathcal{M}')$, realizes $\lambda$ non-principal $1$-types, then $|L(\mathcal{M}')|=2^\lambda$. Thus, the following proposition holds.
\[press5\] For the structure $L(\mathcal{M}')$ the following are equivalent:
$(1)$ $L(\mathcal{M}')$ is a lattice;
$(2)$ $L(\mathcal{M}')$ forms an atomic Boolean algebra;
$(3)$ each type in $S_1({\rm Th}(\mathcal{M}'))$ has at most one realization in $\mathcal{M}'$, and if $\mathcal{M}'$ realizes $\lambda$ non-principal $1$-types, then $|L(\mathcal{M}')|=2^\lambda$.
Proposition \[press5\] admits natural modifications for a series of theories with minimal models, for instance, for models, obtained by replacement of elements in $\mathcal{M}'$ with finite antichains of fixed cardinality marked by unary predicates $P_q$ instead of constants $c_q$. Note that admitting replacement of constants $c_q$ by infinite antichains $P_q$ the structure $L(\mathcal{M}')$ is not a lattice since $P_q$ can be divided by some elementary substructures $\mathcal{M}'_1, \mathcal{M}'_2\prec
\mathcal{M}'$ into two disjoint parts, whence $\mathcal{M}'_1\cap
\mathcal{M}'_2\not\prec \mathcal{M}'$.
Clearly, as above, in the general case if there are separable elements in definable sets $A\subseteq M$ of structure $\mathcal{M}$ then $L(\mathcal{M})$ is not closed under intersections, i.e., $L(\mathcal{M})$ is not even a lower semilattice. Thus, the following proposition holds.
\[press6\] If $L(\mathcal{M})$ is a lattice then $\mathcal{M}$ does not have definable sets $A\subseteq M$ containing elements separable each other, in particular, $\mathcal{M}$ does not contain $\mathcal{H}$-free sets $A\subseteq M$.
In view of Proposition \[press6\] it is natural, for given structure $\mathcal{M}$, along with $L(\mathcal{M})$ to consider for sets $X\subseteq M$ the following [*relative*]{} structures $L_X(\mathcal{M})$. Denote by $H_X(\mathcal{M}$ the family of all sets in $H(\mathcal{M}$ containing the set $X$. Then $L_X(\mathcal{M})\rightleftharpoons\langle
H_X(\mathcal{M};\wedge,\vee\rangle$, where for structures $\mathcal{M}_1,\mathcal{M}_2\prec\mathcal{M}$ containing $X$, $\mathcal{M}_1\wedge\mathcal{M}_2=\mathcal{M}_1\cap\mathcal{M}_2$ and $\mathcal{M}_1\vee\mathcal{M}_2=\mathcal{M}(M_1\cup M_2)$.
Note that if $X$ is a universe of some elementary substructure of structure $\mathcal{M}$ then definable sets $A\subseteq M$ already do not contain elements separable by sets in $L_X(\mathcal{M})$. Then, in any case, $\mathcal{M}_1\wedge\mathcal{M}_2$ is a substructure of $\mathcal{M}$ and the elementarity of that substructure is characterized by Theorem \[thTV2\].
The following example illustrates that apart from the density there are other reasons preventing to consider $L(\mathcal{M})$ as a lattice.
\[ex\_albai\_1\][@albai] Let $\mathcal{M}=\langle M; <,P^1, U^2, c_i \rangle_{i\in\omega}$ be a linearly ordered structure such that $\mathcal{M}$ is a disjoint union of interpretations of unary predicates $P$ and $\neg P$, where $\neg
P(\mathcal{M})<P(\mathcal{M})$. We identify interpretations of $P$ and $\neg P$ with the set $\mathbb{Q}$ of rational numbers with the natural order.
The symbol $U$ interprets the binary relation defined as follows: for any $a\in P(\mathcal{M}), b\in \neg P(\mathcal{M})$ $U(a,b)
\Leftrightarrow b<a+\sqrt{2}$.
The constants $c_i$ interpret an infinite strictly increasing sequence on $P(\mathcal{M})$ as follows: $c_i=i\in \mathbb{Q}$.
Clearly that $Th(\mathcal{M})$ is a weakly o-minimal theory. Let $$p(x):=\{x>c_i\mid i\in \omega\}\cup \{P(x)\},$$ $$q(y):=\{\forall t(U(c_i, t)\to t<y)\mid i\in\omega\}\cup \{\neg P(y)\}.$$
Obviously, $p, q\in S_1(\emptyset)$ are nonisolated types and $p\not\perp^w q$. Since there are no $\emptyset$-definable bijections from $p(\mathcal{M}')$ onto $q(\mathcal{M}')$, where $\mathcal{M}'$ is a model of $Th(\mathcal{M})$ realizing some of these types then $Th(\mathcal{M})$ is not quite o-minimal.
As shown in [@albai], $Th(\mathcal{M})$ has exactly 4 pairwise non-isomorphic countable models: the prime model $\mathcal{M}$, i.e., with $p(\mathcal{M})=\emptyset$ and $q(\mathcal{M})=\emptyset$; the model $\mathcal{M}_1$ such that $p(\mathcal{M}_1)$ has the ordering type $[0,1)\cap\mathbb{Q}$, $q(\mathcal{M}_1)$ has the ordering type $(0,1)\cap \mathbb{Q}$; the model $\mathcal{M}_2$ such that $p(M_2)$ has the ordering type $(0,1)\cap \mathbb{Q}$, $q(M_2)$ has the ordering type $[0,1)\cap\mathbb{Q}$; and the countable saturated model $\mathcal{M}_3$.
Therefore $\mathcal{M}_1\cap \mathcal{M}_2\not\prec
\mathcal{M}_3$. By this reason as well as by the possibility of violation of density in intersections, the structure $L(\mathcal{M}_3)$ does not form a lower semilattice.
\[note\_ehr\_1\] Along with Example \[ex\_albai\_1\] if we consider the known Ehrenfeucht’s example with three models: a prime model $\mathcal{M}_0$, a weakly saturated model $\mathcal{M}_1$, and a countable saturated model $\mathcal{M}_2$, then the structure $L(\mathcal{M}_2)$ is not a lattice in view of presence of dence definable intervals but includes the three-element linearly ordered lattice consisting of the universes $M_0$, $M_1$, $M_2$.
Lattice structures on sets of isomorphism types of models of a theory
=====================================================================
Following Example \[ex\_albai\_1\] and Remark \[note\_ehr\_1\] we consider a question on existence of natural lattices associated with hypergraphs $(M,H(\mathcal{M}))$ which a distinct to $L(\mathcal{M})$. Related lattices are lattices represented by Rudin–Keisler preorders ${\rm RK}(T)$ [@CCMCT14] for isomorphism types of prime models of a theory $T$, over finite sets, or their lattice fragments.
The description [@KulSudRK] of structures ${\rm RK}(T)$ for Ehrenfeucht quite o-minimal theories $T$ implies that these structures, for the considered theories, form finite lattices ${\rm LRK}(T)$ consisting of $2^k\cdot 3^s$ elements and, in view of the main result of the paper [@KS], the number $I(T,\omega)$ of pairwise non-isomorphic countable models of $T$ equals $3^k\cdot 6^s$, $k,s\in\omega$.
The Hasse diagrams illustrating these lattices ${\rm LRK}(T)$ are represented in Fig. \[fig1\]–\[fig9\] for the following values $k$ and $s$:
1\) $k=1$, $s=0$;
2\) $k=0$, $s=1$;
3\) $k=2$, $s=0$;
4\) $k=3$, $s=0$;
5\) $k=0$, $s=2$;
6\) $k=0$, $s=3$;
7\) $k=1$, $s=1$;
3\) $k=2$, $s=1$;
5\) $k=1$, $s=2$.
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(22,7)(4,1.7) [(19.5,2.5)[(0,5)[5]{}]{} (19.5,2.5)[(0,0)\[cc\][$\bullet$]{}]{} (19.5,5)[(0,0)\[cc\][$\bullet$]{}]{} (19.5,7.5)[(0,0)\[cc\][$\bullet$]{}]{} ]{}
(5,1)(-1.14,-0.1)
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(3,1)(-0.69,-0.1)
(0,1)[(0,0)\[cc\][$\bullet$]{}]{}
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(4,3.5)(1.6,0.2)
(3,1)[(0,0)\[cc\][$\bullet$]{}]{} (3,2)[(0,0)\[cc\][$\bullet$]{}]{} (4,0)[(0,0)\[cc\][$\bullet$]{}]{} (4,1)[(0,0)\[cc\][$\bullet$]{}]{} (4.006,2)[(0,0)\[cc\][$\bullet$]{}]{} (4,3)[(0,0)\[cc\][$\bullet$]{}]{} (5,1)[(0,0)\[cc\][$\bullet$]{}]{} (5,2)[(0,0)\[cc\][$\bullet$]{}]{} (3,1)[(0,1)[1]{}]{} (5,1)[(0,1)[1]{}]{} (4,0)[(0,1)[1]{}]{} (4,2)[(0,1)[1]{}]{} (4,0)[(1,1)[1]{}]{} (4,1)[(1,1)[1]{}]{} (4,0)[(-1,1)[1]{}]{} (4,1)[(-1,1)[1]{}]{} (4,3)[(-1,-1)[1]{}]{} (4,2)[(-1,-1)[1]{}]{} (4,3)[(1,-1)[1]{}]{} (4,2)[(1,-1)[1]{}]{} (4,2.5)[(0,0)\[cc\][$\bullet$]{}]{} (3.35,0.65)[(0,0)\[cc\][$\bullet$]{}]{} (3.35,1.15)[(0,0)\[cc\][$\bullet$]{}]{} (3.35,1.65)[(0,0)\[cc\][$\bullet$]{}]{} (3.95,1.75)[(0,0)\[cc\][$\bullet$]{}]{} (3.95,1.25)[(0,0)\[cc\][$\bullet$]{}]{} (3.945,2.25)[(0,0)\[cc\][$\bullet$]{}]{} (4.35,1.65)[(0,0)\[cc\][$\bullet$]{}]{} (4.35,2.15)[(0,0)\[cc\][$\bullet$]{}]{} (4.35,2.65)[(0,0)\[cc\][$\bullet$]{}]{} (3.6,1.6)[(0,0)\[cc\][$\bullet$]{}]{} (3.6,2.1)[(0,0)\[cc\][$\bullet$]{}]{} (3.6,2.6)[(0,0)\[cc\][$\bullet$]{}]{} (4.6,0.6)[(0,0)\[cc\][$\bullet$]{}]{} (4.6,1.1)[(0,0)\[cc\][$\bullet$]{}]{} (4.6,1.6)[(0,0)\[cc\][$\bullet$]{}]{} (4,0.5)[(0,0)\[cc\][$\bullet$]{}]{} (5,1.5)[(0,0)\[cc\][$\bullet$]{}]{} (3,1.5)[(0,0)\[cc\][$\bullet$]{}]{}
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\[th\_qom\_l\] Let $T$ be an Ehrenfeucht quite o-minimal theory, $I(T,\omega)=3^k\cdot 6^s$, $k,s\in\omega$. Then:
$(1)$ ${\rm LRK}(T)$ is a lattice;
$(2)$ ${\rm LRK}(T)$ is a Boolean algebra $\Leftrightarrow$ $k\geq
1$ and $s=0$; in such a case the Boolean lattice ${\rm LRK}(T)$ has a cardinality $2^k$;
$(3)$ ${\rm LRK}(T)$ is linearly ordered $\Leftrightarrow$ $k+s\leq 1$.
Proof of Theorem \[th\_qom\_l\]. Let $\Gamma=\Gamma_1\cup\Gamma_2$ be a maximal independent set of nonisolated types in $S_1(T)$, where realizations of each type in $\Gamma_1$ generate three models, with prime one, and realizations of each type in $\Gamma_2$ generate six models, with prime one, $|\Gamma_1|=k$, $|\Gamma_2|=s$.
$(1)$ We argue to show that ${\rm LRK}(T)$ is a lattice. Indeed, for isomorphism types $\widetilde{\mathcal{M}_1}$ and $\widetilde{\mathcal{M}_2}$ of prime model $\mathcal{M}_1$ and $\mathcal{M}_2$ over some finite sets $A$ and $B$, respectively, we define sets $X,Y\subseteq\Gamma\times\{0,1\}$ defining these isomorphism types such that $X=\{(p,0)\mid \mathcal{M}_1\models
p(a)\mbox{ for some }a\in A,\mbox{ and }|p(\mathcal{M}_1)|=1\mbox{
or }p\in\Gamma_1\}\cup \{(p,1)\mid \mathcal{M}_1\models p(a)\mbox{
for some }a\in A,|p(\mathcal{M}_1)|\geq\omega,p\in\Gamma_2\}$ and $Y=\{(q,0)\mid \mathcal{M}_2\models q(b)\mbox{ for some }b\in
B,\mbox{ and }|q(\mathcal{M}_2)|=1\mbox{ or }q\in\Gamma_1\}\cup
\{(q,1)\mid \mathcal{M}_2\models q(b)\mbox{ for some }b\in
B,|q(\mathcal{M}_2)|\geq\omega,q\in\Gamma_2\}$. Then the isomorphism type for $\widetilde{\mathcal{M}_1}\wedge\widetilde{\mathcal{M}_2}$ corresponds to the set $U\subseteq\Gamma\times\{0,1\}$ consisting of all common pairs of $X$ and $Y$, as well as all possible pairs $(p,0)$, if $(p,0)\in X$ and $(p,1)\in Y$, or $(p,1)\in X$ and $(p,0)\in Y$. And the isomorphism type for $\widetilde{\mathcal{M}_1}\vee\widetilde{\mathcal{M}_2}$ corresponds to the set $V\subseteq\Gamma\times\{0,1\}$ consisting of the following pairs:
i\) all common pairs of $X$ and $Y$,
ii\) all pairs $(p,i)\in X$ such that $Y\cap\{(p,0),(p,1)\}\emptyset$,
iii\) all pairs $(p,i)\in Y$ such that $X\cap\{(p,0),(p,1)\}\emptyset$,
iv\) all pairs $(p,1)$ such that $(p,0)\in X$ and $(p,1)\in Y$, or $(p,1)\in X$ and $(p,0)\in Y$.
The defined correspondence witnesses that ${\rm LRK}(T)$ is a lattice.
$(2)$ If $s\ne 0$ then ${\rm LRK}(T)$ is not a Boolean algebra by Stone Theorem, since the cardinality of each finite Boolean algebra equals $2^n$ for some $n\in\omega$ whereas $|{\rm
LRK}(T)|=2^k\cdot 3^s$. If $s=0$ then ${\rm LRK}(T)$ is a Boolean algebra of a cardinality $2^k$ such that for isomorphism types $\widetilde{\mathcal{M}_1}$ and $\widetilde{\mathcal{M}_2}$ of prime models $\mathcal{M}_1$ and $\mathcal{M}_2$ over some finite sets $A$ and $B$, respectively, and for sets $X,Y\subseteq\Gamma$ such that $X=\{p(x)\in\Gamma\mid \mathcal{M}_1\models p(a)\mbox{
for some }a\in A \}$ and $Y=\{q(x)\in\Gamma\mid
\mathcal{M}_2\models q(b)\mbox{ for some }b\in B \}$, the isomorphism type $\widetilde{\mathcal{M}_1}\wedge\widetilde{\mathcal{M}_2}$ corresponds to the set $X\cap Y$, and the isomorphism type $\widetilde{\mathcal{M}_1}\vee\widetilde{\mathcal{M}_2}$ corresponds to the set $X\cup Y$.
\(3) If $k+s\leq 1$ then ${\rm LRK}(T)$ is linearly ordered as shown in Fig. \[fig1\] and \[fig2\]. If $k+s
>1$ then $|\Gamma|>1$ and for distinct types $p,q\in\Gamma$ the isomorphism types of models $\mathcal{M}_p$ and $\mathcal{M}_q$ are incomparable in ${\rm LRK}(T)$.
The description for distributions of disjoint unions of Ehrenfeucht theories and the arguments for the proof of Theorem \[th\_qom\_l\] allow to formulate the following theorem modifying Theorem \[th\_qom\_l\].
\[th\_2\] Let $T$ be a disjoint union of theories $T_1$ and $T_2$ in disjoint languages and having finite numbers $I(T_1,\omega)$ and $I(T_2,\omega)$ of countable models. Then:
$(1)$ ${\rm LRK}(T)$ is a [(]{}Boolean[)]{} lattice $\Leftrightarrow$ ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ are [(]{}Boolean[)]{} lattices;
$(2)$ ${\rm LRK}(T)$ is linearly ordered $\Leftrightarrow$ ${\rm
LRK}(T_1)$ and ${\rm LRK}(T_2)$ are linearly ordered, and ${\rm min}\{I(T_1,\omega),I(T_2,\omega)\}=1$.
Proof. (1) If ${\rm LRK}(T)$ is a [(]{}Boolean[)]{} lattice, then ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ are [(]{}Boolean[)]{} lattices, since ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ are isomorphic to sublattices $L_1$ and $L_2$ of the lattice ${\rm
LRK}(T)$, and elements/complements of elements in ${\rm LRK}(T)$ are represented as pairs of elements/complements of elements in $L_1$ and $L_2$. If ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ are [(]{}Boolean[)]{} lattices, then ${\rm LRK}(T)$ is a [(]{}Boolean[)]{} lattice again in view of aforesaid representation.
\(2) If ${\rm LRK}(T)$ is linearly ordered then ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ are linearly ordered, being isomorphic to substructures of ${\rm LRK}(T)$. Here $T_1$ or $T_2$ should be $\omega$-categorical, since otherwise prime models over pairs $(p_1,q_1)$ and $(p_2,q_2)$ occur ${\rm LRK}(T)$-incomparable, where $p_1,p_2\in S_1(T_1)$, $q_1,q_2\in S_1(T_2)$, $p_1,q_2$ are isolated, $p_2,q_1$ are nonisolated.
If structures ${\rm LRK}(T_1)$ and ${\rm LRK}(T_2)$ linearly ordered, and ${\rm min}\{I(T_1,\omega),I(T_2,\omega)\}=1$, then ${\rm LRK}(T)$ is linearly ordered, since ${\rm LRK}(T)\simeq {\rm
LRK}(T_1)$ for $I(T_2,\omega)=1$, and ${\rm LRK}(T)\simeq {\rm
LRK}(T_2)$ for $I(T_1,\omega)=1$.
In Fig. \[fig10\] and \[fig11\] we illustrate Theorem \[th\_2\] by structures ${\rm LRK}(T)$ in [@SudRK], for disjoint unions of theories, which are not lattices.
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B.Sh. Kulpeshov, S.V. Sudoplatov, On hypergraphs of prime models for quite o-minimal theories with small number of countable models // Annual Scientific April Conference of Institute of Mathematics and Mathematical Modelling devoted to Science Day and Scientific Seminar ”Differential operators and modelling complex systems” (DOMCS-2017) devoted to 70-th anniversary of professor M.T. Dzhenaliev, Almaty, 7-8 April 2017: abstracts of talks. — Almaty: IMMM, 2017, pp. 30–32.
S.V. Sudoplatov, Derivative Structures in Model Theory and Group Theory // International Conference ”Actual Problems of Pure and Applied Mathematics” devoted to 100-th anniversary of academician A.D. Taimanov, Almaty, 22-25 August 2017: abstracts of talks. — Almaty: IMMM, 2017, pp. 76–79.
B.Sh. Kulpeshov, S.V. Sudoplatov, On decomposition of hypergraphs of models of a theory. Appendix to theories of unars // Sintax and Semantics of Logical Systems: Materials of 5-th Russian School-Seminar. — Ulan-Ude: Buryatsky State University Publishing House, 2017, pp. 52–56.
B.Sh. Kulpeshov, S.V. Sudoplatov, On relative separability in hypergraphs of models of theories // arXiv:1802.08088v1 \[math.LO\]. — 2018. — 11 p.
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, Vaught’s conjecture for quite o-minimal theories // Annals of Pure and Applied Logic. — 2017. — Vol. 168, N 1. — P. 129–149.
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[^1]: This research was partially supported by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No. AP05132546) and Russian Foundation for Basic Researches (Project No. 17-01-00531-a).
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---
abstract: 'Supervised training of neural models to duplicate question detection in community Question Answering ([cQA]{}) requires large amounts of labeled question pairs, which are costly to obtain. To minimize this cost, recent works thus often used alternative methods, e.g., adversarial domain adaptation. In this work, we propose two novel methods: (1) the automatic generation of duplicate questions, and (2) weak supervision using the title and body of a question. We show that both can achieve improved performances even though they do not require any labeled data. We provide comprehensive comparisons of popular training strategies, which provides important insights on how to ‘best’ train models in different scenarios. We show that our proposed approaches are more effective in many cases because they can utilize larger amounts of unlabeled data from [cQA]{}forums. Finally, we also show that our proposed approach for weak supervision with question title and body information is also an effective method to train [cQA]{}answer selection models without direct answer supervision.'
author:
- Andreas Rücklé
- Nafise Sadat Moosavi
- |
Iryna Gurevych\
Ubiquitous Knowledge Processing Lab (UKP)\
Department of Computer Science, Technische Universität Darmstadt\
[[www.ukp.tu-darmstadt.de](www.ukp.tu-darmstadt.de)]{}\
bibliography:
- 'main.bib'
nocite: '[@rueckle:AAAI:2019]'
title: Neural Duplicate Question Detection without Labeled Training Data
---
=1
Introduction
============
The automatic detection of question duplicates in community Question Answering ([cQA]{}) forums is an important task that can help users to more effectively find existing questions and answers [@nakov2017semeval; @Cao2012; @Xue:2008; @Jeon2005], and to avoid posting similar questions multiple times. Neural approaches to duplicate detection typically require large quantities of labeled question pairs for supervised training—i.e., labeled pairs of duplicate questions that can be answered with the same information.[^1]
In practice, it is often difficult to obtain such data because of the immense manual effort that is required for annotation. A large number of [cQA]{}forums thus do not contain enough labeled data for supervised training of neural models.[^2] Therefore, recent works have used alternative training methods. This includes weak supervision with question-answer pairs [@Qiu2015], semi-supervised training [@Uva2018], and adversarial domain transfer [@Shah2018]. An important limitation of these methods is that they still rely on substantial amounts of labeled data—either thousands of duplicate questions (e.g., from a similar source domain in the case of domain transfer) or large numbers of question-answer pairs. Furthermore, unsupervised methods rely on encoder-decoder architectures that impose limitations on the model architectures and they often fall short of the performances that are achieved with supervised training [@Lei2016], or they need to be combined with complex features to achieve state-of-the-art results [@Zhang2018]. To train effective duplicate question detection models for the large number of [cQA]{}forums without labeled duplicates we thus need other methods that do not require any annotations while performing on-par with supervised in-domain training.
In this work, we propose two novel methods for scenarios where we only have access to unlabeled questions (title-body), including (1) automatic duplicate question generation (*DQG*); and (2) weak supervision with the title-body pairs (*WS-TB*). Because a question body typically provides additional important information that is not included in the title [@wu-etal-2018-question], we hypothesize that titles and bodies have similar properties as duplicate questions. For instance, they are only partially redundant but fundamentally describe the same question (see Figure \[fig:introduction:example\] for an example). As a consequence, we can use the information from titles and bodies together with their relations to train our models.
In DQG, we use question generation models to generate a new question title from the body and then consider the generated title as a duplicate to the question’s original title. In WS-TB, we take this one step further and directly train models on title-body pairs—i.e., learning to predict whether both texts belong to the same question. The advantage of our proposed methods, compared to previous work, is that they can make use of the large number of unlabeled questions (titles and bodies) in [cQA]{}forums, which is typically an order of magnitude more data than is available for supervised training.[^3]
In our experiments, we evaluate common question retrieval and duplicate detection models such as RCNN [@Lei2016] and BiLSTM and compare a wide range of training methods: DQG, WS-TB, supervised training, adversarial domain transfer, weak supervision with question-answer pairs, and unsupervised training. We perform extensive experiments on multiple datasets and compare the different training methods in different scenarios, which provides important insights on how to ‘best’ train models with varying amounts training data. We show that:
1. Training models with title-body information is very effective. With larger amounts of unlabeled questions, WS-TB and DQG outperform adversarial domain transfer from similar source domains by more than 5.8pp on average. Because the amounts of labeled question duplicates is often limited, WS-TB and DQG can in some cases achieve better performances than supervised training.
2. DQG transfers well across domains, i.e., question generation models can be applied to novel target domains to obtain generated duplicates that are suitable for model training.
3. Our training methods are effective when being used to fine-tune more recent models such as BERT [@Devlin2018].
4. WS-TB can also be used to train [cQA]{}answer selection models without direct answer supervision. This shows that our methods can have broader impact on related tasks and beyond duplicate question detection.
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(answersheader)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**ANSWER**]{}; (answers)\[draw,below=0.5mm of answersheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgray\] [This can be done using the free AutoHotkey. Create a .ahk text file and enter these contents: ( $\ldots$ )]{};
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Related Work
============
Duplicate question detection is closely related to question-question similarity and question retrieval. Early approaches use translation models [@Jeon2005; @Xue:2008; @Zhou2011] that were further enhanced with question category information [@Cao2012] and topic models [@Ji2012; @Zhang2014]. More recent works in the context of the SemEval [cQA]{}challenges [@nakov2017semeval] improve upon this and use tree kernels (TK) [@Martino2016], TK with neural networks [@Romeo2016], neural networks with multi-task learning [@Bonadiman2017], and encoder-decoder architectures together with shallow lexical matching and mismatching [@Zhang2018]. Common neural models such as CNNs achieved superior performance compared to TK when they were trained on sufficiently large numbers of labeled question pairs [@Uva2018].
Similarly, neural representation learning methods have proved to be most effective in technical [cQA]{}domains. @DosSantos2015, for example, learn representations of questions with CNNs and compare them with cosine similarity for scoring. @Lei2016 propose RCNN, which extends CNN with a recurrent mechanism (adaptive gated decay). This approach was further extended with question-type information [@Gupta2018].
If in-domain training data is scarce—i.e., if the [cQA]{}platform does not offer enough labeled duplicates—alternative training strategies are required. If there exist some labeled question pairs (thousands), one can first train a less data-hungry non-neural model and use it for supervised training of neural models [@Uva2018]. Further, if there exist large numbers of labeled question-answer pairs, we can use them for weakly-supervised training [@Wang2017-cnn; @Qiu2015].
More related to our work are methods that do not rely on any labeled data in the target domain. Existing methods use unsupervised training with encoder-decoder architectures [@Lei2016; @Zhang2018], and adversarial domain transfer where the model is trained on a source domain and adversarially adapted to a target domain [@Shah2018]. However, such approaches typically fall short of the performances that are being achieved with in-domain supervised training. In contrast, we propose two novel methods, DQG and WS-TB, that do not require any annotations for model training and in some cases perform better than in-domain supervised training with duplicate questions. While WS-TB is related to the approaches mentioned before, DQG is is also related to question generation (QG). Most of the previous work in QG is in the context of reading comprehension [e.g., @Du2017:ACL; @Subramanian2018; @Zhao2018; @Du2018:ACL] or QG for question answering [@Duan2017]. They substantially differ from our approach because they generate questions based on specific answer spans, while DQG generates a new title from a question’s body that can be used as a question duplicate.
Training Methods
================
**Method** **Duplicates** **Answers** **Bodies**
----------------- ---------------- ------------- ------------
Supervised -
WS-QA -
Domain Transfer $^*$ -
DQG - -
WS-TB - -
: The different training methods and the data they use. Models typically also use text from the bodies during training and evaluation, which we indicate with . $^*$ = domain transfer requires duplicates from a sufficiently similar source domain.[]{data-label="tbl:methods:overview"}
Given a pair of questions, our goal is to determine whether they are duplicates or not. In practice, the model predictions are often used to rank a list of potentially similar questions in regard to a new user question, e.g., to retrieve the most likely duplicate for automatic question answering. To train models, we obtain a set of examples $\{(x_1,y_1), \ldots, (x_N,y_N)\}$ in which each ${x_n \in \mathcal{X}}$ is an instance, i.e., a tuple containing texts such as two questions, and ${y_n \in \{-1, +1\}}$ is its corresponding binary label, e.g., duplicate/no-duplicate. Obtaining instances with positive labels $\mathcal{X}^+ = \left\{ x_n^+ \in \mathcal{X} | y_n = 1 \right\}$ is generally more difficult than obtaining $\mathcal{X}^-$ because instances with negative labels can be automatically generated (e.g., by randomly sampling unrelated questions). In the following, we outline three existing training methods that use different kinds of instances, and in §\[sec:training:ours\] we present our two novel methods: duplicate question generation, and weak supervision with title-body pairs. Both do not require any annotations in $\mathcal{X}^+$, and can therefore use larger amounts of data from the [cQA]{}forums. Table \[tbl:methods:overview\] gives an overview of the different training methods.
Existing Methods {#sec:training:existing}
----------------
#### Supervised (in-domain) training
is the most common method, which requires labeled question duplicates, i.e., ${x_n^+ = \left(q_n, \tilde{q}_n\right)}$. Unrelated questions can be randomly sampled. With this data, we can train representation learning models [e.g., @Lei2016] or pairwise classifiers [e.g., @Uva2018]. Most models combine the titles and bodies of the questions during training and evaluation (e.g., by concatenation), which can improve performances [@Lei2016; @wu-etal-2018-question].
#### Weak supervision with question-answer pairs (WS-QA)
is an alternative to supervised training for larger platforms without duplicate annotations [@Qiu2015]. WS-QA trains models with questions $q_n$ and accepted answers $a_n$, and therefore ${x_n^+ = \left(q_n, a_n\right)}$. Instances in $X^-$ can be obtained by randomly sampling unrelated answers for a question. An advantage of this method is that there typically exist more labeled answers than duplicate questions. For instance, Yahoo! answers has accepted answers but it does not contain labeled duplicate questions.
#### Domain transfer
performs supervised training in a source domain and applies the trained model to a different target domain in which no labeled duplicate questions exist. @Shah2018 use this method with adversarial training to learn domain-invariant question representations prior to transfer. They show that adversarial training can considerably improve upon direct transfer, but their method requires sufficiently similar source and target domains. For instance, they could not successfully transfer models between technical and other non-technical domains.
Proposed Methods with Unlabeled Data {#sec:training:ours}
------------------------------------
The disadvantage of the existing methods is that they require labeled question duplicates, accepted answers, or similar source and target domains for transfer. We could alternatively use unsupervised training within an encoder-decoder framework, but this imposes important limitations on the network architecture, e.g., a question can only be encoded independently (no inter-attention). Our proposed methods do not suffer from these drawbacks, i.e., they do not require labeled data and they do not impose architectural limitations.
#### Duplicate question generation (DQG)
generates new question titles from question bodies, which we then consider as duplicates to the original titles. Our overall approach is depicted in Figure \[fig:train-inference\].
(title1)\[titlestyle,anchor=north west\] at (0,0) [Title]{}; (body1)\[paragraphstyle\_select,anchor=north west,below=1.5mm of title1.south\] [Body]{}; (title1.south) to\[out=270,in=90\] (body1.north);
(trainlabel)\[draw=none,above left=7mm and 1mm of title1.north west,align=left,anchor=north west,font=\] [**TRAINING**]{};
(title2)\[titlestyle,anchor=north west,right=1.8cm of title1.east\] [Title]{}; (body2)\[paragraphstyle\_select,anchor=north west,below=1.5mm of title2.south\] [Body]{}; (title2.south) to\[out=270,in=90\] (body2.north);
(title3)\[titlestyle,anchor=north west,right=6mm of title2.east\] [Duplicate]{};
(inferencelabel)\[draw=none,above left=7mm and 2mm of title2.north west,align=left,anchor=north west,font=\] [**DATA GENERATION**]{};
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First, we train a question generation model $\mathrm{QG}$ to maximize $P(\mathrm{title}(q_n)|\mathrm{body}(q_n))$. This is similar to news headline generation or abstractive summarization [@Rush2015; @Chopra2016] because $\mathrm{QG}$ needs to identify the most relevant aspects in the body that best characterize the question. However, restoring the exact title is usually not possible because titles and bodies often contain complementary information (see, e.g., Figure \[fig:introduction:example\]). We therefore consider ${\mathrm{QG}(\mathrm{body}(q_n))}$ as a duplicate of $\mathrm{title}(q_n)$ and obtain positive labeled instances ${x_n^+ = \left(\mathrm{title}(q_n), \mathrm{QG}(\mathrm{body}(q_n))\right)}$.
Because DQG requires no annotated data, we can use this method to train duplicate detection models for all [cQA]{}forums that offer a reasonable number of unlabeled title-body pairs to obtain a suitable $\mathrm{QG}$ model (the smallest number of questions we tried for training of question generation models is 23k, see §\[sec:additional:domain-transfer\]). An important advantage is that we can make use of *all questions* (after some basic filtering), which is often an order of magnitude more training data than annotated duplicates.
We can use any sequence to sequence model for $\mathrm{QG}$, and we performed experiments with a Transformer [@Vaswani2017] and MQAN [@McCann2018decaNLP].
#### Weak supervision with title-body pairs (WS-TB)
takes the assumption of DQG one step further. If question titles and question bodies have similar attributes as duplicates, we could also just train duplicate detection models directly on this data without prior question generation. In WS-TB, we thus train models to predict whether a given title and body are related, i.e., whether they belong to the same question. Therefore, ${x^+ = \left(\mathrm{title}(q_n), \mathrm{body}(q_n)\right)}$.
This method considerably simplifies the sourcing of training data because it requires no separate question generation model. However, it also means that the duplicate detection model must be able to handle texts of considerably different lengths during training (for instance, bodies in SuperUser.com have an average length of 125 words). This might not be suitable for some text matching models, e.g., ones that were designed to compare two sentences.
Experiments
===========
Experimental Setup {#sec:experiments:setup}
------------------
**Dataset** **Train / Dev / Test** $|$**Q**$|$ $|$**A**$|$
--------------- ------------------------ ------------- -------------
AskUbuntu-Lei 12,584 / 200 / 200 288k 84k
AskUbuntu 9106 / 1000 / 1000 288k 84k
SuperUser 9106 / 1000 / 1000 377k 142k
Apple - / 1000 / 1000 89k 29k
Android - / 1000 / 1000 47k 14k
: The dataset statistics. Numbers for Train/Dev/Test refer to the number of questions with duplicates. $|$Q$|$ refers to the number of unlabeled questions, and $|$A$|$ refers to the number of accepted answers.[]{data-label="tbl:setup:data"}
We use models and data from previous literature to obtain comparable results for evaluation, and we rely on their official implementations, default hyperparameters, and evaluation measures. An overview of the datasets is given in Table \[tbl:setup:data\], which also shows that they considerably differ in the amounts of data that is available for the different training methods.
The evaluation setup is the same for all datasets: given a user question $q$ and a list of potentially related questions, the goal is to re-rank this list to retrieve duplicates of $q$ (one or more potential related questions are labeled as duplicates). Even though some training methods do not use bodies during training, e.g., WS-DQG, during evaluation they use the same data (annotated pairs of questions with titles and bodies).[^4]
#### AskUbuntu-Lei.
First, we replicate the setup of @Lei2016, which uses RCNN to learn dense vector representations of questions and then compares them with cosine similarity for scoring. Besides supervised training, this also includes unsupervised training with the encoder-decoder architecture. We report precision at 5 (P@5), i.e., how many of the top-5 ranked questions are actual duplicates. The dataset is based on the AskUbuntu data of @DosSantos2015 with additional manual annotations for dev/test splits (user questions have an average of 5.7 duplicates).
#### Android, Apple, AskUbuntu, and Superuser.
Second, we replicate the setup of @Shah2018, which uses BiLSTM to learn question representations. This setup also includes adversarial domain transfer. The data is from the AskUbuntu, Superuser, Android, and Apple sites of StackExchange, and different to AskUbuntu-Lei, each question has only one duplicate. We measure AUC(0.05), which is the area under curve with a threshold for false positives—@Shah2018 argue that this is more stable when there are many unrelated questions.
#### Questions and answers.
To train the models with WS-TB and WS-QA, we use questions and answers from publicly available data dumps[^5] of the StackExchange platforms. We obtain our new training sets as specified in §\[sec:training:ours\]. For instance, for WS-TB we replace every annotated duplicate ${(q_n, \tilde q_n)}$ from the original training split with ${(\mathrm{title}(q_n), \mathrm{body}(q_n))}$, and we randomly sample unrelated bodies to obtain training instances with negative labels.
It is important to note that the number of questions and answers is much larger than the number of annotated duplicate questions. Therefore, we can add more instances to the training splits with these methods. However, if not otherwise noted, we use the same number of training instances as in the original training splits with duplicates.
#### DQG setup.
To train question generation models, we use the same StackExchange data. We filter the questions to ensure that the bodies contain multiple sentences. Further, if a body contains multiple paragraphs, we only keep the one with the highest similarity to the title. Details of the filtering approach are included in the Appendix. Less than 10% of the questions are discarded on average.
We train a MQAN (Multi-task Question Answering Network) model, which was proposed as a very general network architecture to solve a wide variety of tasks as part of the Natural Language Decathlon [@McCann2018decaNLP]. The model first encodes the input with LSTMs and applies different attention mechanisms, including multi-headed self-attention. MQAN also includes pointer-generator networks [@See2017], which allow it to copy tokens from the input text depending on the attention distribution of an earlier layer. We performed the same experiments with a Transformer sequence to sequence model [@Vaswani2017], but on average MQAN performed better because of its ability to copy words and phrases from the body. We include the Transformer results and a comparison with MQAN in the Appendix.
We use all available questions from a [cQA]{}forum to train the question generation model. We perform early stopping using BLEU scores to avoid overfitting. To generate duplicate questions, we then apply the trained model on all questions from the same [cQA]{}forum. We do not use a separate heldout set because this would considerably limit both the question generation training data and the number of generated duplicates. We did not observe negative effects from using this procedure.
Experimental Results {#sec:results}
--------------------
[l|r|rrrrr]{} & **AskUbuntu-Lei** & **Android** & **Apple** & **AskUbuntu** & **Superuser** & Average\
&\
\
Supervised (in-domain) & / 45.0 & - & - & **0.848** & **0.944** & -\
Unsupervised & 42.6 / 42.0 & - & - & - & - & -\
Direct Transfer (best) & - & 0.770 & 0.828 & 0.730 & 0.908 & 0.809\
Adversarial Transfer (best) & - & 0.790 & 0.861 & 0.796 & 0.911 & 0.840\
WS-QA & 47.2 / 45.3 & 0.780 & **0.894** & 0.790 & 0.919 & 0.846\
DQG & 46.4 / 44.8 & 0.793 & 0.870 & 0.801 & 0.921 & 0.846\
WS-TB & 46.4 / & **0.811** & 0.866 & 0.804 & 0.913 & **0.849**\
\
Unsupervised & 43.0 / 41.8 & - & - & - & - & -\
WS-QA & 47.3 / 44.2 & 0.814 & 0.901 & 0.828 & 0.951 & 0.874\
DQG & **47.4** / 44.3 & 0.833 & 0.911 & 0.855 & 0.944 & 0.886\
WS-TB & 47.3 / **45.3** & 0.852 & 0.910 & & & 0.896\
DQG + WS-TB (combined) & 46.4 / 44.0 & & & 0.866 & 0.946 &\
The results are given in Table \[tbl:results:main\]. For domain transfer, we report the best scores from @Shah2018, which reflects an optimal transfer setup from a similar source domain.
#### Supervised training.
As we expect, supervised in-domain training with labeled duplicates achieves better scores compared to other training methods when we consider the same number of training instances. An exception is on AskUbuntu-Lei where DQG, WS-TB, and WS-QA can achieve results that are on the same level on test or marginally worse on dev.
One reason for the better performances with labeled duplicates is that they contain more information, i.e., a pair of questions consist of two titles and two bodies compared to just one title and body for each training instance in WS-TB. However, the results show that all weakly supervised techniques as well as DQG are effective training methods.
#### DQG, WS-TB, and WS-QA.
All methods outperform direct transfer from a similar source domain as well as the encoder-decoder approach on AskUbuntu-Lei. On average, WS-TB is the most effective method, and it consistently outperforms adversarial domain transfer (0.9pp on average).
We otherwise do not observe large differences between the three methods DQG, WS-TB, and WS-QA, which shows that (1) the models we use can learn from different text lengths (title-body, question-answer); and (2) the information that we extract in DQG is suitable for training (examples are given in §\[sec:analysis\]). The good results of WS-TB might suggest that question generation as separate step is not required, however we argue that it can be important in a number of scenarios, e.g., when we need to train sentence matching models that would otherwise not be able to handle long texts.
#### Using all available data.
One of the biggest advantages of our proposed methods is that they can use larger amounts of training data. This greatly improves the model performances for BiLSTM, where we observe average improvements of up to 4.7pp (for WS-TB). In many cases our methods now perform better than supervised training. We observe smaller improvements for WS-QA (2.8pp on avg) because it has access to fewer training instances. The performances for RCNN on AskUbuntu-Lei are mostly unchanged with minor improvements on dev. The reason is that the performances were already close to supervised training with the same data sizes. In Figure \[fig:results:shahplot\] we plot the performance scores of BiLSTM averaged over the four StackExchange datasets in relation to the available training data with WS-TB. We see that the model performance consistently improves when we increase the training data (we observe similar trends for DQG and WS-QA). Thus, it is crucial to make use of all available data from the [cQA]{}forums. We also explored a combination of our two proposed approaches where we merge their respective training sets. We find that this helps mostly for smaller [cQA]{}platforms with fewer questions (where larger training sets would be most necessary), e.g., the performances on Android and Apple improve by 0.6–1.1pp compared to WS-TB. Even though the combination does not introduce new information because both use the same question data, complementing WS-TB with DQG can provide additional variation with the generative component.
In summary, our results show that even when we have access to sufficient numbers of labeled duplicates, the ‘best’ method is not always supervised training. When we use larger numbers of title-body pairs, DQG and WS-TB can achieve better performances.
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Further Application Scenarios {#sec:additional:domain-transfer}
=============================
To test if our methods are applicable to other scenarios with high practical relevance, we explore (1) whether DQG can be used in [cQA]{}forums with fewer unlabeled title-body pairs, (2) if we can use WS-TB to train answer selection models without labeled question-answer pair, and (3) how well large pre-trained language models perform when being fine-tuned with our methods.
DQG for Small-Scale [cQA]{}Forums
---------------------------------
In our previous experiments, we assumed that there exist enough unlabeled questions to train the question generation model (at least 47k questions, see Table \[tbl:setup:data\]). To simulate a more challenging scenario with fewer in-domain questions, we explore the effects of cross-domain question generation. This is highly relevant for DQG because in such scenarios the generated duplicates could be combined with WS-TB to obtain more training data. We replicate the transfer setup of @Shah2018 where they originally transfer the duplicate question detection model from a source to a target domain. For DQG we instead train the question generation model on the source domain and generate duplicates for the target domain, with which we then train the duplicate detection model. To provide a fair comparison against adversarial domain transfer, we always use the same number of 9106 duplicates to train the duplicate detection models. Results for the transfer from SuperUser and AskUbuntu to other domains are given in Table \[tbl:results:transfer\]. They show that the question generation model for DQG can be successfully transferred across similar domains with only minor effects on the performances. Importantly, DQG still performs better than adversarial domain transfer with the same number of training instances. To test an even more extreme case, we also transfer from StackExchange Academia (only 23k title-body pairs to train question generation) to the technical target domains. This could, e.g., be realistic for other languages with fewer [cQA]{}forums. Most notably, the performance of DQG decreases only mildly, which demonstrates its practical applicability in even more challenging scenarios. This is mostly due to the copy mechanism of MQAN, which is stable across domains (see §\[sec:analysis\]).
**Source** **Target** **Adv. DT** **DQG** $\Delta$
------------ ------------ ------------- ----------- ----------
Android 0.790 **0.797** $+$0.004
Apple 0.855 **0.861** $-$0.009
SuperUser 0.911 **0.916** $-$0.005
Android 0.790 **0.794** $+$0.001
Apple **0.861** **0.861** $-$0.009
AskUbuntu 0.796 **0.809** $+$0.008
Android - 0.776 $-$0.017
Apple - 0.854 $-$0.016
SuperUser - 0.912 $-$0.009
AskUbuntu - 0.760 $-$0.039
: The domain transfer performances. $\Delta$ denotes the difference to the setup with in-domain DQG.[]{data-label="tbl:results:transfer"}
Answer Selection {#sec:additional:answer-selection}
----------------
In answer selection we predict whether a candidate answer is relevant in regard to a question [@Tay2017; @nakov2017semeval; @Tan2016; @rueckle:2017:IWCS], which is similar to duplicate question detection. To test whether our strategy to train models with title-body pairs is also suitable for answer selection, we use the data and code of @rueckle:AAAI:2019 and train two different types of models with WS-TB on their five datasets that are based on StackExchange Apple, Aviation, Academia, Cooking, and Travel. We train (1) a siamese BiLSTM, which learns question and answer representations; and (2) their neural relevance matching model COALA. Both are evaluated by how well they re-rank a list of candidate answers in regard to a question.
The results are given in Table \[tbl:results:coala\] where we report the accuracy (P@1), averaged over the five datasets. Interestingly, we do not observe large differences between supervised training and WS-TB for both models when they use the same number of positive training instances (ranging from 2.8k to 5.8k). Thus, using title-body information instead of question-answer pairs to train models without direct answer supervision is feasible and effective. Further, when we use all available title-body pairs, the BiLSTM model substantially improves by 5pp, which is only slightly worse than COALA (which was designed for smaller training sets). We hypothesize that one reason is that BiLSTM can learn improved representations with the additional data. Further, title-body pairs have a higher overlap than question-answer pairs (see §\[sec:analysis\]) which provides a stronger training signal to the siamese network. These results demonstrate that our work can have broader impact to [cQA]{}, e.g., to train models on other tasks beyond duplicate question detection.
**Model** **Supervised** **WS-TB** **WS-TB (all)**
----------- ---------------- ----------- -----------------
BiLSTM 35.3 37.5 42.5
COALA 44.7 45.2 44.5
: Answer selection performances (averaged over five datasets) when trained with question-answer pairs vs. WS-TB.[]{data-label="tbl:results:coala"}
BERT Fine-Tuning {#sec:additional:bert}
----------------
Large pre-trained language models such as BERT [@devlin2018bert] and RoBERTa [@roberta] have recently led to considerable improvements across a wide range of NLP tasks. To test whether our training strategies can also be used to fine-tune such models, we integrate BERT in the setups of our previous experiments.[^6] We fine-tune a pre-trained BERT-base (uncased) model with supervised training, WS-TB (1x), and WS-TB (8x). The results are given in Table \[tbl:results:bert\]. We observe similar trends as before but with overall better results. When increasing the number of training examples, the model performances consistently improve. We note that we have also conducted preliminary experiments with larger BERT models where we observed further improvements.
------------------------ --------------------- -------------------- ----------- --------------- --------------- ----------------------
**AskUbuntu-Lei** **Android** **Apple** **AskUbuntu** **Superuser** **Answer Selection**
Measuring Accuracy
Supervised (in-domain) **54.0** / **52.3** - - 0.862 0.954 56.8
WS-TB (1x) 47.8 / 47.2 0.857 0.908 0.841 0.932 55.5
WS-TB (8x) 50.4 / 49.6 **0.896** **0.933** **0.897** **0.971** **59.7**
------------------------ --------------------- -------------------- ----------- --------------- --------------- ----------------------
Analysis {#sec:analysis}
========
Overlap
-------
To analyze the differences in the training methods we calculate the overlap between the texts of positive training instances (e.g., question-question, title-body, question-answer etc.). For questions, we concatenate titles and bodies.
Figure \[fig:analysis:overlap\] shows the Jaccard coefficient and the TF$*$IDF score averaged over all instances in the four StackExchange datasets of §\[sec:results\]. We observe that the overlap in WS-TB is similar to the overlap of actual duplicate questions in supervised training. The WS-DQG overlap is higher, because generated titles only contain relevant content (e.g., no conversational phrases). We also found that the BLEU scores of the MQAN model for QG are not very high (between 13.3–18.9 BLEU depending on the dataset), which shows that the texts are still sufficiently different. The overlap shows that both our methods use suitable training data with sufficiently similar, but not fully redundant texts.
Interestingly, the overlap scores of question-answer pairs are lower, especially when considering title-answer pairs as it is the case in the answer selection experiments (§\[sec:additional:answer-selection\]). This could explain one factor that may contribute to the better scores that we achieve with WS-TB for BiLSTM in this scenario. Because the overlap of title-body pairs is higher, the siamese network can receive a stronger training signal for positive instances, which could lead to better representations for similarity scoring.
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Qualitative Analysis
--------------------
(title1)\[draw,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [**Title:** 14.10, 15.04 - HDMI audio not working on Dell Vostro 3750 - nVidia card not detected by aplay -l]{}; (mqan1)\[draw,below=-0.2mm of title1.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgray\] [**DQG:** ALSA not detected in nVidia]{};
(title2)\[draw,below=1.5mm of mqan1.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [**Title:** Installing ubuntu 12.04.02 in uefi mode]{}; (mqan2)\[draw,below=-0.2mm of title2.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgray\] [**DQG:** Ubuntu 16.04 LTS boot loader not working]{};
(title3)\[draw,below=1.5mm of mqan2.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [**Title:** Grub2 not updating]{}; (mqan3)\[draw,below=-0.2mm of title3.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgray\] [**DQG:** How to fix Grub2 error]{};
To better understand the results for DQG and WS-QA, we manually checked a random sample of 200 generated questions and title-body pairs from multiple platforms. Three titles and generated duplicates from AskUbuntu are shown in Figure \[fig:analysis:example-1\].
For DQG we found that most of the generated duplicates are sensible, and most of the error cases fall into one of the following two categories:
\(1) Some generated questions are somewhat off-topic because they contain information that was generated from a body that has minimal overlap with the title (see example 4 in the Appendix).
\(2) A number of questions include wrong version numbers or wrong names (see example 5 in the Appendix, or the second example in Figure \[fig:analysis:example-1\]). Generally, however, we find that many of the generated titles introduce novel information, as can be seen in Figure \[fig:analysis:example-1\] (e.g., ‘ALSA’, ‘boot loader’ etc). The same drawbacks and benefits also apply to title-body information in WS-TB, with the exception that they are less noisy (i.e., not generated) but contain conversational phrases and many details. We also checked the training data of the difficult DQG domain transfer case to explore reasons for the small performance decreases when transferring the question generation model. Most importantly, we find that the model often falls back to copying important phrases from the body and sometimes generates additional words from the source domain. We note that this is not the case for models without copy mechanisms, e.g., Transformer often generates unrelated text (examples are in the Appendix).
Conclusion
==========
In this work, we have trained duplicate question detection models without labeled training data. This can be beneficial for a large number of [cQA]{}forums that do not contain enough annotated duplicate questions or question-answer pairs to use existing training methods. Our two novel methods, duplicate question generation and weak supervision with title-body pairs, only use title-body information of unlabeled questions and can thus utilize more data during training. While both are already highly effective when using the same number of training instances as other methods (e.g., outperforming adversarial domain transfer), our experiments have shown that we can outperform even supervised training when using larger amounts of unlabeled questions. Further, we have demonstrated that weak supervision with title-body pairs is well-suited to train answer selection models without direct answer supervision. This shows that our work can potentially benefit a much wider range of related tasks beyond duplicate question detection. For instance, future work could extend upon this by using our methods to obtain more training data in cross-lingual [cQA]{}setups [@Joty2017; @rueckle:WWW:2019], or by combining them with other training strategies, e.g., using our methods for pre-training.
The source code and the data of our experiments are publicly available: <http://github.com/UKPLab/emnlp2019-duplicate_question_detection>.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported by the German Research Foundation (DFG) as part of the QA-EduInf project (grant GU 798/18-1 and grant RI 803/12-1), by the German Federal Ministry of Education and Research (BMBF) under the promotional reference 03VP02540 (ArgumenText), and by the German Federal Ministry of Education and Research (BMBF) as part of the Software Campus program under the promotional reference 01IS17050.
Supplemental Material {#sec:supplemental}
=====================
Filtering and Paragraph Extraction
----------------------------------
Filtering and paragraph selection is necessary to obtain less noisy title-body pairs for QG. This consists of three steps: (1) filtering out questions that are not suitable for QG, (2) extracting paragraphs from bodies, and (3) only keeping one paragraph with the highest similarity to the title. The details are given below.
#### Filtering.
We discard all questions that:
- contain bodies with less than 10 words,
- are downvoted, i.e., have a score on StackExchange that is below zero (‘bad’ questions).
#### Paragraph extraction.
Some questions contain multiple long paragraphs, which is too much information to train suitable question generation or duplicate detection models. We thus extract paragraphs from the text to filter them in a later step.
In StackExchange platforms, users can freely add new lines, new paragraphs (the text then appears in HTML paragraph tags), lists, images, and code. This freedom results in many different ways of writing text. For instance, some users prefer to use paragraph tags and other users separate every sentence with a new-line character and all paragraphs with two or more new-line characters. Further, many users include code and enumerations in their questions.
This makes it difficult to extract actual paragraphs of the text. Thus, we first apply a preprocessing step to remove all HTML tags:
- We remove all code and images from the description.
- We then extract the text of each item from enumerations and append a new-line character.
- Likewise, we extract the text in paragraph tags and append a new-line character. We retain all new-line characters that appear in the paragraph.
We then analyze the new-line characters in the text to form the paragraphs for extraction. We read the input line-by-line:
- If the current line contains only one sentence it is merged with the previous paragraph.
- If the current line contains more than one sentence it is considered as a new paragraph.
#### Paragraph selection.
After extracting N paragraphs $p_1 \ldots p_N$ from the description, we select one paragraph according to $\operatorname*{argmax}_{p_n} f(p_n, \mathrm{title}(q))$. The function $f$ scores each $p_n$ by calculating the maximum cosine similarity of a sentence s in $p_n$ to the question title $\mathrm{title}(q)$ using a sentence encoder ($\mathit{enc}$): $$f(p_i, t) = \max_{s \in p_i} \left[ \cos(\mathrm{enc}(s), \mathrm{enc}(t)) \right]$$ In our experiments, $\mathrm{enc}$ is the (monolingual) encoder of @rueckle:2018, which uses different pooling strategies with multiple types of word embeddings. We calculate the maximum similarity of individual sentences to determine the semantic similarity independent of the paragraph length.
DQG with the Transformer
------------------------
[l|r|rrrrr]{} & **AskUbuntu-Lei** & **Android** & **Apple** & **AskUbuntu** & **Superuser** & Average\
&\
\
Supervised (in-domain) & 48.0 / 45.0 & - & - & 0.848 & 0.944 & -\
Adversarial Transfer (best) & - & 0.790 & 0.861 & 0.796 & 0.911 & 0.840\
DQG w. MQAN & 46.4 / 44.8 & 0.793 & 0.870 & 0.801 & 0.921 & 0.846\
DQG w. Transformer & 47.2 / 44.9 & 0.723 & 0.809 & 0.799 & 0.917 & 0.812\
WS-TB & 46.4 / 45.4 & 0.811 & 0.866 & 0.804 & 0.913 & 0.849\
\
DQG w. MQAN & 47.4 / 44.3 & 0.833 & 0.911 & 0.855 & 0.944 & 0.886\
DQG w. Transformer & 46.4 / 44.7 & 0.783 & 0.876 & 0.836 & 0.942 & 0.859\
WS-TB & 47.3 / 45.3 & 0.852 & 0.910 & 0.871 & 0.952 & 0.896\
------------ ------------ -------- ------------- ------------- ---------- ----------- ----------
**Source** **Target**
Direct Adversarial Transformer $\Delta$ MQAN $\Delta$
Android 0.692 0.790 0.762 $+$0.039 **0.797** $+$0.006
Apple 0.828 0.855 0.821 $+$0.110 **0.861** $-$0.009
SuperUser 0.908 0.911 0.913 $-$0.004 **0.916** $-$0.005
Android 0.770 0.790 0.755 $+$0.028 **0.794** $+$0.001
Apple 0.828 **0.861** 0.833 $+$0.024 **0.861** $-$0.009
AskUbuntu 0.730 0.796 0.797 $-$0.002 **0.809** $+$0.008
------------ ------------ -------- ------------- ------------- ---------- ----------- ----------
------------ ------------ ------------- -------
**Source** **Target**
Transformer MQAN
Android 0.550 0.789
Apple 0.624 0.864
SuperUser 0.856 0.914
AskUbuntu 0.664 0.787
Android 0.530 0.776
Apple 0.576 0.854
SuperUser 0.840 0.912
AskUbuntu 0.672 0.760
------------ ------------ ------------- -------
: The DQG domain transfer performance of different question generation models from more distant source domains that offer smaller numbers of unlabelled questions.[]{data-label="tbl:appendix:transfer-distant"}
In addition to MQAN [@McCann2018decaNLP], we also experimented with the Transformer [@Vaswani2017] for question generation using the Tensor2Tensor library [@tensor2tensor]. The most notable difference to MQAN is that the Transformer does not include a copy mechansim.
For our experiments we use the official implementation of the Transformer and use the same encoder-decoder approach as in machine translation. But instead of translating an input sentence to a target language, we generate a question from a paragraph of the body.
The results are given in Table \[tbl:appendix:results\]. We observe that Transformer performs worse than MQAN in domains that offer fewer unlabeled questions (Android, Apple). In contrast, for domains with more unlabeled questions (AskUbuntu, SuperUser), DQG with Transformer performs on the same level or only mildly worse than DQG with MQAN.
We also tested Transformer in the domain transfer scenarios. Table \[tbl:appendix:transfer\] shows the results when transferring from close domains, and Table \[tbl:appendix:transfer-distant\] shows the results when transferring from more distant domains. In contrast to MQAN, the performance of Transformer substantially decreases. We observe that MQAN is much more robust against domain changes due to its copy mechanism, which allows it to copy words and phrases from the input text. In contrast, Transformer falls back to outputting unrelated (but grammatical) domain-specific text. Examples are given in Appendix \[ap:examples\] below.
Thus, different QG models can have a substantial impact on the performance of DQG. However, this also suggests that better models could have a positive effect on DQG performance, potentially improving upon DQG with MQAN.
BERT Setup {#ap:bert}
----------
For our experiments in Section \[sec:additional:bert\] we add BERT to two experimental frameworks. In both extensions we use the HuggingFace implementation[^7].
We add BERT as a sentence encoder to the experimental software of [@Shah2018] and average over all BERT output states to obtain question representations. The rest of the implementation is the same as for BiLSTM (e.g., loss calculation). We train the models until they do not improve for at least 20 epochs, and we restore the weights of the epoch that obtained the best development score.
For all other datasets (AskUbuntu-Lei and Answer Selection datasets) we add BERT to the experimental software of . We do not include it in the software of because it is tightly coupled to the Theano framework, which is not actively maintained. We add BERT as a pairwise classification model which then directly scores question-question pairs, question-answer pairs, etc. (the labels are binary). The output prediction is then used as a ranking score. We train the models for 10 epochs and restore the weights of the epoch that obtained the best development score.
Additional QG Examples {#ap:examples}
----------------------
Below we show examples of generated questions. The questions generated with MQAN more closely retain the meaning of the body or paragraph, but Transformer questions also contain the relevant keywords (except for the transfer cases). Examples 4 and 5 refer to the error cases mentioned in our analysis (see §\[sec:analysis\]).\
(headline)\[draw=none,align=left,anchor=north west\] at (0,8) [**Example 1**]{};
(titleheader)\[draw=none,below=8mm of headline.south,align=left,anchor=north west,font=\] at (0,8) [**QUESTION**]{};
(title)\[draw,below=0.5mm of titleheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [how to get beep working?]{};
(bodyheader)\[draw=none,below=1.5mm of title.south west,align=left,anchor=north west,font=\] [**RELEVANT PARAGRAPH**]{};
(body)\[draw,below=0.5mm of bodyheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [I have a laptop, i installed the “beep” package. I turned every sound to full, and i: but i can’t hear any “beeping” sound. What am I missing? I just need to run the “beep” when a script is finished. Thank you for any links/howtos!]{};
(title.south) to\[out=270,in=90\] (body.north);
(mqandup\_head)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**IN-DOMAIN QG MODELS**]{}; (mqandup)\[draw,below=0.5mm of mqandup\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** How to fix beep package?]{}; (transformerdup)\[draw,below=1mm of mqandup.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** How to remove “ beep ” from my laptop?]{};
\
(headline)\[draw=none,align=left,anchor=north west\] at (0,8) [**Example 2**]{};
(titleheader)\[draw=none,below=8mm of headline.south,align=left,anchor=north west,font=\] at (0,8) [**QUESTION**]{};
(title)\[draw,below=0.5mm of titleheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [13" MacBook Pro with Win 7 and External VGA gets 640x480]{};
(bodyheader)\[draw=none,below=1.5mm of title.south west,align=left,anchor=north west,font=\] [**RELEVANT PARAGRAPH**]{};
(body)\[draw,below=0.5mm of bodyheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [I have a brand new 13” MacBook Pro - 2.26 GHz and the NVIDIA 9400M Video card. I installed Windows 7 (final) in boot camp and booted up to Windows 7. Installed all the drivers from the Apple disk and it was working great. Then I attached the external VGA adapter (from apple) to connect to a projector and it dropped down at 640x480 resolution. No matter what I did it would n’t let me change to a higher resolution if the external VGA was connected. Once it disconnects then it goes back to the normal resolution. If I am booted into Snow Leopard it works fine. I tried updating the NVIDIA drivers and it behaved exactly the same. Ultimately I want to get 1024x768 or better resolution when connected to an external display. If it isn’t fixable then I am curious if anyone else has seen this, if it is a known issue, and who to contact for support (Apple, Microsoft or NVIDIA?) Update: Just attaching the Mini-DVI to VGA adapter kicks it into 640x480, no projector is required. I tried forcing the display driver from Generic PnP Monitor to one that supported 1024x768 and that didn’t work either.]{};
(title.south) to\[out=270,in=90\] (body.north);
(mqandup\_head)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**IN-DOMAIN QG MODELS**]{}; (mqandup)\[draw,below=0.5mm of mqandup\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** 1024x768 or better resolution when connected to an external display]{}; (transformerdup)\[draw,below=1mm of mqandup.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** How can I get an external display to work?]{};
(headline)\[draw=none,align=left,anchor=south west,text width=7.4cm\] at (0,8) [**Example 3**. This contains a particularly short description with not much overlap to the question. Such examples could be removed with additional preprocessing / filtering.]{};
(titleheader)\[draw=none,below=3mm of headline.south,align=left,anchor=north west,font=\] at (0,8) [**QUESTION**]{};
(title)\[draw,below=0.5mm of titleheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [How to echo contents of file in a DOS/Windows command prompt?]{};
(bodyheader)\[draw=none,below=1.5mm of title.south west,align=left,anchor=north west,font=\] [**RELEVANT PARAGRAPH**]{};
(body)\[draw,below=0.5mm of bodyheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [Like in Unix’ cat filename ... Can this be done?]{};
(title.south) to\[out=270,in=90\] (body.north);
(mqandup\_head)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**IN-DOMAIN QG MODELS**]{}; (mqandup)\[draw,below=0.5mm of mqandup\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** How to make cat filename in Unix’ cat filename ?]{}; (transformerdup)\[draw,below=1mm of mqandup.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** What is the difference between ’cat’ and ‘cat‘ in Unix?]{};
(headline)\[draw=none,align=left,anchor=south west,text width=7.4cm\] at (0,8) [**Example 4** (from Analysis). The duplicates were generated based on a paragraph that does not have much overlap with the question.]{};
(titleheader)\[draw=none,below=3mm of headline.south,align=left,anchor=north west,font=\] at (0,8) [**QUESTION**]{};
(title)\[draw,below=0.5mm of titleheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [How can I remove an autostart service that is not listed in gnome-session-properties?]{};
(bodyheader)\[draw=none,below=1.5mm of title.south west,align=left,anchor=north west,font=\] [**RELEVANT PARAGRAPH**]{};
(body)\[draw,below=0.5mm of bodyheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [Today I upgraded from raring to saucy. This brought the Ubuntu One icon back to my indicator applet, which I had disabled. So I removed the line NoDiplay=true from /etc/xdg/autostart/ubuntuone-launch.desktop. But still Ubuntu One doesn’t show up in gnome-session-properties and I can’t disable it. What gives?]{};
(title.south) to\[out=270,in=90\] (body.north);
(mqandup\_head)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**IN-DOMAIN QG MODELS**]{}; (mqandup)\[draw,below=0.5mm of mqandup\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** Ubuntu One doesn’t show up in gnome-session-properties]{}; (transformerdup)\[draw,below=1mm of mqandup.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** Why doesn’t Ubuntu One sync with an indicator?]{};
(headline)\[draw=none,align=left,anchor=south west,text width=7.4cm\] at (0,8) [**Example 4** (from Analysis). The question generated by the (in-domain) Transformer model is suitable, but it does not contain the correct product name of the printer (“0b” instead of “LBP2900b”). However, even the MQAN model that was trained on StackExchange Travel is able to correctly copy all necessary information from the input. The Transformer trained on StackExchange Travel fails with generic (and grammatical) text from the travel domain.]{};
(titleheader)\[draw=none,below=3mm of headline.south,align=left,anchor=north west,font=\] at (0,8) [**QUESTION**]{};
(title)\[draw,below=0.5mm of titleheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightgreen\] [How to install Canon LBP2900b drivers?]{};
(bodyheader)\[draw=none,below=1.5mm of title.south west,align=left,anchor=north west,font=\] [**RELEVANT PARAGRAPH**]{};
(body)\[draw,below=0.5mm of bodyheader.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [I am trying very hard to install Canon LBP2900b Printer in Ubuntu 13.10. I have searched and googled a lot for the solution over fortnight but none of the site / link gave me the simple solution for me. How can accomplish my goal?]{};
(title.south) to\[out=270,in=90\] (body.north);
(mqandup\_head)\[draw=none,below=2mm of body.south west,align=left,anchor=north west,font=\] [**IN-DOMAIN QG MODELS**]{}; (mqandup)\[draw,below=0.5mm of mqandup\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** How to install Canon LBP2900b Printer in Ubuntu 13.10?]{}; (transformerdup)\[draw,below=1mm of mqandup.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** How to Install Canon 0b Printer on Ubuntu 13.10?]{};
(mqandup\_t\_head)\[draw=none,below=2mm of transformerdup.south west,align=left,anchor=north west,font=\] [**DOMAIN TRANSFER QG MODELS** (from SE Travel)]{}; (mqandup\_t)\[draw,below=0.5mm of mqandup\_t\_head.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**MQAN:** How to install to install Canon LBP2900b in Ubuntu 13.10?]{}; (transformerdup\_t)\[draw,below=1mm of mqandup\_t.south west,align=left,anchor=north west,text width=7.4cm,font=,fill=lightblue\] [**Transformer:** How can I find my boat in Hokkaido?]{};
[^1]: For example, “Nautilus shortcut for new blank files?” and “How do you create a new document keyboard shortcut?” are titles of labeled duplicate questions from AskUbuntu.com.
[^2]: @Shah2018 argue that even larger StackExchange sites do not offer enough duplicates for supervised training. Further, there exist many platforms that do not contain any labeled duplicates (e.g., <https://gutefrage.net>).
[^3]: Question titles and bodies are common in all StackExchange sites, popular platforms in other languages (e.g., GuteFrage.net), and forums such as Reddit. A counterexample is Quora, which only contains question titles. However, there exists a large annotated corpus of question pairs for this forum.
[^4]: It has been shown that including bodies in the experimental setup can lead to improved performances [@Lei2016]. In initial experiments, we found that the performances are mostly impacted by having access to bodies during evaluation.
[^5]: <https://archive.org/download/stackexchange>
[^6]: We add the AskUbuntu-Lei dataset to the framework of @rueckle:AAAI:2019 for our BERT experiments. Details are given in the Appendix.
[^7]: <https://github.com/huggingface/pytorch-transformers>
|
---
abstract: 'We calculate the amplitude of elastic photon-photon scattering via a single quark loop in the Double-Logarithmic Approximation, presuming all external photons to be off-shell and unpolarized.. At the same time we account for the running coupling effects. We consider this process in the forward kinematics at arbitrary relations between $t$ and the external photon virtualities. We obtain explicit expressions for the photon-photon scattering amplitudes in all double-logarithmic kinematic regions. Then we calculate the small-$x$ asymptotics of the obtained amplitudes and compare them with the parent amplitudes, thereby fixing the applicability regions of the asymptorics, i.e. fixing the applicability region for the non-vacuum Reggeons. We find that these Reggeons should be used at $x < 10^{-8}$ only.'
author:
- 'B.I. Ermolaev'
- 'D.Yu. Ivanov'
- 'S.I. Troyan'
title: 'Elastic scattering of virtual photons via quark loop in Double-Logarithmic Approximation'
---
Introduction
============
Since long the light-by-light scattering has been an object of both experimental and theoretical interest. In this paper we consider this process in the high energy limit. The motivation of our study is twofold.
On the one hand, it is well known that, similarly to the $e^+e^-$ annihilation into hadrons, the total cross section of collision of two off-shell photons with large virtualities is an important test ground for perturbative QCD. At a fixed order of in the strong coupling, $\alpha_s$, and at low energies, the dominant contribution comes from the pure QED quark box diagrams, calculated at the leading-order (LO) in Refs. [@Budnev:1974de; @Schienbein:2002wj] and at the next-to-LO (NLO) in $\alpha_s$, see Ref. [@Cacciari:2000cb]. In Ref. [@BL03] the resummation of double logs appearing starting from the first NLO QCD corrections to the quark box was studied. Such contribution are important at high energy where arguments of the logs are large. At even higher energies additional class of QCD diagrams gives important contribution to the cross section. It is a contribution with the two-gluon exchange in the t-channel that overwhelms the quark exchange contribution despite additional $\alpha_s^2$ suppression: it has a different asymptotics in the power of energy and therefore it will exceed the contribution of quark exchange mechanism at sufficiently large c.m.f. energy $\sqrt{s}$. At higher orders in $\alpha_s$, the contributions from t-channel gluons lead to terms with powers of single logarithms of the energy, which must be resummed. The BFKL approach [@BFKL] provides for a consistent theoretical framework for such resummation of the energy logarithms, both in the leading logarithmic approximation (LLA), which means resummation of all terms $\sim (\alpha_s\ln(s))^n$, and in the next-to-leading approximation (NLA), which means resummation of all terms $\sim \alpha_s(\alpha_s\ln(s))^n$. In this approach, the imaginary part of the amplitude (and, hence, the total cross section) for a large-$s$ hard collision process can be written as the convolution of the Green’s function of two interacting Reggeized gluons with the impact factors of the colliding particles.
The study of the $\gamma^* \gamma^*$ total cross section in LLA BFKL has a long history [@photons_BFKL]. For the extension of these results to the NLA level one needs to consider corrections to both the BFKL Green’s function [@NLA-kernel] and to the impact factors of colliding virtual photons.
While its LO expression for the photon impact factor is known since long, the NLO calculation, carried out in the momentum representation, turned out to be rather complicated and was completed only after year-long efforts [@gammaIF], and the results are available only in the form of a numerical code, thus making it of limited practical use. Indeed, until recently, the inclusion of BFKL resummation effects in the NLA calculation of the $\gamma^* \gamma^*$ total cross section was carried out only in approximate way, by taking the BFKL Green’s function in the NLA while using the LO expression for impact factors. This is the case of the pioneer paper in Ref. [@Brodsky:2002ka] (see also Ref. [@Brodsky:1998kn]) and of the later analysis in Refs. [@Caporale2008] and [@Zheng].
The situation changed when the NLO photon impact was calculated analytically in the coordinate space and then transformed to the momentum representation and to the Mellin [@Balitsky2012] (see also Ref. [@Chirilli2014]). This achievement opened a way for a subsequent calculations of $\gamma^* \gamma^*$ total cross section with complete NLA BFKL resummation approach, see [@Chirilli2014; @Ivanov:2014hpa].
In [@Ivanov:2014hpa] a comparison of the NLA BFKL predictions with LEP2 data[@Achard:2001kr; @Abbiendi:2001tv] was made. It was shown that the account of the Balitsky and Chirilli expression for NLO photon impact factor reduces the BFKL contribution to the cross section to very small values making it impossible to describe LEP2 data as a sum of BFKL and LO QED quark box contributions. Note that, as we discussed above, the LO QED quark box itself receives, at higher QCD orders, large corrections enhanced by double logs. Their resummation is important and leads to a considerable enhancement of the quark box contribution (see Ref. [@BL03] for detail), but still these effects are not large enough for a good description of LEP2 data at largest available rapidity without a sizable BFKL contribution. Therefore, in this situation, one of the aim of this paper is to reconsider the derivation of double logs resummation and to confirm results of [@BL03]. Besides, we account for the running QCD coupling effects.
Another motivation for the present paper is related with the possibility to measure amplitude of the light-by-light scattering at non zero angles, i.e. at non-zero values of $t$. Recently, ATLAS Collaboration has reported[@atlas] on evidence for the quasi-real photo-photon scattering scattering in heavy-ion collisions with the ATLAS detector at the LHC. These results proved to be consistent with calculations reported in Ref. [@2; @3; @4], Light-by-light scattering has been an object of both experimental and theoretical interest. For instance, ATLAS Collaboration has recently reported[@atlas] on evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC. These results proved to be consistent with calculations reported in Ref. [@2; @3; @4], where one of the essential ingredients is the amplitudes of the photon-photon elastic scattering studied in the lowest (“Born”) approximation where description of the photon scattering involves a single quark loop only. As accounting for the QCD radiative corrections can essentially change the scattering amplitudes, it is interesting to study their impact. Both technology of accounting for the radiative corrections and their impact strongly depend on the kinematic region of the process. The most interesting kinematics at hight energies is the forward one. Because of that we investigate the photon-photon scattering
$$\label{gamma}
\gamma^* (p)~ \gamma^* (q) \to \gamma^* (p')~ \gamma^*(q'),$$
with all photons being off-shell, via a single quark loop. We consider this reaction in the forward kinematics
$$\label{fkin}
s = (p + q)^2 \gg -t = - (p'- p)^2 .$$
In order to be in agreement with the conventional notations, we denote the photon virtualities as follows:
$$\label{virt}
|p^2| = Q^2_1, |p'^2| = Q'^2, |q^2| = Q^2_2, |q'^2| = Q'^2_2,$$
so that $Q^2_{1,2}, Q'^2_{1,2}$ are positive. We presume that $Q^2_{1,2} \approx Q'^2_{1,2}$. In what follows we consider the case when $s \gg Q^2_{1,2}, Q'^2_{1,2}$, i.e. when $$\label{wq}
s \gg |t|, Q^2_{1,2}.$$ In contrast, we do not fix any hierarchy between $Q^2_{1,2}$ and $t$ and consider all possible situations. Then, throughout the paper we will focus on the unpolarized initial and final photons. We will calculate the amplitude $A_{\gamma\gamma}$ of the reaction (\[gamma\]) in the Double-Logarithmic Approximation (DLA). The imaginary part (with respect to $s$) of this amplitude was calculated in Ref. [@bl] in the collinear kinematics, i.e. in the kinematics (\[fkin\]) with $t =0$, and under the approximation of fixed QCD coupling $\alpha_s$. We check and confirm the results obtained in Ref. [@bl] and, in contrast, we account for the running $\alpha_s$ effects, using the results of Ref. [@egtquark]. In our approach $\alpha_s$ runs in every rung of each involved Feynman ladder graph. Then we consider the process (\[gamma\]) in the forward kinematics, with $t \neq 0$ and obtain a complete expression for the amplitude of this process.
In our calculations we compose and solve Infra-Red Evolution Equations (IREE) for $A_{\gamma\gamma}$. The key point of the IREE method is the property of factorization of the double logarithmic (DL) contributions of the softest partons (i.e. the partons with minimal transverse momenta) out of the scattering amplitudes. This remarkable property of the softest photons was first proved by V.N. Gribov[@g] in the QED context and then its generalization to the non-Abelian theories was obtained in Ref. [@l] and [@kl], where the IREE method was suggested to calculate in DLA amplitudes of quark-antiquark scattering. After that, the IREE method proved to be a simple and effective method to calculate in DLA amplitudes of various inclusive and exclusive processes in QCD and Standard Model, with both fixed and running $\alpha_s$, see e.g. the overviews in Ref. [@egtsum].
The aim of our paper is to calculate the amplitude $M_{\gamma\gamma}$ of the process (\[gamma\]) in the forward kinematics (\[fkin\]) with non-zero value of $t$ and arbitray relations between $t$ and $Q^2_{1,2}$. Throughout the paper we deal with running $\alpha_s$. Technology of composing IREE involves matching $M_{\gamma\gamma}$ with amplitude $A_{\gamma\gamma}$ calculated in the collinear kinematics. Following this pattern, we in the first place calculate amplitude $A_{\gamma\gamma}$ in the collinear kinematics, examining the cases of running and fixed $\alpha_s$ thereby checking results of Ref. [@bl], and then proceed to calculating $M_{\gamma\gamma}$ in the region of non-zero $t$.
Our paper is organized as follows: In Sects. II-V we consider the photon-photon scattering in collinear kinematics, i.e. in kinematics (\[fkin\]) with $t = 0$. In Sect. II we briefly mention the lowest-order results for $A_{\gamma\gamma}$. In Sect. III we compose and solve IREE for $A_{\gamma\gamma}$, expressing it terms of auxiliary amplitudes describing photon-quark scattering. In Sect. IV we compose and solve IREE for the auxiliary amplitudes and express them in terms of amplitudes of the quark-antiquark annihilation in the forward kinematics. Using the obtained results, in Sect. V we express the photon-photon scattering amplitudes through the quark-quark amplitude. Then in Sect. VI we use results of Sect. V in order to calculate the photon-photon scattering amplitude $M_{\gamma\gamma}$ in kinematics (\[fkin\]) at $t \neq 0$. In Sect. VII we discuss the results obtained in Sect. V, VI. Here we consider the high-energy asymptotics of $A_{\gamma\gamma}$ and compare them to the parent amplitudes, thereby defining the applicability region for non-vacuum Reggeons. Finally, Sect. VIII is for our concluding remarks.
Lowest-order amplitudes in the collinear kinematics
===================================================
First of all we consider the “Born”, i.e. the simplest, case, where only quark box diagrams contribute. We also suggest that $t \approx 0$. In this case the amplitude $A_{B}$ of the process (\[gamma\]) in the lowest order, with the quark masses neglected, consists of two terms:
$$\label{ab}
A_B = B + B',$$
where $$\begin{aligned}
\label{aborn}
B =\imath \frac{e^4}{(2 \pi)^4} \int d^4 k \frac{Tr\left[\gamma_{\nu}\left(\hat{q} + \hat{k}\right)
\gamma_{\mu}\hat{k}\gamma_{\lambda}\left(\hat{k}- \hat{p}\right)\gamma_{\rho}
\hat{k}\right]}
{k^2 k^2\left(q + k\right)^2 \left(k - p\right)^2}
\\ \nonumber
l_{\mu} (q)l_{\lambda} (p) l^*_{\nu}(q') l^*_{\lambda} (p')\end{aligned}$$ and $B'$ can be obtained from (\[aborn\]) by replacing $q \to - q$. The important property of (\*\*\* amplitude \*\*\*) $B$ is that $\Im_s B \neq 0$ whereas $\Im_s B' = 0$. By this reason we will not consider $B'$ and focus on $B$ only. In Eq. (\[aborn\]) we have neglected the quark mass and introduced the following notations: $k$ is the loop momentum, $l_{\mu}(q), l_{\lambda}(p)$ are the polarization vectors of the incoming photons and $l^*_{\nu}(q), l^*_{\rho}(p)$ stand for the polarization vectors of the outgoing photons. Throughout the present paper we consider the case of the unpolarized photons and use for them the Feynman gauge where the averaging over the photon polarizations can be done using the following replacements:
$$\label{densmatr}
l_{\mu}(q) l^*_{\nu}(q) = -g_{\mu\nu}/2,~~l_{\lambda}(p)l^*_{\rho}(p) = -g_{\lambda\rho}/2$$
and therefore
$$\begin{aligned}
\label{tcol}
Tr\left[\gamma_{\nu}\left(\hat{q} + \hat{k}\right)
\gamma_{\mu}\hat{k}\gamma_{\lambda}\left(\hat{k}- \hat{p}\right)\gamma_{\rho}
\left(\hat{k} + \hat{p}' - \hat{p}\right)\right]
l_{\mu} (q)l_{\lambda} (p) l^*_{\nu}(q') l^*_{\lambda} (p')
\\ \nonumber
= Tr\left[\left(\hat{q} + \hat{k}\right)
\hat{k}\left(\hat{k}- \hat{p}\right)
\left(\hat{k} + \hat{p}' - \hat{p}\right)\right]
\\ \nonumber
\approx - Tr\left[\hat{q}
\hat{k}\hat{p}\hat{k}\right]
= 2 \left[w k^2 - 2pk 2qk\right],\end{aligned}$$
where we have used the standard notation $w = 2pq$. For the next step, it is convenient to introduce the Sudakov representation[@sud] for the soft momentum $k$:
$$\label{sud}
k = - \alpha \widetilde{q} + \beta \widetilde{p} + k_{\perp},$$
where the light-cone momenta $\widetilde{p},\widetilde{q}$ are made of the photon momenta $p$ and $q$: $$\label{pq}
\widetilde{p} = p - x_p q,~ \widetilde{q} = q - x_q p,~ x_p \approx Q^2_1/w,~x_q \approx Q^2_2/w,$$ so that
$$\label{sudinv}
2pk = - \alpha w + \beta x_p w,~~2qk = \beta w - \alpha x_q w,~
k^2 = - \alpha\beta w - k^2_{\perp}.$$
In terms of the Sudakov variables Eq. (\[tcol\]) looks much simpler:
$$\label{tcolsud}
2 \left[w k^2 - 2pk 2qk\right] \approx - 2 w k^2_{\perp} .$$
Corrections to Eq. (\[tcolsud\]) are $\sim p^2, q^2$. Accounting for them is beyond the DLA accuracy, so we drop them.
Therefore, the DL contribution to amplitude $B$ of Eq. (\[aborn\]) in collinear kinematics is given by the following expression of the Sudakov type:
$$\begin{aligned}
\label{abornsud}
B &=& - \imath \frac{e^4}{16 \pi^3}\int
\frac{d \alpha d \beta d k^2_{\perp} w^2 k^2_{\perp}}
{k^2 k^2\left(x_q w + \beta w - \alpha x_q w + k^2\right) \left(x_p w + \alpha w - \beta x_p w + k^2 \right)}
\\ \nonumber
&\approx& \imath \frac{e^4}{16 \pi^3}\int
\frac{d \alpha d \beta d k^2_{\perp} w^2 }
{k^2\left(x_q w + \beta w + k^2\right) \left(x_p w + \alpha w + k^2 \right)}.\end{aligned}$$
We have used in (\[abornsud\]) that in DLA the integrand does not depend on the azimuthal angle.
Massless external photons
-------------------------
This case is the simplest. Here $p^2 = q^2 = 0$, so the on-shell Born amplitude $B_{on}$ is
$$\begin{aligned}
\label{aborndl}
B_{on} &=& \imath \frac{e^4}{16 \pi^3}\int
\frac{d \alpha d \beta d k^2_{\perp} w^2 }
{k^2\left(\beta w + k^2\right) \left( \alpha w + k^2 \right)}
\\ \nonumber
&=& -\frac{e^4}{8 \pi^2}\int_0^1 d \beta \int_0^s d k^2_{\perp} \frac{w }
{k^2_{\perp}\left(w\beta - k^2_{\perp}\right)}
= -\frac{e^4}{16 \pi^2} \ln^2 (w/\mu^2) \approx -\frac{e^4}{16 \pi^2} \ln^2 (s/\mu^2),\end{aligned}$$
where we have introduced the infrared (IR) cut-off $\mu$ in the transverse space: $k^2_{\perp} \gg \mu^2$. In order to use the cut-off and at the same time neglect the quark masses, $\mu$ should obey the inequality $\mu \gg m_{quark}$. With our accuracy, we have neglected the difference between $s$ and $w$ in Eq. (\[aborndl\]) and will do so throughout the paper.
Off-shell external photons
--------------------------
Here we consider the case of the off-shell photons. After integrating $B$ of Eq. (\[abornsud\]) over $k_{\perp}$, we arrive at
$$\label{abornab}
B = - \frac{e^4}{8 \pi^2}\int_0^1 d \alpha
\int_0^1 d \beta
\frac{\Theta (\alpha\beta - \lambda)}
{\left(\beta + x_q \right) \left(\alpha + x_p \right)},$$
where $\lambda = \mu^2/s$. Depending on the ratio between the photon virtualities and $\mu^2$, there are two different cases:\
**(a) Moderately Virtual photons**\
We call so the case, when virtualities $Q^2_1$ and $Q^2_2$ are sizable but not too great and obey the inequality
$$\label{smallq}
Q^2_1 Q^2_2 \ll s \mu^2.$$
The integration region in this case is depicted in Fig. \[gammafig1\]
![\[gammafig1\] Integration region for Moderately Virtual photons.](gammafig1){width=".4\textwidth"}
and therefore the off-shell Born amplitude $B_{\gamma\gamma}^{(M)}$ in the kinematics (\[smallq\]) is
$$\begin{aligned}
\label{born1}
B_{\gamma\gamma}^{(M)} &=& - \frac{e^4}{8 \pi^2}\int_{x_p}^1 \frac{d \alpha}{\alpha}
\int_{x_q}^1 \frac{d \beta}{\beta} \Theta (\alpha\beta - \lambda)
\\ \nonumber
&=& -\frac{e^4}{16 \pi^2} \left[\ln^2 (s/\mu^2)- \ln^2 (p^2/\mu^2) - \ln^2 (q^2/\mu^2)\right].\end{aligned}$$
**(b) Deeply Virtual photons**\
On the contrary when the photon virtualities are so great that
$$\label{bigq}
Q^2_1Q^2_2 \gg s \mu^2,$$
the integration region does not include or touch the line $s \alpha\beta = \mu^2$ (see Fig. \[gammafig2\]),
![\[gammafig2\] Integration region for Deeply Virtual photons.](gammafig2){width=".4\textwidth"}
so the amplitude $B_{\gamma\gamma}^{(M)}$ does not depend on $\mu$ and becomes IR stable:
$$\label{born2}
B_{\gamma\gamma}^{(D)} = - \frac{e^4}{8 \pi^2}\int_{x_p}^1 \frac{d \alpha}{\alpha}
\int_{x_q}^1 \frac{d \beta}{\beta} = -\frac{e^4}{8 \pi^2} \ln (s/Q^2_1)\ln(s/Q^2_2).$$
Photon-photon amplitudes in DLA
===============================
In this Sect. we account for DL corrections to the Born amplitudes $B^{M}_{\gamma\gamma},~B^{D}_{\gamma\gamma} $ and express the amplitude $A_{\gamma \gamma}(s,Q^2_1, Q^2_2)$ of the process (\[gamma\]) at $t = 0$. We do it with constructing and solving IREE for $A_{\gamma \gamma}(s,Q^2_1, Q^2_2)$. As a result, we represent $A_{\gamma \gamma}(s,Q^2_1, Q^2_2)$ in terms of auxiliary amplitudes $A_{\gamma q}$ and $A_{q \gamma}$ that correspond respectively to the $t-$ channel annihilation of the pair of photons into quarks
$$\label{gammaq}
\gamma^* (p) \gamma^* (q) \to q (p'_1) \bar{q} (p'_2),$$
and to the inverse process. According to the IREE technology, we start with introducing a cut-off $\mu$ in the transverse space:
$$\label{mu}
k_{\perp} \gg \mu,$$
where $k_{\perp }$ refers to the transverse momenta of virtual quarks or gluons. In order to handle virtual quarks and gluons equally, we choose $\mu$ much greater than the masses of involved quarks, which allows us to neglect the quark masses. After that, the amplitude $A_{\gamma \gamma}$ becomes $\mu$-dependent, so we can evolve it with respect to $\mu$ and thereby compose IREE for $A_{\gamma \gamma}$. It is convenient to deal with $A_{\gamma \gamma}$ through its Mellin transformation $F_{\gamma \gamma}$. We will use the Mellin transform as follows:
$$\label{mellin}
A_{\gamma \gamma}(s, Q^2_1, Q^2_2) = \int_{- \imath \infty}^{\imath \infty}
\frac{d \omega}{2 \pi \imath} \left(s/\mu^2\right)^{\omega} F_{\gamma \gamma}(\omega, Q^2_1,Q^2_2)
= \int_{- \imath \infty}^{\imath \infty}
\frac{d \omega}{2 \pi \imath} e^{\omega \rho} F_{\gamma \gamma}(\omega, y_1,y_2),$$
where we have denoted $$\label{y12}
\rho = \ln (s/\mu^2),~~y_1 = \ln (Q^2_1/\mu^2),~~y_2 = \ln (Q^2_2/\mu^2).$$ We will address $F_{\gamma \gamma}(\omega, y_2,y_1)$ and $\left(s/\mu^2\right)^{\omega}$ as the Mellin amplitude and the Mellin factor respectively. We would like to remind that in the context of the Regge processes the Mellin transform in Eq. (\[mellin\]) is actually the asymptotic form of the Sommerfeld-Watson representation for the positive signature amplitudes. Before composing IREE for objects with several $\mu$-dependent variables like $A_{\gamma \gamma}(\rho, y_1, y_2)$, we should order these variables. We use the ordering of Eq. (\[wq\]), complementing it by the restriction $Q^2_1 \gg Q^2_2$ and arriving thereby at
$$\label{rhoy}
\rho > y_1 > y_2.$$
When we obtain expressions for $A_{\gamma \gamma}(\rho, y_1, y_2)$ under the ordering (\[rhoy\]), we will generalize our results on the case of the opposite ordering $y_1 < y_2$ and for $y_1 = y_2$ as well. The general strategy of composing IREE prescribes to start with considering the simplest case: we first compose the IREE for the on-shell amplitude $A_{\gamma \gamma}^{on}$, which describes the process (\[gamma\]) at $y_1 = y_2 = 0$ and therefore depends on the largest variable $\rho$ only. When $A_{\gamma \gamma}^{on}$ is found, we do next step, considering the more involved case of amplitude $\widetilde{A}_{\gamma\gamma} (\rho, y_1)$ of the same process in the kinematics $\rho > y_1 > y_2 = 0$. In order to specify a general solution of the IREE for $\widetilde{A}_{\gamma\gamma} (\rho, y_1)$, we will use matching with the on-shell amplitude $A_{\gamma \gamma}^{on}$, which has been found on the previous step. Then we repeat the same to specify a general solution to the IREE for $A_{\gamma\gamma} (\rho, y_1, y_2)$. Obviously, such procedure can be repeated as many times as one needs, allowing to describe processes with arbitrary number of external kinematic invariants. We suppose that the amplitudes $A_{\gamma \gamma}^{on} (\rho), \widetilde{A}_{\gamma\gamma} (\rho, y_1), A_{\gamma\gamma} (\rho, y_1, y_2)$ are related to the conjugated Mellin amplitudes $f_{\gamma \gamma}(\omega), \widetilde{F}_{\gamma\gamma} (\omega, y_1),
F_{\gamma\gamma} (\omega, y_1, y_2)$ by the Mellin transform (\[mellin\]).
Now we have got all set to compose IREE for the amplitudes of the process (\[gamma\]). The generic form of IREE for $A_{\gamma \gamma}$ is depicted in Fig. \[gammafig3\]. Throughout this paper we will write the IREE directly in the $\omega$-space.
![\[gammafig3\] Infra-Red Evolution Equation for the amplitude $A_{\gamma\gamma}$. The dashed lines denote the external photons, whereas the straight lines correspond to quarks. The blobs stand for amplitudes calculated in DLA. The letters on the blobs denote the IR cut-offs for the involved amplitudes.](gammafig3){width=".4\textwidth"}
All photons are nearly on-shell
-------------------------------
We start with calculation of $A_{\gamma \gamma}^{on}$ in the simplest kinematics where $Q^2_1 \approx Q^2_2 \lesssim \mu^2$. We denote $f_{\gamma \gamma}(\omega)$ the Mellin amplitude for the photon-photon scattering in the case when virtualities $Q^2_{1,2}$ are neglected, i.e. when
$$\label{q12on}
y_2 = y_1 = 0.$$
The IREE for $f_{\gamma \gamma}(\omega)$ is very simple. It represents $f_{\gamma \gamma}(\omega)$ through two auxiliary Mellin amplitudes:
$$\label{fon}
\omega f_{\gamma \gamma}(\omega) = \frac{1}{8 \pi^2} f_{\gamma q}(\omega) f_{q \gamma}(\omega),$$
where $f_{\gamma q}(\omega)$ and $f_{q \gamma}(\omega)$ corresponds to the processes (\[gammaq\]) and the reversal process respectively. In fact, $f_{\gamma q}(\omega) = f_{q \gamma}(\omega)$.
One of the photons is on-shell and the other is off-shell
----------------------------------------------------------
Let us consider the more complicated case when
$$\label{y2mu}
\rho > y_1 > y_2 = 0,$$
i.e. $Q^2_1 \gg Q^2_2 \sim \mu^2$, and denote $\widetilde{F}_{\gamma\gamma} (\omega, y_1)$ the amplitude corresponding to that case. It obeys the following IREE:
$$\label{eq2gen}
\frac{\partial \widetilde{F}_{\gamma\gamma}}{\partial y_1}
+ \omega \widetilde{F}_{\gamma\gamma} = \frac{1}{8 \pi^2} F_{\gamma q} (\omega, y_1)
f_{q \gamma} (\omega).$$
where amplitudes $F_{\gamma q} (\omega, y_1)$ and $f_{q \gamma} (\omega)$ are supposed to be calculated independently. Once they are known, the general solution to Eq. (\[eq2gen\]) is
$$\label{sol2gen}
\widetilde{F}_{\gamma\gamma} = e^{-\omega y_1}
\left[C_2 (\omega) + \frac{1}{8 \pi^2} f_{q \gamma} (\omega) \int_0^{y_1} d y' e^{\omega y'} F_{\gamma q} (\omega, y') \right] \ .$$
In order to specify an unknown function $C_2$ in Eq. (\[sol2gen\]) we use the matching:
$$\label{match2}
\widetilde{F}_{\gamma\gamma} (\omega, y_1)|_{y_1 = 0} = f_{\gamma\gamma} (\omega),$$
where $f_{\gamma\gamma} (\omega)$ is defined in Eq. (\[fon\]). Therefore, $\widetilde{F}_{\gamma\gamma} (\omega, y_1)$ in kinematics (\[y2mu\]) is represented in terms of the photon-quark amplitudes:
$$\label{sol2}
\widetilde{F}_{\gamma\gamma} (\omega, y_1) = e^{-\omega y_1}\left[ \frac{1}{8 \pi^2 \omega} f_{\gamma q} (\omega)
f_{q \gamma} (\omega) + \frac{1}{8 \pi^2}
f_{q \gamma} (\omega) \int_0^{y_1} d y e^{\omega y} F_{\gamma q} (\omega, y)\right],$$
where $F_{\gamma q} (\omega, y)$ is the photon-quark amplitude at $y \neq 0$.
Off-shell photons with moderate virtualities
--------------------------------------------
We call the moderately virtual kinematics the case when $Q^2_1 \gg \mu^2$ and $Q^2_2 \gg \mu^2$ but $Q^2_1 Q^2_2 \ll s \mu^2$. In the logarithmic variables it means that
$$\label{modoffshell}
\rho > y_2 + y_1.$$
The IREE for the amplitude $F_{\gamma\gamma}^{(M)} (\omega, y_1, y_2)$ in the kinematic region (\[modoffshell\]) is
$$\label{eq12gen}
\frac{\partial F_{\gamma\gamma}^{(M)}}{\partial y_2} + \frac{\partial F_{\gamma\gamma}^{(M)}}{\partial y_1}
+ \omega F_{\gamma\gamma}^{(M)} = \frac{1}{8 \pi^2} F_{\gamma q} (\omega, y_1) F_{q \gamma} (\omega, y_2).$$
In order to use the symmetry with respect to $y_1, y_2$ of the differential operator in (\[eq12gen\]) and simplify the IREE, we have introduced new variables $\xi, \eta$:
$$\label{xieta}
\xi = y_1 + y_2,~~\eta = y_1 - y_2.$$
Eq. (\[eq12gen\]) in terms of $\xi, \eta$ takes a simpler form:
$$\label{eq12xieta}
2 \frac{\partial F_{\gamma\gamma}^{(M)}}{\partial \xi}
+ \omega F_{\gamma\gamma}^{(M)} = \frac{1}{8 \pi^2} F_{\gamma q} \left(\omega, y_1 \right)
F_{q \gamma} \left(\omega, y_2\right).$$
A general solution to Eq. (\[eq12xieta\]) is
$$\label{sol12gen}
F_{\gamma\gamma}^{(M)} = e^{- \omega \xi/2} \left[C(\omega, \eta) + \frac{1}{16 \pi^2} \int_0^{\xi} d \xi'
e^{\omega \xi'/2} F_{\gamma q} (\omega, y'_1) F_{q \gamma} (\omega, y'_2) \right],$$
with $C(\omega, \eta)$ being an arbitrary function and the variables $y'_1,y'_2$ are defined as follows:
$$\label{y12prime}
y'_1 = (\xi' + \eta)/2,~~y'_2 = (\xi' -\eta)/2).$$
In order to specify $C(\omega, \eta)$, we use the matching of $F_{\gamma\gamma}^{(M)} (\omega, y_1,y_2)$ with an amplitude $\widetilde{F}_{\gamma\gamma} (\omega, y_1)$ of the same process but in the simpler kinematic regime (\[y2mu\]) considered above:
$$\label{match1}
F_{\gamma\gamma}^{(M)} (\omega, y_1,y_2)|_{y_2 = 0} = \widetilde{F}_{\gamma\gamma} (\omega, y_1),$$
where amplitude $\widetilde{F}_{\gamma\gamma} (\omega, y_1)$ is given by Eq. (\[sol2\]). Combining Eqs. (\[match1\]), (\[sol12gen\]) and (\[sol2\]), we arrive at the following expression for $F_{\gamma\gamma}$:
$$\begin{aligned}
\label{fm}
F_{\gamma\gamma}^{(M)} &=& e^{- \omega \xi/2} \left[e^{\omega \eta/2}
\widetilde{F}_{\gamma\gamma} (\omega, \eta)
+ \frac{1}{16 \pi^2} \int_{\eta}^{\xi} d \xi' e^{\omega \xi'/2}
F_{\gamma q} (\omega, y'_1) F_{q \gamma} (\omega, y'_2)\right].\end{aligned}$$
Substituting Eq. (\[fm\]) in (\[mellin\]), we arrive at the expression for the amplitude $A_{\gamma\gamma}^{(M)}$ at moderate virtualities $Q^2_{1,2}$:
$$\begin{aligned}
\label{am}
A_{\gamma\gamma}^{(M)} &=&
\int_{- \imath \infty}^{\imath \infty}
\frac{d \omega}{2 \pi \imath}e^{\omega (\rho- \xi/2)}
\left[e^{\omega \eta/2}
\widetilde{F}_{\gamma\gamma} (\omega, \eta)
+ \frac{1}{16 \pi^2} \int_{\eta}^{\xi} d \xi' e^{\omega \xi'/2}
F_{\gamma q} (\omega, y'_2) F_{q \gamma} (\omega, y'_1)\right]
\\ \nonumber
&=& \int_{- \imath \infty}^{\imath \infty}
\frac{d \omega}{2 \pi \imath}
\left(\frac{s}{\sqrt{Q^2_1 Q^2_2}}\right)^{\omega}
\left[e^{\omega \eta/2}
\widetilde{F}_{\gamma\gamma} (\omega, \eta)
+ \frac{1}{16 \pi^2} \int_{\eta}^{\xi} d \xi' e^{\omega \xi'/2}
F_{\gamma q} (\omega, y'_2) F_{q \gamma} (\omega, y'_1)\right],\end{aligned}$$
where $\widetilde{F}_{\gamma\gamma}$ is expressed in Eq. (\[sol2\]) through the auxiliary amplitudes. Eqs. (\[fm\],\[am\]) are obtained under the assumption of Eq. (\[modoffshell\]) that $y_1 > y_2$. Writing $A_{\gamma\gamma}^{(M)}$ and $F_{\gamma\gamma}^{(M)}$ in terms of variables $\zeta,\eta$ makes easy to see that the reverse assumption $y_1 > y_2$ leads to the expressions for $A_{\gamma\gamma}^{(M)}, F_{\gamma\gamma}^{(M)}$, with $\eta$ replaced by $- \eta$. Therefore, replacing $\eta$ by $|\eta|$ in Eqs. (\[fm\],\[am\]) allows us to embrace the both cases. After the replacement has been done, Eqs. (\[fm\],\[am\]) are indeed invariant to the exchange $y_1 \leftrightarrows y_2$.
Deeply-virtual photons
----------------------
When $Q^2_1 >> \mu^2$ and $Q^2_2 >> \mu^2$, and their product is also great, $Q^2_1 Q^2_2 >> s \mu^2$, the inequality in Eq. (\[modoffshell\]) is replaced by the opposite one
$$\label{deepy}
\rho < y_2 + y_1.$$
We address such photons as deeply-virtual ones. The principal difference between this case and the case of moderately-virtual photons is that the scattering amplitude $A_{\gamma\gamma}^{(D)} (\rho,y_1,y_2)$ in the kinematics (\[deepy\]) does not depend on $\mu$, so the IREE for it is very simple:
$$\label{eqad}
\frac{\partial A_{\gamma\gamma}^{(D)}}{\partial \rho}
+ \frac{\partial A_{\gamma\gamma}^{(D)}}{\partial y_1 } +
\frac{\partial A_{\gamma\gamma}^{(D)}}{\partial y_2} = 0.$$
A general solution to Eq. (\[eqad\]) can be written in different ways. The most convenient way for our goal is
$$\label{adeepgen}
A_{\gamma\gamma}^{(D)} = M (\rho - y_1, \rho - y_2),$$
with $M$ being an arbitrary analytic function. In order to specify $M$ we use the matching with the amplitude $A_{\gamma\gamma}^{(M)}$ of the same process but in the region (\[modoffshell\]). It means that
$$\label{match12}
A_{\gamma\gamma}^{(D)} (\rho-y_1, \rho - y_2)|_{\rho = y_1 + y_2}
%= A_{\gamma\gamma}^{(D)} (y_2,y_1)
= A_{\gamma\gamma}^{(M)}(\rho,y_1,y_2)|_{\rho = y_1 + y_2}
\equiv \widetilde{A}_{\gamma\gamma}^{(M)}(y_1,y_2).$$
Replacing $y_1 \to \rho - y_2$ and $y_2 \to \rho - y_1$ in $\widetilde{A}_{\gamma\gamma}^{(M)}(y_1,y_2)$ immediately allows us to express $A_{\gamma\gamma}^{(D)}$ through $\widetilde{A}_{\gamma\gamma}^{(M)}$ in the whole the region $\rho \leq y_1 + y_2$:
$$\label{ad}
A_{\gamma\gamma}^{(D)} (\rho,y_1,y_2) = \widetilde{A}_{\gamma\gamma}^{(M)}(\rho - y_2, \rho - y_1),$$
or, in terms of the Mellin transform,
$$\label{afd}
A_{\gamma\gamma}^{(D)} =
\int_{- \imath \infty}^{\imath \infty}
\frac{d \omega}{2 \pi \imath}e^{\omega (\rho- \xi/2)}
\left[e^{\omega \eta/2}
\widetilde{F}_{\gamma\gamma} (\omega, \eta)
+ \frac{1}{16 \pi^2} \int_{\eta}^{2 \rho -\xi} d \xi' e^{\omega \xi'/2}
F_{\gamma q} (\omega, y'_2) F_{q \gamma} (\omega, y'_1)\right] \ .$$
Eq. (\[afd\]) demonstrate that, in contrast to the previous cases, the variable $\rho$ participates not only in the Mallin factor but also in the expression in parentheses. This should be taken as a clear warning not to use the Mellin amplitudes for the matching. Indeed, applying the Mellin transform to Eq. (\[eqad\]) converts it into the following equation for the Mellin amplitude $F_{\gamma\gamma}^{(D)}$:
$$\label{eqfd}
\omega F_{\gamma\gamma}^{(D)}
+ \frac{\partial F_{\gamma\gamma}^{(D)}}{\partial y_1 } +
\frac{\partial F_{\gamma\gamma}^{(D)}}{\partial y_2} = 0,$$
(\*\*\* comment: in the above equation I removes extra $+$ \*\*\*) or
$$\label{eqfdksi}
\omega F_{\gamma\gamma}^{(D)} +
\frac{\partial F_{\gamma\gamma}^{(D)}}{\partial \xi} = 0,$$
with the obvious general solution:
$$\label{fdgen}
F_{\gamma\gamma}^{(D)} = \Phi (\omega, \eta) e^{- \omega \xi},$$
where an unspecified function $\Phi$ is supposed to be found through matching with $F_{\gamma\gamma}^{(M)}$ of Eq. (\[fm\]) at $\rho = \xi$. However, it cannot be done because $\Phi$ by definition does not depend on $\xi$ whereas $F_{\gamma\gamma}^{(M)}$ depends on it. So, the matching can be done for the amplitudes $A_{\gamma\gamma}^{(D)}, A_{\gamma\gamma}^{(M)}$. We consider this issue in more detail in Sect. V.
Auxiliary amplitudes
====================
In the previous Sect. we obtained amplitudes $A_{\gamma\gamma}^{(M,D)}$ in terms of auxiliary amplitudes $A_{\gamma q},A_{q \gamma}$. corresponding to the process of Eq. (\[gammaq\]) and the inverse process respectively. We denote $F_{\gamma q}(\omega,y)$ and $F_{q \gamma}(\omega,y)$ the Mellin amplitudes related to $A_{\gamma q},A_{q \gamma}$ respectively. We remind that throughout the paper we neglect the quark masses. We will compose and solve IREE for them, considering first the simplest kinematics, where the photons are on-shell and then move to the case of off-shell photons. As $A_{\gamma q}$ and $A_{q \gamma}$ are much alike, we consider in detail dealing with $F_{\gamma q}$ only.
Photon-quark amplitude with on-shell photon
-------------------------------------------
We consider the case when $y = 0$ and denote $f_{\gamma q}(\omega)$ the Mellin amplitude of such a process. The IREE for $f_{\gamma q} (\omega) $ is depicted in Fig. \[gammafig4\].
![\[gammafig4\] Infra-Red Evolution Equation for the amplitude $A_{\gamma q}$.](gammafig4){width=".6\textwidth"}
In the $\omega$-space it is $$\label{eqgammaqtilde}
f_{\gamma q}(\omega) =
\frac{a_{\gamma q}}{\omega} + \frac{1}{8 \pi^2 \omega}
f_{\gamma q} (\omega) f_0 (\omega),$$ where $a_{\gamma q}/\omega$, with $a_{\gamma q} = e^2$, is the Born amplitude and $f_0$ is the quark-quark amplitude. It includes the total resummation of DL contributions as well as accounts for the running coupling. The solution to Eq. (\[eqgammaqtilde\]) is
$$\label{gammaqtilde}
f_{\gamma q}(\omega) = \frac{a_{\gamma q}}{\omega - H(\omega)},$$
where, by convenience reason we have introduced the notation $H (\omega) = (1/8 \pi^2) f_0 (\omega)$. In DIS (\*\*\* process \*\*\*) $H$ plays the role of the non-singlet anomalous dimension calculated in DLA.
Off-shell photons
-----------------
IREE for $F_{\gamma q}(\omega, y)$ is
$$\label{eqgq}
\frac{\partial}{\partial y}F_{\gamma q} (\omega, y) + \omega F_{\gamma q}(\omega, y) =
\frac{1}{8 \pi^2}
F_{\gamma q} (\omega, y) f_0 (\omega).$$
A general solution to Eq. (\[eqgq\]) is
$$\label{{solgqgen}}
F_{\gamma q} (\omega, y) = C_{\gamma q}(\omega) e^{-y [\omega - H (\omega)]},$$
with $C_{\gamma g}$ being an arbitrary function. To specify $C_{\gamma g}$ we use the matching
$$\label{matchq}
F_{\gamma q} (\omega,y)|_{y = 0} = \tilde{F}_{\gamma q} (\omega),$$
with $\tilde{F}_{\gamma q} (\omega)$ defined in Eq. (\[gammaqtilde\]). The use of Eq. (\[matchq\]) leads to $$\label{fgammag}
F_{\gamma q}(\omega, y) = \frac{a_{\gamma q}}{\omega - H(\omega)}
e^{-y [\omega - H (\omega)]}$$
Now the auxiliary amplitude $F_{\gamma q} (\omega, y) f_0 (\omega)$ is expressed through the on-shell quark-quark amplitude $f_0$ which is well-known. It was calculated in Ref. [@kl], with $\alpha_s$ being fixed.
Quark-quark amplitude
---------------------
Amplitude $f_0$ was obtained in Ref. [@kl]. It satisfies the simple algebraic equation
$$\label{eqfqq}
f_0 = \frac{a_0}{\omega} + \frac{1}{8 \pi^2 \omega} f_0 f_0,$$
where $a_0/\omega$ is the Born amplitude. Solving Eq. (\[fqq\]), one arrives at the explicit expression for $f_0$:
$$\label{fqq}
f_0 = 4 \pi^2 \left[\omega - \sqrt{\omega^2 - a_0/(2 \pi^2)}\right].$$
Eq. (\[fqq\]) is true for the both cases of fixed and running $\alpha_s$ but $a_0$ in those cases are different. For fixed QCD coupling, $a_0 \equiv a_0^{fix}$ was obtained in Ref. [@kl]:
$$\label{aqqfix}
a_0^{fix} = 4 \pi \alpha_s^{fix} C_F,$$
with $C_F = (N^2-1)/(2N) = 4/3$, while at running $\alpha_s$ it depends on $\omega$ (see [@egtquark] for detail): $$\label{aqqrun}
a_0(\omega) = \frac{4 \pi C_F}{ b} \left[\frac{\zeta}{\zeta^2 + \pi^2}
- \int_0^{\infty}\frac{d \rho e^{- \omega \rho}}
{(\rho + \zeta)^2 + \pi^2}\right] ,$$ where $\zeta = \ln \left(\mu^2/\Lambda^2_{QCD}\right)$ and $b = (11 N - 2 n_f)/(12 \pi^2)$ is the standard notation for first coefficient of the Gell-Mann - Low function.
Representation of the auxiliary amplitudes through the quark amplitude
----------------------------------------------------------------------
Using Eq. (\[eqfqq\]) allows us to simplify Eq. (\[gammaqtilde\]) for the auxiliary amplitude $f_{\gamma q}(\omega)$:
$$\label{gammaqtilde1}
f_{\gamma q}(\omega) = \frac{a_{\gamma q}}{a_0} f_0 (\omega)$$
as well as the expression for $F_{\gamma q} (\omega, y)$ in Eq. (\[fgammag\]):
$$\label{gammaq1}
F_{\gamma q} (\omega, y) =
\frac{a_{\gamma q}}{a_0} f_0 (\omega)
e^{- y \left(\omega - H (\omega)\right)}.$$
The only difference between IREE for $F_{\gamma q} (\omega, y)$ and $F_{q \gamma} (\omega, y)$ is in the use of different the factors $a_{\gamma q}$ and $a_{q \gamma}$ respectively, so expressions for $\tilde{F}_{q \gamma} (\omega, y)$ and $F_{q \gamma} (\omega, y)$ can be immediately obtained from Eqs. (\[gammaqtilde1\]) and (\[gammaq1\]):
$$\label{qgammatilde1}
f_{q\gamma }(\omega) = \frac{a_{q\gamma }}{a_0} f_0 (\omega)$$
$$\label{qgamma1}
F_{q \gamma} (\omega, y) =
\frac{a_{q \gamma}}{a_0} f_0 (\omega)
e^{- y \left(\omega - H (\omega)\right)}.$$
We define the factors $a_{\gamma q}$ and $a_{q \gamma}$ as follows:
$$\label{agammaqborn}
a_{\gamma q} = e^2_q,~~a_{q \gamma} = -e^2_q ,$$
where $e_q$ is the electric charge of the loop quark.
Representation of photon-photon amplitudes through quark-quark amplitudes
=========================================================================
On-shell initial photons
------------------------
Substituting Eqs. (\[qgammatilde1\],\[gammaqtilde1\]) in Eq. (\[fon\]) and using Eq.(\[eqfqq\]), we obtain
$$\label{fonqq}
f_{\gamma \gamma}(\omega) = \kappa
%\frac{a_{\gamma q} a_{q \gamma}}{(a_0)^2}
\left[f_0 (\omega) - \frac{a_0}{\omega} \right],$$
with $$\label{kappa}
\kappa = \frac{a_{\gamma q} a_{q \gamma}}{a_0^2}.$$ According to Eq. (\[aqqrun\]) $\kappa$ depends on $\omega$, when $\alpha_s$ is running, so throughout the paper we will keep it under the Mellin integral sign. The photon-photon scattering amplitude $A^{(on)}_{\gamma\gamma}$, all photons are on-shell, is
$$\label{aon}
A^{(on)}_{\gamma\gamma} (s/\mu^2) =
%\frac{a_{\gamma q} a_{q \gamma}}{(a_0)^2}
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}e^{\omega \rho}~\kappa
\left[f_0 (\omega) - \frac{a_0}{\omega} \right].$$
One of the photons is off-shell
-------------------------------
If $y_1 > y_2 = 0$, amplitude $\widetilde{F}_{\gamma \gamma}$ is given by Eq. (\[sol2\]) where $\widetilde{F}_{\gamma\gamma}(\omega,y_1)$ is represented through the auxiliary amplitudes. Combining Eq. (\[sol2\],\[gammaq1\]) and (\[fonqq\]), we express $\widetilde{F}_{\gamma\gamma}(\omega,y_1)$ in terms of $f_0(\omega)$:
$$\label{fgammatildeqq}
\widetilde{F}_{\gamma\gamma}(\omega,y_1) = \kappa
%\frac{a_{\gamma q} a_{q \gamma}}{(a_0)^2}
e^{- \omega y_1} \left[
f_0 (\omega) e^{y_1H} - \frac{a_0}{\omega} \right].$$
Therefore,
$$\label{agammatildeqq}
\widetilde{A}_{\gamma\gamma}(\omega,y_1) =
%\frac{a_{\gamma q} a_{q \gamma}}{(a_0)^2}
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath} e^{\omega (\rho - y_1)}
~\kappa
\left[f_0 (\omega) e^{y_1 f_0 /(8 \pi^2)} - \frac{a_0}{\omega} \right].$$
Moderately-virtual photons
--------------------------
Combining Eq. (\[fm\]) with Eqs. (\[fgammatildeqq\],\[gammaq1\],\[qgamma1\]) allows us to obtain $F_{\gamma \gamma}^{(M)}$, so we can write the amplitude $A_{\gamma \gamma}^{(M)}$ in the MV region as follows:
$$\label{amw}
A_{\gamma\gamma}^{(M)} =
\int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{\omega \rho} ~\kappa
\left[ W_1 + W_2 \right]
%= \kappa
%\int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
%\left(\frac{s}{\mu^2}\right)^{\omega}
% \left[ W_1 + W_2 \right]
,$$
with
$$\begin{aligned}
\label{wm}
W_1 &=& e^{-\omega (\xi +|\eta|)/2}
\left(- \frac{a_0}{\omega} + f_0 e^{|\eta| H}\right)
\\ \nonumber
W_2 &=& \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} \left[e^{-(\xi + |\eta|)\omega/2 + \eta H} - e^{\xi (-\omega + H)}\right].\end{aligned}$$
We remind that the variables $\xi,\eta$ are defined in Eq. (\[xieta\]). Eqs. (\[amw\],\[wm\]) describe $A_{\gamma\gamma}^{(M)}$ at any ordering between $y_1$ and $y_2$, i.e. at $y_1 > y_2$ and $y_1 < y_2$; they also stand when $y_1 = y_2$.
Deeply-virtual photons
----------------------
According to Eq. (\[match12\]), aamplitude $A_{\gamma\gamma}^{(M)}$ can be found through matching with amplitude $A_{\gamma\gamma}^{(M)}$ at the border between the Deeply-Virtual and Moderately-Virtual regions, where
$$\label{border}
\rho = \xi.$$
As $\rho$ participates in the Mellin factor, the matching should involve the whole amplitudes $A_{\gamma\gamma}^{(M)}, A_{\gamma\gamma}^{(D)}$ rather than $F_{\gamma\gamma}^{(M)}, F_{\gamma\gamma}^{(M)}$. For performing the matching the easiest way, we replace Eq. (\[amw\]) by the following one:
$$\begin{aligned}
\label{amxi}
A_{\gamma\gamma}^{(M)} = \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{(\omega \rho - \omega \xi/2)} ~\kappa F_1 (\omega, \eta)
- \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{(\omega \rho - \omega \xi + \xi H)} \kappa F_2 (\omega)
,
\end{aligned}$$
where
$$\begin{aligned}
\label{f12}
F_1 (\omega, \eta) &=& e^{-\omega \eta/2}
\left[
\left(- \frac{a_0}{\omega} + f_0 e^{\eta H}\right)
+ \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} e^{\eta H}
\right],
\\ \nonumber
F_2 (\omega) &=& \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}}.\end{aligned}$$
Then at $\rho = \xi$ amplitude $A_{\gamma\gamma}^{(M)}$ becomes $\bar{A}_{\gamma\gamma}^{(M)}$:
$$\label{abarxi}
\bar{A}_{\gamma\gamma}^{(M)} = \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{\omega \xi/2}~\kappa F_1 (\omega, \eta)
- \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{\xi H} ~\kappa F_2 (\omega)$$
and therefore in the Deeply-Virtual region
$$\label{adgen}
A_{\gamma\gamma}^{(D)} = \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{\omega (\rho - \xi/2)} ~\kappa F_1 (\omega, \eta)
- \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{ (2\rho - \xi) H} ~\kappa F_2 (\omega).$$
The second integral in Eq. (\[adgen\]) can be dropped because it does not contain the standard Mellin factor $e^{\omega \rho}$ (or $e^{\omega (2\rho- \xi)}$) which would prevent closing the integration contour to the right, where the integrand does not have singularities. Closing the contour to the right, we find that the integration over $\omega$ yields a zero. So, we arrive at the following expression which is true for any ordering of $y_{1,2}$ and for the case $y_1 = y_2$:
$$\begin{aligned}
\label{adsym}
A_{\gamma\gamma}^{(D)} &=& \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
%\left(\frac{s}{\sqrt{Q^2_1Q^2_2}}\right)^{\omega}
e^{\omega (\rho - \xi/2 - |\eta|/2)}
~\kappa \left[
\left(- \frac{a_0}{\omega} + f_0 e^{|\eta| H}\right)
+ \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} e^{|\eta| H}
\right]
\\ \nonumber
&=& \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
\left(\frac{s}{\sqrt{Q^2_1Q^2_2}}\right)^{\omega}
\left(\frac{Q^2_{\max}}{Q^2_{\min}}\right)^{\omega}
~\kappa \left[
\left(- \frac{a_0}{\omega} + f_0 e^{|\eta| H}\right)
+ \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} e^{|\eta| H}
\right],\end{aligned}$$
where we have denoted $Q^2_{\max} = \max [Q^2_{1,2}]$ and $Q^2_{\min} = \min [Q^2_{1,2}]$.
The amplitudes $A_{\gamma\gamma}^{(M)}$ in Eq. (\[amw\]) and $A_{\gamma\gamma}^{(D)}$ in Eqs. (\[adsym\],\[adq\]) are represented in the form different from the expressions for the same amplitudes obtained in Ref. [@bl], which is unessential. The main difference between our approach and Ref. [@bl] is our accounting for the running QCD coupling. In this case $a_0$ depends on $\omega$ (see Eq. (\[aqqrun\])). Now let us remind that $A_{\gamma\gamma}^{(M)}$ and $A_{\gamma\gamma}^{(D)}$ are not complete expressions for amplitudes of the process (\[gamma\]) in the collinear kinematics. In order to account for the missing contributions, we replace $s$ by $u$ in Eqs. (\[amw\]) and (\[adsym\]), obtaining the amplitudes ${A'}_{\gamma\gamma}^{(M)}$ and ${A'}_{\gamma\gamma}^{(D)}$. Adding them to $A_{\gamma\gamma}^{(M)}$ and $A_{\gamma\gamma}^{(D)}$ respectively, we arrive at the complete expressions for the DLA amplitude of the process (\[gamma\]) in the collinear kinematics. In the Moderately-Virtual region (\[modoffshell\]) it is $$\label{amcol}
\widetilde{A}_{\gamma\gamma}^{(M)} = A_{\gamma\gamma}^{(M)} + {A'} _{\gamma\gamma}^{(M)}$$ whereas in the Deeply-Virtual region (\[deepy\])
$$\label{adcol}
\widetilde{A}_{\gamma\gamma}^{(D)} = A_{\gamma\gamma}^{(D)} + {A'} _{\gamma\gamma}^{(D)}.$$
Non-collinear photon-photon scattering
======================================
In this Sect. we extend the results obtained above to the forward Regge kinematics (\[fkin\]) with $t \neq 0$. In order to avoid confusing new amplitudes with $A_{\gamma\gamma}^{(M)}$ and $A_{\gamma\gamma}^{(D)}$ obtained under assumption that $t \sim 0$, we introduce a generic notation $M_{\gamma\gamma}$ for new amplitudes in DLA and will provide this notation with superscripts to specify the kinematics. The Born amplitude, $M_B$ is (cf. Eq. (\[aborn\]))
$$\label{mb}
M_B = \widetilde{B} + \widetilde{B}',$$
with $$\label{mborn}
\widetilde{B} = - \imath \frac{e^4}{16 \pi^3}\int
\frac{d \alpha d \beta d k^2_{\perp} w^2 k^2_{\perp}}
{k^2 (k + q - q')^2\left(x_q w + \beta w - \alpha x_q w + k^2\right) \left(x_p w + \alpha w - \beta x_p w + k^2 \right)}$$ and $\widetilde{B}'$ is obtained from $\widetilde{B}'$ with replacing $q \to - q$. It is obvious that the integration over $k$ in Eq. (\[mborn\]) yields a DL contribution from the region $$\label{kt}
k^2_{\perp} \gg -t = -(q'-q)^2.$$
In other words, $|t|$ acts in Eq. (\[mborn\]) as a new IR cut-off. Therefore in the Born approximation (and beyond it) the amplitude $M_{\gamma\gamma}$ in DLA is IR stable. All results we obtained in the previous Sects., studying the amplitudes in the collinear kinematics, can easily be extended to the region of non-zero $t$ by the simple replacement
$$\label{mut}
\mu^2 \to |t|.$$
A further advancement strongly depends on the hierarchy between $Q^2_{1,2}$ and $|t|$. When
$$\label{q12small}
Q^2_{1,2} < |t|,$$
amplitude $M_B$ does not depend on $Q^2_{1,2}$ under the DL accuracy. In this case
$$\label{qsmall}
M_B = M_B = -\frac{e^4}{16 \pi^2} \ln^2 \left(s/|t|\right).$$
When $s \gg Q^2_{1,2} \gg |t|$, there are again two cases:
$$\label{mbm}
M_B = -\frac{e^4}{16 \pi^2} \left[\ln^2 \left(s/|t|\right) - \ln^2 \left(Q^2_1/|t|\right) - \ln^2 \left(Q^2_2/|t|\right)\right],$$
when $Q^2_1 Q^2_2 \ll s |t|$ and
$$\label{mbd}
M_B = -\frac{e^4}{8 \pi^2} \ln^2 \left(s/Q^2_1\right) \ln \left(s/Q^2_2\right),$$
when $Q^2_1 Q^2_2 \gg s |t|$. Now let us to extend our results for amplitudes $A_{\gamma \gamma}$ to the case of non-zero $|t|$ beyond the Born approximation. To this end, we introduce new logarithmic variables $\bar{\rho}, \bar{y}_1, \bar{y}_2$ instead of the variables $\rho, y_1, y_2$ defined in Eq. (\[y12\]):
$$\label{redef}
\bar{\rho} = \ln \left(s/|t|\right),~ \bar{y}_{1,2} = \ln \left(Q^2_{1,2}/|t|\right).$$
In the case when $s \gg Q^2_1 \gg |t|$ and $Q^2_2 \lesssim |t|$ i.e. when $\bar{\rho} > \bar{y}_1 > \bar{y}_2 \approx 0$, amplitude
$$\label{mgammatildeqq}
\widetilde{M}_{\gamma\gamma}(\omega,y_1) =
%\frac{a_{\gamma q} a_{q \gamma}}{(a_0)^2}
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath} e^{\omega (\bar{\rho} - \bar{y}_1)}
~\kappa
\left[f_0 (\omega) e^{y_1 f_0 /(8 \pi^2)} - \frac{a_0}{\omega} \right].$$
In the more involved case of Moderate Virtualities, when $s \gg Q^2_1, Q^2_2 \gg |t|$ but $s |t| \gg Q^2_1 Q^2_2 $, i.e. when $$\label{mbarkin}
\bar{\rho} > \bar{y}_1 + \bar{y}_2,$$ the scattering amplitude $M_{\gamma\gamma}^{(M)}$ is
$$\label{mmw}
M_{\gamma\gamma}^{(M)} =
\int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
e^{\omega \bar{\rho}}~\kappa
\left[ \bar{W}_1 + \bar{W}_2 \right],$$
with
$$\begin{aligned}
\label{wmbar}
\bar{W}_1 &=& e^{-\omega (\bar{\xi} +|\bar{\eta}|)/2}
\left(- \frac{a_0}{\omega} + f_0 e^{|\bar{\eta}| H}\right)
\\ \nonumber
\bar{W}_2 &=& \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} \left[e^{-(\bar{\xi} + |\bar{\eta}|)\omega/2 + |\bar{\eta}| H} - e^{\bar{\xi} (-\omega + H)}\right].\end{aligned}$$
whereas in the new Deeply-Virtual region
$$\label{dbarkin}
\bar{\rho} < \bar{y}_1 + \bar{y}_2$$
the scattering amplitude $M_{\gamma\gamma}^{(D)}$ is
$$\begin{aligned}
\label{md}
M_{\gamma\gamma}^{(D)} &=& \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath}
%\left(\frac{s}{\sqrt{Q^2_1Q^2_2}}\right)^{\omega}
e^{\omega (\bar{\rho} - \bar{\xi}/2 - |\bar{\eta}|/2)}~\kappa
\left[
\left(- \frac{a_0}{\omega} + f_0 e^{|\bar{\eta}| H}\right)
+ \left(f_0 - \frac{a_0}{\omega}\right)
\frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}} e^{|\bar{\eta}| H}
\right].\end{aligned}$$
Replacing $s$ by $u$ in Eqs. (\[mmw\]) and (\[md\]), we obtain amplitudes ${M'}_{\gamma\gamma}^{(M)}$ and ${M'}_{\gamma\gamma}^{(D)}$. Adding them to the expressions in Eqs. (\[mmw\],\[md\]) and multiplying them by the factor $2 n_f$, with $n_f$ being the number of involved flavors, we arrive at the DL expressions for the scattering amplitudes $\widetilde{M}_{\gamma\gamma}^{(M)}$ (in the Moderately-Virtual region (\[mbarkin\])) and $\widetilde{M}_{\gamma\gamma}^{(D)}$ (in the Deeply-Virtual region (\[dbarkin\])):
$$\label{mmd}
\widetilde{M}_{\gamma\gamma}^{(M)} = 2 n_f [M_{\gamma\gamma}^{(M)} + {M'}_{\gamma\gamma}^{(M)}],~~
\widetilde{M}_{\gamma\gamma}^{(D)} = 2 n_f [M_{\gamma\gamma}^{(D)} + {M'}_{\gamma\gamma}^{(D)}].$$
Amplitudes $\widetilde{M}_{\gamma\gamma}^{(M,D)}$ in Eq. (\[mmd\]) account for the total resuimmation of DL corrections ot the Born amplitude $A_B$ of Eq. (\[ab\]) in the forward kinematic region (\[fkin\]).
Discussion of the obtained results
==================================
In this Sect. we discuss the results obtained in the previous Sects.
Comment on Deeply-Virtual and Moderately-Virtual regions
--------------------------------------------------------
The Deeply-Virtual (DV) and Moderately-Virtual (MV) kinematics are introduced in Eqs. (\[smallq\],\[modoffshell\]) and (\[bigq\],\[deepy\]) respectively. The scattering amplitude $A_{\gamma\gamma}^{(M)}$, being calculated in MV kinematics explicitly depends on the IR cut-off $\mu$. Often, $\mu$ is an artificial parameter with arbitrary value but on the other hand, there are cases, when $\mu$ has a physical meaning. For instance, it can be the heavy quark mass or the masses of $W,Z$ bosons in Standard Model. In such cases the range of $\rho$ in the region (\[deepy\]) is quite restricted:
$$\label{deepreg}
\max [y_1, y_2] < \rho < y_1 + y_2.$$
As a result, $A_{\gamma\gamma}^{(D)}$ cannot be used for calculating the asymptotics of $A_{\gamma\gamma}$, when $s \to \infty$. The asymptotics in this case can be obtained from $A_{\gamma\gamma}^{(M)}$. The same is true also in the case when the running coupling effects for amplitudes in the Regge kinematics are accounted for.
On the contrary, when $\mu$ is not associated with an appropriate mass scale and $\alpha_s$ is fixed or regarded as $\mu$- independent, the value of $\mu$ can be chosen arbitrary small. and, as the DV region ensures the IR stability independently of $\mu$, one can choose $\mu$ very small. This considerably broadens the applicability region for the DV kinematics and makes possible to use it at very high energies. So, despite the kinematics is the Regge one, it is as IR stable as the hard kinematics.
Impact of Higher-loop DL radiative corrections
----------------------------------------------
The $s$-dependent parts, $B_{\gamma\gamma}^{(M,D)}$ of the photon-photon scattering amplitudes in the lowest-order approximation at $t = 0$ are given by Eqs. (\[born1\],\[born2\]) for the MV and DV photons respectively. Accounting for the radiative corrections in DLA converts them into amplitudes $A_{\gamma\gamma}^{(M,D)}$. Let us estimate the impact of the DL radiative corrections on $B_{\gamma\gamma}^{(M,D)}$ in the simplest case when $Q^2_1 \sim Q^2_2 = Q^2$, so the involved amplitudes in this case depend on $x = Q^2/s$ only. In order to be independent of choice of $\mu$, we will do it for the amplitudes with Deeply-Virtual photons, where $A_{\gamma\gamma}^{(D)} (x)$ is given by Eq. (\[adq\]). We define the the ratio $R_{\gamma}$ as follows:
$$\label{rgamma}
R_{\gamma}(x)= A_{\gamma\gamma}^{(D)}/B_{\gamma\gamma}^{(D)}$$
and show the plot of $R_{\gamma}$ against $x$ in Fig. 5.
![\[gammafig5\] Dependence of $R_{\gamma}$ on $x$.](gammafig5){width=".4\textwidth"}
Fig. 5 explicitly demonstrates that the radiative corrections become sizable since $x \sim 10^{-3}$. They double the Bon amplitude at $x \sim 10^{-4} $.
High-Energy asymptotics
-----------------------
At very high energies scattering amplitudes are often approximated by their asymptotics. The asymptotics are much easier to work on than the explicit expressions. However, the applicability region for the asymptotics cannot be deduced from theoretical consideration. Below we first show how to calculate the asymptotics and then outline its applicability region. very small $x$ Asymptotics at $\rho \to \infty$ of all amplitudes we calculated above can be obtained by using the saddle-point method and is given by similar expressions. For the sake of simplicity we consider the small-$x$ asymptotics of amplitudes $A_{\gamma\gamma}^{(M,D)}$ of Eqs. (\[amw\],\[adsym\]) in the particular case when $Q^2 \sim Q^2_2 \equiv Q^2$. In this case Eq. (\[amw\]) is reduced to
$$\label{amq}
A_{\gamma\gamma}^{(M)} (x,y) = \int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath} x^{-\omega}
~\kappa
\left(f_0 - \frac{a_0}{\omega}\right)
\left[1 + \frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}}\left(1 - e^{-y \sqrt{\omega^2 - a_0/(2 \pi^2)}}\right)\right],$$
where $x = Q^2/s$ and $y = Q^2/s$. In contrast, $A_{\gamma\gamma}^{(D)} $ depends on $x$ only:
$$\label{adq}
A_{\gamma\gamma}^{(D)} (x) =
\int_{- \imath \infty}^{\imath \infty} \frac{d \omega}{2 \pi \imath} x^{-\omega}~\kappa
\left(f_0 - \frac{a_0}{\omega}\right) \left[1 + \frac{\omega}{\sqrt{\omega^2 - a_0/(2 \pi^2)}}\right].$$
The saddle-point method, being applied to Eqs. (\[amq\],\[adq\]), immediately yields that the rightmost stationary point $\omega_0$ is the same for the both amplitudes and it is given by the rightmost root of the equation $$\label{omega0}
\omega^2 - a_0/(2 \pi^2) = 0.$$
The further progress depends on the treatment of $\alpha_s$. When $\alpha_s$ is fixed ($\alpha_s = \alpha_s^{fix}$), it is easy to obtain an analytic expressions for $\omega_0$: $$\label{omega0fix}
\omega_0 = \sqrt{2\alpha_s^{fix} C_F/\pi} + 1/(2 z).$$ A numerical estimate for $\alpha_s^{fix}$ in Eq. (\[omega0fix\]) was obtained in Ref. [@egtfrozen]. According to it, $\alpha_s \approx 0.24$.
When the running coupling effects are taken into account, $a_0$ depends on $\omega$ (see Eq. (\[aqqrun\])) and therefore Eq. (\[omega0\]) has to be solved numerically (see Ref. [@egtquark]). It leads to the estimate $\omega_0 \approx 0.4$. The small-$x$ asymptotics $\left[A_{\gamma\gamma}^{(M,D)}\right]_{as}$ of amplitudes $A_{\gamma\gamma}^{(M,D)}$ respectively are:
$$\label{amqas}
\left[A_{\gamma\gamma}^{(M,D)}\right]_{as} = \Pi^{(M,D)} x^{- \omega_0},$$
with the factors $\Pi^{(M)}$ and $\Pi^{(D)}$ being
$$\label{pimd}
\Pi^{(M)} = \frac{e^4}{\pi^2 \omega_0^3} \frac{1}{\sqrt{2 \pi \omega_0 z^3}}
\left[1 + \frac{\omega_0 z}{2} \left(1 - e^{-2y/z}\right)\right],~~
\Pi^{(D)} = \frac{e^4}{4 \pi^2 \omega_0^2} \frac{1}{\sqrt{\pi \omega_0 z}},$$
where we have denoted $z = \ln (1/x)$.
Now let us consider the $x$ dependence of the ratio
$$\label{rasm}
R_{as} = \frac{\left[A_{\gamma\gamma}^{(M)}\right]_{as}}{ A_{\gamma\gamma}^{(M)}}.$$
The $x$-dependence of $R^{(M)}_{as}$ is plotted in Fig. 5. For the sake of simplicity, the graph in Fig. 6 is done for $y \approx 0$. The greater $y$, the lower the graph runs.
![\[gammafig6\] Dependence of $R^{(M)}_{\gamma}$ on $x$.](gammafig6){width=".4\textwidth"}
Similarly to Eq. (\[rasm\]), we define the ratio
$$\label{rasd}
R_{as} =
\frac{\left[A_{\gamma\gamma}^{(D)}\right]_{as}}{ A_{\gamma\gamma}^{(D)}}.$$
The $x$-dependence of $R^{(D)}_{as}$ is plotted in Fig. 7.
![\[gammafig7\] Dependence of $R^{(D)}_{\gamma}$ on $x$.](gammafig7){width=".4\textwidth"}
Figs. 6,7 display that the amplitudes $A_{\gamma\gamma}^{(M,D)}$ are reliably represented by their asymptotics at very small $x$. Indeed, $R_{as}^{(M,D)} \sim 0.8$ at $x \lesssim 10^{-8}$. It perfectly agrees with the results of Ref. [@egtsum] where it was proved that the small-$x$ asymptotics of the non-singlet DIS structure functions $F_1^{NS}$ and $g_1^{NS}$ reliably represent these structure functions at $x \lesssim 10^{-8}$. In terms of the Reggeology, the only difference between $F_1^{NS}$ and $A_{\gamma\gamma}^{(M,D)}$ is the difference in the impact-factors while the Reggeons are the same. This proves that the applicability region for the use of non-vacuum Reggeons is
$$\label{regx}
x \lesssim 10^{-8},$$
i.e. Eq. (\[regx\]) manifests that the non-vacuum Reggeons should not be used for description of hadronic reactions at available energies.
Conclusion
==========
We have calculated the amplitudes of the process (\[gamma\]) in DLA. We considered this process in both the collinear kinematics (amplitudes $A_{\gamma\gamma}^{(M)}$ and $A_{\gamma\gamma}^{(D)})$, where $t=0$, and the forward kinematics (\[fkin\]) (amplitudes $M_{\gamma\gamma}^{(M)}$ and $M_{\gamma\gamma}^{(D)} $) at $t \neq 0$. So as to calculate those amplitudes we constructed and solved the Infrared Evolution Equations for them. According to the general technology of solving IREE, any general solution to IREE for a scattering amplitude in a certain kinematics is specified through matching with the known amplitude of the same process in a simpler kinematics. So, before solving IEEE for the amplitudes $M_{\gamma\gamma}^{(M,D)}$ in the $t \neq 0$ -kinematics, we had to calculate the amplitudes $A_{\gamma\gamma}^{(M,D)}$ of the same process in the collinear kinematics. Doing so, we confirmed the results of Ref. [@bl] and generalized them to the case of running QCD coupling while $\alpha_s$ in Ref. [@bl] was fixed.
At very high energies the scattering amplitudes are often considered in the asymptotic form and such asymptotics are addressed as Reggeons. Such Reggeons are much easier to use than their parent amplitudes. However, applicability regions for the asymptotics (Reggeons) cannot be fixed from theoretical grounds. We do it numerically, calculating the asymptotics of amplitudes $A_{\gamma\gamma}^{(M,D)}$ and comparing the asymptotics to the amplitudes. In order to calculate the asymptotics, we use the saddle-point method. The results are plotted in Figs. 5,6. They outline the applicability region of the non-vacuum Reggeons and perfectly agree with the observation of Ref. [@egtsum] that non-vacuum Reggeons should not be used for describing available experimental data.
Acknowledgements
================
We are grateful to W. Schafer and A. Szczurek for useful communications. The work of D.Yu. Ivanov was supported by the program of fundamental scientific researches of the SB RAS № II.15.1., project № 0314-2016-0021
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abstract: 'We describe our perspective on the Structural Glass Transition (SGT) problem built on the premise that a viable theory must provide a consistent picture of the dynamics and statics, which are manifested by large increase in shear viscosity and thermodynamic anamolies respectively. For the static and dynamic description to be consistent we discovered, using a density functional description without explicit inclusion of quenched random interactions and a mean-field theory, that there be an exponentially large number of metastable states at temperatures less than a critical transition temperature, $T_A$. At a lower temperature ($T_K < T_A$), which can be associated with the Kauzmann temperature, the number of glassy states is non-extensive. Based on this theory we formulated an entropic droplet picture to describe transport in finite dimensions in the temperature range $T_K < T < T_A$. From the finding that glasses are trapped in one of many metastable states below $T_A$ we argue that during the SGT law of large numbers is violated. As a consequence in glasses there are sub sample to sub sample fluctuations provided the system is observed for times longer than the typical relaxation time in a liquid. These considerations, which find support in computer simulations and experiments, also link the notion of dynamic heterogeneity to the violation of law of large numbers. Thus, the finding that there is an extensive number of metastable states in the range $T_K < T < T_A$ offers a coherent explanation of many of the universal features of glass forming materials.'
author:
- 'T. R. Kirkpatrick$^{1,2}$ and D. Thirumalai$^{1,3}$'
bibliography:
- 'Glasses.bib'
title: Random First Order Phase Transition Theory of the Structural Glass Transition
---
I. INTRODUCTION
===============
The Structural Glass Transition (SGT) in a variety of materials whose molecular constituents are chemically different exhibits several universal characteristics. This has prompted a quest for describing a universal mechanism of the nature of the glass transition. Over twenty years ago there were a number of significant contributions to the understanding of the SGT. They included the mode coupling theory of the glass transition [@Leutheusser84PRA; @Gotze92RepProgPhys; @Kirkpatrick85PRA], the random first order transition theory of the glass transition [@Kirk89JPhysA; @Kirk89PRA], and its connection to dynamic theories, and to a lesser extent, the kinetically constrained kinetic models of the glass transition [@Fredrickson84PRL]. Much of this work was, in turn, motivated by research done ten years earlier on the spin glass problem [@Edwards75JPhysF; @Sherrington75PRL; @MezardBook].
In this short perspective we discuss a theory for the SGT that is based on using frozen density fluctuations as an order parameter to characterize the SGT [@Kirk89JPhysA]. The particular type of transition that arises from our considerations has been referred to as a random first order phase transition (RFOPT) [@Kirk89JPhysA; @Kirk89PRA]. Originally this sort of phase transition was theoretically found in certain class of exactly soluble spin glass models [@Kirk87PRL; @Kirk87PRB; @Kirk87PRBa], but it was subsequently realized that it can naturally occur in systems without quenched disorder and in continuum field theory models [@Kirk89JPhysA]. In part we will focus on the connection between a static approach to describe the SGT and a dynamical theory, notion that was first emphasized in [@Kirk87PRA]. Too often these approaches are presented as being distinct from each other while a careful study shows that they are, in fact, closely related. Indeed, establish such a connection is crucial to understanding the nature of the SGT[@Jackle86RepProgPhys; @Martinez01Nature; @Angell95Science]. In linking the static and dynamic theories of the glass transition we discovered that the universal features of glasses are characterized by the nature of an extensive number of metastable states that emerge below a characteristic temperature [@Kirk89JPhysA].
Physically there are obvious similarities between spin and structural glasses [@Kirk95TTSP]. A structural glass is a frozen liquid: A snap shot of a structural glass looks identical to a snapshot of a liquid, i.e., in both cases there is no long range spatial order. It is only when the system develops in time that a glass is obviously different than a liquid since in the former case the molecules are localized to a small region of space on experimental time scales while in the latter case, any molecule diffuses arbitrary far from where it started as $t\rightarrow \infty $. Similarly, a spin glass is a frozen paramagnet: A snapshot of a SG looks identical to a snapshot of a paramagnet, i.e., there is no long ranged magnetic order. When the systems evolves in time the local magnetic moment or spin points in a specific (time) averaged direction in the SG phase while in the paramagnetic phase the spin direction randomly fluctuates in time so that the time averaged magnetic moment is zero. In both the liquid and spin problems it is clear that there is at least an effective phase transition into the glassy state, and that there is broken ergodicity at the glass transition since time averages no longer equal full ensemble averages. Moreover, there are violations of the fluctuation-dissipation relations thus making the dynamics dependent on initial conditions [@Thirum88PRB; @Cugliandolo93PRL], and aging effects become relevant [@Cugliandolo07PhysicaA].
For some time it was thought that there were important conceptional differences between structural and spin glasses. In particular, in SG problems an important input is that there is quenched disorder. In structural glasses, on the other hand, there is quenched disorder only in the glassy phase, and it is self-generated. In other words, the Hamiltonians of glass forming materials are not random whereas the emergence of glassy behavior in SG is due to the presence of quenched random interactions between the spins. We now know that this difference is not so relevant. In fact, the important contribution of [@Kirk89JPhysA] was that it was the first paper to show that the methods that were developed to describe RFOPT in SG models, which appeared to be resticted to models with quenched disorder could, in fact, be used in any system where there are many statistically distributed metastable states [@Kirk89JPhysA]. This crucial discovery that allowed us to produce a consistent static and dynamical theory of the SGT using a density functional Hamiltonian has been subsequently elaborated and expanded by numerous authors (see [@Mezard09CondMat] and references therein).
The plan of this perspective is as follows. In Section II we introduce specific static and dynamic density functional models for the SGT. In Section III the static approach to understanding the SGT is described. In particular, we use a combination of mean-field like approximations and a replica approach to deal with the self-generated randomness in structural glasses to conclude there is a special temperature, conventionally denoted by $T_{A},$ below which there are an exponentially large number of glassy solutions. In Section IV we show the purely dynamical approach yields results for the SGT that are identical to the static approach. In Section IV we describe a related scaling and droplet approach to RFOPT and to the STG transition in particular. We also discuss how to characterize the liquid system below $T_{A}$. We see that a Kauzmann temperature, $T_{K}$, is naturally present in this approach, and that a true glass transition is possible if $T_{K}$ could be reached while maintaining the system in equilibrium. In this Section we also review some speculation on transport at $T_{K}$ is approached. A consequence of the droplet picture is that the law of large numbers is violated in the glassy phase. The implication is that when observed over a period of time various properties in glasses vary from region to region, which naturally explains the emergence of dynamical heterogeneity, and broken ergodicity. We conclude in Section VI with a discussion.
DENSITY FUNCTIONAL MODELS FOR THE GLASS TRANSITION
==================================================
A glass can be characterzied as an amorphous that is described in terms of statistically distributed density field, or as a dynamically frozen density fluctuation. This motivates using the number density, $n(\mathbf{x},t),$as the order parameter for the SGT where $(\mathbf{x},t)$ are space-time points. To be specific we will give a very explicit functional field theory for $n$, and dynamical equations for the density. Once this is done, we will discuss how to characterize the glass transition by the behavior of $n$ as a punitive glass transition is approached within these simple models. More general theories can, of course, be considered.
Density Functional Hamiltonian (DFH)
------------------------------------
The static model for the glass transition reviewed here was motivated by results for spin glass models without reflection symmetry [@Kirk87PRL]. The resulting phase transition is known as a random first order phase transition (RFOPT). Originally it was thought that this sort of unusual transition required the existence of quenched disorder as is the case in spin glass systems [@MezardBook], but not in systems undergoing a structural glass transition, where it is said that the disorder is self-generated. As mentioned in the Introduction this turned out not to the case, as we first showed in 1989 [@Kirk89JPhysA] (referred to from now on as KT). The theoretical ideas in KT form the basis of many subsequent developments in the SGT. Indeed, the generality of the conclusions in KT has subsequently been established by others as well [@Franz95PRL].
The model DFH is,$$\begin{aligned}
\beta \mathcal{H=}&-\mu \int d\bf{x}\delta n(\bf{x}+ \frac{1}{2}\int d \nonumber
\bf{x}_{1}d\bf{x}_{2}\delta n(\bf{x}_{1})\chi^{-1}(\bf{x}_{1}\bf{-x}_{2})\delta n(\bf{x}_{2})+ \frac{g_{3}}{3}\int d\bf{x[\delta n(\bf{x})]^{3}}\\
&+\frac{g_{4}}{4}\int d\bf{x}\bf{[}\delta n(\bf{x})\bf{]}^{4} -\int d\bf{x}H(\bf{x})\delta n(\bf{x})
$$where $\mu $ is the chemical potential, $H(\mathbf{x})$, is a small ($\rightarrow 0$), random, symmetry breaking external field whose role will become clear, and $g_{3}$ and $g_{4}$ are nonlinear coupling constants whose magnitudes are chosen such that a systematic self consistent expansion in density fluctuations is possible. The wavenumber($k$)-dependent $\chi (k)$ is related to the static structure factor and contains information on the short range order in the fluid.
Dynamical model
---------------
Since we are characterizing the SGT in terms of frozen density fluctuations we need a dynamical equation for space and time dependent density fluctuation, $\delta n(\mathbf{x,}t).$ We chose conservative relaxational dynamics, which reflects the fact that at a molecular length scale, density fluctuations are diffusive. The dynamical equation is,$$\frac{1}{\Gamma _{0}}\partial _{t}\delta n(\mathbf{x,}t)=\nabla ^{2}\frac{\delta (\beta \mathcal{H})}{\delta (\delta n(\mathbf{x,}t))}+\xi (\mathbf{x,}t) $$with $\xi (\mathbf{x,}t)$ the usual Gaussian noise term and $\Gamma _{0}$ a bare kinetic coefficient that sets the microscopic time scale. More general dynamics including ’mode coupling’ terms can also be considered.
STATIC THEORY OF THE GLASS TRANSITION
=====================================
The static theory of the glass transition starts with the DFH given by Eq. (1). We introduce two related key notions [@Kirk89JPhysA]. First, we imagine an order parameter description in terms of frozen density fluctuations. Since the glassy state is amorphous or aperiodic [@Singh95PRL] it is most naturally specified by a probability measure $\mathcal{P[}\delta n\mathcal{]}$. Secondly, we allow for a large number of macro states or pure states (at least on a given time scale). These states are characterized as follows. Denote a particular macroscopic state by the label s, with the density field in that state given by $n_{s}=n_{o}+\delta n_{s}.$ We denote the free energy of this state by $F_{s}$. Next compute $F_{s}$ from Eq. (1) by standard loop expansion techniques. We then allow for a possibly large number of statistically similar but different states by using a partition function defined by,$$Z=\sum_{s}\exp (-\beta F_{s})=\int D[\delta n]\Delta (\delta n)\exp (-\beta
F){\displaystyle\prod}\left( \frac{\delta \beta F}{\delta (\delta n(\mathbf{x}))}\right) . $$Here $\Delta (\delta n)=\left\vert \det \delta ^{2}F/\delta n^{2}\right\vert
$ normalizes the delta function in the equation given above. Equation (3) defines a probability measure $\mathcal{P}[\delta n]$ for the field $\delta n$. In usual phase transition problems $\mathcal{P[}\delta n\mathcal{]}$ is a delta function at the unique (or, more generally, at all globally symmetry-related) equilibrium state(s) of the system. However, in general, Eq. (4) is capable of describing a large number of symmetry-unrelated states that are statistically distributed.
We have used these equations to solve for the SGT as follows. We define the correlation function,$$Q(\mathbf{x}_{1},\mathbf{x}_{2})=<\delta n(\mathbf{x}_{1})\delta n(\mathbf{x}_{2})>, $$and an analogous density response function, $R(\mathbf{x}_{1}\mathbf{,x}_{2}).$ Here the angular brackets denote an average with weight $\mathcal{P[}\delta n\mathcal{]}$. We solve for these two correlation functions using a standard loop expansion. We assume that the field $H$ is a small Gaussian random field which statistically breaks the symmetry of the liquid phase to a glassy phase. The random field serves as an external coupling term conjugate to the Edwards-Anderson order parameter, $Q$, which characterizes the glassy phase. The variance of $H$ is set equal to zero at the end of the calculation. Carrying out a self-consistent expansion, using the fact that $g_{3}$ and $g_{4}$ are small, yields closed equations for $R$ and $Q.$ In wavenumber space these equations are,$$R(k)=n_{0}S(k)-Q(k) $$and$$Q(k)=R(k)\left( 2g_{3}^{2}\int_{k_{1}}Q(k-k_{1})Q(k_{1})\right) .
$$An identical nonlinear equation for the glassy order parameter is obtained in the next Section using a dynamical approach. The solution of the resulting equation has been discussed elsewhere ([@Kirk89JPhysA] and references therein). Here we note that nontrivial $Q$ solutions become possible below a temperature, $T_{A}$, and at this tempearture $Q$ jumps discontinuously to a nonzero value [@Kirk89JPhysA]. In giving Eqs.(5) and (6), we explicitly considered random solutions by requiring $V^{-1}\int d\mathbf{x}\delta n(\mathbf{x})\rightarrow 0$, where $V $ is the volume, even though the square (spatial) average of $\delta n$ is nonzero. These two conditions hence lead to the the moniker, random first order phase transition.
We note that the manipulations leading to Eqs.(5) and (6) are similar to those used for mean field spin glasses. Physically, the SGT theory is similar to mean field SG theories because; (1) The term in the brackets in Eq.(6) represents self-generated randomness, and (2) Mean field like approximations were used in deriving Eq.(6). Also, in deriving Eqs.(5) and (6) we used an infinitesimal Gaussian random field to set up the perturbation theory, introduce replicas, and use a replica symmetry breaking scheme that assumed only self-overlap of the metastable glassy states, as one does for SG systems with RFOPT. We stress that at the end of our calculation we set $H=0$, so that there is no ’quenched’ randomness. We note parenthetically that this method of locating a particular pure state has been used in the STG problem studied by replica methods [@Monasson95PRL]
We next discuss the physical significance of the transition temperature where the Eqs.(6) first has a nontrivial solution, $T_{A}$. We define two, in general, distinct free energies,$$F_{c}=\sum_{s}\exp (-\beta F_{s}) $$and$$\overline{F}=\frac{\sum_{s}F_{s}\exp (-\beta F_{s})}{\sum_{s}\exp (-\beta
F_{s})}. $$where $F_{c}$ is the usual canonical free energy while $\overline{F}$ is the component weighted free energy. Direct calculation in the glassy state gives $F_{c}$ =$F_{L}$, with $F_{L}$ the liquid state free energy that does not depend on $Q(k)$, and $\overline{F}>F_{c}$. The inequality $F_{c}\neq
\overline{F}$ can occur if and only if the states leading to Eqs.(5) and (6) are metastable and there are an infinite number of such states. Since $\overline{F}>F_{c}$ (see KT for additional discussions), it follows that at $T_{A}$ the liquid, within our mean-field like approximations, freezes into a metastable glass that is stabilized by an exponentially large solution degeneracy, i.e., there is a complexity, or state entropy associated with the states below $T_{A}$. Mathematically this entropy, $S_{s}$, is defined by, $$TS_{s}=\overline{F}-F_{c}. $$Note that within our approximate calculations $F_{c}$ is not a physically meaningful free energy because it contains an entropic term that is a measure of states not probed in the time scales in which our calculations are valid.
In Section V we argue that $T_{A}$, is not the glass transition temperature. Rather, beacuse of the appearance of many glassy states, it is the temperature below which activated dynamics play an important role and the dynamics become increasingly sluggish. There is a lower Kauzmann temperature, $T_{K}$, where the number of glassy states become nonextensive and where there is a true glass transition.
DYNAMICAL THEORY OF THE GLASS TRANSITION
========================================
The dynamical theory of the SGT starts with Eqs. (1) and (2). We consider the density time correlation function,$$C(\mathbf{k,}t)(2\pi )^{d}\delta (\mathbf{k+k}^{\prime })=<\delta n(\mathbf{k,}t)\delta n(\mathbf{k}^{\prime },0)>. $$The glassy state is defined by frozen density fluctuations or by a non-zero Edwards-Anderson order parameter [@Edwards75JPhysF],$$q(k)=q_{EA}(k)=\lim_{t\rightarrow \infty }C(\mathbf{k},t), $$and we assume that the glassy state is statistically homogeneous and isotropic. The fact that the glassy state has the same statistical properties as the liquid state is necessary to establish the connection between the static and dynamic approaches to the SGT.
Treating the nonlinear terms in Eq. (1) as small, the self-consistent one-loop approximation for $\widehat{C}(\mathbf{k,}\omega )$, the one sided Fourier transform of $C(\mathbf{k,}t),$is, $$\widehat{C}(\mathbf{k},\omega )=C(\mathbf{k},t=0)[-i\omega +\Gamma
_{R}(k,\omega )]^{-1}, $$with $C(\mathbf{k,}t=0)=n_{0}S(k)$ the static structure factor and $\Gamma
_{R}(k,\omega )$ a renormalized kinetic coefficient,$$\Gamma _{R}^{-1}(k,\omega )=\frac{1}{\Gamma _{0}k^{2}}+2g_{3}^{2}\int_{\mathbf{k}_{1}}\int_{0}^{\infty }dt\exp (-i\omega t)C(\mathbf{k-k}_{1}\mathbf{,}t)C(\mathbf{k}_{1},t). $$
Equations (12) and (13) are of a standard form of the equations derived in the so-called mode coupling theory of the glass transition, although the conceptual origin of Eq.(13) is different than in the original mode coupling theory. Equation (13) predicts a continuous slowing down of density fluctuations and a freezing at a temperature denoted by $T_{A}$ [@Gotze92RepProgPhys]. The equation of state for the frozen density fluctuations is obtained by inserting Eq. (11) in Eqs. (12 and 13) and obtaining,$$q(k)=n_{0}S(k)\frac{2g_{3}^{2}\int_{\mathbf{k}_{1}}q(\mathbf{k-k}_{1})q(\mathbf{k}_{1})}{1+2g_{3}^{2}\int_{\mathbf{k}_{1}}q(\mathbf{k-k}_{1})q(\mathbf{k}_{1})}. $$It is easy to show that Eq.(14) is identical to Eq. (6) if we identify $q(k)$ with $Q(k)$.
To summarize, this dynamical approach leads to a continuous freezing of density fluctuation and the frozen density fluctuations can be described by either a static theory, or by the dynamical approach. The static approach has the advantage that one can understand the freezing in terms of the number of states etc. The freezing temperature is denoted by $T_{A}$ because the self-consistent dynamical approach clearly ignore activated dynamics, which dominate transport at low temperatures. This is, in turn, consistent with the static approach where the freezing occurs into metastable glassy states, which can only be precisely defined in some sort of mean-field limit where activated dynamical processes do not occur. In the next Section the temperature region $T<T_{A}$ is considered using non-perturbative scaling and droplet ideas.
SCALING AND DROPLET CONSIDERATIONS
==================================
****
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The mean field theory based on precise calculations using a density functional Hamiltonian without quenched disorder shows that in the temperature range , $T_{A}<T<T_{K}$, the system is frozen in one of the exponentially large number of metastable states. Flow from one of these state (say $\alpha $) to another ($\gamma $) cannot be described within the MFT because $\alpha $ and $\beta $ are two disjoint ergodic states. In order to account for the observed non-Arrhenius slowing down of transport in glassy systems, which is often captured in terms of the Vogel-Fulcher equation, $$\tau (T)=\tau _{0}exp[{\frac{DT}{(T-T_{K})}}] $$Kirkpatrick, Thirumalai, and Wolynes [@Kirk89PRA] (KTW) introduced a new scaling theory based on entropic driving forces, which were argued to be relevant for transport [@Kirk87PRBa]. In Eq. (15) $\tau _{0}$ is a microscopic relaxation time at $T_{A}$ and $D$ is a positive constant. It follows from the KT theory described in Sections III and IV it that the emergence of multiple metastable minima below $T_{A}$ can be quantified in terms of the state entropy $S_{s}$, which is the difference between the canonical and component averaged free energies. In the droplet picture of activated transitions it is $S_{s}$, which is distinct from the configurational entropy in the Adam-Gibbs theory [@Adam65JCP] (for a detailed discussion see [@Kirk89PRA]), is the driving force for activated transition. Consider a region in a glassy state of size $L^{d}$ and let us estimate the probability of nucleating another glassy state inside $L^{d}$. The driving force for being able nucleate a glassy state one inside the other has to be entropic because the various glassy states have roughly the same free energy. Because there are a vast number of accessible (on a long time scale) glassy states such an entropically driven nucleation is possible. Within the droplet picture the driving force for nucleation is $\sim Ts_{s}L^{d}$ where $s_{s}$ is the state entropy per unit volume. The formation of domain within a domain is opposed by surface free energy cost, which can scale at most as $\sigma L^{(d-1)}$ where $\sigma $ is the surface tension (see below for a careful treatment of the scale-dependence of the surface tension between two distinct glassy domains). Balancing these two free energies gives the typical size of the glassy cluster $L\ast \sim \frac{\sigma T}{s_{s}}$ and the barrier to activated transport is $\Delta F\ast \sim (\frac{\sigma T}{s_{s}})^{(d-1)}$. We see that the entropic droplet theory naturally follows from considering the ramifications of the mean-field theory for finite dimensional systems.
The natural generalization of the MFT to describe activated transitions is to assume that flow below $T_{A}$ is triggered by creation and destruction of mosaic states within a large glassy cluster whose size $\xi \sim t^{-\nu }$ ($t=\frac{(T-T_{K})}{T_{K}}$) diverges at $T_{K}$. The entropic droplet picture, that was inspired by the MFT and fluctuation theory [@Kirk89PRA], has been used to show $\nu =\frac{2}{d}$. The time scale associated with these processes increases as the temperature decreases below $T_{A}$ eventually diverging at $T_{K}$ as described in Eq. (15). As long as the size of the glassy domain is large then the entropic driving forces are opposed by surface free energy cost that scales as $$F_{opposing}\approx \gamma L^{\theta } $$where $\theta \leq \frac{d}{2}$ if $\nu =\frac{2}{d}$. In terms of $s_{s}$ or equivalently $t$ (assuming $s_{s}$ varies linearly with $T$ close to $T_{K}$) the entropic driving force for activated transitions is $$F_{driving}\sim -At^{-(\nu d-1)}. $$Similiarly, $F_{opposing}\sim \gamma t^{-\nu \theta }$. The instability of the droplets at large length scale requires that the exponent characterizing the growth of the surface free energy be bounded by $\theta \leq \frac{(\nu
d-1)}{\nu }$.
These considerations can be used to describe the temperature-dependent relaxation time near $T_{K}$. If the typical size of the glassy cluster grows as $L\sim \xi \sim t^{-\nu }\sim t^{-\frac{2}{d}}$ then the typical free energy barrier behaves as $$\Delta F\ast \sim t^{-(\nu d-1)}\sim t^{-1} $$which immediately results in the Vogel-Fulcher law (Eq. (15)). Notice in order to obtain the Vogel-Fulcher equation the free energy cost opposing activated transport must scale as $$F_{opposing}\sim \gamma L^{\frac{d}{2}} $$A more refined treatment that relies on a generalization of Villain’s conjectures [@Villain85JPhys] for the Random Field Ising Model indeed shows that a scale-dependent surface tension which vanishes on length scales greater than $\xi $ shows that Eq. (19) is indeed obeyed in the vicinity of $T_{K}$ [@Kirk89PRA]. Although the KTW scaling picture offers a consistent picture of activated transport that is wholly inspired by the precise theory described in [@Kirk89JPhysA], it still remains heuristic.
****
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The precise connection between the static and dynamical description of the glass transition made using the DFT (see Sections III and IV), which requires the existence of an exponentially large number of metastable states below $T_A$, has been demonstrated by taking long time limits of the order parameter $C(k,t)$. The long time (or more precisely on $t \ge \tau_c$ where $\tau_c$ is a correlation time) limit of $C(k,t)$ is zero in a liquid but persists in glasses. In other words, only when the system develops in time is there an obvious difference between liquids and glasses. In a liquid a given molecule diffuses arbitrarily far from where it started as $t$ grows whereas it is localized in space to a small region. The DFT calculations show that the dynamical treatment and the static treatments give rise to identical physical picture that is manifested by the appearance of a non-zero glassy state Edwards-Anderson order parameter. Thus, within the framework of RFOPT it is only by examining the link between spatial and time correlations can the distinction between liquids and glasses be made.
As argued in the previous section it is fruitful to picture a glass as being partitioned into mosaic states whose characteristic temperature-dependent size is denoted by $\xi_{i}(T)$ where for the sake of generality we consider variations in the sizes of the glassy clusters. Within the $\xi_{i}(T)$ structural rearrangements can be rapid but global relaxation requires activated transitions, which involves nucleating new domains. Slow structural fluctuations in glassy occurs because of entropic driving forces that enable creation and formation of new glassy clusters by fluctuation effects. As the degree of supercooling increases not only does $\xi_{i}(T)$ grow but also the relaxation time associated with particles that cross domain boundaries exceeds the observation time scale ($\tau_{obs})$, thus leading to broken ergodicity. These physical considerations that are embedded in the droplet picture of the RFOPT also lead to violation of law of large numbers as the STG occurs, which we illustrate by using the following arguments. Just as in the scaling theory, we picture the glassy phase as being partitioned into regions with size $\approx \xi_i(T)$ with $(\frac{\xi_i(T)}{a})^3$ being sufficiently large that meaningful average over the number of particle within $\xi_{i}(T)$ can be carried out. In the liquid phase ($T > T_A$) the statistical properties of the liquid (for example the distribution of energies of the particles in a glass forming system $P(\epsilon;
t|\xi_{i}(T))$ in the sub sample is [*independent of $i$*]{} and will coincide with that of the entire sample provided $\xi_{i}(T)$ is large enough and $t
> \tau_c < \tau_{obs}$. This is a consequence of the law of large numbers. In contrast, in the glassy phase each $\xi_i(T)$, which in the MFT corresponds roughly to one of the frozen metastable states, is distinct, and consequently each $P(\epsilon; t|\xi_i(T))$ can be distinct and will depend on $i$. Thus, no single sub sample can characterize the distribution of energies of the entire sample. In other words, in the glassy phase the law of large numbers is violated, and there are sub sample to sub sample fluctuations. Only by examining the *entire sample* on $\tau_{obs}
> \tau(T)$ is ergodicity restored. We see that the so called dynamical heterogeneity, which has been a characteristic of glass forming systems [@Sillescu99JNCS; @Glotzer00JNCS; @Donati99PRL] is seen to be a consequence of the emergence of glassy clusters with the characteristic sizes $\xi(T)$. Because of the variations in both equilibrium and relaxation properties from sub sample to sub sample a glassy phase is inherently heterogeneous.
The preceding arguments were illustrated using simulations of soft-sphere binary mixtures in which the sample was divided into a number of sub samples [@Thirumalai89PRA]. In the liquid phase $P(\epsilon ;t|\xi _{i}(T))$ coincides with the entire sample for all $i$ as long as $t>\tau _{c}$. In contrast there are considerable variations in $P(\epsilon ;t|\xi _{i}(T))$ and are fragments of the entire sample. Thus, the dynamical heterogeneity is really a consequence of the law of large numbers, and very much supports the droplet scenario for activated transition set within the RFOPT context. A corollary of the violation of the law of large numbers is that particles of a specific type (say a large particle in a binary LJ mixture) belonging to two distinct subsamples are not “statistically equivalent” even when $\tau
_{obs} \gg \tau _{c}$. This is in contrast to the liquid phase where on $t\approx \tau _{c}$ all particles of a given type are statistically equivalent. Such a loss in statistical symmetry in the SGT is a time averaged property and can only be inferred by examining the time evolution of the system. The arguments and simulations reported by us [@Thirumalai89PRA] clearly showed that dynamical heterogeneity and broken ergodicity naturally follow from violation of the law of large numbers.
Another consequence of the statistical inequivalence of any two subsamples (whose sizes are on the order of a typical $\xi(T)$) in the glassy phase is that ergodicity is broken in the SGT. To illustrate the concept of ergodicity breaking we introduce a measure referred to as the energy metric, $d(t)$, which is defined as $$Nd(t) = \sum_{i=1}^{N} [\epsilon_i(t|R_{\alpha}(t)) - \epsilon_i(t|R_{\beta}(t)]^2$$ where $\epsilon_i(t|R_{\alpha}(t)) = \frac{1}{t} \int_0^{t} ds E_i(s|R_{\alpha}(t))$. Here, $E_i(s|R_{\alpha}(t))$ is the energy of the $i^{th}$ particle at time $s$ and $R_{\alpha}(t)$ refers to a set of positions of the particles whose initial condition is labeled $\alpha$. Similarly, $\epsilon_i(t|R_{\beta}(t))$ is the corresponding quantity for the trajectory $\beta$. If the system is ergodic on the time scale $\tau_{obs}$ then $d(t)$ vanishes as $t \rightarrow \tau_{obs}$, and therefore $\epsilon_i(\tau_{obs}|R_{\beta}(\tau_{obs})) = \epsilon_i(t|R_{\beta}(t))$ [*independent*]{} of $alpha$ or $\beta$. This is the situation that pertains to the liquid phase. However, if ergodicity is broken, as is expected at the STG, $d(t) \sim C$ ($C$ is a constant) suggesting that the two initial states do not mix on the time scale $\tau_{obs}$. As argued above it is the development in time rather than any equal time correlation functions that distinguishes a glass from a liquid. It can be shown, using scaling-type arguments, that $\frac{d(0)}{d(t)} \approx D_E t$ where the “diffusion” constant $D_E$ is not unrelated to relaxation time set by the shear viscosity [@Thirumalai93PRE]. Thus, $N \frac{d(0)}{d(t)}$, which is extensive in $N$ and $\tau_{obs}$ in the liquid phase, remains only extensive in $N$ in the glassy phase because $\tau_(T) \gg \tau_{obs}$. We demonstrated these ideas using molecular dynamics simulations of two component softly repelling spheres as well Lennard-Jones mixtures with additive diameters chosen to avoid crystallization. At temperatures that are greater than $T_A$ we showed that $\frac{d(0)}{d(t)}$ grows linearly as $t$ increases whereas it saturates in the glassy phase due to the inability to explore distinct regions of the configuration space. The illustrations summarized here, which have been demonstrated by others using different language, follow directly from the physical picture that in the SGT the glass forming system is frozen into one of many disjoint ergodic states that do not mix (or become statistically equivalent) on $\tau_{obs}$.
DISCUSSION
==========
The fundamental goal of any theory of glass forming materials should be to explain both the dramatic viscosity increase and thermodynamic anamolies starting from a theory appropriate for liquids. At the laboratory glass transition temperature $T_g$ the relaxation times far exceed the observation times and the heat capacity has a discontinuity suggesting that providing a kinetic description alone is insufficient. In addition, the goal of any theory of glasses must ultimately be described using quantities that can be measured in experiments. This perspective presents a coherent theory that was advanced by us over twenty years ago, and which was guided by the goals outlined above. The theory and its implications for activated transitions, violation of law of large numbers and the related dynamical heterogeneity, and ergodicity breaking treats both the dynamical and static properties of glasses on equal footing. The major conceptual basis, which was discovered using a density functional description of glasses without quenched disorder, is that at $T \le T_A$ the system is frozen into one of many metastable states. In practical terms $T_A$ ($>T_g$) corresponds to a temperature at which $\eta \sim$ (1-10) poise. Such states are described by frozen density fluctuations from which emerges an Edwards-Anderson order parameter can be obtained from a purely static or a dynamical theory [@Kirk89JPhysA].
There are immediate consequences of the RFOPT of glass transition when applied to finite dimensions. Unlike in the mean field picture the metastable states are not disjoint and transport becomes possible on time scales comparable to $\tau(T)$, which of course, becomes exceedingly long as $T$ decreases. In the temperature range $T_K < T <T_A$ it is fruitful to think of glasses as being composed of a large number of mosaic states on scales on the order of $\xi(T)$. From this picture we draw several significant conclusions.
1. Transport in the temperature range, $T_K < T <T_A$, is driven by activated processes the driving force for which are entropic in nature. Because the entropy vanishes linearly near $T_K$ it follows from our picture that the size of the domains must grow as $\xi \sim (T-T_K)^{-{\frac{2}{d}}}$. The droplet theory [@Kirk89PRA], constructed by balancing the entropic driving force and the opposing cost of creating an interface between two glassy states readily leads to the Vogel-Fulcher equation (Eq. 15. It is useful to comment on the typical values of $\xi(T_g)$ found in practice, Computer simulations of LJ mixtures [@Mountain92PRA] and colloidal glasses composed of mixtures of micron size charged particles [@Rosenberg89JPhysCondMatt] conclude that $t \sim 0.6$ which was used to show that $\xi(T_g) \approx 3 \sigma$ where $\sigma$ is the particle diameter. From the extracted values of $t \sim 0.1$ in several experiments [@Mohanty90JCP] we predict using the $t^{-\frac{2}{d}}$ scaling that $\xi(T_g) \approx 10 \sigma$ [@Berthier05Science]. On these length scales there are in excess of fifty particles so that the activation barrier for transport is large enough that considerations from the scaling theory are appropriate.
2. The partitioning of a glassy state into mosaic states with growing domain size suggests that law of large numbers must be violated, especially at temperatures less than $T_A$ [@Thirumalai89PRA]. This implies that, when observed over a period of time that exceeds $\tau_c$ but is comparable to $\tau_{obs}$, any two mosaic states are statistically inequivalent. As a consequence, glass is dynamically heterogeneous which implies that the statistical properties (averaged over a period of time greater than $\tau_C$) vary from one mosaic state to another. This is not the case in a liquid. These expectations are borne out in computer simulations. The conceptual basis of the origin of dynamic heterogeneity is intimately linked to the violation of law of large numbers [@Thirumalai89PRA].
3. Because of the statistical inequivalence of mosaic states on time scales comparable to $\tau_{obs}$ ergodicity is broken in the STG. This is manifested in the ergodic measure, which is extensive in $\tau_{obs}$ in the liquid phase but becomes essentially independent of $\tau_{obs}$ in the glassy phase [@Thirumalai89PRA].
[**ACKNOWLEDGMENTS**]{}: We are grateful to grants from the National Science Foundation (DMR09-01907 and CHE09-14033) for support of this work.
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abstract: |
In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion. Owners can take cash out of the surplus in the form of “dividends”, subject to transaction costs. However, if the surplus hits 0 then ruin occurs and the business cannot operate any more.
We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) extraordinary dividend payments that can be made immediately at any time (that is, the dividend decision time space is continuous and matches that of the surplus process). Both types of dividends attract proportional transaction costs, and extraordinary distributions also attracts fixed transaction costs, a realistic feature. A dividend strategy that involves both types of distributions (periodic and extraordinary) is qualified as “hybrid”.
We determine which strategies (either periodic, immediate, or hybrid) are optimal, that is, we show which are the strategies that maximise the expected present value of dividends paid until ruin, net of transaction costs. Sometimes, a liquidation strategy (which pays out all monies and stops the process) is optimal. Which strategy is optimal depends on the profitability of the business, and the level of (proportional and fixed) transaction costs. Results are illustrated.
address:
- 'Centre for Actuarial Studies, Department of Economics, University of Melbourne VIC 3010, Australia'
- 'School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia'
author:
- Benjamin Avanzi
- Hayden Lau
- Bernard Wong
bibliography:
- 'libraries.bib'
title: On the optimality of joint periodic and extraordinary dividend strategies
---
Risk analysis ,Dividend decision processes ,Control ,Affine transaction costs
MSC classes: 93E20 ,91G70 ,62P05 ,91B30
Introduction {#S_intro}
============
The literature on risk processes and their optimal control is rich [see, e.g. @AlTh09; @OkSu10 for reviews]. Such processes consider the surplus (or equity) of a risky business. A risky but profitable business will see cash accumulate (on average). They typically would not let their surplus grow to infinity, but to guard against the downside risks, they would retain some cash earnings in order to prevent bankruptcy or financial distress. In this paper, we model such surplus of cash with a stochastic process, and money distributed to shareholders will be interpreted as ‘dividends’; see also @AvTuWo16c for a discussion of such surplus models from a corporate finance perspective. The question then is to determine what the optimal way of distributing surplus is, that is, what the optimal (so-called) ‘dividend’ strategy is. Note that this problem is equivalent to that of determining what the optimal level of retained cash earnings is (noting that inflows come from the business dynamics, and all outflows are labelled as ‘dividends’), but is formulated in function of what owners can *control* (the ‘dividends’).
The natural and usual objective of this optimisation problem is to maximise the expected present value of dividends paid until ruin (which occurs as soon as the surplus becomes negative). Additional historical notes and discussion of dividends in that context can be found in @Ava09. This objective is also a good criterion of “stability” for the company, as it balances profitability (more dividends but earlier ruin) with safety (less dividends but delayed or even absence of ruin); see, e.g. @Buh70. Nevertheless, quantities such as the finite time ruin probability have also been considered [e.g. @DiRo10; @DiKaZh14], and the maximisation can be achieved on more sophisticated objectives, such as involving utility functions [e.g. @BaEg10; @BaJa15].
The recent decade has focused a lot on more *realistic* formulations for the dividends [see @AvTuWo16c for a detailed discussion of what we mean by ‘realistic’]. One of the axes of development recognises that whilst surplus models are continuous, in real life often delays occur [see, e.g., @ChWo17 who consider dividend payments with implementation delays], and also dividend decisions are usually made at periodic intervals [see, e.g., @AlGeSh11].
In this spirit, we consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) immediate dividend payments that can be made at any time (that is, the dividend decision time space is continuous and matches that of the surplus process). This matches the behaviour of companies in real life, as most established firms would pay dividends regularly. If they feel the need to distribute more, then they would clearly label those extra payments as ‘extraordinary’ (and sometimes also do it in a different way, such as with share buy-backs, which is not in contradiction with our framework). One can find real life examples [e.g., @Woodside13; @Wesfarmer14], and was further explained by, for instance, @Morningstar14: “From time to time, companies pay out special dividends when they have had an extraordinarily good period of profitability. These dividends fall outside the scope of the “normal” half-year or full-year result.” This possibly is to avoid signalling the fact that those extra payments should be expected to continue in the future. Furthermore, it does make sense that those extra distributions carry heavier costs than the regular ones (actual costs, but also undesirable signalling costs such as we just explained). We will hence penalise them with heavier fixed transaction costs.
Literature on “periodic” dividends is relatively new, but attracted a lot of attention. @AlChTh11a first proposed to use an erlangisation technique [@AsAvUs02] to approximate the time between dividend decision times. The idea of the erlangisation technique is to set parameters such that the time between decisions is Erlang$(n/\gamma,n)$ distributed (hereafter denoted “Erlang$(n)$”) such that the time between decisions becomes deterministic with mean $1/\gamma$ as $n$ goes to infinity. This convergence was illustrated in the dual model setting (with surplus as a spectrally positive compound Poisson process) in @AvChWoWo13. @AvTuWo14 confirmed that a periodic barrier strategy is optimal in dual model when the inter-dividend decision time is a simpler Erlang$(1)$ variable. @PeYa16 extended those results by considering general spectrally positive Lévy processes as the underlying surplus model. [-@AvTuWo18] studied the optimal problem when the inter-dividend times is a Erlang($n$) random variable. The authors provided a verifying method for Brownian setting and demonstrated the optimality of a periodic barrier strategy when $n=2$. In all those cases, the type of the optimal periodic dividend strategy is that of a barrier strategy, mirroring the analogous result for dividend decisions that can occur at any time [see @BaKyYa13]. Optimal strategies with spectrally negative Lévy processes are considered in @NoPeYaYa17. Of closest relevance to this paper are consideration of optimal periodic (only) dividend strategies with fixed transaction costs, developed in @AvLaWo20d [@AvLaWo20c] for spectrally positive and negative Lévy processes, respectively.
@AvTuWo16, in a dual model framework and with both types of dividends being admissible, show that when transaction costs are moderately cheaper for periodic dividends, then both types of dividends can be optimal, leading to an optimal *hybrid* dividend strategy. These results are extended to spectrally positive Lévy processes by @PeYa18. However, those papers consider proportional transaction costs only, and in reality fixed costs are likely to be the ones that truly differentiate the cost of “periodic” versus “immediate” dividends.
In this paper we extend results on “hybrid” dividend strategies by introducing fixed transaction costs on periodic dividends, which results in a comprehensive, more realistic treatment of optimal hybrid strategies. Furthermore, results are materially different, richer and more realistic as explained below. Furthermore, the cash flow of the company is modeled by a diffusion process, which leads to transparent and many explicit results, and is sufficient to get insights about the optimal strategies.
When the company is profitable, an optimal strategy is a hybrid $(a_p,a_c,b)$ strategy which (1) pays non-regular dividends only when the surplus is too high (2) pays regular (periodic) dividends when the surplus is moderate. This strategy has some desired properties. Namely, the regular dividends are either zero or bounded. When the regular dividend is zero, either the company is at risk of bankruptcy or a recent special dividend has been paid. In either case, such behaviour is reasonable. When the company is non-profitable, the model has a different (and no less interesting) interpretation. The main results of the paper are summarised in Section \[S\_map\], after our notation is introduced.
This paper is organised as follows. Section \[S.the.model\] introduces our mathematical framework. Section \[S.Verification\] proposes a set of sufficient conditions for a strategy to be optimal, regardless of whether the business is profitable. From there until Section \[S.Optimality\], it is assumed that when the business is profitable. As an application of the results developed in Section \[S.Verification\], an optimal strategy is formulated when the proportional cost is higher than a certain threshold. Section \[S.hybridG\] introduces the class of hybrid $(a_p,a_c,b)$ strategy and calculates the value function of a general hybrid $(a_p,a_c,b)$ strategy. Section \[S.hybridG\] shows constructively that our candidate strategy exists among the class of hybrid $(a_p,a_c,b)$ strategy, when the proportional cost is low (lower than a certain threshold). Section \[S.Optimality\] proves that our candidate strategy is optimal, when the proportional cost is low. Section \[S.mu.neg\] studies the remaining case when the business is strictly non-profitable. Section \[S.Convergence\] discusses how the different optimal strategies “connect” (i.e., across the Table in Section \[S\_map\]). Finally, \[S.Numerical\] presents numerical illustrations, and Section \[S.conclusion\] concludes.
The model {#S.the.model}
==========
Surplus model before dividends
------------------------------
We define the surplus process $X=\{X(t);t\geq 0\}$ under the family of laws $(\mathbb{P}_x;x\in\mathbb{R})$ to be a diffusion process that starts at $x\geq0$, i.e. $$\label{Def.Diffusion}
X(t)=x+\mu t+\sigma W(t),$$ where $W=\{W(t);t\geq 0\}$ is a standard Brownian motion. This surplus process is to be interpreted as the excess, discretionary equity available to the company to pay dividends. It is assumed that it is sufficiently liquid to pay dividends immediately when it is so decided.
We denote the expected profit per unit of time of the business as ${\ensuremath{\mathbb{E}}}[X(t+1)-X(t)]:=\mu$. Unless stated otherwise, we assume that $$\label{Ass.mu.positive}
\mu \geq 0,$$ which means that the business is profitable. The opposite case will be studied in Section \[S.mu.neg\], and the connection of the optimal strategies between the cases $\mu$ greater than, equal to, and small than $0$ is conducted in Section \[S.VaryingMu\] (continuity of the barriers).
The introduction of dividends
-----------------------------
In this paper, a dividend strategy is comprised of two components. Dividends can be paid at any time, but there are periodic opportunities to pay dividends at lower transaction costs. A *dividend strategy* must hence determine how much periodic dividends to pay and how much “immediate” (extraordinary) dividends to pay and when. For a dividend strategy $\pi$, we denote the accumulated periodic “regular” dividend process as $D^\pi_p=\{D^\pi_p(t);t\geq 0\}$ and the accumulated non-periodic “immediate” dividend process as $D^\pi_c=\{D^\pi_c(t);t\geq 0\}$. The strategy $\pi$ is then specified through $(D^\pi_p,D^\pi_c)$, and the accumulated total dividend process under strategy $\pi$ is denoted as $D^\pi=\{D^\pi(t);t\geq 0\}$. This means $$D^\pi(t)=D^\pi_p(t)+D^\pi_c(t),~t\geq 0.$$ Note that the subscripts $p$ and $c$ refer to the timing of the dividend decision process, be it ‘periodic’ or ‘continuous’, in line with previous literature.
We need to clarify mathematically how the “regular”, or periodic payment times are defined. Define $N_\gamma=\{N_\gamma(t);t\geq 0\}$ as a Poisson process (independent of $W$) with rate $E[N_\gamma(1)]=\gamma>0$, which serves as our periodic *dividend decision times*. In other words, periodic dividends can only be paid when $N_\gamma$ has increments. Such times are denoted as ${\mathbb{T}}=\{T_i;i\in\mathbb{N}\}$ with $$\label{E_Ti}
T_i=\inf\{t\geq 0: N_\gamma(t)=i\}.$$ This implies that [$T_1$]{} and $T_{i+1}-T_i$[, $i\in\mathbb{N}$]{}, are i.i.d. exponential random variable with mean $1/\gamma$, for all $i\in\mathbb{N}$.
A Markovian *stationary* strategy is a strategy where the control at time $t$ is a deterministic function of $X^\pi(t-)$ known at time $0-$ which maps the surplus and its characteristics into a dividend payment, i.e. $(\Delta D_p^\pi(t),\Delta D_c^\pi(t))=(f_p(X^\pi(t-))1_{\{t\in \mathbb{T}\}},f_c(X^\pi(t-))1_{\{t\notin \mathbb{T}\}})$ for a given function $f=(f_p,f_c)$. For such a strategy $\pi$, if $D^\pi_c(t)\equiv 0$, we call it a (pure) periodic strategy (with regular payments only). If $D^\pi_p(t)\equiv 0$, we call it a (pure) continuous strategy (with immediate payments only). Otherwise, we refer it as a hybrid strategy, as there is a non-zero probability that both components are present.
Note that if dividends can only be paid after every $n$ increment then the time between dividend decision times is Erlang distributed with shape parameter $n$ and rate parameter [$n\gamma$]{} (the sum of $n$ independent exponential[($n\gamma$)]{} random variables). This random variable can have arbitrarily small variance for appropriate choice of parameters. This is what led to the so-called “Erlangisation” technique as discussed in @AsAvUs02 [@AlChTh11a]. Indeed, letting the parameter $n$ increase to infinity leads to the variance of the Erlang($n$) random variable to vanish, which means arbitrary large $n$ can approximate deterministic quantities.
This motivates model setups with ‘simple’ Poissonian distribution strategies (whereby inter-dividend decision times are exponentially distributed), such as in this paper. These are an important first step to solving the more general Erlang with $n\ge2$ case. Showing optimality for $n\ge 2$ is surprisingly difficult, but not impossible; see @AvTuWo18.
Now, the surplus process after the dividend payments is $X^\pi=\{X^\pi(t);t\geq 0\}$ with $$X^\pi(t)=X(t)-D^\pi(t).$$ We define $\tau^\pi$ to be the ruin time of the process $X^\pi$, i.e. $$\tau^\pi=\inf\{t\geq 0:X^\pi(t)<0\},$$ that is, the company must stop operations as soon as its surplus hits zero, and no further dividends will be paid.
By defining the filtration generated by the process $(X,N_\gamma)$ by $\mathbb{F}=\{\mathscr{F}_t:t\geq 0\}$, we say a (hybrid) dividend strategy $\pi:=\{(D^\pi_p(t),D^\pi_c(t));t\geq 0\}$ is admissible if both $D^\pi_p$ and $D^\pi_c$ are non-decreasing, right continuous and $\mathbb{F}$-adapted process where the sample path of the process $D^\pi_c$ is an increasing step function in time (as a fixed cost will be incurred at each payment), and where the cumulative amount of periodic dividends $D^\pi_p$ admits the form $$D^\pi_p(t)=\int_{[0,t]}\nu^\pi(s)dN_\gamma(s),~t\geq 0.$$ Furthermore, note by definition the sample paths of $X$ are continuous ($X(t)=X(t-)$) and hence we require $$\Delta D^\pi(t)\leq X^\pi(t-),\quad t\leq \tau^\pi$$ that is, the dividend paid at $T_i$—denoted $\xi^\pi_i:=\nu^\pi(T_i)$—cannot exceed the current value of the surplus. Denote this set of admissible strategies $\Pi$.
The expected present value of dividends until ruin
--------------------------------------------------
To measure the performance of the strategies, we will focus on the expected present value of dividends until ruin $$\label{E_EPVD}
V_{1-\beta,\chi}(x;\pi)=V(x;\pi):={\mathbb{E}_x}\int_{0}^{\tau^\pi}e^{-\delta t} \Big(dD^\pi_p(t)+(\beta dD^\pi_c(t)-\chi)1_{\{\Delta D^\pi_c(t)>0\}}\Big),$$ where ${\mathbb{E}_x}[\cdot]:=\mathbb{E}[\cdot|X(0)=x]$ is the mathematical expectation under the law $\mathbb{P}_x$ (for each $x\in\mathbb{R}$), and where $\delta>0$ is a time-preference parameter (or discount factor). Furthermore, non-periodic “immediate” dividend payments of amount $\xi>0$ incur a transaction costs $(1-\beta)\xi+\chi$. In other words, there is a proportional transaction rate of $1-\beta$, and fixed transaction costs of $\chi$.
We seek to maximise the expected present value of dividends, which means that we will look for an optimal strategy $\pi^*\in\Pi$ such that $$\label{Def.Optimal}
V(x;\pi^*)=\sup_{\pi\in\Pi}V(x;\pi):=v(x)=v_{1-\beta,\chi}(x),\quad x\geq 0.$$ Because the process is ruined immediately when it reaches $0$, we have $$\label{E_V0}
V(0;\pi)=0 \quad \text{for}\quad \pi\in\Pi.$$ Note that we will also write $\mathbb{P}$ and $\mathbb{E}$ for $\mathbb{P}_0$ and $\mathbb{E}_0$ respectively.
\[Remark.Rational\] An optimal strategy should demonstrate the following 2 rational behaviours:
1. The non-periodic dividend payment at time $t$, $\Delta D^\pi(t)$, is either $0$ or strictly greater than ${\chi}/{\beta}$. This is because any strategy that pays a non-periodic dividend less that ${\chi}/{\beta}$ does not contribute positively to the value function and therefore has at most the same value function as the same strategy without negative contributions.
2. At periodic dividend times $t=T_i$ for some $i\in\mathbb{N}$, we do not pay non-periodic dividends. Otherwise, a higher transaction cost is paid, yielding at most the same value function.
\[R\_betacp\] Note that in periodic dividends do not attract any transaction costs. With respect to *proportional* transaction costs this is without of loss of generality, as long as proportional transaction costs on periodic dividends (say, $1-\beta_p$) are smaller than that on immediate dividends (say, $1-\beta_c$), which is what you would expect in practice as discussed earlier. In this case, would become $$\begin{aligned}
V(x;\pi)&=&{\mathbb{E}_x}\int_{0}^{\tau^\pi}e^{-\delta t} \Big(\beta_p dD^\pi_p(t)+(\beta_cdD^\pi_c(t)-\chi)1_{\{\Delta D^\pi_c(t)>0\}}\Big)\label{prob.1} \\
&\equiv& \beta_p{\mathbb{E}_x}\int_{0}^{\tau^\pi}e^{-\delta t} \Big(dD^\pi_p(t)+(\beta dD^\pi_c(t)-\chi/\beta_p)1_{\{\Delta D^\pi_c(t)>0\}}\Big) \quad \text{with }\beta=\frac{\beta_c}{\beta_p}\le1.\label{prob.2}
\end{aligned}$$ That is, the objective is simply scaled by a constant ($\beta_p$), which will not affect the generality of our set-up. In other words, an optimal strategy in problem is optimal in problem . However, note that the fixed transaction cost amount $\chi$ needs to be appropriately scaled if one wants to obtain accurate numerical valued for one problem from the other.
On the other hand, introduction of fixed transaction costs $\chi_p$ on periodic dividends would likely alter the form of the optimal dividend strategy fundamentally. We expect that the optimal periodic barrier would be split into a higher trigger barrier, and lower dividend payment barrier, as is often the case in classical impulse cases (because of the reason explained under item 1 in Remark \[Remark.Rational\]). Furthermore, we postulate that ascertaining which type of dividends attracts higher transaction costs *on average* or *in an expected sense* would be critical in determining the optimal dividend strategy. We believe the optimal dividend strategy would depend on some sort of ‘expected’ overall transaction costs for each type, which is not trivial to determine as the number and timing of dividends are random in both cases, and do not match. That being said, if one assume that both proportional *and* fixed transaction costs are lower on regular dividends (as opposed to immediate dividends), then extension of the current paper should be relatively straightforward.
Definition of relevant dividend strategies
------------------------------------------
In this section, we define all dividend strategies that we will refer to in this paper. Note that they are all Markovian *stationary* strategies as defined above just after .
\[Def.periodic.b\] A periodic $b$ strategy, denoted $\pi_b$, is a periodic dividend strategy which pays a dividend $$\Delta D^\pi_p(t)=(X^{\pi_b}(T_i-)-b)1_{(X^\pi(T_i-)\geq b)},\quad \Delta D^\pi_c(t)\equiv 0.$$ at time $T_i$, as long as ruin has not occurred yet, that is, for all $T_i\le\tau^{\pi_b}$, $i\in\mathbb{N}$.
We now define the class of strategies that we prove optimal in some cases later in the paper.
\[Def.hybrid.apacb\] A hybrid $(a_p,a_c,b)$ strategy with $0\leq a_p\leq a_c\leq b$, denoted as $\pi_{a_p,a_c,b}$, is a strategy which
1. pays (before ruin) periodic dividend that brings the surplus down to $a_p$ whenever the (controlled) surplus $X^{\pi_{a_p,a_c,b}}$ is above or equal to [$a_p$]{} right before the dividend payment times,
2. pays (before ruin) an immediate dividend that brings the surplus down to $a_c$ whenever the surplus $X^{\pi_{a_p,a_c,b}}$ is above or equal to $b$ outside the periodic dividend times.
In mathematical notation, it means $$\begin{cases}
\Delta D^\pi_p(T_i)=(X^\pi(T_i-)-a_p)1_{(X^\pi(T_i-)\geq b)}1_{\{T_i\leq \tau^\pi\}}\\
\Delta D^\pi_c(t)=(X^\pi(t-)-a_c)1_{(X^\pi(t-)\geq b)}1_{(t\neq T_i)}1_{\{t\leq \tau^\pi\}}
\end{cases},$$ with $\pi=\pi_{a_p,a_c,b}$.
Figure \[fig.apacb\] illustrates the strategy described in Definition \[Def.hybrid.apacb\]. It charts a typical sample path when a hybrid $(a_p,a_c,b)$ is applied, where $a_p=1$, $a_c=2$, and $b=4$. The dotted vertical lines indicate periodic dividend decision times $T_i$’s. From the graph, we see that there is an immediate payment just before $T_1$, of amount $b-a_c=2$ (before transaction costs). On the other hand, all $T_i$’s in the graph trigger periodic payments. Note that Definition \[Def.hybrid.apacb\] indicates that $a_p\leq a_c\leq b$, but in fact Remark \[Remark.Rational\] implies that only the cases $b>a_c+{\chi}/{\beta}$ (and hence $b\geq a_p+{\chi}/{\beta}$ since $a_c>a_p$) make sense in order to avoid negative contribution to the value function. Note also that the solid vertical lines are not part of the processes. They are displayed “artificially” to illustrate the “jump” of the processes.
![Illustrations of the main optimal strategies[]{data-label="fig.strategies"}](apacb.pdf "fig:"){width="100.00000%"} \[fig.apacb\]
![Illustrations of the main optimal strategies[]{data-label="fig.strategies"}](b1b2.pdf "fig:"){width="100.00000%"} \[fig.b1b2\]
\[def.Liq\] A liquidation $(b_1,b_2)$ strategy, characterised by 2 parameters $0<b_1<b_2\leq \infty$, denoted as $\pi_{b_1,b_2}$, is the strategy that
1. pays (before ruin) non-periodic dividend $X(\theta)$ [(surplus just before ruin caused by this final dividend)]{}, where $\theta=\inf\{t\geq 0: X(t)\in(b_1,b_2)\}$, the first time the surplus is within the open interval $(b_1,b_2)$;
2. pays (before ruin) periodic dividend of size $X^\pi(T_1-)$ [(surplus just before ruin caused by this dividend)]{} when $X^\pi(T_1-)\leq b_1$ or $X^\pi(T_1-)\geq b_2$, where $\pi=\pi_{b_1,b_2}$.
In mathematical notation, it means $$\begin{cases}
\Delta D^\pi_c(t)=X(\theta)1_{(t=\theta< T_1)}1_{\{t\leq \tau^\pi\}},\quad \theta=\inf\{t\geq 0: X(t)\in(b_1,b_2)\}, \\
\Delta D^\pi_p(T_1)=X^\pi(T_1-)1_{\{T_1\leq \tau^\pi\}},
\end{cases}$$ with $\pi=\pi_{b_1,b_2}$.
The class of liquidation strategies as defined in Definition \[def.Liq\] will be sometimes optimal when $\mu<0$, and we distinguish two cases here for notation purposes:
1. Liquidation $(b_1,b_2)$ strategy, with $b_2<\infty$.
2. Liquidation $(b,\infty)$ strategy, denoted as $\pi_{b,\infty}$, which is the liquidation $(b,b_2)$ strategy with $b_2=\infty$.
[Figure \[fig.b1b2\] illustrates the strategy of Definition \[def.Liq\]. It shows different sample paths from when $X(0)=2.5$ (upper part), and $X(0)=0.5$ (lower part), with $b_1=1$ and $b_2=2$. When $X(0)=2.5$, the path in grey represents the case when the first dividend decision time ($\widetilde{T}_1$ in the figure) comes before the surplus reaches $b_2$. As a result, the company is liquidating at $\widetilde{T}_1$ at that surplus level. With the other scenario (represented by the path in black), the first dividend decision time (not shown in the figure) comes after the surplus hits $b_2$ and therefore it immediately liquidates at that time, and a surplus of $b_2=2$ (before transaction costs) is distributed. Similar behaviour can be seen when $X(0)=0.5$. When to liquidate depends on whether the surplus touches $b_1$ first (black path) or $T_1$ comes first (grey path), conditioning on survival. Otherwise, the company is ruined (the lowest black path).]{}
The strategies mentioned above are related:
1. The periodic $0$ strategy, denoted as $\pi_0$ (see Definition \[Def.periodic.b\]) will sometimes also be optimal when $\mu<0$; see Section \[S\_map\]. This strategy pays $X^{\pi_0}(T_1-)$ when $T_1\leq \tau^{\pi_0}$, and can be seen as the limit of a liquidation $(b_1,b_2)$ strategy when $b_2-b_1\downarrow 0$, or simply $b_1\uparrow \infty$. It is also denoted as $\pi_{a,a}$, or $\pi_{\infty,\infty}$.
2. The liquidation $(b,\infty)$ strategy can be seen as a hybrid $(0,0,b)$ strategy, i.e. $\pi_{b,\infty}=\pi_{0,0,b}$.
Further convergence results are developed and illustrated in Section \[S.Convergence\].
Main results {#S_map}
------------
The nature of the optimal strategy will depend on the value of some key parameters, as is shown in this paper. Our main results are summarised in Table \[T\_roadmap\], which can also be used as a road map for reading the paper. In addition, the transition between cells is “continuous”, except for the cells in the second row for $\mu<0$, as they are disjoint unless $\beta=\gamma/(\gamma+\delta)$; see Section \[S.Convergence\] for details and proofs of that statement.
--------------------------------------------------- --------------------------------------------------- ---------------------------------------------- --------------------------------------------
$\mu \ge 0$
$\chi \ge 0$ $\chi/\beta \ge -\frac{\mu}{\gamma+\delta} $ $\chi/\beta < -\frac{\mu}{\gamma+\delta} $
$0\le \beta \le \beta_0$ Periodic barrier $\pi_b$ (Thm \[Thm.Small.Beta\])
$\beta_0< \beta \le \frac{\gamma}{\gamma+\delta}$ Periodic barrier $\pi_b$ (Thm \[Thm.Small.Beta\])
$\frac{\gamma}{\gamma+\delta} < \beta \le 1 $ Hybrid $\pi_{a_p,a_c,b}$ (Thm \[Thm\])
--------------------------------------------------- --------------------------------------------------- ---------------------------------------------- --------------------------------------------
: Map of the dividend strategies proven as optimal in the different cases considered in the paper[]{data-label="T_roadmap"}
The results can be interpreted as follows.
Recall that $\gamma/(\gamma+\delta)$ is the expected present value at time 0 of a payment 1 paid after an $\gamma$-exponentially distributed random amount of time, discounted with a continuous force of interest $\delta$. At a very high level, this explains why this ratio is involved in most thresholds in the table: at any point in time, the model balances the choice between (i) a dollar of dividend paid immediately, with net value involving $\beta$ and $\chi$, and (ii) a dollar paid at the next periodic time (without transaction costs), with expected present value $\gamma/(\gamma+\delta)$.
Let us first focus first on the threshold $\gamma/(\gamma+\delta)$ for $\beta$. Ruin is unlikely in the next instant, and if $\mu\geq 0$ then we do not want to liquidate at first opportunity, so we ignore ruin for now. If a dividend of size of $\xi$ is to be paid, the decision between paying now (as immediate dividend) or paying later (as a periodic dividend) should depend on whether $$\beta\xi-\chi > \frac{\gamma}{\gamma+\delta}\xi,$$ in an expected sense. This would be the case if and only if $$\xi>\frac{\chi}{\beta-\frac{\gamma}{\gamma+\delta}}\quad\text{and}\quad \beta>\frac{\gamma}{\gamma+\delta}.$$ This condition will indeed re-appear later when we construct our candidate strategy (which will be proved optimal); for instance in Proposition \[prop.2\] where we use $\alpha$ to denote $\beta-\gamma/(\gamma+\delta)$. It plays a significant role in determining the minimum distance between barriers $a_c$ and $b$.
Now, when $\mu<0$, we must liquidate as soon as possible so only $\xi>\chi/\beta$ is required for immediate dividend, [because this is the amount of fixed transaction costs that need to be paid, and the optimisation won’t require extra to compensate for future possible gains (since the business is not profitable)—we only need the dividend to be admissible]{}. This explains the distinction between the two right columns. This can also be interpreted as follows. The quantity $$-\frac{\mu}{\gamma+\delta} = -\frac{\mu}{\gamma}\frac{\gamma}{\gamma+\delta}$$ is in fact the expected present value of the expected loss $-\mu/\gamma$ (in absolute terms) that will be accumulated until the next periodic payment. Whether $\beta$ times this quantity is more or less than the fixed transaction cost $\chi$ impacts the optimal strategy, which makes intuitive sense. This is especially the case when $\chi$ is large, that is, when immediate dividends are very expensive. In this case, even for sufficiently high $\beta_0< \beta \le \gamma/(\gamma+\delta)$, it will be optimal to liquidate with a periodic payment at first opportunity ($\pi_0$), but not immediately. The threshold $\beta_0$ will be defined in Section \[S.mu.neg\].
The expected present value of a Periodic barrier $\pi_b$ can be found in @AvTuWo16 or @PeYa16b, that of a hybrid strategy $(a_p,a_c,b)$ in Section \[S.hybridG\] (with optimal parameters in Section \[S.hybridG\]), an that of a Liquidation $(b_1,\infty)$ strategy in Section \[S.mu.neg\]. Optimality of those strategies is established thanks to the Verification lemma in Section \[S.Verification\] through the referenced Theorems in Sections \[S\_betasucks\], \[S.Optimality\], and \[S.mu.neg\], respectively. Further illustrations are provided in Section \[S.Numerical\].
A verification lemma {#S.Verification}
====================
In this section, we provide a set of sufficient conditions for a strategy $\pi\in\Pi$ to be optimal, in the sense of (\[Def.Optimal\]). Recall that for a real-valued function $F$, the extended generator for the stochastic process $X$ on a real-valued function $F$ is defined to be $$\mathscr{A}F(x):=\frac{\sigma^2}{2}F''(x)+\mu F'(x)$$ for $x\in\mathbb{R}$ such that the above makes sense. Throughout this paper, we will repeatedly use the following lemma to prove the optimality of different dividend strategies in different cases.
\[Verification.lemma\] For a strategy $\pi^*\in\Pi$, denote its value function $H(x):=V(x;\pi^*)$. Suppose there is a finite set $E\subseteq \mathbb{R}_+$ such that $H$ satisfies
1. $H\geq 0$,
2. $H\in \mathscr{C}^1(\mathbb{R}_+)\cap\mathscr{C}^2(\mathbb{R}_+\backslash E)$,
3. $H'$ is bounded on sets $[1/n,n]$ for all $n\in\mathbb{N}$,
4. On $\mathbb{R}_+\backslash E$, $H$ satisfies $$\label{eqt.HJB1}
(\mathscr{A}-\delta)H(x)+\gamma \sup_{\xi\in[0,x]}\Big(\xi+H(x-\xi)-H(x)\Big)
\leq 0,$$
5. On $\mathbb{R}_+$, $H$ satisfies $$\label{eqt.HJB2}
\sup_{\xi\in[0,x]}\Big((\beta\xi-\chi)1_{\{\xi>0\}}+H(x-\xi)-H(x)\Big)
= 0,$$
then $\pi^*$ is optimal, i.e. $V(x;\pi^*)=v(x)$ for all $x\geq 0$.
The proof which is provided in Appendix \[A.ver.lemma\] is standard; see for instance @PeYa16. However, it requires careful treatment of (1) the different types of strategies (2) immediate dividend at time $0$ (3) the approximation for Itô’s lemma at the points when $H$ is not smooth.
Optimality of a periodic barrier strategy when proportional transaction costs $1-\beta$ are high {#S_betasucks}
================================================================================================
In this section, we show that a periodic $b$ strategy (see Definition \[Def.periodic.b\]) is optimal when $\beta\leq {\gamma}/{(\gamma+\delta)}$ and $\mu\geq 0$. This case corresponds to the top left cell of Table \[T\_roadmap\].
From @PeYa16 it follows that there exists a optimal barrier $b_0^*\geq 0$ such that the periodic $b_0^*$ strategy, $\pi_{b_0^*}$, is optimal when dividends are only allowed to paid at the dividend payment times. Note that this strategy is also admissible in our setting and our definition of value functions for $\pi_{b_0^*}$ are the same, as there are no dividends to be paid outside the (Poissonian) dividend payment times and periodic dividends do not attract transaction costs. Therefore, we can borrow the results from @PeYa16 regarding the behaviour of the value function $V(\cdot;\pi_{b_0^*})$, which is summarised as follows:
1. The first 4 conditions in Lemma \[Verification.lemma\] hold, with the finite set $E=\{b_0^*\}$.
2. When $b_0^*>0$, the function is concave. In particular, we have $$V'(x;\pi_{b_0^*})\begin{cases}
>1,\quad&x\in(0,b_0^*)\\
=1,\quad&x=b_0^*\\
\in(\frac{\gamma}{\gamma+\delta},1),\quad&x\in(b_0^*,\infty)
\end{cases}.$$
3. When $b_0^*=0$, we have
1. If $\mu>0$, then $1\geq V'(0+;\pi_{b_0^*})>V'(x;\pi_{b_0^*})>{\gamma}/{(\gamma+\delta)}\geq \beta>0$, for $x>0$.
2. If $\mu=0$, then $1>V'(x;\pi_{b_0^*})={\gamma}/{(\gamma+\delta)}\geq \beta>0$, for $x>0$.
Hence, to show that $\pi_{b_0^*}$ is optimal, it suffices to show (\[eqt.HJB2\]), i.e. $$\sup_{\xi\in[0,x]}\Big((\beta\xi-\chi)1_{\{\xi>0\}}+V(x-\xi;\pi_{b_0^*})-V(x;\pi_{b_0^*})\Big)\leq 0,\quad x>0.$$ Denote $$H_0(\xi)=\beta\xi-\chi+V(x-\xi;\pi_{b_0^*})-V(x;\pi_{b_0^*}),\quad x>0,$$ and by taking derivative w.r.t. $\xi$, we get $$H_0^\prime(\xi)=\beta-V'(x-\xi;\pi_{b_0^*})$$ which is always non-positive when $\mu\geq 0$. Hence, the supremum of $H_0$ on $[0,x]$ is attained at $\xi=0$ with value $H_0(0)=-\chi<0$. This shows that the left hand side of (\[eqt.HJB2\]) is $\max(0,-\chi)=0$.
The above result is restated as the following theorem.
\[Thm.Small.Beta\] When $\mu\geq 0$ and $0\leq \beta\leq {\gamma}/{(\gamma+\delta)}$, the periodic $b_0^*$ strategy is optimal, where $b_0^*$ is specified in the third item in Proposition \[Lemma.convergence.beta\].
The hybrid $(a_p,a_c,b)$ strategy {#S.hybridG}
=================================
In this section, we calculate the expected present value of dividends of a general hybrid $(a_p,a_c,b)$ strategy and then pick a candidate strategy from the class using a “maximisation principle”. We will first use scale functions to derive some general results then use the classical PDE method to specialise to the case of diffusions. We make use of the fluctuation theory for Lévy processes [see, e.g. @PeYa16b Section 6, and references therein].
Value function
--------------
Denote $\Psi(\theta)=1/t \log{\ensuremath{\mathbb{E}}}(e^{\theta X(t)})$, $\theta\in\mathbb{R}$, as the Laplace exponent of a spectrally negative Lévy process $X$. Then for $q\geq 0$, the $q$-scale function $W_q$ is the mapping from $\mathbb{R}$ to $[0,\infty)$ that takes the value zero on the negative half-line, while on the positive half-line, it is a strictly increasing function that is defined by its Laplace transform: $$\int_0^\infty e^{-\theta x}W_q(x)dx=\frac{1}{\Psi(\theta)-q},\quad \theta>\phi(q),$$ (q)={0:()=q}. In particular, when $X$ is a diffusion process (defined by (\[Def.Diffusion\])) and $q>0$, we have $$\label{W.scalefcn.Diffusion}
W_q(x)=\frac{e^{r^{(q)} x}-e^{s^{(q)} x}}{\frac{\sigma^2}{2}(r^{(q)}-s^{(q)})},$$ where $r^{(q)}>0$ and $s^{(q)}<0$ are the two distinct roots of $$\Psi(\theta)-q=0\iff \frac{\sigma^2}{2}\theta^2+\mu\theta-q=0.$$
In addition, we also define $$\begin{aligned}
W_{r,q,a}(x):=~&W_q(x)+r\int_0^{x-a}W_{q+r}(x-y-a)W_q(y+a)dy,\quad x\geq a,\\
\overline{W}_q(x):=~&\int_0^x W_q(y)dy,\quad x\geq 0,\\
\overline{\overline{W}}_q(x):=~&\int_0^x \overline{W}_q(y)dy,\quad x\geq 0.\end{aligned}$$
In the following, we slightly abuse the notation and assume that the barriers $(a_p,a_c,b)$ are given as $(a,a_c,b)$ and therefore denote the value function $V(x):=V(x;\pi_{a,a_c,b})$. If the dependence on the strategy $\pi$ or costs $1-\beta$ and $\chi$ needs to be stressed, we will write the full version $V(x;\pi_{a,a_c,b})$, or $V_{1-\beta,\kappa}(\cdot)$, respectively. The value function $V$ is given by the following lemma.
\[lemma.value.function.hybrid.scale\] For a given hybrid $(a_p,a_c,b)$ strategy (with barrier levels $0< a_p=a\leq a_c<b$), its value function is continuous and is given by $$\label{value.hybrid}
V(x)=\begin{cases}
\frac{V(a)}{{W_{\delta}}(a)}{W_{\delta}}(x)&x\in(-\infty,a),\\
\frac{V(a)}{{W_{\delta}}(a)}G(a,x)-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(x-a)&x\in(a,b),\\
\beta(x-a_c-\kappa)+V(a_c)&x\in(b,\infty),
\end{cases}$$ with $$G(a,x):={W_{\delta}}(x)+\int_0^{x-a}{W_{\gamma+\delta}}(x-a-y)\Big({W_{\delta}}(a+y)-{W_{\delta}}(a)\Big)dy$$ where $V(a)$, $V(a_c)$ and $V(b)$ can be found by solving three linear equations in them.
For barrier levels $0= a_p=a\leq a_c<b$, we use $$\label{def.VW0}
\frac{V(a)}{{W_{\delta}}(a)}:=\frac{\beta(y-\kappa)+\gamma({\overline{\overline{W}}_{\gamma+\delta}}(y+l)-{\overline{\overline{W}}_{\gamma+\delta}}(l))}{G(a,y+l)-G(a,l)},$$ which also holds for the above case when $a_p=a>0$.
Using the notations introduced above, with a minor modification of the proofs in Sections 5.1, 6.1 and 6.2 in @PeYa16b, we can deduce that for $a>0$ $$V(x)=\frac{{W_{\delta}}(x)}{{W_{\delta}}(a_p)}V(a_p),\quad x\in(-\infty,a_p]$$ and $$V(x)=V(a_p)(\frac{{W_{\gamma,\delta,a_p}}(x)}{{W_{\delta}}(a_p)}-\gamma{\overline{W}_{\gamma+\delta}}(x-a_p))-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(x-a_p),\quad x\in[a_p,b],$$ which is the same as after some rearrangement. The case for $a=0$ can be carried over by a limit argument as in @NoPeYaYa17.
From we see that $$\begin{aligned}
V(x)=~&\beta(x-\kappa)+\Big(V(a_c)-\beta a_c\Big),\quad x>b,\\
V(x)=~&{W_{\delta}}(x)\Big(\frac{V(a)}{{W_{\delta}}(a)}\Big),\quad x\leq a.\end{aligned}$$ Hence it is reasonable to attempt maximisation of $V(a_c)-\beta a_c$ or $V(a)/{W_{\delta}}(a)$ w.r.t. the parameters $(a,a_c,b)$ (and we will see both approaches are equivalent).
Choice of candidate strategies
------------------------------
We now proceed to pick a candidate strategy from the class of hybrid $(a_p,a_c,b)$ strategies. A “nice” hybrid $(a_p,a_c,b)$ strategy is characterised by the derivatives of its value function at the boundaries, see e.g. @AvLaWo20d [Remark 9.2] for an intuitive explanation. We postulate (and later show) that those “nice” properties will lead to the optimal set of strategies, and hence refer to those as candidates.
\[Def.Nice.hybrid\] A strategy is said to be a nice hybrid $(a_p,a_c,b)$ strategy if the following are satisfied:
1. It is a hybrid $(a_p,a_c,b)$ strategy (see Definition \[Def.hybrid.apacb\]);
2. $b\geq a_c+{\chi}/{\beta}$ and $V'(b)=\beta$;
3. Either $a_c=a_p$ and $V'(0)\leq \beta$, or $V'(a_c)=\beta$;
4. Either $a_p=0$ and $V'(0)\leq 1$, or $V'(a_p)=1$.
In the following, we will re-parametrise $(a,a_c,b)$ using $(a,l,y)$ with $l:=a_c-a$ and $y:=b-a_c$. The support of $(a,l,y)$ is $[0,\bar{a}]\times [0,\infty)\times [\kappa,\infty)$, where $\bar{a}$ is the unique solution for ${W_{\delta}}''(x)=0$ if it exists, otherwise $\bar{a}=0$. We chose to maximise $V(a_c)-\beta a_c$. Regarding the auxiliary function $G$, it is easy to see $${\frac{\partial}{\partial a}}G(a,d)={\frac{\partial}{\partial d}}G(a,d)-\gamma {W_{\delta}}'(a){\overline{W}_{\gamma+\delta}}(d)
\quad
\text{and}
\quad
{\frac{\partial}{\partial d}}G(a,d)>0.$$ For the derivatives of the value function at $a_p=a$, $a_c$ and $b$, we have $$\begin{aligned}
V'(a)=~&\frac{V(a)}{{W_{\delta}}(a)}{W_{\delta}}'(a),\label{eq1a}\\
V'(a_c)=~&\frac{V(a)}{{W_{\delta}}(a)}{\frac{\partial}{\partial l}}G(a,l)-\gamma {\overline{W}_{\gamma+\delta}}(l),\\
V'(b)=~&\frac{V(a)}{{W_{\delta}}(a)}{\frac{\partial}{\partial (y+l)}}G(a,y+l)-\gamma {\overline{W}_{\gamma+\delta}}(y+l).\label{eq1c}\end{aligned}$$
We will first show that the derivative conditions are satisfied, provided a maximiser exists for our objective function $$\label{obj.fcn}
V(a_c)-\beta a_c=\frac{V(a)}{{W_{\delta}}(a)}\Big({W_{\delta}}(x)+\int_0^{l}{W_{\gamma+\delta}}(l-y)\Big({W_{\delta}}(a+y)-{W_{\delta}}(a)\Big)dy\Big)-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(l)-\beta(a+l).$$ We will then show the existence of the maximiser. The following proposition illustrate the properties of a set of optimal parameters $(a,l,y)$, assuming its existence. For the moment, we make the following assumption which will be lifted by Proposition \[Prop.3\].
\[Ass0\] For any $a\geq 0$, we have $${\frac{\partial}{\partial x}}\frac{G(a,x)}{\gamma{\overline{W}_{\gamma+\delta}}(x)}<0,\quad x\geq 0.$$
\[prop.1\] Denote $(a^*,l^*,y^*)$ a maximiser of the objective function $V(a_c)-\beta a_c$ and we recall the support is a subset of $a\in[0,\bar{a}]$. Under Assumption \[Ass0\], if $(a^*,l^*,y^*)$ lie in the interior of the support, then we can conclude that with such choice of parameters, we have $$\label{dev.con}
V'(a)=1,\quad
V'(a_c)=\beta,\quad
V'(b)=\beta.$$ Otherwise, if $a^*=0$, then $V'(0)\leq 1$; if $l^*=0$, we have $a^*=l^*=0$ and $V'(0)\leq 1$. These are the only boundary cases.
Since $(a^*,l^*,y^*)$ is a maximiser and the objective function is differentiable in the arguments, all the partial derivatives are zero (except in the boundary which requires extra care). In summary, the proof requires a direct checking in the argument $y$, then $l$ then $a$, assisted with the help of equations -; see Appendix \[A\_Prop55\] for details.
Although the proof of Proposition \[prop.1\] is simple and is similar to existing proofs in the literature [e.g., @Loe08a], it presents the main ingredients in showing the existence of a candidate strategy characterised by *three* non-zero parameters. This is generally a difficult problem since explicit calculation is often impossible. To our best knowledge, this is the first time such problem has been solved.
From the proof in Appendix \[A\_Prop55\], we see that maximising $V(a_c)-\beta a_c$ is the same as maximising $V(a)/{W_{\delta}}(a)$. From the formula of $V(a)/{W_{\delta}}(a)$ in , we see that the $a$-argument of maximiser of $V(a)/{W_{\delta}}(a)$ cannot live outside $[0,\bar{a}]$, which justifies our choice of a narrower support $a\in[0,\bar{a}]$.
We now proceed to show the existence of a local maximiser $(a^*,l^*,y^*)$. Due to the complexity of the calculation using scale functions [generally one assumes completely monotonic Lévy density and proceed with complicated calculations, see e.g. @NoPeYaYa17], we specialise our calculations using the classical PDE methods. Denote the following functions: $$\begin{aligned}
\psi(\theta):=~&\frac{\sigma^2}{2}\theta^2+\mu \theta\\
f(x):=~&e^{r_0 x}-e^{s_0 x},\\
g(x):=~&e^{r_1 x}-e^{s_1 x},\\
J(x):=~&-s_1g(x)+(r_1-s_1)(e^{s_1x}-1),\\
J'(x)=~&-r_1s_1g(x),\end{aligned}$$ where $(r_0,s_0)$ and $(r_1,s_1)$ are the positive and negative roots of equations $\psi(\theta)-\delta=0$ and $\psi(\theta)-\gamma-\delta=0$ respectively, with $|s_i|>|r_i|$, $i=0,1$ (since $\mu>0$). Note $f$, $g$ and $J$ are proportional to the *scale functions* ${W_{\delta}}$, ${W_{\gamma+\delta}}$ and ${\overline{W}_{\gamma+\delta}}$, respectively, see equation (\[W.scalefcn.Diffusion\]).
Before stating the value function in terms of $f$, $g$ and $J$, we shall discuss the smoothness of the value function for the PDEs to be solved. From the proof of Lemma \[lemma.value.function.hybrid.scale\], we can conclude that the value function $V$ is continuous, continuously differentiable except at $\{0\}$, and twice differentiable except at $0$ and at $b$, i.e. $V\in\mathscr{C}(\mathbb{R})\cap\mathscr{C}^1(\mathbb{R}\backslash\{0\})\cap\mathscr{C}^2(\mathbb{R}\backslash\{0,b\})$.
The following proposition provides an alternative characterisation for the value function of a hybrid $(a_p,a_c,b)$ strategy.
\[Prop.Vfcn\] For given ($a,a_c,b$) with $b>a_c+\chi/\beta$ and $ a_c\geq a\geq 0$, the value function of the hybrid $(a,a_c,b)$ strategy is given by $$\label{eq.PDE}
V(x;\pi_{a,a_c,b})=\begin{cases}
0,&x\in(-\infty,0)\\
C(e^{r_0 x}-e^{s_0 x}),&x\in[0,a)\\
A(e^{r_1(x-a)}-e^{s_1(x-a)})+Be^{s_1(x-a)}+\frac{\gamma}{\gamma+\delta}(x-a+\frac{\mu}{\gamma+\delta}+V(a)),&x\in[a,b)\\
\beta(x-a_c)-\chi+V(a_c),&x\in[b,\infty)
\end{cases},$$ with $l=a_c-a$, $d=b-a$, $g(d,l)=g(d)-g(l)$, $J(d,l)=J(d)-J(l)$, $$\label{C.formula}
C=\frac{(r_1-s_1)\big((\beta-\frac{\gamma}{\gamma+\delta})(d-l)-\chi\big)+\frac{\gamma}{\gamma+\delta}g(d,l)+\frac{\gamma\mu}{(\gamma+\delta)^2}J(d,l)}{\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)},$$ $$\label{B.formula}
B=\frac{\delta}{\gamma+\delta}Cf(a)-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta},$$ $$\label{A.formula}
A=\frac{1}{r_1-s_1}\Big(Cf'(a)-Bs_1-\frac{\gamma}{\gamma+\delta}\Big)$$ and $$\begin{aligned}
V(a)=~&Cf(a)\\
V(a_c)=~&Ag(l)+Be^{s_1 l}+\frac{\gamma}{\gamma+\delta}(l+\frac{\mu}{\gamma+\delta}+Cf(a)).
\end{aligned}$$ We also adopt the (unusual) convention that $[0,0)=\emptyset$ in *(\[eq.PDE\])*.
Formulas can be derived by either directly substitute the formula of in Lemma \[lemma.value.function.hybrid.scale\], or by a classical PDE approach.
So far, Proposition \[prop.1\] holds for general spectrally negative Lévy processes (where Assumption \[Ass0\] may or may not hold). We now specialise our results in the diffusion setting, and show in Proposition \[Prop.3\] that Assumption \[Ass0\] always holds for diffusion processes, so we can use the conclusion of Proposition \[prop.1\] freely.
\[Prop.3\] Assumption \[Ass0\] holds for diffusion processes.
The result stems directly from the explicit formula given by Proposition \[Prop.Vfcn\]; see Appendix \[A.1\] for details.
Thanks to Proposition \[Prop.Vfcn\], we have an explicit formula for the value function. We are now ready to show the following proposition.
\[prop.2\] There exists a triplet $(a^*,l^*,y^*)\in\mathscr{B}$ such that the “derivative conditions” hold.
Thanks to Propositions \[Prop.3\] and \[prop.1\], it remains to construct a large enough box to contain the maximum of the objective $V(a_c)-\beta a_c$. See Appendix \[A.prop2\] for details.
Sufficient conditions for liquidation strategies {#Remark.suff.cons}
------------------------------------------------
We are now able to derive some sufficient conditions for liquidation strategies to be optimal. Denote the functions $$\label{Q.function}
Q(a):=1-\frac{f(a)/f'(a)}{\mu/\delta}$$ and $$\label{I.def}
I(x,q):=\frac{\beta-\frac{\gamma}{\gamma+\delta}+\Big(\frac{\gamma}{\gamma+\delta}-\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+q(r_1+s_1)}\Big)
e^{s_1 x}}{g'(x)+g(x)(-r_1s_1)\frac{\mu}{\gamma+\delta}(1-q)},$$ where $Q$ maps the periodic lower barrier $a_p\in[0,\bar{a}]$ to a number $q\in[0,1]$ in an decreasing manner, whereas $I(\cdot,q)$ is a function decreasing to $0$ at infinity after it achieves its maximum.
For a hybrid $(a_p,a_c,b)$ strategy, if $V'(b)=\beta$, we have (using the formula for $C$ given by )
$$\begin{aligned}
V'(a_p)-\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+Q(a_p)(r_1+s_1)}=~&(r_1-s_1)
I(y+l,q)\\
<~&(r_1-s_1)\frac{\beta}{g'(g_0)}=:\varepsilon_k,
\end{aligned}$$
with $g_0$ is a constant representing the minimum of the denominator of w.r.t. $(x,q)$.
Hence, if $$\frac{-s_1\frac{\gamma}{\gamma+\delta}}{r_1}\leq 1-\varepsilon_k,$$ we will choose $a=0$ ($Q(a)=1$). If we consider $\varepsilon_k=0$, this condition is equivalent to the condition in Remark 4.1(i) in @NoPeYaYa17, i.e. $\gamma\leq({\sigma^2}/{2})r_1^2$.
Likewise, if $$\frac{-s_1\frac{\gamma}{\gamma+\delta}}{r_1}\leq \beta-\varepsilon_k,$$ we will choose $a=l=0$ ($a_p=a_c=0$). In fact, for any $q$ such that $\frac{-s_1\frac{\gamma}{\gamma+\delta}}{r_1+q(r_1+s_1)}\geq \beta,$ we have $I(0,q)\leq 0$ and therefore for any $q$ there are always $(l,y+l)$ with $l>0$ such that $I(l,q)=I(y+l,q)$. This implies that when $$1-\varepsilon_k\geq \frac{-s_1\frac{\gamma}{\gamma+\delta}}{r_1}\geq \beta,$$ we will choose $a=0$ but $l>0$ ($a_p=0<a_c$).
In practice, $\varepsilon_k$ is usually negligible, compared to the size of $\beta$. Likewise, we can see that once we have the smoothness condition $V'(b)=\beta$, the right hand side is (almost) negligible and therefore the derivative at $a$ is (almost) independent of both $y$ and $l$.
In terms of the parameters in the model, we have $$\frac{-s_1}{r_1}=1+\frac{1+\sqrt{1+2\nu}}{\nu},\quad \text{ with }\quad \nu:=\Big(\frac{\sigma}{\mu}\Big)^2(\gamma+\delta)$$ and therefore the above two sufficient conditions can be rewritten as $$\frac{\gamma}{\gamma+\delta}\frac{1+\sqrt{1+2\nu}}{\nu}+\varepsilon_k\leq \frac{\delta}{\gamma+\delta}$$ +\_k-.$$ Note further $({1+\sqrt{1+2x}})/{x}$ is decreasing in $x$, hence there are thresholds $\nu_1(\gamma,\delta)$ and $\nu_\beta(\gamma,\delta)$ (with $\nu_\beta>\nu_1$ unless $\beta\geq 1-\varepsilon_k$) such that $$\begin{cases}
\nu_\beta(\gamma,\delta)\geq\Big(\frac{\sigma}{\mu}\Big)^2(\gamma+\delta)\geq \nu_1(\gamma,\delta)&\implies\quad a=0, ~l>0\\
\Big(\frac{\sigma}{\mu}\Big)^2(\gamma+\delta)\geq \nu_\beta(\gamma,\delta)&\implies\quad a=0,~l=0\\
\end{cases}.$$
Note the sufficient conditions for liquidation at first opportunity ($a=0$) do not depend on the transaction costs $1-\beta$ and $\chi$ as one can always ignore the opportunities to pay immediate dividends. However, it does depend on the time parameters for the frequency periodic payments and discounting ($\gamma$ and $\delta$), as well as the “coefficient of variation” ${\sigma}/{\mu}$, a measurement of the riskiness of the business.
The derivative of the value function of our candidate strategy {#S.VD}
==============================================================
From the previous section, we see that there exists a nice hybrid $(a_p,a_c,b)$ strategy (see Definition \[Def.Nice.hybrid\] and Propositions \[prop.1\], \[prop.2\]). In other words, there are $(a_p,a_c,b)$ such that $a_p\leq \bar{a}$, $V'(a_p)=1$ (or $a_p=0$ and $V'(0)\leq 1$), $V'(a_c)=\beta$ (or $a_p=a_c=0$ and $V'(0)\leq \beta$) and $V'(b)=\beta$. We will pick this strategy and refer it as our candidate strategy, and use $V$ to denote its value function.
In this section, we first establish some results regarding the derivative of the value function of our candidate strategy. They will then be used to verify the optimality of our strategy in Section \[S.Optimality\].
As explained before (e.g., Sections \[S\_map\] and \[S\_betasucks\]) we must have $$\label{Ass.betaLarge}
\frac{\gamma}{\gamma+\delta}<\beta\leq 1,$$ which becomes apparent in some areas of the proof.
Our goal in this section is to establish Lemma \[Lemma.VD.Nice\]. In order to do that, we need to first establish Lemma \[Lemma.VD.opt\] below, which concerns the behaviour of the derivative of the value function of our candidate strategy.
\[Lemma.VD.opt\] Regarding the derivative of the value function, we have the following:
1. Suppose $a_p>0$, then $$\label{eqt.VD.opt1}
V'(x)\begin{cases}
>1,~&x\in[0,a_p)\\
=1,~&x=a_p\\
\in (\beta,1),~&x\in[a_p,a_c)\\
=\beta,~&x=a_c\\
\in(0,\beta),~&x\in(a_c,b)\\
=\beta,~&x\in[b,\infty)
\end{cases}.$$
2. Suppose $a_p=0$ and $a_c>0$, then $$\label{eqt.VD.opt2}
V'(x)\begin{cases}
\in (\beta,1],~&x=0\\
\in (\beta,1),~&x\in(0,a_c)\\
=\beta,~&x=a_c\\
\in(0,\beta),~&x\in(a_c,b)\\
=\beta,~&x\in[b,\infty)
\end{cases}.$$
3. Suppose $a_p=a_c=0$, then $$\label{eqt.VD.opt3}
V'(x)\begin{cases}
\in (0,\beta],~&x=0\\
\in(0,\beta),~&x\in(0,b)\\
=\beta,~&x\in[b,\infty)
\end{cases}.$$
In any case, we have $V'>0$ on $[0,\infty)$.
The proof requires analysing the functional form of the value function with the derivative conditions imposed for the candidate strategy; see Appendix \[A.Lemma6.1\] for details.
The next Lemma shows that our candidate strategy satisfies the last 2 conditions in Lemma \[Verification.lemma\].
\[Lemma.VD.Nice\] The value function of a nice hybrid $(a_p,a_c,b)$ strategy, $V$, satisfies $$(\mathscr{A}-\delta)V(x)+\gamma \sup_{\xi\in[0,x]}\Big(\xi+V(x-\xi)-V(x)\Big)
\leq 0, \quad x\in\mathbb{R}^+\backslash\{b\}$$ and $$\sup_{\xi\in[0,x]}\Big((\beta\xi-\chi)1_{\{\xi>0\}}+V(x-\xi)-V(x)\Big)
= 0,\quad x\in\mathbb{R}^+.$$
The result comes as a straightforward consequence of Lemma \[Lemma.VD.opt\]; see Appendix \[A.Lemma6.2\] for details.
Optimality in case of profitable business ($\mu \ge 0$) {#S.Optimality}
=======================================================
In this section, we show the optimality of a nice hybrid $(a_p,a_c,b)$ strategy, which is the following theorem.
\[Thm\] Suppose $\mu\geq 0$. Denote $V$ the value function of a nice hybrid $(a_p,a_c,b)$ strategy. Suppose $\pi\in\Pi$, then $V(x)\geq V(x;\pi)$ for all $x\geq 0$. In other words, any nice hybrid $(a_p,a_c,b)$ strategy is optimal.
Thanks to Lemma \[Verification.lemma\], we only need to check that $V$ satisfies all conditions proposed, which is essentially Lemma \[Lemma.VD.opt\] and \[Lemma.VD.Nice\], with the finite set $E=\{b\}$.
We now present a corollary regarding the uniqueness of nice hybrid $(a_p,a_c,b)$ strategies.
\[Corrolary.unique.barriers\] Suppose $\mu\geq 0$. There is one and only one nice hybrid $(a_p,a_c,b)$ strategy. Denote its parameters $(a_p^*,a_c^*,b^*)$. Hence, the hybrid $(a_p^*,a_c^*,b^*)$ strategy is optimal.
Lemma \[Lemma.VD.opt\] characterised the derivative of the value function, which together with the optimality implies uniqueness. A similar proof can be found in @AvLaWo20d [Lemma 9.3].
Optimality in case of unprofitable business ($\mu<0$) {#S.mu.neg}
=====================================================
In this section, we discuss the optimal strategy when the business is strictly unprofitable, i.e. $\mu<0$. As such, solely in this section, we make the following assumption.
\[Ass.Mu.Neg\] We assume $\mu<0$.
Recall that $\mu<0$ implies that we want to liquidate the business in the most (cost-)efficient way. In the following, we focus on the case when $\chi>0$ and the case for $\chi=0$ can be seen as the case when $\chi\downarrow 0$ in terms of the structure of the optimal strategy.
Our candidate strategies are the Liquidation $(b_1,b_2)$ strategy, characterised by 2 parameters $0<b_1<b_2\leq \infty$, denoted as $\pi_{b_1,b_2}$ (see Definition \[Def.periodic.b\]) and the periodic $0$ strategy, denoted as $\pi_0$ (see Definition \[def.Liq\]). The latter pays $X^{\pi_0}(T_1-)$ when $T_1\leq \tau^{\pi_0}$, and it can also be seen as the limit of a liquidation $(b_1,b_2)$ strategy when $b_2-b_1\downarrow 0$, or simply $b_1\uparrow \infty$. Therefore, it is also denoted as $\pi_{a,a}$, or $\pi_{\infty,\infty}$.
For $\beta> {\gamma}/{(\gamma+\delta)}$, it should be intuitively clear that the form of $\pi_{b,\infty}$ is optimal if we can choose the lower barrier $b$ nicely. On the other hand, for $\beta<{\gamma}/{(\gamma+\delta)}$, we proceed the following. It is known that [e.g. from @PeYa16] that $V'(x;\pi_0)$ is increasing in $x$ to ${\gamma}/{(\gamma+\delta)}$. Therefore, for $V'(0;\pi_0)<\beta<{\gamma}/{(\gamma+\delta)}$, there is a unique $a_\beta>0$ such that $V'(a_\beta;\pi_0)=\beta$. If moreover $$\label{Condition.Liqb1b2.Optimal}
V(a_\beta;\pi_0)<\beta a_\beta-\chi,$$ then there is a unique $c_{\beta,\chi}$ such that $0<c_{\beta,\chi}<a_\beta$ with $V(c_{\beta,\chi};\pi_0)=\beta c_{\beta,\chi}-\chi.$ Note that (\[Condition.Liqb1b2.Optimal\]) is equivalent to $$\label{Condition.Liqb1b2.Optimal2}
\frac{\chi}{\beta}<a_\beta-\frac{V(a_\beta;\pi_0)}{\beta}.$$ Denote the right hand side as a function of $\beta$, i.e. $$\label{Def.Lambda}
\Lambda(\beta)=a_\beta-\frac{V(a_\beta;\pi_0)}{\beta},$$ then we have that $$\label{Property.Lambda}
\text{$\Lambda$ is increasing from $0$ to the limit $\frac{-\mu}{\gamma+\delta}$ when $\beta$ increases on the interval $[V'(0;\pi_0),\frac{\gamma}{\gamma+\delta})$;}$$ a proof of which is provided in Appendix \[A.Proof.Lambda\]. Hence, (\[Condition.Liqb1b2.Optimal\]) is only possible when $\chi<\beta({-\mu}/({\gamma+\delta}))$. To further explain what it means, note when the surplus is $x$ and we can choose either (1) liquidate now, or (2) liquidate in the next Poissonian time, we need to consider the trade-off. If we liquidate now, the fixed cost is $\chi$. If we wait, assuming ruin is not an issue, the *expected* (discounted) loss in surplus is then $${\ensuremath{\mathbb{E}}}(-\mu T_1 e^{-\delta T_1})=\frac{-\gamma\mu}{(\gamma+\delta)^2},$$where we recall that $T_1$ is an exponential random variable with mean $1/\gamma$. Hence, when $\chi\geq \beta({-\mu}/({\gamma+\delta}))$ and $\beta\leq {\gamma}/{(\gamma+\delta)}$, we shall never liquidate immediately. On the other hand, if $\chi<\beta({-\mu}/({\gamma+\delta}))$, then in view of (\[Condition.Liqb1b2.Optimal2\]) and (\[Property.Lambda\]), there is a $\beta_0\in(V'(0;\pi_0),{\gamma}/{(\gamma+\delta)})$ defined by $\Lambda(\beta_0)={\chi}/{\beta}$ such that (\[Condition.Liqb1b2.Optimal\]) does not hold whenever $\beta\in(V'(0),\beta_0)$ and (\[Condition.Liqb1b2.Optimal\]) holds whenever $\beta\in(\beta_0,{\gamma}/{(\gamma+\delta)})$.
In light of the above analysis, we should not be surprised with the results in this section. They are summarised by the following theorem.
\[Thm.mu.neg\] For $\mu<0$, we have the following:
1. $\chi\geq \beta\frac{-\mu}{\gamma+\delta}$. We have
1. For $\beta\in(0,{\gamma}/{(\gamma+\delta)}]$, the periodic $0$ strategy is optimal.
2. For $\beta\in({\gamma}/{(\gamma+\delta)},1]$, a liquidation $(b,\infty)$ strategy is optimal with $b>0$ characterised by $$V'(b-;\pi_{b,\infty})=\beta=V'(b+;\pi_{b,\infty}).$$
2. $\chi<\beta\frac{-\mu}{\gamma+\delta}$. Denote $\beta_0:=\Lambda^{-1}(\frac{\chi}{\beta})$, we have
1. For $\beta\in(0,\beta_0]$, the periodic $0$ strategy is optimal.
2. For $\beta\in(\beta_0,{\gamma}/{(\gamma+\delta)})$, a liquidation $(b_1,b_2)$ strategy is optimal, with $(b_1,b_2)$ such that $0<c_{\beta,\chi}<b_1<a_\beta<b_2<\infty$ and $$V'(b_1-;\pi_{b_1,b_2})=V'(b_1+;\pi_{b_1,b_2})=\beta=V'(b_2-;\pi_{b_1,b_2})=V'(b_2+;\pi_{b_1,b_2}).$$
3. For $\beta\in[{\gamma}/{(\gamma+\delta)},1]$, a liquidation $(b,\infty)$ strategy is optimal, with $b>0$ characterised by $$V'(b-;\pi_{b,\infty})=\beta=V'(b+;\pi_{b,\infty}).$$
In order to prove Theorem \[Thm.mu.neg\], we will need the following lemmas. Our first lemma calculates the value function for each strategy.
\[Lemma.ValueFunction\] The value function of a periodic $0$ strategy is given by $$V(x;\pi_0)=-\frac{\gamma\mu}{(\gamma+\delta)^2}e^{s_1x}+\frac{\gamma}{\gamma+\delta}(x+\frac{\mu}{\gamma+\delta}),\quad x\geq 0.$$
The value function of a liquidation $(b_1,b_2)$ strategy (with $0<b_l<b_2<\infty$) is given by $$\label{value.Liq.b1b2}
V(x;\pi_{b_1,b_2})=\begin{cases}
Ag(x)-\frac{\gamma\mu}{(\gamma+\delta)^2}e^{s_1x}+\frac{\gamma}{\gamma+\delta}(x+\frac{\mu}{\gamma+\delta}),\quad& x\in[0,b_1)\\
\beta x-\chi,\quad& x\in[b_1,b_2)\\
Be^{s_1 x}+\frac{\gamma}{\gamma+\delta}(x+\frac{\mu}{\gamma+\delta}),\quad& x\in[b_2,\infty)
\end{cases},$$ with $$\label{ADef}
A=A(b_1):=\frac{(\beta-\frac{\gamma}{\gamma+\delta})b_1-\chi-\frac{\gamma\mu}{(\gamma+\delta)^2}(1-e^{s_1b_1})}{g(b_1)}$$ and $B$ can be determined using $V(b_2-)=V(b_2)$, in the case where $b_2<\infty$.
The value function of a liquidation $(b,\infty)$ strategy, with $b>0$ is simply (\[value.Liq.b1b2\]) without the $x\in[b_2,\infty)$ branch, where the formula for $A$ is still the same and we do not need to compute $B$.
We omit the proof for Lemma \[Lemma.ValueFunction\] as it can be obtained easily by solving PDE with the continuity of the value functions on the boundaries being the boundary conditions.
\[Remark.mu.neg.obs\] The following results will be repeatedly used in what follows, $$V(x;\pi_{b_1,b_2})=V(x;\pi_0)+Ag(x),\quad \text{and} \quad
V''(x;\pi_{b_1,b_2})=Ag''(x)-\frac{\gamma\mu}{(\gamma+\delta)^2}s_1^2e^{s_1x},\quad x\leq b_1,$$ where $b_2$ can possibly be infinity. Therefore, we have $V''(x)>0$ if $A\geq 0.$
The next lemma is also an argument which will be used over time.
\[Lemma.A.increasing\] For a liquidation $(b,b_2)$ strategy, with $0<b<b_2\leq \infty$, we have $${\frac{\partial}{\partial b}}A(b)>0\text{ if and only if }V'(b-;\pi_{b,b_2})<\beta.$$ We can replace the inequalities by equalities simultaneously.
This can be derived through direct computation using (\[ADef\]).
The next 2 lemmas establish the existence of the candidate strategy described in Theorem \[Thm.mu.neg\] (for every possible case).
\[Lemma.b1b2.Exist\] If $V'(0;\pi_0)<\beta<\frac{\gamma}{\gamma+\delta}$ and $V(a_\beta;\pi_0)<\beta a_\beta-\chi$, then there are $(b_1,b_2)$ with $0<c_\beta<b_1<a_\beta<b_2<\infty$ such that $V'(b_1-;\pi_{b_1,b_2})=V'(b_1+;\pi_{b_1,b_2})=\beta=V'(b_2-;\pi_{b_1,b_2})=V'(b_2+;\pi_{b_1,b_2}).$
The proof is based on continuity arguments. For example, for $b_1$, if one denotes the objective function $b_1\mapsto \widetilde{O}(b_1):=V'(b_1+)-\beta$, it suffices to show that $\widetilde{O}(c_\beta)\widetilde{O}(a_\beta)<0$. Details are provided in Appendix \[A.Lemma8.6\].
\[Lemma.b.Exist\] If ${\gamma}/{(\gamma+\delta)}<\beta\leq 1$, then there is a $b>{\chi}/{\beta}$ such that $V'(b-;\pi_{b,\infty})=\beta$.
The proof is similar to that of Lemma \[Lemma.b1b2.Exist\]; see Appendix \[A.Lemma8.7\]
Lemmas \[Lemma.b1b2.Exist\]-\[Lemma.b.Exist\] shows that our candidate strategy (as described in Theorem \[Thm.mu.neg\]) exists. Our last lemma below shows that the derivative of its value functions is increasing, which essentially completes the proof of Theorem \[Thm.mu.neg\].
In each case considered in Theorem \[Thm.mu.neg\], the derivative of the value function is increasing for our candidate strategy.
We only need to prove the case when the liquidation $(b_1,b_2)$ strategy (with $b_2\leq \infty$) is optimal.
From the proof of Lemma \[Lemma.b1b2.Exist\], if we choose $b_1$ to be the smallest one, then we have that $A(b_1)>0$, as $A$ is increasing from $0$ at $c_\beta$ to $b_1$. This implies that $V''(x;\pi_{b_1,b_2})>0$ and hence $V'(x;\pi_{b_1b_2})$ is an increasing function on $[0,b_2]$. Now, from $$V'(b_2-;\pi_{b_1,b_2})=Bs_1e^{s_1b}+\frac{\gamma}{\gamma+\delta}=\beta,$$ we can deduce that $B>0$ and consequently that $V'(x;\pi_{b_1,b_2})$ is also increasing on $[b_2,\infty)$. This completes the proof for the case when $V'(0;\pi_0)<\beta<{\gamma}/{(\gamma+\delta)}$ and $V(a_\beta;\pi_0)<\beta a_\beta-\chi$.
From the proof of Lemma \[Lemma.b.Exist\], if we choose $b$ to be the smallest one, using the same argument, we have that $A(b)>0$, which shows that $V'(x;\pi_{b,\infty})$ is increasing on $[0,b]$ and hence on $[0,\infty)$.
Note the value function of the candidate strategy in each case is continuously differentiable and twice differentiable except at the boundary points. Therefore the first 3 conditions of Lemma \[Verification.lemma\] are automatically satisfied. What is yet to be shown are the last 2 conditions of Lemma \[Verification.lemma\], which can be done in a similar way to what has been done in Sections \[S.Verification\] and \[S.VD\]. Hence, Theorem \[Thm.mu.neg\] holds.
On the convergence of strategies across solution thresholds {#S.Convergence}
===========================================================
In this section, we discuss the convergence of strategies when $\mu\uparrow 0$ and when $\beta\downarrow{\gamma}/{(\gamma+\delta)}$. These correspond to major thresholds in Table . By convergence in strategy, we mean point-wise converge of the value function at each $x\geq 0$. Since barriers determine the value function, convergence in the barriers implies convergence in strategy.
When $\mu\uparrow 0$ {#S.VaryingMu}
--------------------
[The objective of this section is to investigate how the strategies and their parameters “connect” when $\mu$ goes from negative to positive.]{}
[We start by some numerical exploration.]{} Our baseline parameters are $(\sigma,\chi)=(0.3,0.15)$ (dollar parameters), and $(\gamma,\delta)=(1,0.15)$ (time parameters). Note that the crucial constant $\gamma/(\gamma+\delta)\approx 0.87$, and the form of optimal strategies (rather, their connection here) will potentially differ on either side of that constant. Hence, we will illustrate the cases $\beta=0.7$ and $\beta=0.9$ separately.
![Continuity of optimal barriers at $\mu=0$[]{data-label="fig.TB"}](muSbeta.pdf "fig:"){width="100.00000%"} \[fig.a\]
![Continuity of optimal barriers at $\mu=0$[]{data-label="fig.TB"}](muLbeta.pdf "fig:"){width="100.00000%"} \[fig.b\]
From Figure \[fig.a\] (high proportional transaction cost $1-\beta$), we can see that around the neighbourhood of $0$, the periodic $0$ strategy is optimal. On the other hand, when the proportional cost is low, we can see from Figure \[fig.b\] that a liquidation $(b,\infty)$, or equivalently a hybrid $(0,0,b)$ strategy is optimal around the neighbourhood of $0$. This observation is true in general thanks to Lemma \[L\_conv\] below.
Further analysis of Figure \[fig.a\] is interesting. The vertical grey dashed line in Figure \[fig.a\] corresponds to the threshold between the second-last and last columns of Table \[S\_map\] (in the second row), so that obviously $\beta_0<0.7$ here. The right-hand side (where $b_0$ becomes strictly positive) corresponds to the first column.
\[L\_conv\] When $\beta>{\gamma}/{(\gamma+\delta)}$, a hybrid $(0,0,b)$ strategy is optimal for $\mu=0$. Moreover, when $\mu\uparrow 0$, the lower barrier of the optimal $(b_1,\infty)$ strategy, denoted as $b_1=b_1(\mu)$ converges to $b$. When $\beta\leq {\gamma}/{(\gamma+\delta)}$, the periodic barrier strategy with barrier level $0$, $\pi_0$, is optimal on the neighbourhood of $\mu=0$.
For each case, one needs to check the corresponding barrier converges to the barrier at $\mu=0$ when $\mu\uparrow 0$; see details in Appendix \[A.Lemma9.1\].
When $\beta\downarrow \gamma/(\gamma+\delta)$ {#SubS.Convergence.Beta}
---------------------------------------------
The convergence of the barriers when $\beta\downarrow{\gamma}/{(\gamma+\delta)}$ is described in the following proposition.
\[Lemma.convergence.beta\] Recall the function $Q$ in . When $\beta\downarrow\frac{\gamma}{\gamma+\delta}$, we have the following:
1. If $\frac{-s_1}{r_1}\frac{\gamma}{\gamma+\delta}>1$, $$a_p=Q^{-1}\Big(\frac{s_1\frac{\delta}{\gamma+\delta}}{r_1+s_1}\Big),\quad a_c=\infty,\quad b=\infty,$$ where “$=$” is in limit sense.
2. If $\frac{-s_1}{r_1}\frac{\gamma}{\gamma+\delta}\leq 1$, $$a_p=0,\quad a_c=\infty,\quad b=\infty,$$ where “$=$” is in limit sense.
3. $Q^{-1}\Big(\frac{s_1\frac{\delta}{\gamma+\delta}}{r_1+s_1}\Big)=b_0^*$ and $\lim_{a_c\rightarrow \infty,b\rightarrow\infty}V(x;\pi_{a_p,a_c,b})=V(x;\pi_{b_0^*})$ for all $x\geq 0$. That is, both the barrier and the value function exhibit continuity behaviour at $\beta=\frac{\gamma}{\gamma+\delta}$.
The proof requires functions $Q$ and $I(x,q)$ introduced in Section \[Remark.suff.cons\]. The first two parts requires an investigation of the condition $I(l,Q(a))=I(l+y,Q(a))$, whereas the last part calculates require a verification of the condition $V'''(b^*_0+)=V'''(b^*_0-)$ with the given formula for $b^*_0$. Details are provided in Appendix \[A.proof.Convergence.beta\].
Continuity for different cases in Theorem \[Thm.mu.neg\]
--------------------------------------------------------
Note when $\mu<0$, similar continuity results hold, as shown in Appendix \[A\_Thm.mu.neg.cont\] sequentially for the $4$ different cases enumerated in Theorem \[Thm.mu.neg\].
Numerical illustrations {#S.Numerical}
=======================
In this section, We illustrate numerically some results from previous sections. The first and the second sections are devoted to the case when $\mu>0$ and $\mu<0$ respectively. The [connection between both cases (when $\mu=0$) has been discussed in Section \[S.VaryingMu\].]{}
When the business is profitable ($\mu> 0$) {#S.numericalP}
------------------------------------------
Our baseline setting includes: scale parameters $(\mu,\sigma,\chi)=(1,0.3,0.01)$, time parameters $(\gamma,\delta)=(1,0.15)$ and proportional transaction cost parameter $\beta=0.0$. In particular, we have $\beta>{\gamma}/{(\gamma+\delta)}$ which guarantees that the hybrid $(a_p^*,a_c^*,b^*)$ strategy is optimal. In the following, we will study the impact of the parameters on the optimal barriers.
### Transaction costs
\[fig.muP.costs\]
![Impact of transaction costs. Solid vertical line: $\beta={\gamma}/{(\gamma+\delta)}$. Dotted line: approximation line (see Appendix \[S\_comput\])](chi.pdf "fig:"){width="100.00000%"} \[fig.muP.a\]
![Impact of transaction costs. Solid vertical line: $\beta={\gamma}/{(\gamma+\delta)}$. Dotted line: approximation line (see Appendix \[S\_comput\])](costP1.pdf "fig:"){width="100.00000%"} \[fig.muP.b\]
![Impact of transaction costs. Solid vertical line: $\beta={\gamma}/{(\gamma+\delta)}$. Dotted line: approximation line (see Appendix \[S\_comput\])](costP2.pdf "fig:"){width="100.00000%"} \[fig.muP.c\]
Figure \[fig.muP.a\] plots the barriers $(a_p^*,a_c^*,b^*)$ when the fixed cost $\chi$ increases from $0.001$ to $0.1$. As we can see, the increase in $\chi$ is compensated primarily by the increase in $b^*$, with almost insignificant drops in both $a_p^*$ and $a_c^*$. This makes sense because with an increased difficulty in paying dividends outside the periodic times, one would simply choose to pay more often at the periodic times. Although not obvious in the figure, one should expect that $a_p^*$ and $b^*$ coincide when $\chi=0$. This corresponds to the special case described in @AvTuWo16 [without fixed transaction costs].
Figure \[fig.muP.b\] plots the barriers $(a_p^*,a_c^*,b^*)$ when the proportional cost rate $1-\beta$ increases from $0$ to $0.15$, i.e. across and beyond the threshold ${\delta}/({\gamma+\delta})$. As $1-\beta$ increases from $0$, the two barriers $a_p^*$ and $a_c^*$ split. In addition, from Figure \[fig.muP.c\], we can see that while $a_p^*$ decreases with a converging behaviour (to $b_0^*$), $a_c^*$ is increasing with a diverging behaviour, as predicted and due to Lemma \[Lemma.convergence.beta\]. As another illustration of Lemma \[Lemma.convergence.beta\], there is a continuity behaviour between $a_p^*$ and the optimal periodic barrier $b_0^*$ at $\beta={\gamma}/{(\gamma+\delta)}$.
### Volatility
\[fig.muP.volatility\]
{width="100.00000%"} \[fig.muP.d\]
{width="100.00000%"} \[fig.muP.d2\]
Figure \[fig.muP.d\] plots the barriers $(a_p^*,a_c^*,b^*)$ when the volatility parameter $\sigma$ increases from $0.01$ to $20$. When the volatility is small, the business is virtually riskless and excess capital is not needed as a buffer. Therefore, both $a_p^*$ and $a_c^*$ are close to zero. When $\sigma$ increases, the business becomes more risky and hence all 3 barriers increase. However, as $\sigma$ further increases beyond a certain level, the business is deemed too risky and early exit would be a better choice. This is reflected by the decrease in the lower barrier $a_p^*$. Furthermore, we can see from Figure \[fig.muP.d2\] that $a_c^*$ is also going down eventually but the behaviour of $b^*$ is unclear. Heuristically we expect that $b^*\uparrow\infty$ so that the optimal strategy converges to a “liquidation at first opportunity” strategy. The idea is that $\sigma\uparrow\infty$ is equivalent to $\mu,\kappa\downarrow 0$ (after scaling), where a hybrid $(0,0,b)$ strategy is optimal (Lemma \[L\_conv\]). Converting to the original scale, we have $b^*\uparrow\infty$.
### Time parameters
![Sensitivities to the time parameters Solid line: $\beta={\gamma}/{(\gamma+\delta)}$. Dotted line: approximation line (see Appendix \[S\_comput\])](gammaP.pdf "fig:"){width="100.00000%"} \[fig.muP.e\]
![Sensitivities to the time parameters Solid line: $\beta={\gamma}/{(\gamma+\delta)}$. Dotted line: approximation line (see Appendix \[S\_comput\])](deltaP.pdf "fig:"){width="100.00000%"} \[fig.muP.f\]
Figure \[fig.muP.e\] plots the barriers $(a_p^*,a_c^*,b^*)$ when the dividend frequency parameter $\gamma$ increases from $0.01$ to beyond $1.5$. It is clear that all three barriers are increasing with $\gamma$. This is consistent with the intuition that with more frequent chances to pay periodic dividends (which attract no fixed costs), one does not have the urgency to pay more which puts the company at risk. When $\gamma$ increases to the point that ${\gamma}/{(\gamma+\delta)}$ approaches $\beta$, both $a_c^*$ and $b^*$ should increase to infinity while $a_p^*$ increases to $b_0^*$, which resemble a periodic $b_0^*$ strategy. This behaviour is similar to the change in $1-\beta$ studied in Figure \[fig.muP.b\].
Finally, Figure \[fig.muP.f\] plots the barriers $(a_p^*,a_c^*,b^*)$ when the time preference parameter $\delta$ ranges from $0.01$ to $3$. Unsurprisingly, the effect is qualitatively the reverse of that of $\gamma$ in Figure \[fig.muP.e\], with also a smooth connection with the periodic $b_0^*$ strategy.
When the business is unprofitable ($\mu<0$) {#S.NumericalN}
-------------------------------------------
Our baseline setting includes: scale parameters $(\mu,\sigma,\chi)=(-1,0.3,0.15)$, time parameters $(\gamma,\delta)=(1,0.15)$ and proportional transaction cost parameter $\beta=0.7$. Section \[SubS.cost.N\] explores the impact of the 2 types of costs to the optimal barriers. Following that, Section \[SubS.sens.N\] illustrates the sensitivities of other parameters to the optimal barriers.
### Transaction costs {#SubS.cost.N}
We start by discussing the impact of the two types of transaction costs (proportional and fixed) on the optimal barriers. Remember that $\beta\le 1$ is the ratio of net dividends of immediate dividends, as compared to periodic (see Remark \[R\_betacp\]). This means that $1-\beta$ is the net level of proportional transaction costs and high levels further penalise the immediate dividends as compared to the periodic ones. Recall as well that immediate liquidation occurs as soon as the surplus level its the area between $b_1$ and $b_2$, and at first opportunity otherwise.
![Interplay between proportional and fixed transaction costs. An empty region means $\pi_0$ is optimal.[]{data-label="fig.muN.betaKappa"}](CostNeg1.pdf "fig:"){width="100.00000%"} \[fig.muN.betaKappaSmall\]
![Interplay between proportional and fixed transaction costs. An empty region means $\pi_0$ is optimal.[]{data-label="fig.muN.betaKappa"}](CostNeg2.pdf "fig:"){width="100.00000%"} \[fig.muN.betaKappaLarge\]
Figure \[fig.muN.betaKappa\] illustrates the change in the optimal barriers $(b_1^*,b_2^*)$ with increasing proportional transaction cost $1-\beta$ and different fixed transaction costs $\chi$. First, when $\chi$ is relatively large, the periodic $0$ strategy is optimal for $1-\beta>{\delta}/{(\gamma+\delta)}$ (high proportional transaction cost), which is evident in Figure \[fig.muN.betaKappaLarge\], compared to Figure \[fig.muN.betaKappaSmall\]. If we imagine the periodic $0$ strategy as a liquidation $(b_1,b_2)$ strategy with both barriers being infinity, then the two graphs are consistent. Hence, we can focus on Figure \[fig.muN.betaKappaSmall\].
From Figure \[fig.muN.betaKappaSmall\], we can see that when the proportional transaction cost $1-\beta$ decreases, immediate dividends become optimal, and the associated two barriers appear and diverge. The upper barriers increase to infinity when $1-\beta$ approaches ${\delta}/{(\gamma+\delta)}$ from above, and the lower barrier stabilises to a certain level. On the other hand, we can see that the two barriers degenerate to one level which corresponds to the periodic $0$ strategy when the proportional transaction cost $1-\beta$ is large. This continuity feature is quite surprising and remarkable, especially the continuity of the lower barrier $b_1^*$ at $\beta={\gamma}/{(\gamma+\delta)}$.
The collapse of the the area between the two barriers is quite intuitive as an increase in the cost $1-\beta$ makes the decision to liquidate the company immediately very expensive compared to waiting for the next dividend decision time and liquidate at that first opportunity. Further, when $1-\beta$ is too large, we totally ignore the option to liquidate the company immediate and choose to wait. Similarly, when the fixed cost $\chi$ increases, the option to liquidate the company now becomes less favourable. This is indicated by the smaller area covered by the 2 dotted lines compared to the solid lines, when $\chi$ increases from $0.15$ to $0.3$. Obviously, when $\chi$ increases, we should expect an increase in $b_1^*$, as displayed in Figure \[fig.muN.betaKappaSmall\].
### Sensitivities {#SubS.sens.N}
![Sensitivities to parameters. Empty region means $\pi_0$ is optimal.[]{data-label="fig.muN"}](gammaN.pdf "fig:"){width="100.00000%"} \[fig.muN.a\]
![Sensitivities to parameters. Empty region means $\pi_0$ is optimal.[]{data-label="fig.muN"}](deltaN.pdf "fig:"){width="100.00000%"} \[fig.muN.b\]
![Sensitivities to parameters. Empty region means $\pi_0$ is optimal.[]{data-label="fig.muN"}](sigmaN.pdf "fig:"){width="100.00000%"} \[fig.muN.c\]
Figure \[fig.muN\] displays the sensitivities of the barriers to the parameters $\gamma$, $\delta$ and $\sigma$. Note that the (baseline) fixed cost $\chi$ is chosen to be “small” to showcase the presence of two barriers. When $\gamma$ increases, the chance of being able to liquidate the company at low cost improves. This favours the option to wait instead of liquidating the company now and is clearly indicated in Figure \[fig.muN.a\] where the area between the two barriers is shrinking. The opposite effect is present for the impatience parameter $\delta$. Note if we chose a larger base value for $\gamma$, we will see both barriers meet just as in Figure \[fig.muN.a\]. This is because $\delta$ and $\gamma$ have somewhat inverse roles, and are both functions of how time is defined.
Because the company is non-profitable (negative $\mu$), waiting is speculative because there is nothing left if the company gets ruined before it is liquidated. Figure \[fig.muN.c\] shows that increased volatility makes such a speculation increasingly worthwhile. When $\sigma$ is low, the lower barrier to be very close to ${\chi}/{\beta}$, which means we will liquidate the company as long as the outcome gives us a positive value, since there is no chance of recovering: indeed the negative drift $\mu<0$ will occur with little chance of being compensated by a positive random diffusion path because $\sigma$ is too low. On the other hand a very high $\sigma$ means it is worth trying one’s luck and wait. Note that Figure \[fig.muN.c\] uses $\beta\leq {\gamma}/{\gamma+\delta}$; the case when $\beta>{\gamma}/{\gamma+\delta}$ is similar, except we do not have $b_2^*$ as it is infinity.
Conclusion {#S.conclusion}
==========
In this paper, we considered a diffusion model for the retained cash earnings of a risk business, and studied comprehensively its optimal control via dividends (cash payments) of two different types as observed in real life. Under realistic transaction cost assumptions, we were able to replicate dividend payment behaviour actually observed as optimal. In particular, for realistic ranges of parameters a *hybrid* dividend strategy is optimal, whereby periodic dividends are paid regularly, and extraordinary dividends are paid when the surplus becomes too high. All results summarised in Section \[S\_map\] and Table \[T\_roadmap\] are rigorously shown in the paper and its online supplements.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper was presented at the 23rd International Congress on Insurance: Mathematics and Economics (IME) in July 2019 (Munich, Germany) and at the 54th Actuarial Research Conference (ARC) in August 2019 (Purdue University, USA). The authors are grateful for constructive comments received from colleagues who attended those events.
This research was supported under Australian Research Council’s Linkage (LP130100723) and Discovery (DP200101859) Projects funding schemes. Hayden Lau acknowledges financial support from an Australian Postgraduate Award and supplementary scholarships provided by the UNSW Australia Business School. The views expressed herein are those of the authors and are not necessarily those of the supporting organisations.
References {#references .unnumbered}
==========
Proof of Lemma \[Verification.lemma\] {#A.ver.lemma}
=====================================
By the definition of $v$, it suffices to show that under the hypothesis, we have $H(x)\geq V(x;\pi)$ for all $\pi\in\Pi$.
We first prove the case when $D^\pi(0)=0$, i.e. there is no dividend at time $0$.
Since $H\in \mathscr{C}^1(\mathbb{R}^+)\cap\mathscr{C}^2(\mathbb{R}^+\backslash E)$, we need to use Itō-Meyer [e.g. Thm IV.70 in @Pro05], where @Pes05 shows that $H\in\mathscr{C}^1$ is enough to kill the local time at $E$. As a result, we can still apply the Itō Lemma in its standard form.
There is nothing to prove when $x=0$, see . Hence, we assume $x>0$. For each $n\in\mathbb{N}$, we define a family of increasing stopping time $(T_n,n\in\mathbb{N})$ with $T_n:=\inf\{t> 0: X^\pi(t)> n\text{ or }X^\pi(t)< \frac{1}{n}\}$. By applying the Itō Lemma to the semi-martingale $\{e^{-\delta(t\wedge T_n)}H(X^\pi(t\wedge T_n));t\geq 0\}$ (with $a\wedge b=\min(a,b)$ for $a,b\in\mathbb{R}$), conditioning on the event $\{X(0)=x\}$, we have $$\begin{aligned}
&e^{-\delta(t\wedge T_n)}H(X^\pi(t\wedge T_n))-H(x)\\
=~&\int_0^{t\wedge T_n}-\delta e^{-\delta s}H(X^\pi(s-))ds+\int_0^{t\wedge T_n}e^{-\delta s} H'(X^\pi(s-))dX^\pi(s)\\&+\frac{1}{2}\int_0^{t\wedge T_n}e^{-\delta s}H''(X^\pi(s-))1_{\{X^\pi(s-)\notin E\}}d[X^\pi,X^\pi]^c(s)\\&+\sum_{0<s\leq t\wedge T_n}e^{-\delta s}\Big(H(X^\pi(s))-H(X^\pi(s-))-H'(X^\pi(s-))\Delta X^\pi(s)\Big),\end{aligned}$$ where for a function $F$, $\Delta F(t)=F(t)-F(t-)=F(t)-\lim_{s\uparrow t}F(s)$.
As $X^\pi=X-D^\pi$, $X$ is a diffusion process and $D^\pi$ is a finite variation (FV) process, we have that $d[X^\pi,X^\pi]^c(s)=d[X,X]^c(s)=\sigma^2 ds$. On the other hand, $X$ being a diffusion implies that all the jumps in $X^\pi$ come from $D^\pi$. Therefore, we can rewrite the above as [$$\begin{aligned}
&e^{-\delta(t\wedge T_n)}H(X^\pi(t\wedge T_n))-H(x)\\
=~&\int_0^{t\wedge T_n}-\delta e^{-\delta s}H(X^\pi(s-))ds+
\int_0^{t\wedge T_n} e^{-\delta s}H'(X^\pi(s-))dX(s)\\&-\int_0^{t\wedge T_n}e^{-\delta s}H'(X^\pi(s-))dD^\pi(s)+\frac{\sigma^2}{2}\int_0^{t\wedge T_n}e^{-\delta s}H''(X^\pi(s-))1_{\{X^\pi(s-)\notin E\}} ds\\&+\sum_{0<s\leq t\wedge T_n}e^{-\delta s}\Big(H(X^\pi(s-)-\Delta D^\pi(s))-H(X^\pi(s-))+H'(X^\pi(s-))\Delta D^\pi(s)\Big)\\
=~&\int_0^{t\wedge T_n}e^{-\delta s}(\mathscr{A}-\delta)H(X^\pi(s-))1_{\{X^\pi(s-)\notin E\}}ds+
\int_0^{t\wedge T_n} e^{-\delta s}H'(X^\pi(s-))\sigma dW(s)\\&+\sum_{0<s\leq t\wedge T_n}e^{-\delta s}\Big(H(X^\pi(s-)-\Delta D^\pi(s))-H(X^\pi(s-))\Big),\end{aligned}$$]{} where $W=\{W(t);t\geq 0\}$ is a standard Brownian motion. We now decompose $D^\pi$ into $D^\pi_p(t)=\int_0^tdD^\pi_p(s)dN_\gamma(s)$ and $D^\pi_c(t)$, where we denote the jump times of $D^\pi_c$ as ${\mathcal{T}}$. In this sense, after some algebraic effort, we can rewrite the above as $$\begin{aligned}
&e^{-\delta(t\wedge T_n)}H(X^\pi(t\wedge T_n))-H(x)\\
=~&\int_0^{t\wedge T_n}e^{-\delta s}\Big((\mathscr{A}-\delta)H(X^\pi(s-))+\gamma\Big(\Delta D^\pi_p(s)+H(X^\pi(s-)-\Delta D^\pi_p(s))-H(X^\pi(s-))\Big)\Big)1_{\{X^\pi(s-)\notin E\}}ds\\&+\int_0^{t\wedge T_n}e^{-\delta s}\Big(\Delta D^\pi_p(s)+H(X^\pi(s-)-\Delta D^\pi_p(s))-H(X^\pi(s-))\Big)(dN_\gamma(s)-\gamma ds)\\
&+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}\Big(\beta \Delta D^\pi_c(s)-\chi+H(X^\pi(s-)-\Delta D^\pi_c(s))-H(X^\pi(s-))\Big)\\&+
\int_0^{t\wedge T_n} e^{-\delta s}H'(X^\pi(s-))\sigma dW(s)\\
&-\Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta \Delta D^\pi_c(s)-\chi)\Big).\end{aligned}$$ By denoting [$$M(t):=\int_0^{t}e^{-\delta s}\Big(\Delta D^\pi_p(s)+H(X^\pi(s-)-\Delta D^\pi_p(s))-H(X^\pi(s-))\Big)(dN_\gamma(s)-\gamma ds)+\int_0^{t} e^{-\delta s}H'(X^\pi(s-))\sigma dW(s),$$]{} we can rewrite the above as [$$\begin{aligned}
&H(x)\\=~&e^{-\delta(t\wedge T_n)}H(X^\pi(t\wedge T_n))\\
&-\int_0^{t\wedge T_n}e^{-\delta s}\Big((\mathscr{A}-\delta)H(X^\pi(s-))+\gamma\Big(\Delta D^\pi_p(s)+H(X^\pi(s-)-\Delta D^\pi_p(s))-H(X^\pi(s-))\Big)\Big)1_{\{X^\pi(s-)\notin E\}}ds\\
&-\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}\Big(\beta\Delta D^\pi_c(s)-\chi+H(X^\pi(s-)-\Delta D^\pi_c(s))-H(X^\pi(s-))\Big)\\
&+\Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big)-M(t\wedge T_n).\end{aligned}$$]{} Now, by hypothesis (Conditions 1,2,4,5), the first 3 lines on the right hand side of the equation are non-negative, which implies $$H(x)\geq \Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big)-M(t\wedge T_n)$$ Note that $M$ is a zero-mean martingale as all the terms in the inegral are finite, by hypothesis (Condition 3). Hence, by taking expectation, we have $$H(x)\geq {\mathbb{E}_x}\Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big).$$ Finally, we observe that $T_n\rightarrow \tau^\pi$ a.s. and the terms inside the expectation are non-negative. Hence, by applying Fatou’s Lemma, we get $$\begin{aligned}
H(x)\geq~&\liminf_{t,n\uparrow \infty} {\mathbb{E}_x}\Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big)\\
\geq ~&{\mathbb{E}_x}\Big(\liminf_{t,n\uparrow \infty}\Big(\int_0^{t\wedge T_n}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0, t\wedge T_n]\cap {\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big)\Big)\\
=~&{\mathbb{E}_x}\Big(\int_0^{\tau^\pi}e^{-\delta s}\Delta D^\pi_p(s)dN_\gamma(s)+\sum_{s\in(0,\tau^\pi]\cap{\mathcal{T}}}e^{-\delta s}(\beta\Delta D^\pi_c(s)-\chi)\Big)\\
=~&V(x;\pi),\end{aligned}$$ which completes the proof for strategies $\pi\in\Pi$ such that $D^\pi(0)=0$. For strategies $\pi\in\Pi$ such that $D^\pi(0)>0$, we denote $\widetilde{\pi}$ the same strategy for $t>0$, i.e. $D^{\widetilde{\pi}}(t)=\{D^\pi_p(t),D^\pi_c(t)-D^\pi(0)\}$. Then we have $$V(x;\pi)={\mathbb{E}_x}(\beta D^\pi(0)-\chi+V(x-D^\pi(0);\widetilde{\pi}))\leq{\mathbb{E}_x}( \sup_{\xi\in(0,x]}\Big(\beta \xi-\chi+H(x-\xi)\Big))\leq H(x)$$ by an application of the previous result for $\widetilde{\pi}$ and Condition 5.
Proof of Proposition \[prop.1\] {#A_Prop55}
===============================
Note Existence is established in Proposition \[prop.2\] and here we assume $(a^*,l^*,y^*)$ exists.
It is clear from that the objective function is differentiable w.r.t. $(a,l,y)$. Therefore, being optimal implies the partial derivatives are zero (except at the boundary). It is straight-forward to show $${\frac{\partial}{\partial y}}\frac{V(a)}{{W_{\delta}}(a)}=0 \iff V'(b)=\beta.$$ From this, we see that at $(a^*,l^*,y^*)$ $$\begin{aligned}
&0={\frac{\partial}{\partial y}}\Big(V(a_c)-\beta a_c\Big)={\frac{\partial}{\partial y}}\Big(\frac{V(a)}{{W_{\delta}}(a)}G(a,l)\Big)=\Big({\frac{\partial}{\partial y}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,l)\\
\implies ~&{\frac{\partial}{\partial y}}\frac{V(a)}{{W_{\delta}}(a)}=0\iff V'(b)=\beta.
\end{aligned}$$ A further calculation (to appear later in ) shows that it is never optimal to have $y=\kappa$ (i.e. at the boundary) so the equality above always hold.
Now, using $$V(a_c)+\beta(y-\kappa)=V(b)=\frac{V(a)}{{W_{\delta}}(a)}G(a,y+l)-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(y+l),$$ we get $$\begin{aligned}
{\frac{\partial}{\partial l}}\Big(V(a_c)-\beta a_c\Big)
=~&{\frac{\partial}{\partial l}}\Big(\frac{V(a)}{{W_{\delta}}(a)}G(a,y+l)-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(y+l)\Big)-\beta\\
=~&\Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,y+l)+V'(b)-\beta\\
=~&\Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,y+l)
\end{aligned}$$ so we have $$\begin{aligned}
{\frac{\partial}{\partial l}}\Big(V(a_c)-\beta a_c\Big)={\frac{\partial}{\partial l}}V(a_c)-\beta
=~&{\frac{\partial}{\partial l}}\Big(\frac{V(a)}{{W_{\delta}}(a)}G(a,l)-\gamma{\overline{\overline{W}}_{\gamma+\delta}}(l)\Big)-\beta\\
=~&\Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,l)+\frac{V(a)}{{W_{\delta}}(a)}{\frac{\partial}{\partial l}}G(a,l)-\gamma{\overline{W}_{\gamma+\delta}}(l)-\beta\\
=~&\Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,l)+V'(a_c)-\beta\\
\implies \Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)\Big(G(a,y+l)-G(a,l)\Big)=~&V'(a_c)-\beta
\end{aligned}$$ so we have $$V'(a_c)=\beta.$$ if $l^*>0$. Otherwise, if $l^*=0$ (i.e. at the boundary), we see that $l\mapsto V(a_c)-\beta a_c$ is decreasing in $l$ near zero, i.e. $$0\geq{\frac{\partial}{\partial l}}\Big(V(a_c)-\beta a_c\Big)=\Big({\frac{\partial}{\partial l}}\frac{V(a)}{{W_{\delta}}(a)}\Big)G(a,y+l)$$ and therefore by noting $G(a,y+l)>G(a,l)$ we get $$V'(a_c)\leq \beta.$$
Similarly, we have $$\begin{aligned}
&{\frac{\partial}{\partial a}}\Big(V(a_c)-\beta a_c\Big)={\frac{\partial}{\partial a}}V(b)-\beta\\
=~&G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}{\frac{\partial}{\partial a}}G(a,y+l)-\beta\\
=~&G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}\Big({\frac{\partial}{\partial (y+l)}}G(a,y+l)-\gamma{W_{\delta}}'(a){\overline{W}_{\gamma+\delta}}(y+l)\Big)-\beta\\
=~&G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}\Big({\frac{\partial}{\partial (y+l)}}G(a,y+l)-\gamma{\overline{W}_{\gamma+\delta}}(y+l)\Big)-\beta\\&+\gamma{\overline{W}_{\gamma+\delta}}(y+l)\Big(1-\frac{V(a)}{{W_{\delta}}(a)}{W_{\delta}}'(a)\Big)\\
=~&G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\Big(V'(b)-\beta\Big)+\gamma{\overline{W}_{\gamma+\delta}}(y+l)\Big(1-V'(a)\Big)\\
=~&G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\gamma{\overline{W}_{\gamma+\delta}}(y+l)\Big(1-V'(a)\Big)
\end{aligned}$$ and $$\begin{aligned}
&{\frac{\partial}{\partial a}}\Big(V(a_c)-\beta a_c\Big)\\
=~&G(a,l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}{\frac{\partial}{\partial a}}G(a,l)-\beta\\
=~&G(a,l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}\Big({\frac{\partial}{\partial l}}G(a,l)-\gamma{W_{\delta}}'(a){\overline{W}_{\gamma+\delta}}(l)\Big)-\beta\\
=~&G(a,l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\frac{V(a)}{{W_{\delta}}(a)}\Big({\frac{\partial}{\partial l}}G(a,l)-\gamma{\overline{W}_{\gamma+\delta}}(l)\Big)-\beta+\gamma{\overline{W}_{\gamma+\delta}}(l)\Big(1-\frac{V(a)}{{W_{\delta}}(a)}{W_{\delta}}'(a)\Big)\\
=~&G(a,l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\Big(V'(a_c)-\beta\Big)+\gamma{\overline{W}_{\gamma+\delta}}(l)\Big(1-V'(a)\Big).
\end{aligned}$$
To summarise, we have $$\begin{aligned}
{\frac{\partial}{\partial a}}\Big(V(a_c)-\beta a_c\Big)=~&\Big(V'(b)-\beta\Big)+G(a,y+l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\gamma{\overline{W}_{\gamma+\delta}}(y+l)\Big(1-V'(a)\Big),\label{eq1}\\
{\frac{\partial}{\partial a}}\Big(V(a_c)-\beta a_c\Big)=~&\Big(V'(a_c)-\beta\Big)+G(a,l){\frac{\partial}{\partial a}}\frac{V(a)}{{W_{\delta}}(a)}+\gamma{\overline{W}_{\gamma+\delta}}(l)\Big(1-V'(a)\Big).\label{eq2}
\end{aligned}$$ Now, Assumption \[Ass0\] implies $$\label{use.ass0}
\Delta:=G(a,y+l)\gamma{\overline{W}_{\gamma+\delta}}(l)- G(a,l)\gamma{\overline{W}_{\gamma+\delta}}(y+l)<0.$$ Hence, we can eliminate the term with ${\frac{\partial}{\partial a}}(V(a)/{W_{\delta}}(a))$ to get $$\label{eq.con}
\Big(G(a,y+l)-G(a,l)\Big){\frac{\partial}{\partial a}}\Big(V(a_c)-\beta a_c\Big)=G(a,y+l)\Big(V'(a_c)-\beta\Big)+|\Delta| (V'(a)-1).$$
Now, suppose $a^*=\bar{a}$, then we have $$\frac{V(\bar{a})}{{W_{\delta}}(\bar{a})}\leq \frac{\frac{{W_{\delta}}(\bar{a})}{{W_{\delta}}'(\bar{a})}}{{W_{\delta}}(\bar{a})}=\frac{1}{{W_{\delta}}'(\bar{a})}\implies V'(a)=V'(\bar{a})=\frac{V(\bar{a})}{{W_{\delta}}(\bar{a})}{W_{\delta}}(\bar{a})\leq 1,$$ because the value function (of our strategy) is smaller in the current setting than the setting when there is no transaction costs (e.g. in [@Loe08]) and the optimal value function at $\bar{a}$ is given above. Hence the right hand side of the above equation is negative and so as the left hand side. This means it is impossible for $a^*=\bar{a}$ to be a maximiser for $V(a_c)-\beta a_c$. On the other hand, it is possible for $a^*=0$. In that case, we have ${\frac{\partial}{\partial a}}(V(a_c)-\beta a_c)\leq 0$. If furthermore $l^*>0$, we have $V'(a_c)=\beta $ and therefore we can conclude $V'(a)\leq 1$.
Suppose $a^*>0$, i.e. ${\frac{\partial}{\partial a}}(V(a_c)-\beta a_c)=0$. If $l^*=0$, we have $V'(a)=V'(a_c)\leq \beta$ which is a contradiction in view of . Therefore, we must have $l^*>0$ and therefore we can conclude $V'(a)=1$.
This completes the proof.
Proof of Proposition \[Prop.3\] {#A.1}
===============================
Note $\gamma{\overline{W}_{\gamma+\delta}}(x)=kH(x)$ for some positive constant $k$, and recall from equation and the definition of $J$ that $$\begin{aligned}
G(a,x)=~&g(x)\frac{f'(a)-s_1\frac{\delta}{\gamma+\delta}f(a)}{r_1-s_1}+e^{s_1x}\frac{\delta}{\gamma+\delta}f(a)+\frac{\gamma}{\gamma+\delta}f(a),\\
J(x)=~&g(x)(-s_1)+e^{s_1}(r_1-s_1)+(-(r_1-s_1)),\end{aligned}$$ we want to show that (for any $a\geq 0$) $$J(x){\frac{\partial}{\partial x}}G(a,x)-G(a,x){\frac{\partial}{\partial x}}J(x)<0,\quad x>0.$$ By direct computation, we see that $$\begin{aligned}
&J(x){\frac{\partial}{\partial x}}G(a,x)-G(a,x){\frac{\partial}{\partial x}}J(x)\nonumber\\
=~&-f'(a)\Big(g'(x)-(r_1-s_1)e^{(r_1+s_1)x}\Big)\label{eq.a}
\\&+s_1f(a)r_1g(x)\nonumber.\end{aligned}$$ Denote the function $F_1$ with $$F_1(x):=g'(x)-(r_1-s_1)e^{(r_1+s_1)x}.$$ It is easy to see $F_1(0)=0$ and $F_1'(x)>0$ for $x>0$ and hence we have $F_1(x)>0$ for $x>0$.
In view of the above equality $$J(x){\frac{\partial}{\partial x}}G(a,x)-G(a,x){\frac{\partial}{\partial x}}J(x)=-f'(a)F_1(x)+s_1f(a)r_1g(x),$$ we can conclude that $$J(x){\frac{\partial}{\partial x}}G(a,x)-G(a,x){\frac{\partial}{\partial x}}J(x)<0.$$
This completes the proof.
Proof of Proposition \[prop.2\] {#A.prop2}
===============================
The statement is a directly consequence of Propositions \[Prop.3\] and \[prop.1\]. Therefore, we are left to show the hypothesis in Proposition \[prop.1\].
Using the formulas in Proposition \[Prop.Vfcn\], we have $$\begin{aligned}
(r_1-s_1)A=~&\frac{\alpha(r_1-s_1)(y-\frac{\chi}{\alpha})(f'(a)-s_1\frac{\delta}{\gamma+\delta}f(a))+\frac{\gamma}{\gamma+\delta}(J(d,l)+s_1g(d,l)))}{\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)},\\
{\frac{\partial}{\partial l}}A=~&-A\frac{\frac{\delta}{\gamma+\delta}f(a)J'(d,l)+f'(a)g'(d,l)}{\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)},\\ (\text{with }J'(d,l)=~&J'(d)-J'(l),~g'(d,l)=g'(d)-g'(l))\\
\lim_{l\rightarrow \infty} C=~&\frac{\frac{\gamma\mu}{(\gamma+\delta)^2}-\frac{\gamma}{\gamma+\delta}\frac{1}{s_1}}{\frac{\delta}{\gamma+\delta}f(a)-\frac{f'(a)}{s_1}}<\infty,\\
\lim_{l\rightarrow\infty}\Big(g(l)(1+s_1\frac{g(d,l)}{J(d,l)})\Big)=~&\lim_{l\rightarrow\infty}\frac{(r_1-s_1)g(l)(e^{s_1d}-e^{s_1l})}{J(d,l)}=0,\\
\lim_{l\rightarrow\infty}Ag(l)=~&\frac{\alpha}{e^{r_1y}-1}\big(y-\frac{\chi}{\alpha}\big)<\infty,\\
\lim_{l\rightarrow\infty}\frac{g(d,l)}{J(d,l)}=~&\frac{-1}{s_1},\quad
\lim_{l\rightarrow\infty}\frac{g'(d,l)}{J(d,l)}=\frac{-r_1}{s_1},\quad
\lim_{l\rightarrow\infty}\frac{J'(d,l)}{J(d,l)}=r_1,\quad
\lim_{l\rightarrow\infty}{\frac{\partial}{\partial l}}C=0.\end{aligned}$$ This implies $$\begin{aligned}
\lim_{l\rightarrow\infty}{\frac{\partial}{\partial l}}(Ag(l))=~&\lim_{l\rightarrow\infty}\Big(Ag'(l)+g(l){\frac{\partial}{\partial l}}A\Big)\\=~&\lim_{l\rightarrow\infty}Ag(l)\Big(\lim_{l\rightarrow\infty}\frac{g'(l)}{g(l)}-\lim_{l\rightarrow\infty}\frac{\frac{\delta}{\gamma+\delta}f(a)J'(d,l)+f'(a)g'(d,l)}{\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)}\Big)\\
=~&\lim_{l\rightarrow\infty}Ag(l)(r_1-r_1)=0.\end{aligned}$$
Therefore, we have $$\begin{aligned}
\lim_{l\rightarrow\infty}{\frac{\partial}{\partial l}} (V(a_c)-\frac{\gamma}{\gamma+\delta}l)=~&\lim_{l\rightarrow\infty}{\frac{\partial}{\partial l}}(Ag(l))=0\end{aligned}$$ and hence $$\lim_{l\rightarrow\infty}{\frac{\partial}{\partial l}} (V(a_c)-\beta l)=\frac{\gamma}{\gamma+\delta}-\beta<0.$$ From this, we see that $V(a_c)-\beta a_c$ is decreasing for large enough $l$ (independent of $(a,y)$), say $\bar{l}$, i.e. $l\mapsto (V(a_c)-\beta a_c)$ cannot attain its local maximum for $l>\bar{l}$. Since we have already chosen $a\in[0,\bar{a}]$, we do not worry about the $a$ dimension.
On the other hand, we have $$\begin{aligned}
V(a_c)=~&C\Bigg(f(a)\Big(\frac{\gamma}{\gamma+\delta}+\frac{\delta}{\gamma+\delta}(e^{s_1l}-s_1\frac{g(l)}{r_1-s_1})\Big)+f'(a)\frac{g(l)}{r_1-s_1}\Bigg)\nonumber\\
&+\frac{g(l)}{r_1-s_1}(s_1\frac{\gamma\mu}{(\gamma+\delta)^2}-\frac{\gamma}{\gamma+\delta})-e^{s_1l}\frac{\gamma\mu}{(\gamma+\delta)^2}+\frac{\gamma}{\gamma+\delta}(l+\frac{\mu}{\gamma+\delta}).\label{Vac}\end{aligned}$$
It is easy to see that $C=V(a)/{W_{\delta}}(a)$, the terms inside the bracket after $C$ is $G(a,l)$, and the terms in the second line correspond to $\gamma{\overline{\overline{W}}_{\gamma+\delta}}(l)$.
We are left to work with the $y$ dimension. From , it is clear that in terms of $y$, the objective function $V(a_c)-\beta a_c$ solely depends on $C$, as we have also discovered before using scale functions. There is no shortcut but to compute the derivative w.r.t. $y$. From $$C=\frac{(r_1-s_1)\big((\beta-\frac{\gamma}{\gamma+\delta})(d-l)-\chi\big)+\frac{\gamma}{\gamma+\delta}g(d,l)+\frac{\gamma\mu}{(\gamma+\delta)^2}J(d,l)}{\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)},$$ we get (after some tedious algebric operations) $$\begin{aligned}
&{\frac{\partial}{\partial y}}C\times (\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l))^2\\
=~&\alpha(r_1-s_1)\Big(\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l)-(y-\frac{\chi}{\alpha})(\frac{\delta}{\gamma+\delta}f(a)J'(d)+f'(a)g'(d))\Big)\\
&+\frac{\gamma}{\gamma+\delta}f'(a)\Big(\frac{\mu}{\gamma+\delta}-\frac{\frac{\delta}{\gamma+\delta}f(a)}{f'(a)}\Big)\Big(g(d,l)J'(d)-J(d,l)g'(d)\Big),\end{aligned}$$ where $$\frac{\mu}{\gamma+\delta}-\frac{\frac{\delta}{\gamma+\delta}f(a)}{f'(a)} \geq 0$$ for $a\in[0,\bar{a}]$ and it can be checked by taking derivative w.r.t. $y$ that $$g(d,l)J'(d)-J(d,l)g'(d)=g'(d)e^{s_1l}-s_1g(l)e^{s_1d}-(r_1-s_1)e^{(r_1+s_1)d}> 0$$ for $y\geq {\chi}/{\alpha}$. This implies $$\label{PC.geq.0.min}
{\frac{\partial}{\partial y}}\frac{V(a)}{{W_{\delta}}(a)}\Big|_{y=\chi/\alpha}={\frac{\partial}{\partial C}}\Big|_{y=\chi/\alpha}>0,\quad (a,l)\in[0,\bar{a}]\times[0,\bar{l}].$$
Next, we take the limit $y\rightarrow\infty$, and see (after some algebraic operations) that $$\begin{aligned}
&{\frac{\partial}{\partial y}}C\times (\frac{\delta}{\gamma+\delta}f(a)J(d,l)+f'(a)g(d,l))^2\\
=~&f'(a)
\alpha(r_1-s_1)\Big(g(d,l)-(y-\frac{\chi}{\alpha})g'(d)+\frac{1}{\alpha(r_1-s_1)}\frac{\gamma\mu}{(\gamma+\delta)^2}\big(g'(d)e^{s_1l}-s_1g(l)e^{s_1d}-(r_1-s_1)e^{(r_1+s_1)d}\big)\Big)\\
&+\frac{\delta}{\gamma+\delta}f(a)\alpha(r_1-s_1)\Big(J(d,l)-(y-\frac{\chi}{\alpha})J'(d)+\frac{\gamma}{\gamma+\delta}\big(g'(d)e^{s_1l}-s_1g(l)e^{s_1d}-(r_1-s_1)e^{(r_1+s_1)d}\big)\Big)\\
\leq~&f'(a)
\alpha(r_1-s_1)\Big(g(d)-(y-\frac{\chi}{\alpha}-\frac{1}{\alpha(r_1-s_1)}\frac{\gamma\mu}{(\gamma+\delta)^2})g'(d)\Big)\\
&+\frac{\delta}{\gamma+\delta}f(a)\alpha(r_1-s_1)\Big(J(d)-(y-\frac{\chi}{\alpha})J'(d)+\frac{\gamma}{\gamma+\delta}g'(d)\Big)\end{aligned}$$ which drifts to $-\infty$ when $y\rightarrow\infty$. Hence, we can choose $\underline{d}$ such that $d>\underline{d}$ implies ${\frac{\partial}{\partial y}}C$ is decreasing for all $(a,l)\in[0,\bar{a}]\times[0,\bar{l}]$. In particular, we can choose $\bar{y}=(\bar{l}+\underline{d})\vee 2\chi/\alpha$ such that the same holds for $y\geq \bar{y}$.
To conclude, we have find a box for $\mathscr{B}:=[0,\bar{a}]\times[0,\bar{l}]\times[\chi/\alpha,\bar{y}]$ for $(a,l,y)$ such that
1. The objective function $V(a_c)-\beta a_c$ attains its maximum inside $\mathscr{B}$,
2. Its maximum $(a^*,l^*,y^*)$ either occurs in the interior of $\mathscr{B}$, or we have $a^*=0$ or $l^*=0$ or both, but not other cases.
This concludes the hypothesis in Proposition \[prop.1\] and hence completes the proof.
Proof of Lemma \[Lemma.VD.opt\] {#A.Lemma6.1}
================================
Denote $\widetilde{A}=A$ and $\widetilde{B}=B-A$ so that the derivative of the value function on $[a_p,b]$ is $$\label{eqt.VDnice.mid}
V'(a_p+x)=\widetilde{A}r_1e^{r_1x}+\widetilde{B}s_1e^{s_1 x}+\frac{\gamma}{\gamma+\delta},~x\in[0,d].$$ From $V'(b)=\beta$, we have $$V'(b)=\widetilde{A}r_1e^{r_1d}+\widetilde{B}s_1e^{s_1d}+\frac{\gamma}{\gamma+\delta}=\beta,$$ or $$\label{eqt.vbD.eq1}
\widetilde{A}r_1e^{r_1d}+\widetilde{B}s_1e^{s_1d}=\alpha.$$ Moreover, we have $$\label{eqt.vDD}
V''(a_p+x)=\widetilde{A}r_1^2e^{r_1x}+\widetilde{B}s_1^2e^{s_1x}$$ and $$\label{eqt.vDDD}
V'''(a_p+x)=\widetilde{A}r_1^3e^{r_1x}+\widetilde{B}s_1^3e^{s_1x}.$$
We first show that $\widetilde{A}>0$ by contradiction. Suppose $\widetilde{A}\leq0$ and $\widetilde{B}\geq0$, then the L.H.S. of (\[eqt.vbD.eq1\]) is negative, which is impossible. On the other hand, if we assume $\widetilde{A}\leq0$ and $\widetilde{B}<0$, then we have from (\[eqt.vDD\]) $V''< 0$, which implies that $V'$ is decreasing on $[a_p,b]$. However, from $V\in\mathscr{C}^1(\mathbb{R}_+)$, we have $$\int_{a_c}^{b-}(\beta-{V'(x)})dx=\chi>0,$$ which implies that $V'\geq\beta$ on $[a_c,b]$, which is also impossible.
Now we have established $\widetilde{A}>0$. If we further assume $\widetilde{B}\geq0$, then we have from (\[eqt.vDD\]) $V''\geq0$, which implies that $V'$ is increasing on $[a_p,b]$. Note that this would not be possible unless $V'(0)<\beta\implies a_c=a_p=0$ because otherwise we have $V'(a_c)=\beta=V'(b)$. Regardless, as $V'$ increases to $V'(b)=\beta$, we have $V'<\beta$ on $[0,b)$ and $V'\equiv\beta$ on $[b,\infty)$, which also holds if $b=0$. Furthermore, the fact that $V$ is positive (by the definition of the value function) implies that $V'(0)>0$, which in turn implies that $V'>0$ on $[0,\infty)$.
For the last case, $\widetilde{A}>0$ and $\widetilde{B}<0$, we can deduce from (\[eqt.vDDD\]) that $V'''\geq0$ on $[a_p,b]$, or equivalently, $V'$ is convex on $[a_p,b]$. This together with $V'(a_c)\leq \beta=V'(b)$ gives $V'\leq\beta$ on $[a_c,b]$. This fact combining with $V'(a_p)\leq 1$ shows that $V'$ is decreasing from $a_p$ to $a_c$, then further decreasing and finally increasing to $\beta$ at $b$, as $V'$ is convex on $[a_p,b]$, or simply increasing to $\beta$ if $a_p=a_c=0$ and $V'(0)<\beta$. Furthermore, in view of (\[eqt.VDnice.mid\]), we have that $V'>0$ on $[0,\infty)$.
\[Remark.VDDatB\] Note in any cases, we have $V''(b-)>0$ as $V'$ is increasing at $b-\varepsilon$ for all small enough $\varepsilon>0$. On the other hand, we have $V''(b+)=0$.
Proof of Lemma \[Lemma.VD.Nice\] {#A.Lemma6.2}
=================================
For $x> b$, we have $$\begin{aligned}
&(\mathscr{A}-\delta)V(x)+\gamma \Big(x-a_p+V(a_p)-V(x)\Big)\\
=~&(\mathscr{A}-\delta)V(b+)+\gamma \Big(b-a_p+V(a_p)-V(b)\Big)-(\gamma+\delta)(V(x)-V(b))+\gamma(x-b)\\
=~&(\mathscr{A}-\delta)V(b+)+\gamma \Big(b-a_p+V(a_p)-V(b)\Big)-(\gamma+\delta)\Big(\beta (x-b)-\frac{\gamma}{\gamma+\delta}(x-b)\Big)\\
\leq ~&(\mathscr{A}-\delta)V(b+)+\gamma \Big(b-a_p+V(a_p)-V(b)\Big)
\end{aligned}$$ as $\beta>{\gamma}/({\gamma+\delta})$ by the assumption in .
Together with Remark \[Remark.VDDatB\] and the fact that $V\in\mathscr{C}^1(\mathbb{R}_+)$, we have $$\begin{aligned}
0=~&(\mathscr{A}-\delta)V(b-)+\gamma \Big(x-a_p+V(a_p)-V(b)\Big)>(\mathscr{A}-\delta)V(b+)+\gamma \Big(x-a_p+V(a_p)-V(b)\Big)\nonumber\\
\geq ~&(\mathscr{A}-\delta)V(x)+\gamma \Big(x-a_p+V(a_p)-V(x)\Big)\label{eq.HJB1.bGeq0}
\end{aligned}$$ for $x>b$. Now, denote $$H_1(\xi):=\xi+V(x-\xi)-V(x)$$ and by taking derivative with respect to $\xi$, we have $$H_1^\prime(\xi)=1-V'(x-\xi).$$ In view of Lemma \[Lemma.VD.opt\], $V'(x-\xi)<1$ is equivalent to $x-\xi> a_p$, or equivalently $\xi< x-a_p$. Therefore, we can deduce that $H_1$ is increasing on $[0,a_p]$ if $a_p>0$ then decreasing on $(a_p,\infty)$, which implies that in any case it attains its maximum at $\xi=x-a_p$. Therefore, we have $$\begin{aligned}
&(\mathscr{A}-\delta)V(x)+\gamma \sup_{\xi\in[0,x]}\Big(\xi+V(x-\xi)-V(x)\Big)\\=~&\begin{cases}
(\mathscr{A}-\delta)V(x)+\gamma \Big(x-a_p+V(a_p)-V(x)\Big),\quad&x>b\\
(\mathscr{A}-\delta)V(x)+\gamma \Big(x-a_p+V(a_p)-V(x)\Big),\quad&a_p\leq x<b\\
(\mathscr{A}-\delta)V(x),\quad&x<a_p
\end{cases}\\
\leq ~&0
\end{aligned}$$ in view of (\[eq.HJB1.bGeq0\]) and the PDEs satisfied by the value function.
For the second equation, clearly the left hand side cannot be positive when $x\leq {\chi}/{\beta}$, since $V$ is increasing. For $x>{\chi}/{\beta}$, we denote $$H_2(\xi):=\beta\xi-\chi+V(x-\xi)-V(x),\quad \xi\geq 0.$$ Clearly, we have $H_2(0)=-\chi<0$. Taking derivative w.r.t. $\xi$, we get $$H_2^\prime(\xi)=\beta-V'(x-\xi).$$ In view of Lemma \[Lemma.VD.opt\], $V'(x-\xi)<\beta$ is equivalent to $x-\xi\in(a_c,b)$, or equivalently $x-b<\xi<x-a_c$. Therefore, we can deduce that $H_2$ is decreasing on $[0,x-b]$ if $x-b\geq 0$ then increasing on $[\max(x-b,0),x-a_c]$ if $x-a_c\geq 0$, then decreasing on $(a_c,\infty)$. Hence, on $[0,x]$, its maximum is attained at either $0$, or $x-a_c$ if $x-a_c>0$. Suppose $\xi=x-a_c>0$, we have $$H_2(x-a_c)=\beta(x-a_c)-\chi+V(a_c)-V(x),$$ which as a function of $x$, is increasing on $[a_c+{\chi}/{\beta},b]$. This implies that $$H_2(x-a_c)\leq H_2(b-a_c)=\beta(b-a_c)-\chi+V(a_c)-V(b)=0>H_2(0).$$ Therefore, $\sup_{\xi\in[0,x]}H_2(\xi)\leq 0$. This completes the proof as $H_2$ and the term inside the bracket in the second component differs only at $\xi=0$, where both of them have value less than or equal to $0$.
Proof of Equation {#A.Proof.Lambda}
==================
First, we have $\Lambda(V'(0;\pi_0))=0$ as $\beta=V'(0;\pi_0)$ implies that $a_\beta=0$. In addition, $\Lambda$ is an increasing function because $$\begin{aligned}
{\frac{\partial}{\partial \beta}}\Lambda(\beta)=~&{\frac{\partial}{\partial \beta}}a_\beta-\frac{V'(a_\beta;\pi_0){\frac{\partial}{\partial \beta}}a_\beta}{\beta}+\frac{V(a_\beta;\pi_0)}{\beta^2}\\
=~&\Big({\frac{\partial}{\partial \beta}}a_\beta\Big)\Big(1-\frac{V'(a_\beta;\pi_0)}{\beta}\Big)+\frac{V(a_\beta;\pi_0)}{\beta^2}\\
>~&0.\end{aligned}$$ Moreover, when $\beta\uparrow\frac{\gamma}{\gamma+\delta}$, we have $a_\beta\uparrow\infty$ and therefore $$\begin{aligned}
\lim_{\beta\uparrow\frac{\gamma}{\gamma+\delta}}\Lambda(\beta)=~&\lim_{x\uparrow\infty}(x-\frac{V(x;\pi_0)}{(\frac{\gamma}{\gamma+\delta})})\\
=~&\frac{\gamma+\delta}{\gamma}\lim_{x\uparrow \infty}(\frac{\gamma}{\gamma+\delta}x-\frac{\gamma\mu}{(\gamma+\delta)^2}(1-e^{s_1x})-\frac{\gamma}{\gamma+\delta}x)\\
=~&\frac{-\mu}{\gamma+\delta},\end{aligned}$$ where we have used Lemma \[Lemma.ValueFunction\] to compute $V(x;\pi_0)$.
Proof of Lemma \[Lemma.b1b2.Exist\] {#A.Lemma8.6}
====================================
Using the fact that the value function is continuous at $b_2$, we get $$\label{eqt.cont.at.b} Be^{s_1b_2}=\beta b_2-\chi-\frac{\gamma}{\gamma+\delta}b_2-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta},$$ which implies that $$V'(b_2+;\pi_{b_1,b_2})=\frac{\gamma}{\gamma+\delta}+Bs_1e^{s_1b_2}=(\beta b_2-\chi)s_1-\frac{\gamma}{\gamma+\delta}b_2s_1-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta}s_1+\frac{\gamma}{\gamma+\delta}$$ Therefore $V'(b_2+;\pi_{b_1,b_2})=\beta$ is equivalent to $$\label{eqt.for.b2}
s_1(\beta-\frac{\gamma}{\gamma+\delta}) b_2-\Big(\chi s_1+s_1\frac{\gamma\mu}{(\gamma+\delta)^2} -(\frac{\gamma}{\gamma+\delta}-\beta)\Big)=0.$$ Similarly, by replacing the equality of (\[eqt.cont.at.b\]) by “$\leq$” and $B$ by $\widetilde{B}=\frac{-\gamma\mu}{(\gamma+\delta)^2}$, we have $$\beta=V'(a_\beta;\pi_0)\geq \frac{\gamma}{\gamma+\delta}+\widetilde{B}s_1e^{s_1a_\beta}=(\beta a_\beta-\chi)s_1-\frac{\gamma}{\gamma+\delta}a_\beta s_1-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta}s_1+\frac{\gamma}{\gamma+\delta},$$ which shows that when $b_2=a_\beta$, the left hand side of (\[eqt.for.b2\]) is negative. As the left hand side of (\[eqt.for.b2\]) is linear in $b_2$ with positive coefficient, we establish the existence (and uniqueness) of $b_2\in(a_\beta,\infty)$.
For $b_1$, we note (from definitions of $c_{\beta,\chi}$, $\pi_{c_{\beta,\chi},b_2}$ and Remark \[Remark.mu.neg.obs\]) that $$A(c_{\beta,\chi})g(c_{\beta,\chi})+\beta c_{\beta,\chi}-\chi=V(c_{\beta,\chi};\pi_{c_{\beta,\chi},b_2})=\beta c_{\beta,\chi}-\chi,$$ which shows that $A(c_{\beta,\chi})=0$. Therefore, we have $$V'(c_{\beta,\chi};\pi_{c_{\beta,\chi},b_2})=A(c_{\beta,\chi})g'(c_{\beta,\chi})+V'(c_{\beta,\chi};\pi_0)=V'(c_{\beta,\chi};\pi_0)<V'(a_\beta;\pi_0)=\beta.$$ Similarly, we have $$V'(a_\beta-;\pi_{a_\beta,b_2})=A(a_\beta)g'(a_\beta)+V'(a_\beta;\pi_0)=A(a_\beta)g'(a_\beta)+\beta.$$ As $g'(a_\beta)>0$, if $A(a_\beta)\leq 0$, then Lemma \[Lemma.A.increasing\] implies that there is an interval $I\subsetneq (c_{\beta,\chi},a_\beta)$ where $A(b)$ is decreasing, since $A(b)$ is increasing on the neighbourhood of $c_\beta$. Again from Lemma \[Lemma.A.increasing\], we can deduce that $V'(b-;\pi_{b,b_2})>\beta$ for $b\in I$, which is what we have to show. If $A(a_\beta)>0$, then from the above equation, we have $V'(a_\beta-;\pi_{a_\beta,b_2})>\beta$ and hence by continuity, there is a $c_{\beta,\chi}<b<a_\beta$ such that $V'(b-;\pi_{b,b_2})=\beta$. For uniqueness, we will choose $b_1$ to be the smallest one if there are more than one such $b$.
Proof of Lemma \[Lemma.b.Exist\] {#A.Lemma8.7}
=================================
We first need to establish that for $\kappa>0$, it holds that $$\frac{-\mu}{\gamma+\delta}(1-e^{s_1\kappa})-\kappa<0.$$ Denote $h$ the left hand side of the above inequality, then we have obviously $h(0)=0$. By taking derivative with respect to $\kappa$ and note $r_1s_1({\mu}/({\gamma+\delta}))=r_1s_1$, we can have $$\begin{aligned}
h'(\kappa)=\frac{-\mu}{\gamma+\delta}(-s_1)e^{s_1\kappa}-1 =(e^{s_1\kappa}-1)+\frac{s_1}{r_1}e^{s_1k}
<0.
\end{aligned}$$
From Lemma \[Lemma.ValueFunction\], we have $$V'(b-;\pi_{b,\infty})=A(b)g'(b)-\frac{\gamma\mu}{(\gamma+\delta)^2}s_1e^{s_1b}+\frac{\gamma}{\gamma+\delta}$$ with $$A(b)=\frac{(\beta-\frac{\gamma}{\gamma+\delta})b-\chi-\frac{\gamma\mu}{(\gamma+\delta)^2}(1-e^{s_1b})}{g(b)}.$$ This implies that there is a $\widetilde{b}\geq 0$ such that $b>\widetilde{b}$ implies $A(b)>0$, and $$\lim_{b\rightarrow\infty}A(b)g'(b)=+\infty$$ and hence we are done if $$V'(\frac{\chi}{\beta}-;\pi_{\frac{\chi}{\beta},\infty})\leq \beta.$$ In particular, this would be the case if $A({\chi}/{\beta})<0$, since $g'({\chi}/{\beta})>0$ and from Lemma \[Lemma.ValueFunction\] we have $$V'(\frac{\chi}{\beta}-;\pi_{\frac{\chi}{\beta},\infty})=A(\frac{\chi}{\beta})g'(\frac{\chi}{\beta})+V'(\frac{\chi}{\beta};\pi_0)\leq A(\frac{\chi}{\beta})g'(\frac{\chi}{\beta})+\frac{\gamma}{\gamma+\delta}<A(\frac{\chi}{\beta})g'(\frac{\chi}{\beta})+\beta.$$ This is indeed the case, thanks to very first inequality developed in this section, $$A(\frac{\chi}{\beta})=\frac{\frac{\gamma}{\gamma+\delta}(\frac{-\mu}{\gamma+\delta}(1-e^{s_1\frac{\chi}{\beta}})-\frac{\chi}{\beta})}{g(\frac{\chi}{\beta})}<0.$$
Proof of Lemma \[L\_conv\] {#A.Lemma9.1}
==========================
Suppose $\beta\leq {\gamma}/{(\gamma+\delta)}$, then from Theorem \[Thm.Small.Beta\], we know that a periodic barrier strategy $\pi_b$ is optimal. Based on the observation in e.g. @AvTuWo14, we can conclude that $\pi_0$ is optimal for $\mu=0$ (we omit the proof here although a separate check is possible). Now, in view of Theorem \[Thm.mu.neg\], we are in Case 1: $\chi\geq\beta({-\mu}/({\gamma+\delta}))$ for small enough $|\mu|$ (note $\mu<0$). Hence, when $\beta\leq {\gamma}/{(\gamma+\delta)}$, we conclude that a periodic barrier strategy $\pi_0$ is optimal. Hence, the continuity is established.
For $\beta>{\gamma}/{(\gamma+\delta)}$, again in view of Theorem \[Thm.mu.neg\] Case 1, we can conclude that a liquidation $(b_1,\infty)$ is optimal, with the barrier $b_1$ such that the derivative of the value function at the barrier is $\beta$. Note that such strategy is also the same as a hybrid $(a_p,a_c,b)$ strategy with $a_p=a_c=0$ and $b=b_1$. Since the *optimal* hybrid $(a_c,a_p,b)$ strategy imposes that the derivative of the value function at the upper barrier $b$ is $\beta$ (which is unique by Corollary \[Corrolary.unique.barriers\]), the continuity of the barriers would be established if we can show that for $\mu=0$, the hybrid $(0,0,b)$ strategy is optimal.
The value function of a hybrid $(0,0,b)$ strategy, denoted by $V$ (instead of $V(\cdot;\pi_{0,0,b})$ for convenience), is given by $$V(x)=\begin{cases}
Ag(x)+\frac{\gamma}{\gamma+\delta}x,&x\leq b\\\beta x-\chi,&x>b
\end{cases},$$ which can be derived with ease. When $\mu=0$, the function $g(x)$ can be rewritten as $$g(x)=e^{r_1x}-e^{-r_1x},$$ which has the property $g''(x)=r_1^2g(x)>0$ for $x>0$. This implies that $$\label{eq.mu0.1}
g'(b)>g'(0).$$ On the other hand, by direct computation, we have $$V'(b)=Ag'(b)+\frac{\gamma}{\gamma+\delta}=\beta$$ which shows that $A>0$. Therefore, we have $$\begin{aligned}
V'(0)=~&Ag'(0)+\frac{\gamma}{\gamma+\delta}\\
\leq ~&Ag'(b)+\frac{\gamma}{\gamma+\delta}=\beta.
\end{aligned}$$ This shows that the hybrid $(0,0,b)$ strategy is optimal for $\mu=0$ (see Definition \[Def.Nice.hybrid\] and Theorem \[Thm\]) and completes the proof.
Proof of Lemma \[Lemma.convergence.beta\] {#A.proof.Convergence.beta}
=========================================
For a hybrid $(a_p,a_c,b)$ strategy, if $V'(a_c)=V'(b)=\beta$, we have $$V'(a)-\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+q(r_1+s_1)}=(r_1-s_1)I(y+l,q)=(r_1-s_1)I(l,q),$$ with $$\begin{aligned}
I(x,q)=~&
\frac{\beta-\frac{\gamma}{\gamma+\delta}+\Big(\frac{\gamma}{\gamma+\delta}-\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+q(r_1+s_1)}\Big)
e^{s_1 x}}{g'(x)+g(x)(-r_1s_1)\frac{\mu}{\gamma+\delta}(1-q)},\\
q:=~&Q(a)=1-\frac{f(a)/f'(a)}{\mu/\delta}\in[0,1].\end{aligned}$$
When $\beta\downarrow \frac{\gamma}{\gamma+\delta}$, from the fact that $q>0$ for our candidate strategy, it is easy to see that $y\uparrow \infty$. This implies that $I(y+l,q)\downarrow 0$. Suppose $\frac{-s_1}{r_1}\frac{\gamma}{\gamma+\delta}>1$, we get $V'(a_p)=\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+q(r_1+s_1)}= 1$, and Therefore, in order for $I(l,q)=I(y+l,q)$, we must have $$l\uparrow\infty,\quad a_p=Q^{-1}\Big(\frac{s_1\frac{\delta}{\gamma+\delta}}{r_1+s_1}\Big),\quad y\uparrow\infty.$$ Similarly, if $\frac{-s_1}{r_1}\frac{\gamma}{\gamma+\delta}\leq 1$, we have $$l\uparrow \infty,\quad a_p=0,\quad y\uparrow\infty.$$
If $a>0$, after some tedious algebra, we can show that $$\begin{aligned}
&(r_1-s_1)A+\frac{\gamma}{\gamma+\delta}=\frac{\gamma}{\gamma+\delta},\end{aligned}$$ implying that $A=0$.
Otherwise, if $a=0$, we have $q=1$ and therefore $\frac{-s_1\frac{\gamma}{\gamma+\delta}}{-s_1+q(r_1+s_1)}=\frac{\gamma}{\gamma+\delta}\frac{-s_1}{r_1}$. Similar to the above calculation, we also have $$\begin{aligned}
&(r_1-s_1)A+\frac{\gamma}{\gamma+\delta}=\frac{\gamma}{\gamma+\delta},\end{aligned}$$ implying that $A=0$. Therefore, when $\beta\downarrow{\gamma}/({\gamma+\delta})$, the value function $V(x;\pi_{a_p,a_c,b\uparrow\infty})$ converges to $V(x;\pi_{a_p})$, the value function of the periodic barrier strategy with barrier level $a_p$. We are left to verify that $a_p$ is the optimal barrier in the pure periodic setting, e.g. @NoPeYaYa17.
Recall from Remark \[Remark.suff.cons\] that the condition for $a=0$ are the same when $\beta\downarrow{\gamma}/({\gamma+\delta})$. For $a>0$, we have $Cf'(a)=\frac{-s_1\frac{\gamma}{\gamma+\delta}}{r_1+q(r_1+s_1)}=1$, which is the same as $$\frac{f(a)}{f'(a)}=(1-\frac{s_1\frac{\delta}{\gamma+\delta}}{r_1+s_1})\frac{\mu}{\delta}.$$ Use these, we get $$\begin{aligned}
V'''(a+)=Bs_1^3=~&\frac{\delta}{\gamma+\delta}Cf(a)s_1^3-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta}s_1^3\\
=~&\frac{\delta}{\gamma+\delta}\frac{f(a)}{f'(a)}Cf'(a)s_1^3-\frac{\gamma}{\gamma+\delta}\frac{\mu}{\gamma+\delta}s_1^3\\
=~&\frac{\delta}{\gamma+\delta}s_1^2\end{aligned}$$ and $$\begin{aligned}
f'''(a)=~&\frac{\gamma+\delta}{\sigma^2/2}\Big(\frac{\delta}{\gamma+\delta}f'(a)-\frac{\mu}{\gamma+\delta}f''(a)\Big)\\
=~&(-r_1s_1)\Big((\frac{\delta}{\gamma+\delta}-(r_1+s_1)\frac{\mu}{\gamma+\delta})f'(a)+(r_1+s_1)\frac{\delta}{\gamma+\delta}f(a)\Big).\end{aligned}$$ Therefore, we have $$\begin{aligned}
\frac{f'''(a)}{f'(a)(-r_1s_1)}=~&\frac{\delta}{\gamma+\delta}-(r_1+s_1)\frac{\mu}{\gamma+\delta}+(r_1+s_1)(1-\frac{s_1}{r_1+s_1}\frac{\delta}{\gamma+\delta})\frac{\mu}{\gamma+\delta}\\
=~&\frac{\delta}{\gamma+\delta}\frac{-s_1}{r_1},\end{aligned}$$ which further implies $$V'''(a-)=Cf'''(a)=Cf'(a)\frac{f'''(a)}{f'(a)}=\frac{\delta}{\gamma+\delta}s_1^2=V'''(a+),$$ which is the “smoothness condition”, equation (4.1) in @NoPeYaYa17, which characterises the unique periodic barrier.
Computational considerations {#S_comput}
============================
In the computation of the barriers for hybrid $(a_p,a_c,b)$ strategies, instead of using the maximisation of $V(a_c)-\beta a_c$, we solved the derivative conditions directly. The barriers were efficiently calculated without difficulties.
1. For each $l\geq 0$, denote the *unique* $y$ such that $V'(b)=\beta$ as $y_{\bar{a}}(l)$ and $y_0(l)$ for $a=\bar{a}$ and $a=0$ respectively.
2. For each $l\geq 0$ and $y\in[y_{\bar{a}}(l),y_0(l)]$, we can find a corresponding $q=Q(a)$ such that $V'(b)=\beta$. As $Q$ is monotone in $a$, we can recover $a$.
3. For each $l\geq 0$, $(y,a)$ pair in the previous step indexed by $y$, we evaluate the derivative of the value function at $a$. For each $l\geq 0$, we can find a $(y,a)$ pair such that both $V'(a)=1$ (or $a=0$ if $V'(0)\leq 1$) and $V'(a+y+l)=\beta$.
4. For $l=0$, using the corresponding $(y,a)$ pair to compute $V'(a)$. If $V'(a)\leq \beta$, then set $l=0$. Otherwise, write a function to output the corresponding $V'(a+l)$ using the $(y,a)$ pair from the previous step. By increasing $l$ following by a bisection method, we can find a corresponding $l$ such that $V'(a+l)=\beta$.
All equations can be solved by for example bisection method combining with some searching technique. In case of numerical overflow, we can rescale the scale parameters $(\mu,\sigma,\chi)$ to find the barriers then scale back. It should be clear that rescaling should not change the optimality of an optimal strategy.
In order to make sure that the possibilities of multiple solutions for some equations would not result in some disruptive impact to the numerical procedure, we can verify that the final output indeed satisfies all conditions proposed.
Finally, when $\beta$ is close to the asymptote $\gamma/(\gamma+\delta)$, we have numerical overflow as $b^*\uparrow\infty$. In this case, an approximation is used based on Proposition \[Lemma.convergence.beta\], where we treat $b^*=\infty$ and calculate $a_p^*$ and $a_c^*-a_p^*$ independently. Sometimes, $b^*$ may not be large enough to validate such approximation and a “bias” is resulted. In this case, we will adjust the bias term such that the approximation piece glues to the piece without approximation. We decreases the bias term (linearly for convenience) so that it eventually vanishes at the asymptote $\beta=\gamma/(\gamma+\delta)$. The region where we employ such approximation is indicated between the dotted line (where the numerical overflow starts) and the solid line (the asymptote).
For $\mu<0$, it is straightforward from Theorem \[Thm.mu.neg\]. Specifically, we proceed the following:
1. First check whether $\chi\geq \beta({-\mu}/({\gamma+\delta}))$.
2. Suppose $\chi\geq \beta({-\mu}/({\gamma+\delta}))$. If $\beta\leq {\gamma}/({\gamma+\delta})$, $\pi_0$ is optimal. No numerical method is needed. On the other hand, if $\beta> {\gamma}/({\gamma+\delta})$, we express $V'(b-;\pi_{b,\infty})$ as a function of $b$ and search for $V'(b-;\pi_{b,\infty})=\beta$.
3. Suppose $\chi< \beta({-\mu}/({\gamma+\delta}))$. Invert $\Lambda$ at ${\chi}/{\beta}$ to output $\beta_0=\Lambda^{-1}({\chi}/{\beta})$. If $\beta\leq \beta_0$, then $\pi_0$ is optimal and no numerical method is needed. If $\beta\in(\beta_0,{\gamma}/({\gamma+\delta}))$, first output $c_{\beta,\chi}$ by solving $V'(a_\beta)=\beta$. It is then followed by solving 2 equations: $V'(b_1-;\pi_{b_1,b_2})=\beta$ in $b_1$ and $V'(b_2+;\pi_{b_1,b_2})=\beta$ in $b_2$, respectively. Note that for both equations, the other parameter is not used. Finally, if $\beta\in[{\gamma}/({\gamma+\delta}),1]$, as in the previous case, we express $V'(b-;\pi_{b,\infty})$ as a function of $b$ and search for $V'(b-;\pi_{b,\infty})=\beta$.
Again, all equations can be solved by for example bisection method combining with some search techniques.
To compute $b_0^*$, we do not use the results directly from @NoPeYaYa17. Instead, we use the formula given by the third item in Proposition \[Lemma.convergence.beta\], which holds for any $\beta$.
Continuity for different cases in Theorem \[Thm.mu.neg\] {#A_Thm.mu.neg.cont}
========================================================
We consider here the $4$ different cases enumerated in Theorem \[Thm.mu.neg\] sequentially.
$\frac{\chi}{\beta}<\frac{-\mu}{\gamma+\delta}$, $\beta\uparrow\frac{\gamma}{\gamma+\delta}$.
---------------------------------------------------------------------------------------------
When $\beta\uparrow\gamma/(\gamma+\delta)$, we have (from $V'(\cdot;\pi_0)\uparrow\gamma/(\gamma+\delta)$) that $a_\beta\rightarrow \infty$ which implies that $b_2\rightarrow \infty$ and therefore from Theorem \[Thm.mu.neg\] case 2b we can conclude that $b_1$ is continuous at $\beta=\gamma/(\gamma+\delta)$, i.e. $$\pi_{b_1,b_2\rightarrow\infty}\rightarrow\pi_{b_1,\infty}$$ when $\beta\uparrow\gamma/(\gamma+\delta)$.
$\frac{\chi}{\beta}\geq \frac{-\mu}{\gamma+\delta}$, $\beta\downarrow\frac{\gamma}{\gamma+\delta}$.
---------------------------------------------------------------------------------------------------
Recall the optimal strategy in the setting $\pi_{b,\infty}$ is characterised by the derivative condition at $b$, i.e. $V'(b;\pi_{b,\infty})=\beta$. In view of the first $2$ equations in the proof of Lemma \[Lemma.b.Exist\], we have $$\label{eq.VDb}
V'(b-;\pi_{b,\infty})=\frac{(\beta-\frac{\gamma}{\gamma+\delta})b-\chi-\frac{\gamma\mu}{(\gamma+\delta)^2}(1-e^{s_1b})}{g(b)}g'(b)-\frac{\gamma\mu}{(\gamma+\delta)^2}s_1e^{s_1b}+\frac{\gamma}{\gamma+\delta}.$$ Therefore, we have $${\frac{\partial}{\partial \beta}}V'(b-;\pi_{b,\infty})=b\frac{g'(b)}{g(b)}>0.$$ This implies that when $\beta$ decreases from $\beta_1$ to $\beta_2$, the original $b=b(\beta_1)$ yields $V'(b-;\pi_{b,\infty})<\beta_2$ and therefore from the proof of Lemma \[Lemma.b.Exist\] we need to use a larger $b$. This implies that $b(\beta_2)>b(\beta_1)$. In other words, when $\beta\downarrow\gamma/(\gamma+\delta)$, the corresponding $b=b(\beta)$ is increasing. It remains to show that $b(\beta)$ is not converging so that we have $$\pi_{b\uparrow\infty,\infty}\rightarrow \pi_0.$$ Suppose $b(\beta)\uparrow b<\infty$ as $\beta\downarrow\gamma/(\gamma+\delta)$. Then by taking the limit, we have $$V'(b-;\pi_{b,\infty})=\beta=\frac{\gamma}{\gamma+\delta}.$$ Thus, from we have $$-\chi=\frac{\gamma\mu}{(\gamma+\delta)^2}\Big(s_1e^{s_1b}+(1-e^{s_1b})\frac{g'(b)}{g(b)}\Big)=\frac{\gamma\mu}{(\gamma+\delta)^2}s_1,$$ which is impossible since $-\chi$ is negative but the very last term is positive.
$\frac{\chi}{\beta}\uparrow\frac{-\mu}{\gamma+\delta}$, $\beta\in[\beta_0,\frac{\gamma}{\gamma+\delta}]$.
---------------------------------------------------------------------------------------------------------
Here, we want to show that if $\beta<\gamma/(\gamma+\delta)$ then when $\chi/\beta$ is “close” to $-\mu/(\gamma+\delta)$, we have $\beta_0>\beta$. This means the two conditions $\chi/\beta\uparrow-\mu/(\gamma+\delta)$ and $\beta\in[\beta_0,\gamma/(\gamma+\delta)]$ cannot be satisfied simultaneously unless $\beta=\gamma/(\gamma+\delta)$. In other words, the cells in the second row of the Table \[T\_roadmap\] are continuous only at $\beta=\gamma/(\gamma+\delta)$, which has already been taken care of.
Recall that $\beta_0$ is defined by the inverse of the increasing function $\Lambda$ at ${\chi}/{\beta}$, i.e. $\beta_0=\Lambda^{-1}(\chi/\beta)$. In addition, $\Lambda$ maps $\beta\in[V'(0;\pi_0),\gamma/(\gamma+\delta))$ to $[0,-\mu/(\gamma+\delta))$. Therefore, when $\chi/\beta\uparrow\gamma/(\gamma+\delta)$, we have $\beta_0\uparrow\gamma/(\gamma+\delta)$, which implies that $\beta\rightarrow\gamma/(\gamma+\delta)$.
$\frac{\chi}{\beta}<\frac{-\mu}{\gamma+\delta}$, $\beta\downarrow\beta_0$.
--------------------------------------------------------------------------
Recall $\beta_0=\Lambda^{-1}(\chi/\beta)$ and therefore $\beta\downarrow\beta_0$ implies that is an equality at the limit and we have $a_\beta-c_{\beta,\chi}\rightarrow 0$. Consequently, from $c_{\beta,\chi}\leq b_1\leq a_\beta$ we can conclude that $b_1\rightarrow a_\beta$.
It remains to show $b_2\rightarrow a_\beta$. In view of Lemma \[Lemma.b1b2.Exist\], we can establish $b_2\rightarrow a_\beta$ if we can show that at $\beta=\beta_0$, we have with $b_2=a_\beta$, i.e. $$s_1(\beta-\frac{\gamma}{\gamma+\delta}) a_\beta-\Big(\chi s_1+s_1\frac{\gamma\mu}{(\gamma+\delta)^2} -(\frac{\gamma}{\gamma+\delta}-\beta)\Big)=0.$$
Using $$V(x;\pi_0)=\frac{-\gamma\mu}{(\gamma+\delta)^2}e^{s_1x}+\frac{\gamma}{\gamma+\delta}\Big({x+\frac{\mu}{\gamma+\delta}}\Big),$$ we can re-express $V'(a_\beta;\pi_0)=\beta$ as $$\label{dev.eqvuiv}
s_1\frac{-\gamma\mu}{(\gamma+\delta)^2}e^{s_1a_\beta}=\beta-\frac{\gamma}{\gamma+\delta}$$ and (with inequality replaced by equality) as $$\begin{aligned}
&s_1\Big(\frac{-\gamma\mu}{(\gamma+\delta)^2}e^{s_1a_\beta}+\frac{\gamma}{\gamma+\delta}\Big({a_\beta+\frac{\mu}{\gamma+\delta}}\Big)\Big)=s_1\beta a_\beta-s_1\chi\\\iff~& s_1(\beta-\frac{\gamma}{\gamma+\delta})a_\beta-s_1\chi-s_1\frac{\gamma\mu}{(\gamma+\delta)^2}=s_1\frac{-\gamma\mu}{(\gamma+\delta)^2}e^{s_1a_\beta}=\beta-\frac{\gamma}{\gamma+\delta},\end{aligned}$$ where the last equality is from and the last line is essentially what we are trying to show, i.e. .
Since we have $b_1,b_2\rightarrow a_\beta$ when $\beta\downarrow\beta_0$, we have $$\pi_{b_1\rightarrow a_\beta,b_2\rightarrow a_\beta}\rightarrow \pi_0.$$
|
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abstract: 'In this paper, we extend earlier work by showing that if $X$ and $Y$ are simplicial complexes (i.e. simplicial sets whose nondegenerate simplices are determined by their vertices), an *isomorphism* $\cfn X\cong\cfn Y$ of $\s$-coalgebras implies that the 3-skeleton of $X$ is weakly equivalent to the 3-skeleton of $Y$, also implying that $\pi_{1}(X)=\pi_{1}(Y)$.'
author:
- 'Justin R. Smith'
title: '$\mathfrak{S}$-coalgebras determine fundamental groups'
---
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Introduction
============
In [@Smith:1994], the author defined the functor $\cf *$ on simplicial sets — essentially the chain complex equipped with the structure of a coalgebra over an operad $\s$. This coalgebra structure determined all Steenrod and other cohomology operations. Since these coalgebras are not *nilpotent*[^1] ** they have a kind of “transcendental” structure that contains much more information.
In section \[sec:The-functor-cfn\], we define a variant of the $\cf *$-functor, named $\cfn *$. It is defined for simplicial complexes — semi-simplicial sets whose simplices are uniquely determined by their vertices. The script-N emphasizes that its underlying chain-complex is *normalized* and $\cf *$ can be views as an extension of $\cfn X$ to general simplicial sets (see [@smith-cellular]).
In [@smith-cellular], we showed that if $X$ and $Y$ are pointed, reduced simplicial sets, then a quasi-isomorphism $\cf X\to\cf Y$ induces one of their $\ints$-completions $\ints_{\infty}X\to\ints_{\infty}Y$. It follows that that the $\cf *$-functor determine a *nilpotent* space’s weak homotopy type.
In the present paper, we extend this by showing:
Corollary. \[cor:cellular-determines-pi1\]. *If $X$ and $Y$ are simplicial complexes with the property that there exists an isomorphism $$g:\cfn X\to\cfn Y$$ then their 3-skeleta are isomorphic and $$\pi_{1}(X)\cong\pi_{1}(Y)$$*
This implies that the functors $\cf *$ and $\cfn *$ encapsulate “non-abelian” information about a simplicial set — such as its (possibly non-nilpotent) fundamental group. The requirement that $g$ be an *isomorphism* is stronger than needed for this (but *quasi*-isomorphism is not enough).
The proof actually requires $X$ and $Y$ to be simplicial *complexes* rather than general simplicial sets. The author conjectures that the $\cf *$-functor determines the integral homotopy type of an arbitrary simplicial set.
Since the transcendental portion of $\cf X$ can be mapped to a power series ring (see the proof of lemma \[lem:diagonals-linearly-independent\]), the analysis of this data may require methods of analysis and algebraic geometry.
Definitions and assumptions
===========================
\[defr:chaincat\]Let $\chaincat$ denote the category of $\ints$-graded $\ints$-free chain complexes and let $\chaincatp\subset\chaincat$ denote the subcategory of chain complexes concentrated in positive dimensions.
If $c\in\chaincat$, $$C^{\otimes n}=\underbrace{C\otimes_{\ints}\otimes\cdots\otimes_{\ints}C}_{n\,\text{times }}$$
We also have categories of spaces:
\[def:simplicial-set-complex\]Let $\spaces$ denote the category of simplicial sets and $\simpc$ that of simplicial *complexes.* A simplicial *complex* is a simplicial set without degeneracies (i.e., a semi-simplicial set) with the property that simplices are uniquely determined by their vertices.
Following [@may-finite], we can define a simplicial set to have *Property A* if every face of a nondegenerate simplex is nondegenerate. Theorem 12.4.4 of [@may-finite] proves that simplicial sets with property A have *second subdivisions* that are simplicial complexes. The bar-resolution $\rs 2$ is an example of a simplicial set that does *not* have property A.
On the other hand, it is well-known that *all* topological spaces are weakly homotopy equivalent to simplicial complexes — see, for example, Theorem 2C.5 and Proposition 4.13 of [@hatcher-alg-top].
We make extensive use of the Koszul Convention (see [@Gugenheim:1960]) regarding signs in homological calculations:
\[def:koszul-1\] If $f:C_{1}\to D_{1}$, $g:C_{2}\to D_{2}$ are maps, and $a\otimes b\in C_{1}\otimes C_{2}$ (where $a$ is a homogeneous element), then $(f\otimes g)(a\otimes b)$ is defined to be $(-1)^{\deg(g)\cdot\deg(a)}f(a)\otimes g(b)$.
If $f_{i}$, $g_{i}$ are maps, it isn’t hard to verify that the Koszul convention implies that $(f_{1}\otimes g_{1})\circ(f_{2}\otimes g_{2})=(-1)^{\deg(f_{2})\cdot\deg(g_{1})}(f_{1}\circ f_{2}\otimes g_{1}\circ g_{2})$.
\[def:homcomplex-1\]Given chain-complexes $A,B\in\chaincat$ define $$\homz(A,B)$$ to be the chain-complex of graded $\ring$-morphisms where the degree of an element $x\in\homz(A,B)$ is its degree as a map and with differential $$\partial f=f\circ\partial_{A}-(-1)^{\deg f}\partial_{B}\circ f$$ As a $\ring$-module $\homz(A,B)_{k}=\prod_{j}\homz(A_{j},B_{j+k})$.
Given $A,B\in\mathbf{Ch}^{S_{n}}$, we can define $\homzs n(A,B)$ in a corresponding way.
\[def:tmap\] Let $\alpha_{i}$, $i=1,\dots,n$ be a sequence of nonnegative integers whose sum is $|\alpha|$. Define a set-mapping of symmetric groups $$\tlist{\alpha}n:S_{n}\to S_{|\alpha|}$$ as follows:
1. for $i$ between 1 and $n$, let $L_{i}$ denote the length-$\alpha_{i}$ integer sequence:
2. ,where $A_{i}=\sum_{j=1}^{i-1}\alpha_{j}$ — so, for instance, the concatenation of all of the $L_{i}$ is the sequence of integers from 1 to $|\alpha|$;
3. $\tlist{\alpha}n(\sigma)$ is the permutation on the integers $1,\dots,|\alpha|$ that permutes the blocks $\{L_{i}\}$ via $\sigma$. In other words, $\sigma$ s the permutation $$\left(\begin{array}{ccc}
1 & \dots & n\\
\sigma(1) & \dots & \sigma(n)
\end{array}\right)$$ then $\tlist{\alpha}n(\sigma)$ is the permutation defined by writing $$\left(\begin{array}{ccc}
L_{1} & \dots & L_{n}\\
L_{\sigma(1)} & \dots & L_{\sigma(n)}
\end{array}\right)$$ and regarding the upper and lower rows as sequences length $|\alpha|$.
Do not confuse the $T$-maps defined here with the transposition map for tensor products of chain-complexes. We will use the special notation $T_{i}$ to represent $T_{1,\dots,2,\dots,1}$, where the 2 occurs in the $i^{\mathrm{th}}$ position. The two notations don’t conflict since the old notation is never used in the case when $n=1$. Here is an example of the computation of $\tmap{2,1,3}((1,3,2))=\tmap{2,1,3}\left(\begin{array}{ccc}
1 & 2 & 3\\
3 & 1 & 2
\end{array}\right)$:$L_{1}=\{1\}2$, $L_{2}=\{3\}$, $L_{3}=\{4,5,6\}$. The permutation maps the ordered set $\{1,2,3\}$ to $\{3,1,2\}$, so we carry out the corresponding mapping of the sequences $\{L_{1},L_{2},L_{3}\}$ to get $\left(\begin{array}{ccc}
L_{1} & L_{2} & L_{3}\\
L_{3} & L_{1} & L_{2}
\end{array}\right)=\left(\begin{array}{ccc}
\{1,2\} & \{3\} & \{4,5,6\}\\
\{4,5,6\} & \{1,2\} & \{3\}
\end{array}\right)=\left(\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6\\
4 & 5 & 6 & 1 & 2 & 3
\end{array}\right)$ (or $((1,4)(2,5)(3,6))$, in cycle notation).
\[def:operad\] A sequence of differential graded $\mathbb{Z}$-free modules, $\{\mathcal{V}_{i}\}$, will be said to form an *operad* if:
1. there exists a *unit map* (defined by the commutative diagrams below) $$\eta:\mathbb{Z}\to\mathcal{V}_{1}$$
2. for all $i>1$, $\mathcal{V}_{i}$ is equipped with a left action of $S_{i}$, the symmetric group.
3. for all $k\ge1$, and $i_{s}\ge0$ there are maps $$\gamma:\mathcal{V}_{i_{1}}\otimes\cdots\otimes\mathcal{V}_{i_{k}}\otimes\mathcal{V}_{k}\to\mathcal{V}_{i}$$ where $i=\sum_{j=1}^{k}i_{j}$.
The $\gamma$-maps must satisfy the conditions:
[Associativity]{}
: the following diagrams commute, where $\sum j_{t}=j$, $\sum i_{s}=i$, and $g_{\alpha}=\sum_{\ell=1}^{\alpha}j_{\ell}$ and $h_{s}=\sum_{\beta=g_{s-1}+1}^{g_{s}}i_{\beta}$:
[Units]{}
: the diagrams $$\begin{array}{cc}\xymatrix{{{\integers}^{k}\otimes\mathcal{V}_{k}}\ar[r]^{\cong}\ar[d]_{{\eta}^{k}\otimes\text{Id}}&{\mathcal{V}_{k}}\\
{{\mathcal{V}_{1}}^{k}\otimes{\mathcal{V}_{k}}}\ar[ur]_{\gamma}&}&\xymatrix{{\mathcal{V}_{k}\otimes\integers}\ar[r]^{\cong}\ar[d]_{\text{Id}\otimes\eta}&{\mathcal{V}_{k}}\\
{\mathcal{V}_{k}\otimes\mathcal{V}_{1}}\ar[ur]_{\gamma}&}\end{array}$$ commute.
[Equivariance]{}
: the diagrams $$\xymatrix@C+20pt{{\mathcal{V}_{j_{1}}\otimes\cdots\otimes\mathcal{V}_{j_{k}}\otimes\mathcal{V}_{k}}\ar[r]^-{\gamma}\ar[d]_{\sigma^{-1}\otimes\sigma}&{\mathcal{V}_{j}}\ar[d]^{\tmap{j_{1},\dots,j_{k}}(\sigma)}\\
{\mathcal{V}_{j_{\sigma(1)}}\otimes\cdots\otimes\mathcal{V}_{j_{\sigma(k)}}\otimes\mathcal{V}_{k}}\ar[r]_-{\gamma}&{\mathcal{V}_{j}}}$$commute, where $\sigma\in S_{k}$, and the $\sigma^{-1}$ on the left permutes the factors $\{\mathcal{V}_{j_{i}}\}$ and the $\sigma$ on the right simply acts on $\mathcal{V}_{k}$. See \[def:tmap\] for a definition of $\tmap{j_{1},\dots,j_{k}}(\sigma)$. $$\xymatrix@C+20pt{{\mathcal{V}_{j_{1}}\otimes\cdots\otimes\mathcal{V}_{j_{k}}\otimes\mathcal{V}_{k}}\ar[r]^-{\gamma}\ar[d]_{\tau_{1}\otimes\cdots\tau_{k}\otimes\text{Id}}&{\mathcal{V}_{j}}\ar[d]^-{\tau_{1}\oplus\cdots\oplus\tau_{k}}\\
{\mathcal{V}_{j_{\sigma(1)}}\otimes\cdots\otimes\mathcal{V}_{j_{\sigma(k)}}\otimes\mathcal{V}_{k}}\ar[r]_-{\gamma}&{\mathcal{V}_{j}}}$$ where $\tau_{s}\in S_{j_{s}}$ and $\tau_{1}\oplus\cdots\oplus\tau_{k}\in S_{j}$ is the block sum.
The alert reader will notice a discrepancy between our definition of operad and that in [@Kriz-May] (on which it was based). The difference is due to our using operads as parameters for systems of *maps*, rather than $n$-ary operations. We, consequently, compose elements of an operad as one composes *maps*, i.e. the second operand is to the *left* of the first. This is also why the symmetric groups act on the *left* rather than on the right.
\[def:unitaloperad\]An operad, $\mathcal{V}$, will be called *unital* if $\mathcal{V}$ has a $0$-component $\mathcal{V}_{0}=\integers$, concentrated in dimension $0$ and augmentations $$\epsilon_{n}:\mathcal{V}_{0}\otimes\cdots\otimes\mathcal{V}_{0}\otimes\mathcal{V}_{n}=\mathcal{V}_{n}\to\mathcal{V}_{0}=\integers$$ induced by their structure maps.
The term “unital operad” is used in different ways by different authors. We use it in the sense of Kriz and May in [@Kriz-May], meaning the operad has a $0$-component that acts like an arity-lowering augmentation under compositions.
We will frequently want to think of operads in other terms:
\[def:operadcomps\] Let $\mathcal{V}$ be an operad, as defined above. Given $i\le k_{1}>0$, define the $i^{\mathrm{th}}$ *composition*
$$\circ_{i}:\mathcal{V}_{k_{2}}\otimes\mathcal{V}_{k_{1}}\to\mathcal{V}_{k_{1}+k_{2}-1}$$ as the composite $$\begin{gathered}
\underbrace{\integers\otimes\cdots\otimes\integers\otimes\mathcal{V}_{k_{2}}\otimes\integers\otimes\cdots\otimes\integers}_{\text{\ensuremath{i^{\text{th}}}factor}}\otimes\mathcal{V}_{k_{1}}\\
\to\underbrace{\mathcal{V}_{1}\otimes\cdots\otimes\mathcal{V}_{1}\otimes\mathcal{V}_{k_{2}}\otimes\mathcal{V}_{1}\otimes\cdots\otimes\mathcal{V}_{1}}_{\text{\ensuremath{i^{\text{th}}}factor}}\otimes\mathcal{V}_{k_{1}}\to\mathcal{V}_{k_{1}+k_{2}-1}\end{gathered}$$ where the final map on the right is $\gamma$.
These compositions satisfy the following conditions, for all $a\in\mathscr{U}_{n}$, $b\in\mathscr{U}_{m}$, and $c\in\mathscr{U}_{t}$:
[Associativity]{}
: $(a\circ_{i}b)\circ_{j}c=a\circ_{i+j-1}(b\circ_{j}c)$
[Commutativity]{}
: $a\circ_{i+m-1}(b\circ_{j}c)=(-1)^{mn}b\circ_{j}(a\circ_{i}c)$
[Equivariance]{}
: $a\circ_{\sigma(i)}(\sigma\cdot b)=\tunderi ni(\sigma)\cdot(a\circ_{i}b)$
I am indebted to Jim Stasheff for pointing out to me that operads were originally defined this way and called *composition algebras.* Given this definition of operad, we recover the $\gamma$ map in definition \[def:operad\] by setting: $$\gamma(u_{i_{1}}\otimes\cdots\otimes u_{i_{k}}\otimes u_{k})=u_{i_{1}}\circ_{1}\cdots\circ_{k-1}u_{i_{k}}\circ_{k}u_{k}$$ (where the implied parentheses associate to the right). It is left to the reader to verify that the two definitions are equivalent (the commutativity condition, here, is a special case of the equivariance condition). Given a *unital* operad, we can use the augmentation maps to recover the composition operations.
A simple example of an operad is:
\[example:frakS0\]For each $n\ge0$, $X$, the operad $\s_{0}$ has $\s_{0}(n)=\integers S_{n}$, concentrated in dimension $0$, with structure-map induced by $$\begin{aligned}
\gamma_{\alpha_{1},\dots,\alpha_{n}}:S_{\alpha_{1}}\times\cdots\times S_{\alpha_{n}}\times S_{n} & \to & S_{\alpha_{1}+\cdots+\alpha_{n}}\\
\sigma_{\alpha_{1}}\times\cdots\times\sigma_{\alpha_{n}}\times\sigma_{n} & \mapsto & \tlist{\alpha}n(\sigma_{n})\circ(\sigma_{\alpha_{1}}\oplus\cdots\oplus\sigma_{\alpha_{n}})\end{aligned}$$ In other words, each of the $S_{\alpha_{i}}$ permutes elements within the subsequence $\{\alpha_{1}+\cdots+\alpha_{i-1}+1,\dots,\alpha_{1}+\cdots+\alpha_{i}\}$ of the sequence $\{1,\dots,\alpha_{1}+\cdots+\alpha_{n}\}$ and $S_{n}$ permutes these $n$ blocks.
For the purposes of this paper, the main example of an operad is
\[def:coend\]Given any $C\in\chaincat$, the associated *coendomorphism operad*, $\coend(C)$ is defined by $$\coend(C)(n)=\homz(C,C^{\otimes n})$$ Its structure map $$\begin{gathered}
\gamma_{\alpha_{1},\dots,\alpha_{n}}:\homz(C,C^{\otimes n})\otimes\homz(C,C^{\otimes\alpha_{1}})\otimes\cdots\otimes\homz(C,C^{\otimes\alpha_{n}})\to\\
\homz(C,C^{\otimes\alpha_{1}+\cdots+\alpha_{n}})\end{gathered}$$ simply composes a map in $\homz(C,C^{\otimes n})$ with maps of each of the $n$ factors of $C$.
This is a non-unital operad, but if $C\in\chaincat$ has an augmentation map $\varepsilon:C\to\ring$ then we can regard $\epsilon$ as the only element of $\homz(C,C^{\otimes n})=\homz(C,C^{\otimes0})=\homz(C,\ring)$.
Morphisms of operads are defined in the obvious way:
\[def:operadmorphism\] Given two operads $\mathcal{V}$ and $\mathcal{W}$, a *morphism* $$f:\mathcal{V}\to\mathcal{W}$$ is a sequence of chain-maps $$f_{i}:\mathcal{V}_{i}\to\mathcal{W}_{i}$$ commuting with all the diagrams in \[def:operad\].
Verification that this satisfies the required identities is left to the reader as an exercise.
\[def:sfrakfirstmention\]Let $\s$ denote the operad defined in [@Smith:1994] — where $\s_{n}=\rs n$ is the normalized bar-resolution of $\integers$ over $\zs n$ for all $n>0$. This is similar to the Barratt-Eccles operad defined in [@Barratt-Eccles-operad], except that the latter is composed of *unnormalized* bar-resolutions. See [@Smith:1994] or appendix A of [@smith-cellular], for the details.
Appendix A of [@smith-cellular] contains explicit computations of some composition-operations in $\s$.
Now we are ready to define the all-important concept of *coalgebras* over an operad:
\[def:coalgebra-over-operad-1\]A chain-complex $C$ is a *coalgebra over the operad* $\mathcal{V}$ if there exists a morphism of operads $$\mathcal{V}\to\coend(C)$$
A coalgebra, $C$, over an operad, $\mathcal{V}$, is a sequence of maps $$f_{n}:\mathcal{V}_{n}\otimes C\to C^{\otimes n}$$ for all $n>0$, where $f_{n}$ is $\zs n$-equivariant and $S_{n}$ acts by permuting factors of $C^{\otimes n}$. The maps, $\{f_{n}\}$, are related in the sense that they fit into commutative diagrams: $${\makeatletter \xydef@\xymatrixcolsep@{20pt} \makeatother }\xymatrix{{\mathscr{\mathcal{V}}_{n}\otimes\mathscr{\mathcal{V}}_{m}\otimes C}\ar[r]^{\circ_{i}} & {\mathscr{\mathcal{V}}_{n+m-1}\otimes C}\ar[r]^{f_{n+m-1}} & {C^{\otimes n+m-1}}\\
{\mathscr{\mathcal{V}}_{n}\otimes\mathscr{\mathcal{V}}_{m}\otimes C}\ar[r]_{1\otimes f_{m}}\ar@{=}[u] & {\mathscr{\mathcal{V}}_{n}\otimes C^{\otimes m}}\ar[r]_{Z_{i-1}\qquad\quad} & {C^{i-1}\otimes\mathscr{\mathcal{V}}_{n}\otimes C\otimes C^{\otimes m-i}}\ar[u]_{1\otimes\dots\otimes f_{n}\otimes\dots\otimes1}
}
\label{dia:coalgebra-over-operad}$$ for all $n,m\ge1$ and $1\le i\le m$. Here $Z_{i-1}:\mathcal{V}_{n}\otimes C^{m}\to C^{\otimes i-1}\otimes\mathcal{V}_{n}\otimes C\otimes C^{\otimes m-i}$ is the map that shuffles the factor $\mathcal{V}_{n}$ to the right of $i-1$ factors of $C$. In other words: The abstract composition-operations in $\mathcal{V}$ exactly correspond to compositions of maps in $\{\homz(C,C^{\otimes n})\}$. We exploit this behavior in applications of coalgebras over operads, using an explicit knowledge of the algebraic structure of $\mathcal{V}$.
The structure of a coalgebra over an operad can also be described in several equivalent ways:
1. $f_{n}:\mathcal{V}(n)\otimes C\to C^{\otimes n}$
2. $g:C\to\prod_{n=0}^{\infty}\homzs n(\mathcal{V}(n),C^{\otimes n})$
where both satisfy identities that describe how composites of these maps are compatible with the operad-structure.
\[def:coalgebra-over-operad\]A chain-complex $C$ is a *coalgebra over the operad* $\mathcal{V}$ if there exists a morphism of operads $$\mathcal{V}\to\coend(C)$$
The structure of a coalgebra over an operad can be described in several equivalent ways:
1. $f_{n}:\mathcal{V}(n)\otimes C\to C^{\otimes n}$
2. $g:C\to\prod_{n=0}^{\infty}\homzs n(\mathcal{V}(n),C^{\otimes n})$
where both satisfy identities that describe how composites of these maps are compatible with the operad-structure.
\[def:s-coalgebra-morphism\]Using the second description, $$\alpha_{C}:C\to\prod_{n=1}^{\infty}\homzs n(\rs n,C^{\otimes n})$$ an $\s$-coalgebra morphism $$f:C\to D$$ is a chain-map that makes the diagram $$\xymatrix{{\forgetful C}\ar[d]_{\forgetful f}\ar[r]^{\alpha_{C}\qquad\qquad\quad} & {\prod_{n=1}^{\infty}\homzs n(\rs n,\forgetful C^{\otimes n})}\ar[d]^{\prod_{n=1}^{\infty}\homzs n(1,\forgetful f^{\otimes n})}\\
{\forgetful D}\ar[r]_{\alpha_{D}\qquad\qquad\quad} & {\prod_{n=1}^{\infty}\homzs n(\rs n,\forgetful D^{\otimes n})}
}$$ commute, where $\forgetful *$ is the forgetful functor that turns a coalgebra into a chain-complex.
We also need
morphisms of $\s$-coalgebras\[sec:morphisms\]
=============================================
Proposition \[pro:simplicespropertyS\] proves that if $e_{n}=\underbrace{[(1,2)|\cdots|(1,2)]}_{n\text{ terms}}\in\rs 2$ and $x\in\cfn X$ is the image of a $k$-simplex, then $$f_{2}(e_{k}\otimes x)=\xi_{k}\cdot x\otimes x$$ where $\xi_{k}=(-1)^{k(k-1)/2}$.
\[def:gamma-m-map\]If
We have
\[cor:simplex-image\]If $X$ is a simplicial set and $c\in\cf X$ is an element generated by an $n$-simplex, then the image of $c$ under the composite $$\cfn X_{n}\xrightarrow{\alpha_{\cf X}}\prod_{k=1}^{\infty}\homzs k(\rs k,\cfn X^{\otimes k})\xrightarrow{\gamma_{n}}\prod_{k=1}^{\infty}\cfn X^{\otimes k}$$ is $$e(c)=(c,c\otimes c,\dots)$$
This follows immediately from proposition \[pro:simplicespropertyS\] and the fact that operad-compositions map to compositions of coproducts.
Lemma \[lem:diagonals-linearly-independent\] implies that
\[cor:n-simplices-map-to-simplices\]Let $X$ be a simplicial set and suppose $$f:\ns n=\cfn{\Delta^{n}}\to\cfn X$$ is a $\s$-coalgebra morphism. Then the image of the generator $\Delta^{n}\in\cfn{\Delta^{n}}$ is a generator of $\cfn X$ defined by an $n$-simplex of $X$.
Suppose $$f(\Delta^{n})=\sum_{k=1}^{t}c_{k}\cdot\sigma_{k}^{n}\in\cfn X$$ where the $\sigma_{k}^{n}$ are images of $n$-simplices of $X$. If $f(\Delta^{n})$ is not equal to one of the $\sigma_{k}^{n}$, lemma \[lem:diagonals-linearly-independent\] implies that its image is linearly independent of the $\sigma_{k}^{n}$, a *contradiction.* The statement about sub-simplices follows from the main statement.
We also conclude that:
\[cor:automorphisms-trivial\]If $f:\cfn{\Delta^{n}}\to\cfn{\Delta^{n}}$ is
1. an isomorphism of $\s$-algebras in dimension $n$ and
2. an endomorphism in lower dimensions
then $f$ must be an isomorphism. If $n\le3$, then $f$ must also be the identity map.
The final statement actually works for some larger values of $n$, but the arguments become vastly more complicated (requiring the use of higher coproducts). It would have extraordinary implications if it were true for *all* $n$.
We first show that $f$ must be an isomorphism. We are given that $f$ is an isomorphism in dimension $n$. We use downward induction on dimension to show that it is an isomorphism in lower dimensions:
Suppose $f$ is an isomorphism in dimension $k$ and $\Delta^{k}\subset\Delta^{n}$ is a simplex. The boundary of $\Delta^{k}$ is a linear combination of $k+1$ distinct faces which corollary
It follows that $f$ is actually an *automorphism* of $\cfn{\Delta^{n}}$. Now we assume that $n\le3$ and show that $f$ is the *identity map:*
If $n=1$ then corollary \[cor:n-simplices-map-to-simplices\] implies that $f|\cfn{\Delta^{1}}_{1}=1$. Since the $0$-simplices must map to $0$-simplices (by corollary \[cor:n-simplices-map-to-simplices\]) with a $+1$ sign it follows that the only possible non-identity automorphism of $\cfn{\Delta^{1}}$ swaps the ends of $\Delta^{1}$ — but this would violate the condition that $f$ is a chain-map.
In dimension 2, let $\Delta^{2}$ be a $2$-simplex. Similar reasoning to that used in the one-dimensional case implies that a non-identity automorphism of $\cfn{\Delta^{2}}$ would (at most) involve permuting some of its faces. Since $$\partial\Delta^{2}=F_{0}\Delta^{2}-F_{1}\Delta^{2}+F_{2}\Delta^{2}$$ the only non-identity permutation compatible with the boundary map swaps $F_{0}\Delta^{2}$ and $F_{2}\Delta^{2}$. But the coproduct of $\Delta^{2}$ is given by $$[\,]_{2}\otimes\Delta^{2}\mapsto\Delta^{2}\otimes F_{0}F_{1}\Delta^{2}+F_{2}\Delta^{2}\otimes F_{0}\Delta^{2}+F_{1}F_{2}\Delta^{2}\otimes\Delta^{2}$$ (see proposition \[prop:e1timesdelta2\]) where $[\,]$ is the 0-dimensional generator of $\rs 2$ — the bar-resolution of $\ints$ over $\zs{_{2}}$. It follows that swapping $F_{0}\Delta^{2}$ and $F_{2}\Delta^{2}$ would violate the condition that $f$ must preserve coproducts. The case where $n=1$ implies that the vertices cannot be permuted.
When $n=3$, we have $$\partial\Delta^{3}=F_{0}\Delta^{3}-F_{1}\Delta^{3}+F_{2}\Delta^{3}-F_{3}\Delta^{3}$$ so, in principal, we might be able to swap $F_{0}\Delta^{3}$ and $F_{2}\Delta^{3}$ or $F_{1}\Delta^{3}$ and $ $$F_{3}\Delta^{3}$. The coproduct does not rule any of these actions out since it involves multiple face-operations. The first “higher coproduct” does, however — see \[prop:e1timesdelta3\]:
A similar line of reasoning implies that:
\[cor:cf-gives-simplices\]Let $X$ be a simplicial complex and let $$f:\cfn{\Delta^{n}}\to\cfn X$$ map $\Delta^{n}$ to a simplex $\sigma\in\cf X$ defined by the inclusion $\iota:\Delta^{n}\to X$. Then $$f(\cfn{\Delta^{n}})\subset\cfn{\iota}(\cfn{\Delta^{n}})$$ so that $f=\alpha\circ\cfn{\iota}$, where $\alpha:\cfn{\Delta^{n}}\to\cfn{\Delta^{n}}$ is an automorphism. If $n\le3$, then $f=\cfn{\iota}$.
Since $X$ is a simplicial complex, the map $\iota$ is an inclusion.
Suppose $\Delta^{k}\subset\Delta^{n}$ and $f(\cfn{\Delta^{k}})_{k}\subset\cfn{\Delta^{k}}_{k}$. Since the boundary of $\Delta^{k}$ is an alternating sum of $k+1$ faces, and since they must map to $k-1$-dimensional simplices of $\cfn{f(\Delta^{k})}$ with the same signs (so no cancellations can take place) we must have $f(F_{i}\Delta^{k})\subset\cfn{f(\Delta^{k})}$ and the conclusion follows by downward induction on dimension. The final statements follow immediately from corollary \[cor:automorphisms-trivial\].
The functor $\nfc *$
====================
We define a complement to the $\cfn *$-functor:
\[def:fc\]Define a functor $$\nfc *:\ircoalgcat\to\spaces$$ to the category of semi-simplicial sets, as follows:
If $C\in\ircoalgcat$, define the $n$-simplices of $\nfc C$ to be the $\mathfrak{S}$-coalgebra morphisms $$\ns n\to C$$ where $\ns n=\cfn{\Delta^{n}}$ is the normalized chain-complex of the standard $n$-simplex, equipped with the $\s$-coalgebra structure defined in theorem \[thm:ns-construct\].
Face-operations are duals of coface-operations $$d_{i}:[0,\dots,i-1,i+1,\dots n]\to[0,\dots,n]$$ with $i=0,\dots,n$ and vertex $i$ in the target is *not* in the image of $d_{i}$.
Compare this to the functor $\fc *$ defined in [@smith-cellular]. The subscript $\mathbf{n}$ emphasizes that we do not take *degeneracies* into account.
\[prop:ux-map\]If $X$ is a simplicial complex (i.e., its simplices are uniquely determined by their vertices) there exists a natural map $$u_{X}:X\to\nfc{\cfn X}$$
To prove the first statement, note that any simplex $\Delta^{k}$ in $X$ comes equipped with a canonical inclusion $$\iota:\Delta^{k}\to X$$ The corresponding order-preserving map of vertices induces an $\s$-coalgebra morphism $$\cfn{\iota}:\cfn{\Delta^{k}}=\ns k\to\cfn X$$ so $u_{X}$ is defined by $$\Delta^{k}\mapsto\cfn{\iota}$$ It is not hard to see that this operation respects face-operations.
\[thm:simplicial-complexes-determined\]If $X\in\simpc$ is a simplicial complex then the canonical map $$u_{X}:X\to\nfc{\cfn X}$$ defined in proposition \[prop:ux-map\] is an isomorphism of the 3-skeleton.
This follows immediately from corollary \[cor:n-simplices-map-to-simplices\], which implies that simplices map to simplices and corollary \[cor:cf-gives-simplices\], which implies that these maps are *unique.*
\[cor:cellular-determines-pi1\]If $X$ and $Y$ are simplicial complexes with the property that there exists an isomorphism $$\cfn X\to\cfn Y$$ then their 3-skeleta are weakly equivalent and $$\pi_{1}(X)\cong\pi_{1}(Y)$$
Any morphism $g:\cfn X\to\cfn Y$ induces a morphism of simplicial sets $$\fc g:\nfc{\cfn X}\to\nfc{\cfn Y}$$ and this is an isomorphism (and homeomorphism) of simplicial complexes if $g$ is an isomorphism. The conclusion follows from theorem \[thm:simplicial-complexes-determined\] which implies that the canonical maps $$\begin{aligned}
u_{X}:X\to & \nfc{\cfn X}\\
u_{Y}:Y\to & \nfc{\cfn Y}\end{aligned}$$ are isomorphisms of the 3-skeleta, and the fact that fundamental groups are determined by the 2-skeleta.
The functor $\cfn *$\[sec:The-functor-cfn\]
===========================================
We begin with the elementary but powerful Cartan Theory of Constructions, originally described in [@Cartan3; @Cartan4; @Cartan5; @Cartan6]:
\[lem:cartanconstruction\]Let $M_{i}$, $i=1,2$ be DGA-modules, where:
1. $M_{1}=A_{1}\otimes N_{1}$, where $N_{1}$ is $\integers$-free and $A_{1}$ is a DGA-algebra (so $M_{1}$, merely regarded as a DGA-algebra, is free on a basis equal to a $\integers$-basis of $N_{1}$)
2. $M_{2}$ is a left DGA-module over a DGA-algebra $A_{2}$, possessing
1. a sub-DG-module, $N_{2}\subset M_{2}$, such that $\partial_{M_{2}}|N_{2}$ is injective,
2. a contracting chain-homotopy $\varphi:M_{2}\to M_{2}$ whose image lies in $N_{2}\subset M_{2}$.
Suppose we are given a chain-map $f_{0}:M_{1}\to M_{2}$ in dimension $0$ with $f_{0}(N_{1})\subseteq N_{2}$ and want to extend it to a chain-map from $M_{1}$ to $M_{2}$, subject to the conditions:
- $f(N_{1})\subseteq N_{2}$
- $f(a\otimes n)=g(a)\cdot f(n)$, where $g:A_{1}\to A_{2}$ is some morphism of DG-modules such that $a\otimes n\mapsto g(a)\cdot f(n)$ is a chain-map.
Then the extension $f:M_{1}\to M_{2}$ exists and is unique.
In applications of this result, the morphism $g$ will often be a morphism of DGA-algebras, but this is not necessary.
The *existence* of $f$ immediately follows from basic homological algebra; the interesting aspect of it is its *uniqueness* (not merely uniqueness up to a chain-homotopy). We will use it repeatedly to prove associativity conditions by showing that two apparently different maps satisfying the hypotheses must be *identical*.
The Theory of Constructions formed the cornerstone of Henri Cartan’s elegant computations of the homology and cohomology of Eilenberg-MacLane spaces in [@Cartan11].
The uniqueness of $f$ follows by induction and the facts that:
1. $f$ is determined by its values on $N_{1}$
2. the image of the contracting chain-homotopy, $\varphi$, lies in $N_{2}\subset M_{2}$.
3. the boundary map of $M_{2}$ is *injective* on $N_{2}$ (which implies that there is a *unique* lift of $f$ into the next higher dimension).
Now construct a contracting cochain on the normalized chain-complex of a standard simplex:
\[def:simplex-contracting-cochain\]Let $\Delta^{k}$ be a standard $k$-simplex with vertices $\{[0],\dots,[k]\}$ and $j$-faces $\{[i_{0},\dots,i_{j}]\}$ with $i_{0}<\cdots<i_{j}$ and let $s^{k}$ denote its normalized chain-complex with boundary map $\partial$. This is equipped with an augmentation $$\epsilon:s^{k}\to\ints$$ that maps all vertices to $1\in\ints$ and all other simplices to $0$. Let $$\iota_{k}:\ints\to s^{k}$$ denote the map sending $1\in\ints$ to the image of the vertex $[n]$. Then we have a contracting cochain **$$\varphi_{k}([i_{0},\dots,i_{t}]=\left\{ \begin{array}{cc}
(-1)^{t+1}[i_{0},\dots,i_{t},k] & \mathrm{if}\, i_{t}\ne k\\
0 & \mathrm{if}\, i_{t}=k
\end{array}\right.\label{eq:simplex-contracting-cochain}$$ and $1-\iota_{k}\circ\epsilon=\partial\circ\varphi_{k}+\varphi_{k}\circ\partial$.**
\[thm:ns-construct\]The normalized chain-complex of $[i_{0},\dots,i_{k}]=\Delta^{k}$ has a $\s$-coalgebra structure that is natural with respect to order-preserving mappings of vertex-sets $$[i_{0},\dots,i_{k}]\to[j_{0},\dots,j_{\ell}]$$ with $j_{0}\le\cdots\le j_{\ell}$ and $\ell\ge k$. This $\s$-coalgebra is denoted $\ns k$.
If $X$ is a simplicial complex (a semi-simplicial set whose simplices are uniquely determined by their vertices), then the normalized chain-complex of $X$ has a natural $\s$-coalgebra structure $$\cfn X=\dlimit\ns k$$ for $\Delta^{n}\in\boldsymbol{\Delta}\downarrow X$ — the simplex category of $X$.
The author has a Common LISP program for computing $f_{n}(x\otimes C(\Delta^{k}))$ — the number of terms is exponential in the dimension of $x$.
Compare this with the functor $\cf *$ defined in [@Smith:1994] and [@smith-cellular]. For simplicial complexes, $\cf X=\cfn X$.
**If $C=s^{k}=C(\Delta^{k})$ — the (unnormalized) chain complex — we can define a corresponding contracting homotopy on $C^{\otimes n}$ via $$\begin{aligned}
\Phi= & \varphi_{k}\otimes1\otimes\cdots\otimes1+\iota_{k}\circ\epsilon\otimes\varphi_{k}\otimes\cdots\otimes1+\\
& \cdots+\iota_{k}\circ\epsilon\otimes\cdots\otimes\iota_{k}\circ\epsilon\otimes\varphi_{k}\end{aligned}$$ where $\varphi_{k}$, $\iota_{k}$, and $\epsilon$ are as in definition \[def:simplex-contracting-cochain\].** Above dimension $0$, $\Phi$ is effectively equal to $\varphi_{k}\otimes1\otimes\cdots\otimes1$**. Now set $M_{2}=C^{\otimes n}$ and $N_{2}=\img(\Phi)$. In dimension $0$, we define $f_{n}$ for all $n$ via: $$f_{n}(A\otimes[0])=\left\{ \begin{array}{ll}
[0]\otimes\cdots\otimes[0] & \mathrm{if}\, A=[\,]\\
0 & \mathrm{if}\,\dim A>0
\end{array}\right.$$ This clearly makes $s^{0}$ a coalgebra over $\s$.**
**Suppose that the $f_{n}$ are defined below dimension $k$. Then $\cf{\partial\Delta^{k}}$ is well-defined and satisfies the conclusions of this theorem. We define $f_{n}(a[a_{1}|\dots|a_{j}]\otimes[0,\dots,k])$ by induction on $j$, requiring that:**
\[cond:invariant-condition\] $$f_{n}(A(S_{n},1)\otimes s^{k})\subseteq[i_{1},\dots,k]\otimes\mathrm{other}\,\mathrm{terms}\label{eq:invariant-condition}$$
— in other words, the *leftmost factor* must be in $\img\varphi_{k}$. This is the same as the leftmost factors being “rearward” faces of $\Delta^{k}$.
Now we set $$\begin{aligned}
f_{n}(A\otimes s^{k}) & = & \Phi\circ f_{n}(\partial A\otimes s^{k})\nonumber \\
& + & (-1)^{\dim A}\Phi\circ f_{n}(A\otimes\partial s^{k})\label{eq:high-diag-comp}\end{aligned}$$ where $A\in A(S_{n},1)\subset\rs n$ and the term $f_{n}(A\otimes\partial s^{k})$ refers to the coalgebra structure of $\cf{\partial\Delta^{k}}$.
The term $f_{n}(A\otimes\partial s^{k})$ is defined by induction and diagram \[dia:coalgebra-over-operad\] commutes for it. The term $f_{n}(\partial A\otimes s^{k})$ is defined by induction on the dimension of $A$ and diagram \[dia:coalgebra-over-operad\] for it as well.
The composite maps in both branches of diagram \[dia:coalgebra-over-operad\] satisfy condition \[cond:invariant-condition\] since:
1. any composite of $f_{n}$-maps will continue to satisfy condition \[cond:invariant-condition\].
2. $\circ_{i}(1\otimes A(S_{n},1)\otimes\cdots\otimes A(S_{m},1))\subseteq1\otimes A(S_{n+m-1},1)$ so that composing an $f_{n}$-map with $\circ_{i}$ results in a map that still satisfies condition \[cond:invariant-condition\].
3. the diagram commutes in lower dimensions (by induction on $k$)
Lemma \[lem:cartanconstruction\] implies that the composites one gets by following the two branches of diagram \[dia:coalgebra-over-operad\] must be *equal,* so the diagram commutes.
We ultimately get an expression for $f_{n}(x\otimes[0,\dots,k])$ as a sum of tensor-products of sub-simplices of $[0,\dots,k]$ — given as ordered lists of vertices.
We claim that this $\s$-coalgebra structure is natural with respect to ordered mappings of vertices. This follows from the fact that the only significant property that the vertex $k$ *has* in equation \[eq:simplex-contracting-cochain\], condition \[cond:invariant-condition\] and equation \[eq:high-diag-comp\] is that it is the *highest numbered* vertex.
We conclude this section some computations of higher coproducts:
\[example:e1timesdelta2\]If $[0,1,2]=\Delta^{2}$ is a $2$-simplex, then
$$f_{2}([\,]\otimes\Delta^{2})=\Delta^{2}\otimes F_{0}F_{1}\Delta^{2}+F_{2}\Delta^{2}\otimes F_{0}\Delta^{2}+F_{1}F_{2}\Delta^{2}\otimes\Delta^{2}\label{eq:delta-2-coproduct-1}$$
— the standard (Alexander-Whitney) coproduct — and
$$\begin{aligned}
f_{2}([(1,2)]\otimes\Delta^{2})= & [0,1,2]\otimes[1,2]-[0,2]\otimes[0,1,2]\\
& -[0,1,2]\otimes[0,1]\end{aligned}$$
or, in face-operations
$$\begin{aligned}
f_{2}([(1,2)]\otimes\Delta^{2})= & \Delta^{2}\otimes F_{0}\Delta^{2}-F_{1}\Delta^{2}\otimes\Delta^{2}\label{eq:e1timesdelta2-1}\\
& -\Delta^{2}\otimes F_{2}\Delta^{2}\nonumber \end{aligned}$$
If we write $\Delta^{2}=[0,1,2]$, we get $$f_{2}([\,]\otimes\Delta^{2})=[0,1,2]\otimes[2]+[0,1]\otimes[1,2]+[0]\otimes[0,1,2]$$
To compute $f_{2}([(1,2)]\otimes\Delta^{2})$ we have a version of equation \[eq:high-diag-comp\]: $$\begin{aligned}
f(e_{1}\otimes\Delta^{2}) & =\Phi_{2}(f_{2}(\partial e_{1}\otimes\Delta^{2})-\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})\\
& =-\Phi_{2}(f_{2}((1,2)\cdot[\,]\otimes\Delta^{2})+\Phi_{2}(f_{2}([\,]\otimes\Delta^{2})-\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})\end{aligned}$$ Now $$\begin{aligned}
\Phi_{2}(1,2)\cdot(f_{2}([\,]\otimes\Delta^{2})= & (\varphi_{2}\otimes1)\bigl([2]\otimes[0,1,2]-[1,2]\otimes[0,1]\\
& +[0,1,2]\otimes[0]\bigr)\\
& +(i\circ\epsilon\otimes\varphi_{2})\bigl([2]\otimes[0,1,2]\\
& -[1,2]\otimes[0,1]+[0,1,2]\otimes[0]\bigr)\\
= & 0\end{aligned}$$ and $$\begin{aligned}
\Phi_{2}(f_{2}([\,]\otimes\Delta^{2})= & (\varphi_{2}\otimes1)\bigl([0,1,2]\otimes[2]+[0,1]\otimes[1,2]\\
& +[0]\otimes[0,1,2]\bigr)\\
= & [0,1,2]\otimes[1,2]-[0,2]\otimes[0,1,2]\end{aligned}$$ In addition, proposition \[pro:simplicespropertyS\] implies that $$\begin{aligned}
f_{2}(e_{1}\otimes\partial\Delta^{2})= & [1,2]\otimes[1,2]-[0,2]\otimes[0,2]\\
& +[0,1]\otimes[0,1]\end{aligned}$$ so that $$\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})=[0,1,2]\otimes[0,1]$$
We conclude that $$\begin{aligned}
f_{2}([(1,2)]\otimes\Delta^{2})= & [0,1,2]\otimes[1,2]-[0,2]\otimes[0,1,2]\\
& -[0,1,2]\otimes[0,1]\end{aligned}$$ which implies equation \[eq:e1timesdelta2-1\].
We end this section with computations of some “higher coproducts.” We have a $\zs 2$-equivariant chain-map $$f_{2}(\rs 2\otimes C)\to C\otimes C$$
\[prop:e1timesdelta2\]If $\Delta^{2}$ is a $2$-simplex, then:
$$f_{2}([\,]\otimes\Delta^{2})=\Delta^{2}\otimes F_{0}F_{1}\Delta^{2}+F_{2}\Delta^{2}\otimes F_{0}\Delta^{2}+F_{1}F_{2}\Delta^{2}\otimes\Delta^{2}\label{eq:delta-2-coproduct}$$
Here $e_{0}=[\,]$ is the 0-dimensional generator of $\rs 2$ and this is just the standard (Alexander-Whitney) coproduct.
In addition, we have: $$\begin{aligned}
f_{2}([(1,2)]\otimes\Delta^{2})= & \Delta^{2}\otimes F_{0}\Delta^{2}-F_{1}\Delta^{2}\otimes\Delta^{2}\label{eq:e1timesdelta2}\\
& -\Delta^{2}\otimes F_{2}\Delta^{2}\nonumber \end{aligned}$$
If we write $\Delta^{2}=[0,1,2]$, we get $$f_{2}([\,]\otimes\Delta^{2})=[0,1,2]\otimes[2]+[0,1]\otimes[1,2]+[0]\otimes[0,1,2]$$
To compute $f_{2}([(1,2)]\otimes\Delta^{2})$ we have a version of equation \[eq:hdiag-comp\]: $$\begin{aligned}
f_{2}(e_{1}\otimes\Delta^{2}) & =\Phi_{2}(f_{2}(\partial e_{1}\otimes\Delta^{2})-\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})\\
& =-\Phi_{2}(f_{2}((1,2)\cdot[\,]\otimes\Delta^{2})+\Phi_{2}(f_{2}([\,]\otimes\Delta^{2})-\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})\end{aligned}$$ Now $$\begin{aligned}
\Phi_{2}(1,2)\cdot(f_{2}([\,]\otimes\Delta^{2})= & (\varphi_{2}\otimes1)([2]\otimes[0,1,2]-[1,2]\otimes[0,1]+[0,1,2]\otimes[0])\\
& +(i\circ\epsilon\otimes\varphi_{2})([2]\otimes[0,1,2]-[1,2]\otimes[0,1]+[0,1,2]\otimes[0])\\
= & 0\end{aligned}$$ and $$\begin{aligned}
\Phi_{2}(f_{2}([\,]\otimes\Delta^{2}) & =(\varphi_{2}\otimes1)\left([0,1,2]\otimes[2]+[0,1]\otimes[1,2]+[0]\otimes[0,1,2]\right)\\
& =[0,1,2]\otimes[1,2]-[0,2]\otimes[0,1,2]\end{aligned}$$ In addition, proposition \[pro:simplicespropertyS\] implies that $$f_{2}(e_{1}\otimes\partial\Delta^{2})=[1,2]\otimes[1,2]-[0,2]\otimes[0,2]+[0,1]\otimes[0,1]$$ so that $$\Phi_{2}f_{2}(e_{1}\otimes\partial\Delta^{2})=[0,1,2]\otimes[0,1]$$
We conclude that $$\begin{aligned}
f_{2}([(1,2)]_{2}\otimes\Delta^{2})= & [0,1,2]\otimes[1,2]-[0,2]\otimes[0,1,2]\\
& -[0,1,2]\otimes[0,1]\end{aligned}$$ which implies equation \[eq:e1timesdelta2\].
We continue this computation one dimension higher:
\[prop:e1timesdelta3\]If $\Delta^{3}$ is a $3$-simplex, then:
As before, $\Delta^{3}=[0,1,2,3]$, and we have $$\begin{aligned}
f_{2}(e_{1}\otimes\Delta^{3}) & =\Phi_{3}(f_{2}(\partial e_{1}\otimes\Delta^{3})-\Phi_{3}f_{2}(e_{1}\otimes\partial\Delta^{3})\\
& =-\Phi_{3}(f_{2}((1,2)\cdot[\,]\otimes\Delta^{3})+\Phi_{3}(f_{2}([\,]\otimes\Delta^{3})-\Phi_{3}f_{2}(e_{1}\otimes\partial\Delta^{3})\end{aligned}$$ and $$\Phi_{3}(f_{2}((1,2)\cdot[\,]\otimes\Delta^{3})=0$$ We also conclude
With this in mind, note that images of simplices in $\cfn *$ have an interesting property:
\[pro:simplicespropertyS\]Let $X$ be a simplicial set with $C=\cfn X$ and with coalgebra structure $$f_{n}:RS_{n}\otimes\cfn X\to\cfn X^{\otimes n}$$ and suppose $RS_{2}$ is generated in dimension $n$ by $e_{n}=\underbrace{[(1,2)|\cdots|(1,2)]}_{n\text{ terms}}$. If $x\in C$ is the image of a $k$-simplex, then $$f_{2}(e_{k}\otimes x)=\xi_{k}\cdot x\otimes x$$ where $\xi_{k}=(-1)^{k(k-1)/2}$.
This is just a chain-level statement that the Steenrod operation $\operatorname{Sq}^{0}$ acts trivially on mod-$2$ cohomology. A weaker form of this result appeared in [@Davis:mco].
Recall that $(\rs 2)_{n}=\ints[\ints_{2}]$ generated by $ $$e_{n}=[\underbrace{(1,2)|\cdots|(1,2)}_{n\text{ factors}}]$. Let $T$ be the generator of $\ints_{2}$ — acting on $C\otimes C$ by swapping the copies of $C$.
We assume that $f_{2}(e_{i}\otimes C(\Delta^{j}))\subset C(\Delta^{j})\otimes C(\Delta^{j})$ so that $$i>j\implies f_{2}(e_{i}\otimes C(\Delta^{j}))=0\label{eq:big-diag-condition}$$
As in section 4 of [@Smith:1994], if $e_{0}=[\,]\in\rs 2$ is the $0$-dimensional generator, we define $$f_{2}:\rs 2\otimes C\to C\otimes C$$ inductively by $$\begin{aligned}
f_{2}(e_{0}\otimes[i]) & = & [i]\otimes[i]\nonumber \\
f_{2}(e_{0}\otimes[0,\dots,k]) & = & \sum_{i=0}^{k}[0,\dots,i]\otimes[i,\dots,k]\label{eq:big-diag1}\end{aligned}$$ Let $\sigma=\Delta^{k}$ and inductively define $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma) & =\Phi_{k}(f_{2}(\partial e_{k}\otimes\sigma)+(-1)^{k}\Phi_{k}f_{2}(e_{k}\otimes\partial\sigma)\nonumber \\
& =\Phi_{k}(f_{2}(\partial e_{k}\otimes\sigma)\label{eq:hdiag-comp}\end{aligned}$$ because of equation \[eq:big-diag-condition\].
Expanding $\Phi_{k}$, we get $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma) & =(\varphi_{k}\otimes1)(f_{2}(\partial e_{k}\otimes\sigma))+(i\circ\epsilon\otimes\varphi_{k})f_{2}(\partial e_{k}\otimes\sigma)\nonumber \\
& =(\varphi_{k}\otimes1)(f_{2}(\partial e_{k}\otimes\sigma))\label{eq:big-diag2}\end{aligned}$$ because $\varphi_{k}^{2}=0$ and $\varphi_{k}\circ i\circ\epsilon=0$.
Noting that $\partial e_{k}=(1+(-1)^{k}T)e_{k-1}\in\rs 2$, we get $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma) & =(\varphi_{k}\otimes1)(f_{2}(e_{k-1}\otimes\sigma)+(-1)^{k}(\varphi_{k}\otimes1)\cdot T\cdot f_{2}(e_{k-1}\otimes\sigma)\\
& =(-1)^{k}(\varphi_{k}\otimes1)\cdot T\cdot f_{2}(e_{k-1}\otimes\sigma)\end{aligned}$$ again, because $\varphi_{k}^{2}=0$ and $\varphi_{k}\circ\iota_{k}\circ\epsilon=0$. We continue, using equation \[eq:big-diag2\] to compute $f(e_{k-1}\otimes\sigma)$: $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma)= & (-1)^{k}(\varphi_{k}\otimes1)\cdot T\cdot f_{2}(e_{k-1}\otimes\sigma)\\
= & (-1)^{k}(\varphi_{k}\otimes1)\cdot T\cdot(\varphi_{k}\otimes1)\biggl(f_{2}(\partial e_{k-1}\otimes\sigma)\\
& +(-1)^{k-1}f_{2}(e_{k-1}\otimes\partial\sigma)\biggr)\\
= & (-1)^{k}\varphi_{k}\otimes\varphi_{k}\cdot T\cdot\biggl(f_{2}(\partial e_{k-1}\otimes\sigma)\\
& +(-1)^{k-1}f_{2}(e_{k-1}\otimes\partial\sigma)\biggr)\end{aligned}$$ If $k-1=0$, then the left term vanishes. If $k-1=1$ so $\partial e_{k-1}$ is $0$-dimensional then equation \[eq:big-diag1\] gives $f(\partial e_{1}\otimes\sigma)$ and this vanishes when plugged into $\varphi_{k}\otimes\varphi_{k}$. If $k-1>1$, then $f_{2}(\partial e_{k-1}\otimes\sigma)$ is in the image of $\varphi_{k}$, so it vanishes when plugged into $\varphi_{k}\otimes\varphi_{k}$.
In *all* cases, we can write $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma) & =(-1)^{k}\varphi_{k}\otimes\varphi_{k}\cdot T\cdot(-1)^{k-1}f_{2}(e_{k-1}\otimes\partial\sigma)\\
& =-\varphi_{k}\otimes\varphi_{k}\cdot T\cdot f_{2}(e_{k-1}\otimes\partial\sigma)\end{aligned}$$ If $f_{2}(e_{k-1}\otimes\Delta^{k-1})=\xi_{k-1}\Delta^{k-1}\otimes\Delta^{k-1}$ (the inductive hypothesis), then $$\begin{gathered}
f_{2}(e_{k-1}\otimes\partial\sigma)=\\
\sum_{i=0}^{k}\xi_{k-1}\cdot(-1)^{i}[0,\dots,i-1,i+1,\dots k]\otimes[0,\dots,i-1,i+1,\dots k]\end{gathered}$$ and the only term that does not get annihilated by $\varphi_{k}\otimes\varphi_{k}$ is $$(-1)^{k}[0,\dots,k-1]\otimes[0,\dots,k-1]$$ (see equation ). We get $$\begin{aligned}
f_{2}(e_{k}\otimes\sigma) & =\xi_{k-1}\cdot\varphi_{k}\otimes\varphi_{k}\cdot T\cdot(-1)^{k-1}[0,\dots,k-1]\otimes[0,\dots,k-1]\\
& =\xi_{k-1}\cdot\varphi_{k}\otimes\varphi_{k}(-1)^{(k-1)^{2}+k-1}[0,\dots,k-1]\otimes[0,\dots,k-1]\\
& =\xi_{k-1}\cdot(-1)^{(k-1)^{2}+2(k-1)}\varphi[0,\dots,k-1]\otimes\varphi[0,\dots,k-1]\\
& =\xi_{k-1}\cdot(-1)^{k-1}[0,\dots,k]\otimes[0,\dots,k]\\
& =\xi_{k}\cdot[0,\dots,k]\otimes[0,\dots,k]\end{aligned}$$ where the sign-changes are due to the Koszul Convention. We conclude that $\xi_{k}=(-1)^{k-1}\xi_{k-1}$.
Proof of lemma \[lem:diagonals-linearly-independent\]
=====================================================
\[lem:diagonals-linearly-independent\]Let $C$ be a free abelian group, let $$\hat{C}=\ints\oplus\prod_{i=1}^{\infty}C^{\otimes i}$$
Let $e:C\to\hat{C}$ be the function that sends $c\in C$ to $$(1,c,c\otimes c,c\otimes c\otimes c,\dots)\in\hat{C}$$ For any integer $t>1$ and any set $\{c_{1},\dots,c_{t}\}\in C$ of distinct, nonzero elements, the elements $$\{e(c_{1}),\dots,e(c_{t})\}\in\rats\otimes_{\ints}\hat{C}$$ are linearly independent over $\rats$. It follows that $e$ defines an injective function $$\bar{e}:\ints[C]\to\hat{C}$$
We will construct a vector-space morphism $$f:\rats\otimes_{\ints}\hat{C}\to V\label{eq:diagonals-linearly-independent1}$$ such that the images, $\{f(e(c_{i}))\}$, are linearly independent. We begin with the “truncation morphism” $$r_{t}:\hat{C}\to\ints\oplus\bigoplus_{i=1}^{t-1}C^{\otimes i}=\hat{C}_{t-1}$$ which maps $C^{\otimes1}$ isomorphically. If $\{b_{i}\}$ is a $\ints$-basis for $C$, we define a vector-space morphism $$g:\hat{C}_{t-1}\otimes_{\ints}\rats\to\rats[X_{1},X_{2},\dots]$$ by setting $$g(c)=\sum_{\alpha}z_{\alpha}X_{\alpha}$$ where $c=\sum_{\alpha}z_{\alpha}b_{\alpha}\in C\otimes_{\ints}\rats$, and extend this to $\hat{C}_{t-1}\otimes_{\ints}\rats$ via $$g(c_{1}\otimes\cdots\otimes c_{j})=g(c_{1})\cdots g(c_{j})\in\rats[X_{1},X_{2},\dots]$$ The map in equation \[eq:diagonals-linearly-independent1\] is just the composite $$\hat{C}\otimes_{\ints}\rats\xrightarrow{r_{t-1}\otimes1}\hat{C}_{t-1}\otimes_{\ints}\rats\xrightarrow{g}\rats[X_{1},X_{2},\dots]$$ It is not hard to see that $$p_{i}=f(e(c_{i}))=1+f(c_{i})+\cdots+f(c_{i})^{t-1}\in\rats[X_{1},X_{2},\dots]$$ for $i=1,\dots,t$. Since the $f(c_{i})$ are *linear* in the indeterminates $X_{i}$, the degree-$j$ component (in the indeterminates) of $f(e(c_{i}))$ is precisely $f(c_{i})^{j}$. It follows that a linear dependence-relation $$\sum_{i=1}^{t}\alpha_{i}\cdot p_{i}=0$$ with $\alpha_{i}\in\rats$, holds if and only if $$\sum_{i=1}^{t}\alpha_{i}\cdot f(c_{i})^{j}=0$$ for all $j=0,\dots,t-1$. This is equivalent to $\det M=0$, where $$M=\left[\begin{array}{cccc}
1 & 1 & \cdots & 1\\
f(c_{1}) & f(c_{2}) & \cdots & f(c_{t})\\
\vdots & \vdots & \ddots & \vdots\\
f(c_{1})^{t-1} & f(c_{2})^{t-1} & \cdots & f(c_{t})^{t-1}
\end{array}\right]$$ Since $M$ is the transpose of the Vandermonde matrix, we get $$\det M=\prod_{1\le i<j\le t}(f(c_{i})-f(c_{j}))$$ Since $f|C\otimes_{\ints}\rats\subset\hat{C}\otimes_{\ints}\rats$ is *injective,* it follows that this *only* vanishes if there exist $i$ and $j$ with $i\ne j$ and $c_{i}=c_{j}$. The second conclusion follows.
[10]{}
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[to3em]{}, *Op[é]{}rations dans les constructions acycliques*, S[é]{}minaire Henri Cartan, vol. 7, Secr[é]{}tariat math[é]{}matique, Paris, 1954-1955, pp. 1–11.
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1)$]{}’s*, Lecture Notes in Mathematics, vol. 1126, Springer-Verlag, 1983, pp. 51–61.
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[^1]: In a nilpotent coalgebra, iterated coproducts of elements “peter out” after a finite number of steps. See
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---
abstract: 'In this paper, we will study the deformation of a three dimensional $\mathcal{N} = 2$ supersymmetry gauge theory. We will deform this theory by imposing non-anticommutativity. This will break the supersymmetry of the theory from $\mathcal{N} = 2$ supersymmetry to $\mathcal{N} = 1$ supersymmetry. We will address the problem that occurs in the Landau gauge due to the existence of multiple solutions to the gauge fixing condition. This will be done by generalizing the Faddeev-Popov method. This formalism is motivated from the Nicolai map in topological field theories. Finally, we will study the extended BRST symmetry that occurs in this theory.'
author:
- |
Paul Weinreb$^1$ and Mir Faizal$^2$\
$^1$Department of Mathematics King’s College London,\
Strand London WC2R 2LS, UK\
$^2$Department of Physics and Astronomy,\
University of Waterloo, Waterloo,\
Ontario N2L 3G1, Canada
title: 'Generalized Faddeev-Popov Method for a Deformed Supersymmetric Yang-Mills Theory'
---
Introduction
============
The studies done on closed strings in the presence of a constant two form field, and the gravitational action induced by the bosonic string theory on a space-filling D-brane with a constant magnetic field, have motivated a noncommutative deformation of field theories [@stri]-[@stri1]. The noncommutative field theory can be constructed by replacing all the products of fields in the action by Moyal products of those fields. This lead to the mixing of ultraviolet and infrared divergences [@UV]. These theories are also non-local, but the non-locality is introduced in a controlled manner [@NL]. The existence of such noncommutative deformations have motivated the study of a wider class of deformations for supersymmetric field theories. As the coordinate space of supersymmetric theories has Grassmann coordinates, it is possible to impose a noncommutative deformation between such coordinates and ordinary spacetime coordinates, along with introducing a non-anticommutative deformation between the Grassmann coordinates [@non]-[@non1]. It is possible to deform the supersymmetric gauge theories using such deformations. [@ng]-[@gn]. It may be noted that the non-anticommutative deformation of field theories can be related to the existence of $R-R$ backgrounds fields [@r1]-[@1r].
In this paper, we will analyse such a deformation for a three dimensional supersymmetric gauge theory. So, we will promote the Grassmann coordinates to non-anticommutating coordinates. This will break half the supersymmetry of the original theory. Thus, if we start from a four dimensional theory with $\mathcal{N} = 1$ supersymmetry, this deformation would break the supersymmetry down to $\mathcal{N} = 1/2$ supersymmetry [@12]-[@21]. This is because the four dimensional gauge theories have enough degrees of freedom to partially break the supersymmetry of the theory. However, if we tried to deform a three dimensional theory with $\mathcal{N} = 1$ supersymmetry, we would break all the supersymmetry of the theory. So, we will consider a theory with $\mathcal{N} = 2$ supersymmetry in three dimensions, and the non-anticommutativity will break half of this supersymmetry. Thus, the theory will have $\mathcal{N} = 1$ supersymmetry after this deformation. It may be noted that such deformation of supersymmetric field theories has been studied, and it has observed that the non-anticommuativity does break the supersymmetry from $\mathcal{N} = 2$ supersymmetry to $\mathcal{N} = 1$ supersymmetry [@az]-[@za].
As a gauge theory has a non-physical degrees of freedom, it cannot be quantized without fixing a gauge. The gauge fixing term can be incorporated at a quantum level by adding a ghost term and a gauge fixing term to the original classical action. This new action, which is obtained by the adding of the gauge fixing term and the ghost term to the original action is invariant under a symmetry called the BRST symmetry. [@BRST]-[@brst1]. It is also invariant under another symmetry which dual to the BRST symmetry called the anti-BRST symmetry [@antibrst]. However, for non-perturbative gauge-fixing the Gribov ambiguity causes a problem [@gr]-[@gr1]. This is because it is not possible to obtain a unique representative on gauge orbits once large scale field fluctuations in gauges like the Landau gauge. The Gribov ambiguity for supersymmetry theories has been recently studied [@rg]-[@rg1].
It may be noted that the Gribov ambiguity is related to the existence of topological properties of field theories [@1]. In fact, it has also been demonstrated that gauge theories can be expressed in terms of topologically quantities [@2]-[@4]. This construction has motivated the study of two different kind of topological field theories called the Schwarz type theories [@2] and Witten type theories [@b]. The Witten type theories can be related to the existence of Gribov ambiguity in field theory. This is because the Witten type theories are related to the cohomology, and can have a direct connection with the BRST symmetry in the gauge theory. In fact, it has been demonstrated that the Nicolai map is can be constructed in Witten type theories [@ni]. The Nicolai map can be used to restricted the path integral to the moduli space of classical solutions [@ni]. So, the Nicolai map has also been used to address the Gribov ambiguity for usual gauge theories [@GKW]. In this analysis, the BRST symmetry of the gauge theory was extended to include an extended BRST symmetry. We will apply this formalism to three dimensional non-anticommutative Yang-Mills theory.
The remaining of the paper is organized as follows. In section \[a\], we will deform a three dimensional supersymmetric Yang-Mills theory theory in $\mathcal{N} =2$ supersaturate formalism. This deformation will break the supersymmetry of the theory to $\mathcal{N} =1$ supersymmetry. In section \[b\], we will analyse the quantization of this theory. We will also address the problem that occurs in the Landau gauge due to the existence of multiple solutions to the gauge fixing condition. A formalism that has been motivated by the Nicolai map in topological field theories will be used to address this problem. In section \[c\], we will analyse the extended BRST symmetry for this formalism. Finally, in section \[d\], we will summarize our results and discuss some possible extensions of this work.
Non-anticommutativity {#a}
=====================
We will analyse a three dimensional gauge theory in $\mathcal{N} = 2$ superspace formalism. This space is parameterized by the coordinates, $(x^\mu, \theta_{1\alpha}, \theta_{2\alpha})$, where $\theta_{1\alpha} = \theta_{1 +}, \theta_{1-} , $ and $ \theta_{2\alpha} = \theta_{2 +}, \theta_{2-} $. The generators of $\mathcal{N} = 2$ supersymmetry can be written as $$\begin{aligned}
Q_{1a} &=& \partial_{1\alpha} - (\gamma^\mu \theta_1)_\alpha \partial_\mu,
\nonumber \\
Q_{2a} &=& \partial_{2\alpha} - (\gamma^\mu \theta_2)_\alpha \partial_\mu.
\end{aligned}$$ The super-derivatives which commute with these generators of supersymmetry can be written as $$\begin{aligned}
D_{1\alpha} &=& \partial_{1\alpha} + (\gamma^\mu \theta_1)_\alpha \partial_\mu,
\nonumber \\
D_{2\alpha} &=& \partial_{2\alpha} + (\gamma^\mu \theta_2)_\alpha \partial_\mu.\end{aligned}$$ It is possible to transform the $\theta_1$ and $\theta_2$ to another set of Grassman coordinates $$\begin{aligned}
\left( \begin{array}{ccc}
\theta_{ a} \\
\bar \theta_{ a} \\
\end{array} \right) =
\left( \begin{array}{ccc}
x_{11} & x_{12} \\
x_{21} & x_{22} \\
\end{array} \right)
\left( \begin{array}{ccc}
\theta_{1a} \\
\theta_{2a} \\
\end{array} \right),
\end{aligned}$$ as long as the $\det (x_{ij}) \neq 0$. Now we choose $x_{ij}$, such that $ \theta_{a} = (\theta_{1a} + i \theta_{2a})/2, \, \,
\bar \theta_{a} = (\theta_{1a} - i \theta_{2a})/2$, and define another set of covariant derivatives as [@berm] $$\begin{aligned}
D_\alpha= \frac{1}{2} ( D_{1\alpha} + i D_{2\alpha}),&&
\bar D_\alpha = \frac{1}{2} ( D_{1\alpha} - i D_{2\alpha}). \end{aligned}$$ These covariant derivatives satisfy, $$\begin{aligned}
\{ D_\alpha, \bar D_\beta\} = i(\gamma^\mu \partial_\mu)_{\alpha\beta},
&&\{\bar D_\alpha, \bar D_\beta\} = 0,\nonumber \\
\{ D_\alpha, D_\beta\} = 0.
&&\end{aligned}$$ We also define $D^2 = D^\alpha D_\alpha/2$ and $\bar D^2 = \bar D^\alpha \bar D_\alpha/2$. This superspace is parameterized by the coordinates, $(x^\mu, \theta^\alpha, \bar \theta^{ {\alpha}})$, where $\mu = 0, 1,2, 3$, and $\alpha, {\alpha} = 1, 2$. We will impose the non-anticommutative deformation between $\theta^\alpha$ as $$\{\theta^\alpha, \theta^{\beta} \} = C^{\alpha \beta}.$$ The product of superfields of $\theta^\alpha$ can be Weyl ordered. This is done by the ordinary product of superfields by a star product, which is a fermionic version of the Moyal product $$V(x, \theta, \bar \theta) \star V'(x, \theta, \bar \theta)= V(x, \theta, \bar \theta)
\exp \left(-\frac{ C^{\alpha\beta}}{2} \overleftarrow{\frac{\partial}{\partial \theta^\alpha }}
\overrightarrow{\frac{\partial}{\partial \theta^\beta}} \right) V'(x, \theta, \bar \theta),$$ where $V(x, \theta, \bar \theta)$, and $V' (x, \theta, \bar \theta)$ are supervector fields. It may be noted that the Grassmann coordinate $\bar \theta_{ \alpha}$ satisfy, $$\begin{aligned}
\{ \bar \theta_{ {\alpha}}, \bar \theta_{\beta}\} =0, &&
\{ \bar \theta_{ {\alpha}}, \theta_\beta\} =0, \nonumber \\
{ [\bar \theta_{ {\alpha}}, x^\mu] }=0, &&\end{aligned}$$ and the bosonic coordinates satisfy $$\begin{aligned}
[x^\mu, x^\nu] = \bar \theta \bar \theta C^{\mu \nu} , && [x^\mu, \theta^\alpha ]
= i C^{\alpha\beta}\sigma^\mu_{\beta\delta}\bar\theta^{\delta}, \end{aligned}$$ where $ C^{\mu\nu} = C^{\alpha\beta} \epsilon_{\beta \delta}(\sigma^{\mu\nu})_\alpha^\delta $. It is possible to write $$\begin{aligned}
[ \theta_{\alpha }, y^\mu] =0, &&
[ \bar \theta_{\alpha}, y^\mu] =0 , \nonumber \\
{[y^\mu, y^\nu]} =0, &&\end{aligned}$$ where $
y^\mu = x^\mu + i \theta^\alpha \gamma^\mu_{\alpha\beta} \bar \theta^{\beta},
$ The superfields can be expressed as functions of $(y^\mu, \theta^\alpha , \bar \theta^{ {\alpha}})$ [@non]-[@non1]. The supervector field $V(y, \theta, \bar \theta)$ in the Wess-Zumino gauge is given by $$\begin{aligned}
V(y, \theta, \bar \theta) &=& - \theta \sigma^\mu \bar \theta A_\mu + i \theta\theta \bar \theta \bar \lambda
- i \bar \theta \bar \theta \theta^\alpha \left( \lambda_\alpha + \frac{1}{4}\epsilon_{\alpha \beta}
C^{\beta \delta}
\sigma^\mu_{\delta \rho }
[\bar\lambda^{\rho}, A_\mu] \right) \nonumber \\ && + \frac{1}{2}
\theta\theta \bar \theta \bar \theta (D - i \partial_\mu A^\mu), \end{aligned}$$ where $V^a(y, \theta, \bar \theta)T_a = V(y, \theta, \bar \theta)$. Here $T_a$ are generators of $SU(N)$ Lie algebra, $$[T_a, T_b] = i f_{ab}^cT_c.$$ We can define the Chiral and anti-Chiral field strength for this theory as $$\begin{aligned}
W_\alpha &=& - \frac{1}{4} \bar D \bar D e^{-V}_\star \star D_\alpha e^V_\star, \nonumber \\
\bar W_{\dot{\alpha}} &=& \frac{1}{4} D D e^{-V}_\star \star \bar D_{\dot{\alpha}} e^V_\star.
\end{aligned}$$ The action for $\mathcal{N} = 1$ gauge theory can be written as $$S_{DSYM} = Tr \int d^3 x d^2 \theta \, W^\alpha \star W_\alpha + Tr \int d^3 x d^2\bar \theta \,
\bar W^{\dot{\alpha}} \star \bar W_{\dot{\alpha}}.$$ It is possible to expand this in component form as $$\begin{aligned}
S_{DSYM} &=& Tr \int d^3 x \left[ ( - 4 i \bar \lambda \sigma^\mu D_\mu \lambda - F^{\mu\nu} F_{\mu\nu} + 2 D^2)
\right. \nonumber \\ && \left. + Tr \int d^3 x \left( - 2 iC^{\mu\nu} F_{\mu\nu}\bar\lambda\bar \lambda +
\frac{C^{\mu\nu}C_{\mu\nu}}{2} (\bar \lambda\bar \lambda)^2 \right) \right]. \end{aligned}$$
Quantization {#b}
============
In the previous section, we analysed the non-anticommutative deformation of a gauge theory in $\mathcal{N} = 2$ superspace formalism. This deformation broke the supersymmetry of the gauge theory from $\mathcal{N} = 2$ supersymmetry to $\mathcal{N} = 1$ supersymmetry. In this section, we will analyse the quantization of this $\mathcal{N} = 1$ supersymmetry. We will perform this analysis using a covariant formalism. We can also express this deformed three dimensional theory using the covariant derivative [@1001] $$\begin{aligned}
\nabla_A &=& (-i \{\mathcal{D}_\alpha , \bar D_{ {\beta}}\}_\star ,
\mathcal{D}_\alpha , \bar D_{ {\alpha}}),
\nonumber \\
\exp ( V )_\star \star \nabla_A \star \exp( -V )_\star &=& (-i \{ D_\alpha,\bar \mathcal{D}_{
\beta} \}_\star ,
D_\alpha , \bar \mathcal{D}_{ {\alpha}} ), \end{aligned}$$ where $
\mathcal{D}_\alpha = \exp (-V)_\star \star D_\alpha \exp (V )_\star, $ and $
\bar \mathcal{D}_{ {\alpha}} =
\exp (V)_\star \star \bar D_{ {\alpha}} \exp(- V )_\star $. We can express it using a covarient formalism as [@1001] $$\begin{aligned}
\nabla_A = D_A - i \Gamma_A, \end{aligned}$$ where $$\begin{aligned}
D_A &=& ( \partial_{\alpha\beta}, D_\alpha, \bar D_{\beta}),
\nonumber \\
\Gamma_A&=& (\Gamma_{\alpha \beta},\Gamma_\alpha,\bar \Gamma_{ {\alpha}} ). \end{aligned}$$ It is possible to express the Bianchi identity as $ [\nabla_{[A}, H_{BC)}\}_\star =0$, where $H_{AB}= [ \nabla_A, \nabla_B \}_\star = T^C_{AB}\nabla_C - i F_{AB} $. The gauge transformation of this covariant derivative are given by $\nabla_A \to e^{ i \Lambda}_\star \star \nabla_A \star e^{-i\Lambda}_\star$, and $ e^{V}_\star \star \nabla_A \star e^{-V}_\star
\to e^{i \bar \Lambda}_\star \star e^{V}_\star \star \nabla_A \star e^{-V}_\star \star e^{ -i \bar\Lambda}_\star$. However, it is is possible to construct a covariant derivative in a different representation, and the gauge transformations of this covariant derivative is given by $$\nabla_A \to u \star \nabla_A \star u^{-1},$$ where $u$ is defined as $ u = e^{iK}_\star $, and the parameter $K = K^A T_A$ is a real superfield [@1001]. The transformation of the spinor fields can be written as $$\begin{aligned}
\Gamma_\alpha &\to& i u \star \nabla_\alpha \star u^{-1},\nonumber \\
\bar \Gamma_{ {\alpha}} &\to& i u\star \bar \nabla_{ {\alpha}}\star u^{-1},\nonumber \\
\Gamma_{\alpha\beta} &\to& i u \star \nabla_{ \alpha\beta } \star u^{-1}.
\end{aligned}$$
As this theory has gauge symmetry, we will have to introduce a ghost term and a gauge fixing term to the original action. However, to calculate non-perturbative effects, we will have to deal with the Gribov ambiguity. So, we will apply the extended BRST symmetry to this theory [@GKW]. Thus, for a gauge fixing condition $F[^g \Gamma] =0, $ and $ \bar F[^g \bar \Gamma] =0 $, where $g$ is an element of $SU(N)$. For the Landau gauge, we can write $$\begin{aligned}
F[\Gamma] = D^a \Gamma_a =0, && \bar F[\bar \Gamma] = \bar D^a \bar \Gamma_a =0.
\end{aligned}$$ The Faddeev-Popov operator can be written as $$\begin{aligned}
M_F[\Gamma] = \left(\frac{\delta F[^g \Gamma]}{\delta g}\right), &&
\bar M_F[\bar\Gamma] = \left(\frac{\delta \bar F[^g\bar \Gamma]}{\delta g}\right).\end{aligned}$$ In the standard Faddeev-Popov method, the following expression is used, $$\begin{aligned}
1&=&\int \mathcal{D}_g\Delta_F[^g\Gamma]\delta[F[^g\Gamma]] + \int \mathcal{D}_g\bar \Delta_F[^g\bar
\Gamma]\delta[\bar F[^g\bar\Gamma]],
\end{aligned}$$ However, as we can have multiple solutions in the Landau gauge, we can generalize the standard Faddeev-Popov to multiple solutions as follows [@GKW], $$\begin{aligned}
N_F[\Gamma, \bar \Gamma]&=&\int \mathcal{D}_g\Delta_F[^g\Gamma]\delta[F[^g\Gamma]] +
\int \mathcal{D}_g\bar \Delta_F[^g\bar \Gamma]\delta[\bar F[^g\bar \Gamma]]
\end{aligned}$$ where $ N_F[\Gamma,\bar \Gamma]$ denotes the number solutions corresponding to a gauge fixing condition.
It is known that for usual gauge theories, the fundamental modular region is a unique representation of every gauge orbit [@x]-[@y]. Even though the theory considered here is a supersymmetric Yang-Mills theory, it is a gauge theory and hence we expect such a unique representation of every supergauge orbit. Furthermore, it is possible to generalize the arguments used for obtaining these results for a supersymmetric theory. It may be noted that even though this theory has $\mathcal{N} = 1$ supersymmetry, the gauge symmetry for this theory is different from both the un-deformed Yang-Mills theory with $\mathcal{N} = 1$ supersymmetry and the un-deformed Yang-Mills theory with $\mathcal{N} = 2$ supersymmetry. This is because the gauge transformations are defined in terms of the deformed superspace coordinates, and thus involve fermionic version of the Moyal product. However, as this theory has a gauge symmetry, we can in principle define unique representation of every supergauge orbit on deformed superspace. So, we will denote the fundamental modular region $\Lambda$, as the set of absolute minima of the functional $$\begin{aligned}
V_{\Gamma, \bar \Gamma} =\int d^3x d ^2 \theta (^g\Gamma)^2_\star + \int d^3x d ^2 \bar \theta (^g\bar \Gamma)^2_\star.
\end{aligned}$$ The stationary points of $V_{\Gamma, \bar \Gamma}$ satisfy the Landau gauge condition, and the boundary of the fundamental modular region $\partial \Lambda$ is the set of degenerate absolute minimum of $V_{\Gamma, \bar \Gamma}$. The fundamental modular region lies within the Gribov region, and the operators $M_F [\Gamma]$ and $\bar M_F [\bar \Gamma]$ obtain zero modes in the boundary of the Gribov region, which is called the Gribov horizon. The gauge orbits can be labeled using this fundamental modular region, and denoted $\Gamma (v), \bar \Gamma(v)$ as configurations in the fundamental modular region. So, we have $N_F[\Gamma, \bar \Gamma]$, as every orbit crosses the fundamental modular region only once. Since the integrand is positive the minima of $V_{\Gamma, \bar \Gamma}$ are those $\Gamma_\alpha, \bar \Gamma_\alpha$ satisfying the Landau gauge condition $D^\alpha\Gamma^\alpha=0$ and $\bar D^\alpha\bar \Gamma^\alpha=0$. The boundary $\partial \Lambda$ is the set of degenerate absolute minima of $V_{\Gamma, \bar \Gamma}$.
The expectation value of a gauge invariant operator $\mathcal{O}_\star$ defined on the deformed superspace, can be written as $$\begin{aligned}
\langle\mathcal{O}{[\Gamma, \bar \Gamma]_\star}\rangle
&=&\frac{\int \mathcal{D} \mathcal{O}
[\Gamma,\bar\Gamma]_\star e^{-S_{DSYM}}}{\int \mathcal{D} e^{-S_{DSYM}}}, \end{aligned}$$ where $\mathcal{D} =\mathcal{D}[ \Gamma (v) \bar \Gamma (v)] $ is a suitably defined measure for the path integral. This is well defined if there are unique solutions to the gauge fixing condition. Since $N_F $ is finite, we obtain we write $$\begin{aligned}
\langle\mathcal{O}{[\Gamma, \Gamma]_\star}\rangle&=&
\left[\int \mathcal{D} {N_F[\Gamma, \bar \Gamma]}^{-1}
\int \mathcal{D} g \delta(F[^g\Gamma])|\det M_F[^g\Gamma]\right.\nonumber \\ &&\left.
\delta(\bar F[^g\bar\Gamma])|\det \bar M_F[^g\bar\Gamma]|
\mathcal{O}[\Gamma, \bar \Gamma]_\star e^{-S_{DSYM[\Gamma, \bar \Gamma]}
[\Gamma, \bar \Gamma]}\right]_\star\nonumber \\ && \times \left[\int \mathcal{D} {N_F[\Gamma, \bar \Gamma]}^{-1}
\int \mathcal{D}g\delta(F[^g\Gamma])|\det M_F[^g\Gamma]
\right.\nonumber \\ && \left. \delta(\bar F[^g\bar \Gamma])|\det \bar M_F[^g\bar\Gamma]
|e^{-S_{DSYM}[\Gamma, \bar \Gamma]}\right]_\star^{-1}.\end{aligned}$$ So, we have $N_F[\Gamma (v), \bar \Gamma (v)] = N_F[^g\Gamma (v), ^g \bar \Gamma (v)]=N_F[\Gamma, \bar \Gamma] $. So, we can write $$\begin{aligned}
\langle\mathcal{O}{[\Gamma, \bar \Gamma]_\star}\rangle&=&
\left[\int \mathcal{D} \delta(F[\Gamma])|\det M_F[\Gamma] |\delta(\bar F[\bar\Gamma])\right.\nonumber \\
&& \left.|\det \bar M_F[\bar\Gamma]|\mathcal{O}[\Gamma, \bar \Gamma]
e^{-S_{DSYM}[\Gamma, \bar \Gamma]} \right]_\star \nonumber \\ && \times \left[\int \mathcal{D} \delta(F[\Gamma])
|\det M_F[\Gamma]|\delta(\bar F[\bar\Gamma])\right.\nonumber \\ && \left.
|\det \bar M_F[\bar\Gamma]|e^{-S_{DSYM}[\Gamma, \bar \Gamma]}\right]_\star^{-1}.\end{aligned}$$ It may be noted that this expression involves the fermionic version of the Moyal product. This expression can be written in terms of the power series involving the deformation matrix $C^{\alpha \beta}$. The first term in the series will correspond to the usual un-deformed supersymmetric Yang-Mills theory. Thus, the expectation value of a gauge invariant operator also receives corrections from the deformation of the Yang-Mills theory. Furthermore, in absence of supersymmetry, this expression reduces to the expression for the usual Yang-Mills case. This can be seen by setting by making all the fermionic fields in the expression to vanish. However, this will be different from breaking all the supersymmetry by imposing two non-anticommutative deformations. The expectation value of gauge invariant operators in this case, will be different from the expectation value of gauge invariant operators of the usual Yang-Mills theory.
Symmetries {#c}
==========
In the previous section, we analysed the quantization of a three dimensional non-anticommutative gauge theory. We generalized the standard Faddeev-Popov method to incorporate the existence of multiple solutions to the gauge fixing condition. This was done to address the existence of non-perturbative effects. In this section, we will analyse the BRST symmetry for this generalized Faddeev-Popov method. The partition function for this generalized Faddeev-Popov method, can be written as $$\begin{aligned}
Z_{gf}&=&\left[\int \mathcal{D} \delta(F[\Gamma])|\det M_F[\Gamma] |\delta(\bar F[\bar\Gamma])\right.\nonumber \\
&& \left.|\det \bar M_F[\bar\Gamma]|
e^{-S_{DSYM}[\Gamma, \bar \Gamma]} \right]_\star.\end{aligned}$$ This is valid for non-perturbative field theory, as it takes the modulus of the determinant into account. Thus, we can write [@GKW] $$\begin{aligned}
|\det M_F [\Gamma] |_\star = [\rm{sgn}(\det M_F [\Gamma]) \det M_F [\Gamma]]_\star,
\nonumber \\ |\det \bar M_F [\bar \Gamma]_\star = [\rm{sgn}(\det \bar M_F [\bar \Gamma]) \det \bar M_F [\bar\Gamma]]_\star. \end{aligned}$$ We can express the action corresponding to $ \det M_F [\Gamma] $ and $\det \bar M_F [\bar\Gamma]$ as follows, $$\begin{aligned}
S_{det}&=&\int d^3 x d^2 \theta \left[-b^a \star D^\alpha\Gamma^a_\alpha
+\frac{\xi}{2}b^a \star b^a+ \tilde c^a \star M_F^{ab} \star c^b\right]
\nonumber \\&& + \int d^3 x d^2\bar \theta \left[
-\bar b^a \star \bar D^\alpha \bar \Gamma^a_\alpha +\frac{\xi}{2} \bar b^a \star
\bar b^a+ \bar{\tilde c}^a \star \bar M_F^{ab}\star c^b\right]. \end{aligned}$$ Here the ghosts and anti-ghosts are denoted by $c^a, \bar c^a$ and $\tilde c^a, \bar{\tilde c}^a$, respectively. The Nakanishi-Lautrup auxiliary fields are denoted by $b^a, \bar b^a$. So, we obtain $$\begin{aligned}
\lim_{\xi\rightarrow0}\int \mathcal{D} e^{-S_{det}}&=&[\delta (F[\Gamma]) \det M_F [\Gamma]
\delta (\bar F[\bar \Gamma]) \det \bar M_F [\bar \Gamma] ]_\star\end{aligned}$$ where the measure $\mathcal{D}$ includes an integral over the ghosts, anti-ghosts and auxiliary fields. We can also write the action corresponding to $\rm{sgn}(\det M_F [\Gamma])$ and $\rm{sgn}(\det \bar M_F [\Gamma])$ as $$\begin{aligned}
S_{sgn} &=&\int d^3 x d^2 \theta \left[
i B^a \star M_F^{ab}\star \phi^b-i\tilde d^a \star M_F^{ab}\star d^b+ \frac{1}{2}B^a\star B^b\right]
\nonumber \\
&& +\int d^3 x d^2 \bar \theta \left[
i \bar B^a \star \bar M_F^{ab}\star \bar \phi^b
-i\bar{\tilde d}^a \star\bar M_F^{ab}\star \bar d^b + \frac{1}{2}\bar B^a\star \bar B^b \right] .
\end{aligned}$$
Here $ d^a, \bar d^a, , \tilde d^a, \bar {\tilde d}^a$ are new Grassman odd superfields and $\phi^a, \bar \phi^a$, $B^a,\bar B^a$ are new auxiliary Grassman even superfields. Completing the square, the $B$ field can be integrated out, and so we can define the effective action as $$\begin{aligned}
S'_{sgn}&=& \int d^3 x d^2 \theta \left[\frac{1}{2}\phi^a\star ((M_F)^T)^{ab}\star M_F^{bc}\star
\phi^c-i\tilde d^a\star M_F^{ab}\star d^b\right]
\nonumber \\ &&
\int d^3 x d^2 \bar\theta \left[\frac{1}{2}\bar \phi^a\star ((\bar M_F)^T)^{ab}\star \bar M_F^{bc}\star
\bar \phi^c-i\bar{\tilde{d}}^a\star\bar M_F^{ab}\star \bar d^b\right].\end{aligned}$$ Thus, the partition function can be written as $$\begin{aligned}
\nonumber
\mathcal{Z}_{gf}=\int \mathcal{D} N_F[\Gamma, \bar \Gamma]^{-1} e^{-S_{DSYM}-S_{det}-S_{sgn}}. \end{aligned}$$
It may be noted that all these superfields are defined on deformed superspace. So, just like the gauge transformations, the BRST and the anti-BRST transformations of these superfields will also involve the fermionic version of the Moyal product. However, apart from the difference the expression in the superspace take similar forms. In fact, these deformed expression can be expressed as of power series involving the deformation matrix $C^{\alpha \beta}$. The first term in this series will correspond to the usual BRST and the usual anti-BRST transformations. So, even though they appear to have similar form in the superspace, the component form of the BRST and the anti-BRST transformations will look very different for the deformed and the un-deformed Yang-Mills theory. Furthermore, the BRST and the anti-BRST transformation for the usual Yang-Mills theory can be obtained by making all the fermionic field to vanish in the supersymmetric Yang-Mills theory. However, it is also possible to obtain a Yang-Mills theory without supersymmetry by breaking all the supersymmetry by imposing two non-anticommutative deformations. In this case, the BRST and the anti-BRST transformation will be different from the BRST and the anti-BRST transformations for the usual Yang-Mills theory.
The standard BRST transformations for the gauge fields $s$ and $\bar s$ can be written as $s \Gamma_\alpha^a = \nabla_\alpha^{ab} \star c^b$ and $s \bar \Gamma_\alpha^a = \bar \nabla_\alpha^{ab} \star \bar c^b$. We can also write the anti-BRST transformations for the gauge fields as $\tilde s \Gamma_\alpha^a = \nabla_\alpha^{ab} \star \tilde c^b$ and $s \bar \Gamma_\alpha^a = \bar \nabla_\alpha^{ab} \star \bar{\tilde c}^b$. The BRST transformation of the ghosts is given by $s c^a = - f_{bc}^a c^b \star c^c/2$ and $s \bar c^a = - f_{bc}^a \bar c^b \star \bar c^c/2$. The anti-BRST transformation of anti-ghosts is given by $\tilde s \tilde c^a = - f_{bc}^a \tilde c^b \star \tilde c^c/2$ and $\tilde s \bar {\tilde c}^a =
- f_{bc}^a \bar {\tilde c}^b \star \bar {\tilde c}^c/2$. The BRST transformation of anti-ghosts is given by $s c^a = b^a $ and $s \bar c^a = \bar b^a$. The anti-BRST transformation of ghosts is given by $s \tilde c^a = \tilde b^a $ and $s \bar {\tilde{c}}^a
= -\bar b^a$. Apart from this the BRST and anti-BRST transformation of all the other auxiliary fields vanishes, $s b^a = 0$, $ s \bar b^a =0$ and $\tilde s b^a = 0$, $ \tilde s \bar b^a =0$. Apart from these BRST and anti-BRST transformations, this action is also invariant under a double BRST and an anti-BRST transformations. The double BRST transformations are given by $t \phi^a = d^a$, $t \bar \phi^a = \bar d^a$ and $t \tilde d^a = B^a$, $t \bar {\tilde d}^a = \bar B^a$. The double BRST transformation of all the other fields vanishes, $t d^a =0$, $t\bar d^a = 0$ and $t B^a =0$, $ t \bar B^a =0$. The double anti-BRST transformations are given by $\tilde t \phi^a = \tilde d^a$, $\tilde t \bar {\phi}^a = \bar {\tilde d}^a$ and $\tilde
t d^a = - B^a$, $\tilde t \bar { d}^a =- \bar B^a$. The double BRST transformation of the all the other fields vanishes, $\tilde t \tilde
d^a =0$, $\tilde t\bar {\tilde d}^a = 0$ and $\tilde t B^a =0$, $ \tilde t \bar B^a =0$. So, we can define fields $\Gamma^a_i = (\Gamma^a_\alpha, \phi^a),
\bar \Gamma^a_i = (\bar \Gamma^a_\alpha, \bar\phi^a), \mathcal{C}^a_i = (c^a, d^a), \bar \mathcal{C}^a_i =
(\bar c^a, \bar d^a), \tilde \mathcal{C}^a_i = (\tilde c^a, \tilde d^a), \bar {\tilde \mathcal{C}}^a_i =
(\bar {\tilde{c}}^a, \bar {\tilde{d}}^a), \mathcal{B}^a_i = (b^a, B^a), \bar \mathcal{B}^a_i = (\bar b^a,\bar B^a)$. The BRST transformations can be written as $\mathcal{S} \Gamma^a_i = (\nabla^{ab} \star \mathcal{C}^b)_i,
\mathcal{S} \bar \Gamma^a_i = (\bar \nabla^{ab} \star \bar \mathcal{C}^b)_i,
\mathcal{S} \mathcal{C}_i^a = Y^{jk}_i f^a_{bc} \mathcal{C}^b_j \star \mathcal{C}^c_k,
\mathcal{S} \bar \mathcal{C}_i^a = Y^{jk}_i f^a_{bc} \bar \mathcal{C}^b_j \star \bar \mathcal{C}^c_k,
\mathcal{S}\tilde \mathcal{C}^a = \mathcal{B}^a_i, \mathcal{S}\bar {\tilde \mathcal{C}}^a_i = \mathcal{B}^a_i,
\mathcal{S} \mathcal{B}^a_i =0, \mathcal{S} \bar \mathcal{B}^a_i =0,
$ where $Y^1_{11} = 1, Y^i_{jk} =0, $ if $i, j, k \neq 1$. It is possible to write [@GKW] $$\begin{aligned}
S_{det} + S_{sgn} = Tr \int d^3 x d^2 \theta \mathcal{S} \mathcal{U}_\star +
Tr \int d^3 x d^2 \bar \theta \mathcal{S} \bar \mathcal{U}_\star , \end{aligned}$$ where $\mathcal{U}_\star = \rm{diag}(\tilde c^a \star F^a, \tilde d^a \star (i M_F^{ab} \star \phi^b + B^a/2 ) ) $ and $\bar \mathcal{U}_\star = \rm{diag}(\bar{\tilde c}^a \star \bar F^a, \bar {\tilde d}^a \star
(i \bar M_F^{ab} \star \bar \phi^b + \bar B^a/2) ) $. The anti-BRST transformation can be written as $\tilde \mathcal{S} \Gamma^a_i = (\nabla^{ab} \star \tilde \mathcal{C}^b)_i,
\tilde \mathcal{S} \bar \Gamma^a_i = (\bar \nabla^{ab} \star \bar {\tilde\mathcal{C}}^b)_i,
\tilde \mathcal{S} \tilde \mathcal{C}_i^a = Y^{jk}_i f^a_{bc} \tilde \mathcal{C}^b_j \star \tilde \mathcal{C}^c_k,
\tilde \mathcal{S} \bar {\tilde\mathcal{C}}_i^a = Y^{jk}_i f^a_{bc} \bar{\tilde \mathcal{C}}^b_j \star \bar {\tilde\mathcal{C}}^c_k,
\tilde \mathcal{S}\tilde \mathcal{C}^a_i = -\mathcal{B}^a_i, \tilde \mathcal{S}\bar { \mathcal{C}}^a_i = -\mathcal{B}^a_i,
\tilde \mathcal{S} \mathcal{B}^a_i =0, \tilde \mathcal{S} \bar \mathcal{B}^a_i =0. $ We can write [@GKW] $$\begin{aligned}
S_{det} + S_{sgn} = Tr \int d^3 x d^2 \theta \mathcal{S} \tilde \mathcal{S}\mathcal{W}_\star +
Tr \int d^3 x d^2 \bar \theta \mathcal{S}\tilde \mathcal{S} \bar \mathcal{W}_\star , \end{aligned}$$ where $\mathcal{W}_\star =\rm{diag}( \Gamma^{\alpha a} \star \Gamma^a_\alpha, \phi^a \star M_F^{ab} \star \phi^b,
\tilde d^a \star d^a)$ and $\bar \mathcal{W}_\star =\rm{diag}( \bar\Gamma^{\alpha a} \star \bar \Gamma^a_\alpha,
\bar\phi^a \star \bar M_F^{ab} \star \bar \phi^b,
\bar {\tilde{d}}^a \star \bar d^a)$. Thus, we are able to formulate the modulus of the determinant in Landau gauge in terms of a Lagrangian. However, this procedure also holds for non-perturbative phenomena.
Conclusion {#d}
==========
In this paper, we analysed a three dimensional supersymmetric Yang-Mills theory on deformed superspace. We deformed the superspace by imposing non-anticommutativity. The original theory has $\mathcal{N} =2 $ supersymmetry. The deformation of this theory broke half of the supersymmetry of the original theory. Thus, the final theory had only $\mathcal{N} = 1$ supersymmetry. We analysed the quantization of this theory, and addressed the problem that occurs due to the Gribov ambiguity. This was done by generalizing the standard Faddeev-Popov method. This generalized Faddeev-Popov method was used for analysing the existence of multiple solutions to the gauge fixing condition. Thus, non-perturbative effects could be address using this generalized Faddeev-Popov method. We derived an expression for calculating the expectation values of gauge invariant operators. A partition function for this deformed theory was also constructed. We analysed the BRST and the anti-BRST symmetries this partition function. It was demonstrated that apart from being invariant under the usual BRST and the usual anti-BRST transformations, this partition function also invariant under double BRST and double anti-BRST transformations. We were able to combine the usual BRST and anti-BRST transformation, with these new BRST and anti-BRST transformations. These new BRST and anti-BRST transformations, take into account the existence of multiple solutions to the gauge fixing conditions, and so the results of this paper can be used for analysing different aspects of non-perturbative phenomena.
It may be noted that a similar analysis can be done for theories with higher amount of supersymmetry. It would be interesting to apply this formalism for studying supersymmetric theories with boundaries. This is because it is possible to construct a three dimensional theory with $\mathcal{N} = 1/2$ supersymmetry by combining the boundary effects with non-anticommuativity [@za]. However, the quantization of this theory has not been studied. It would be interesting to analyse the effect of boundaries on the results obtained in this paper. It may be noted that the existence of Gribov ambiguity is related to the gauge fixing procedure for quantizing Yang-Mills theories, and this problem is usually addressed in the Gribov-Zwanziger formalism [@z1]-[@z]. It is possible to analyse effects coming from the existence of the Gribov copies in a local way using this formalism. This formalism has been used to analyse the infrared behavior of the gluon and ghost propagator. In fact, the Gribov-Zwanziger formalism has been used for studding the zero momentum value of the gluon propagator. These propagators have been used for analysing the the spectrum of gauge theories [@z2]-[@2z]. The supersymmetric generalization of the Gribov-Zwanziger formalism has also been performed [@supe]. This has been used for analysing existence of the condensate and vanishing of the vacuum energy. The renormalization of supersymmetric Yang-Mills theory with $\mathcal{N}=1$ supersymmetry was analysed using the Gribov-Zwanziger formalism [@super]. This was done by using the Landau condition. The proof of renormalizability of this theory to all orders was studied using an algebraic renormalization procedure. It was demonstrated that only three renormalization constants are needed for this theory. In fact, the non-renormalization theorem in the Landau gauge were analysed using the Gribov-Zwanziger formalism. The renormalization factors for a non-linear realization of the supersymmetry were also studied in this formalism. It will be interesting to analyse the effect of non-anticommutative deformation on these results. This can be done by analysing a deformed supersymmetric Yang-Mills theory in Gribov-Zwanziger formalism. Thus, we can study a non-anticommutative a four dimensions supersymmetric Yang-Mills theory in $\mathcal{N}=1$ superspace formalism. The non-anticommutativity will break the supersymmetry of the theory from $\mathcal{N}=1$ supersymmetry to $\mathcal{N}=1/2$ supersymmetry. It will also be interesting to perform a similar analysis for a deformed three dimensional theory in $\mathcal{N}=2$ superspace formalism. Here the non-anticommuativity will break the supersymmetry of the theory from $\mathcal{N}=2$ supersymmetry to $\mathcal{N}=1$ supersymmetry. It will be interesting to analyse the effect of this supersymmetry breaking on the existence of the condensate and vanishing of the vacuum energy.
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abstract: 'The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today’s Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten.'
author:
- |
Holger Schellwat\
\
[`[email protected]`](mailto:[email protected]), Örebro universitet, Sweden\
Universidade Eduardo Mondlane, Moçambique\
date: '12 January, 2017'
title: A Gentle Introduction to a Beautiful Theorem of Molien
---
Introduction {#sintro .unnumbered}
============
We present some memories of a visit to the ring zoo in 2004. This time we met an animal looking like a unicorn, known by the name of invariant theory. It is rare, old, and very beautiful. The purpose of this note is to give an almost self contained introduction to and clarify the proof of the amazing theorem of Molien, as presented in [@Sloane1]. An introduction into this area, and much more, is contained in [@Sturmfels]. There are many very short proofs of this theorem, for instance in [@Stanley], [@Hu2], and [@Tambour].
Informally, Moliens Theorem is a power series generating function formula for counting the dimensions of subrings of homogeneous polynomials of certain degree which are invariant under the action of a finite group acting on the variables. As an apetizer, we display this stunning formula: $$\Phi_G(\lambda) := \frac{1}{|G|} \sum_{g\in G} \frac{1}{\det({\mathrm{id}}- \lambda T_g)}$$ We can immediately see elements of linear algebra, representation theory, and enumerative combinatorics in it, all linked together. The paper [@Sloane1] nicely shows how this method can be applied in Coding theory. For Coding Theory in general, see [@jubi].
Before we can formulate the Theorem, we need to set the stage by looking at some Linear Algebra (see [@Roman]), Group Theory (see [@Hu]), and Representation Theory (see [@Sagan] and [@Tambour]).
Preliminaries {#spreliminaries}
=============
Let $V{\cong}{{\mathbf{C}}}^n$ be a finite dimensional complex inner product space with orthonormal basis ${\mathcal{B}}= ({\mathbf{e}}_1,\dots,{\mathbf{e}}_n)$ and let ${\mathbf{x}}= (x_1,\dots,x_n)$ be the orthonormal basis of the algebraic dual space $V^\ast$ satisfying $\forall 1\le i,j \le n : x_i({\mathbf{e}}_j) = \delta_{ij}$. Let $G$ be a finite group acting unitarily linear on $V$ from the left, that is, for every $g\in G$ the mapping $V \to V, {\mathbf{v}}\mapsto g{.}{\mathbf{v}}$ is a unitary bijective linear transformation. Using coordinates, this can be expressed as $[g{.}{\mathbf{v}}]_{\mathcal{B}}= [g]_{{\mathcal{B}},{\mathcal{B}}} [{\mathbf{v}}]_{\mathcal{B}}$, where $[g]_{{\mathcal{B}},{\mathcal{B}}}$ is unitary. Thus, the action is a unitary representation of $G$, or in other words, a $G$–module. Note that we are using [left composition]{} and column vectors, i.e. ${\mathbf{v}}= (v_1, \dots , v_n) {\overset{convention}{=}} [v_1 \, v_2 \, \dots \, v_n]^\top$, c. f. [@Anton].
The elements of $V^\ast$ are [linear forms]{}(linear functionals), and the elements $x_1,\dots, x_n$, looking like variables, are also linear forms, this will be important later.
Thinking of $x_1,\dots, x_n$ as variables, we may view (see [@Tambour]) $S(V^\ast)$, the [*symmetric algebra*]{} on $V^\ast$ as the algebra $R := {{\mathbf{C}}}[{\mathbf{x}}] := {{\mathbf{C}}}[x_1,\dots, x_n] $ of polynomial functions $V\to {{\mathbf{C}}}$ or polynomials in these variables (linear forms). It is naturally graded by degree as $R = \bigoplus_{d \in {{\mathbf{N}}}} R_d$, where $R_d$ is the vector space spanned by the polynomials of (total) degree $d$, in particular, $R_0 = {{\mathbf{C}}}$, and $R_1 = V^\ast$.
The action of $G$ on $V$ can be lifted to an action on $R$.
\[pindact\] Let $V$, $G$, $R$ as above. Then the mapping ${.}: G \times R \to R, (g,f) \mapsto g{.}f$ defined by $(g{.}f) ({\mathbf{v}}) := f(g{^{-1}}{.}{\mathbf{v}})$ for ${\mathbf{v}}\in V$ is a left action.
For ${\mathbf{v}}\in V$, $g,h \in G$, and $f \in R$ we check
1. $(1{.}f)({\mathbf{v}}) = f(1{^{-1}}{.}{\mathbf{v}}) = f(1{.}{\mathbf{v}}) = f({\mathbf{v}})$
2. $$\begin{gathered}
((hg){.}f)({\mathbf{v}}) = f((hg){^{-1}}{.}{\mathbf{v}}) =
f((g{^{-1}}h{^{-1}}){.}{\mathbf{v}}) =\\ f (g{^{-1}}{.}(h{^{-1}}{.}{\mathbf{v}}))
= (g{.}f)(h{^{-1}}{.}{\mathbf{v}}) = (h{.}(g{.}f))({\mathbf{v}})
\end{gathered}$$
In fact, we know more.
\[pindact2\] Let $V$, $G$, $R$ as above. For every $g \in G$, the mapping $T_g: R \to R, f \mapsto g{.}f$ is an algebra automorphism preserving the grading, i.e. $g.R_d \subset R_d$ (here we do not bother about surjectivity).
For ${\mathbf{v}}\in V$, $g\in G$, $c\in {{\mathbf{C}}}$, and $f,f' \in R$ we check
1. $$\begin{gathered}
(g{.}(f+f'))({\mathbf{v}}) =
(f+f')(g{^{-1}}{.}{\mathbf{v}}) =
f(g{^{-1}}{.}{\mathbf{v}}) + f'(g{^{-1}}{.}{\mathbf{v}}) =\\
(g{.}f)({\mathbf{v}}) + (g{.}f')({\mathbf{v}}) =
(g{.}f + g{.}f')({\mathbf{v}})
\textrm {, thus } g{.}(f+f') =g{.}f + g{.}f'
\end{gathered}$$
2. $$\begin{gathered}
(g{.}(f\cdot f'))({\mathbf{v}}) =
(f\cdot f')(g{^{-1}}{.}{\mathbf{v}}) =
f(g{^{-1}}{.}{\mathbf{v}}) \cdot f'(g{^{-1}}{.}{\mathbf{v}}) =\\
(g{.}f)({\mathbf{v}}) \cdot (g{.}f')({\mathbf{v}}) =
(g{.}f \cdot g{.}f')({\mathbf{v}})
\textrm {, thus } g{.}(f\cdot f') =g{.}f \cdot g{.}f'
\end{gathered}$$
3. $
(g{.}(cf))({\mathbf{v}}) = (cf)(g{^{-1}}{.}{\mathbf{v}}) = c (f(g{^{-1}}{.}{\mathbf{v}})) =
c ((g{.}f)({\mathbf{v}})) = (c (g{.}f))({\mathbf{v}})
$
4. By part $2.$ it is clear that the grading is preserved.
5. To show that $f \mapsto g{.}f$ is bijective it is enough to show that this mapping is injective on the finite dimensional homogeneous components $R_d$. Let us introduce a name for this mappig, say $T_g^d : R_d \to R_d, f \mapsto g{.}f$. Now $f \in \ker(T_g^d)$ implies that $g{.}f = 0 \in R_d$, i.e. $g{.}f$ is a polynomial mapping from $V$ to ${{\mathbf{C}}}$ of degree $d$ vanishing identically, $\forall {\mathbf{v}}\in V: (g{.}f)({\mathbf{v}}) = 0$. By definition of the extended action we have $\forall {\mathbf{v}}\in V: f(g{^{-1}}{.}{\mathbf{v}}) = 0$. Since $G$ acts on $V$ this implies that $\forall {\mathbf{v}}\in V : f({\mathbf{v}}) = 0$, so $f$ is the zero mapping. Since our ground field has characteristic $0$, this implies that $f$ is the zero polynomial, which we may view as an element of every $R_d$. See for instance [@Cox], proposition 5 in section 1.1.
6. Note that every $T_g^d$ is also surjective, since all group elements have their inverse in $G$.
Both propositions together give us a homomorphism from $G$ into ${\ensuremath{\mathrm{Aut}}}(R)$. They also clarify the rôle of the *induced* matrices, which are classical in this area, as mentionend in [@Sloane1]. Since the monomials $x_1,\dots,x_n$ of degree one form a basis for $R_1$, it follows from the proposition that their products ${\mathbf{x}}_2 := (x_1^2,x_1 x_2,x_1 x_3,\dots,x_1 x_n, x_2^2,x_2 x_3,\dots)$ form a basis for $R_2$, and, in general, the monomials of degree $d$ in the linear forms (!) $x_1,\dots,x_n$ form a basis ${\mathbf{x}}_d$ of $R_d$. Clearly, they certainly span $R_d$, and by the last observation in the last proof they are linearly independent.
\[dinduced\] In the context from above, that is $g \in G$, $f \in R^d$, and ${\mathbf{v}}\in V$, we define $$T_g^d : R_d \to R_d, f \mapsto g{.}f : R^d \to {{\mathbf{C}}}, {\mathbf{v}}\mapsto f(g{^{-1}}{.}{\mathbf{v}}) = f(T_{g{^{-1}}} ({\mathbf{v}}))
.$$
\[rinduced\] In particular, we have $(T_g^1 (f))({\mathbf{v}}) = f(T_{g{^{-1}}}({\mathbf{v}}) ),$ see proposition \[pprop0\] below.
Keep in mind that a function $f \in R_d$ maps to $T_g^d (f) = g{.}f$. Setting $A_g := [T_g^1]_{{\mathbf{x}},{\mathbf{x}}}$, then $A_g^{[d]} := [T_g^d]_{{\mathbf{x}}_d,{\mathbf{x}}_d}$ is the $d$–th induced matrix in [@Sloane1], because $T_g^1(f\cdot f') = T_g^1(f)\cdot T_g^1(f')$. Also, if $f,f'$ are eigenvectors of $T_g^1$ corresponding to the eigenvalues $\lambda,\lambda'$, then $f\cdot f'$ is an eigenvector of $T_g^2$ with eigenvalue $\lambda \cdot \lambda'$, because $T_g(f\cdot f') = T_g(f) \cdot T_g(f') = (\lambda f) \cdot (\lambda' f')
= (\lambda \lambda')(f \cdot f')$. All this generalizes to $d>2$, we will get back to that later.
We end this section by verifying two little facts needed in the next section.
\[ppropd\] The [*first induced operator*]{} of the inverse of a group element $g\in G$ is given by $T_{g{^{-1}}}^1 = (T_g^1){^{-1}}$.
Since $\dim(V^\ast) < \infty$, it is sufficient to prove that $T_{g{^{-1}}}^1 \circ T_{g}^1 = {\mathrm{id}}_{V^\ast}$. Keep in mind that $(T_g^1 (f))({\mathbf{v}}) = f (T_{g{^{-1}}} ({\mathbf{v}}))$. For arbitrary $f \in V^\ast$ we see that $$\begin{aligned}
(T_{g{^{-1}}}^1 \circ T_{g}^1 )(f) = T_{g{^{-1}}}^1 ( T_{g}^1 (f)) = T_{g{^{-1}}}^1 ( g{.}f)
= g{^{-1}}{.}( g{.}f) = (g{^{-1}}g){.}f = f.\end{aligned}$$
We will be mixing group action notation and composition freely, depending on the context. The following observation is a translation device.
\[pprop0\] For $g \in G$ nd $f \in V^\ast$ the following holds: $$T^1(f) = g{.}f = f \circ T_{g{^{-1}}}.$$
For ${\mathbf{v}}\in V$ we see $(T^1(f))({\mathbf{v}}) = (g{.}f)({\mathbf{v}}) {\overset{def}{=}} f(g{^{-1}}{.}{\mathbf{v}}) = f( T_{g{^{-1}}}({\mathbf{v}}) ).$
The Magic Square {#ssquare}
================
Remember that we require a unitary representation of $G$, that is the operators $T_g : V \to V$ need to be unitary, i.e. $\forall g \in G : (T_g){^{-1}}= (T_g)^\ast$. The first goal of this sections is to show that this implies that the induced operators $T_g^d : R_d \to R_d, f \mapsto g{.}f$ are also unitary. We saw that $T_g^1 = V^\ast$, the algebraic dual of $V$. In order to understand the operator duals of $V$ and $V^\ast$ we need to look on their inner products first. We may assume that the operators $T_g$ are unitary with respect to the standard inner product ${\ensuremath\left\langle {\mathbf{u}}\,, {\mathbf{v}}\right \rangle} = [{\mathbf{u}}]_{{\mathcal{B}}, {\mathcal{B}}} \bullet \overline{[{\mathbf{v}}]_{{\mathcal{B}}, {\mathcal{B}}}}$, where $\bullet$ denotes the dot product.
Before we can speak of unitarity of the induced operators $T_g^d$ we have to make clear which inner product applies on $R^1 = V^\ast$. Quite naively, for $f,g \in V^\ast$ we are tempted to define ${\ensuremath\left\langle f \,, g \right \rangle} = [f]_{{\mathbf{x}}, {\mathbf{x}}} \bullet \overline{[g]_{{\mathbf{x}}, {\mathbf{x}}}}$.
We will motivate this in a while, but first we take a look at the diagram in [@Roman], chapter10, with our objects:
$$\begin{CD}
\quad @<T_g^\times<< \quad\\
R^1 = V^\ast @>T_g^1>>V^\ast = R^1 \\
@VVPV@VVPV\\
V@>T_g>>V\\
\quad @<T_g^\ast<< \quad\\
\end{CD}$$
Here $P$ (Rho ) denotes the [Riesz map]{}, see [@Roman], Theorem 9.18, where it is called $R$, but $R$ denotes already our big ring. We started by looking at the operator $T_g$, which is unitary, so its inverse is the Hilbert space adjoint $T_g^\ast$. Omiting the names of the bases we have $[T_g^\ast] = [T_g]^\ast $. We also see the operator adjoint $T_g^\times$ with matrix $[T_g^\times] = [T_g]^\top$, the transpose. However, the arrow for $T_g^1$ is not in the original diagram, but soon we will see it there, too.
Fortunately, the Riesz map $P$ turns a linear form into a vector and its inverse $\tau : V \to V^\ast$ maps a vector to a linear form, both are conjugate isomorphisms. This is mostly all we need in order to show that $T_g^1$ is unitary. In the following three propositions we use that $V$ has the orthonormal basis ${\mathcal{B}}$ and that $V^\ast$ has the orthonormal basis ${\mathbf{x}}$.
\[ppropa\] For every $f \in V^\ast$ the coordinates of its Riesz vector are given by $$[P(f)]_{\mathbf{e}}= (\overline{f({\mathbf{e}}_1)}, \dots , \overline{f({\mathbf{e}}_n)}).$$
Writing $\tau$ for the inverse of $P$, we need to show that $$P(f) = {\ensuremath \sum_{i = 1}^n \, \overline{f({\mathbf{e}}_i)} }{\mathbf{e}}_i$$ which is equivalent to $$f = \tau \left ( {\ensuremath \sum_{i = 1}^n \, \overline{f({\mathbf{e}}_i)} }{\mathbf{e}}_i \right ).$$ It is sufficient to show the latter for values of $f$ on the basis vectors ${\mathbf{e}}_j$, $1 \le j \le n$. We obtain $$\begin{aligned}
\left (\tau \left ( {\ensuremath \sum_{i = 1}^n \, \overline{f({\mathbf{e}}_i)} }{\mathbf{e}}_i \right )\right ) ({\mathbf{e}}_j) &=
{\ensuremath\left\langle {\mathbf{e}}_j \,, \left ( {\ensuremath \sum_{i = 1}^n \, \overline{f({\mathbf{e}}_i)} }{\mathbf{e}}_i \right ) \right \rangle}
=
{\ensuremath \sum_{i = 1}^n \, {\ensuremath\left\langle {\mathbf{e}}_j \,, \left ( {\overline{f({\mathbf{e}}_i)}}{\mathbf{e}}_i \right ) \right \rangle} } \\
&= \overline{\overline{f({\mathbf{e}}_i)}} {\ensuremath \sum_{i = 1}^n \, {\ensuremath\left\langle {\mathbf{e}}_j \,, {\mathbf{e}}_i \right \rangle} }
= f({\mathbf{e}}_i) \cdot 1.\end{aligned}$$
In particular, this implies that $P(x_i) = {\mathbf{e}}_i$.
\[ppropb\] Our makeshift inner product on $V^\ast$ satisfies $${\ensuremath\left\langle f \,, g \right \rangle} = {\ensuremath\left\langle P(f) \,, P(g) \right \rangle}
,$$ where $f,g \in V^\ast$.
By our vague definition we have ${\ensuremath\left\langle f \,, g \right \rangle} = [f]_{{\mathbf{x}}, {\mathbf{x}}} \bullet \overline{[g]_{{\mathbf{x}}, {\mathbf{x}}}}$. It is enough to show that ${\ensuremath\left\langle x_i \,, x_j \right \rangle} = {\ensuremath\left\langle P(x_i) \,, P(x_j) \right \rangle}$. From the comment after the proof of Proposition \[ppropa\] we obtain $${\ensuremath\left\langle P(x_i) \,, P(x_j) \right \rangle} = {\ensuremath\left\langle {\mathbf{e}}_i \,, {\mathbf{e}}_j \right \rangle} = \delta_{ij} = {\mathbf{e}}_i \bullet {\mathbf{e}}_j
= [x_i]_{{\mathbf{x}}, {\mathbf{x}}} \bullet \overline{[x_j]_{{\mathbf{x}}, {\mathbf{x}}}}
.$$
Hence, our guess for the inner product on $V^\ast$ was correct. We will now relate the Riesz vector of $f \in V^\ast$ to the Riesz vector of $f \circ T_g{^{-1}}$. Recall that the Riesz vector of $f \in V^\ast$ is the unique vector ${\mathbf{w}}= P(f)$ such that $f({\mathbf{v}}) = {\ensuremath\left\langle {\mathbf{v}}\,, {\mathbf{w}}\right \rangle}$ for all ${\mathbf{v}}\in V$. If $f \ne 0$ it can be found by scaling any nonzero vector in the cokernel of $f$, which is one–dimensional, see [@Roman], in particular Theorem 9.18.
\[pprope\] Let $T_g : V \to V$ be unitary, $f \in V^\ast$, ${\mathbf{w}}= P(f)$ the vector of $f \in V^\ast$. Then $T_g({\mathbf{w}})$ is the Riesz vector of $f \circ T_g{^{-1}}$, i.e. the Riesz vector of $T^1_g(f)$.
We may assume that $f \ne 0$. Using the notation ${\left\langle {\mathbf{w}}\right\rangle}$ for the one–dimensional subspace spanned by ${\mathbf{w}}$, we start with a little diagram: $${\left\langle {\mathbf{w}}\right\rangle} \odot \ker(f) \overset{T_g}{\longrightarrow} {\left\langle T_g({\mathbf{w}}) \right\rangle} \odot \ker(f \circ T_g{^{-1}}),$$ wheere $\odot$ denotes the orthogonal direct sum.
We need to show that $f \circ T_g{^{-1}}= {\ensuremath\left\langle \cdot \,, T_g({\mathbf{w}}) \right \rangle}$, i.e. that $(f \circ T_g{^{-1}})({\mathbf{v}}) = {\ensuremath\left\langle {\mathbf{v}}\,, T_g({\mathbf{w}}) \right \rangle}$ for all ${\mathbf{v}}\in V$. Since ${\mathbf{w}}= P(f)$ the vector of $f$, we have $f({\mathbf{v}}) = {\ensuremath\left\langle {\mathbf{v}}\,, {\mathbf{w}}\right \rangle}$ for all ${\mathbf{v}}\in V$. We obtain $$\begin{aligned}
(f \circ T_g{^{-1}})({\mathbf{v}}) &= {\ensuremath\left\langle T_g{^{-1}}({\mathbf{v}}) \,, {\mathbf{w}}\right \rangle} {\overset{T_g \,\,\mathrm{unitary} }{=}}
{\ensuremath\left\langle {\mathbf{v}}\,, T_g({\mathbf{w}}) \right \rangle}. \end{aligned}$$ From remark \[rinduced\] we conclude that $f \circ T_g{^{-1}}= T^1_g(f)$.
Observe that proposition \[pprope\] implies the commutativity of the following two diagrams. $$\begin{CD}
V^\ast @>T_g^1>>V^\ast\\
@VVPV@VVPV\\
V@>T_g>>V\\
\end{CD}
\qquad \mathrm{and } \qquad
\begin{CD}
V^\ast @>(T_g^1){^{-1}}>>V^\ast\\
@VVPV@VVPV\\
V@>(T_g){^{-1}}>>V\\
\end{CD}$$ Indeed, \[pprope\] implies $$\begin{aligned}
P \circ T_g^1 &= T_g \circ P \\
P \circ (T_g^1){^{-1}}&= (T_g){^{-1}}\circ P\end{aligned}$$
\[pproplink\] The first induced operator $T_g^1$ is unitary.
We may use that $T_g$ is unitary, that is, $${\ensuremath\left\langle T_g({\mathbf{v}}) \,, {\mathbf{w}}\right \rangle} = {\ensuremath\left\langle {\mathbf{v}}\,, (T_g){^{-1}}({\mathbf{w}}) \right \rangle}
= {\ensuremath\left\langle {\mathbf{v}}\,, (T_{g{^{-1}}})({\mathbf{w}}) \right \rangle} \qquad (\ast)
.$$ Let $f,h \in V^\ast$ arbitrary, ${\mathbf{w}}:= P(f)$, and ${\mathbf{u}}:= P(h)$. We need to check that ${\ensuremath\left\langle (T_g^1)(f) \,, h \right \rangle} = {\ensuremath\left\langle f \,, (T_g^1){^{-1}}(h) \right \rangle}$. We see that $$\begin{aligned}
{\ensuremath\left\langle (T_g^1)(f) \,, h \right \rangle} &{\overset{\mathrm{proposition }\ref{ppropb}}{=}}
{\ensuremath\left\langle (P\circ T_g^1)(f) \,, P(h) \right \rangle} {\overset{(1)}{=}} {\ensuremath\left\langle (T_g\circ P )(f) \,, P(h) \right \rangle} \\
&= {\ensuremath\left\langle (T_g( P ))(f) \,, P(h) \right \rangle} = {\ensuremath\left\langle T_g({\mathbf{w}}) \,, {\mathbf{u}}\right \rangle} {\overset{\ast}{=}} {\ensuremath\left\langle {\mathbf{w}}\,, T_g{^{-1}}({\mathbf{u}}) \right \rangle} \\
&= {\ensuremath\left\langle P(f) \,, T_g{^{-1}}(P(h)) \right \rangle} = {\ensuremath\left\langle P(f) \,, (T_g{^{-1}}\circ P ) (h) \right \rangle}\\
&{\overset{(2)}{=}} {\ensuremath\left\langle P(f) \,, ( P \circ (T_g^1){^{-1}}) (h) \right \rangle} = {\ensuremath\left\langle P(f) \,, P ((T_g^1){^{-1}}(h)) \right \rangle}\\
&= {\ensuremath\left\langle f \,, (T_g^1){^{-1}}(h) \right \rangle}\end{aligned}$$
After having looked at eigenvalues we will see that this generalizes to higher degree, that $T_g^d$ is diagonalizable for all $d\in {{\mathbf{Z}}}^+$. But first let us look at the matrix version of proposition \[pproplink\].
\[ppropf\] $$[T^1_g]_{{\mathbf{x}},{\mathbf{x}}} = \overline{[T_g]_{{\mathbf{e}},{\mathbf{e}}}}$$
Let $A := [T_g]_{{\mathcal{B}}, {\mathcal{B}}} = [A_1| \cdots |A_i| \cdots | A_n] = [a_{i,j}]$ and $B := [T_g^1]_{{\mathbf{x}},{\mathbf{x}}} = [B_1| \cdots |B_i| \cdots | B_n] = [b_{i,j}]$. We will use the commutativity of the diagram, i.e. $P{^{-1}}\circ T_g \circ P = T_g$, which we will mark as $\square$. No, the proof is not finished here. We get $T_g({\mathbf{e}}_i) = A_i = {\ensuremath \sum_{k = 1}^n \, a_{k,i} } {\mathbf{e}}_k$ and $$\begin{aligned}
T_g^1 (x_i) &{\overset{\square}{=}} (P{^{-1}}\circ T_g \circ P)(x_i) = P{^{-1}}( T_g ( P (x_i)) \\
&{\overset{\ref{ppropa}}{=}} P{^{-1}}( T_g ( {\mathbf{e}}_i)) = P{^{-1}}\left ( {\ensuremath \sum_{k = 1}^n \, a_{k,i} } {\mathbf{e}}_k \right)
{\overset{\textrm{konj.}}{=}} {\ensuremath \sum_{k = 1}^n \, \overline{a_{k,i}} P{^{-1}}\left ( {\mathbf{e}}_k \right) }\\
&{\overset{\ref{ppropa}}{=}} {\ensuremath \sum_{k = 1}^n \, \overline{a_{k,i}} x_k } \end{aligned}$$ On the other hand, $[T^1_g(x_i)]_{{\mathbf{x}}} = [T^1_g]_{{\mathbf{x}},{\mathbf{x}}} {\mathbf{e}}_i = B_i$ implies $T^1_g(x_i) = {\ensuremath \sum_{k = 1}^n \, b_{k,i} }{\mathbf{e}}_k$. Together we obtain $b_{k,i} = \overline{a_{k,i}}$, and the proposition follows.
Averaging over the Group {#sreynolda}
========================
Now we apply averaging to obtain self-adjoint operators.
\[dreynolds\] We define the following operators:
1. $\displaystyle \hat{T} : V\to V, {\mathbf{v}}\mapsto \hat{T}({\mathbf{v}})
:= {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_g({\mathbf{v}}) }$
2. $\displaystyle \hat{T^1} : V^\ast\to V^\ast, f \mapsto \hat{T^1}(f)
:= {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T^1_g(f) }$
These are sometimes called the [*Reynolds*]{} operator of $G$.
\[preynolds\] The operators $ \hat{T}$ and $\hat{T^1}$ are self-adjoint (Hermitian).
The idea of the averaging trick is that if $g\in G$ runs through all group element and $g' \in G$ is fixed, then the products $g'g$ run also through all group elements. We will make use of the facts that every $T_g$ and every $T^1_g$ is unitary.
1. We need to show that ${\ensuremath\left\langle \hat{T}({\mathbf{v}}) \,, {\mathbf{w}}\right \rangle} = {\ensuremath\left\langle {\mathbf{v}}\,, \hat{T}({\mathbf{w}}) \right \rangle}$ for arbitrary ${\mathbf{v}},{\mathbf{w}}\in V$. We obtain $$\begin{aligned}
{\ensuremath\left\langle \hat{T}({\mathbf{v}}) \,, {\mathbf{w}}\right \rangle} &={\ensuremath\left\langle {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_g({\mathbf{v}}) } \,, {\mathbf{w}}\right \rangle} = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, {\ensuremath\left\langle T_g({\mathbf{v}}) \,, {\mathbf{w}}\right \rangle} }\\
&{\overset{unit.}{=}} {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, {\ensuremath\left\langle {\mathbf{v}}\,, (T_g){^{-1}}({\mathbf{w}}) \right \rangle} } = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, {\ensuremath\left\langle {\mathbf{v}}\,, (T_{g{^{-1}}})({\mathbf{w}}) \right \rangle} } \\
&= {\ensuremath \frac{1}{|G|}\sum_{g' \in G} \, {\ensuremath\left\langle {\mathbf{v}}\,, (T_{g'})({\mathbf{w}}) \right \rangle} } = {\ensuremath\left\langle {\mathbf{v}}\,, \hat{T}({\mathbf{w}}) \right \rangle}
\end{aligned}$$
2. The same proof, *mutitis mutandis*, replacing $\hat{T} \leftrightarrow \hat{T^1}$, $T_g \leftrightarrow T_g^1$, ${\mathbf{v}}\leftrightarrow f$, and ${\mathbf{w}}\leftrightarrow h$ shows that ${\ensuremath\left\langle \hat{T^1}(f) \,, h \right \rangle} = {\ensuremath\left\langle f \,, \hat{T^1}(h) \right \rangle}.$
Consequently, $ \hat{T}$ and $\hat{T^1}$ are unitarily diagonalizable with real spectrum.
\[pproph\] The operators $ \hat{T}$ and $\hat{T^1}$ are [idempotent]{}, i.e.
1. $ \hat{T} \circ \hat{T} = \hat{T} $
2. $ \hat{T^1} \circ \hat{T^1} = \hat{T^1} $ .
In particular, the eigenvalues of both operators are either $0$ or $1$.
Again, we show only one part, the other part is analog. To begin with, let $s \in G$ be fixed. Then $$\begin{aligned}
T_s \circ \hat{T} &= T_s \circ {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_g } = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_s \circ T_g } \\
&= {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_{sg} } = {\ensuremath \frac{1}{|G|}\sum_{g' \in G} \, T_{g'} } = \hat{T}.
\end{aligned}$$ From this it follows that $$\begin{aligned}
\hat{T} \circ \hat{T} &= \left ({\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_g } \right) \circ \hat{T}
= {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T_g\circ \hat{T} } {\overset{above}{=}}
= {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, \hat{T} } \\
&= \frac{1}{|G|} \cdot |G| \cdot\hat{T} = \hat{T}.
\end{aligned}$$ From $ \hat{T} \circ \hat{T} = \hat{T} $ we conclude that $ \hat{T} \circ (\hat{T} - {\mathrm{id}}) = 0 $. Thus the minimal polynomial of $T$ divides the polynomial $\lambda (\lambda - 1)$, so all eigenvalues are contained in ${\left\{0,1\right\}}$.
We will now look at the eigenvalues of $T_g$ and $T^1_g$ and their interrelation. Since both operators are unitary, their eigenvalues have absolute value $1$.
\[pvictor\]
1. If ${\mathbf{v}}\in V$ is an eigenvector of $T_g$ for the eigenvalue $\lambda$, then ${\mathbf{v}}$ is an eigenvector of $T_{g{^{-1}}}$ for the eigenvalue $\overline{\lambda} = \frac{1}{\lambda}$.
2. If $f\in V^\ast$ is an eigenvector of $T^1_g$ for the eigenvalue $\lambda$, then $f$ is an eigenvector of $T^1_{g{^{-1}}}$ for the eigenvalue $\frac{1}{\lambda}$.
3. If $f\in V^\ast$ is an eigenvector of $T^1_g$ for the eigenvalue $\lambda$, then $P(f) \in V$ is an eigenvector of $T_g$ for the eigenvalue $\overline{\lambda} = \frac{1}{\lambda}$.
4. If ${\mathbf{v}}\in V$ is an eigenvector of $T_g$ for the eigenvalue $\lambda$, then $P{^{-1}}({\mathbf{v}}) \in V^\ast$ is an eigenvector of $T^1_g$ for the eigenvalue $\overline{\lambda}=\frac{1}{\lambda}$.
We will make use of the commutativity of Proposition \[pprope\]. Observe that $g{.}{\mathbf{v}}= T_g({\mathbf{v}})$ and $g{.}f = f \circ T_g$.
1. $$\begin{aligned}
T_g({\mathbf{v}}) &= g{.}{\mathbf{v}}= \lambda {\mathbf{v}}\implies g{^{-1}}{.}g{.}{\mathbf{v}}= g{^{-1}}{.}\lambda {\mathbf{v}}\implies g{^{-1}}{.}g{.}{\mathbf{v}}= \lambda g{^{-1}}{.}{\mathbf{v}}\\
& \implies {\mathbf{v}}= \lambda g{^{-1}}{.}{\mathbf{v}}\implies T_{g{^{-1}}}({\mathbf{v}}) = g{^{-1}}{.}{\mathbf{v}}= \frac{1}{\lambda} {\mathbf{v}}\end{aligned}$$
2. $$\begin{aligned}
T^1_g(f) &= g{.}f = \lambda f \implies g{^{-1}}{.}g{.}f = g{^{-1}}{.}\lambda f
\implies g{^{-1}}{.}g{.}f = \lambda g{^{-1}}{.}f \\
& \implies f = \lambda g{^{-1}}{.}f \implies T^1_{g{^{-1}}}(f) = g{^{-1}}{.}f = \frac{1}{\lambda} f
\end{aligned}$$
3. $$\begin{aligned}
T^1_g(f) = \lambda f &{\overset{P\circ}{\Longrightarrow}} P(T^1_g(f)) = P(\lambda f) {\overset{(1)}{\Longrightarrow}} T_g(P(f)) = P(\lambda f) \\
&\implies T_g(P(f)) = \overline{\lambda} P( f) =\frac{1}{\lambda} P( f)
\end{aligned}$$
4. $$\begin{aligned}
T_g({\mathbf{v}}) = \lambda {\mathbf{v}}&{\overset{P{^{-1}}\circ}{\Longrightarrow}} P{^{-1}}(T_g({\mathbf{v}})) = P{^{-1}}(\lambda {\mathbf{v}})
{\overset{\square}{\Longrightarrow}} (T_g^1 \circ P{^{-1}})({\mathbf{v}}) = \overline{\lambda} P{^{-1}}({\mathbf{v}}) \\
&\implies T_g^1 ( P{^{-1}}({\mathbf{v}})) = \frac{1}{\lambda} P{^{-1}}({\mathbf{v}})
\end{aligned}$$
This implies that if we consider the union of the spectra over all $g\in G$, then we obtain the same (multi)set, no matter if we take $T_g$ or $T^1_g$.
Eigenvectors and eigenvalues {#svictor}
============================
Now we continue from where we left at the end of section \[spreliminaries\], fixing one group element $g \in G$ and compare $T_g^1$ with $T_g^d$ for $d > 1$. By a method called [*stars and bars*]{} it is easy to see that $$\tilde{d} := \dim_{{\mathbf{C}}}(R_d)
= \frac{(n+d+1)!}{(n-1)!d!} .$$
Remember that every $T_g^1$ is unitarily diagonalizable with eigenvalues of absolute value $1$. If ${\ensuremath \mathrm{spec}}(T_g^1) = (\omega_1,\dots , \omega_n) \in U(1)^n $, then $V^\ast$ has an orthonormal basis ${\mathbf{y}}_g^1 := (y_{1}, \dots ,y_{n} )$, such that $T_g^1 (y_{i}) = \omega_i \cdot y_{i} $ for all $1 \le i \le n$, and $[T_g^1]_{{\mathbf{y}}_g^1,{\mathbf{y}}_g^1} = {\ensuremath \mathrm{diag}}(\omega_1,\dots , \omega_n)$. Moreover, $$[T_g^1]_{{\mathbf{y}}_g^1,{\mathbf{y}}_g^1} = [{\mathrm{id}}]_{{\mathbf{y}}_g^1, {\mathbf{x}}} \cdot [T_g^1]_{{\mathbf{x}},{\mathbf{x}}} \cdot [{\mathrm{id}}]_{ {\mathbf{x}}, {\mathbf{y}}_g^1}
= {\ensuremath \mathrm{diag}}(\omega_1,\dots , \omega_n) ,$$ where $[{\mathrm{id}}]_{{\mathbf{y}}_g^1, {\mathbf{x}}} = [{\mathrm{id}}]_{ {\mathbf{x}}, {\mathbf{y}}_g^1}^\ast$ is unitary.
For $d>1$ put $${\mathbf{x}}^d := (x_1^d, x_2^d, \dots , x_n^d, x_1^{d-1}x_2 ,x_1^{d-1}x_3 , \dots ,x_1^{d-1}x_n, \dots )
=: (\tilde{x_1}, \dots, \tilde{x}_{\tilde{d}})
,$$ all monomials in the $x_i$ of total degree $d$, numbered from $1$ to $\tilde{d}$.
These are certainly linear independent, since we have no relations amongst the variables, and span $R_d$, since every monomial of total degree $d$ can be written as a linear combination of these. So the form a basis for $R_d$. We will not require that this can be made into an orthonormal basis, we do not even consider any inner product on $R_d$ for $d>1$.
We rather want to establish that $${\mathbf{y}}^d := (y_1^d, y_2^d, \dots , y_n^d, y_1^{d-1}y_2 ,y_1^{d-1}y_3 , \dots ,y_1^{d-1}y_n, \dots )
=: (\tilde{y_1}, \dots, \tilde{y}_{\tilde{d}})$$ is a basis of eigenvectors of $T_g^d$ diagonalizing $T_g^d$, using the same numbering.
Arranging the eigenvalues of $T_g^1$ in the sam way we put $$\mathbf{\omega}^d := (\omega_1^d, \omega_2^d, \dots , \omega_n^d, \omega_1^{d-1}\omega_2 ,\omega_1^{d-1}\omega_3 ,
\dots ,\omega_1^{d-1}\omega_n, \dots )
=: (\tilde{\omega_1}, \dots, \tilde{\omega}_{\tilde{d}}).$$
Now we establish that the $\tilde{y_i}$, $1\le i \le \tilde{d}$ are the eigenvectors for the eigenvalues $\tilde{\omega_1}$ of $T_g^d$.
\[pinducedeigen\] In the context above, $$T_g^d (\tilde{y_i}) = \tilde{\omega_i} \cdot \tilde{y_i}$$ for all $1\le i \le \tilde{d}$.
The key is proposition \[pindact2\], as in the preliminary observations at the end of section \[spreliminaries\]. Let $$\tilde{y_i} = \prod_{j=1}^{n} y_j^{\epsilon_j}$$ and $$\tilde{\omega_i} = \prod_{j=1}^{n} \omega_j^{\epsilon_j}
,$$ where $\epsilon_j \in {{\mathbf{N}}}$ and the sum of these exponents is $d$. Then $$\begin{aligned}
T_g^d (\tilde{y_i}) &= T_g^d \left ( \prod_{j=1}^{n} y_j^{\epsilon_j} \right )
= \prod_{j=1}^{n} T_g^1 \left ( y_j^{\epsilon_j} \right )
= \prod_{j=1}^{n} \omega_j^{\epsilon_j} y_j^{\epsilon_j}
= \tilde{\omega_i} \cdot \tilde{y_i}\end{aligned}$$
As a consequence, $R_d$ has a basis of eigenvectors of $T_g^d$ and $T_g^d$ is similar to the [diagonal matrix]{} ${\ensuremath \mathrm{diag}}(\tilde{\omega_1}, \dots, \tilde{\omega}_{\tilde{d}})$.
Moliens Theorem {#sstart}
===============
We will now make some final preparations and then present the proof of Moliens Theorem.
For $f \in R$ and $g \in G$ we say that $f$ is an [*invariant*]{} of $g$ if $g{.}f = f$ and that $f$ is a (simple) invariant of $G$ if $\forall g \in G : g{.}f = f$. The method of averaging from section \[sreynolda\] can also be applied to create invariants:
\[ppropg\] For $f\in V^\ast$ put $\hat{f} := \hat{T^1} (f)$. Then $\hat{f}$ is an invariant of $G$.
Let $g \in G$ be arbitrary. We will show that $g.\hat{f} = \hat{f}$. Clearly, from proposition \[pprop0\] we get that $$\begin{aligned}
g.\hat{f} &= \hat{f} \circ T_{g{^{-1}}} = (\hat{T^1} (f)) \circ T_{g{^{-1}}} \\
&= \left( {\ensuremath \frac{1}{|G|}\sum_{s \in G} \, T_s^1(f) }\right )\circ T_{g{^{-1}}} = \left( {\ensuremath \frac{1}{|G|}\sum_{s \in G} \, f \circ T_{s{^{-1}}} }\right )\circ T_{g{^{-1}}}\\
&= {\ensuremath \frac{1}{|G|}\sum_{s \in G} \, f \circ T_{s{^{-1}}} \circ T_{g{^{-1}}} } = {\ensuremath \frac{1}{|G|}\sum_{t \in G} \, f \circ T_{t{^{-1}}} } = \hat{f}.\end{aligned}$$
Now, we call $$R^G := {\left\{\,f\in R{\,\, : \,\,}\forall g \in G : g{.}f = f\,\right\}}$$ the [*algebra of invariants*]{} of $G$.
\[pinvalg\] $R^G$ is a subalgebra of $R$.
Since the mapping $f \mapsto g.f$ is linear for every $g\in G$, $R^G$ is the intersection of subspaces, and hence a subspace. Let us check the subring conditions in more detail. For arbritrary $g \in G$, $f,h \in R^G$, and ${\mathbf{v}}\in V$ we have $g{.}f = f$, $g{.}h = h$
1. For the zero $0 \in R$ we obtain $(g{.}0)({\mathbf{v}}) = 0(g{^{-1}}{.}{\mathbf{v}}) = 0({\mathbf{v}})$, so $0 \in R^G$.
2. We see $$\begin{aligned}
g{.}(f-h)({\mathbf{v}}) &= (f-h)(g{^{-1}}{.}{\mathbf{v}}) = f(g{^{-1}}{.}{\mathbf{v}}) - h(g{^{-1}}{.}{\mathbf{v}}) \\
&= (g {.}f)({\mathbf{v}}) - (g {.}h)({\mathbf{v}}) = f({\mathbf{v}}) - h({\mathbf{v}}) = (f-h)({\mathbf{v}})
\end{aligned}$$
3. Likewise, $$\begin{aligned}
g{.}(f\cdot h)({\mathbf{v}}) &= (f\cdot h)(g{^{-1}}{.}{\mathbf{v}}) = f(g{^{-1}}{.}{\mathbf{v}}) \cdot h(g{^{-1}}{.}{\mathbf{v}}) \\
&= (g {.}f)({\mathbf{v}}) \cdot (g {.}h)({\mathbf{v}}) = f({\mathbf{v}}) \cdot h({\mathbf{v}}) = (f\cdot h)({\mathbf{v}}).
\end{aligned}$$
Our subalgebra $R^G$ is graded in the same way as $R$.
\[pinvalggraded\] The algebra of invariants of $G$ is naturally graded as $$R^G = \bigoplus_{d \in {{\mathbf{N}}}} R^G_d,$$ where $R^G_d = {\left\{\,f\in R_d{\,\, : \,\,}\forall g \in G : g{.}f = f\,\right\}}$, called the $d$–th [*homogeneous component*]{} of $R^G$.
This follows directly from proposition \[pindact\] and proposition \[pindact2\].
\[dmolien\] Viewing $R^G_d$ as a vector space, we define $$a_d := \dim_{{\mathbf{C}}}R^G_d,$$ the number of linearly independent homogeneous invariants of degree $d\in {{\mathbf{N}}}$, and $$\Phi_G(\lambda) := \sum_{d\in{{\mathbf{N}}}} a_d \lambda^d,$$ the [*Molien series*]{} of $G$.
Thus, the Molien series of $G$ is an ordinary power series generating function whose coefficients are the numbers of linearly independent homogeneous invariants of degree $d$. The following beautiful formula gives these numbers, its proof is the aim of this paper.
\[tmolien\] $$\Phi_G(\lambda) := \frac{1}{|G|} \sum_{g\in G} \frac{1}{\det({\mathrm{id}}- \lambda T_g)}$$
Following [@Sloane1] we first look the number $a_1$ of linearly independent homogeneous invariants of degree $d$.
\[t13\] $$a_1 = {\ensuremath{\mathrm{Tr}}}(\hat{T}) = {\ensuremath{\mathrm{Tr}}}(\hat{T^1})$$
First, we note that the equation ${\ensuremath{\mathrm{Tr}}}(\hat{T}) = {\ensuremath{\mathrm{Tr}}}(\hat{T^1}) $ follows from the remark at the end of section \[sreynolda\], since the sum for the trace runs over all group elements. Remember that the trace is independent of the choice of basis. From proposition \[pproph\] we know that both operators are idempotent hermitian and $V^\ast$ has a an orthornormal basis ${\mathbf{f}}= ({\mathbf{f}}_a,\dots, {\mathbf{f}}_n)$ of eigenvectors of $\hat{T^1}$, corresponding to the eigenvalues $\lambda_1, \dots , \lambda_n \in {\left\{0,1\right\}}$, so $$[\hat{T^1}]_{{\mathbf{f}},{\mathbf{f}}} = {\ensuremath \mathrm{diag}}(\lambda_1, \dots , \lambda_n).$$
Let us say that this matrix has $r$ entries $1$ and the remaining $n-d$ entries $0$. By rearranging the eigenvalues and eigenvectors we may assume that the first $r$ entries are $1$ and the remaining $n-d$ are $0$, i.e. $$\left ([\hat{T^1}]_{{\mathbf{f}},{\mathbf{f}}}\right )_{i,i} =
\begin{cases}
1 & : 1 \le i \le r\\
0 & : r+1 \le i \le n.
\end{cases}$$ Hence $\hat{T^1} (f_i) = f_i$ for $1 \le i \le r$ and $\hat{T^1} (f_i) = 0$ for $r+1 \le i \le n$. Any linear invariant of $G$ is certainly fixed by $\hat{T^1}$, so $a_1 \le r$. On the other hand, by proposition \[ppropg\], $\hat{f_i} := \hat{T^1} (f_i) = \lambda_i f_i$ is an invariant of $G$ for every $1\le i\le r$, so $a_1 \ge r$. Together, $a_1 = r$.
Before the final proof, let us introduce a handy notation.
\[dcorfficient\] Let $p(\lambda) \in {{\mathbf{C}}}[\lambda]$ or $p(\lambda) \in {{\mathbf{C}}}[[\lambda]]$. Then $[\lambda^i]:p(\lambda)$ denotes the [coefficient]{} of $\lambda^i$ in $p(\lambda)$.
So, for example $[x^2]: 2x^3 + 42x^2 - 6 = 42$ and $[\lambda^d]:\Phi_G(\lambda) = a_d$.
[(Moliens Theorem)]{} We just established the case $d = 1$, so the reader is probably expecting a proof by induction over $d$. But this is *not* the case. Rather, the case $d = 1$ applies to all $d > 1$. Note that $a_d$ is equal to the number of linearly independent invariants of all of the $T_g^d$. So Theorem \[t13\] gives us $$\begin{aligned}
a_1 &= {\ensuremath{\mathrm{Tr}}}(\hat{T}) = {\ensuremath{\mathrm{Tr}}}(\hat{T^1}) \qquad \mathrm{ and} \qquad\\
a_d &= {\ensuremath{\mathrm{Tr}}}(\hat{T^d}),\end{aligned}$$ where the latter includes the first. From definition \[dreynolds\] we also have $$\hat{T^1} = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T^1_g }
\quad
\textrm{and in general}
\quad
\hat{T^d} = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, T^d_g } ,$$ so we already know that $$a_d = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, {\ensuremath{\mathrm{Tr}}}(T^d_g) }.$$ So all we need to show is $$[\lambda^d]:\frac{1}{|G|} \sum_{g\in G} \frac{1}{\det({\mathrm{id}}- \lambda T^1_g)} = {\ensuremath \frac{1}{|G|}\sum_{g \in G} \, {\ensuremath{\mathrm{Tr}}}(T^d_g) }.$$ We will show that for every summand (group element) the equation $$[\lambda^d]: \frac{1}{\det({\mathrm{id}}- \lambda T^1_g)} = {\ensuremath{\mathrm{Tr}}}(T^d_g)$$ holds. From proposition \[pinducedeigen\] we get for every $g\in G$ that $$\begin{aligned}
{\ensuremath{\mathrm{Tr}}}(T^d_g) &= {\ensuremath{\mathrm{Tr}}}({\ensuremath \mathrm{diag}}(\tilde{\omega_1}, \dots, \tilde{\omega}_{\tilde{d}})) \\&=
\tilde{\omega_1} + \dots + \tilde{\omega}_{\tilde{d}} =\end{aligned}$$ sum of the products of the $\omega_1, \omega_2, \dots ,\omega_n $, taken $d$ of them at a time. On the other hand, for the same $g\in G$ we obtain from section \[svictor\] that $[T_g^1]_{{\mathbf{y}}_g^1,{\mathbf{y}}_g^1} = {\ensuremath \mathrm{diag}}(\omega_1,\dots , \omega_n)$ so that
$$\begin{aligned}
\det({\mathrm{id}}- \lambda T^1_g) &= \det({\mathrm{id}}- \lambda \cdot {\ensuremath \mathrm{diag}}(\omega_1,\dots , \omega_n) ) \\
&= (1 - \lambda \omega_1 )(1 - \lambda \omega_2 )\dots(1 - \lambda \omega_n ),\end{aligned}$$
so $$\begin{aligned}
\quad & \frac{1}{\det({\mathrm{id}}- \lambda T^1_g)} = \frac{1}{(1 - \lambda \omega_1 )(1 - \lambda \omega_2 )\dots(1 - \lambda \omega_n )} \\
&= \frac{1}{(1 - \lambda \omega_1) } \cdot \frac{1}{(1 - \lambda \omega_2)} \cdot \dots \frac{1}{(1 - \lambda \omega_n)} \\
&= (1 + \lambda \omega_1 + \lambda^2 \omega_1^2 + \dots )(1 + \lambda \omega_2 + \lambda^2 \omega_2^2 + \dots ) \dots
(1 + \lambda \omega_n + \lambda^2 \omega_n^2 + \dots ) \end{aligned}$$ and here the coefficient of $\lambda^d$ is also sum of the products of $\omega_1, \omega_2, \dots ,\omega_n $, taken $d$ of them at a time.
Again, the last claim $$\frac{1}{|G|} \sum_{g\in G} \frac{1}{\det({\mathrm{id}}- \lambda T_g)} =
\frac{1}{|G|} \sum_{g\in G} \frac{1}{\det({\mathrm{id}}- \lambda T^1_g)}$$ follows from the remark at the end of section \[preynolds\], since the sum runs over all group elements.
Symbol table {#ssymbol}
============
[2]{}
$a_d$
: number of linearly independent homogeneous invariants of degree $d$
$\tilde{d}$
: Dimension of $R_d$
${\mathcal{B}}$
: ON basis for $V$
$G$
: Finite group
$\omega_i$
: eigenvalue of $T_g^1$ ([@Sloane1] $= w_i$ )
$P(f)$
: Rho Riesz vector of $f$.
$\rho$
: Unitary representation $\rho : G \to U(V), g \mapsto T_g$
$R$
: Big algebra, direct sum of
$R_d$
: Direct summand of degree $d$
$R^G$
: Ring of invariants of $d$
$R^G_d$
: Degree $d$ summand
$T_g$
: representation of $g$ on $V$, ([@Sloane1] $ A_\alpha= [T_{g_\alpha}]_{{\mathcal{B}}, {\mathcal{B}}} $ )
$V$
: Complex inner product space
$V^\ast$
: Algebraic dual of $V$
Lost and found {#slostfound}
==============
Some things to explore from here:
- If we know the conjugacy classes of $G$, we may be able to say more, since every unitary representation splits into irreducible components.
- There seems to be a link to Pólya enumeration.
- We have GAP code, see [@GAP4].
- An example would be nice.
- Relations on the generators in $S$ of the Cayley graph $\Gamma(G,S)$ should lead to conditions of the minimal polynomial of its adjacency operator $Q(\Gamma(G,S))$.
- Also, Cayley graphs of some finite reflection groups [@Hu2] should become accessible.
- Check some more applications, as mentioned in [@Sloane1].
- For finding invariants, check also [@Cox], Gröbner bases.
[ABC 99]{}
Howard Anton, *Elementary Linear Algebra*, $6^{th}$ ed., John Wiley and Sons, New York, 1973. Jürgen Bierbrauer, *Introduction to Coding Theory*, Discrete Mathematics and Its Applications, Volume: 28, CRC Press Inc, Boca Raton, 2004. D. Cox, J. Little, D. O’Shea, *Ideals, Varieties, and Algorithms*, Springer-Verlag, New York, 1991. The GAP Group, *GAP – Groups, Algorithms, and Programming, Version 4.4*; 2004, `(http://www.gap-system.org)`.
John F. Humphreys, *A Course in Group Theory*, Oxford University Press, Oxford, 1994. James E. Humphreys, *Reflection Groups and Coxeter Groups*, Cambridge University Press, Cambridge, 1990. Steven Roman, *Advanced linear algebra, $3^{rd}$ Edition*, Springer-Verlag, New York, 2008. Bruce E. Sagan, *The Symmetric Group*, Wadsworth & Brooks, Pacific Grove, 1991. Neil J. A. Sloane, “Error Correcting Codes and Invariant Theory: New Applications of a Nineteenth–Century Technique”, *American Mathematical Monthly*, **84**,(1977), 82–107. Richard P. Stanley, “Invariants of Finite Groups and their Applications to Combinatorics”, *Bulletin (New Series) of the American Mathematical Society*, **1**, No. 3 (1979), 475–511. B. Sturmfels, *Algorithms in Invariant Theory*, Springer-Verlag, Wien, New York, 1993. Torbjörn Tambour, *Introduction to Finite Groups and their Representations*, Lecture notes, Lund, 1991.
[^1]
[^1]: `.tex` Typeset:
|
---
abstract: 'We develop a general framework to assess capabilities and limitations of the Gaussian toolbox in continuous variable quantum information theory. Our framework allows us to characterize the structure and properties of quantum resource theories specialized to Gaussian states and Gaussian operations, establishing rigorous methods for their description and yielding a unified approach to their quantification. We show in particular that, under a few intuitive and physically motivated assumptions on the set of free states, no Gaussian quantum resource can be distilled with free Gaussian operations, even when an unlimited supply of the resource state is available. This places fundamental constraints on state manipulations in all such Gaussian resource theories. We discuss in particular the applications to quantum entanglement, where we extend previously known results by showing that Gaussian entanglement cannot be distilled even with Gaussian operations preserving the positivity of the partial transpose, as well as to other Gaussian resources such as steering and optical nonclassicality. A comprehensive semidefinite programming representation of all these resources is explicitly provided.'
author:
- Ludovico Lami
- Bartosz Regula
- Xin Wang
- Rosanna Nichols
- Andreas Winter
- Gerardo Adesso
bibliography:
- 'main.bib'
title: Gaussian quantum resource theories
---
Introduction
============
Continuous-variable (CV) systems of quantum harmonic oscillators play a prominent role in quantum science, due to their ubiquitous presence, outstanding theoretical importance, and practical relevance in many quantum technologies [@braunstein_2005; @cerf_2007; @adesso_2014]. Among them, so-called *Gaussian states* are privileged as being remarkably affordable to produce and control in laboratory, while retaining, together with Gaussian operations, a significant part of the power of quantum information processing [@adesso_2007-2; @wang_2007; @weedbrook_2012; @adesso_2014]. However, such a restricted set of resources is insufficient to realize fundamental tasks like fault-tolerant quantum computation [@knill_2001], entanglement distillation [@eisert_2002-1; @fiurasek_2002; @giedke_2002], error correction [@niset_2009], or optimal metrology [@adesso_2009], and needs to be supplemented by nonlinear elements, e.g., photon detectors [@duan_2000; @takahashi_2010; @knill_2001], to achieve universality. A thorough investigation of properties and limitations of the Gaussian paradigm is thus crucial to deepen our theoretical understanding of quantum optics and information and to set suitable experimental benchmarks in practical tasks.
![A general framework is developed to assess capabilities and limitations of the Gaussian toolbox in continuous variable quantum information theory. It is shown that under a few assumptions on the set of free states, no Gaussian quantum resource can be distilled with free Gaussian operations, even when an unlimited supply of the resource state is available. We refer to this general no-go result as the [*Gaussbusters*]{} Theorem.[]{data-label="busterfig"}](Gaussbusters.png){width="5.5cm"}
The formalism of *quantum resource theories* [@horodecki_2012; @sperling_2015; @brandao_2015; @delrio_2015; @coecke_2016; @gour_2017; @regula_2018; @anshu_2017; @lami_2018] lends itself well to the investigation of such features and restrictions. Different quantum phenomena have been recently recognized and characterized as resources, including entanglement [@horodecki_2009], asymmetry [@gour_2008; @lostaglio_2015], athermality [@brandao_2013], purity [@horodecki_2003], nonlocality [@gallego_2012; @vicente_2014], coherence [@aberg_2006; @gour_2008; @baumgratz_2014; @streltsov_2017], nonclassicality [@vogel_2014; @killoran_2016; @theurer_2017; @das_2017], Einstein-Podolsky-Rosen (EPR) steering [@gallego_2015], contextuality [@grudka_2014], magic [@veitch_2014; @ahmadi_2017], non-Markovianity [@wakakuwa_2017], and noiseless classical or quantum communication in quantum Shannon theory [@Devetak_2005]. However, since each such resource may require a completely different approach to describe it, the alluring task of establishing a unified framework is very challenging. Although some general statements about quantum resource theories can be derived from suitable assumptions [@horodecki_2012; @brandao_2015; @gour_2017; @liu_2017; @regula_2018; @anshu_2017], most research to date focused on finite-dimensional scenarios, and despite an increasing interest in developing a resource-theoretic approach to quantum optics [@eisert_2003-1; @idel_2016; @zhang_2016; @brown_2016; @tan_2017; @friis_2017], there are no results that apply to a large class of resources in infinite dimensions.
In this paper we extend the general formalism of resource theories to CV systems, by introducing a framework for the study of resources whose free states (Sec. \[sec:freestates\]) and operations (Sec. \[sec:freeoperations\]) are Gaussian. This allows us to exploit the general resource-theoretic formalism to ultimately assess how powerful Gaussian states and operations are in CV quantum information theory. We show that all Gaussian resource theories which satisfy a set of physically-motivated conditions share a common structure, allowing us to simplify the description and quantification of many fundamental resources such as entanglement, EPR steering, and nonclassicality (squeezing). We establish universal constraints on state transformations under free Gaussian operations in any such resource theory, showing in Sec. \[sec:nogo\] that the operational task of resource distillation is impossible if one is restricted to Gaussian states and free Gaussian operations (see Fig. \[busterfig\]). In particular, we generalize the no-go theorem of Ref. [@giedke_2002] by showing that Gaussian entanglement cannot be distilled by Gaussian operations preserving the positivity of partial transpose — a larger class than previously considered — and we prove equivalent no-go results for other relevant Gaussian resources. Detailed examples and applications are illustrated in Sec. \[sec examples\].We discuss our main results below, deferring technical derivations to the Appendix.
Free states {#sec:freestates}
===========
Let us briefly recall the basics of Gaussian states [@weedbrook_2012; @adesso_2014; @serafini_2017]. Mathematically, a $n$-mode CV system is identified by a collection of canonical operators $x_1,p_1,\ldots, x_n, p_n$, which we can arrange in a vector $r\coloneqq (x_1,\ldots, x_n, p_1, \ldots, p_n)^T$. The canonical commutation relations $[x_j,p_k]=i\delta_{jk}$ can then be cast as $[r,r^T]= i\Omega$, where $\Omega \coloneqq \left(\begin{smallmatrix} 0 & \mathds{1} \\ -\mathds{1} & 0 \end{smallmatrix} \right)$ is the symplectic form. Denoting by $\mathcal{G}_n$ the set of $n$-mode Gaussian states, any $\rho \coloneqq \rho_G[V,s] \in \mathcal{G}_n$ is fully specified by its (real) displacement vector $s \coloneqq \operatorname{Tr}[r\, \rho]$ and its (real, symmetric) covariance matrix $V \coloneqq \operatorname{Tr}[ \{r - s, r^T - s^T\}\,\rho]$ with $\{\cdot, \cdot\}$ being the anticommutator. Letting ${\mathcal{M}}_{2n}({\mathbb{R}})$ denote the set of all real $2n \times 2n$ matrices, we will call $${\text{\rm QCM}}_n \coloneqq {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\mathcal{M}}_{2n}({\mathbb{R}}) {\;\rule{0pt}{9.5pt}\right|\;}V = V^T,\, V \geq i \Omega \rset$$ the set of quantum covariance matrices, i.e., those $V$ that satisfy the Robertson-Schrödinger uncertainty principle [@simon_1994]. Note that any such $V \in {\text{\rm QCM}}_n$ is strictly positive definite [@serafini_2017].
Resource theories are built upon two main notions [@brandao_2015]: (i) the subset ${\mathcal{F}}$ of free states, i.e., those which do not possess the given resource; and (ii) the subset ${\mathcal{O}}$ of free operations, i.e., those quantum channels unable to generate the resource, specified by the physical constraints of the theory. As free states can be prepared by free operations at no cost, during a protocol one may add ancillary systems to one’s original system; following [@brandao_2015], we will refer to such systems as spatially separated. In any resource theory, there may be different ways to define the set of free states (think e.g. of entanglement theory, in which one needs to specify a partition to identify separable states). To address this, we assume that each of the spatially separated subsystems $j=1,\ldots, l$ is fully specified by a set $\lambda_j$ of variables, which can then be grouped in a single vector ${\boldsymbol{\lambda}}\coloneqq (\lambda_1, \ldots, \lambda_l)$. For instance, one such variable will be the total number of modes $n_j$ of each subsystem $j$. For a CV system made of $l$ spatially separated subsystems identified by a vector of variables ${\boldsymbol{\lambda}}$, we then denote by ${\mathcal{F}}({\boldsymbol{\lambda}})$ the subset of free states, and by ${\mathcal{F}}\coloneqq \bigcup_{{\boldsymbol{\lambda}}} {\mathcal{F}}({\boldsymbol{\lambda}})$ the set of all free states over arbitrary collections of spatially separated subsystems. Since we care about the Gaussian restriction of any resource theory, we will focus on the set ${\mathcal{F}}_G\coloneqq \bigcup_{{\boldsymbol{\lambda}}} {\mathcal{F}}_G({\boldsymbol{\lambda}})$ of free *Gaussian* states, where ${\mathcal{F}}_G({\boldsymbol{\lambda}})\coloneqq {\mathcal{F}}({\boldsymbol{\lambda}})\cap \mathcal{G}_N$, with $N=\sum_j n_j$, for a fixed ${\boldsymbol{\lambda}}$, and the corresponding set of free covariance matrices is $$\begin{aligned}
{\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}) \coloneqq {\left\{\left.}
\newcommand{\rset}{\right\}}V\in \mathcal{M}_{2N}({\mathbb{R}}) {\;\rule{0pt}{9.5pt}\right|\;}\exists\, s\in\mathds{R}^{2N}:\ \rho_G[V,s]\in {\mathcal{F}}({\boldsymbol{\lambda}}) \rset.
\end{aligned}$$ There are some standard assumptions about the set of free states, formalized as Postulates \[post tensor\]-\[post weak closed\] in [@brandao_2015], that have a sound theoretical basis and apply to a wide range of theories. We will therefore regard them as a safe starting point to establish a set of fundamental requirements for Gaussian resource theories. Before proceeding further, let us make a first working assumption that will simplify the following analysis considerably:
[0]{} \[post D invariance\] The set of free states is invariant under displacement operations.
To justify this assumption, note that displacement operations can be applied to any system by adding an ancillary system in a highly excited coherent state, and combining the two systems at a low-transmissivity beam splitter [@paris_1996]. From an experimental standpoint, coherent states and beam splitters are relatively cheap tools. Crucially, Postulate \[post D invariance\] implies that the set of free Gaussian states is now fully described by the corresponding covariance matrices, so we can write ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}) = {\left\{\left.}
\newcommand{\rset}{\right\}}V\in \mathcal{M}_{2N}({\mathbb{R}}) {\;\rule{0pt}{9.5pt}\right|\;}\rho_G[V,0]\in {\mathcal{F}}({\boldsymbol{\lambda}})\rset$ and ${\mathcal{F}}_G({\boldsymbol{\lambda}}) = {\left\{\left.}
\newcommand{\rset}{\right\}}\rho_G[V,s] {\;\rule{0pt}{9.5pt}\right|\;}V\in {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}),\ s\in\mathds{R}^{2N} \rset$. The following assumptions define the structure of the theory for composite systems.
[I]{}\[post tensor\] The set of free states is closed under tensor products of spatially separated subsystems.
[II]{}\[post partial trace\] The set of free states is closed under partial traces over spatially separated subsystems.
[III]{}\[post permutation\] The set of free states is closed under permutations of spatially separated subsystems.
These three properties carry over to the restricted set of free Gaussian states ${\mathcal{F}}_G$, since it is well known that Gaussian states are also closed under the above operations. Postulate \[post tensor\] can be rewritten symbolically as ${\mathcal{F}}({\boldsymbol{\lambda}}) \otimes {\mathcal{F}}({\boldsymbol{\lambda'}}) \subseteq {\mathcal{F}}({\boldsymbol{\lambda}} \oplus {\boldsymbol{\lambda'}})$, where for ${\boldsymbol{\lambda}}=(\lambda_1,\ldots, \lambda_l)$ and ${\boldsymbol{\lambda'}}=(\lambda_1, \ldots, \lambda_{l'})$ one sets ${\boldsymbol{\lambda}}\oplus {\boldsymbol{\lambda'}} \coloneqq (\lambda_1, \ldots, \lambda_l, \lambda'_1, \ldots, \lambda'_{l'})$. At the level of covariance matrices, this translates to ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}) \oplus {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda'}}) \subseteq {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}\oplus {\boldsymbol{\lambda'}})$. Similarly, we can formulate Postulate \[post partial trace\] as $\Pi\, {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}\oplus {\boldsymbol{\lambda'}})\, \Pi^T \subseteq {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$, where $\Pi$ is the projector onto the subsystems corresponding to ${\boldsymbol{\lambda}}$.
Apart from the above Postulates, one of the the most basic assumptions that we can make about the unrestricted set (i.e., before the intersection with $\mathcal{G}_N$) of free states is undoubtedly *convexity*. This is justifiable as in most physically relevant cases we do not expect an increase in quantum resources under classical mixing [^1]. At the level of all states, we have then:
[IV]{} \[post convexity\] For all ${\boldsymbol{\lambda}}$, the set of free states ${\mathcal{F}}({\boldsymbol{\lambda}})$ is convex.
Our final Postulate will pertain to the closedness of the set of free states in the Banach space ${\mathcal{T}}({\mathcal{H}}_N)$ of trace-class operators acting on the Hilbert space ${\mathcal{H}}_N$, endowed with the trace norm $\|\cdot\|_1$.
[V]{} \[post weak closed\] For all ${\boldsymbol{\lambda}}$, the set of free states ${\mathcal{F}}({\boldsymbol{\lambda}}) \subseteq {\mathcal{T}}({\mathcal{H}}_N)$ is norm-closed.
Since infinite-dimensional spaces admit many legitimate linear topologies with respect to which we can define closedness, the above choice of the norm topology may seem rather arbitrary. However, it turns out that in the present context any other reasonable choice still yields the same result. In fact, Mackey’s theorem [@aliprantis_2007 Thm. 8.9] ensures that all linear topologies on ${\mathcal{T}}({\mathcal{H}}_N)$ that agree on the set of continuous linear functionals possess the same closed convex sets. To see why this is physically relevant, remember that the norm-continuous linear functionals on ${\mathcal{T}}({\mathcal{H}}_N)$ can be written as $\rho\mapsto \operatorname{Tr}[\rho A]$, where $A\in {\mathcal{B}}({\mathcal{H}}_N)$ is a generic bounded operator, i.e., an observable. Summarizing the above discussion: Postulates \[post convexity\] and \[post weak closed\] together imply that all sets of free states ${\mathcal{F}}({\boldsymbol{\lambda}})$ and ${\mathcal{F}}_G({\boldsymbol{\lambda}})$ are closed with respect to any linear topology whose corresponding continuous linear functionals are (all) the observables. For further details on the issue of closedness of the relevant sets, refer to Appendix \[app topoogy\].
Let us now discuss some consequences of the above assumptions in the Gaussian setting. In order to do so, it is important to understand the topology of the set of Gaussian states in some detail. In Appendix \[app topoogy\], we show that Gaussian states form a closed set with respect to the trace norm topology (Lemma \[lemma G closed\]), and that the map $(V,s)\mapsto \rho_G[V,s]$ that sends a pair formed by a quantum covariance matrix and a real vector to the corresponding Gaussian state is continuous with respect to the same topology (Lemma \[lemma V to rho 1-continuous\]). A key difference between a Gaussian resource theory satisfying Postulates \[post tensor\]-\[post weak closed\] and a corresponding finite-dimensional theory is that the set of Gaussian states is non-convex, hence ${\mathcal{F}}_G({\boldsymbol{\lambda}})$ can not be expected to be convex either. We have instead a weaker property that we dub *Gaussian convexity*: if a trace norm limit of convex combinations of free Gaussian states is a Gaussian state, then it must be free. Importantly, this implies the *upward closedness* of the set of free covariance matrices ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$, formalized as follows.
\[prop V\] When Postulates \[post convexity\] and \[post weak closed\] hold, the set ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ is topologically closed as well as upward closed, in the sense that, if $V\in {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ and $W\geq V$, then $W\in{\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$.
Another desirable property of the set of free covariance matrices ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ is for itself to be convex. Interestingly, this does not follow directly from our Postulates, but is indeed implied by an additional natural assumption, i.e., that the given set of free Gaussian states ${\mathcal{F}}_G({\boldsymbol{\lambda}})$ is closed under mode-by-mode mixing with 50:50 beam splitters (Prop. \[prop convexity\]).
Free operations and quantification {#sec:freeoperations}
==================================
In any resource theory, a free operation can be any channel which always maps free states into free states. However, the physical setting of the given resource can further restrict the allowed free operations: for instance, in entanglement theory, the distant laboratories paradigm leads to the set of local operations and classical communication (LOCC). To keep our results as general as possible, we will consider the *maximal* set of resource non-generating operations, and only impose the natural restriction that free operations should also be Gaussian, i.e., such that they always map a Gaussian state to a Gaussian state [@giedke_2002; @depalma_2015].
A quantum channel $\Lambda:{\mathcal{T}}({\mathcal{H}}_N)\rightarrow {\mathcal{T}}({\mathcal{H}}_M)$ is called *resource non-generating* if $\Lambda \left[ {\mathcal{F}}({\boldsymbol{\lambda}}) \right] \subseteq {\mathcal{F}}({\boldsymbol{\mu}})$ for systems described by variables ${\boldsymbol{\lambda}},{\boldsymbol{\mu}}$. The set of all resource non-generating operations is denoted by ${\mathcal{O}}({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})$, and the restriction to Gaussian operations by ${\mathcal{O}}_G({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})$. In particular, $\Lambda \left[ {\mathcal{F}}_G({\boldsymbol{\lambda}}) \right] \subseteq {\mathcal{F}}_G({\boldsymbol{\mu}})$ for all $\Lambda \in {\mathcal{O}}_G({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})$.
A fundamental question in any resource theory is, given several resourceful states, to quantify the degree of their resourcefulness and thus compare the usefulness of the states in operational tasks [@plenio_2007; @brandao_2015; @regula_2018; @anshu_2017]. For this, one needs a measure $\mu : {\mathcal{T}}({\mathcal{H}}_N) \rightarrow {\mathbb{R}}_+$ which satisfies two basic criteria: faithfulness, i.e., being minimum on all (and only on) free states, $\mu (\rho) = \inf_{\sigma\in{\mathcal{T}}({\mathcal{H}}_{N})} \mu(\sigma) \iff \rho \in {\mathcal{F}}({\boldsymbol{\lambda}})$, as well as monotonicity, i.e., $\mu(\Lambda(\rho)) \leq \mu(\rho)$ for all free operations $\Lambda$. Here we stress that we can consider the maximal set of free operations ${{\mathcal{O}}({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})}$ without loss of generality, since any measure monotonic under ${{\mathcal{O}}({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})}$ will also be a monotone under any smaller subset of free operations. Analogously, in the setting of Gaussian resources, we will be interested in quantifiers $\mu_G : {\text{\rm QCM}}_N \rightarrow {\mathbb{R}}_+$ defined at the level of covariance matrices and monotonic under the free Gaussian operations, so that $\mu_G(V') \leq \mu_G(V)$ where $V'$ is the covariance matrix corresponding to the state $\Lambda(\rho_G[V,s])$ with $\Lambda \in {\mathcal{O}}_G({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})$.
A general instance of such a measure — a variant of which has been considered in the characterization of entanglement before [@giedke_2002] — can be defined for any $V\in {\text{\rm QCM}}_N$ as $$\kappa_{\mathcal{F}}(V) \coloneqq \min{\left\{\left.}
\newcommand{\rset}{\right\}}t \geq 1 {\;\rule{0pt}{9.5pt}\right|\;}t V\in {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}) \rset.
\label{kappa}$$ The measure can be easily seen to be faithful in the sense that $\kappa_{\mathcal{F}}(V) \geq 1$ and $\kappa_{\mathcal{F}}(V) = 1$ iff $V \in {\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$, and the fact that the set on the right-hand side of Eq. is non-empty is ensured by the upward closedness of ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$. The properties and monotonicity of the above quantifier can be summarized as follows.
\[prop:kappa\] The function $\kappa_{\mathcal{F}}(\cdot)$ is finite and well-defined on all covariance matrices, faithful, continuous, and monotonic under ${{\mathcal{O}}_G({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})}$.
We defer the proof to Appendix \[app G resources\] (see Prop. \[prop:kappa-reformul\]). Note that, if membership of the set ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ can be decided by semidefinite constraints at the level of covariance matrices, the evaluation of $\kappa_{\mathcal{F}}(V)$ reduces to a semidefinite program. We discuss such cases in Section \[sec examples\] and in Appendix \[app SDP\].
No-go theorem for Gaussian resource distillation {#sec:nogo}
================================================
At the heart of every resource theory lies the problem of characterizing state transformations which are allowed by the given set of free operations. In particular, the operational task of [*resource distillation*]{} deals with using free operations to convert multiple copies of a given quantum state into a smaller number of target states, usually representing maximally resourceful states. This task was first considered in the resource theory of entanglement with LOCC [@bennett_1996-4; @bennett_1996-5], and has been later extended to more general settings [@rains_1999; @rains_2001; @brandao_2011; @buscemi_2010-2; @fang_2017] and other quantum resources [@bravyi_2005; @winter_2016; @regula_2017-2]. Entanglement distillation has also been considered for Gaussian states [@giedke_2002; @fiurasek_2002; @eisert_2002-1], where the task can be expressed as using LOCC to transform multiple copies of a bipartite state $\rho_{AB}^{\otimes n}$ into a state which approaches a maximally entangled state as $n \to \infty$. An archetypal example of the analysis of the limitations of the Gaussian paradigm in this context has been carried out in [@giedke_2002], where it was shown that Gaussian LOCC protocols are not sufficient to distill Gaussian entanglement.
Since the existence of a “golden unit” or a unique maximally resourceful state is not guaranteed in arbitrary quantum resource theories, we can consider the more general task of approximately converting multiple copies of a quantum state into another state which is more resourceful; that is, given a Gaussian state with covariance matrix $V$, we ask about the existence of free operations that implement the transformations $V^{\oplus n} \rightarrow W_{n}$, for some sequence of covariance matrices $W_{n}$ that approach a fixed target $W$ such that $\kappa_{\mathcal{F}}(W) > \kappa_{\mathcal{F}}(V)$. A central result of this work is a general [*no-go*]{} result entailing that, in any resource theory in our framework, the distillation of the given resource with free Gaussian operations is [*de facto*]{} impossible, as illustrated in Fig. \[busterfig\]. This result is in stark contrast with the main finding of [@brandao_2015], which instead implies the complete reversibility of the considered resource theory. Such a dramatic difference in the conclusions is even more surprising when one considers that the starting postulates are quite similar in the two cases, and illustrates clearly the intrinsic limitations of the Gaussian framework.
\[thm:no-go\] Consider an arbitrary Gaussian resource theory satisfying Postulates \[post D invariance\]-\[post weak closed\] and two covariance matrices $V,W\in \text{\emph{QCM}}_N$. If $\kappa_{\mathcal{F}}(W) > \kappa_{\mathcal{F}}(V)$, then it is impossible to find a sequence $(W_{n})_{n\in\mathds{N}}\subset\text{\emph{QCM}}_{N}$ such that $\lim_{n\rightarrow\infty} W_{n}=W$ and the transformations $V^{\oplus n}\rightarrow W_{n}$ are possible with Gaussian resource non-generating operations for all $n$.
The proof of the above theorem relies on a special property of the measure $\kappa_{\mathcal{F}}$ that we could call, borrowing terminology from classical probability theory, *tensorization property* [@beigi_2016]: for all resource theories in consideration, $\kappa_{\mathcal{F}}$ does not change when multiple copies of a quantum state are considered; more generally, we have $\kappa_{\mathcal{F}}\left(V \oplus W\right) = \max\{ \kappa_{\mathcal{F}}(V), \kappa_{\mathcal{F}}(W)\}$ for any two covariance matrices $V,W$ (Lemma \[kappa multi-copy\]). This, together with the monotonicity of $\kappa_{\mathcal{F}}$, immediately implies that distillation is impossible since we cannot increase $\kappa_{\mathcal{F}}$ with free Gaussian operations. In the following, we present explicit applications of our framework to a broad set of continuous variable resources, namely squeezing (equivalently, nonclassicality), quantum entanglement manipulated via local operations and classical communication or via operations preserving the positivity of the partial transpose, and steering.
Examples and applications {#sec examples}
=========================
Quite remarkably, it turns out that in many — if not all — physically relevant resource theories, ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ is not only a convex set, but can even be described by means of semidefinite programming (SDP) constraints. Although we leave open the question of whether a general principle can be found from which the existence of such a description follows naturally, we will now characterize the quantification of all resources for which such SDP structure is known to exist. In particular, our results apply to any resource theory satisfying Postulates \[post D invariance\]–\[post weak closed\] whose set of free states can be described by constraints of the kind $$\begin{aligned}
{\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}}) = {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\text{\rm QCM}}_{N} {\;\rule{0pt}{9.5pt}\right|\;}V \geq f(Q) + C,\ g(Q)\geq D \rset
\end{aligned} \label{V as SDP}$$ where $Q$ is a Hermitian matrix variable of some fixed size, $f$ and $g$ are linear functions, and $C$ and $D$ are constant Hermitian matrices. The main advantage of the representation in Eq. is that the associated quantifier $\kappa_{\mathcal{F}}$ in Eq. can then be evaluated via an efficiently computable semidefinite program: $$\label{kappa SDP}
\begin{aligned}
\hspace{-5em}\kappa_{\mathcal{F}}(V) = \ & \underset{\xi,\, Q}{\text{minimize}}
& & \xi \\
& \text{subject to} & & \xi \, V \geq f(Q) + C\\
&&& g(Q)\geq D \\
&&& \xi \geq 1.
\end{aligned}$$
Alternatively, one can choose to introduce the quantity $$\begin{aligned}
\upsilon_{\mathcal{F}}(V) \coloneqq \max_{\zeta,\, Q} {\left\{\left.}
\newcommand{\rset}{\right\}}\zeta {\;\rule{0pt}{9.5pt}\right|\;}V \geq \zeta (f(Q) + C),\ g(Q)\geq D \rset
\end{aligned}$$ in which case $\kappa_{\mathcal{F}}(V) = \max \left\{ 1,\, 1/\upsilon_{\mathcal{F}}(V)\right\}$. The advantage this formulation is that the dual of the optimization problem $\upsilon_{\mathcal{F}}$ can be expressed by means of the so-called resource witnesses based on second moments [@hyllus_2006-1], that is, as an optimization over the expectation values $\operatorname{Tr}(W V)$ at the level of the covariance matrix. Assuming that strong duality for the problem holds (which can be straightforwardly verified for all of the considered resource theories), we then have the SDP $$\begin{aligned}
\hspace{-5em}\upsilon_{\mathcal{F}}(V) = \ & \underset{W,\, Y}{\text{minimize}}
& & \operatorname{Tr}( W V )\\
& \text{subject to} & & \operatorname{Tr}(W C) + \operatorname{Tr}(Y D) = 1\\
&&& f^\dagger (W) = g^\dagger(Y) \\
&&& W,Y \geq 0
\end{aligned}$$ where $f^\dagger, g^\dagger$ are the adjoint maps, that is, the unique linear maps satisfying $\operatorname{Tr}(f(A)B)=\operatorname{Tr}(A f^\dagger(B))$ for any Hermitian $A,B$.
Many common Gaussian resources can indeed be expressed and quantified in this way — we will now provide some representative examples of such resources, which we have also collected in Table \[tab:resources\].
Resource $f(Q)$ $g(Q)$ $C$ $D$ Further constraints on $Q$
-------------------------------------------------- ------------------ ------------------ --------------------------------- ----------------------------------- ------------------------------
Bipartite entanglement [@werner_2001] $Q_A \oplus Q_B$ $Q_A \oplus Q_B$ — $i \Omega_{A} \oplus i\Omega_{B}$ $Q_A = Q_A^T$, $Q_B = Q_B^T$
Bipartite entanglement (simplified) [@lami_2016] $Q_A \oplus 0_B$ $Q_A$ $0_A \oplus i \Omega_B$ $i \Omega_A$ $Q_A = Q_A^T$
Negative partial transpose [@werner_2001] — — $i\Omega_A \oplus (-i\Omega_B)$ —
Steerability ($A\to B$) [@wiseman_2007] — — $0_A \oplus i\Omega_B$ —
Nonclassicality [@simon_1994] — — $\mathbbm{1}$ —
Squeezing (nonclassicality)
---------------------------
We start by looking at the simplest Gaussian resource theory of all, namely that of squeezing or nonclassicality [@lee_1991; @simon_1994; @serafini_2005; @idel_2016]. The free states of this theory, also called *classical* states from now on, are simply convex mixtures of coherent states. Within this framework, the goal is usually that of preparing squeezed states, which may be useful for some practical (e.g. metrological) tasks [@SchnabelPhysRep]. It is not difficult to see that the continuous variable resource theory of squeezing obeys all the Postulates we presented. The free operations include in particular passive transformations, obtained by concatenating [@yadin_2018]: (i) the addition of ancillae in classical Gaussian states; (ii) passive unitaries, defined as those symplectic unitaries that preserve the total photon number; and (iii) destructive Gaussian measurements. These operations are relatively cheap to realize experimentally, as passive unitaries can always be implemented by combining beam splitters and phase shifters [@reck_1994].
Restricting to the Gaussian setting, free states in this theory admit a remarkably simple description in terms of their covariance matrices, for $\rho_G[V,s]$ is a classical state if and only if $V\geq {\mathds{1}}$ [@simon_1994]. This gives us the simple form $$\begin{aligned}
\hspace{-5em}\kappa_{\mathcal{C}}(V) = \,& \underset{\xi\, \geq\, 1}{\text{minimize}}
& & \xi \\
& \text{subject to} & & \xi \, V \geq {\mathds{1}}\end{aligned}$$ which can be easily seen to be exactly computable as $\kappa_{\mathcal{C}}(V) = \max \left\{ 1,\, 1/\lambda_{\min}(V) \right\}$, where $\lambda_{\min}$ denotes the minimal eigenvalue. Our main result in Thm. \[thm:no-go\] then establishes a no-go result about the convertibility of nonclassical Gaussian states under all operations preserving the set of classical states, and in particular passive operations.
Entanglement
------------
The resource theory of quantum entanglement is another example of a theory for which all of the Postulates hold [@brandao_2015; @adesso_2014]. Focusing on bipartite entanglement between parties $A$ and $B$ for simplicity, the set of free states is formed by the separable states ${\mathcal{S}}(A|B)$. The most operationally relevant set of free operations includes all transformations that are implementable as local operations assisted by classical communication (LOCC), and is a strict subset of all the resource non-generating (separability-preserving) operations.
In our Gaussian setting, the set ${\mathcal{V}}_{\mathcal{S}}$ can be described by semidefinite constraints of the form $$\begin{aligned}
{\mathcal{V}}_{\mathcal{S}}(A|B) = {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\text{\rm QCM}}_{N_{AB}} {\;\rule{0pt}{9.5pt}\right|\;}V \geq \gamma_A \oplus \gamma_B,\ \gamma_{i} \in {\text{\rm QCM}}_{N_{i}} \rset.
\end{aligned}$$ which can be easily expressed in the form of Eq. (see Table \[tab:resources\]). The associated measure $\kappa_{\mathcal{S}}$ can then be computed as a semidefinite program [@hyllus_2006-1], and corresponds to the inverse of a quantifier studied in [@giedke_2002].
Notice that Thm. \[thm:no-go\] includes as a particular case the result of [@giedke_2002], showing the impossibility of entanglement distillation with Gaussian LOCC: in fact, it readily generalizes the result by showing that distillation with Gaussian separability-preserving operations is also impossible.
We can strengthen the result even further by relating the resource theory of entanglement to the one of negative partial transpose, in which the free states ${\mathcal{P}}(A|B)$ are those with positive partial transpose across the cut $A|B$. This set can also be obtained from Eq. as $$\begin{aligned}
{\mathcal{V}}_{\mathcal{P}}(A|B) = {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\text{\rm QCM}}_{N_{AB}} {\;\rule{0pt}{9.5pt}\right|\;}V \geq i\Omega_A \oplus (-i\Omega_B) \rset.
\end{aligned}$$ Here, the quantifier $\kappa_{\mathcal{P}}$ admits an analytical characterization as $\kappa_{\mathcal{P}}(V_{AB}) = \max\{ 1, 1/\nu_{\min} (\widetilde{V}_{AB})\}$ with $\nu_{\min}(\widetilde{V}_{AB})$ being the smallest symplectic eigenvalue of the partially transposed covariance matrix [@lami_2016]. We then notice that, for any sequence of states $\rho(n)$ which approaches the maximally entangled state in the limit $n \to \infty$, we have $\lim_{n\to\infty} \kappa_{{\mathcal{S}}}(\rho(n)) =\infty = \lim_{n\to\infty} \kappa_{{\mathcal{P}}}(\rho(n))$, and therefore the distillation of entanglement would necessarily involve increasing $\kappa_{{\mathcal{P}}}$. By Thm. \[thm:no-go\], we get that Gaussian entanglement distillation is impossible even with Gaussian operations preserving the positivity of the partial transpose. Among those operations — which can be strictly more powerful than LOCC alone — there are for instance those transformations implementable by means of Gaussian LOCC assisted by an *unlimited* supply of bound entangled Gaussian states [@werner_2001].
We further remark that the characterization of the set of separable Gaussian states and their corresponding covariance matrices can be simplified to [@lami_2016] $$\begin{aligned}
{\mathcal{V}}_{\mathcal{S}}(A|B) = {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\text{\rm QCM}}_{N_{AB}} {\;\rule{0pt}{9.5pt}\right|\;}V \geq \gamma_A \oplus i\Omega_B,\ \gamma_{A} \in {\text{\rm QCM}}_{N_{A}} \rset
\end{aligned}$$ which in particular means that the computation of the quantifier $\kappa_{\mathcal{S}}$ can be performed by optimizing only over one of the subsystems — this has particular implications for the case where one of the subsystems has a larger dimension than the other, simplifying the computation of the relevant quantities. For completeness, we give the full forms of the measures $\kappa_{\mathcal{S}}$ and $\upsilon_{\mathcal{S}}$ simplified in this way in Appendix \[app SDP\].
Steering
--------
Another fundamental resource theory is based on the phenomenon of EPR steering [@schrodinger_1935; @reid_1989], in which party $A$ can exploit quantum correlations to influence the state of another party $B$ by only performing measurements on $A$’s subsystem. In resource-theoretic approaches to steering [@wiseman_2007; @gallego_2015; @piani_2015], the free states are referred to as $A\!\to\! B$ unsteerable, and free operations are commonly chosen to be one-way LOCC, reflecting the asymmetric nature of steering. Steering admits a simplified characterization when restricted to Gaussian measurements, allowing for a dedicated resource theory of Gaussian steering to be established [@wiseman_2007; @kogias_2015; @ji_2015; @lami_2016-1; @xiang_2017]. It turns out that the set of free states ${\mathcal{T}}_{A \to B}$ that are unsteerable by Gaussian measurements on $A$ can be described as [@wiseman_2007] $$\begin{aligned}
{\mathcal{V}}_{\mathcal{T}}({\boldsymbol{\lambda}}) = {\left\{\left.}
\newcommand{\rset}{\right\}}V \in {\text{\rm QCM}}_{N_{AB}} {\;\rule{0pt}{9.5pt}\right|\;}V \geq 0_A\oplus i\Omega_B \rset.
\end{aligned}$$ It is then easy to verify that our Postulates are satisfied, and the no-go result of Thm. \[thm:no-go\] holds also for the Gaussian resource theory of steering — that is, the distillation of steering from Gaussian states is impossible by Gaussian steering non-generating operations, with the latter including all relevant classes of free operations such as one-way Gaussian LOCC. We remark that in this case the quantifier $\kappa_{\mathcal{T}}$ can be computed as $$\begin{aligned}
\hspace{-5em}\kappa_{\mathcal{T}}(V_{AB}) = \,& \underset{\lambda \geq 1}{\text{minimize}}
& & \lambda\\
& \text{subject to} & & \lambda \, V_{AB} / V_A \geq i \Omega_B,
\end{aligned}$$ with $V_{AB}/V_A$ denoting the Schur complement, which admits an analytical characterization as $\kappa_{\mathcal{T}}(V_{AB}) = \max \{ 1, \, 1/\nu_{\min}(V_{AB}/V_A) \}$. In the particular case when system $B$ consists of only one mode, $\log \kappa_{\mathcal{T}}$ is equal to a previously introduced quantifier of Gaussian steering [@kogias_2015].
Conclusions {#sec conclusions}
===========
We have introduced a framework for the characterization of general CV Gaussian quantum resource theories satisfying a set of intuitive constraints on their set of free states. The approach allowed us to describe many important resources such as entanglement, steering, and nonclassicality together in a common formalism, obtaining novel results in the characterization of the resources as well as shedding light onto their properties. In particular, we showed that the task of resource distillation is impossible with free Gaussian operations in the given resource theories, by proving specifically that, by such operations, one cannot convert (even infinitely many copies of) a Gaussian state into another Gaussian state with a higher resource content as quantified by the resource monotone defined in this paper. This establishes fundamental limitations of the Gaussian paradigm for state transformations. An interesting open question is whether some sort of converse of Thm. \[thm:no-go\] holds. Namely, given any Gaussian resource theory and two covariance matrices $V,W$ such that $\kappa_{\mathcal{F}}(V) \geq \kappa_{\mathcal{F}}(W)$, is it always possible to convert a large number of copies of $V$ into a single copy of $W$ with Gaussian resource non-generating operations? Even more ambitiously, can the transformation $V\rightarrow W$ happen with asymptotic non-zero rate, if one allows for vanishing errors? These questions will be explored in further work.
In summary, our results are a step forward in the characterization of general quantum resources, bridging the gap between the different approaches to finite- and infinite-dimensional settings, and elucidating the power of Gaussian states and operations in quantum information processing. Our work opens an avenue for further investigation of many aspects of CV resources, including a complete characterization of state transformations as well as operational tasks and protocols such as resource distillation and dilution beyond Gaussianity.
*Acknowledgements.* We are grateful to Giacomo De Palma, Vittorio Giovannetti and Krishna Kumar Sabapathy for useful discussions on this and related topics. We acknowledge financial support from the European Research Council (ERC) under the Starting Grant GQCOP (Grant No. 637352) and from the Foundational Questions Institute under the Physics of the Observer Programme (Grant No. FQXi-RFP-1601). AW was supported by the ERC Advanced Grant IRQUAT, and the Spanish MINECO, project FIS2016-86681-P. LL and BR contributed equally to this work.
Topology of Gaussian states {#app topoogy}
===========================
Notation and definitions
------------------------
For completeness, we recall the relevant definitions and concepts. Consider a continuous variable system of $n$ modes, for which we adopt the so-called real notation. In what follows, we reserve the letter $r$ for the column vector formed by the $n$ pairs of canonically conjugated field operators, sorted as $$r\coloneqq (x_{1}, \ldots, x_{n}, p_{1}, \ldots, p_{n})^{T}\, .
\label{r}$$ Here, the transposition sign refers only to the phase space degrees of freedom, and does not act on the Hilbert space. With the help of this notation, the *canonical commutation relations* $[x_j, p_k] = i \delta_{jk}$ can be rewritten in a compact vector form as $$[r,r^T] = i \Omega \coloneqq i \begin{pmatrix} 0 & {\mathds{1}}\\ - {\mathds{1}}& 0 \end{pmatrix}\, .
\label{Omega}$$ The displacement operator associated with $\xi\in \mathds{R}^{2n}$ is given by $D(\xi) \coloneqq e^{i \xi^{T} \Omega r}$ and satisfies the identity $$D(\xi_{1}) D(\xi_{2}) = e^{-\frac{i}{2} \xi_{1}^{T}\Omega \xi_{2}} D(\xi_{1}+\xi_{2})\, ,
\label{Weyl}$$ referred to as the *Weyl form of the canonical commutation relations.* Observe that $D(\xi)^\dag=D(-\xi)$ for all real vectors $\xi$.
The displacement operators can be used to generate the notable set of coherent states. For $u\in\mathds{R}^{2n}$, one defines $$\ket{u} \coloneqq D(u) \ket{0}\, ,
\label{coherent}$$ where $\ket{0}$ is the vacuum state. Applying the Campbell-Baker-Hausdorff formula to the exponential that defines the displacement operator, it is not too difficult to show that $$\braket{0|u} = \braket{0|D(u)|0} = e^{-\frac14 u^T u}\, .
\label{D on 0}$$
Coherent states are just particular examples of Gaussian states, defined as thermal states of quadratic Hamiltonians. We denote the set of Gaussian states of an $n$-mode system by $\mathcal{G}_n$. Remember that Gaussian states can be uniquely identified by their first and second moments, respectively given by $$\begin{aligned}
s &\coloneqq \operatorname{Tr}[\rho\, r]\, \label{first} \\
V_{jk} &\coloneqq \operatorname{Tr}\left[ \rho \left\{(r-s)_j, (r-s)_k \right\} \right] . \label{second}\end{aligned}$$ Here, the anticommutator $\{H,K\}\coloneqq HK+KH$ is needed in order to make the above expression real. While any vector $s\in \mathds{R}^{2n}$ can represent the first moments of an $n$-mode Gaussian state, it is well-known that the entries of a real symmetric matrix $V$ are the second moments of some Gaussian state if and only if $$V\geq i\Omega\, ,
\label{Heisenberg}$$ the above relation encoding the constraints coming from Heisenberg’s uncertainty principle in this context. Real symmetric matrices satisfying Eq. are called quantum covariance matrices in what follows. It can be shown that every such matrix is necessarily strictly positive, i.e., Eq. implies that $V>0$.
For every trace class operator $T$, it is convenient to define its *characteristic function* $$\chi_{T}(\xi) \coloneqq \operatorname{Tr}[T D(\xi)]\, .
\label{chi}$$ The operator can be reconstructed from its characteristic functions by means of the following relation [@holevo_2011 Cor. 5.3.5]: $$T = \int \frac{d^{2n}\xi}{(2\pi)^{n}} \, \chi_{T}(\xi)\, D(-\xi)\, ,
\label{integral}$$ where the integral converges in the weak topology, see for instance [@holevo_2011 Cor. 5.3.5]. For more on what this means, see below.
It can be shown that the characteristic function of a Gaussian state $\rho_G[V,s]$ takes the form [@serafini_2017 Eq. (4.48)] $$\chi_{\rho_G[V,s]}(\xi) = \operatorname{Tr}[\rho_G[V,s]\, D(\xi)] = e^{-\frac14 \xi^T \Omega^T V \Omega \xi + i s^T \Omega \xi}\, .
\label{chi G}$$ Up to a change of variables, Eq. can then be rewritten as follows: $$\rho_G[V,s] = \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T V \xi - i s^T\xi} D(\Omega \xi)\, ,
\label{integral G}$$ where the integral converges weakly, see again [@holevo_2011 Cor. 5.3.5]. Among the other things, from Eq. and it can be appreciated, that Gaussian states are exactly those quantum states whose characteristic function is a (multivariate) Gaussian.
A useful formula that we will employ in what follows gives the action of a random displacement on a Gaussian state: for all $K>0$, one has $$\int d^{2n}\xi\, \frac{e^{-\xi^T K^{-1} \xi}}{\pi^n \sqrt{\det K}}\ D(\xi)\, \rho_G [V,s]\, D(\xi)^\dag\, =\, \rho_G[V+K, s]\, ,
\label{classical mixing}$$ again in the sense of weak convergence. This can be seen as an immediate consequence of Eq. .
Closedness and continuity results
---------------------------------
Let $\mathcal{H}_n$ be the Hilbert space associated with a finite number $n$ of harmonic oscillators, and let $\mathcal{T}(\mathcal{H}_n)$ be the set of trace class operators over $\mathcal{H}_n$. Observe that $\mathcal{T}(\mathcal{H}_n)$ becomes a Banach space once it is equipped with the trace norm $\|\cdot\|_1$, and its Banach dual is well known to be identifiable with the set of bounded operators, denoted by $\mathcal{B}(\mathcal{H}_n)$. Let us stress here that this is by no means a mathematical concept only. On the contrary, in quantum mechanics $\mathcal{B}(\mathcal{H}_n)$ has a physical interpretation as the set of all observables on the system.
In general, given a Banach space $E$ it is always possible to consider its (Banach) *dual*, i.e., the space $E^*$ of all continuous linear functionals $\varphi: E\rightarrow \mathds{C}$. Remember that a linear functional is continuous if and only if it is bounded, i.e., if and only if $\sup_{x\in E,\, \|x\|\leq 1} |\varphi(x)|$ is finite. The Banach dual can be used to induce another topology which is of interest, i.e., the *weak topology*, defined as the coarsest topology that makes all the functionals in $E^*$ continuous. As a matter of fact, the topologies on $E$ such that the corresponding continuous dual is $E^*$ are exactly those that are coarser than the norm topology (induced by the norm on $E$) and finer than the weak topology. This is a special case of the Mackey-Arens theorem [@aliprantis_2007 Thm. 8.14]. For a discussion of these concepts, see [@megginson_1998 Sec. 2.5] or [@rudin_1991 Sec. 3.11].
If $E$ is infinite-dimensional it can be shown that the weak topology is always different (in fact, as the name suggests, strictly coarser) than the norm topology. Hence, when it comes to taking closures (something we shall be concerned with) one has to specify which topology is used, as in general the weak closure will be larger than the norm closure. However, this is not always the case. Indeed, there is an important class of sets for which weak and norm closure always coincide, i.e., that of convex sets (see [@megginson_1998 Thm. 2.5.16] or [@rudin_1991 Sec. 3.12]). By the above discussion, it should be clear by now that all topologies on a Banach space $E$ such that the corresponding continuous dual coincides with the Banach dual $E^*$ have in fact the same closed convex sets.
The Banach space we care about here is $\mathcal{T}(\mathcal{H}_n)$, hence the norm topology is induced by the trace norm $\|\cdot\|_1$, and the weak topology is nothing but the the coarsest topology that makes all linear functionals $\operatorname{Tr}[A (\cdot)] : \mathcal{T}(\mathcal{H}_n)\rightarrow \mathds{C}$ continuous, for all $A\in \mathcal{B}(\mathcal{H}_n)$. Inside $\mathcal{T}(\mathcal{H}_n)$ lies the set of Gaussian states, denoted by $\mathcal{G}_n$, where $n$ is the number of modes. It is not completely trivial to show that $\mathcal{G}_n$ is norm-closed, and so we first show this result below.
\[lemma G closed\] The set of Gaussian states $\mathcal{G}_n\subset \mathcal{T}(\mathcal{H}_n)$ is closed with respect to the topology induced by the trace norm.
We have to show that given a sequence $\rho^{(k)}_G$ of Gaussian states with the property that $\lim_k \|\rho^{(k)}_G - \rho\|_1$ for some trace class operator $\rho$, we have that $\rho$ itself is a Gaussian state. In what follows, we denote by $V(k)$ and $s(k)$ the covariance matrix and displacement vector of $\rho_G^{(k)}$, respectively, so that $\rho_G^{(k)}=\rho_G[V(k), s(k)]$.
The first step in the proof consists in showing that $V(k)$ and $s(k)$ are bounded sequences, i.e., that there exists $M\in\mathds{R}$ such that $\|V(k)\|_\infty, |s(k)|_2 \leq M$ for all $k$ (where $\|\cdot\|_\infty$ is the operator norm, and $|\cdot|_2$ the Euclidean norm for vectors). In order to see why, write $$\begin{aligned}
&\braket{u| \rho_G[V(k), s(k)] |u} \\
&\qquad = \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T V(k) \xi - i s(k)^T\xi} \braket{u|D(\Omega \xi)|u} \\
&\qquad{\stackrel{\mathclap{\mbox{\text{\scriptsize (1)}}}}{=}} \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T V(k) \xi - i s(k)^T\xi} \braket{0| D(-u) D(\Omega \xi) D(u)|0} \\
&\qquad{\stackrel{\mathclap{\mbox{\text{\scriptsize (2)}}}}{=}} \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T V(k) \xi - i s(k)^T\xi} e^{-i u^T \xi} \braket{0| D(\Omega \xi) |0} \\
&\qquad{\stackrel{\mathclap{\mbox{\text{\scriptsize (3)}}}}{=}} \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T V(k) \xi - i s(k)^T\xi} e^{-i u^T \xi} e^{-\frac14 \xi^T \xi} \\
&\qquad= \int \frac{d^{2n}\xi}{(2\pi)^n}\, e^{-\frac14 \xi^T \left( V(k) + {\mathds{1}}\right) \xi - i \left(s(k) + u\right)^T \xi} \\
&\qquad{\stackrel{\mathclap{\mbox{\text{\scriptsize (4)}}}}{=}} \frac{2^n e^{- (s(k)+u)^T (V(k)+{\mathds{1}})^{-1} (s(k)+u)}}{\sqrt{\det\left( V(k)+{\mathds{1}}\right)}}\, .\end{aligned}$$ The justification of the above steps is as follows: (1) we used the definition of coherent states, Eq. ; (2) we applied Eq. twice; (3) we made use of Eq. ; (4) we performed the Gaussian integral. Now, we take the limit $k\rightarrow \infty$ on both sides of the equality $$\braket{u| \rho_G[V(k),s(k)] |u} = \frac{2^n e^{- (s(k)+u)^T (V(k)+{\mathds{1}})^{-1} (s(k)+u)}}{\sqrt{\det\left( V(k)+{\mathds{1}}\right)}}\, .
\label{G on coherent}$$
On the left-hand side we get $\braket{u|\rho|u}$ because of the properties of the convergence in norm. Let us now look at the right-hand side. Observe that $\det\left( V(k)+{\mathds{1}}\right)\geq \|V(k)\|_\infty+1$, and that the exponential term is at most $1$. If the sequence $V(k)$ were unbounded, then there would exist a subsequence $k_m$ on which $\|V(k_m)\|_\infty\rightarrow \infty$, which implies by the above equality that $\braket{u|\rho|u}=0$. Since this would happen for all $u\in\mathds{R}^{2n}$, we would deduce that $\braket{\psi|\rho|\psi}=0$ for all vectors $\ket{\psi}\in\mathcal{H}_n$, because coherent states are dense, and $\rho$ is a bounded (even trace class) operator. It is elementary to verify that this would imply that $\rho=0$ identically, a contradiction. Hence, we are led to conclude that $V(k)$ must be bounded, i.e., $V(k)\leq M{\mathds{1}}$ for some $M\in\mathds{R}$.
This implies immediately that $(V(k)+{\mathds{1}})^{-1}\geq (M+1)^{-1} {\mathds{1}}$, hence if the sequence $s(k)$ were unbounded, for every fixed $u$ we could find a subsequence $k_m$ on which $|s(k_m)+u|_2\rightarrow\infty$, which implies that $$e^{-(s(k_m)+u)^T (V(k_m) + {\mathds{1}})^{-1} (s(k_m)+u)}\leq e^{-\frac{1}{M+1} |s(k_m)+u|_2^2}\underset{m\rightarrow\infty}{\longrightarrow} 0\, .$$ Since the determinant appearing in Eq. is always at least $1$, we would deduce that the whole r.h.s. of Eq. tends to $0$ on that subsequence, hence that $\braket{u|\rho|u}=0$ for all $u\in\mathds{R}^{2n}$, again a contradiction.
This shows that $V(k)$ and $s(k)$ form bounded sequences. Since they live in finite-dimensional spaces, they will admit two simultaneously convergent subsequences $$\begin{aligned}
V(k_m) &\underset{m\rightarrow\infty}{\longrightarrow} V\, , \\
s(k_m) &\underset{m\rightarrow\infty}{\longrightarrow} s\, .\end{aligned}$$ Clearly, one still has $\lim_{m\rightarrow\infty} \|\rho_G[V(k_m), s(k_m)] - \rho\|_1=0$. Now, we use this to take the limit $m\rightarrow \infty$ on both sides of the equality $$\operatorname{Tr}[\rho_G[V(k_m), s(k_m)]\, D(\xi) ] = e^{-\frac14 \xi^T \Omega^T V(k_m) \Omega \xi + i s(k_m)^T \Omega \xi}\, ,$$ which is just a rewriting of Eq. (here, $\xi\in\mathds{R}^{2n}$ is fixed). On the left-hand side we have $$\lim_{m\rightarrow \infty} \operatorname{Tr}[\rho_G[V(k_m), s(k_m)]\, D(\xi) ] = \operatorname{Tr}[\rho D(\xi)] = \chi_\rho(\xi)$$ because the convergence of the sequence of states is in trace norm, and $D(\xi)$ is a bounded (even unitary) operator. On the right-hand side, by our hypotheses $$\lim_{m\rightarrow\infty} e^{-\frac14 \xi^T \Omega^T V(k_m) \Omega \xi + i s(k_m)^T \Omega \xi} = e^{-\frac14 \xi^T \Omega^T V \Omega \xi + i s^T \Omega \xi}\, .$$ The equality above then implies that $$\chi_\rho(\xi) = e^{-\frac14 \xi^T \Omega^T V \Omega \xi + i s^T \Omega \xi}\, ,$$ from which we see that the limit state $\rho$ has a Gaussian characteristic function, hence it is Gaussian.
There is another continuity result that we shall need in what follows. In a way, this can be considered as a strengthening of [@depalma_2015 Lemma 1].
\[lemma V to rho 1-continuous\] Consider a continuous variable system with $n$ degrees of freedom. The map $$\begin{array}{rcl}
\text{\emph{QCM}}_n \oplus \mathds{R}^{2n} & \longrightarrow & \mathcal{T}(\mathcal{H}_n) \\[1ex]
(V,s) & \longmapsto & \rho_G[V,s]\, ,
\end{array}
\label{V to rho}$$ which sends a pair $(V,s)$, where $V$ is a QCM and $s$ a real vector, is continuous with respect to the trace norm. Here, the topology on $$\text{\emph{QCM}}_n \oplus \mathds{R}^{2n} \subset \mathcal{M}_{2n}(\mathds{R}) \oplus \mathds{R}^{2n} \simeq \mathds{R}^{(2n)^2+2n}$$ is understood to be the standard one.
We have to show that whenever $\lim_{k\rightarrow \infty} V(k) = V$ and $\lim_{k\rightarrow \infty} s(k)=s$ one has also $\lim_{k\rightarrow\infty} \|\rho_G[V(k),s(k)]-\rho_G[V,s]\|_1=0$. At first glance we seem to have a problem here, as the trace distance of two Gaussian states is not a handy object when dealt with from the phase space perspective. However, we can exploit the Fuchs-van de Graaf’s inequality $\|\rho-\sigma\|_1\leq 2\sqrt{1-F(\rho,\sigma)^2}$ to upper bound the trace distance by means of a fidelity-based quantity. The fidelity between two Gaussian states happens to have an explicit expression in terms of their first and second moments [@banchi_2015 Eq. (9)-(14)]. One can verify by direct inspection that this is continuous with respect to the involved covariance matrices and displacement vectors, and of course it reduces to $1$ when the first and second moments of the first state coincide with those of the second state. Hence, $$\begin{aligned}
&\lim_{k\rightarrow\infty} \|\rho_G[V(k),s(k)]-\rho_G[V,s]\|_1 \\
&\qquad \leq \lim_{k\rightarrow\infty} 2\sqrt{1-F(\rho_G[V(k), s(k)],\rho_G[V,s])^2} \\
&\qquad= 2\sqrt{1- \left( \lim_{k\rightarrow\infty} F(\rho_G[V(k), s(k)],\rho_G[V,s]) \right)^2} \\
&\qquad= 2\sqrt{1- \left( F(\rho_G[V, s],\rho_G[V,s]) \right)^2} \\
&\qquad= 2\sqrt{1- \left( 1 \right)^2} \\
&\qquad= 0\, ,\end{aligned}$$ as claimed.
Gaussian resources {#app G resources}
==================
Free states {#free-states}
-----------
\[lemma F closed\] Let $\tau$ be a linear topology on $\mathcal{T}(\mathcal{H}_N)$ (the space of trace-class operators) such that the corresponding continuous dual is $\left(\tau, \mathcal{T}(\mathcal{H}_N)\right)' =\mathcal{B}(\mathcal{H}_N)$ (the space of bounded operators). If Postulates \[post convexity\] and \[post weak closed\] hold, then all sets of free states $\mathcal{F}({\boldsymbol{\lambda}})$ are closed with respect to $\tau$.
Since the weak topology on $\mathcal{T}(\mathcal{H}_N)$ is by definition the coarsest topology that makes all functionals $\operatorname{Tr}[A(\cdot)]:\mathcal{T}(\mathcal{H}_N)\rightarrow\mathds{C}$ continuous (where $A\in\mathcal{B}(\mathcal{H}_N)$ is generic), any topology $\tau$ that satisfies the hypothesis will be finer than the weak topology. Thus, it suffices to show that all sets $\mathcal{F}({\boldsymbol{\lambda}})$ are weakly closed. This follows since $\mathcal{F}({\boldsymbol{\lambda}})$ are norm-closed and convex by assumption, and weak closure and norm closure always coincide for convex sets by Mazur’s theorem (see e.g. [@megginson_1998 Thm. 2.5.16] or [@rudin_1991 Sec. 3.12]).
\[lemma FG closed\] When Postulate \[post weak closed\] holds, the set of Gaussian free states $\mathcal{F}_G({\boldsymbol{\lambda}})$ is norm-closed.
By definition $\mathcal{F}_G({\boldsymbol{\lambda}}) = \mathcal{F}({\boldsymbol{\lambda}}) \cap \mathcal{G}_N$. The set $\mathcal{F}({\boldsymbol{\lambda}})$ is norm-closed by Postulate \[post weak closed\], and the set $\mathcal{G}_N$ of all Gaussian states is also norm-closed by Lemma \[lemma G closed\]. Since the intersection of closed sets is closed, we conclude.
If Postulate \[post weak closed\] holds, then the set $\mathcal{V}({\boldsymbol{\lambda}})$ is topologically closed. If also Postulate \[post convexity\] holds, then $\mathcal{V}({\boldsymbol{\lambda}})$ becomes ‘upward closed’, in the sense that $V\in \mathcal{V}({\boldsymbol{\lambda}})$ and $W\geq V$ implies $W\in\mathcal{V}({\boldsymbol{\lambda}})$.
We first show that $\mathcal{V}({\boldsymbol{\lambda}})$ is topologically closed. By Lemma \[lemma V to rho 1-continuous\], we know that the map $\Gamma: \text{QCM}_N\rightarrow \mathcal{T}(\mathcal{H}_N)$ whose action is defined by $\Gamma(V) \coloneqq \rho_G[V,0]$ is continuous with respect to the trace norm. With this notation, the set $\mathcal{V}({\boldsymbol{\lambda}})$ can be rewritten as $$\mathcal{V}({\boldsymbol{\lambda}}) = \Gamma^{-1} \left( \mathcal{F}_G({\boldsymbol{\lambda}}) \right)\, .$$ Since $\mathcal{F}_G({\boldsymbol{\lambda}})$ is norm-closed by Lemma \[lemma FG closed\], and the preimages of closed sets via continuous maps are closed, we conclude that $\mathcal{V}({\boldsymbol{\lambda}})$ is closed as well.
We now move on to the second claim. Since we already showed that $\mathcal{V}({\boldsymbol{\lambda}})$ is topologically closed, it is enough to show that is *strictly* upward closed, i.e., that for all $V\in\mathcal{V}({\boldsymbol{\lambda}})$ and $W>V$ one has also $W\in\mathcal{V}({\boldsymbol{\lambda}})$. This is an easy consequence of Eq. . If we substitute there $K=W-V$, on the left-hand side we get a state in $\operatorname{cl}\left(\operatorname{co}\mathcal{F}_G({\boldsymbol{\lambda}})\right)$, the closed convex hull of the set of free Gaussian states. From the right-hand side we learn that this state is actually a Gaussian state, hence by Gaussian convexity of the set $\mathcal{F}_G({\boldsymbol{\lambda}})$ it must be also free. Finally, its covariance matrix is $V+K=W$, which leads us to conclude that $W\in\mathcal{V}({\boldsymbol{\lambda}})$.
\[prop convexity\] Assume that Postulates \[post tensor\], \[post partial trace\] and \[post weak closed\] hold. Moreover, let the set of Gaussian free states be invariant under local mixing with 50:50 beam splitters, i.e., assume that for any pair of states $\rho,\sigma\in \mathcal{F}_G({\boldsymbol{\lambda}})$ of a system with total number of modes $N$ one has $$\left( \bigotimes_{j=1}^N U(\pi/4)_{j,j} \right) (\rho\otimes \sigma) \left( \bigotimes_{j=1}^N U(\pi/4)_{j,j} \right)^\dag \in \mathcal{F}_G({\boldsymbol{\lambda}})\, ,$$ where $U(\pi/4)_{j,j}$ is the unitary that implements the action of a 50:50 beam splitter on the $j$-th mode of $\rho$ and the same mode of $\sigma$. Then the corresponding set of free covariance matrices $\mathcal{V}({\boldsymbol{\lambda}})$ is convex.
Since $\mathcal{V}({\boldsymbol{\lambda}})$ is topologically closed by Prop. \[prop V\], it is convex if and only if it is midpoint convex, meaning that $\frac12 (V+W)\in\mathcal{V}({\boldsymbol{\lambda}})$ whenever $V,W\in\mathcal{V}({\boldsymbol{\lambda}})$. Hence, let us show that $\mathcal{V}({\boldsymbol{\lambda}})$ is midpoint convex. Picking $V,W$ as above, construct the state $\rho_G[V,0]\otimes \rho_G[W,0]$, which is free by Postulate \[post tensor\], and whose covariance matrix is $V\oplus W = \left( \begin{smallmatrix} V & \\ & W \end{smallmatrix}\right)$. By hypothesis, mode-by-mode mixing with a 50:50 beam splitter yields another free state, whose covariance matrix will be $$\frac12 \begin{pmatrix} V+W & V-W \\ V-W & V +W \end{pmatrix} .$$ Tracing away one of the two spatially separated subsystems leaves the other in a state with covariance matrix $\frac12 (V+W)$. Such a state must be free by Postulate \[post partial trace\], hence we conclude that $\frac12 (V+W)\in\mathcal{V}({\boldsymbol{\lambda}})$, as claimed.
Quantification and distillation
-------------------------------
We remind the reader that in general a Gaussian completely positive map $\Lambda$ from $A$ to $B$ acts on covariance matrices as follows [@giedke_2002; @fiurasek_2002]: $$\Lambda: V_{A}\longmapsto \big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma)\, .
\label{G map}$$ Here, $\Gamma_{AB}$ is the quantum covariance matrix associated with the Choi state of the map, and $\Sigma$ is the matrix that reverses the signs of all the momenta of the system on which it is acting, i.e., $$\Sigma \coloneqq \begin{pmatrix} {\mathds{1}}& \\ & -{\mathds{1}}\end{pmatrix}
\label{Sigma}$$ according to the block decomposition of Eq. . The Schur complement of a $2\times 2$ block matrix $M = \left( \begin{smallmatrix} P & X \\ Y & Q \end{smallmatrix} \right)$ with respect to one of its square invertible blocks is given by $$M/P \coloneqq Q - YP^{-1} X\, .
\label{Schur}$$ It is elementary to verify that the above quantity behaves well under scalar multiplication, in the sense that $(\lambda M) / (\lambda P) = \lambda (M/P)$ for all scalars $\lambda\neq 0$. Furthermore, it is known that the Schur complement admits the following variational representation: $$M/P = \max \big\{ R:\ M \geq 0 \oplus R \big\} \, ,
\label{Schur var}$$ the ordering of the set on the right-hand side being the positive semidefinite (aka Löwner) ordering. From Eq. it follows in particular that $M/P$ is monotonically non-decreasing in $M>0$. For more details on the properties of Schur complements we refer the reader to the excellent monograph [@zhang_2005]. A straightforward consequence of the above discussion is the following result.
\[Gamma free\] If $\Gamma_{AB}$ represents a Gaussian free operation $\Lambda\in \mathcal{O}_G({\boldsymbol{\lambda}}_A\rightarrow{\boldsymbol{\lambda}}_B)$, then $$\big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma)\, \in\, \mathcal{V}({\boldsymbol{\lambda}}_B)\qquad \forall\ V_A\in\mathcal{V}({\boldsymbol{\lambda}}_A)\, .
\label{Gamma free eq1}$$ Equivalently, $$\forall\ V_A\in\mathcal{V}({\boldsymbol{\lambda}}_A)\ \ \exists\ W_B\in\mathcal{V}({\boldsymbol{\lambda}}_B) : \quad \Gamma_{AB} \geq (- \Sigma V_A \Sigma) \oplus W_B\, .
\label{Gamma free eq2}$$
The first claim is a direct reformulation of the definition of resource non-generating operations, obtained via the explicit action of a Gaussian completely positive map as given by Eq. . As for the second, let us observe that the inequality $\Gamma_{AB} \geq (- \Sigma V_A \Sigma) \oplus W_B$ implies, by Eq. , that $\big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma)\geq W_B$. Since the right-hand side is the covariance matrix of a free state by hypothesis, and Prop. \[prop V\] holds, we deduce that the left-hand side is a free covariance matrix as well. The converse inequality is proved similarly, by realizing that Eq. implies that $$\Gamma_{AB} + \Sigma V_A \Sigma \geq 0_A \oplus \big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma)\eqqcolon 0_A \oplus W_B\, ,$$ which leads immediately to $\Gamma_{AB}\geq (-\Sigma V_A \Sigma) \oplus W_B$, as claimed.
We now come to the discussion of the properties of the $\kappa_{{\mathcal{F}}}$ function defined in Eq. . We first state some elementary facts.
\[elementary properties kappa\] The set $T_{{\mathcal{F}}}(V) \coloneqq {\left\{\left.}
\newcommand{\rset}{\right\}}t \geq 1 {\;\rule{0pt}{9.5pt}\right|\;}t V\in \mathcal{V}({\boldsymbol{\lambda}}) \rset$ is non-empty and topologically closed for all $V\in\text{QCM}_{N}$ as long as $\mathcal{V}({\boldsymbol{\lambda}})$ is non-empty.
We first show that $T_{{\mathcal{F}}}(V)\neq \emptyset$ for all $V\in\text{QCM}_{N}$. Picking $W\in \mathcal{V}({\boldsymbol{\lambda}})\neq \emptyset$, it is easy to see that $$\|W\|_\infty \|V^{-1}\|_\infty\, V \geq W\, ,
\label{upper bound eigenvalues}$$ where $\|\cdot\|_\infty$ denotes the operator norm. Observe that quantum covariance matrices are always strictly positive, hence $V^{-1}$ exists. We then write $$\begin{aligned}
\|W\|_\infty \|V^{-1}\|_\infty\, V &\geq \|W\|_\infty \|V^{-1}\|_\infty\, \lambda_{\min}(V) {\mathds{1}}\\
&= \|W\|_\infty \|V^{-1}\|_\infty\, \|V^{-1}\|_\infty^{-1} {\mathds{1}}\\
&= \|W\|_\infty {\mathds{1}}\\
&\geq W\, .\end{aligned}$$ By the upward closedness of ${\mathcal{V}}_{\mathcal{F}}({\boldsymbol{\lambda}})$ (Prop. \[prop V\]), one deduces immediately that $\max\left\{1, \|W\|_\infty \|V^{-1}\|_\infty \right\} \in T_{{\mathcal{F}}}(V)$, showing that the set is non-empty. To show that it is also topologically closed, just observe that $$T_{{\mathcal{F}}}(V) = \Big([1,\infty)\cdot V\Big)\cap \mathcal{V}({\boldsymbol{\lambda}})\, .$$ The left-hand side of the above identity is the intersection of two closed sets, thanks to Prop. \[prop V\], hence it is itself closed.
The following is a refinement of Prop. \[prop:kappa\] from the main text.
\[prop:kappa-reformul\] The function $\kappa_{{\mathcal{F}}}(\cdot)$ defined by Eq. is:
(a) finite and well-defined for all $V\in\text{QCM}_N$;
(b) faithful, in the sense that $\kappa_{{\mathcal{F}}}(V)=1$ if and only if $V\in\mathcal{V}({\boldsymbol{\lambda}})$;
(c) such that $\kappa_{{\mathcal{F}}}(s V)\geq s^{-1}\kappa_{{\mathcal{F}}}(V)$ for all $s\geq 1$;
(d) monotonically non-increasing under ${{\mathcal{O}}_G({\boldsymbol{\lambda}}\rightarrow{\boldsymbol{\mu}})}$; and
(e) continuous.
Claim (a) follows directly from Lemma \[elementary properties kappa\], while (b) is obvious from the definition. As for (c), one can distinguish two cases: if $s V\in \mathcal{V}({\boldsymbol{\lambda}})$, then on the one hand $s \geq \kappa_{{\mathcal{F}}}(V)$, while on the other hand $\kappa_{{\mathcal{F}}}(s V)=1\geq s^{-1}\kappa_{{\mathcal{F}}}(V)$; if $s V\notin \mathcal{V}({\boldsymbol{\lambda}})$, then $$\begin{aligned}
\kappa_{{\mathcal{F}}}(s V) &= \min {\left\{\left.}
\newcommand{\rset}{\right\}}t\geq 1 {\;\rule{0pt}{9.5pt}\right|\;}t s V\in \mathcal{V}({\boldsymbol{\lambda}}) \rset \\
&= s^{-1} \min{\left\{\left.}
\newcommand{\rset}{\right\}}t'\geq s {\;\rule{0pt}{9.5pt}\right|\;}t' V\in \mathcal{V}({\boldsymbol{\lambda}}) \rset \\
&= s^{-1} \min{\left\{\left.}
\newcommand{\rset}{\right\}}t'\geq 1{\;\rule{0pt}{9.5pt}\right|\;}t' V\in \mathcal{V}({\boldsymbol{\lambda}}) \rset \\
&= s^{-1} \kappa_{{\mathcal{F}}}(V)\, .\end{aligned}$$
We now turn to the proof of (d), i.e., the monotonicity of $\kappa_{\mathcal{F}}$ under Gaussian free operations. Call $\xi\coloneqq \kappa_{\mathcal{F}}(V)$. Then, by virtue of Eq. , all we have to show is that $\kappa_{\mathcal{F}}\left( ( \Gamma_{AB} + \Sigma V_A \Sigma ) \big/ (\Gamma_A + \Sigma V_A \Sigma) \right)\leq \xi$, for all free Gaussian operations represented by covariance matrices $\Gamma_{AB}$ as in Lemma \[Gamma free\]. This amounts to proving that $\xi \big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma) \in \mathcal{V}({\boldsymbol{\lambda}}_B)$. We write $$\begin{aligned}
&\xi \big( \Gamma_{AB} + \Sigma V_A \Sigma \big) \big/ (\Gamma_A + \Sigma V_A \Sigma) \\
&\qquad {\stackrel{\mathclap{\mbox{\text{\scriptsize (1)}}}}{=}} \big( \xi \Gamma_{AB} + \xi \Sigma V_A \Sigma \big) \big/ (\xi \Gamma_A + \xi \Sigma V_A \Sigma) \\
&\qquad{\stackrel{\mathclap{\mbox{\text{\scriptsize (2)}}}}{\geq}} \big( \Gamma_{AB} + \xi \Sigma V_A \Sigma \big) \big/ ( \Gamma_A + \xi \Sigma V_A \Sigma) \\
&\qquad\overset{(3)}{\in} \mathcal{V}({\boldsymbol{\lambda}}_B)\, .\end{aligned}$$ The justification of the above steps is as follows: (1) comes from homogeneity; (2) uses the monotonicity of the Schur complement, together with the observation that since $\xi=\kappa_{\mathcal{F}}(V)\geq 1$ one has $\xi \Gamma_{AB}\geq \Gamma_{AB}$; (3) is an elementary consequence of Eq. applied to the free covariance matrix $\xi V$.
A useful observation that follows from the just established property (d) is that $\kappa_{\mathcal{F}}(\cdot)$ is also monotonically non-increasing with respect to the positive semidefinite ordering. In fact, adding some positive semidefinite matrix to the input never creates a resource state out of a free state, i.e., it is always a free operation.
What is left to show is claim (e). We will break the proof into two steps: first, we will show that $\limsup_{\Delta \rightarrow 0} \kappa_{{\mathcal{F}}} (V+\Delta) \leq \kappa_{{\mathcal{F}}}(V)$ for all $V>0$ (upper semicontinuity); second, we will complement this bound by means of the inequality $\liminf_{\Delta\rightarrow 0} \kappa_{{\mathcal{F}}} (V+\Delta) \geq \kappa_{{\mathcal{F}}}(V)$ (lower semicontinuity). Clearly, the two statements together imply that $\lim_{\Delta\rightarrow 0} \kappa_{{\mathcal{F}}} (V+\Delta) = \kappa_{{\mathcal{F}}}(V)$, which is claim (e). Now, the upper semicontinuity rests on the upward closedness of $\mathcal{V}({\boldsymbol{\lambda}})$. For a sufficiently small perturbation $\Delta$, write $$\begin{aligned}
V+\Delta &\geq V - \|\Delta\|_{\infty} {\mathds{1}}\\
&\geq V - \|\Delta\|_{\infty} \|V^{-1}\|_{\infty} V \\
&= \left(1- \|\Delta\|_{\infty} \|V^{-1}\|_{\infty}\right) V\, ,\end{aligned}$$ we deduce that $$\frac{V+\Delta}{1-\|\Delta\|_{\infty} \|V^{-1}\|_{\infty}} \geq V\, .$$ Using properties (c) and (d), this in turn implies that $$\begin{aligned}
\left( 1-\|\Delta\|_{\infty} \|V^{-1}\|_{\infty}\right) \kappa_{{\mathcal{F}}} (V+\Delta) &\leq \kappa_{{\mathcal{F}}} \left( \frac{V+\Delta}{1-\|\Delta\|_{\infty} \|V^{-1}\|_{\infty}} \right) \\
&\leq \kappa_{{\mathcal{F}}}(V)\, ,\end{aligned}$$ from which it follows that $$\kappa_{{\mathcal{F}}}(V+\Delta) \leq \frac{\kappa_{{\mathcal{F}}}(V)}{1-\|\Delta\|_\infty \|V^{-1}\|_\infty}\, .$$ In particular, $$\limsup_{\Delta\rightarrow 0} \kappa_{{\mathcal{F}}} (V+\Delta) \leq \lim_{\Delta\rightarrow 0} \frac{\kappa_{{\mathcal{F}}}(V)}{1-\|\Delta\|_{\infty} \|V^{-1}\|_{\infty}} = \kappa_{\mathcal{F}}(V) \, ,$$ which proves upper semicontinuity. The lower semicontinuity comes instead from the topological closedness of the set $\mathcal{V}({\boldsymbol{\lambda}})$, as established by Prop. \[prop V\]. To see why, consider a $V>0$ and a sequence of sufficiently small perturbation matrices $(\Delta_{n})_{n\in\mathds{N}}$ such that $\lim_{n\rightarrow\infty} \Delta_{n} = 0$. Since $\kappa_{{\mathcal{F}}} (V+ \Delta_{n})\, (V+\Delta_n)\in \mathcal{V}({\boldsymbol{\lambda}})$ for all $n$, taking a subsequence $(n_{k})_{k\in\mathds{N}}$ such that $\lim_{k\rightarrow \infty} \kappa_{{\mathcal{F}}} (V+\Delta_{n_{k}}) = \liminf_{n\rightarrow \infty} \kappa_{{\mathcal{F}}} (V+\Delta_{n})$, we obtain by closedness that $$\begin{aligned}
\mathcal{V}({\boldsymbol{\lambda}}) &\ni \lim_{k\rightarrow\infty} \left( \kappa_{{\mathcal{F}}} (V+\Delta_{n_{k}})\, (V+\Delta_{n_{k}})\right) \\
&= \left( \lim_{k\rightarrow\infty} \kappa_{\mathcal{F}}(V+\Delta_{n_k}) \right) V \\
&= \left( \liminf_{n\rightarrow \infty} \kappa_{{\mathcal{F}}} (V+\Delta_{n}) \right) V\, ,\end{aligned}$$ which implies in turn that $\kappa_{{\mathcal{F}}} (V) \leq \liminf_{n\rightarrow \infty} \kappa_{{\mathcal{F}}} (V+\Delta_{n})$, proving lower semicontinuity and hence claim (e).
In fact, we have shown that $$\kappa_{{\mathcal{F}}} (V) \leq \max\left\{ 1,\,\|V^{-1}\|_\infty \min_{W\in \mathcal{V}({\boldsymbol{\lambda}})} \|W\|_\infty\right\} .$$ for all $V\in\text{QCM}_{N}$.
An inspection of the above proof of the monotonicity result (Prop. \[prop:kappa\](d)) reveals that the only property of the Choi covariance matrix $\Gamma_{AB}$ we have made use of is its positive semidefiniteness. This observation shows that $\kappa_{{\mathcal{F}}}$ is monotonic under any operation of the form specified by Eq. with $\Gamma_{AB}\geq 0$.
The fundamental property of the $\kappa_{\mathcal{F}}$ measure we employ here concerns its behaviour when multiple copies of the same state are considered.
\[kappa multi-copy\] For all ${\boldsymbol{\lambda}},{\boldsymbol{\mu}}$, consider the $\kappa_{\mathcal{F}}$ functions identified via Eq. by the sets of free covariance matrices $\mathcal{V}({\boldsymbol{\lambda}})$, $\mathcal{V}({\boldsymbol{\mu}})$, and $\mathcal{V}({\boldsymbol{\lambda}}\oplus{\boldsymbol{\mu}})$. Then for all $V\in\text{\emph{QCM}}_N$ and $W\in\text{\emph{QCM}}_M$, where $N=\sum_j n_j$ and $M=\sum_j m_j$, it holds that $$\kappa_{\mathcal{F}}\left(V \oplus W\right) = \max\{ \kappa_{\mathcal{F}}(V), \kappa_{\mathcal{F}}(W)\}\, .$$
Call $\eta\coloneqq \max\{ \kappa_{\mathcal{F}}(V), \kappa_{\mathcal{F}}(W)\}$. From Prop. \[prop V\] and from the inequalities $\eta\geq \kappa_{\mathcal{F}}(V),\kappa_{\mathcal{F}}(W)$ we deduce immediately that $\eta V\in\mathcal{V}({\boldsymbol{\lambda}})$, $\eta W\in \mathcal{V}({\boldsymbol{\mu}})$. By Postulate \[post tensor\], we deduce that $$\eta \left( V\oplus W \right) = (\eta V)\oplus (\eta W) \in \mathcal{V}({\boldsymbol{\lambda}})\oplus \mathcal{V}({\boldsymbol{\mu}})\subseteq \mathcal{V}({\boldsymbol{\lambda}}\oplus {\boldsymbol{\mu}})\, ,$$ which implies by definition that $\kappa_{\mathcal{F}}(V\oplus W)\leq \eta = \max\{ \kappa_{\mathcal{F}}(V)_{\mathcal{F}}, \kappa_{\mathcal{F}}(W)\}$. As for the opposite inequality, call $\zeta \coloneqq \kappa_{\mathcal{F}}\left(V \oplus W\right)$. Then $\zeta (V\oplus W)\in\mathcal{V}({\boldsymbol{\lambda}}\oplus {\boldsymbol{\mu}})$, and by Postulate \[post partial trace\] we can generate a free state of the first system by tracing away the second. At the level of covariance matrices this amounts to performing a local projection, for which we adopt the same notation as in the characterization of Postulate \[post partial trace\] in the manuscript. We then obtain $$\zeta V = \Pi \left( \zeta (V\oplus W) \right) \Pi^T \in \Pi\, \mathcal{V}({\boldsymbol{\lambda}}\oplus{\boldsymbol{\mu}})\, \Pi^T \subseteq \mathcal{V}({\boldsymbol{\lambda}})\, .$$ This shows that $\kappa_{\mathcal{F}}(V) \leq \zeta = \kappa_{\mathcal{F}}(V\oplus W)$. Repeating the reasoning with $W$ instead of $V$ we get also $\kappa_{\mathcal{F}}(W)\leq \kappa_{\mathcal{F}}(V\oplus W)$, and putting the two inequalities together we have $\max\{ \kappa_{\mathcal{F}}(V), \kappa_{\mathcal{F}}(W)\} \leq \kappa_{\mathcal{F}}(V\oplus W)$, which completes the proof.
Consider an arbitrary Gaussian resource theory satisfying Postulates \[post D invariance\]-\[post weak closed\] and two covariance matrices $V,W\in \text{\emph{QCM}}_N$. If $\kappa_{\mathcal{F}}(W) > \kappa_{\mathcal{F}}(V)$, then it is impossible to find a sequence $(W_{n})_{n\in\mathds{N}}\subset\text{\emph{QCM}}_{N}$ such that $\lim_{n\rightarrow\infty} W_{n}=W$ and the transformations $V^{\oplus n}\rightarrow W_{n}$ are possible with Gaussian resource non-generating operations for all $n$.
If said transformation were possible, by combining Prop. \[prop:kappa\] and Lemma \[kappa multi-copy\] one would obtain $$\kappa_{\mathcal{F}}(V) = \kappa_{\mathcal{F}}\left( V^{\oplus n} \right) \geq \kappa_{\mathcal{F}}(W_n)\, .$$ Since $\kappa_{\mathcal{F}}$ is continuous, one has $\lim_{n\rightarrow\infty} \kappa_{\mathcal{F}}(W_n) = W$ and hence also $\kappa_{\mathcal{F}}(V)\geq \kappa_{\mathcal{F}}(W)$, which is a contradiction.
The remark after Prop. \[prop:kappa\] has an important consequence here. Namely, we now see that the above no-go result still holds if one allows as free operations all resource non-generating maps of the form given by Eq. with $\Gamma_{AB}\geq 0$. Remember that a map acting on the second moments as in Eq. is a valid physical transformation (completely positive map) if and only if $\Gamma_{AB}$ is a quantum covariance matrix, i.e., if and only if $\Gamma_{AB}\geq i\Omega_{AB}$. Since this is a strictly stronger constraint than simply requiring that $\Gamma_{AB}\geq 0$, this observation extends the validity of Thm. \[thm:no-go\] even further. For instance, its domain of applicability now includes the maps considered in [@depalma_2015 Eq. (24)-(26)], since the corresponding Choi covariance matrices can be shown to be positive semidefinite provided [@depalma_2015 Eq. (27)] is obeyed. However, as some of these maps will be unphysical, the extension discussed here may be regarded mainly as a mathematical curiosity.
Semidefinite programming representation of Gaussian resources {#app SDP}
=============================================================
Quantum entanglement
--------------------
Recall that the characterization of the set of separable states $\rho_G[V_{AB},s] \in {\mathcal{S}}_{A|B}$ can be simplified to [@lami_2016] $$\label{eq:sep_simplified}
\rho_G[V_{AB},s] \in {\mathcal{S}}_{A|B} \iff V_{AB} \geq \gamma_A \oplus i\Omega_B,$$ which gives the following semidefinite representation of the quantifier $\kappa_{\mathcal{S}}$: $$\begin{aligned}
\hspace{-5em}\kappa_{\mathcal{S}}(V_{AB}) = \,& \underset{\lambda, \gamma_{A}}{\text{minimize}}
& & \lambda\\
& \text{subject to} & & \lambda \, V_{AB} \geq \gamma_A \oplus i \Omega_B\\
&&& \gamma_A = \gamma_A^T\\
&&& \gamma_A \geq i\Omega_A\\
&&& \lambda \geq 1
\end{aligned}$$ where one can equivalently consider the subsystem $B$ instead. The Lagrange dual of $\upsilon_{\mathcal{S}}$ can be obtained as $$\begin{aligned}
\upsilon_{\mathcal{S}}(V_{AB}) = \,& \underset{W, X}{\text{minimize}}
& & {\left\langle}W, V_{AB}{\right\rangle}\\
& \text{subject to} & & {\left\langle}W_{22}, i\Omega_B {\right\rangle}+ {\left\langle}X, i\Omega_A {\right\rangle}= 1\\
&&& \operatorname{Re}(W_{11}) = \operatorname{Re}(X)\\
&&& W, X \geq 0\\
\end{aligned}$$ where $W = \left(\begin{smallmatrix}W_{11} & W_{12}\\ W_{12}^\dagger & W_{22}\end{smallmatrix}\right)$ and we use the Hilbert-Schmidt inner product ${\left\langle}X,Y{\right\rangle}= \operatorname{Tr}(XY)$. With respect to the dual problem in Ref. [@hyllus_2006-1] which requires an optimization over the spaces of Hermitian matrices ${\mathbb{H}}_{2n} \oplus {\mathbb{H}}_{2n}$, using the simplified the condition for separability in Eq. reduces the optimization space to ${\mathbb{H}}_{2n}\oplus{\mathbb{H}}_{2 n_A}$.
To see that we were justified in claiming that the optimal value of $\upsilon_{\mathcal{S}}$ is equal to the optimal value of the dual, we will show that strong duality holds. Take $W^\star\oplus X^\star = \mathbbm{1}_{2n+2n_A} + \frac{i}{2n} \Omega\oplus\Omega_A$, and notice that $W^\star \oplus X^\star > 0$ since it is Hermitian and all of its eigenvalues are given by $\frac{2n\pm 1}{2n}>0$, and $\operatorname{Tr}\left(W_{22}^\star i \Omega_B \right) + \operatorname{Tr}\left(X i \Omega_A\right) = 1$. This means that $W^\star$ and $X^\star$ form a strictly feasible solution to the dual problem, and so Slater’s condition is satisfied and strong duality holds [@boyd_2004].
Steering
--------
The primal problem corresponding to the quantifier of $A\to B$ steerability $\kappa_{\mathcal{T}}$ is then given by $$\begin{aligned}
\hspace{-5em}\kappa_{\mathcal{T}}(V_{AB}) = \,& \underset{\lambda\geq 1}{\text{minimize}}
& & \lambda\\
& \text{subject to} & & \lambda \, V_{AB} \geq 0_A \oplus i \Omega_B.
\end{aligned}$$ An important property of the Schur complement is that, given a Hermitian matrix $M = \left( \begin{smallmatrix} P & X \\ X^\dagger & Q \end{smallmatrix} \right)$ such that $P > 0$, we have $M \geq 0 \iff M/P \geq 0$ [@zhang_2005]. Now, since $V_A > 0$, we can equivalently write $$\begin{aligned}\label{eq:steering_dual}
\hspace{-5em}\kappa_{\mathcal{T}}(V_{AB}) = \,& \underset{\lambda \geq 1}{\text{minimize}}
& & \lambda\\
& \text{subject to} & & \lambda \, V_{AB} / V_A \geq i \Omega_B.
\end{aligned}$$ The corresponding inverse dual is given as $$\begin{aligned}
\hspace{-31em}{\upsilon_{\mathcal{T}}}(V_{AB}) = \,& \underset{W}{\text{minimize}}
& & {\left\langle}W, V_{AB} {\right\rangle}\\
& \text{subject to} & & {\left\langle}W_{22}, i\Omega_B {\right\rangle}= 1\\
&&& W \geq 0\\
\hphantom{\hspace{-31em}{\upsilon_{\mathcal{T}}}(V_{AB})} = \,& \underset{W}{\text{minimize}}
& & {\left\langle}W, V_{AB} / V_A {\right\rangle}\\
& \text{subject to} & & {\left\langle}W, i\Omega_B {\right\rangle}= 1\\
&&& W \geq 0.
\end{aligned}$$ Taking $W^\star = \mathbbm{1}_{2n} + \frac{i}{2n} \Omega$, we have that $W^\star > 0$ since it is Hermitian and its eigenvalues are given by $\frac{2n\pm 1}{2n}>0$, and $\operatorname{Tr}\left(W^\star i \Omega\right) = 1$. This means that $W^\star$ is a strictly feasible solution to the latter dual problem, so Slater’s condition is satisfied and strong duality holds.
In fact, Eq. \[eq:steering\_dual\] suggests an interesting alternative characterization of this quantifier in terms of a symplectic eigenvalue problem. To see this, consider first the following result (see also [@bhatia_2015]).
The smallest symplectic eigenvalue $\nu_{\min}(V)$ of any $V > 0$ can be expressed as $$\begin{aligned}\label{eq:smin}
\nu_{\min}(V) &= \max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda \geq 0 {\;\rule{0pt}{9.5pt}\right|\;}V \geq i\lambda\Omega \rset\\
&= \min {\left\{\left.}
\newcommand{\rset}{\right\}}{\left\langle}W, V {\right\rangle}{\;\rule{0pt}{9.5pt}\right|\;}{\left\langle}W, i\Omega {\right\rangle}= 1 \rset.
\end{aligned}$$
Recall that a matrix $S$ is called symplectic if $S \Omega S^T = \Omega$. By Williamson’s theorem [@williamson_1936; @simon_1999], there exists a symplectic matrix $S$ such that $S V S^T = D \oplus D$ with $D=\operatorname{diag}\left(\nu_1(V),\ldots,\nu_n(V)\right) > 0$ being a diagonal matrix of the symplectic eigenvalues of $V$. We then have $$\begin{aligned}
\max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda {\;\rule{0pt}{9.5pt}\right|\;}V \geq i \lambda \Omega \rset &{\stackrel{\mathclap{\mbox{\text{\scriptsize (1)}}}}{=}} \max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda {\;\rule{0pt}{9.5pt}\right|\;}SVS^T \geq i \lambda S \Omega S^T \rset\\
&= \max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda {\;\rule{0pt}{9.5pt}\right|\;}D \oplus D \geq i \lambda \Omega \rset\\
&{\stackrel{\mathclap{\mbox{\text{\scriptsize (2)}}}}{=}} \max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda {\;\rule{0pt}{9.5pt}\right|\;}D - \lambda^2 D^{-1} \geq 0\rset\\
&{\stackrel{\mathclap{\mbox{\text{\scriptsize (3)}}}}{=}} \max {\left\{\left.}
\newcommand{\rset}{\right\}}\lambda {\;\rule{0pt}{9.5pt}\right|\;}\nu_j(V)^2 - \lambda^2 \geq 0 \; \forall j \rset\\
&= \nu_{\min}(V)
\end{aligned}$$ where (1) follows since any symplectic matrix is non-singular, (2) follows from the Schur complement condition for positive semidefiniteness, and (3) follows since both $D$ and $D^{-1}$ are diagonal with $\nu_j(V) > 0 \;\forall j$. The second line of Eq. then follows by strong Lagrange duality.
This leads to the following simple representation: $$\begin{aligned}
\upsilon_{\mathcal{T}}(V_{AB}) = \nu_{\min}\left(V_{AB} / V_A\right).
\end{aligned}$$ The quantifier can thus be related to a commonly used measure, the Gaussian $A\to B$ steerability [@kogias_2015] $N_{\mathcal{T}}(V_{AB}) \coloneqq -\sum_k \log \min\{1, \nu_k(V_{AB} / V_A) \}$. In particular, in the case of a bipartite system where $n_B = 1$, $V_{AB}/V_A$ has only one symplectic eigenvalue and therefore we have $N_{\mathcal{T}}(V_{AB}) = \log \kappa_{\mathcal{T}}(V_{AB})$.
[^1]: Note, however, that one may define also non-convex resources — notably, we obtain one such resource theory by letting the set of all Gaussian states to be free — but we will not study them here.
|
---
abstract: 'We consider the problem of portfolio optimization in a simple incomplete market and under a general utility function. By working with the associated Hamilton-Jacobi-Bellman partial differential equation (HJB PDE), we obtain a closed-form formula for a trading strategy which approximates the optimal trading strategy when the time horizon is small. This strategy is generated by a first order approximation to the value function. The approximate value function is obtained by constructing classical sub- and super-solutions to the HJB PDE using a formal expansion in powers of horizon time. Martingale inequalities are used to sandwich the true value function between the constructed sub- and super-solutions. A rigorous proof of the accuracy of the approximation formulas is given. We end with a heuristic scheme for extending our small-time approximating formulas to approximating formulas in a finite time horizon.'
address:
- |
Dept. Mathematics,\
Wayne State University,\
Detroit, MI 48202
- |
Dept. Mathematics,\
Wayne State University,\
Detroit, MI 48202\
\
\
author:
- '[^1]'
- '[^2]'
title: Asymptotic approximation of optimal portfolio for small time horizons
---
Introduction
============
In this paper, we study the problem of portfolio optimization when the time horizon is small. For simplicity, we consider a financial market with two assets, one risky and one risk-free. Given a pair $(t,x) \in [0,T]\times(0,\infty)$ of initial time $t$ and initial wealth $x$, an investor wishes to invest in such a way as to maximize her expected utility of wealth at time $T$ given today’s information. Specifically, if $U_{T}(x)$ is a function modeling the investor’s utility of wealth at the terminal time $T$, then the investor wishes to choose a portfolio $\pi$ which maximizes $E[U_{T} | \Ft]$ ($\Ft$ being the sigma-algebra that informally represents information up to time $t$).
Under Markovian assumptions on the price process of the risky asset, this optimization problem can be studied via the associated HJB equation, as, for example, in [@Lorig2016LSV; @Merton69Lifetime; @Nadtochiy13ApproxOIP; @Zariphopoulou01Val]. This portfolio optimization problem was first studied in a continuous time setting by Merton [@Merton69Lifetime; @Merton71Optimum] in a complete market. The utility maximization problem can also be studied using duality arguments as in [@CvitanicKaratzas92ConvexDuality; @KaratzasEtAl91MartingaleAndDuality; @KramkovSchachermayer99AsymptoticElasticity; @Schachermayer01NegativeWealth].
Portfolio optimization has also been studied under the assumption of an infinite time horizon, for example in [@Merton69Lifetime; @Merton71Optimum; @Pang04InfiniteTimePower; @Pang06InfiniteTimeLog]. In [@Pang04InfiniteTimePower; @Pang06InfiniteTimeLog], the author studies the problem of optimal investment and consumption in an infinite time horizon assuming a model with stochastic interest rate and where risky asset price is a geometric Brownian motion.
Tehranchi [@Tehranchi04ExpInc] studied the problem under the assumption of an incomplete market, where the market is driven by two Brownian motions and the asset price is not necessarily Markovian. The absence of the Markovian structure precludes the use of the dynamic programming principle. By proving Hölder-type inequalities for functionals of correlated Brownian motions, Tehranchi [@Tehranchi04ExpInc] was able to study the portfolio optimization problem when the utility function is a product of a function of the wealth and a function of a correlated stochastic factor. Explicit formulas were obtained in [@Tehranchi04ExpInc] when the function of wealth is an exponential function, a logarithmic function, and a power function.
In an incomplete market, few explicit formulas for the optimal portfolio exist in the literature and attempts have been made to obtain approximating formulas. We mention some of the work in this direction where the risky asset price model is Markovian and has correlated stochastic factors. In [@Zariphopoulou01Val], the utility function is assumed to be of Constant Relative Risk Aversion (CRRA) type, i.e., a product of a power function in wealth and a function depending on the stochastic factor. Under this assumption, Zariphopoulou in [@Zariphopoulou01Val] is able to obtain the value function in terms of the solution to a linear parabolic PDE. The results in [@Zariphopoulou01Val] have proved useful for computing explicit formulas for specific examples. In [@Lorig2016LSV], Lorig and Sircar consider the problem of portfolio optimization in finite horizon assuming a local stochastic volatility model for a risky asset. They use a Taylor series expansion of the model coefficients to obtain approximating formulas for the value function and optimal portfolio. While approximating formulas are obtained for general utility functions, accuracy of the approximation is established only in the case of power utilities. Fouque et al., in [@Fouque2015Portfolio], assume a model with multiscale stochastic factors and by asymptotic analysis obtain approximating formulas for the optimal portfolio.
In [@Fouque2015Portfolio; @Lorig2016LSV], well-posedness of the associated HJB equation is not established and the authors work under the assumption that the value function is the classical solution of the HJB equation with a sufficient degree of regularity. In [@Nadtochiy13ApproxOIP], as well as in our paper, no such assumption is made. In [@Nadtochiy13ApproxOIP], Nadtochiy and Zariphopoulou state that their model is the “simplest and most direct extension" [@Nadtochiy13ApproxOIP] to an incomplete market of the model introduced by Merton in [@Merton69Lifetime; @Merton71Optimum]. In this model, the utility function depends only on wealth. Instead of working directly with the associated HJB equation, the authors work with the marginal HJB equation, which they prove has a unique viscosity solution (see [@Crandall92UGvisc] for more information on viscosity solutions). Without assuming the value function satisfies the HJB equation, Nadtochiy and Zariphopoulou, in [@Nadtochiy13ApproxOIP], prove that the integral of the viscosity solution of the marginal HJB equation is indeed the value function. In addition, the authors derive approximations to the optimal portfolio which they term “$\epsilon$-optimal portfolios" [@Nadtochiy13ApproxOIP].
In this paper, we find a closed-form formula for an approximation to the optimal portfolio in a small time horizon under a stochastic volatility model for the risky asset price. As the well-posedness of the associated HJB equation is not established, we do not assume the value function is a classical solution of the HJB equation. Additionally, we do not assume a specific form for the utility function. We only assume that the asymptotic behavior of our utility function as wealth approaches $0$ or $\infty$ is as a logarithmic utility, or sum of power utilities. Accuracy of the approximation is established. We then use the small time approximation to iteratively build an approximation on longer time horizons.
Our discussion and results are organized as follows: we state our model assumptions, as well as our assumptions on the behavior of the utility function and its derivatives, in section 2. The main theorem is proved in section 3. In section 4, we build our close-to-optimal portfolio and verify the degree of closeness. In section 5, we graphically illustrate our small time approximation by an example. The results in section 4 are then extended to longer time horizons in section 6 and applied to an example.
Model and Assumptions
=====================
We consider the following simple incomplete market model, as in [@Nadtochiy13ApproxOIP], however our assumptions on the terminal utility function will be different from those considered in [@Nadtochiy13ApproxOIP].
Consider a market consisting of one risky asset (e.g., a stock) with price process $S_{t}$ and one riskless asset (e.g., a bond). The price process of the risky asset satisfies $$\label{stockprice}
dS_{t} = \mu(Y_{t}) S_{t}\,dt + \sigma(Y_{t}) S_{t} \,dW^{1}_{t},$$ where $Y_{t}$ is a stochastic factor which evolves as $$\label{stochasticfactor}
dY_{t} = b(Y_{t}) \,dt + a(Y_{t}) (\rho dW^{1}_{t} + \sqrt{1 - \rho^{2}} \,dW^{2}_{t}).$$ The vector $W_{t} = (W^{1}_{t}, W^{2}_{t})$ is a two-dimensional standard Brownian motion adapted to the natural filtration $(\mathcal{F}_{t})_{t \in [0,T]}$ given by $\mathcal{F}_{t} = \sigma(W_{s} : 0 \leq s \leq t )$, and $\rho$ satisfies $-1 < \rho < 1$. We also define the Sharpe ratio $\lambda(Y_{t}) := \frac{\mu(Y_{t}) - r}{\sigma(Y_{t})}$, where $r$ is the risk free interest rate.
\[modelassumptions\] Denote as $C(\Bbb R)$ the space of continuous functions $f: \Bbb R \to \Bbb R$, while $C^{k}(\Bbb R)$ is the space of $k$-times continuously differentiable functions $g: \Bbb R \to \Bbb R$ (for $k \geq 1, k \in \Bbb N$). The coefficients in the stochastic differential equations (SDEs) and , as well as $\lambda$, satisfy the following conditions (as in Assumption 1 of [@Nadtochiy13ApproxOIP]):
1. $\mu, \sigma \in C(\Bbb R)$ with $\sigma$ strictly positive.
2. $b \in C^{1}(\Bbb R)$ and $\lambda, a \in C^{2}(\Bbb R)$ with $a$ strictly positive.
3. There exists a constant ${c_{1}}>0$ such that $$|a| + |\frac{1}{a}| + |a'| + |a''| + |b| + |b'| + |\lambda| + |\lambda'| + |\lambda''| \leq {c_{1}}.$$
\[utilityassumptions\] We denote the investor’s terminal utility function by $U_{T}(x)$. We assume $U_{T}(x)$ is a strictly increasing, concave function belonging to $C^{5}(\Bbb R)$. In addition, we make the following assumptions on the asymptotic growth of the utility function:\
$U_T(x)$ is such that, either
Conditions - hold for $M(x) := \log(x)$;
or
Conditions - hold for $M(x) :=\frac{x^{1-\alpha}}{1-\alpha} +\frac{x^{1-\beta}}{1-\beta}$, $\alpha, \beta \neq 1$ and positive.
Asymptotic growth conditions: $$\label{u(x)}
0 < \inf \limits_{x > 0} \left (\frac{U_{T}'(x)}{M'(x)}\right ) \leq \sup \limits_{x > 0} \left (\frac{U_{T}'(x)}{M'(x)}\right ) < \infty \\$$
$$\label{uprime(x)}
0 < \inf \limits_{x > 0} \left (\frac{U_{T}''(x)}{M''(x)}\right ) \leq \sup \limits_{x > 0} \left (\frac{U_{T}''(x)}{M''(x)}\right ) < \infty \\$$
$$\label{Rprime(x)}
0 < \inf \limits_{x > 0} \left (\frac{U_{T}^{(3)}(x)}{M^{(3)}(x)}\right ) \leq \sup \limits_{x > 0} \left (\frac{U_{T}^{(3)}(x)}{M^{(3)}(x)}\right ) < \infty \\$$
$$\label{uprime2(x)}
0 < \inf \limits_{x > 0} \left (\frac{U_{T}^{(4)}(x)}{M^{(4)}(x)}\right ) \leq \sup \limits_{x > 0} \left (\frac{U_{T}^{(4)}(x)}{M^{(4)}(x)}\right ) < \infty. \\$$
These assumptions allow for any strictly increasing, concave utility function in $C^5(\mathbb R)$ that behaves as a logarithmic function, or as different power functions, asymptotically as wealth approaches $0$ and $\infty$. Three examples of such utility functions are:
1. [*Power utility:*]{} $U_{T}(x) = \frac{x^{1-\gamma}}{1-\gamma}$, for some positive $\gamma \neq 1$.
2. [*Mixture of power utilities: $U_{T}(x) = {c_{1}}\frac{x^{1-\alpha}}{1-\alpha} + {c_{2}}\frac{x^{1-\beta}}{1-\beta}$, for ${c_{1}}, {c_{2}} > 0$ and for positive $\alpha, \beta \neq 1$*]{}.
3. [*Log utility:*]{} $U_{T}(x) = \log(x)$.
While the utility assumptions in Assumption \[utilityassumptions\] are similar to those in [@Nadtochiy13ApproxOIP], these assumptions allow for logarithmic utility functions, as well as utility functions described by mixtures of power functions; these two examples are not covered by the results in paper [@Nadtochiy13ApproxOIP].
As the investor will be investing in one risky and one risk-free asset, $\pi_{t}$ will denote the discounted amount of wealth invested into the risky asset at time $t$. We wish to consider only self-financing trading strategies, and thus, denoting by $\pi^{0}_{t}$ the discounted amount of wealth invested in the risk-free asset, choosing $\pi_{t}$ necessarily implies the value of $\pi^{0}_{t}$. Because of this, as in [@Nadtochiy13ApproxOIP], we will identify each trading strategy with the amount $\pi_{t}$ invested in the risky asset. From this we can define the discounted wealth process $X^{\pi}_{t} := \pi_{t} + \pi^{0}_{t}$. This process evolves in the following way $$\label{dscw}
dX^{\pi}_{t} = \sigma(Y_{t})\pi_{t}(\lambda(Y_{t})\,dt + \,dW^{1}_{t}) \text{ for } t \in [0,T].$$ Henceforth, we will denote $\sigma(Y_{s})$ by $\sigma_{s}$ and $\lambda(Y_{s})$ by $\lambda_{s}$.
\[admissibledefinition\] The only strategies considered will be [*admissible*]{} strategies, meaning:
1. $\pi_{t}$ is progressively measurable with respect to the natural filtration of our two-dimensional Brownian motion.
2. $E \displaystyle \int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} \,ds < \infty$.
3. Given an initial wealth $x \in (0,\infty)$, the discounted wealth process is strictly positive for all $t \in [0,T]$.
4. $E \left [\int \limits_{0}^{T} (X^{\pi}_{s})^{-2\gamma} \sigma_{s}^{2}\pi_{s}^{2} \,ds \right ] < \infty$, where $\gamma = 1$ under Case 1 of Assumption \[utilityassumptions\], and $\gamma := \max\{\alpha, \beta\} > 1$ under Case 2 of Assumption \[utilityassumptions\].
Denoting the set of admissible trading strategies as $\mathcal{A}$, we define the value function, $J(t,x,y)$, as $$\label{valuefunction}
J(t,x,y) := \operatorname*{ess\,sup}\limits_{\pi \in \mathcal{A}} E[ U_{T}(X^{\pi}_{T}) | X^{\pi}_{t} = x, Y_{t} = y ].$$ The value function $J$ is formally a solution to the Hamilton-Jacobi-Bellman (HJB) equation given by $$\label{hjb}
\begin{cases}
\begin{split}
U_{t} &+ \max \limits_{\pi} \left ( \frac{1}{2} \sigma^{2}(y)\pi^{2}U_{xx} + \pi(\sigma(y) \lambda(y) U_{x} + \rho \sigma(y) a(y) U_{xy}) \right )\\& + \frac{1}{2}a^{2}(y)U_{yy} + b(y)U_{y} = 0, \qquad\qquad \text{for }(t,x,y)\in (0,T)\times (0,\infty)\times \mathbb R, \end{split}\\
U(T,x,y) = U_{T}(x), \qquad\qquad \text{for }(x,y)\in (0,\infty)\times \mathbb R.
\end{cases}$$ It is easy to see that the expression being maximized in achieves its maximum at the portfolio given by $$\label{maxport}
\pi(t,x,y) = \frac{-\lambda(y)U_{x}(t,x,y) - \rho a(y) U_{xy}(t,x,y)}{\sigma(y)U_{xx}(t,x,y)}.$$ The optimal strategy is thus $\pi_{t}:= \pi(t,X^{\pi}_{t},Y_{t})$ for $t \in [0,T]$. Substituting the maximizing portfolio in equation gives $$\label{hjb2}
U_{t} - \frac{1}{2} \frac{(\lambda(y) U_{x} + \rho a(y) U_{xy})^{2}}{U_{xx}} + \frac{1}{2}a^{2}(y)U_{yy} + b(y)U_{y} = 0.$$
Main theorem {#shorttimesection}
============
In this section, we introduce the main result of this paper, that the value function defined in can be approximated in such a way as to yield an error measured in terms of time to horizon. This result is achieved by constructing classical sub- and super-solutions to the HJB equation which have the form of a second order expansion in powers of $T-t$, the time to horizon. The sub- and super-solutions will coincide up to the first order terms, and this will serve as the value function approximation. The second order terms in the expansions of the sub- and super-solution will yield the error. A probabilistic argument using martingale inequalities will show that the value function lies between the constructed sub- and super-solutions.
\[maintheorem\] Let $J(t,x,y)$ be the value function defined in , and let $U_{T}(x)$ denote the terminal utility function. Define $$\label{valueapproximation}
\Uh(t,x,y) := U_{T}(x) - (T-t) \frac{\lambda^{2}(y)}{2} \frac{U_{T}'(x)^{2}}{U_{T}''(x)}.$$ Suppose Assumptions \[modelassumptions\] and \[utilityassumptions\] hold .Then there exists constants ${c_{2}} > 0$ and $0 < \delta < \min\{1, T\}$ such that $$\label{mainresultinequality}
\left|J(t,x,y) - \Uh(t,x,y) \right| \leq {c_{2}} (T-t)^{2} h(x), \qquad \text{ for } (t,x,y) \in (T-\delta,T)\times (0,\infty) \times \Bbb R,$$ where $h(x) \equiv 1$ under Case 1 of Assumption \[utilityassumptions\], and $h(x) = x^{1-\alpha} + x^{1-\beta}$ under Case 2 of Assumption \[utilityassumptions\]; the constants ${c_{2}}$ and $\delta$ are independent of $t,x$ and $y$.
We prove this result in two parts: first, we will construct the classical sub- and super-solutions $\ul{U} (t,x,y)$ and ${\overline{U}}(t,x,y)$, respectively, to the HJB equation. Once established, we will then show that the value function lies between the sub- and super-solutions, i.e., $\ul{U}(t,x,y) \leq J(t,x,y) \leq {\overline{U}}(t,x,y)$.
We begin by constructing $\ul{U}$ and ${\overline{U}}$. Consider the HJB equation given in : $$\label{shorthjb}
\begin{cases}
\begin{split}
U_{t} + \H(U) = 0, \qquad\qquad \text{for }(t,x,y)\in (0,T)\times (0,\infty)\times \mathbb R, \end{split}\\
U(T,x,y) = U_{T}(x), \qquad\qquad \text{for }(x,y)\in (0,\infty)\times \mathbb R,
\end{cases}$$ where $$\begin{split}
\H(U) := H(y,U,U_{x},U_{y},U_{xx},U_{xy},U_{yy}) &:= - \frac{1}{2} \frac{(\lambda(y) U_{x} + \rho a(y) U_{xy})^{2}}{U_{xx}} + \frac{1}{2}a^{2}(y)U_{yy} + b(y)U_{y}.
\end{split}$$ We consider sub- and super-solutions having the following expansion in terms of powers of $T-t$: $$U(t,x,y) := {U^{(0)}}(x,y) + (T-t) {U^{(1)}}(x,y) + (T-t)^{2} {U^{(2)}}(x,y).$$ So that our first order approximation coincides with the value function at the terminal time $T$, we choose the terminal condition of for the value of ${U^{(0)}}$, i.e., $${U^{(0)}}(x,y) := U_{T}(x) \text{ for all } (x,y) \in (0,\infty) \times \Bbb R.$$ We now substitute $U$ into to obtain: $$\label{expansion-PDE}\begin{split}
&-{U^{(1)}} - 2(T-t) {U^{(2)}} \\
&-\frac{1}{2} \frac{\left(\lambda(y) ({U^{(0)}}_{x} + (T-t) {U^{(1)}}_{x} + (T-t)^{2} {U^{(2)}}_{x}) + \rho a(y) ({U^{(0)}}_{xy} + (T-t) {U^{(1)}}_{xy} + (T-t)^{2} {U^{(2)}}_{xy}) \right)^{2}}{{U^{(0)}}_{xx} + (T-t) {U^{(1)}}_{xx} + (T-t) {U^{(2)}}_{xx}} \\
& + \frac{1}{2} a^{2}(y) \left({U^{(0)}}_{yy} + (T-t) {U^{(1)}}_{yy} + (T-t) {U^{(2)}}_{yy} \right) + b(y) \left({U^{(0)}}_{y} + (T-t) {U^{(1)}}_{y} + (T-t) {U^{(2)}}_{y} \right) = 0.
\end{split}$$ We choose ${U^{(1)}}$ such that terms of order $O(1)$ on the left-hand side of equation disappear. To do this, we clear fractions and collect terms of order $O(1)$. Equating these with zero and then removing any terms equivalent to zero (i.e., terms containing partial derivatives of ${U^{(0)}}$ in the variables $t$ and $y$) yields the following formula for ${U^{(1)}}$: $${U^{(1)}}(x,y) = -\frac{\lambda^{2}(y)}{2} \frac{U_{T}'(x)^{2}}{U_{T}''(x)} \text{ for all } (x,y) \in (0,\infty) \times \Bbb R.$$ Finally, we choose two different functions for ${U^{(2)}}$ so as to obtain a sub- and super-solution. This is done by analyzing the formula for ${U^{(2)}}$ which results from equating the coefficients of the $T-t$ terms in , after clearing fractions, to zero. Removing any terms with factors of zero, as before, yields $$\label{U2formula}
\begin{split}
&{U^{(2)}}(x,y) = \\ &\frac{U_{T}'(x)^{2}}{U_{T}''(x)}\left( \frac{1}{4}\lambda^{4}(y) - \frac{1}{2}b(y) \lambda(y) \lambda'(y) + \rho a(y) \lambda^{2}(y) \lambda'(y) - \frac{1}{4} a^{2}(y) (\lambda'(y))^{2} - \frac{1}{4} a^{2}(y) \lambda(y) \lambda''(y) \right) \\ &- \frac{U_{T}'(x)^{3}}{U_{T}''(x)^{3}} \left( \frac{1}{2}\rho a(y) \lambda^{2}(y) \lambda'(y) U_{T}^{(3)}(x) + \frac{\lambda^{4}(y)}{4}\frac{U_{T}'(x)(U_{T}^{(3)}(x))^{2}}{(U_{T}''(x))^{2}} - \frac{\lambda^{4}(y)}{8}\frac{U_{T}'(x)U_{T}^{(4)}(x)}{U_{T}''(x)} \right).
\end{split}$$ We abbreviate this expression by expanding the terms on the right hand side of and enumerating the resulting terms as $a_{1}, \dots, a_{8}$, giving as $${U^{(2)}}(x,y) = \sum \limits_{i = 1}^{8} a_{i}(x,y).$$ We note that $a_{i} \sim h(x)$, where $h(x) \equiv 1$ if $M(x) = \log(x)$ (i.e., if in Case 1 of Assumption \[utilityassumptions\]), and $h(x) = x^{1-\alpha} + x^{1-\beta}$ if $M(x) = \frac{x^{1-\alpha}}{1-\alpha}+ \frac{x^{1-\beta}}{1-\beta}$ (i.e., if in Case 2 of Assumption \[utilityassumptions\]) and bounded in $y$ for $1 \leq i \leq 8$. Set $${c_{2}} := \left (8\max \limits_{1 \leq i \leq 8} \sup \limits_{ \substack{ x > 0 \\ y \in \Bbb R} } \frac{|a_{i}|}{h(x)} \right ) + 1$$ and define ${\overline{u}}_{2} := {c_{2}} h(x)$ and $\ul{u}_{2} := - {\overline{u}}_{2}$. Then substituting $$\label{subsolution}
\ul{U} := {U^{(0)}} + (T-t) {U^{(1)}} + (T-t)^{2} \ul{u}_{2}$$ (and similarly $$\label{supersolution}
{\overline{U}} := {U^{(0)}} + (T-t) {U^{(1)}} + (T-t)^{2} {\overline{u}}_{2})$$ in the left-hand side of equation and clearing fractions, terms of $O(1)$ disappear, while the terms that comprise the coefficient $T-t$ are strictly positive (respectively, strictly negative).
We now observe that the following inequality holds: $$\label{orderhjbnofrac}
\ul{U}_{xx}^{2} | \ul{U}_{t} + \H(\ul{U})| \leq c(T-t) \tilde{h}(x),$$ where $\tilde{h}(x) = x^{-4}$ under Case 1 of Assumption \[utilityassumptions\], and $\tilde{h}(x) = (x^{1-\alpha} + x^{1-\beta})(x^{-2 - 2\alpha} + x^{-2 -2\beta})$ under Case 2 of Assumption \[utilityassumptions\]. We use Assumption \[utilityassumptions\] to verify , and we note that holds for ${\overline{U}}$ in place of $\ul{U}$. Also note that $\frac{1}{c} x^{-2} \leq \ul{U}_{xx} \leq cx^{-2}< 0$ in Case 1 and $\frac{1}{c}(x^{-1 - \alpha} + x^{-1-\beta}) \leq \ul{U}_{xx} \leq c (x^{-1 - \alpha} + x^{-1-\beta}) < 0$ in Case 2 of Assumption \[utilityassumptions\], for some $c < 0$. So $(\ul{U}_{xx})^2$ is bounded away from $0$.
To argue that $\ul{U}$ is a sub-solution to , we recall that the coefficient of $T-t$ in the expression $\ul{U}_{xx}^{2}(\ul{U}_{t} + \H(\ul{U}))$ is strictly positive (by choice of $\ul{u}_{2}$). Inequality implies this coefficient has growth in $x$ on the order of $\tilde{h}(x)$ and growth in $y$ bounded. also implies the $o(T-t)$ terms of $\ul{U}_{xx}^{2}(\ul{U}_{t} + \H(\ul{U}))$ also have growth in $x$ on the order of $\tilde{h}(x)$ and growth in $y$ bounded. Thus, for $t$ near $T$, the positive coefficient of $T-t$ dominates the $o(T-t)$ terms uniformly in $x$ and $y$, implying $\ul{U}_{t} + \H(\ul{U})>0$, i.e., $\ul{U}$ is a classical sub-solution of . A mirror of this argument proves ${\overline{U}}$ is a classical super-solution of .
It remains to be shown that the value function $J(t,x,y)$ given in $\eqref{valuefunction}$ lies between the sub- and super-solution, i.e., $$\ul{U}(t,x,y) \leq J(t,x,y) \leq {\overline{U}}(t,x,y) \qquad \text{ for all } (t,x,y) \in [0,T] \times (0,\infty) \times \Bbb R.$$ We will first show that $\ul{U}(t,x,y) \leq J(t,x,y)$, and then we will show $J(t,x,y) \leq {\overline{U}}(t,x,y)$. To prove $\ul{U}(t,x,y) \leq J(t,x,y)$, we first consider the trading strategy $\ul{\pi}(t,X^{\ul{\pi}}_{t},Y_{t})$ generated by the sub-solution $\ul{U}$, where $\ul{\pi}(t,x,y)$ is the function obtained by substituting $\ul{U}$ into , i.e., $$\ul{\pi}(t,x,y) = \frac{-\lambda(y)\ul{U}_{x}(t,x,y) - \rho a(y) \ul{U}_{xy}(t,x,y)}{\sigma(y)\ul{U}_{xx}(t,x,y)}.$$ Applying Ito’s formula to $\ul{U}(t,X^{\ul{\pi}}_{t},Y_{t})$ gives $$\label{Itosubsolution}
\begin{split}
\underbrace{\ul{U}(T, X^{\ul{\pi}}_{T}, Y_{T})}_{U_{T}(X^{\ul{\pi}}_{T})} - \ul{U}(t, X^{\ul{\pi}}_{t}, Y_{t}) =& \int \limits_{t}^{T} \left ( \ul{U}_{t} + \sigma \ul{\pi} \lambda \ul{U}_{x} + b\ul{U}_{y} + \frac{1}{2} \sigma^{2} \ul{\pi}^{2} \ul{U}_{xx} + \sigma \ul{\pi} a \rho \ul{U}_{xy} + \frac{1}{2} a^{2} \ul{U}_{yy} \right ) \,ds \\
+& \underbrace{\int \limits_{t}^{T} \left ( \sigma \ul{\pi}\ul{U}_{x} + a \rho \ul{U}_{y} \right ) \, dW^{1}_{s} + \int \limits_{t}^{T} a \sqrt{1 - \rho^{2}}\ul{U}_{y} \, dW^{2}_{s}}_{\text{local martingales}}.
\end{split}$$ Since the stochastic integrals on the right hand side are local martingales, we can find a sequence $\{ \tau_{n} \}_{n = 1}^{\infty}$ of stopping times such that $\tau_{n} \in [t,T]$, $\tau_{n} \leq \tau_{n + 1}$ a.s. for all $n$, and $\tau_{n} \to T$ a.s. as $n \to \infty$. In particular, if we replace $T$ with $T \wedge \tau_{n}$, the local martingales will become martingales: $$\label{localizedIto}
\begin{split}
&\ul{U}(T\wedge \tau_{n}, X^{\ul{\pi}}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}}) - \ul{U}(t, X^{\ul{\pi}}_{t}, Y_{t}) \\
&=\int \limits_{t }^{T\wedge \tau_{n}} \left ( \ul{U}_{t} + \sigma \ul{\pi} \lambda \ul{U}_{x} + b\ul{U}_{y} + \frac{1}{2} \sigma^{2} \ul{\pi}^{2} \ul{U}_{xx} + \sigma \ul{\pi} a \rho \ul{U}_{xy} + \frac{1}{2} a^{2} \ul{U}_{yy} \right ) \,ds \\& + \int \limits_{t}^{T\wedge \tau_{n}} \left ( \sigma \ul{\pi}\ul{U}_{x} + a \rho \ul{U}_{y} \right ) \, dW^{1}_{s} + \int \limits_{t}^{T\wedge \tau_{n}} a \sqrt{1 - \rho^{2}}\ul{U}_{y} \, dW^{2}_{s}.
\end{split}$$ Note that the integrand of the first term on the right hand side of is the left-hand side of the HJB equation with the sub-solution $\ul{U}$ substituted in, thus making the term non-negative. Taking the conditional expectation of both sides of equation we get $$\ul{U}(t, x, y) \leq E[ \ul{U}(T\wedge \tau_{n}, X^{\ul{\pi}}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}}) | X^{\ul{\pi}}_{t} = x, Y_{t} = y].$$
Now, clearly, $\ul{U}(T \wedge \tau_{n}, X^{\ul{\pi}}_{T\wedge \tau_{n}}, Y_{T\wedge \tau_{n}}) \to \ul{U}(T, X^{\ul{\pi}}_{T}, Y_{T}) = U_{T}(X^{\ul{\pi}}_{T})$ a.s. as $n \to \infty$. Also, we have $$\label{dominating_subsol}
\begin{split}
&|\ul{U}(T \wedge \tau_{n}, X^{\ul{\pi}}_{T\wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| \\
&= \left |U_{T}(X^{\ul{\pi}}_{T\wedge \tau_{n}}) - (T-T \wedge \tau_{n}) \frac{\lambda^{2}(Y_{T\wedge \tau_{n}})}{2} \frac{U_{T}'(X^{\ul{\pi}}_{T\wedge \tau_{n}})^{2}}{U_{T}''(X^{\ul{\pi}}_{T\wedge \tau_{n}})} - c_{2}(T\wedge\tau_{n} - t)^{2} h(X^{\ul{\pi}}_{T\wedge\tau_{n}})\right | \\
&\leq \left |U_{T}(X^{\ul{\pi}}_{T\wedge \tau_{n}}) \right | + T \frac{\lambda^{2}(Y_{T\wedge \tau_{n}}))}{2} \left |\frac{U_{T}'(X^{\ul{\pi}}_{T\wedge \tau_{n}})^{2}}{U_{T}''(X^{\ul{\pi}}_{T\wedge \tau_{n}})} \right | + c_{2}Th(X^{\ul{\pi}}_{T \wedge \tau_{n}}) \\
& \leq {c_{3}} G(X^{\ul{\pi}}_{T\wedge \tau_{n}}),
\end{split}$$ for some constant ${c_{3}}$, where $G(x) = \log(x) + 1$ under Case 1 of Assumption \[utilityassumptions\], and $G(x) = x^{1-\alpha} + x^{1-\beta}$ under Case 2 of Assumption \[utilityassumptions\].
To see that $\{G(X^{\ul{\pi}}_{T\wedge \tau_{n}})\}_{n =1}^{\infty}$ is dominated by an integrable random variable, we refer the reader to Lemma \[boundedintfunc\] which is proved in the appendix. Thus, we are in a position to apply the dominated convergence theorem, which yields $$E[\ul{U}(T\wedge \tau_{n}, X^{\ul{\pi}}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| X^{\ul{\pi}}_{t} = x, Y_{t} = y ] \to E[ U_{T}(X^{\ul{\pi}}_{T}) | X^{\ul{\pi}}_{t} = x, Y_{t} = y ] \ a.s.$$ as $n \to \infty$. This implies $$\ul{U}(t, x, y) \leq E[ U_{T}(X^{\ul{\pi}}_{T}) | X^{\ul{\pi}}_{t} = x, Y_{t} = y ].$$ From the admissibility of $\ul{\pi}(t,X^{\ul{\pi}}_{t},Y_{t})$ (which follows from a similar argument used to prove Lemma \[admissibilitypihat\]), it immediately follows that $\ul{U}(t,x,y) \leq J(t,x,y)$.
We now verify that $J(t,x,y) \leq {\overline{U}}(t,x,y)$. To begin, let $\tilde{\pi}$ be any admissible trading strategy. Note that because ${\overline{U}}$ is a super-solution of the HJB equation , we have that $${\overline{U}}_{t} + \frac{1}{2} \sigma^{2}(y)\tilde{\pi}^{2}{\overline{U}}_{xx} + \tilde{\pi}(\sigma(y) \lambda(y) {\overline{U}}_{x} + \rho \sigma(y) a(y) {\overline{U}}_{xy}) + \frac{1}{2}a^{2}(y){\overline{U}}_{yy} + b(y){\overline{U}}_{y} \leq 0.$$ Thus, applying Ito’s formula to ${\overline{U}}(t,X^{\tilde{\pi}}_{t},Y_{t})$, followed by localizing and taking conditional expectations, we have $$E[ {\overline{U}}(T\wedge \tau_{n}, X^{\tilde{\pi}}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}}) | X^{\tilde{\pi}}_{t} = x, Y_{t} = y] \leq {\overline{U}}(t, x, y) \text{ for each }n.$$ As in , we can show that $|{\overline{U}}(T\wedge \tau_{n}, X^{\tilde{\pi}}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}})|\leq {c_{3}} G(X^{\tilde{\pi}}_{T\wedge \tau_{n}})$ where $G(X^{\tilde{\pi}}_{T\wedge \tau_{n}})$ is dominated by an integrable random variable (shown in Lemma \[boundedintfunc\]). The dominated convergence theorem then implies $E[ U_{T}(X^{\tilde{\pi}}_{T}) | X^{\tilde{\pi}}_{t} = x, Y_{t} = y ] \leq {\overline{U}}(t,x,y)$. As $\tilde{\pi}_{t}$ is an arbitrary admissible portfolio, this implies that $J(t,x,y) \leq {\overline{U}}(t,x,y)$, as desired.
Having established that the value function lies between the sub- and super-solution, i.e., $\ul{U}(t,x,y) \leq J(t,x,y) \leq {\overline{U}}(t,x,y)$, it follows immediately from the definitions of $\ul{U}$ and ${\overline{U}}$ in and , respectively, that for $\Uh(t,x,y)$ as in , $|J(t,x,y) - \Uh(t,x,y) | \leq c_{2}(T-t)^{2}h(x)$.
Building the approximating portfolio {#approxportsec}
====================================
In section \[shorttimesection\], we showed that the value function $J(t,x,y)$ given in can be approximated by a first order expansion in powers of the time to horizon $T-t$, namely, $\hat{U}(t,x,y)$ given in . In addition, we showed that the error between $J(t,x,y)$ and $\hat{U}(t,x,y)$ is controlled by the square time to horizon $(T-t)^{2}$.
In this section, we will show that our first order approximation generates a close-to-optimal trading strategy near horizon. To show this, we first recall formula , which the Verification Theorem tells us would represent the optimal trading strategy in the case that the HJB equation were well-posed. As we do not assume a classical solution to the HJB equation, the conclusion of the Verification Theorem may not be applied. Formula will still be useful in our analysis, however. We will show that our smooth approximation $\hat{U}(t,x,y)$, when substituted into , produces a portfolio which yields an expected utility close to the maximum expected utility, with the error measured in terms of the square time to horizon $(T-t)^{2}$. This result is stated in the following lemma.
\[utilityapproxlemma\] Let $J(t,x,y)$ be the value function defined in , $\hat{U}(t,x,y)$ be as in , and let $X^{\pih}_{s}$ be the wealth process with evolution described by under portfolio $\pih(s,X^{\pih}_{s},Y_{s})$, where $\pih(s,x,y)$ the function given by $$\label{approxoptimalport}
\pih(s,x,y) := -\frac{\lambda(y)}{\sigma(y)} \frac{\Uh_{x}(s,x,y)}{\Uh_{xx}(s,x,y)} - \frac{\rho a(y)}{\sigma(y)} \frac{\Uh_{xy}(s,x,y)}{\Uh_{xx}(s,x,y)}, \,\,\,\,\,\,\, s\in[t,T], \,\,x \in (0,\infty), \,\, y \in \Bbb R.$$ Under Assumptions \[modelassumptions\] and \[utilityassumptions\], there exists a constant $C > 0$ and $0 < \delta < \min\{1,T\}$ such that $$|J(t,x,y) - E[ U_{T}(X^{\pih}_{T}) |X^{\pih}_{t} = x, Y_{t} = y ]| \leq C (T-t)^{2}h(x), \qquad \text{ for } (t,x,y) \in (T-\delta,T)\times (0,\infty) \times \Bbb R$$ where $h(x) \equiv 1$ under Case 1 of Assumption \[utilityassumptions\], and $h(x) = x^{1-\alpha} + x^{1-\beta}$ under Case 2 of Assumption \[utilityassumptions\]; the constants $C$ and $\delta$ are independent of $t$, $x$ and $y$.
We begin by referring the reader to Lemma \[admissibilitypihat\], which is proven in the Appendix and asserts the strategy $\pih_{t}$ is indeed admissible as in Definition \[admissibledefinition\].
We now prove that the expected utility of terminal wealth under $\pih_{t}$ is near the maximal expected utility. To do this, we start by applying Ito’s formula to $\Uh(s, X^{\pih}_{s}, Y_{s})$ and obtain $$\label{ItoUhat}
\begin{split}
\Uh(T, X^{\pih}_{T}, Y_{T}) - \Uh(t, X^{\pih}_{t}, Y_{t}) =& \int \limits_{t}^{T} \left ( \Uh_{t} + \sigma \pih \lambda \Uh_{x} + b\Uh_{y} + \frac{1}{2} \sigma^{2} \pih^{2} \Uh_{xx} + \sigma \pih a \rho \Uh_{xy} + \frac{1}{2} a^{2} \Uh_{yy} \right ) \,ds \\
+& \int \limits_{t}^{T} \left ( \sigma \pih\Uh_{x} + a \rho \Uh_{y} \right ) \, dW^{1}_{s} + \int \limits_{t}^{T} a \sqrt{1 - \rho^{2}}\Uh_{y} \, dW^{2}_{s}.
\end{split}$$ Recall that by definition of $\Uh$ and Assumption \[utilityassumptions\], we get $|\partial_{t} \Uh + \H(\Uh)| = O(T-s)O(h(X^{\pih}_{s}))$, which shows the integrand of the drift term is $O(T-s)O(h(X^{\pih}_{s}))$. Parallel to the proof of Theorem \[maintheorem\], we use the sequence of stopping times $\{\tau_{n}\}_{n=1}^{\infty}$ to localize , converting the local martingale terms to martingales. Taking the conditional expectation of both sides then yields $$\begin{split}
&E[\Uh(T \wedge \tau_{n}, X^{\pih}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| X^{\pih}_{t} = x, Y_{t} = y ] - \Uh(t, x,y) \\
&=\int \limits_{t }^{T\wedge \tau_{n}} E \left [ O(T-s)O(h(X^{\pih}_{s})) | X^{\pih}_{t} = x, Y_{t} = y \right ] \,ds.
\end{split}$$
Using the uniform bound of $h(X^{\pih})$ which can be obtained from the proof of Lemma \[admissibilitypihat\], it follows that $$\label{stoppedbound}
|E[\Uh(T \wedge \tau_{n}, X^{\pih}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| X^{\pi}_{t} = x, Y_{t} = y ] - \Uh(t, x, y)| \leq C_{1}(T-t)^{2}h(x),$$ for some constant $C_1>0$. Now, parallel to the proof of Theorem \[maintheorem\], we have $\Uh(T \wedge \tau_{n}, X^{\pih}_{T\wedge \tau_{n}}, Y_{T\wedge \tau_{n}}) \to \Uh(T, X^{\pih}_{T}, Y_{T}) = U_{T}(X^{\pih}_{T})$ a.s. as $n \to \infty$. Also, we have $|\Uh(T \wedge \tau_{n}, X^{\pih}_{T\wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| \leq {c_{3}} G(X^{\pih}_{T\wedge \tau_{n}})$ for some constant ${c_{3}}$, where $G(x) = \log(x) + 1$ under Case 1 of Assumption \[utilityassumptions\], and $G(x) = x^{1-\alpha} + x^{1-\beta}$ under Case 2 of Assumption \[utilityassumptions\].
By Lemma \[boundedintfunc\], the sequence $\{G(X^{\pih}_{T\wedge \tau_{n}})\}_{n =1}^{\infty}$ is dominated by an integrable function, implying by the dominated convergence theorem that $$E[\Uh(T\wedge \tau_{n}, X^{\pih}_{T \wedge \tau_{n}}, Y_{T\wedge \tau_{n}})| X^{\pih}_{t} = x, Y_{t} = y ] \to E[ U_{T}(X^{\pih}_{T}) | X^{\pih}_{t} = x, Y_{t} = y ] \text{ a.s. as } n \to \infty.$$ As a result, gives $$\label{notstoppedbound}
|E[U_{T}(X^{\pih}_{T})| X^{\pi}_{t} = x, Y_{t} = y ] - \Uh(t, x, y)| \leq C_{1}(T-t)^{2}h(x).$$ By Theorem \[maintheorem\] and inequality , it follows that $$\begin{split}
|E[U_{T}(X^{\pih}_{T})| X^{\pi}_{t} = x, Y_{t} = y ] - J(t, x, y)| &\leq |E[U_{T}(X^{\pih}_{T})| X^{\pi}_{t} = x, Y_{t} = y ] - \Uh(t, x, y)| + |\Uh(t, x,y) - J(t, x, y)| \\
&\leq C_{1}(T-t)^{2} h(x) + c_{2} (T-t)^{2} h(x) \\
&\leq C(T-t)^{2}h(x),
\end{split}$$ for $0 < T-t < \delta$.
Example {#smalltimeexamplesec}
=======
We consider the following stochastic volatility model, which was used in [@Fouque2015Portfolio], with parameter estimations taken from [@ChackoViceira05Dynamic]. The time horizon considered is $[0,T]$. The risky asset satisfies with $\mu(y) = \mu$ a constant function and $\sigma(y) = \frac{1}{\sqrt{y}}$. The stochastic factor satisfies with $b(y) = (m - y)$ and $a(y) = \beta \sqrt{y}$ where $m$ and $\beta$ are constants. In [@Fouque2015Portfolio], the authors assume their model has a slow factor (hence the presence of a factor $\delta$ in their model). As we do not assume the factor in our model is a slow factor, we have set $\delta = 1$. We set: $\mu = 0.0811$, $m = 27.9345$, and $\beta = 1.12$; the variable $y$ is fixed at $27.9345$; $T=2$; the correlation coefficient between our two Brownian motions is $\rho = 0.5241$; and $\gamma = 3$. Under this model, we consider the power utility function $U_{T}(x) = \frac{x^{(1-\gamma)}}{1-\gamma} = -\frac{1}{2x^{2}}$. Note that while this utility function satisfies Assumption \[utilityassumptions\], not all of the model assumptions in Assumption \[modelassumptions\] are satisfied (e.g., $\lambda(y)$ is not absolutely bounded). Nevertheless, our results are shown in this section to be good approximations under this model as well.
The authors in [@Fouque2015Portfolio] obtained an explicit formula for the value function under the assumed model by solving a linear PDE derived in [@Zariphopoulou01Val]. We now restate the formula for the value function found in [@Fouque2015Portfolio]. If $f(r) := \frac{ \beta^{2}}{2} r^{2} + (\frac{(1-\gamma) \beta \mu \rho - \gamma}{\gamma})r + \frac{(\gamma + (1-\gamma)\rho^{2})(1-\gamma)\mu^{2}}{2\gamma^{2}}$, where we substitute the above values we have assumed for the variables in $f(r)$, then solving $f(r) =0$ gives one positive root and one negative root of $f$, denoted $a_{+}$ and $a_{-}$, respectively. In addition, we set $\alpha$ to be the square root of the discriminant of the quadratic polynomial $f(r)$. Then, if we define $A(t,T) := \frac{(1 - e^{-\alpha (T-t)})a_{-}}{1- \frac{a_{-}}{a_{+}}e^{-\alpha(T-t)}}$ and $B(t,T) := m \left( (T-t)a_{-} - \frac{2 }{ \beta^{2}} \log \left(\frac{1 - \frac{a_{-}}{a_{+}}e^{-\alpha(T-t)}}{1 - \frac{a_{-}}{a_{+}}} \right) \right )$, the value function is given by $$\label{exvalueformula}
U(t,x,y) = -\frac{1}{2x^{2}} e^{\left (\frac{\gamma}{\gamma + (1-\gamma) \rho^{2}} \right ) ( yA(t,T) + B(t,T) ) }.$$
Recall that our approximation of the value function is given by $$\label{exapproxformula}
\hat{U}(t,x,y) = U_{T}(x) - (T-t) \frac{\lambda^{2}(y)}{2} \frac{U_{T}'(x)^{2}}{U_{T}''(x)},$$ We can now substitute and into to obtain the optimal and approximating portfolios, $\pi_{U}$ and $\pih$, respectively. For the parameter values assumed in the beginning of this section, we obtain the following formulas for the value function and its approximation, the optimal portfolio and its approximation, and the respective errors:
[47em]{}[ c c c c c c c c ]{} $t$ & $T$ & $U(t,x,y)$ & $\Uh(t,x,y)$ & $|U - \Uh|$ & $\pi_{U}(t,x,y)$ & $\pih(t,x,y)$ & $|\pi_{U} - \pih|$\
$1.5$ & $2$ & $\approx -\dfrac{0.485022}{x^{2}}$ & $\approx -\dfrac{0.484689}{x^{2}}$ & $\approx \dfrac{0.000333}{x^{2}}$ & $\approx 0.750482x$ & $\approx 0.748982x$ & $\approx 0.0015x $\
1.9 & 2 & $\approx -\dfrac{0.496952}{x^{2}}$ & $\approx -\dfrac{0.496938}{x^{2}}$ & $\approx \dfrac{0.000014}{x^{2}}$ & $\approx 0.754024x$ & $\approx 0.753957x$ & $\approx 0.000067x$\
In figures \[fig:valuefullonehalf\], \[fig:valuetimeonehalf\], and \[fig:valuetimepointone\], we graph the value function against the zero and first order approximations. In figures \[fig:portfoliosonehalf\] and \[fig:portfoliospointone\], we graph the optimal portfolio against our approximating portfolio.
![($t = 1.5, \,T=2$) The value function is plotted against the zero order approximation, $U_{T}(x)$, and the zero order approximation with the additional correction term. It is difficult to distinguish between the value function and the first order approximation (i.e. approximation with correction term).[]{data-label="fig:valuefullonehalf"}](valueapproxfullxrestrict.pdf)
![($t=1.9, \,T=2$) When the time interval is shortened from a length of $0.5$ to a length of $0.1$, the approximation with correction is much closer to the value function.[]{data-label="fig:valuetimepointone"}](valueapproxtimeonehalf.pdf)
![($t=1.9, \,T=2$) When the time interval is shortened from a length of $0.5$ to a length of $0.1$, the approximation with correction is much closer to the value function.[]{data-label="fig:valuetimepointone"}](valueapproxtimepointone.pdf)
![($t=1.9, \,T=2$) When the time interval is shortened from a length of $0.5$ to a length of $0.1$, the approximating portfolio is much closer to the optimal portfolio.[]{data-label="fig:portfoliospointone"}](portfoliosonehalf.pdf)
![($t=1.9, \,T=2$) When the time interval is shortened from a length of $0.5$ to a length of $0.1$, the approximating portfolio is much closer to the optimal portfolio.[]{data-label="fig:portfoliospointone"}](portfoliospointone.pdf)
Portfolio optimization on a finite time horizon
===============================================
Approximation scheme {#schemesec}
--------------------
In section \[shorttimesection\], we approximated the value function $J(t,x,y)$, given in , for values of time $t$ near the terminal time $T$. We then used this approximation in section \[approxportsec\] to generate a trading strategy $\pih_{t} := \pih(t,X^{\pih}_{t}, Y_{s})$ (with the function $\pih(t,x,y)$ given in ) which was shown to be close-to-optimal when the time to horizon $T-t$ is small.
In this section, we present a heuristic scheme to approximate the value function for all times $t$ in some finite horizon $[0,T]$, and then utilize this approximation in tandem with the function $\pi(t,x,y)$ from to generate a close-to-optimal trading strategy on $[0,T]$. To begin, we partition the interval $[0,T]$ into small sub-intervals, given by $\{0 = t_{0} < t_{1} < \dots < t_{n - 1} < t_{n} = T\}$. The scheme is then given by $$\label{valueapproximationscheme}
\begin{split}
\Uh(t,x,y) := \Uh(t_{k+1},x,y) + (t_{k+1}-t) \biggl [ - \frac{1}{2}\frac{(\lambda(y) \Uh_{x}(t_{k+1},x,y) + \rho a(y) \Uh_{xy}(t_{k+1},x,y))^{2}}{\Uh_{xx}(t_{k+1},x,y)} \\
+ \frac{1}{2} a^{2}(y) \Uh_{yy}(t_{k+1},x,y) + b(y) \Uh_{y}(t_{k+1},x,y) \biggr],
\end{split}$$ for $t_{k} \leq t \leq t_{k+1}$, $(x,y) \in (0,\infty) \times \Bbb R$, with $\Uh(T,x,y) = U_{T}(x)$. The close-to-optimal trading strategy will then be given by $\pih_{t} := \pih(t,X^{\pih}_{t},Y_{t})$, where the function $\pih(t,x,y)$ is the function $$\label{finitehorizonpi}
\pih(t,x,y) := \frac{-\lambda(y)\Uh_{x}(t,x,y)}{\sigma(y)\Uh_{xx}(t,x,y)} - \frac{\rho a(y) \Uh_{xy}(t,x,y)}{\sigma(y)\Uh_{xx}(t,x,y)},$$ for $t_{k} \leq t < t_{k+1}$, $(x,y) \in (0,\infty) \times \Bbb R$, with $\Uh(T,x,y) = U_{T}(x)$.
A formal justification of these formulae is given as follows. Approximating formula is obtained by recursively implementing the technique for constructing the sub- and super-solutions to the HJB equation in section \[shorttimesection\]. We first apply the result in Theorem \[maintheorem\] to the horizon $[t_{n-1},T]$. We remind the reader that the zero order term in the approximation formula will be the terminal condition $U_{T}(x)$ on the interval $[t_{n-1},T]$. For the earlier time interval, say, $[t_{n-2}, t_{n-1}]$, $\ul{U}(t_{n-1},x,y)$ and ${\overline{U}}(t_{n-1},x,y)$ will serve as the terminal conditions for the sub- and super-solutions constructed on the interval $[t_{n-2}, t_{n-1}]$. Note the $y$-independence of the terminal condition $U_{T}(x)$ on the subinterval $[t_{n-1},T]$ reduces formula to . The dependence of the terminal conditions on $y$ at earlier time intervals introduces the additional terms in the first order term of .
The accuracy of approximations and will be rigorously proved in future work. This can be accomplished by repeating the procedure used to verify the accuracy in section \[shorttimesection\]. On a fixed, finite horizon, however, this will require a higher degree of regularity of the terminal condition. In addition, establishing an inequality in the spirit of will be more involved and is beyond the scope of this paper.
Example {#example}
-------
In this section, we consider the model and utility function described in Section \[smalltimeexamplesec\], and graphically analyze the accuracy of approximation of our scheme on the finite horizon $[0,T]$, where $T=2$. In Section \[smalltimeexamplesec\], the value function was calculated at times $t =1.5$ and $t = 1.9$, which were close to T = 2. However, following the approximation scheme in Section \[schemesec\], we can now approximate the value function at time $t = 0$.
For comparison, we also compute a Merton approximation to the optimal portfolio. A naive Merton-like approximation for the value function is given by $$\label{mertonscheme}
\Umer(x):= -e^{-0.0001569674298} \frac{1}{2x^{2}},$$ where we have taken the process $Y_{t}$ to be fixed at the value $y=27.9345$ for all $t$. This was obtained by solving the Merton HJB equation $$\begin{cases}
v_{t} - \frac{1}{2} \lambda^{2}(y) \frac{v_{x}^{2}}{v_{xx}} = 0 \\
v(T,x) = -\frac{1}{2x^{2}},
\end{cases}$$ with $\lambda^{2}(y) = 0.0002354511446$ at $y = 27.9345$. The corresponding Merton trading strategy is then given by $$\pi^{\text{Mer}}(x) = -\frac{\lambda(y)\Umer_{x}}{\sigma(y)\Umer_{xx}},$$ which simplifies to $$\label{MertonPortfolio}
\pi^{\text{Mer}}(x) = 0.755162649999999x.$$
We now compare, at time $t = 0$, our approximation to the optimal portfolio, given by , against the actual optimal portfolio $\pi_{U}(x) \approx 0.745029x$ (obtained by substituting into and evaluating at $t=0$), and the Merton portfolio (see figures \[fig:schemeportsunrestricted\] and \[fig:schemeportsrestricted\]). We also graph the value function against our approximation to the value function given by (see figures \[fig:schemewealthunrestricted\] and \[fig:schemewealthrestricted\]).
![($t=0, \,T=2, \,n=4$) When the wealth interval of figure \[fig:schemewealthunrestricted\] is shortened. []{data-label="fig:schemewealthrestricted"}](schemeOneWealthUnrestricted.pdf)
![($t=0, \,T=2, \,n=4$) When the wealth interval of figure \[fig:schemewealthunrestricted\] is shortened. []{data-label="fig:schemewealthrestricted"}](schemeOneWealthRestricted.pdf)
![($t=0, \,T=2, \,n=4$) When the wealth interval of figure \[fig:schemeportsunrestricted\] is shortened, the accuracy of the approximation is more apparent.[]{data-label="fig:schemeportsrestricted"}](portfoliosSchemeZMOUT.pdf)
![($t=0, \,T=2, \,n=4$) When the wealth interval of figure \[fig:schemeportsunrestricted\] is shortened, the accuracy of the approximation is more apparent.[]{data-label="fig:schemeportsrestricted"}](portfoliosSchemeZMIN.pdf)
Appendix
========
Admissibility of $\pih$
-----------------------
\[admissibilitypihat\] Let $X^{\pih}_s$ be the wealth process given by the SDE under portfolio $\pih_{t} := \pih(t,X^{\pih}_{t},Y_{t})$ (with $\pih(t,x,y)$ as defined in ) and assume $X^{\pih}_{t} = x \in (0,\infty)$. Under Assumptions \[modelassumptions\] and \[utilityassumptions\], the strategy $\pih_{t}$ is admissible as defined in Definition \[admissibledefinition\].
We begin by noting that progressive measurability of $\pih_{t}$ follows from the continuity of this function in the variable $t$. In addition, Assumptions \[modelassumptions\] and \[utilityassumptions\] and the definition of $\pih(t,x,y)$ in imply $$\label{linearboundpihat}
\left | \sigma(y)\frac{\pih(t,x,y)}{x} \right| \leq C_{1}$$ for some constant $C_{1}$, which, along with , immediately implies that $\pih_{t}$ yields a strictly positive wealth process. It remains to be shown that $\pih_{t}$ satisfies $E \left [\displaystyle \int \limits_{0}^{T} \sigma_{s}^{2} \pih_{s}^{2} \,ds \right ] + E \left [\displaystyle \int \limits_{0}^{T} (X^{\pih}_{s})^{-2\gamma} \sigma_{s}^{2}\pih_{s}^{2} \,ds \right ] < \infty$, with $\gamma$ as in Definition \[admissibledefinition\]. Note that by , we have $|\sigma_{s}^{2} \pih_{s}^{2}| \leq c_{1}(X^{\pih}_{s})^{2}$ for some constant $c_{1}$, so it is enough to show that $(X^{\pih}_{\cdot})^{p}$ is integrable, where $p= 2$ or $p = -2\gamma$.
We apply Ito’s formula to $\log (X^{\pih}_{\cdot})^{p}$ to get $$\begin{aligned}
\log (X^{\pih}_u)^{p}&= p\log X^{\pih}_u\\
&=\log x^{p} + p\int_t^u\left[\sigma(Y_s)\lambda(Y_s)\frac{\pih_s}{X^{\pih}_s}-\frac{1}{2}\sigma^2(Y_s)\left(\frac{\pih_s}{X^{\pih}_s}\right)^2 \right]ds + p\int_t^u\sigma(Y_s)\frac{\pih_s}{X^{\pih}_s}dW^{1}_s.
\end{aligned}$$ Let $M_u:=p\int_t^u\sigma(Y_s)\frac{\pih_s}{X^{\pih}_s}dW^{1}_s$, for $t\leq u\leq T$. From the boundedness of $\left|\frac{\sigma(y)\pih(t,x,y)}{x}\right|$ and $\lambda(y)$, it follows that there exists $c_{2}>0$ such that $$\begin{aligned}
\log (X^{\pih}_u)^{p}&\leq \log x^{p}+c_{2}(u-t)+M_u\\
&\leq \log x^{p}+c_{2}T+M_u.
\end{aligned}$$ So $$(X^{\pih}_u)^{p}\leq x^{p}e^{c_{2}T}e^{M_u}.$$ Define $Z_u:=e^{M_u-\frac{1}{2}[M]_u}$ where $[M]_u=p^{2}\int_t^u \sigma_{s}^{2}\left(\frac{\pih_s}{X^{\pih}_s}\right)^2ds$. Since $\sigma(y)\frac{\pih(t,x,y)}{x}$ is a bounded function, $Z$ is a square-integrable martingale. Then for a constant $c_{3} > 0$, we can write $$\label{uniformbound}
(X^{\pih}_u)^{p}\leq x^{p}e^{{c_{3}} T}Z_u\leq x^{p}e^{c_{3} T}\sup_{t\leq u\leq T}Z_u.$$ Define $K:=x^{p}e^{c_{3} T}\sup_{t\leq u\leq T}Z_u$, then $$E|K|\leq x^{p}e^{c_{3} T}(1+4EZ^2_T)<\infty.$$ This shows $\{(X^{\pih}_u)^{p}\}_{u\in[t,T]}$ is bounded uniformly in $u$ by an integrable random variable, and thus establishes $$E \left [\displaystyle \int \limits_{0}^{T} \sigma_{s}^{2} \pih_{s}^{2} \,ds \right ] + E \left [\displaystyle \int \limits_{0}^{T} (X^{\pih}_{s})^{-2\gamma} \sigma_{s}^{2}\pih_{s}^{2} \,ds \right ] < \infty.$$
Uniform bound of $G(X^{\pi}_{T\wedge \tau_{n}})$
------------------------------------------------
\[boundedintfunc\] Let $X^{\pi}_s$ be the wealth process given by the SDE under the arbitrary admissible portfolio $\pi(t,X^{\pi}_{t},Y_{t})$ and assume $X^{\pi}_{t} = x$. Under the assumptions of Theorem \[maintheorem\], $\{G(X^{\pi}_{T\wedge \tau_{n}})\}_{n =1}^{\infty}$ is uniformly bounded by an integrable random variable, where $G(x) = \log(x) + 1$ under Case 1 of Assumption \[utilityassumptions\], and $G(x) = x^{1-\alpha} + x^{1-\beta}$ for positive $\alpha, \beta \neq 1$, under Case 2 of Assumption \[utilityassumptions\].
We will first consider case 1 where $G(x) = x^{1-\alpha} + x^{1-\beta}$. Note that it is enough to prove $\{(X^{\pi}_{T\wedge \tau_{n}})^{1-\gamma}\}_{n=1}^{\infty}$, for some positive $\gamma \neq 1$, is uniformly bounded by an integrable random variable.
If $0 < \gamma < 1$, then Young’s inequality gives $(X^{\pi}_{T\wedge \tau_{n}})^{1-\gamma} \leq C \left(1 + (X^{\pi}_{T\wedge \tau_{n}})^{2} \right)$ for some constant $C$. If we set $M_{u} := \int \limits_{t}^{u} \sigma_{s} \pi_{s} \,dW^{1}_{s}$, then $M_{u}$ is a martingale, and gives $$(X^{\pi}_{T\wedge \tau_{n}})^{2} \leq {c_{1}}\left (x^{2} + \int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} \lambda_{s}^{2} \,dt + \left(\sup \limits_{u \in [t,T]} M_{u}\right)^{2} \right )$$ for some constant ${c_{1}}$. In particular, taking expectation yields, by Doob’s maximal inequality, $$E\left [(X^{\pi}_{T\wedge \tau_{n}})^{2} \right ] \leq {c_{2}} \left (1 + E \left [(M_{T})^{2} \right ] \right) = {c_{2}}\left (1 + E\left [\int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} \,ds \right ] \right) < \infty,$$ for some constant ${c_{2}}$. Therefore, $\{(X^{\pi}_{T\wedge \tau_{n}})^{2}\}_{n=1}^{\infty}$ is uniformly bounded by the integrable random variable $\xi := {c_{1}}\left (1 + x^{2} + \displaystyle \int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} \lambda_{s}^{2} \,ds + \left(\sup \limits_{u \in [t,T]} M_{u}\right)^{2} \right )$, implying $(X^{\pi}_{T\wedge \tau_{n}})^{1-\gamma} \leq C \left(1 + (X^{\pi}_{T\wedge \tau_{n}})^{2} \right)$ is uniformly bounded in $n$ by an integrable random variable.
In the case of $\gamma > 1$, we apply Ito’s formula to $(X^{\pi}_{u})^{1-\gamma}$ to obtain $$(X^{\pi}_{u})^{1-\gamma} = x^{1-\gamma} + (1-\gamma) \int \limits_{t}^{u} \lambda_{s} \sigma_{s} \pi_{s} (X^{\pi}_{s})^{-\gamma} - \frac{(1-\gamma)\gamma}{2} \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-\gamma -1} \,ds + (1-\gamma)\int \limits_{t}^{u} \sigma_{s} \pi_{s} (X^{\pi}_{s})^{-\gamma} \,dW^{1}_{s}.$$ Thus, letting $Z_{u} := \int \limits_{t}^{u} \sigma_{s} \pi_{s} (X^{\pi}_{s})^{-\gamma} \,dW^{1}_{u}$, which is a martingale by the admissibility of $\pi_{s}$ (see Definition \[admissibledefinition\]), we see that $$\label{ItoBound}
(X^{\pi}_{u})^{1-\gamma} \leq {c_{3}}\left(1 + \int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-2\gamma} + \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-2} \,ds + \left (\sup \limits_{s \in [0,T]} Z_{u}\right)^{2}\right),$$ for some constant ${c_{3}}$. Taking the expectation of both sides and applying Doob’s maximal inequality gives $$E[(X^{\pi}_{u})^{1-\gamma}] \leq {c_{4}}\left(1 + E\left[\int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-2\gamma} + \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-2} \,ds\right] + E[Z_{T}^{2}] \right).$$ As $E[Z_{T}^{2}] = E\left [ \displaystyle \int \limits_{0}^{T} \sigma_{s}^{2} \pi_{s}^{2} (X^{\pi}_{s})^{-2\gamma} \,ds \right] < \infty$, the right hand side of the above inequality is finite. Thus, the right hand side of inequality serves as the uniform bound for $(X^{\pi}_{T\wedge \tau_{n}})^{1-\gamma}$ when $\gamma > 1$.
To prove the result in the case that $G(x) = 1 + \log(x)$, it is enough to show $\{\log(X^{\pi}_{T\wedge \tau_{n}})\}_{n=}^{\infty}$ is uniformly bounded by an integrable random variable. We apply Ito’s formula to $\log (X^{\pi}_{\cdot})$ to get $$\log (X^{\pi}_u) =\log x + \int_t^u\left[\sigma(Y_s)\lambda(Y_s)\frac{\pi_s}{X^{\pi}_s}-\frac{1}{2}\sigma^2(Y_s)\left(\frac{\pi_s}{X^{\pi}_s}\right)^2 \right]ds + \int_t^u \sigma(Y_s)\frac{\pi_s}{X^{\pi}_s}dW^{1}_s.$$ As in the power case, we let $\E_{u} := \displaystyle \int_{t}^{u} \sigma(Y_{s})\frac{\pi_{s}}{X_{s}^{\pi}}dW^{1}_s$. Then, by the admissibility of $\pi_{s}$ in Definition \[admissibledefinition\], $\E_{u}$ is a martingale, and so we can apply Doob’s maximal inequality as above. In particular, we have $$\label{ItoBound2}
\log(X^{\pi}_{u}) \leq {c_{5}}\left(1 + \int \limits_{0}^{T} \sigma^{2} \pi^{2} (X^{\pi}_{s})^{-2}\,ds + \left (\sup \limits_{s \in [t,T]} \E_{s}\right )^{2}\right)$$ for some constant ${c_{5}}$. Thus, taking expectations gives $$E[\log(X^{\pi}_{u})] \leq {c_{5}}\left (1 + E \left [\displaystyle \int \limits_{0}^{T} \sigma^{2} \pi^{2} (X^{\pi}_{s})^{-2} \,ds \right ] + E\left [\left (\sup \limits_{s \in [t,T]} \E_{s}\right )^{2} \right] \right ),$$ and Doob’s inequality gives, for some constant ${c_{6}}$, $$E \left [\left (\sup \limits_{s \in [t,T]} \E_{s}\right)^{2} \right ] \leq {c_{6}} E[\E_{T}^{2}] = {c_{6}}E\left [ \displaystyle \int \limits_{t}^{T} \sigma^{2} \pi^{2} (X^{\pi}_{s})^{-2} \,ds \right] < \infty,$$ with the last inequality following by the definition of admissibility given in Definition \[admissibledefinition\]. This establishes the right hand side of as the integrable random variable which uniformly bounds $\{\log\left(X^{\pi}_{T \wedge \tau_{n}}\right)\}_{n = 1}^{\infty}$.
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[^1]: [[email protected]]{}
[^2]: [[email protected]]{}
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---
abstract: 'We perform a fully nonlinear analysis of superhorizon perturbation in Hořava-Lifshitz gravity, based on the gradient expansion method. We present a concrete expression for the solution of gravity equations up to the second order in the gradient expansion, and prove that the solution can be extended to any order. The result provides yet another example for analogue of the Vainshtein effect: the nonlinear solution is regular in the limit $\lambda\to 1$ and recovers general relativity coupled to dark matter at low energy. Finally, we propose a definition of nonlinear curvature perturbation ${\cal R}$ in Hořava-Lifshitz gravity and show that it is conserved up to the first order in the gradient expansion.'
author:
- Keisuke Izumi
- Shinji Mukohyama
title: 'Nonlinear superhorizon perturbations in Hořava-Lifshitz gravity'
---
Introduction
============
The new theory of gravitation proposed recently by Hořava [@Horava:2009uw; @Mukohyama:2010xz] is expected to be renormalizable and unitary. For this reason, it has been attracting significant amount of attention [@Charmousis:2009tc; @Koyama:2009hc; @Mukohyama:2009mz; @Mukohyama:2009gg]. The theory is power-counting renormalizable because of the anisotropic scaling in the ultraviolet (UV), $$\begin{aligned}
t \to b^z t, \qquad \Vec x \to b \Vec x,
\label{scalling}\end{aligned}$$ with the dynamical critical exponent $z\ge 3$. Since this scaling is called Lifshitz scaling, the theory is often called Hořava-Lifshitz gravity.
Because of the anisotropic scaling (\[scalling\]), the time and the space in this theory must be treated separately. Thus, we must abandon the $4$-dimensional diffeomorphism invariance. Instead, the fundamental symmetry of the theory is the invariance under the so-called foliation-preserving diffeomorphism: $$t \to t'(t), \quad \vec{x}\to\vec{x}'(t,\vec{x}), \label{foliation}$$ which preserves the way the spacetime is foliated by constant-time hypersurfaces. Since this symmetry is smaller than the counterpart of general relativity i.e. the $4$-dimensional diffeomorphism invariance, the structure of the action is less restrictive and allows more parameters. For example, the kinetic part of the action is a linear combination of $K^2$ and $K^{ij}K_{ij}$ with arbitrary coefficients, where $K_{ij}$ is the extrinsic curvature of the constant-time hypersurface and $K=K^i_{\ i}$. Hence, the kinetic part can be written as $$\begin{aligned}
I_{kin}=\frac{M_{Pl}^2}{2}\int Ndt \, \sqrt{g}d^3\vec{x}
\left(K^{ij} K_{ij} - \lambda K^2\right),\end{aligned}$$ where $M_{Pl}$ is the reduced Planck mass and $\lambda$ is an arbitrary parameter. While in general relativity the value of $\lambda$ is fixed to unity due to the 4-dimensional diffeomorphism invariance, in Hořava-Lifshitz gravity any value of $\lambda$ is consistent with the foliation-preserving diffeomorphism invariance.
In Hořava-Lifshitz gravity, in addition to the usual tensor gravitons, there is an extra physical degree of freedom called the scalar graviton [@Horava:2009uw]. In order for the scalar graviton not to be a ghost, the regime $1/3<\lambda<1$ should be excluded [@Charmousis:2009tc]. Outside this forbidden interval, the scalar graviton has negative sound speed squared. Therefore, in order for the theory to be observationally viable, we need to impose the condition under which the associated long-distance instability does not show up [@Mukohyama:2010xz]. The condition essentially says that $\lambda$ must be sufficiently close to $1$ in the infrared (IR), and should be considered as a phenomenological constraint on properties of the renormalization group (RG) flow. Since the value of $\lambda$ continuously changes by the RG flow, only the regime $\lambda>1$ is allowed.
When $\lambda$ is very close to $1$, the scalar graviton gets strongly coupled [@Koyama:2009hc]. If we adopt the usual metric perturbation method then we find that higher order terms in the time kinetic part of the action for the scalar graviton become larger. This indicates the breakdown of the perturbative expansion in the scalar graviton sector. Note that this does not necessarily imply the loss of predictability since all coefficients of infinite number of terms can be written in terms of finite parameters in the action if the theory is renormalizable. However, because of the breakdown of the perturbative expansion, we need to employ a more or less non-perturbative method to analyze the fate of the scalar graviton in the limit $\lambda\to 1$. Such an analysis was performed in [@Mukohyama:2010xz] for spherically symmetric, static, vacuum configurations and it was shown that the limit is continuous and recovers general relativity. This may be considered as an analogue of the Vainshtein effect [@Vainshtein:1972sx].
The main purpose of this paper is to provide yet another example of the analogue of the Vainshtein effect in Hořava-Lifshitz gravity. For this purpose, we perform a fully nonlinear analysis of superhorizon cosmological perturbation, adopting the so-called gradient expansion method [@Salopek:1990jq]. The result is obviously continuous in the limit $\lambda\to 1$ and recovers general relativity coupled to dark matter.
We also propose a definition of nonlinear curvature perturbation ${\cal R}$ in Hořava-Lifshitz gravity and show that it is conserved up to the first order in the gradient expansion.
The paper is organized as follows. In § \[basic\], we briefly review the basic equations in Hořava-Lifshitz gravity. In § \[solutions\], we introduce the gradient expansion method in this theory and present the solution to equations of motion. In § \[CP\], we proposed a definition of nonlinear curvature perturbation ${\cal R}$ and show that it is conserved up to the first order in the gradient expansion. § \[sec:summary\] is devoted to a summary of this paper. In appendix \[proof\], we prove that the solution obtained in § \[solutions\] satisfies the momentum constraint in any order of the gradient expansion.
Basic equations {#basic}
===============
In this section we review the basic equations of Hořava-Lifshitz gravity, following the notation in [@Mukohyama:2010xz], and reformulate them in a way suitable for gradient expansion. Basic quantities of Hořava-Lifshitz gravity are the lapse $N(t)$, the shift $N^i(t,\vec{x})$ and the $3$-dimensional spatial metric $g_{ij}(t,\vec{x})$. Combining these quantities, we can construct $4$-dimensional spacetime metric in the ADM form as $$\begin{aligned}
ds^2= -N^2dt^2+g_{ij}(dx^i+N^idt)(dx^j+N^jdt).
\label{metric}\end{aligned}$$ The fundamental symmetry of the theory is the invariance under the so called foliation preserving diffeomorphism: $$t \to t'(t), \quad \vec{x}\to\vec{x}'(t,\vec{x}),$$ which preserves the way the spacetime is foliated by constant-time hypersurfaces.
By requiring invariance under spatial parity and time reflection, the gravitational action is specified as $$\begin{aligned}
I_g=\frac{M_{Pl}^2}{2}\int Ndt \, \sqrt{g}d^3\vec{x}
\left(K_{ij} K^{ij} - \lambda K^2 -2\Lambda+R+L_{z>1}\right) ,
\label{eq:action}\end{aligned}$$ where $\sqrt{g}$ is the determinant of $g_{ij}$, $K_{ij}$ is the extrinsic curvature defined as $$\begin{aligned}
K_{ij}=\frac{1}{2N}\left(
\partial_t g_{ij}-D_i N_j-D_j N_i
\right) ,\end{aligned}$$ $K$ ($=g^{ij}K_{ij}$) is the trace of $K_{ij}$, $D_i$ is the spatial covariant derivative compatible with $g_{ij}$, and $R$ is the Ricci scalar constructed from $g_{ij}$. To lower and raise an index, $g_{ij}$ and its inverse $g^{ij}$ are used. We impose the so called projectability condition, namely we require that the lapse function should depend only on time. In order to realize the power-counting renormalizability, the higher curvature Lagrangian $L_{z>1}$ should include up to sixth or higher spatial derivatives. For our analysis in this paper, we do not need to specify the concrete form of $L_{z>1}$. Moreover, for simplicity we shall not include matter action and analyze the pure gravity described by the action (\[eq:action\]).
The equation of motion for $g_{ij}$ is ${\cal E}_{g ij}=0$, where $$\begin{aligned}
{\cal E}_{g ij} & \equiv &
g_{ik}g_{jl}\frac{2}{N\sqrt{g}}
\frac{\delta I_g}{\delta g_{kl}}\nonumber\\
& = & M_{Pl}^2
\left[
-\frac{1}{N}(\partial_t-N^kD_k)p_{ij}
+ \frac{1}{N}(p_{ik}D_jN^k+p_{jk}D_iN^k)
\right.\nonumber\\
& & \left.
- Kp_{ij} + 2K_i^kp_{kj}
+ \frac{1}{2}g_{ij}K^{kl}p_{kl}-\Lambda g_{ij}
- G_{ij}\right]
+ {\cal E}_{z>1 ij}.\end{aligned}$$ Here, $p_{ij}\equiv K_{ij}-\lambda Kg_{ij}$, ${\cal E}_{z>1 ij}$ is the contribution from $L_{z>1}$ and $G_{ij}$ is Einstein tensor of $g_{ij}$. The trace part and traceless part of this equation are, respectively, $$(3\lambda-1)\left(\partial_{\perp}K+\frac{1}{2}K^2\right)
+ \frac{3}{2}A^i_{\ j}A^j_{\ i} + Z = 0,
\label{eq:Tr}$$ and $$\partial_{\perp}A^i_{\ j} + KA^i_{\ j}
+ \frac{1}{N}(A^k_{\ j}\partial_kN^i-A^i_{\ k}\partial_jN^k)
-\left(Z^i_{\ j}-\frac{1}{3}Z\delta^i_j\right) = 0,
\label{eq:Trless}$$ where $$A^i_{\ j} \equiv K^i_{\ j} - \frac{1}{3}K\delta^i_j$$ is the traceless part of $K^i_{\ j}$, $$\partial_{\perp} \equiv \frac{1}{N}(\partial_t-N^k\partial_k),$$ and $$Z^i_{\ j} \equiv -\Lambda\delta^i_j - G^i_{\ j}
+ M_{Pl}^{-2} g^{ik}{\cal E}_{z>1 kj}, \quad
Z = Z^i_{\ i} = -3\Lambda + \frac{1}{2}R
+ M_{Pl}^{-2}g^{ij}{\cal E}_{z>1 ij}.
\label{eq:defZ}$$ The foliation preserving diffeomorphism includes the three dimensional spatial diffeomorphism as a part of it. As a result, $Z^i_{\ j}$ satisfies the generalized Bianchi identity, $$\begin{aligned}
D_j Z^j_{\ i}=0.\label{eq:DZ}\end{aligned}$$
For convenience, we decompose the spatial metric and the extrinsic curvature as $$\begin{aligned}
g_{ij} & = & a^2(t) e^{2\zeta(t,\vec{x})} \gamma_{ij}(t,\vec{x}), \\
K^i_{\ j} & = & \frac{1}{3} K(t,\vec{x}) \delta^i_{\ j}
+A^i_{\ j}(t,\vec{x}),
\label{eq:decomposition}\end{aligned}$$ where we define $\zeta(t,\vec{x})$ so that $\det\gamma =1$, and $a(t)$ is defined in eq.(\[defa\]) later. The trace part and the traceless part of the definition of the extrinsic curvature lead, respectively, to $$\partial_{\perp} \zeta+ \frac{\partial_t a}{Na}
= \frac{1}{3}\left(K +\partial_iN^i\right),
\label{eq:TrK}$$ and $$\partial_{\perp}\gamma_{ij}
= 2\gamma_{ik}A^k_{\ j}
+ \frac{1}{N}
\left(\gamma_{jk}\partial_iN^k
+ \gamma_{ik}\partial_jN^k
-\frac{2}{3}\gamma_{ij}\partial_kN^k
\right).
\label{eq:TrlessK}$$
The momentum constraint, i.e. the equation of motion for $N^i$, is $$D_jK^j_{\ i}-\lambda \partial_iK = 0.$$ According to the decomposition (\[eq:decomposition\]), the momentum constraint is rewritten as $$\partial_jA^j_{\ i} + 3A^j_{\ i}\partial_j\zeta
- \frac{1}{2}A^j_{\ l}\gamma^{lk}\partial_{i}\gamma_{jk}
- \frac{1}{3}\left(3\lambda-1\right)\partial_iK = 0.
\label{eq:momentum-constraint}$$
As a consistency check, it is instructive to calculate time derivatives of $A^i_{\ i}$, $\ln\det\gamma$, $\gamma_{ij}-\gamma_{ji}$ and $\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i}$ without using the fact that they actually vanish. The results are $$\begin{aligned}
\partial_{\perp} A^i_{\ i} & = & -KA^i_{\ i}, \nonumber\\
\partial_{\perp}(\ln\det\gamma) & = &
2 A^i_{\ i}, \nonumber\\
\partial_{\perp}(\gamma_{ij}-\gamma_{ji})
& = & 2(\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i}),
\nonumber\\
\partial_{\perp}(\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i})
& = & -K(\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i})
+ 2(\gamma_{il}A^l_{\ k}-\gamma_{kl}A^l_{\ i})A^k_{\ j}
- 2(\gamma_{jl}A^l_{\ k}-\gamma_{kl}A^l_{\ j})A^k_{\ i}
\nonumber\\
& &
+ 2(\gamma_{kl}-\gamma_{lk})A^k_{\ i}A^l_{\ j}
+\frac{1}{N}(\gamma_{ik}A^k_{\ l}-\gamma_{lk}A^k_{\ i})
\partial_jN^l
-\frac{1}{N}(\gamma_{jk}A^k_{\ l}-\gamma_{lk}A^k_{\ j})
\partial_iN^l \nonumber\\
& &
- \frac{2}{3}(\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i})
\partial_lN^l
- \frac{1}{3}Z(\gamma_{ij}-\gamma_{ji}).\end{aligned}$$ The right hand side of each equation vanishes when $A^i_{\ i}$, $\ln\det\gamma$, $\gamma_{ij}-\gamma_{ji}$ and $\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i}$ vanish. Therefore, the evolution equations we have derived are consistent with vanishing $A^i_{\ i}$, $\ln\det\gamma$, $\gamma_{ij}-\gamma_{ji}$ and $\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i}$.
Gradient expansion {#solutions}
==================
The gradient expansion is the method to analyze the full non-linear dynamics at large scale. In the gradient expansion, we consider nonlinear perturbation around a flat Friedmann-Robertson-Walker background and suppose that the characteristic spatial scale $L$ of the perturbation is much larger than the Hubble horizon scale $1/H$. To make the argument transparent, we introduce a small parameter $\epsilon$ defined by $\epsilon\sim 1/(HL)$ and expand all relevant quantities and equations with respect to $\epsilon$. For example, a spatial derivative acted on a relevant quantity raises the order of $\epsilon$ and thus is counted as $O(\epsilon)$. We then solve the equations order by order in gradient expansion.
Gauge fixing
------------
The foliation preserving diffeomorphism invariance is, like all other gauge symmetries, redundancy of descriptions. In order to extract physical quantities and statements, we thus need to eliminate gauge freedom by imposing appropriate gauge condition. In this paper we adopt the synchronous gauge, or the Gaussian normal coordinate system, by setting the lapse to unity and the shift to zero. $$\begin{aligned}
N=1, \ N^i= 0.\label{eq:gaugefix}\end{aligned}$$ This fixes the time coordinates but does not completely fix the spatial coordinates. There still remains gauge freedom of time-independent spatial diffeomorphism, corresponding to the change of coordinates on the initial constant-time hypersurface. This residual gauge degree of freedom will be discussed later.
In this gauge our basic equations (\[eq:Tr\]), (\[eq:Trless\]), (\[eq:TrK\]) and (\[eq:TrlessK\]) are simplified as $$\begin{aligned}
(3\lambda-1) \partial_t K & = &
-\frac{1}{2}(3\lambda-1)K^2 -\frac{3}{2}A^i_{\ j}A^j_{\ i} - Z,
\label{eq:K}\\
\partial_t A^i_{\ j} & = &
-KA^i_{\ j}+Z^i_{\ j}-\frac{1}{3}Z\delta^i_{\ j},\label{eq:A}\\
\partial_t \zeta & = &
-\frac{\partial_t a}{a}+\frac{1}{3}K,\label{eq:psi}\\
\partial_t \gamma_{ij} & = &
2\gamma_{ik}A^k_{\ j}.\label{eq:evogama}\end{aligned}$$ Hereafter we assume that $\lambda\ne 1/3$. Actually, as already explained in the introduction, the regime of physical interest is $\lambda>1$.
Order analysis
--------------
In order to expand the equations and to write down equations in each order of gradient expansion, we need to know the orders of all relevant variables. Therefore, we begin with the order analysis to determine them.
Since we are interested in the spacetime which is not so much different from the exact Friedmann universe, we suppose that $$\begin{aligned}
\partial_t \gamma_{ij} = O(\epsilon).\end{aligned}$$ Substituting this into eq.(\[eq:evogama\]), we obtain $$\begin{aligned}
A^i_{\ j}=O(\epsilon).\end{aligned}$$ Then, the constraint equation (\[eq:momentum-constraint\]) implies that $$\begin{aligned}
\partial_i K=O(\epsilon^2).\end{aligned}$$ In other words, $K^{(0)}$ depends on $t$ only. This fact enables us to define $a(t)$ by $$\begin{aligned}
3 \frac{\partial_t a(t)}{a(t)}= K^{(0)} (\equiv 3 H(t)).
\label{defa}\end{aligned}$$ With this definition of $a(t)$, eq. (\[eq:psi\]) leads to $$\begin{aligned}
\partial_t \zeta=O(\epsilon). \end{aligned}$$ In summary we have the following expansion. $$\begin{aligned}
\zeta & = & \zeta^{(0)}(\vec{x}) +\epsilon \zeta^{(1)}(t,\vec{x})+
\epsilon^2 \zeta^{(2)}(t,\vec{x})+\cdots , \\
\gamma_{ij} & = & f_{ij}(\vec{x})
+\epsilon \gamma_{ij}^{(1)}(t,\vec{x})+
\epsilon^2 \gamma_{ij}^{(2)}(t,\vec{x})+\cdots ,\\
K & = & 3 H(t)+\epsilon K^{(1)}(t,\vec{x})
+ \epsilon^2 K^{(2)}(t,\vec{x})+ \cdots, \\
A^i_{\ j} & = & \epsilon A^{(1)\, i}_{\ \ \ \ \, j}(t,\vec{x}) +\epsilon^2
A^{(2)\, i}_{\ \ \ \ \, j}(t,\vec{x}) + \cdots, \end{aligned}$$ where a quantity with the upper index $(n)$ is $n$-th order in gradient expansion.
Equations in each order
-----------------------
We have found the orders of all physical quantities. Substituting this into the evolution equations (\[eq:K\]-\[eq:evogama\]), we can obtain the evolution equations in each order.
In the zero-th order of gradient expansion we have $$(3\lambda-1)\left(\partial_t H + \frac{3}{2}H^2\right)
= \Lambda.
\label{0thii}$$ The first integral of this equation leads to $$3H^2 = \frac{2\Lambda}{3\lambda-1} + \frac{\tilde{C}}{a^3},$$ where $\tilde{C}$ is an integration constant. The second term in the right hand side of this equation is the “dark matter as an integration constant” [@Mukohyama:2009mz].
The $n$-th ($n\geq 1$) order equations are written as $$\begin{aligned}
a^{-3}\partial_t \left(a^3 K^{(n)}\right)
& = &
-\frac{1}{2}\sum_{p=1}^{n-1} K^{(p)}K^{(n-p)}
-\frac{3}{2(3\lambda-1)}\sum_{p=1}^{n-1}A^{(p)\, i}_{\ \ \ \ \, j}
A^{(n-p)\, j}_{\qquad\ \ i}- \frac{Z^{(n)}}{3\lambda-1},
\label{eq:n-K}\\
a^{-3}\partial_t\left( a^3 A^{(n)\, i}_{\ \ \ \ \, j}\right)
& = &
-\sum_{p=1}^{n-1}K^{(p)}A^{(n-p)\, i}_{\qquad\ \ j}
+Z^{(n)\, i}_{\ \ \ \ \, j}-\frac{1}{3}Z^{(n)}\delta^i_{\ j},
\label{eq:n-A}\\
\partial_t \zeta^{(n)}
& = &
\frac{1}{3} K^{(n)},\label{eq:n-zeta}\\
\partial_t \gamma^{(n)}_{ij}
& = &
2 \sum_{p=0}^{n-1}\gamma^{(p)}_{ik}
A^{(n-p)\, k}_{\qquad\ \ j},
\label{eq:n-gamma}\end{aligned}$$ In a similar way, from eq.(\[eq:momentum-constraint\]) we obtain the ($n+1$)-th ($n\geq 1$) order momentum constraint equation as $$\begin{aligned}
\partial_j A^{(n)\, j}_{\ \ \ \ \ i}
+3 \sum_{p=1}^n A^{(p)\, j}_{\ \ \ \ \ i}\partial_j\zeta^{(n-p)}
- \frac{1}{2}\sum_{p=1}^n\sum_{q=0}^{n-p}
A^{(p)\, j}_{\ \ \ \ \ l}(\gamma^{-1})^{(q)\, lk}
\partial_{i}\gamma^{(n-p-q)}_{jk}
-\frac{1}{3}(3\lambda-1)\partial_i K^{(n)}
=0,
\label{eq:n-const}\end{aligned}$$ where $(\gamma^{-1})^{(n)\, ij}$ is the $n$-th order part of the inverse of $\gamma_{ij}$, i.e. the inverse $(\gamma^{-1})^{ij}$ is expanded as $$(\gamma^{-1})^{ij} =
f^{ij}
+ \epsilon (\gamma^{-1})^{(1)\, ij}
+ \epsilon^2 (\gamma^{-1})^{(2)\, ij}
+ \cdots,$$ where $f^{ij}=(\gamma^{-1})^{(0)\, ij}$ is the inverse of $f_{ij}$. It is easy to show that $(\gamma^{-1})^{(n)\, ij}$ ($n\geq 1$) satisfies the following differential equation. $$\partial_t (\gamma^{-1})^{(n)\, ij} =
-2\sum_{p=1}^n
A^{(p)\, i}_{\quad\ \ k}
(\gamma^{-1})^{(n-p)\, kj}.
\label{eqn:gamma-inv-nth-eq}$$
There are some useful identities. The generalized Bianchi identity (\[eq:DZ\]) leads to $$\partial_j Z^{(n)\, j}_{\quad\ \ i}
+ 3\sum_{p=0}^n
\left(Z^{(p)\, j}_{\quad\ \ i}
-\frac{1}{3}Z^{(p)}\delta^j_{\ i}\right)\partial_j\zeta^{(n-p)}
-\frac{1}{2}\sum_{p=0}^n\sum_{q=0}^{n-p}
Z^{(p)\, j}_{\quad\ \ l}(\gamma^{-1})^{(q)\, lk}
\partial_i\gamma^{(n-p-q)}_{jk} = 0, \label{eq:nth-DZ}$$ where $Z^{(n)\, i}_{\quad\ \ j}$ and $Z^{(n)}$ are the $n$-th order parts of $Z^i_{\ j}$ and $Z$, respectively. The conditions $A^i_{\ i}=0$, $\partial_i\ln\det\gamma=0$, $\gamma_{ij}-\gamma_{ji}=0$, $\gamma_{ik}A^k_{\ j}-\gamma_{jk}A^k_{\ i}=0$ and $A^i_{\ j}-\gamma_{jk}A^k_{\ l}(\gamma^{-1})^{li}=0$ lead to the following identities. $$\begin{aligned}
& &
A^{(n)\, i}_{\quad\ \ i} = 0, \quad
\sum_{p=0}^n(\gamma^{-1})^{(p)\, jk}
\partial_i\gamma^{(n-p)}_{jk} = 0, \quad
\gamma^{(n)}_{ij}-\gamma^{(n)}_{ji}=0, \nonumber\\
& &
\sum_{p=0}^{n-1}
\left( \gamma^{(p)}_{ik}A^{(n-p)\, k}_{\qquad\ \ j}
-\gamma^{(p)}_{jk}A^{(n-p)\, k}_{\qquad\ \ i}
\right) = 0, \quad
A^{(n)\, i}_{\quad\ \ j} -
\sum_{p=0}^{n-1} \sum_{q=0}^{n-p-1}
\gamma^{(p)}_{jk}A^{(n-p-q)\, k}_{\qquad\quad\quad l}
(\gamma^{-1})^{(q)\, li}
= 0.
\label{eqn:nth-identities}\end{aligned}$$
$O(\epsilon)$ solution {#orderepsilon}
----------------------
For the first order ($n=1$), eqs.(\[eq:n-K\]-\[eq:n-gamma\]) are reduced to $$\begin{aligned}
\partial_t \left( a^3 K^{(1)} \right) & = & 0,
\label{eq:leading-Tr-EoM} \\
\partial_t \left( a^3 A^{(1)\, i}_{\ \ \ \ \, j} \right) & = & 0,
\label{eq:leading-Trless-EoM}\\
\partial_t \zeta^{(1)} & = & \frac{1}{3} K^{(1)},
\label{eq:leading-TrK}\\
\partial_t \gamma^{(1)}_{ij} & = &
2 f_{ik} A^{(1)\, k}_{\ \ \ \ \, j}.
\label{eq:leading-TrlessK}\end{aligned}$$ Note that $Z^{(1)i}_{\quad\ j}=0$. Integrating these equations, we obtain $$\begin{aligned}
K^{(1)} & = & \frac{C^{(1)}}{a(t)^3},
\label{K1}\\
A^{(1)\, i}_{\ \ \ \ \, j} & = &
\frac{C^{(1)\, i}_{\ \ \ \ \, j}}{a(t)^3},\\
\zeta^{(1)} & = &
\frac{C^{(1)}}{3}
\int^t_{t_{in}} \frac{dt'}{a^3(t')} + \zeta^{(1)}_{in},\\
\label{zeta1}
\gamma^{(1)}_{ij} & = &
2 f_{ik}C^{(1)\, k}_{\ \ \ \ \, j}
\int^t_{t_{in}}\frac{dt'}{a^3(t')}
+ \gamma^{(1)}_{in\, ij},
\label{gamma1}\end{aligned}$$ where the integration constants $C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(1)}_{in}$ and $\gamma^{(1)}_{in\, ij}$ depend on the spatial coordinates $\vec{x}^i$ only, and satisfy $$C^{(1)\, i}_{\ \ \ \ \, i}=0,\quad
f_{ik}C^{(1)\, k}_{\ \ \ \ \, j} = f_{jk}C^{(1)\, k}_{\ \ \ \ \, i}.$$ The two integration constants, $\zeta^{(1)}_{in}$ and $\gamma^{(1)}_{in\, ij}$, can be absorbed into the zero-th order counterparts, $\zeta^{(0)}_{in}$ and $\gamma^{(0)}_{in\, ij}$. Thus, without loss of generality, we can set $$\zeta^{(1)}_{in} = 0, \quad \gamma^{(1)}_{in\, ij}=0.$$ Finally, the momentum constraint equation (\[eq:n-const\]) with $n=1$ leads to the following relation among the remaining integration constants, $C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(0)}$ and $f_{ij}$. $$\partial_j C^{(1)\, j}_{\ \ \ \ \, i}
+ 3C^{(1)\, j}_{\ \ \ \ \, i}\partial_j\zeta^{(0)}
- \frac{1}{2}C^{(1)\, j}_{\ \ \ \ \, l}f^{lk}\partial_i f_{jk}
- \frac{1}{3}\left(3\lambda-1\right)\partial_iC^{(1)} = 0,
\label{const-1}$$ where $f^{ij}$ is the inverse of $f_{ij}$.
$O(\epsilon^2)$ solution {#subsec:1storder}
------------------------
With the first order solution obtained in the previous subsection, we can solve the second order equations. In the second order, eqs.(\[eq:n-K\]-\[eq:n-gamma\]) become $$\begin{aligned}
a^{-3}\partial_t \left(a^3 K^{(2)}\right)
& = & -\frac{1}{2}(K^{(1)})^2
-\frac{3}{2((3\lambda-1))}
A^{(1)\, i}_{\ \ \ \ \, j}A^{(1)\, j}_{\ \ \ \ \, i}
-\frac{1}{2(3\lambda-1)}a^{-2}\tilde{R} ,\\
a^{-3}\partial_t\left( a^3 A^{(2)\, i}_{\ \ \ \ \, j}\right)
& = &
-K^{(1)}A^{(1)\, i}_{\ \ \ \ \, j}
-a^{-2}
\left(\tilde{R}^i_{\ j}-\frac{1}{3}\tilde{R}\delta^i_{\ j}\right),\\
\partial_t \zeta^{(2)} & = & \frac{1}{3} K^{(2)} ,\\
\partial_t \gamma^{(2)}_{ij} & = &
2 \left( f_{ik} A^{(2)\, k}_{\ \ \ \ \, j}
+\gamma^{(1)}_{ik} A^{(1)\, k}_{\ \ \ \ \, j}\right),\end{aligned}$$ where $\tilde{R}^i_{\ j}$ and $\tilde{R}$ are Ricci tensor and Ricci scalar constructed from the 0-th order conformally-transformed metric $a^{-2}g^{(0)}_{ij}=e^{2\zeta^{(0)}} f_{ij}$, and we have used the fact that $Z^{(2)\ i}_{\qquad j}=-a^{-2}
\left(\tilde{R}^i_{\ j}-\frac{1}{2}\tilde{R}\delta^i_{\ j}\right)$. By integrating these equations we obtain $$\begin{aligned}
K^{(2)} & = & -\frac{1}{2a^3(t)}
\left\{
\left[\left(C^{(1)}\right)^2
+ \frac{3}{3\lambda-1}
C^{(1)\, i}_{\ \ \ \ \, j}C^{(1)\, j}_{\ \ \ \ \, i}
\right]\int^t_{t_{in}}\frac{dt'}{a^3(t')}
+ \frac{\tilde{R}}{3\lambda-1}
\int^t_{t_{in}} a(t') dt'
\right\},\label{eq:2ndK}\\
A^{(2)\, i}_{\ \ \ \ \, j} & = &
-\frac{1}{a^3(t)}
\left\{
C^{(1)}C^{(1)\, i}_{\ \ \ \ \, j}
\int^t_{t_{in}}\frac{dt'}{a^3(t')}
+ \left(\tilde R^{i}_{\ j}
-\frac{1}{3}\tilde R \delta^i_{\ j}\right)
\int^t_{t_{in}} a(t') dt'
\right\},\\
\zeta^{(2)} & = &
-\frac{1}{6}
\left\{
\left[\left(C^{(1)}\right)^2
+\frac{3}{3\lambda-1}
C^{(1)\, i}_{\ \ \ \ \, j}C^{(1)\, j}_{\ \ \ \ \, i}
\right]
\int^t_{t_{in}}\frac{dt'}{a^3(t')}\int^{t'}_{t_{in}}\frac{dt''}{a^3(t'')}
+\frac{\tilde{R}}{3\lambda-1}
\int^t_{t_{in}}\frac{dt'}{a^3(t')}
\int^{t'}_{t_{in}} a(t'') dt''\right\}, \label{zeta2}
\\
\gamma^{(2)}_{ij} & = & 2f_{ik}
\left[
\left(
2C^{(1)\, k}_{\ \ \ \ \, l}C^{(1)\, l}_{\ \ \ \ \, j}-
C^{(1)}C^{(1)\, k}_{\ \ \ \ \, j} \right)
\int^t_{t_{in}}\frac{dt'}{a^3(t')}\int^{t'}_{t_{in}}\frac{dt''}{a^3(t'')}
-\left(\tilde R^{ k}_{\ j}
-\frac{1}{3}\tilde R \delta^k_{\ j}\right)
\int^t_{t_{in}}\frac{dt'}{a^3(t')}\int^{t'}_{t_{in}} a(t'') dt''
\right], \label{eq:2ndgamma}\end{aligned}$$ where we have set $$\left. K^{(2)}\right|_{t=t_{in}} =
\left. A^{(2)\, i}_{\ \ \ \ \, j}\right|_{t=t_{in}} =
\left. \zeta^{(2)}\right|_{t=t_{in}} =
\left. \gamma^{(2)}_{ij}\right|_{t=t_{in}} = 0$$ by redefinition of $C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(0)}$ and $f_{ij}$, respectively.
Provided that the redefined integration constants ($C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(0)}$, $f_{ij}$) satisfy (\[const-1\]) up to $O(\epsilon^3)$, one can show that the solution (\[eq:2ndK\]-\[eq:2ndgamma\]) automatically satisfies the third-order momentum constraint equation $$\begin{aligned}
& & \partial_j A^{(2)\, j}_{\ \ \ \ \, i}
+ 3A^{(1)\, j}_{\ \ \ \ \, i}\partial_j\zeta^{(1)}
+ 3A^{(2)\, j}_{\ \ \ \ \, i}\partial_j\zeta^{(0)}
\nonumber\\
& & \quad
- \frac{1}{2}A^{(1)\, j}_{\ \ \ \ \, l}f^{lk}
\partial_i\gamma^{(1)}_{jk}
- \frac{1}{2}A^{(1)\, j}_{\ \ \ \ \, l}(\gamma^{-1})^{(1)\, lk}
\partial_if_{jk}
- \frac{1}{2}A^{(2)\, j}_{\ \ \ \ \, l}f^{lk}
\partial_if_{jk}
- \frac{1}{3}\left(3\lambda-1\right)\partial_iK^{(2)}
= 0. \end{aligned}$$ The proof is given in Appendix \[proof\].
$O(\epsilon^n)$ solution ($n\geq 2$)
------------------------------------
For general $n$ ($\geq 2$), the solution to eqs.(\[eq:n-K\]-\[eq:n-gamma\]) is $$\begin{aligned}
K^{(n)}
& = &
\frac{1}{a^3(t)}\int^t_{t_{in}}dt' a^3(t')
\left[
-\frac{1}{2}\sum_{p=1}^{n-1} K^{(p)}K^{(n-p)}
-\frac{3}{2(3\lambda-1)}\sum_{p=1}^{n-1}A^{(p)\, i}_{\ \ \ \ \, j}
A^{(n-p)\, j}_{\qquad\ \ i}- \frac{Z^{(n)}}{3\lambda-1}
\right], \label{eq:nthK} \\
A^{(n)\, i}_{\ \ \ \ \, j}
& = &
\frac{1}{a^3(t)}\int^t_{t_{in}}dt' a^3(t')
\left[
-\sum_{p=1}^{n-1}K^{(p)}A^{(n-p)\, i}_{\qquad\ \ j}
+Z^{(n)\, i}_{\ \ \ \ \, j}-\frac{1}{3}Z^{(n)}\delta^i_{\ j}
\right], \label{eq:nthA} \\
\zeta^{(n)}
& = &
\frac{1}{3} \int^t_{t_{in}}dt' K^{(n)},\\
\gamma^{(n)}_{ij}
& = &
2 \int^t_{t_{in}}dt' \sum_{p=0}^{n-1}\gamma^{(p)}_{ik}
A^{(n-p)\, k}_{\qquad\ \ j}, \label{eq:nthgamma}\end{aligned}$$ where we have set $$\left. K^{(n)}\right|_{t=t_{in}} =
\left. A^{(n)\, i}_{\ \ \ \ \, j}\right|_{t=t_{in}} =
\left. \zeta^{(n)}\right|_{t=t_{in}} =
\left. \gamma^{(n)}_{ij}\right|_{t=t_{in}} = 0$$ by redefinition of $C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(0)}$ and $f_{ij}$, respectively. Note that, in subsection \[subsec:1storder\], we have already set $$\left. \zeta^{(1)}\right|_{t=t_{in}} =
\left. \gamma^{(1)}_{ij}\right|_{t=t_{in}} = 0$$ by redefinition of $\zeta^{(0)}$ and $f_{ij}$, respectively.
The initial condition for $\gamma^{(n)}_{ij}$ ($n\geq 1$) implies that $\left.\gamma_{ij}\right|_{t=t_{in}}=f_{ij}$, that $\left.(\gamma^{-1})^{ij}\right|_{t=t_{in}}=f^{ij}$ and that $\left.(\gamma^{-1})^{(n)\, ij}\right|_{t=t_{in}}=0$ ($n\geq 1$). Therefore, for $n\geq 1$, the solution to (\[eqn:gamma-inv-nth-eq\]) is $$(\gamma^{-1})^{(n)\, ij} =
-2\int_{t_{in}}^t dt'\sum_{p=1}^n
A^{(p)\, i}_{\quad\ \ k}
(\gamma^{-1})^{(n-p)\, kj}.$$
As shown in Appendix \[proof\], the solution (\[eq:nthK\]-\[eq:nthgamma\]) automatically satisfies the ($n+1$)-th order momentum constraint equation (\[eq:n-const\]), provided that the redefined integration constants ($C^{(1)}$, $C^{(1)\, i}_{\ \ \ \ \, j}$, $\zeta^{(0)}$, $f_{ij}$) satisfy (\[const-1\]) up to $O(\epsilon^{n+1})$.
Number of physical degrees of freedom
-------------------------------------
Our solution involves functions $\zeta^{(0)}(\vec{x})$, $f_{ij}(\vec{x})$, $C^{(1)}(\vec{x})$ and $C^{(1)\, i}_{\ \ \ \ \, j}(\vec{x})$ of spatial coordinates as integration ‘constant‘. They are subject to the constraint (\[const-1\]). Also, as stated just after (\[eq:gaugefix\]), our gauge fixing condition (\[eq:gaugefix\]) leaves time-independent spatial diffeomorphism as residual gauge freedom. Therefore, the number of physical degrees of freedom included in each integration ‘constant‘ is $$\begin{aligned}
\zeta^{(0)}(\vec{x}) & \cdots &
1 \mbox{ scalar growing mode }
= 1 \mbox{ component }, \nonumber\\
f_{ij}(\vec{x}) & \cdots &
2 \mbox{ tensor growing modes }
= 5 \mbox{ components } - 3 \mbox{ gauge }, \nonumber\\
C^{(0)}(\vec{x}) & \cdots &
1 \mbox{ scalar decaying mode }
= 1 \mbox{ component }, \nonumber\\
C^{(1)\, i}_{\ \ \ \ \, j}(\vec{x}) & \cdots &
2 \mbox{ tensor decaying modes }
= 5 \mbox{ components } - 3 \mbox{ constraints }. \end{aligned}$$ This is consistent with the fact that the Hořava-Lifshitz gravity includes not only a tensor graviton ($2$ propagating degrees of freedom) but also a scalar graviton ($1$ propagating degree of freedom).
Conserved curvature perturbation {#CP}
================================
In this section we propose a definition of nonlinear curvature perturbation and show that it is conserved up to the first order in the gradient expansion. This statement holds for any values of $\lambda$. Because of the conservation, this quantity is expected to be useful for the analysis of superhorizon evolution of nonlinear perturbation.
Definition
----------
In general relativity, it is known that the curvature perturbation in the uniform density slice conserves up to the first order in the gradient expansion [@Salopek:1990jq]. Motivated by this fact, we define the quantity ${\cal R}$ in Hořava-Lifshitz gravity by $${\cal R}(t,\vec{x}) \equiv \zeta(\tilde{t},\vec{x})
+ \ln\left[\frac{a(\tilde{t})}{a(t)} \right],
\label{Rdef}$$ where $\tilde{t}(t,\vec{x})=t+O(\epsilon)$ is the solution to $$\rho_{dm}^{(0)}(\tilde{t}) +
\delta\rho_{dm}(\tilde{t},\vec{x})
= \rho_{dm}^{(0)}(t),$$ and $$\rho_{dm}^{(0)}(t) \equiv
3M_{Pl}^2H^2-\frac{2M_{Pl}^2}{3\lambda-1}\Lambda, \quad
\delta\rho_{dm}(t,\vec{x})
\equiv \frac{M_{Pl}^2}{2}
\left[ R + \frac{2}{3}(K^2-9H^2) - A^i_jA^j_i\right].$$
In the following we shall show that ${\cal R}$ is indeed conserved up to the first order in the gradient expansion by explicitly calculating it.
Concrete expression and conservation up to $O(\epsilon)$
--------------------------------------------------------
According to the gradient expansion, we expand $\delta\rho_{dm}$, $\tilde{t}$ and ${\cal R}$ as $$\delta\rho_{dm} = \sum_{n=1}^{\infty}\rho_{dm}^{(n)}, \quad
\tilde{t} = t + \sum_{n=1}^{\infty}\tilde{t}^{(n)}, \quad
{\cal R} = \sum_{n=0}^{\infty}{\cal R}^{(n)},$$ where a quantity with the superscript $(n)$ is of order $O(\epsilon^n)$.
At the order $O(\epsilon^0)$, we obtain $${\cal R}^{(0)} = \zeta^{(0)}(\vec{x}),$$ and this is constant in time.
At the order $O(\epsilon)$, we obtain $$\rho_{dm}^{(1)} = 2M_{Pl}^2 H K^{(1)}, \quad
\tilde{t}^{(1)} = -\frac{\rho_{dm}^{(1)}}{\partial_t \rho^{(0)}}
= -\frac{K^{(1)}}{3\partial_t H}, \quad
{\cal R}^{(1)} = \zeta^{(1)} + H\tilde{t}^{(1)}
= \frac{C^{(1)}}{3}
\left[\int^t_{t_{in}}\frac{dt'}{a^3(t')}- \frac{H}{a^3\partial_t H}
\right].$$ Thus, the time derivative of ${\cal R}^{(1)}$ is shown to vanish as $$\partial_t {\cal R}^{(1)} = \frac{C^{(1)}H}{3a^3(\partial_t H)^2}
(\partial_t^2 H+3H\partial_t H) = 0,$$ where we have used the time derivative of (\[0thii\]) to show the last equality.
We therefore conclude that ${\cal R}$ defined in (\[Rdef\]) is conserved up to the first order in the gradient expansion. This statement holds for any values of $\lambda$.
Summary and Discussion {#sec:summary}
======================
We have performed a fully nonlinear analysis of superhorizon perturbation in Hořava-Lifshitz gravity by using the gradient expansion technique. In § \[solutions\] we have presented a concrete expression for the solution of gravity equations up to the second order in the gradient expansion. We have also proven that the solution can be extended to any order of the gradient expansion, by showing that the solution to the dynamical equation satisfies the constraint equation at each order.
Based on the result, in § \[CP\] we have proposed a definition of nonlinear curvature perturbation ${\cal R}$ in Hořava-Lifshitz gravity and have shown that it is conserved up to the first order in the gradient expansion.
It is known that, in the limit $\lambda\to 1$, the scalar graviton gets strongly coupled and the usual metric perturbation breaks down in the scalar graviton sector [@Charmousis:2009tc]. Here, we stress that the breakdown of the perturbative expansion does not necessarily lead to a loss of predictability since all coefficients of infinite number of terms in the perturbative expansion can be written in terms of finite parameters in the action if the theory is renormalizable. Indeed, for spherically symmetric, static, vacuum configurations, one can perform fully nonlinear analysis to show that the limit $\lambda\to 1$ is continuous and the general relativity is recovered in the limit [@Mukohyama:2010xz]. This result may be considered as an analogue of Vainshtein effect and suggests the possibility that the scalar graviton may safely be decoupled from the rest of the world, i.e. the tensor graviton and the matter sector, in the limit.
The result of the present paper is based on the fully nonlinear analysis and may be considered as yet another example of the analogue of Vainshtein effect. Up to any order of the gradient expansion, the equations of motion and their solutions are manifestly regular in the limit $\lambda\to 1$. The solutions reduce to those in general relativity coupled to dark matter in the limit at low energy.
In the present paper we have concentrated on the pure gravity system in the projectable Hořava-Lifshitz theory. Because of the existence of the scalar graviton, this simple system is still rich enough as a testing ground for the analogue of Vainshtein effect. Indeed, the so called “dark matter as integration constant” [@Mukohyama:2009mz] drives non-trivial cosmological dynamics in this system, and thus the nonlinear analysis presented in the present paper provides a convincing evidence for the analogue of Vainshtein effect. It is certainly interesting and important to extend the nonlinear analysis to more general situations with matter contents [@GMW].
K.I. acknowledges supports by the Grant-in-Aid for Scientific Research (A) No. 21244033. Part of this work was done during S.M.’s participation in YITP molecule-type workshop (T-10-05): Cosmological Perturbation and Cosmic Microwave Background. He thanks YITP for stimulating atmosphere and warm hospitality. The work of S.M. is supported by Grant-in-Aid for Scientific Research 17740134, 19GS0219, 21111006, 21540278, and by World Premier International Research Center Initiative (WPI Initiative). The authors are also supported by Japan-Russia Research Cooperative Program.
($n+1$)-th order momentum constraint ($n\geq 2$) {#proof}
================================================
In this appendix, by induction we prove that the $n$-th order solution (\[eq:nthK\]-\[eq:nthgamma\]) satisfies the ($n+1$)-th order momentum constraint equation (\[eq:n-const\]) for $n\geq 2$.
The basic logic of the proof is to rewrite the left hand side of the ($n+1$)-th order constraint (\[eq:n-const\]) as a linear combination of lower order constraints by using the explicit solution (\[eq:nthK\]-\[eq:nthgamma\]). For this purpose we shall use the generalized Bianchi identity (\[eq:nth-DZ\]) and other identities (\[eqn:nth-identities\]). We also use the following identity for functions $f(t)$ and $g(t)$ satisfying $a^3(t_{in})f(t_{in})g(t_{in})=0$: $$f(t)g(t) = \frac{1}{a^3(t)}\int_{t_{in}}^t dt'a^3(t')
\left[a(t')^{-3}\partial_{t'}(a^3(t')f(t'))\cdot g(t')
+ f(t')\cdot \partial_{t'}g(t')\right]. \label{eqn:integration-by-part}$$
By applying the identity (\[eqn:integration-by-part\]) to $(f(t),g(t))=(A^{(p)\, j}_{\ \ \ \ \ i},\partial_j\zeta^{(n-p)})$ and $(f(t),g(t))=(A^{(p)\, j}_{\ \ \ \ \ l},
(\gamma^{-1})^{(q)\, lk}\partial_{i}\gamma^{(n-p-q)}_{jk})$, the left hand side of the ($n+1$)-th order momentum constraint equation (\[eq:n-const\]) is rewritten as $$\begin{aligned}
{\cal C}^{(n+1)}_i & \equiv &
\partial_j A^{(n)\, j}_{\ \ \ \ \ i}
+3 \sum_{p=1}^n A^{(p)\, j}_{\ \ \ \ \ i}\partial_j\zeta^{(n-p)}
- \frac{1}{2}\sum_{p=1}^n\sum_{q=0}^{n-p}
A^{(p)\, j}_{\ \ \ \ \ l}(\gamma^{-1})^{(q)\, lk}
\partial_{i}\gamma^{(n-p-q)}_{jk}
-\frac{1}{3}(3\lambda-1)\partial_i K^{(n)} \nonumber\\
&=&
\partial_j A^{(n)\, j}_{\ \ \ \ \ i}
+ \frac{1}{a^3(t)}\int_{t_{in}}^t dt' a^3(t')
\left\{
3\sum_{p=1}^n\left[
a^{-3}\partial_{t'}
\left( a^3A^{(p)\, j}_{\ \ \ \ \ i}\right)
\partial_j\zeta^{(n-p)}
+ A^{(p)\, j}_{\ \ \ \ \ i}
\partial_j\left(\partial_{t'}\zeta^{(n-p)}\right)
\right]
\right. \nonumber\\
& &
- \frac{1}{2}\sum_{p=1}^n\sum_{q=0}^{n-p}
\left[ a^{-3}\partial_{t'}
\left( a^3A^{(p)\, j}_{\ \ \ \ \ l}\right)
(\gamma^{-1})^{(q)\, lk}
\partial_{i}\gamma^{(n-p-q)}_{jk}
+ A^{(p)\, j}_{\ \ \ \ \ l}
\partial_{t'}\left((\gamma^{-1})^{(q)\, lk}\right)
\partial_{i}\gamma^{(n-p-q)}_{jk}\right.
\nonumber\\
& & \left.\left.
+ A^{(p)\, j}_{\ \ \ \ \ l}
(\gamma^{-1})^{(q)\, lk}
\partial_{i}\left(\partial_{t'}\gamma^{(n-p-q)}_{jk}\right)
\right]\right\}
-\frac{1}{3}(3\lambda-1)\partial_i K^{(n)}. \nonumber\\\end{aligned}$$ Using (\[eq:nthK\]-\[eq:nthA\]), (\[eq:n-zeta\]-\[eq:n-gamma\]) and (\[eqn:gamma-inv-nth-eq\]), this is further rewritten as $$\begin{aligned}
{\cal C}^{(n+1)}_i
&=&
\frac{1}{a^3(t)}\int_{t_{in}}^t dt' a^3(t')
\left\{
\partial_j
\left( -\sum_{p=1}^{n-1}K^{(p)}A^{(n-p)\, j}_{\qquad\ \ i}
\right)
\right. \nonumber\\
& &
+3\left[
\sum_{p=2}^n
\left(-\sum_{q=1}^{p-1}K^{(q)}A^{(p-q)\, j}_{\qquad\ \ i}\right)
\partial_j\zeta^{(n-p)}
+ \sum_{p=1}^{n-1}A^{(p)\, j}_{\ \ \ \ \ i}
\partial_j\left(\frac{1}{3}K^{(n-p)}\right)
\right]
\nonumber\\
& &
- \frac{1}{2}\sum_{p=1}^n\sum_{q=0}^{n-p}
\left[
\left(-\sum_{r=1}^{p-1}K^{(r)}A^{(p-r)\, j}_{\qquad\ \ l}\right)
(\gamma^{-1})^{(q)\, lk}
\partial_{i}\gamma^{(n-p-q)}_{jk}
\right.\nonumber\\
& &
\left.
+ A^{(p)\, j}_{\ \ \ \ \ l}
\left(-2\sum_{r=1}^q
A^{(r)\, l}_{\quad\ \ m}
(\gamma^{-1})^{(q-r)\, mk}\right)
\partial_{i}\gamma^{(n-p-q)}_{jk}\right.
\nonumber\\
& & \left.
+ A^{(p)\, j}_{\ \ \ \ \ l}
(\gamma^{-1})^{(q)\, lk}
\partial_{i}
\left(
2 \sum_{r=1}^{n-p-q-1}\gamma^{(r)}_{jm}
A^{(n-p-q-r)\, m}_{\qquad\qquad\quad k}
\right)\right]
\nonumber\\
& &
-\frac{1}{6}(3\lambda-1)\partial_i
\left(
-\sum_{p=1}^{n-1} K^{(p)}K^{(n-p)}
\right)
-\frac{1}{2}\partial_i
\left(
-\sum_{p=1}^{n-1}A^{(p)\, j}_{\ \ \ \ \, k}
A^{(n-p)\, k}_{\qquad\ \ j}
\right)
\nonumber\\
& & \left.
+ \frac{1}{6}\sum_{p=1}^nZ^{(p)}
\sum_{q=0}^{n-p}
(\gamma^{-1})^{(q)\, jk}
\partial_{i}\gamma^{(n-p-q)}_{jk}
\right\},\end{aligned}$$ where we have used the generalized Bianchi identity (\[eq:nth-DZ\]). By using the identities (\[eqn:nth-identities\]) we finally obtain $${\cal C}^{(n+1)}_i =
-\frac{1}{a^3(t)}\int_{t_{in}}^t dt' a^3(t')
\sum_{p=1}^{n-1}K^{(n-p)}{\cal C}^{(p+1)}_i.$$ Since we already know that ${\cal C}^{(2)}_i=0$ under the condition (\[const-1\]), this is enough to prove ${\cal C}^{(n+1)}=0$ for $n\geq 2$.
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|
---
abstract: 'Let ${\mathcal{A}}_2$ be the moduli stack of principally polarized abelian surfaces. Let ${\mathbb{V}}$ be a smooth $\ell$-adic sheaf on ${\mathcal{A}}_2$ associated to an irreducible rational finite dimensional representation of $\operatorname{Sp}(4)$. We give an explicit expression for the cohomology of ${\mathbb{V}}$ in any degree in terms of Tate type classes and Galois representations attached to elliptic and Siegel cusp forms. This confirms a conjecture of Faber and van der Geer. As an application we prove a dimension formula for vector-valued Siegel cusp forms for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ of weight three, which had been conjectured by Ibukiyama.'
address:
- |
Departement Mathematik\
ETH Zürich\
Rämistrasse 101\
8092 Zürich\
Switzerland
- |
Institutionen för Matematik\
KTH Royal Institute of Technology\
100 44 Stockholm\
Sweden
author:
- Dan Petersen
bibliography:
- '../database.bib'
title: Cohomology of local systems on the moduli of principally polarized abelian surfaces
---
Introduction
============
Let $Y=\Gamma \backslash \mathfrak{H}$ be a modular curve, given by the quotient of the upper half plane by a congruence subgroup $\Gamma \subset \operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$. An irreducible rational representation ${\mathbb{V}}$ of $\operatorname{SL}(2)$ defines a local system on $Y$, since ${\mathbb{V}}$ is in particular a representation of $\pi_1(Y) \cong \Gamma \subset \operatorname{SL}(2)$. After work of Eichler, Shimura, Ihara, Deligne, and many others after them, we understand extremely well the cohomology groups $H^\bullet(Y,{\mathbb{V}})$. The cohomology classes can be described group-theoretically in terms of modular forms for the group $\Gamma$, and it has a (split) mixed Hodge structure in which the pure part corresponds to cusp forms and its complement to Eisenstein series. We can think of ${\mathbb{V}}$ also as a smooth $\ell$-adic sheaf (and $Y$ as defined over a number field, or a deeper arithmetic base), in which case the étale cohomology $H^\bullet(Y,{\mathbb{V}})$ can be expressed in terms of Galois representations attached to the same modular forms [@deligne69].
There is a vast theory describing the generalization of the above to moduli spaces of higher-dimensional abelian varieties with some extra structure (polarization, endomorphism, and level), and to more general Shimura varieties. But there is not a single example where our understanding is as complete as in genus one.
In this article we consider one of the simplest higher-genus examples and give a quite explicit description of the cohomology in this case. Namely, consider the moduli space ${\mathcal{A}}_2$ of principally polarized abelian surfaces, and let ${\mathbb{V}}$ be a smooth $\ell$-adic sheaf associated to an irreducible representation of $\operatorname{Sp}(4)$. The main theorem of this article is an explicit expression for the (semi-simplification of the) $\ell$-adic Galois representation $H^k_c({\mathcal{A}}_2,{\mathbb{V}})$ for any $k$ and any ${\mathbb{V}}$ in terms of Tate type classes and Galois representations attached to level $1$ elliptic/Siegel cusp forms.
These cohomology groups are natural objects of study for algebraic geometers, in particular because of applications to moduli of curves. In particular, the results of this paper are used in [@m28ct] to prove that the Gorenstein conjecture fails for the tautological rings of the spaces $\mathcal M_{2,n}^{\mathsf{ct}}$ for $n \geq 8$. There is some history of algebraic geometers studying the cohomology of ${\mathbb{V}}_{a,b}$ for small values of $a+b$ by ad hoc methods for such applications, see e.g. [@getzlergenustwo Section 8], [@bergstrom09], [@petersentommasi Section 3]. Let us also mention [@fvdg1] who used point counts over finite fields to conjecture an expression for the virtual $\ell$-adic Galois representation $$\sum_k (-1)^k [H^k_c({\mathcal{A}}_2,{\mathbb{V}})] \in K_0(\mathsf{Gal})$$ for any ${\mathbb{V}}_{a,b}$; see also [@bfg11 Section 6] for a more detailed description. The results in this paper confirm Faber and van der Geer’s conjecture. When ${\mathbb{V}}$ has regular highest weight, their conjecture was proven in [@weissauer] (and later independently in [@tehraniendoscopy]).
Using the BGG-complex of Faltings, one can relate the results of this paper to the coherent cohomology of the bundles of Siegel modular forms for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, as we explain at the end of Section \[mainthm\]. A direct consequence of our main theorem is a proof of a dimension formula for vector-valued Siegel modular forms for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ of weight $3$, which had been conjectured in [@ibukiyama1]. This result has been independently obtained in [@taibi] using Arthur’s trace formula.
The strategy of our proof is as follows. Up to semi-simplification, the cohomology is the direct sum of the *Eisenstein cohomology* and the *inner cohomology*. The Eisenstein cohomology on ${\mathcal{A}}_2$ of an arbitrary local system was determined in [@harder], so we need only to find the inner cohomology. Now we use that the inner cohomology contains the cuspidal cohomology and is contained in the intersection cohomology, and both of these can be understood in terms of data attached to discrete spectrum automorphic representations for ${\mathrm{GSp}}(4)$. There is a very large body of work dealing with automorphic representations on ${\mathrm{GSp}}(4)$ (due to Piatetski-Shapiro, Soudry, Arthur, Weissauer, Taylor, Hales, Waldspurger and many others) since it is one of the first test cases for the general Langlands program. Since we will only work in level $1$, we can work with ${\mathrm{PGSp}}(4)$, in which case all necessary information on the discrete spectrum automorphic representations is worked out and described very explicitly in [@flicker]. These results allow us to determine both the cuspidal and the intersection cohomology of these local systems, and to deduce after comparing with Harder’s results that the inner cohomology coincides with the cuspidal cohomology in these cases.
In Section 2 of this article I state the main theorem and explain the applications to vector-valued Siegel cusp forms. Section 3 contains a brief review of automorphic representations and the cohomology of Shimura varieties. I hope that this will help make the arguments accessible for algebraic geometers without this background. Section 4 specializes to ${\mathrm{PGSp}}(4)$ and contains the proof of the main theorem.
I am grateful to Jonas Bergström for many useful discussions on these topics and for his interest in this work, and Tomoyoshi Ibukiyama for several helpful pointers to the literature.
Statement of results {#mainresults}
====================
Let ${\mathcal{A}}_2$ denote the moduli stack of principally polarized abelian surfaces. Let $f \colon {\mathscr X}\to {\mathcal{A}}_2$ be the universal family. We have a local system (smooth $\ell$-adic sheaf) ${\mathbb{V}}= \mathrm R^1 f_\ast {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell$ on ${\mathcal{A}}_2$ of rank $4$ and weight $1$, and there is a symplectic pairing $$\wedge^2 {\mathbb{V}}\to {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-1).$$ Here ${\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-1)$ denotes the Tate twist of the constant local system on ${\mathcal{A}}_2$. Recall Weyl’s construction of the irreducible representations of $\operatorname{Sp}(4)$ [@fh91 Section 17.3]: if $V$ is the standard $4$-dimensional symplectic vector space, then the irreducible representation with highest weight $a \geq b \geq 0$ is a constituent of $V^{\otimes (a+b)}$, where it is ‘cut out’ by Schur functors and by contracting with the symplectic form. For instance, the representation of highest weight $(2,0)$ is $\operatorname{Sym}^2(V)$, and the representation $(1,1)$ is the complement of the class of the symplectic form inside $\wedge^2 V$. Weyl’s construction works equally well in families, and so for each $a \geq b \geq 0$ we obtain a local system ${\mathbb{V}}_{a,b}$ which is a summand in ${\mathbb{V}}^{\otimes (a+b)}$. In this paper we determine the cohomology of ${\mathbb{V}}_{a,b}$ considered as an $\ell$-adic Galois representation up to semi-simplification.
Note that every point of ${\mathcal{A}}_2$ has the automorphism $(-1)$, given by inversion on the abelian variety. This automorphism acts as multiplication by $(-1)^{a+b}$ on the fibers of ${\mathbb{V}}_{a,b}$. This shows that the local system has no cohomology when $a+b$ is odd. Hence we restrict our attention to the case when $a+b$ is even.
Before we can state our main results we need to introduce some notation. For any $k$, let $s_k$ denote the dimension of the space of cusp forms for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ of weight $k$. Similarly for any $j\geq 0$, $k\geq 3$ we denote by $s_{j,k}$ the dimension of the space of vector-valued Siegel cusp forms for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, transforming according to the representation $\operatorname{Sym}^{ j} \otimes \det^{ k}$.
To each normalized cusp eigenform $f$ for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ of weight $k$ is attached a $2$-dimensional $\ell$-adic Galois representation $\rho_f$ of weight $k-1$ [@deligne69]. We define ${\mathsf{S}}_k = \bigoplus_{f} \rho_f$ to be the direct sum of these Galois representation for fixed $k$. By the main theorem of [@weissauer4d] there are also $4$-dimensional Galois representations attached to vector-valued Siegel cusp eigenforms for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ of type $\operatorname{Sym}^j \otimes \det^k$ with $k \geq 3$, and we define ${\mathsf{S}}_{j,k}$ analogously. So $\dim {\mathsf{S}}_k = 2s_k$ and $\dim {\mathsf{S}}_{j,k} = 4s_{j,k}$.
Moreover, we introduce $s_k'$: this is the cardinality of the set of normalized cusp eigenforms $f$ of weight $k$ for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, for which the central value $L(f,\frac 1 2)$ vanishes. In this paper all $L$-functions will be normalized to have a functional equation relating $s$ and $1-s$. The functional equation shows that the order of $L(f,s)$ at $s=\frac 1 2$ is always odd if $k \equiv 2 \pmod 4$ and even if $k \equiv 0 \pmod 4$. Hence in the former case $s_k = s_k'$; in the latter case $0 \leq s_k' \leq s_k$. In our results, the quantity $s_k'$ will only occur in the case $k \equiv 0 \pmod 4$, and in this case it is conjectured that $s_k' = 0$. Indeed, [@conreyfarmer] proved that this vanishing is implied by Maeda’s conjecture; Maeda’s conjecture has been verified numerically for weights up to $14000$ [@ghitza-mcandrew].
Finally we define $\overline{\mathsf{S}}_{j,k} = {\mathfrak{gr}}^W_{j+2k-3}{\mathsf{S}}_{j,k}$; in other words, we consider only the part of ${\mathsf{S}}_{j,k}$ which satisfies the Ramanujan conjecture. Counterexamples to the Ramanujan conjecture arise from the Saito–Kurokawa lifting: for a cusp eigenform $f$ of weight $2k$ for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, where $k$ is odd, there is attached a scalar valued Siegel cusp form of weight $k+1$ for $\operatorname{Sp}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ whose attached $\ell$-adic Galois representation has the form $${\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell (-k+1) \oplus \rho_f \oplus {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-k)$$ where $\rho_f$ is the Galois representation of weight $2k-1$ attached to $f$. By [@weissauer Theorem 3.3], these are in fact the only Siegel cusp forms violating the Ramanujan conjecture. Thus $\overline{{\mathsf{S}}}_{j,k} = {\mathsf{S}}_{j,k}$ unless $j=0$ and $k$ is even, in which case $\overline{{\mathsf{S}}}_{j,k}$ is obtained from ${\mathsf{S}}_{j,k}$ by removing the two summands of Tate type from each Saito–Kurokawa lift.
Note that the definitions of $s_k$, ${\mathsf{S}}_k$, $s_{j,k}$ and ${\mathsf{S}}_{j,k}$ used in [@fvdg1] are different from ours: theirs is not only a sum over cusp forms, but includes in the case $k=2$ (resp. $j=0$, $k=3$) the contribution from the trivial automorphic representation. This allows for a compact expression for the virtual Galois representation $\sum_i (-1)^i [H^i_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b})]$ but will not be used here.
\[mainthm\]Suppose $(a,b) \neq (0,0)$, and that $a+b$ is even. Then:
1. $H^k_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ vanishes for $k \notin \{2,3,4\}$.
2. In degree $4$ we have $$H^4_c ({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) = \begin{cases} s_{a+b+4} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-b-2) & a=b \text{ even,}
\\ 0 & \text{otherwise.} \end{cases}$$
3. In degree $3$ we have, up to semi-simplification, $$\begin{aligned}
H^3_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) &= \overline{\mathsf{S}}_{a-b,b+3} \\
& + s_{a+b+4}{\mathsf{S}}_{a-b+2}(-{b-1}) \\
&+ {\mathsf{S}}_{a+3} \\
&+ \begin{cases}
s_{a+b+4}' {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-{b-1}) & a=b \text{ even,} \\
s_{a+b+4} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-{b-1}) & \text{otherwise,} \end{cases}\\
&+ \begin{cases} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell & a=b \text{ odd,} \\ 0 & \text{otherwise,}\end{cases} \\
&+ \begin{cases} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-1) & b=0, \\ 0 & \text{otherwise.} \end{cases} .\end{aligned}$$
4. In degree $2$ we have, again up to semi-simplification, that $$\begin{aligned}
H^2_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) &=
{\mathsf{S}}_{b+2} \\
&+ s_{a-b+2} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell\\
&+ \begin{cases}
s_{a+b+4}' {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-{b-1}) & a=b \text{ even,} \\
0 & \text{otherwise,} \end{cases} \\
&+ \begin{cases} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell & a > b > 0 \text{ and } a, b \text{ even,} \\ 0 & \text{otherwise.} \end{cases} \end{aligned}$$
To exemplify the notation: $s_{a+b+4}{\mathsf{S}}_{a-b+2}(-{b-1})$ means a direct sum of $s_{a+b+4}$ copies of the Galois representation ${\mathsf{S}}_{a-b+2}$, Tate twisted $b+1$ times.
As remarked earlier, it is conjectured that both occurrences of $s_k'$ in the above theorem can be replaced by $0$.
\[hodgetheory\] It will be clear from the proof that the result is valid (and even a bit easier) also in the category of mixed Hodge structures. Harder’s computation of the Eisenstein cohomology is valid in this category, and our computation of the inner cohomology identifies it with the cuspidal cohomology, which obtains a natural Hodge structure from the bigrading on $({\mathfrak{g}},K)$-cohomology. This bigrading is compatible with the one obtained using the ‘filtration bête’ and the BGG-complex of [@faltingschai Theorem VI.5.5.].
\[motivicremark\] It is conjectured that the Galois representations $H^k_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ are not semisimple in general. Suppose that $a=b = 2k-1$. Then our expression for the semisimplification of $H^3_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ contains the terms $s_{4k+2} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-2k)$ and ${\mathsf{S}}_{4k+2}$, the latter being the ‘Saito–Kurokawa’ summand of $\overline {\mathsf{S}}_{0,2k+2}$. According to a conjecture in [@harderbook 81–82], these should form a nontrivial extension: $$0 \to s_{4k+2}{\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-2k) \to M \to {\mathsf{S}}_{4k+2} \to 0.$$ Note that if $f$ is a Hecke eigenform of weight $4k+2$ and $\rho_f$ is the attached Galois representation (or ‘motive’), then conjectures of Deligne–Bloch–Beilinson [@grosscentralvalue Section 1] predict that $$\dim \mathrm{Ext}^1({\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-2k),\rho_f) = \mathrm{ord}_{s=\tfrac 1 2} L(f,s),$$ and the functional equation for $L(f,s)$ forces it to vanish at $s=\tfrac 1 2$. Here the Ext-group is computed either in the category of $\ell$-adic Galois representations, or (even better) in the category of mixed motives. I do not know whether there exists a cusp form for the full modular group whose $L$-function vanishes to more than first order at the central point.
Application to dimension formulas for Siegel modular forms
----------------------------------------------------------
A consequence of Remark \[hodgetheory\] is that our main theorem can be applied to produce dimension formulas for vector-valued Siegel modular forms. Let $i \colon {\mathcal{A}}_2 \hookrightarrow {\widetilde{{\mathcal{A}}}}_2$ be a toroidal compactification. Let ${\mathscr V}_{j,k}$ for $j,k \in {\ensuremath{{\ensuremath{\mathbf{Z}}}}}$, $j \geq 0$ be the vector bundle on ${\widetilde{{\mathcal{A}}}}_2$ whose global sections are vector-valued Siegel modular forms of type $\operatorname{Sym}^{ j} \otimes \det^{ k}$. Let similarly ${\mathscr V}_{j,k}(-D_\infty)$ be the vector bundle of Siegel cusp forms. The *BGG-complex* (resp. the *dual BGG-complex*) is a resolution of $i_\ast {\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}$ (resp. $i_! {\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}$) in terms of the vector bundles ${\mathscr V}_{j,k}$ (resp. ${\mathscr V}_{j,k}(-D_\infty)$). Then [@faltingschai Theorem VI.5.5] asserts that the hypercohomology spectral sequence of the BGG-complex degenerates, and that the Hodge filtration on the cohomology of ${\mathbb{V}}_{a,b}$ can be defined in terms of a filtration of the BGG-complex. There is also an analogous statement for the dual BGG-complex and the compactly supported cohomology. Specialized to our case, their theorem (in the case of the dual BGG-complex) asserts the following (see [@getzlergenustwo Theorem 17]):
The cohomology groups $H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}})$ have a Hodge filtration with Hodge numbers in the set $\{a+b+3,a+2,b+1,0\}$. The associated graded pieces satisfy $$\begin{aligned}
{\mathfrak{gr}}_F^0 H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) & \cong H^\bullet({\widetilde{{\mathcal{A}}}}_2,{\mathscr V}_{a-b,-a}(-D_\infty)), \\
{\mathfrak{gr}}_F^{b+1} H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) & \cong H^{\bullet-1}({\widetilde{{\mathcal{A}}}}_2,{\mathscr V}_{a+b+2,-a}(-D_\infty)), \\
{\mathfrak{gr}}_F^{a+2} H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) & \cong H^{\bullet-2}({\widetilde{{\mathcal{A}}}}_2,{\mathscr V}_{a+b+2,1-b}(-D_\infty)), \\
{\mathfrak{gr}}_F^{a+b+3} H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) & \cong H^{\bullet-3}({\widetilde{{\mathcal{A}}}}_2,{\mathscr V}_{a-b,b+3}(-D_\infty)). \end{aligned}$$
We record three immediate consequences of this theorem combined with our main theorem. The first of these is a proof of a conjecture of Ibukiyama, whereas the second two are new proofs of results which are already known (by admittedly much more direct arguments).
1. [The bundles ${\mathscr V}_{j,k}(-D_\infty)$ have no higher cohomology for any $j \geq 0$, $k \geq 3$, with the sole exception of $H^3({\widetilde{{\mathcal{A}}}}_2,{\mathscr V}_{0,3}(-D_\infty)) \cong {\ensuremath{{\ensuremath{\mathbf{C}}}}}$.]{} (To prove this, consider ${\mathfrak{gr}}_F^{a+b+3}$.) An explicit formula for the Euler characteristic of the vector bundles ${\mathscr V}_{j,k}(-D_\infty)$ was calculated in [@tsushima] using Hirzebruch–Riemann–Roch; thus, we obtain a dimension formula for vector-valued Siegel cusp forms for all $j \geq 0$, $k \geq 3$. Tsushima himself proved that these bundles have no higher cohomology when $k \geq 5$ using the Kawamata–Viehweg vanishing theorem, and conjectured that it can be improved to $k \geq 4$. The fact that this vanishing result can be extended to $k \geq 3$ is particular to the case of the full modular group and was conjectured in [@ibukiyama1 Conjecture 2.1]. The resulting dimension formula for $k=3$ can be stated as $$\sum_{j \geq 0} s_{j,3}x^j = \frac{x^{36}}{(1-x^6)(1-x^8)(1-x^{10})(1-x^{12})}.$$ This result has also been proven in [@taibi Section 5].
2. There are no vector-valued Siegel modular forms of weight $1$ for the full modular group. (Put $b=0$ and consider ${\mathfrak{gr}}_F^{a+2}$ to prove the case $\operatorname{Sym}^j \otimes \det$ with $j \geq 2$; the cases $j < 2$ require a separate \[easy\] argument.) This result was previously known by [@ibukiyamaproceedings Theorem 6.1].
3. The Siegel $\Phi$-operator is surjective for any $j \geq 0, k \geq 3$. Recall that the $\Phi$-operator maps Siegel modular forms of type $\operatorname{Sym}^j \otimes \det^k$ to elliptic modular forms of weight $j+k$, and that the image of $\Phi$ consists only of cusp forms if $j > 0$. Now, the dimension of the part of $ {\mathfrak{gr}}_F^{a+b+3} H^3({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}})$ given by Eisenstein cohomology is exactly the dimension of the image of the $\Phi$-operator for $\operatorname{Sym}^{a-b} \otimes \det^{b+3}$, since the part given by inner cohomology coincides with the dimension of the space of cusp forms. But the dimension of this part of Eisenstein cohomology is $s_{a+3}$ unless $a=b$ is odd, in which case it is $s_{a+3}+1$. The result follows from this. Surjectivity of the $\Phi$-operator is known more generally for arbitrary level when $k \geq 5$ and $j > 0$ by [@arakawa]. The scalar valued case is a classical theorem of Satake. The case $k=4$ (and $k=2$) is [@ibukiyamawitt Theorem 5.1].
Only Siegel modular forms of weight two are inaccessible via the cohomology of local systems. In a sequel to this paper we will use similar arguments to derive dimensional results for Siegel modular forms with nontrivial level.
Résumé of automorphic representations
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In this section I briefly recall some (mostly standard) facts from the theory of automorphic representations that are needed for this paper. Rather than providing detailed references everywhere, I will give general references at the beginning of each subsection.
Automorphic representations {#auto}
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[@boreljacquet; @cogdellkimmurty] Let $G$ be a reductive connected group over ${\ensuremath{{\ensuremath{\mathbf{Q}}}}}$. Let ${{\ensuremath{\mathbf{A}}}}= {{\ensuremath{\mathbf{A}}}}_{\mathrm{fin}}\times {\ensuremath{{\ensuremath{\mathbf{R}}}}}$ be the ring of (rational) adèles. Let $Z$ be the center of $G$, and $\omega$ a unitary character of $Z({{\ensuremath{\mathbf{A}}}})/Z({\ensuremath{{\ensuremath{\mathbf{Q}}}}})$. We define $L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}),\omega)$ to be the space of measurable functions $f$ on $G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}})$ which are square integrable with respect to a translation invariant measure, and which satisfy $f(zg) = \omega(z)f(g)$ for any $z \in Z({{\ensuremath{\mathbf{A}}}})$. The group $G({{\ensuremath{\mathbf{A}}}})$ acts on this space by right translation. A representation of $G({{\ensuremath{\mathbf{A}}}})$ is called *automorphic* if it is a subquotient of $L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}),\omega)$, for some $\omega$. We call $\omega$ the *central character* of the automorphic representation.
The space $L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}),\omega)$ contains a maximal subspace which is a direct sum of irreducible representations. This subspace is called the *discrete spectrum*, and an automorphic representation occuring here is called *discrete*. The orthogonal complement of this subspace is the *continuous spectrum*. Langlands identified the continuous spectrum with ‘Eisenstein series’; it is the direct integral of families of representations induced from parabolic subgroups of $G({{\ensuremath{\mathbf{A}}}})$. The discrete spectrum, in turn, also decomposes as the direct sum of the *cuspidal* and the *residual* spectrum. The cuspidal spectrum is defined as the subspace spanned by functions $f$ such that the integral over $N({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash N({{\ensuremath{\mathbf{A}}}})$ of $f$, and all its translates under $G({{\ensuremath{\mathbf{A}}}})$, vanishes, for $N$ the unipotent radical of any proper parabolic subgroup. Langlands proved that the residual spectrum is spanned by the residues of Eisenstein series, and that all residual representations are quotients of representations induced from a parabolic subgroup.
Any irreducible automorphic representation $\pi$ of $G({{\ensuremath{\mathbf{A}}}})$ is a completed (restricted) tensor product of local representations $\pi_p$ of $G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}_p)$, where $p$ ranges over the prime numbers, and an archimedean component $\pi_\infty$. Let $K_p \subset G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}_p)$ be a special maximal compact subgroup. We say that $\pi$ is *spherical* at $p$ if $\pi_p$ contains a nonzero vector fixed by $K_p$, in which case this vector will be unique up to a nonzero scalar. The representation $\pi$ is spherical at all but finitely many primes. The word ‘restricted’ in the first sentence of this paragraph means that the component of the representation at $p$ should be equal to the spherical vector for all but finitely many $p$.
The archimedean component $\pi_\infty$ can be identified with an irreducible $({\mathfrak{g}},K_\infty)$-module, where ${\mathfrak{g}}$ is the Lie group of $G({\ensuremath{{\ensuremath{\mathbf{R}}}}})$ and $K_\infty \subset G({\ensuremath{{\ensuremath{\mathbf{R}}}}})$ is a maximal compact subgroup. The center of the universal enveloping algebra of ${\mathfrak{g}}$ acts by a scalar on $\pi_\infty$. The resulting map $Z(\mathcal U {\mathfrak{g}}) \to {\ensuremath{{\ensuremath{\mathbf{C}}}}}$ is called the *infinitesimal character* of $\pi$.
Local factors
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[@borelLfunctions] Suppose $\pi$ is spherical at $p$. We define the *spherical Hecke algebra* ${\mathscr H}_{G,K_p}$ to be the convolution algebra of $K_p$-bi-invariant ${\ensuremath{{\ensuremath{\mathbf{Q}}}}}$-valued functions on $G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}_p)$. This algebra acts on the one-dimensional space of spherical vectors, and $\pi_p$ is uniquely determined by this action. Hence specifying a spherical representation is equivalent to specifying a homomorphism ${\mathscr H}_{G,K_p} \to {\ensuremath{{\ensuremath{\mathbf{C}}}}}$. We should therefore understand the ring ${\mathscr H}_{G,K_p}$, and this we can do via the *Satake isomorphism*. For this we need the notion of the *dual group*. If $G$ is defined by a root datum, then the dual group $\widehat{G}$ is obtained by switching roots and co-roots, and characters and $1$-parameter subgroups. The Satake isomorphism states that the Hecke algebra ${\mathscr H}_{G,K_p}$ and the ring of virtual representations $K_0({\mathrm{Rep}}(\widehat{G}))$ become isomorphic after an extension of scalars: one has $${\mathscr H}_{G,K_p} \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}\cong K_0({\mathrm{Rep}}(\widehat{G})) \otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}.$$ In particular a homomorphism ${\mathscr H}_{G,K_p} \to {\ensuremath{{\ensuremath{\mathbf{C}}}}}$ is identified with a homomorphism $K_0({\mathrm{Rep}}(\widehat{G})) \to {\ensuremath{{\ensuremath{\mathbf{C}}}}}$. But the latter is determined by a semisimple conjugacy class $c_p$ in $\widehat{G}({\ensuremath{{\ensuremath{\mathbf{C}}}}})$. (You evaluate such a class on a representation $V$ via $\operatorname{Tr}(c_p \mid V)$.)
Now suppose instead we have an $\ell$-adic (or $\lambda$-adic) representation $\rho \colon {\mathrm{Gal}}(\overline {\ensuremath{{\ensuremath{\mathbf{Q}}}}}/{\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \to \widehat{G}(\overline {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell)$. For all but finitely many primes, $\rho$ is going to be unramified, which means in particular that the expression $\rho(\mathrm{Frob}_p)$ is well defined up to conjugacy. If we choose an isomorphism ${\ensuremath{{\ensuremath{\mathbf{C}}}}}\cong \overline {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell$, then it makes sense to ask whether $c_p$ and $\rho(\mathrm{Frob}_p)$ are conjugate for almost all $p$. If this holds, then we say that $\rho$ is *attached* to the automorphic representation $\pi$. We remark that this definition would make sense also if we replace ${\mathrm{Gal}}(\overline {\ensuremath{{\ensuremath{\mathbf{Q}}}}}/{\ensuremath{{\ensuremath{\mathbf{Q}}}}})$ by the absolute Weil group, or the conjectural global Langlands group, as in both cases $\rho(\mathrm{Frob}_p)$ should be well defined up to conjugacy at unramified primes.
By the Chebotarev density theorem, there is at most one Galois representation attached to a given automorphic representation. The Strong Multiplicity One theorem shows the converse when $G = \operatorname{GL}(n)$, but in general there will be several automorphic representations with the same attached Galois representation. Conjecturally, two automorphic representations will have the same attached Galois representation if and only if they lie in the same ‘$L$-packet’. However, the notion of a packet has not been rigorously defined in general.
One of many conjectures within the Langlands program says roughly that there should in fact be a bijection between packets of automorphic representations for $G$ and $\ell$-adic Galois representations into the dual group. As stated this conjecture is however false, and making the conjecture precise is a rather delicate matter. For a formulation in terms of the hypothetical Langlands group, see [@arthurlgroup], and for a more restrictive formulation only in terms of Galois representations, see [@buzzardgee].
Often one fixes once and for all $r \colon \widehat{G} \hookrightarrow \operatorname{GL}(n)$. Then the conjugacy class $c_p$ can be described by specifying an $n \times n$ diagonal matrix $\mathrm{diag}(t_1,\ldots,t_n)$. The numbers $t_i$ are called the *Langlands parameters* of $\pi$ at $p$. Moreover, one can then attach an $L$-function to any automorphic representation. At a prime $p$ where $\pi$ is spherical, the local $L$-factor is given by $$\det(\mathbf 1_n - p^{-s} r(c_p))^{-1}.$$ On the other hand, given $r$ we also obtain from $\rho$ an $n$-dimensional $\ell$-adic Galois representation, which also has an attached $L$-function. Thus the Langlands parameters can be identified with the Frobenius eigenvalues of the attached Galois representations. Usually the notion of $\rho$ being attached to $\pi$ is defined in terms of an equality of $L$-functions, but $L$-functions will play only a minor role in this paper.
Shimura varieties
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[@deligne-shimuravarieties; @harderbook Kapitel II]\[shimura\] For $G$ as above, suppose that $h \colon \mathrm{Res}_{{\ensuremath{{\ensuremath{\mathbf{C}}}}}/{\ensuremath{{\ensuremath{\mathbf{R}}}}}} \mathbb G_m \to G_{/{\ensuremath{{\ensuremath{\mathbf{R}}}}}}$ is a homomorphism satisfying the axioms 2.1.1.1–2.1.1.3 of [@deligne-shimuravarieties]. Let $K_\infty$ be the stabilizer of $h$ in $G({\ensuremath{{\ensuremath{\mathbf{R}}}}})$. Let $K_{\mathrm{fin}}$ be any compact open subgroup of $G({{\ensuremath{\mathbf{A}}}}_{\mathrm{fin}})$. For $K = K_{\mathrm{fin}}\times K_\infty$ we can consider the quotient $$S_{K} = G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \backslash G({{\ensuremath{\mathbf{A}}}}) / K = G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \backslash X \times G({{\ensuremath{\mathbf{A}}}}_{\mathrm{fin}}) / K_{\mathrm{fin}},$$ the *Shimura variety* associated to $K$. Here $X = G({\ensuremath{{\ensuremath{\mathbf{R}}}}})/K_\infty$. For $K_{\mathrm{fin}}$ small enough $S_K$ is, in fact, a smooth algebraic variety which is naturally defined over a number field (the reflex field), but in the case we will consider in this paper we will actually need to think of $S_K$ as an orbifold or Deligne–Mumford stack.
\[Ag\] Siegel modular varieties are Shimura varieties. Let $G= {\mathrm{GSp}}(2g)$ and put $$h(x+iy) = \small\begin{bmatrix}
x I_g & y I_g \\ -yI_g & xI_g
\end{bmatrix}.$$ Then $X = \mathfrak H_g \sqcup \overline{\mathfrak H}_g$ is the union of Siegel’s upper half space and its complex conjugate. If we choose $K_{\mathrm{fin}}= G(\widehat {\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, then $G({{\ensuremath{\mathbf{A}}}}_{\mathrm{fin}}) = G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \cdot K_{\mathrm{fin}}$ and $$S_K = G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \backslash X \times G({{\ensuremath{\mathbf{A}}}}_{\mathrm{fin}}) / K_{\mathrm{fin}}\cong (G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}) \cap K_{\mathrm{fin}}) \backslash X = G({\ensuremath{{\ensuremath{\mathbf{Z}}}}}) \backslash X.$$ Now $G({\ensuremath{{\ensuremath{\mathbf{Z}}}}}) \backslash X$ is naturally isomorphic to the stack ${\mathcal{A}}_g$ parametrizing principally polarized abelian varieties of dimension $g$. Had we chosen $K_{\mathrm{fin}}$ smaller, $S_K$ would instead be a disjoint union of finite covers of ${\mathcal{A}}_g$, parametrizing abelian varieties with ‘$K_{\mathrm{fin}}$-level structure’.
Let ${\mathbb{V}}$ be an irreducible finite dimensional rational representation of $G$. To ${\mathbb{V}}$ we can attach a local system on $S_K$, which we also denote ${\mathbb{V}}$. As the reader may already have noticed, we (sloppily) use ‘local system’ as a catch-all term to describe several different structures: we obtain a locally constant sheaf of ${\ensuremath{{\ensuremath{\mathbf{Q}}}}}$-vector spaces on the topological space $S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}})$ which in a natural way underlies a variation of Hodge structure; moreover, ${\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell$ can (for any $\ell$) be identified with the base change of a smooth $\ell$-adic sheaf on ${S_K}$ over the reflex field. The étale cohomology groups of said $\ell$-adic sheaves are (after base changing to ${\ensuremath{{\ensuremath{\mathbf{C}}}}}$) related to the ordinary singular cohomology groups by a comparison isomorphism, and we may think informally of ${\mathbb{V}}$ as a ‘motivic sheaf’ and $H^\bullet(S_K,{\mathbb{V}})$ as a ‘mixed motive’ with a compatible system of $\ell$-adic and Hodge-theoretic realizations.
Decomposing cohomology
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[@arthurL2] In this subsection we will find the need to compare several different cohomology theories. We will use the phrase ‘ordinary cohomology’ to refer to the usual cohomology of the topological space $S_K(\mathbf C)$.
The spectral decomposition of $L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}))$ contains much information about the cohomology of Shimura varieties for $G$. The connection to automorphic representations is most transparent if we work transcendentally and consider the sheaf ${\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}$ on $S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}})$. Then one can consider instead of the usual de Rham complex the complex of forms $\omega$ such that $\omega$ and $d\omega$ are square integrable; the cohomology of this complex is called the *$L^2$-cohomology*. The $L^2$-cohomology has an interpretation in terms of $({\mathfrak{g}},K_\infty)$-cohomology: $$H^\bullet_{(2)}(S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),{\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) \cong H^\bullet({\mathfrak{g}},K_\infty; {\mathbb{V}}\otimes L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}))^{K_{\mathrm{fin}}}.$$ According to [@borelcasselman Section 4], the contribution from the continuous spectrum to the $(\mathfrak g, K_\infty)$-cohomology vanishes in many natural cases (including all Shimura varieties); in fact, the contribution is non-zero if and only if the $L^2$-cohomology is infinite dimensional. In particular, we may in our case replace $L^2(G({\ensuremath{{\ensuremath{\mathbf{Q}}}}})\backslash G({{\ensuremath{\mathbf{A}}}}))$ by the direct sum $\bigoplus_\pi m(\pi) \pi$ over the discrete spectrum, giving instead the expression $$H^\bullet_{(2)}(S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),{\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) \cong \bigoplus_{\pi \text{ disc.}} m(\pi) \pi_{\mathrm{fin}}^{K_{\mathrm{fin}}} \otimes H^\bullet({\mathfrak{g}},K_\infty;{\mathbb{V}}\otimes \pi_\infty).$$ In this decomposition, each $\pi_{\mathrm{fin}}^{K_{\mathrm{fin}}}$ is a module over the Hecke algebra, giving the cohomology a Hecke action. Each $H^\bullet({\mathfrak{g}},K_\infty;{\mathbb{V}}\otimes \pi_\infty)$ has a natural $(p,q)$-decomposition, defining a pure Hodge structure on each cohomology group $H^k_{(2)}(S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),{\mathbb{V}})$.
We say that an automorphic representation $\pi$ is *cohomological* if there exists a representation ${\mathbb{V}}$ for which $H^\bullet({\mathfrak{g}},K_\infty;{\mathbb{V}}\otimes \pi_\infty) \neq 0$. Wigner’s lemma gives a necessary condition for this nonvanishing of $({\mathfrak{g}},K_\infty)$-cohomology, namely that $\pi_\infty$ and ${\mathbb{V}}^\vee$ (denoting the contragredient) have the same infinitesimal character. For cohomological representations, the infinitesimal character is the book-keeping device that tells you to which local system the automorphic representation will contribute $L^2$-cohomology.
The natural map from $L^2$-cohomology to ordinary cohomology is in general neither injective nor surjective. One can however also define the *cuspidal cohomology* as the direct summand $$H^\bullet_{{{\mathrm{cusp}}}} (S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),{\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) \cong \bigoplus_{\pi \text{ cusp.}} m(\pi) \pi_{\mathrm{fin}}^{K_{\mathrm{fin}}} \otimes H^\bullet({\mathfrak{g}},K_\infty;{\mathbb{V}}\otimes \pi_\infty)$$ of the $L^2$-cohomology, and it injects naturally into the ordinary cohomology [@borelstablereal Corollary 5.5].
Finally one can consider the *inner cohomology*, which is defined as $$H^\bullet_!(S_K,{\mathbb{V}}) = \mathrm{Image}(H^\bullet_c(S_K,{\mathbb{V}}) \to H^\bullet(S_K,{\mathbb{V}})).$$ When we extend scalars to ${\ensuremath{{\ensuremath{\mathbf{C}}}}}$, the inner cohomology is sandwiched between the cuspidal and the $L^2$-cohomology. Indeed, the map from compactly supported cohomology to ordinary cohomology always factors through the $L^2$-cohomology, since the orthogonal projection of a closed compactly supported form to the space of harmonic forms is square integrable. This shows that the inner cohomology is a subquotient of the $L^2$-cohomology. On the other hand, the aforementioned result of Borel shows that the cuspidal cohomology injects into the inner cohomology.
The ‘complement’ of the inner cohomology is called the *Eisenstein cohomology*. Formally, it is defined as the cokernel of $H^\bullet_c(S_K,{\mathbb{V}}) \to H^\bullet(S_K,{\mathbb{V}})$. One could also consider the kernel, which gives the *compactly supported Eisenstein cohomology*. We denote these $H^\bullet_{\mathrm{Eis}}$ and $H^\bullet_{c,{\mathrm{Eis}}}$, respectively. We will consider $G = {\mathrm{GSp}}(2g)$, in which case each local system ${\mathbb{V}}$ is isomorphic to its dual, up to a twist by the multiplier. Indeed, the restriction of the representation ${\mathbb{V}}$ to $\operatorname{Sp}(2g)$ satisfies ${\mathbb{V}}\cong {\mathbb{V}}^\vee$, with the symplectic pairing providing the isomorphism. In this case, we see that either one of $H^\bullet_{\mathrm{Eis}}$ and $H^\bullet_{c,{\mathrm{Eis}}}$ determines the other via Poincaré duality.
The Zucker conjecture, proven independently in [@zuckerconj1; @zuckerconj2], gives an isomorphism between the $L^2$-cohomology of $S_K$ and the intersection cohomology of the Baily–Borel–Satake compactification $\overline S_K$: $$H^\bullet_{(2)}(S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),{\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}) \cong {H}^\bullet(\overline S_K({\ensuremath{{\ensuremath{\mathbf{C}}}}}),j_{!\ast} {\mathbb{V}}\otimes {\ensuremath{{\ensuremath{\mathbf{C}}}}}),$$ where $j : S_K \to \overline S_K$ is the inclusion, and $j_{!\ast}$ denotes the intermediate extension. This isomorphism is compatible with the Hecke algebra action. But the intersection cohomology makes sense algebraically, and we can decompose the intersection cohomology of ${\mathbb{V}}$ into irreducible Hecke modules already over some number field $F$. We thus get a decomposition $${H}^\bullet(\overline S_K,j_{!\ast}{\mathbb{V}}\otimes F) = \bigoplus_{\pi_{\mathrm{fin}}} \pi_{\mathrm{fin}}^{K_{\mathrm{fin}}} \otimes H^\bullet(\pi_{\mathrm{fin}}).$$ Here the sum runs over the finite parts of all discrete automorphic representations, and $H^\bullet(\pi_{\mathrm{fin}}) \otimes_F {\ensuremath{{\ensuremath{\mathbf{C}}}}}$ is isomorphic to $\bigoplus_{\pi_\infty} m(\pi_{\mathrm{fin}}\otimes \pi_\infty) H^\bullet({\mathfrak{g}},K_\infty;{\mathbb{V}}\otimes \pi_\infty)$. For any non-archimedean place $\lambda$ of $F$ we also get a structure of $\lambda$-adic Galois representation on each $H^\bullet(\pi_{\mathrm{fin}}) \otimes F_\lambda$ by a comparison isomorphism with the étale intersection cohomology. If we do not insist on a decomposition into absolutely irreducible Hecke modules we can take $F={\ensuremath{{\ensuremath{\mathbf{Q}}}}}$, as in Theorem \[mainthm\], where we e.g. consider a summand corresponding to all cusp forms of given weight, instead of a decomposition into Galois representations attached to individual cusp forms. See [@blasiusrogawskimotives Conjecture 5.2] for a conjectural formula expressing $H^\bullet(\pi_{\mathrm{fin}}) \otimes F_\lambda$ in terms of Galois representations attached to $\pi$.
The case of ${\mathcal{A}}_2$
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Consider again the stack ${\mathcal{A}}_2$ of principally polarized abelian surfaces. As in Example \[Ag\], we may think of it as a Shimura variety for ${\mathrm{GSp}}(4)$. However, we would prefer to work with $G = {\mathrm{PGSp}}(4)$, and there is a minor issue here. If we put $K_{\mathrm{fin}}= G(\widehat{{\ensuremath{{\ensuremath{\mathbf{Z}}}}}})$, then the corresponding Shimura variety is $S_K = {\mathrm{PGSp}}(4,{\ensuremath{{\ensuremath{\mathbf{Z}}}}}) \backslash (\mathfrak{H}_2 \coprod \overline{\mathfrak{H}}_2)$, which fails to be isomorphic to ${\mathcal{A}}_2$ as a *stack*. Indeed every point of ${\mathcal{A}}_2$ has $\pm 1$ in its isotropy group, but a general point of $S_K$ has trivial isotropy. The projection ${\mathrm{GSp}}(4) \to {\mathrm{PGSp}}(4)$ defines a map $\pi \colon {\mathcal{A}}_2 \to S_K$ which induces an isomorphism on coarse moduli spaces, but which is a $\mu_2$-gerbe in the sense of stacks.
The finite dimensional irreducible representations of $G$ are indexed by integers $a \geq b \geq 0$ for which $a+b$ is even. The local systems on $S_K$ obtained in this way are strongly related to the local systems ${\mathbb{V}}_{a,b}$ that we defined in Section \[mainresults\]. Specifically, if $a+b$ is even, then we may Tate twist the local system ${\mathbb{V}}_{a,b}$ on ${\mathcal{A}}_2$ to be a weight zero variation of Hodge structure/$\ell$-adic sheaf; its pushforward under $\pi$ is the one that is naturally attached to an irreducible representation of ${\mathrm{PGSp}}(4)$. Since $\mathrm R \pi_ \ast {\mathbb{V}}_{a,b} = \pi_\ast {\mathbb{V}}_{a,b}$, it will suffice to compute the cohomology of the local systems on $S_K$. From now on we tacitly identify the local systems on ${\mathcal{A}}_2$ and on $S_K$ with each other.
In this section we will see how the results in [@flicker] allow the computation of the cuspidal and intersection cohomology of these local systems on ${\mathcal{A}}_2$. Let me emphasize that as mentioned in the above paragraph, by our definition the ${\mathbb{V}}_{a,b}$ are Weil sheaves of weight $a+b$; this is the *cohomological normalization*, which is the most natural from the point of view of algebraic geometry. There is also the *unitary normalization*, where ${\mathbb{V}}_{a,b}$ has weight $0$, which is used in Flicker’s work. If $a+b$ is even, as in our case, then the two differ only by a Tate twist. We will from now on always make this Tate twist whenever we quote results from Flicker’s book, without explicitly mentioning it.
Since ${\mathcal{A}}_2$ is the complement of a normal crossing divisor in a smooth proper stack over $\operatorname{Spec}({\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, and the local systems ${\mathbb{V}}_{a,b}$ are also defined over $\operatorname{Spec}({\ensuremath{{\ensuremath{\mathbf{Z}}}}})$, the cohomology groups $H^\bullet({\mathcal{A}}_2,{\mathbb{V}}_{a,b} \otimes {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell)$ must define Galois representations of a very special kind: they are unramified at every prime $p \neq \ell$ and crystalline at $\ell$. The same phenomenon is clear also on the automorphic side. If $\pi_{\mathrm{fin}}^{K_{\mathrm{fin}}} \neq 0$ and $K_{\mathrm{fin}}= G(\widehat{{\ensuremath{{\ensuremath{\mathbf{Z}}}}}})$, then $\pi_{\mathrm{fin}}$ must be spherical at all primes by definition since $G({\ensuremath{{\ensuremath{\mathbf{Z}}}}}_p)$ is a special maximal compact subgroup of $G({\ensuremath{{\ensuremath{\mathbf{Q}}}}}_p)$. Conversely if $\pi_{\mathrm{fin}}$ is spherical everywhere then $\pi_{\mathrm{fin}}^{K_{\mathrm{fin}}}$ is exactly $1$-dimensional.
Considering ${\mathrm{PGSp}}(4)$ rather than ${\mathrm{GSp}}(4)$ is the same as only considering automorphic representations of ${\mathrm{GSp}}(4)$ with trivial central character. The reason we can do this is that we are considering only the completely unramified case (i.e. the case of the full modular group); in general, the image of a congruence subgroup of ${\mathrm{GSp}}(4)$ in ${\mathrm{PGSp}}(4)$ will no longer be a congruence subgroup. We restrict ourselves to ${\mathrm{PGSp}}(4)$ in this paper as this is the situation considered in [@flicker].
We note that Flicker’s work assumes that all automorphic representations $\pi$ occuring are elliptic at at least three places. This is explained in Section I.2g of Part 1 of the book. This assumption is present in order to replace Arthur’s trace formula with the simple trace formula of [@flickerkazhdan]. However, he also notes that this assumption is only present in order to simplify the exposition — the same results can be derived assuming only that $\pi$ is elliptic at a single real place, using the same ideas used to derive the simple trace formula in [@flickerkazhdan], as detailed in [@laumoncompositio; @laumonasterisque]. In particular Flicker’s classification of the cohomological part of the discrete spectrum carries through (an archimedean component which is cohomological is elliptic).
We begin by determining ${H}^\bullet(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})$. This amounts to determining all representations in the discrete spectrum of ${\mathrm{PGSp}}(4)$ which are spherical at every finite place and cohomological, and for each of them the corresponding Galois representation $H^\bullet(\pi_f)$. All these things are described very precisely by Flicker. Then we shall see that $H^\bullet_{{\mathrm{cusp}}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is well defined as a subspace of the étale intersection cohomology, and that it coincides with the inner cohomology.
The Vogan–Zuckerman classification
----------------------------------
Recall that an automorphic representation $\pi_{\mathrm{fin}}\otimes \pi_\infty$ is cohomological if $\pi_\infty$ has nonzero $({\mathfrak{g}},K_\infty)$-cohomology with respect to some finite dimensional representation ${\mathbb{V}}$. If $\pi_\infty$ is in the discrete series, then $\pi$ is always cohomological. The cohomological representations which are not in the discrete series can be determined by [@vz]. We recall from [@taylor3fold 293] the result for ${\mathrm{GSp}}(4)$:
In the regular case there are no cohomological ones apart from the two discrete series representations, which we denote by $\pi^H$ and $\pi^W$ (we omit the infinitesimal character from the notation). The former is in the holomorphic discrete series and the latter has a Whittaker model. Both have $2$-dimensional $({\mathfrak{g}},K_\infty)$-cohomology, concentrated in degree 3: their Hodge numbers are $(3,0)$ and $(0,3)$, and $(2,1)$ and $(1,2)$, respectively.
The representations $\pi$ with $\pi_\infty = \pi^H$ are correspond bijectively to cuspidal Siegel modular eigenforms. If $F$ is a holomorphic modular form on Siegel’s upper half space of genus $g$, then by the Strong Approximation theorem it defines a function on $G({{\ensuremath{\mathbf{A}}}})$, where $G={\mathrm{GSp}}(2g)$. If it is modular for the full modular group, then we obtain a function with trivial central character. The subspace spanned by all right translates of this function is the sought for automorphic representation (or a sum of several copies of it). Conversely, any automorphic representation $\pi$ with archimedean component in the holomorphic discrete series determines uniquely a holomorphic vector-valued cusp form by considering the one-dimensional space of lowest $K_\infty$-type in $\pi_\infty$, and $\pi_{\mathrm{fin}}^{K_{\mathrm{fin}}}$ being one-dimensional forces it to be an eigenvector for all Hecke operators. See [@asgarischmidt] for more details.
For singular weights there are further possibilities. If $b=0$ there is a unitary representation $\pi^1$ whose $({\mathfrak{g}},K_\infty)$-cohomology is 2-dimensional in degrees $2$ and $4$, with Hodge types $(2,0)$, $(0,2)$, $(3,1)$ and $(1,3)$.
If $a=b$ there are two unitary representations $\pi^{2+}$ and $\pi^{2-}$. One is obtained from the other by tensoring with the sign character. Both have $1$-dimensional $({\mathfrak{g}},K_\infty)$-cohomology in degrees $2$ and $4$, with Hodge types $(1,1)$ and $(2,2)$.
Finally if $a=b=0$ we must in addition consider one-dimensional representations, which have cohomology in degrees $0$, $2$, $4$ and $6$: we will ignore this case.
Packets and multiplicities
--------------------------
In Flicker’s book, the discrete spectrum of ${\mathrm{PGSp}}(4)$ is partitioned into ‘packets’ and ‘quasi-packets’, and he conjectures that these coincide with the conjecturally defined $L$-packets and $A$-packets. However, in the totally unramified case the situation simplifies. In general, the (conjectural) $A$-packets are products of *local* $A$-packets, which specify the possible local components $\pi_v$. The local packets at non-archimedean $v$ are expected to have exactly one spherical member. Since we are only going to consider representations which are spherical at *every* finite place, we thus see that $\pi$ and $\pi'$ will be in the same $A$-packet if and only if they are in the same $L$-packet if and only if $\pi_{\mathrm{fin}}\cong \pi_{\mathrm{fin}}'$. For this reason we simply write *packet* everywhere in what follows.
In Flicker’s classification there are five types of automorphic representations in the discrete spectrum. In the first three types, the corresponding packets are *stable*: each representation in the packet occurs with multiplicity exactly $1$ in the discrete spectrum. Types $4$ and $5$, however, are *unstable*. This means that the multiplicities are not constant over the packets: in general some representations in the packet occur with multiplicity $0$ and others with multiplicity $1$. Flicker [@flicker Section 2.II.4] gives explicit formulas for the multiplicities of the representations in the packet.
In general there are local packets at each prime $p$ in the unstable case, which consist of either one or two elements. We write such local packets as $\{\Pi_p^+\}$ and $\{\Pi_p^+,\Pi_p^-\}$, respectively. An element of the global packet is specified by choosing an element of the local packet at each $p$. All but finitely many of the local packets will be singletons, so each packet is finite. When $p=\infty$ we always have $\Pi_p^- = \pi^H$. If $\pi$ lies in an unstable packet, its multiplicity in the discrete spectrum depends only on the parity of the number of places $p$ where $\pi_p = \Pi_p^-$.
However, as we have already mentioned, the local packet contains only one element for a prime $p$ where $\pi$ is spherical. More generally, certain local representations need to be discrete series in order for $\Pi_p^-$ to be nonzero. Since we are in the level $1$ case, this means that the representations in the packet can differ only in their archimedean component, and the multiplicity formulas simplify significantly: they depend only on whether or not $\pi_\infty = \pi^H$.
To each discrete spectrum automorphic representation $\pi$ one can attach a $4$-dimensional Galois representation whose Frobenius eigenvalues at $p$ are given by the Langlands parameters at $p$ of $\pi$. (Here we fix the 4-dimensional spin representation of $\mathrm{Spin}(5)$, the dual group of ${\mathrm{PGSp}}(4)$.) If $\pi$ is in a stable packet, then $H^\bullet(\pi_{\mathrm{fin}})$ is $4$-dimensional and coincides with this attached representation. In the unstable case, the attached Galois representation is always a sum of two $2$-dimensional pieces, and $H^\bullet(\pi_{\mathrm{fin}})$ is given by one of these two summands. Which of the two halves contributes nontrivially is decided by a formula similar to the multiplicity formula, see [@flicker Part 2, V.2]. In particular it again has the feature that it depends on the parity of the number of places where $\pi_p = \Pi_p^-$, and simplifies significantly in the completely unramified case.
The discrete spectrum of ${\mathrm{PGSp}}(4)$
---------------------------------------------
The discrete spectrum automorphic representations which can contribute nontrivially to $H^\bullet_{(2)}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ have an archimedean component with infinitesimal character $(a,b)+(2,1)$. A complete classification into five types is given in [@flicker Theorem 2, pp. 213–216]. We deal with each type separately. This classification is the same as the one announced by Arthur for ${\mathrm{GSp}}(4)$ [@arthurclassification], except that the ones of Howe–Piatetski-Shapiro-type do not appear.
In what follows we write in parentheses the names assigned to these families by Arthur.
### Type 1 (General type)
These are exactly the ones that lift to cuspidal representations of ${\mathrm{PGL}}(4)$.
Each of these lies in a packet of cardinality $2$, where the elements in the packet are distinguished by their archimedean component: one is in the holomorphic discrete series and the other has a Whittaker model. Both elements of the packet occur with multiplicity $1$ in the discrete spectrum. Packets of this type correspond bijectively to vector valued cuspidal Siegel eigenforms which are neither endoscopic (a Yoshida-type lifting) nor CAP (a Saito–Kurokawa-type lifting). The contribution from this part of the discrete spectrum to ${H}^\bullet(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})$ is concentrated in degree $3$ and is the sum of the Galois representations attached to the Siegel cusp forms. We shall see that the Yoshida-type liftings do not occur in level 1. We denote this contribution to the cohomology by ${\mathsf{S}}_{a-b,b+3}^{\mathrm{gen}}$.
### Type 2 (Soudry type)
These packets are singletons, and the archimedean component is $\pi^1$, and will therefore not occur unless $b=0$. Every packet is obtained by a lifting from a cuspidal representation $\Pi$ of $\operatorname{GL}(2)$, corresponding to a cusp eigenform of weight $a+1$ whose central character $\xi$ is quadratic, $\xi \neq 1$, and $\xi \Pi = \Pi$. This is obviously impossible in level $1$ for several reasons: for one, $a$ must be even, and there are no modular forms of odd weight for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$.
### Type 3 (One-dimensional type)
These are the representations with $\pi_\infty$ $1$-dimensional and will only occur when $a=b=0$; for our purposes this case can clearly be ignored.
### Type 4 (Yoshida type)
This is the first unstable case. All these $\pi$ have $\pi_\infty \in \{\pi^H,\pi^W\}$ and their $L$-function is the product of $L$-functions attached to cusp forms for $\operatorname{GL}(2)$. For each pair of cuspidal automorphic representations $\Pi_1$ and $\Pi_2$ of ${\mathrm{PGL}}(2)$ whose weights are $a+b+4$ and $a-b+2$, respectively, there is a packet $\{\pi\}$ of Yoshida type. As explained earlier, the fact that we are in the unramified case implies that members of the packet can only differ in their archimedean component, so we should consider only $\pi_{{\mathrm{fin}}} \otimes \pi^H$ and $\pi_{\mathrm{fin}}\otimes \pi^W$. The multiplicity formula simplifies (since we are in the unramified case) to $$m(\pi_{\mathrm{fin}}\otimes \pi^H) = 0, \qquad m(\pi_{\mathrm{fin}}\otimes \pi^W)=1,$$ and so $\pi_{\mathrm{fin}}\otimes \pi^W$ will contribute a $2$-dimensional piece of the cohomology in degree 3. The trace of Frobenius on this part of cohomology is also calculated by Flicker and we find the Galois representation $\rho_{\Pi_2} \otimes {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-b-1)$, where $\rho_{\Pi_2}$ is the $2$-dimensional representation attached to $\Pi_2$. Summing over all $\Pi_1 $ and $\Pi_2$ this part therefore contributes $$s_{a+b+4}{\mathsf{S}}_{a-b+2} (-b-1)$$ to ${H}^3(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})$.
We note in particular that there are no Yoshida-type liftings to Siegel cusp forms in level 1: these would correspond to a $\pi$ with $\pi_\infty = \pi^H$ and multiplicity $1$.
The required liftings and multiplicity formulas for the endoscopic case have also been established for ${\mathrm{GSp}}(4)$ in [@weissauerbook Theorem 5.2].
### Type 5 (Saito–Kurokawa type)
This case appears only when $a=b$. Here there are four possible archimedean components: $\pi^H, \pi^W, \pi^{2+}$ and $\pi^{2-}$. Every packet contains precisely one of $\pi^{2+}$ and $\pi^{2-}$. For each cuspidal automorphic representation $\Pi$ of ${\mathrm{PGL}}(2)$ of weight $a+b+4$ and for $\xi \in \{1,\operatorname{sgn}\}$ we get a Saito–Kurokawa packet $\{\pi\}$. Since we are in level $1$, we can ignore the character $\xi$ (it must be trivial), which means that $\pi^{2-}$ will not appear.
I should also say that there is a minor error at this place of Flicker’s book. Flicker states that the Langlands parameters at a place $u$ are (his notation) $$\mathrm{diag}(\xi_u q_u^{1/2}z_{1u}, \xi_u q_u^{1/2}z_{2u}, \xi_u q_u^{-1/2}z_{2u}, \xi_u q_u^{-1/2}z_{1u})$$ when they should be $$\mathrm{diag}(z_{1u}, \xi_u q_u^{1/2}, \xi_u q_u^{-1/2}, z_{2u}).$$
Let us then consider the multiplicities, which again simplify since we are in the level 1 case: we find $$m(\pi_{\mathrm{fin}}\otimes \pi_\infty) = \frac 1 2 \left( 1 + \varepsilon(\Pi,{\tfrac 1 2} )\cdot (-1)^{n}\right)$$ where $n = 1$ if $\pi_\infty = \pi^H$ and $n=0$ otherwise, and $\varepsilon(\Pi, \frac{1}{2}) = (-1)^k$ if $\Pi$ is attached to a cusp form of weight $2k$.
We thus see that if $a=b$ is odd, the only representation in the packet with nonzero multiplicity is $\pi_{\mathrm{fin}}\otimes \pi^H$, which should correspond to a Siegel modular form. The Siegel modular forms obtained in this way are precisely the classical Saito–Kurokawa liftings, and the contribution in this case is exactly ${\mathsf{S}}_{a+b+4}$.
For $a = b$ even we could a priori have both $\pi_{\mathrm{fin}}\otimes\pi^W$ and $\pi_{\mathrm{fin}}\otimes \pi^{2+}$ with nonzero multiplicity. But we can see by studying the Frobenius eigenvalues that $\pi^W$ will not appear. Indeed, the representation $\pi_{\mathrm{fin}}\otimes \pi^W$ would contribute to the intersection cohomology in degree $3$, as we see from the $({\mathfrak{g}},K_\infty)$-cohomology of $\pi^W$. Then its Frobenius eigenvalues are pure of weight $a+b+3$. But the Frobenius eigenvalues at $p$ will be $p^{b+1}$ and $p^{b+2}$, as determined by Flicker, a contradiction. On the other hand, we know that $\pi_{\mathrm{fin}}\otimes \pi^{2+}$ is automorphic: it is the Langlands quotient of $$\mathrm{Ind}_{P({{\ensuremath{\mathbf{A}}}})}^{{\mathrm{PGSp}}(4,{{\ensuremath{\mathbf{A}}}})} \left( \Pi \otimes 1 \right),$$ where $P$ is the Siegel parabolic (whose Levi component is ${\mathrm{PGL}}(2) \times \operatorname{GL}(1)$), and the multiplicity formula shows that it has multiplicity $1$ in the discrete spectrum. The representations of this form will contribute a term $s_{a+b+4} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-{b-1}) $ to ${H}^2(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})$ and $s_{a+b+4}{\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-{b-2}) $ to ${H}^4(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})$.
That $\pi_\infty=\pi^W$ does not occur in the Saito–Kurokawa case is mentioned as a conjecture of Blasius and Rogawski in [@tilouinesiegelthreefold Section 6]. The argument above will prove this conjecture for ${\mathrm{PGSp}}(4)$. Probably a proof for ${\mathrm{GSp}}(4)$ in general can be obtained by a similar argument, or by considering the possible Hodge numbers of $H^\bullet(\pi_{\mathrm{fin}})$.
The inner cohomology and the proof of the main theorem
------------------------------------------------------
From what we have seen so far, we can completely write down the $L^2$-cohomology and the intersection cohomology of any local system on ${\mathcal{A}}_2$. Summing up the contributions from all parts of the discrete spectrum, we see that $${H}^3(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})\cong {\mathsf{S}}^{{\mathrm{gen}}}_{a-b,b+3} +s_{a+b+4} {\mathsf{S}}_{a-b+2} +
\begin{cases} {\mathsf{S}}_{a+b+4} & a=b \text{ odd,} \\ 0 & \text{otherwise.}\end{cases}$$ The cohomology vanishes outside the middle degree in all cases except when $a=b$ is even, when we have $${H}^2(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b}) \cong \begin{cases} s_{a+b+4} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}_\ell(-b-1) & a=b \text{ even,} \\ 0 & \text{otherwise,}\end{cases}$$ and ${H}^4(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b}) \cong {H}^2(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b})(-1).$
Note that the sum $${\mathsf{S}}^{{\mathrm{gen}}}_{a-b,b+3} +
\begin{cases} {\mathsf{S}}_{a+b+4} & a=b \text{ odd,} \\ 0 & \text{otherwise,}\end{cases}$$ is exactly what was denoted $\overline {\mathsf{S}}_{a-b,b+3}$ in Theorem \[mainthm\], since there are no Yoshida-type liftings in our case.
If we wish to determine in addition the cuspidal cohomology, then we need to understand which of the above representations are in the residual spectrum. The residual spectrum of ${\mathrm{GSp}}(4)$ is completely described in [@kimresidual Section 7]. We see that there is exactly one case above where the representation is residual: namely, the Langlands quotient of $ \mathrm{Ind}_{P({{\ensuremath{\mathbf{A}}}})}^{{\mathrm{PGSp}}(4,{{\ensuremath{\mathbf{A}}}})} \left( \Pi \otimes 1 \right)$ is residual if and only if $L(\Pi, \frac 1 2 ) $ is nonzero. We deduce that the cuspidal cohomology coincides with the $L^2$-cohomology except in degrees $2$ and $4$ when $a=b$ is even, where we have $$H^2_{{\mathrm{cusp}}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \cong s'_{a+b+4} {\ensuremath{{\ensuremath{\mathbf{Q}}}}}(-b-1)$$ (so conjecturally, it vanishes) and similarly for $H^4_{\mathrm{cusp}}$. We also observe that for all packets, either all discrete representations are cuspidal or all are residual, so that the cuspidal cohomology makes sense also as a summand of the étale intersection cohomology (a priori it is only a summand in the $L^2$-cohomology), and we can talk about the Galois representation on the cuspidal cohomology.
The Eisenstein cohomology of any local system on ${\mathcal{A}}_2$ has been completely determined in any degree, considered as an $\ell$-adic Galois representation up to semi-simplification, in [@harder]. From loc. cit. and the above discussion we may deduce the following.
The natural map $H^\bullet_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \to H^\bullet_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is an isomorphism for any $a,b$.
Recall that one has $${H}^k(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b}) = H^k_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \oplus H^k_{\mathrm{res}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$$ and $$W_{k+a+b}H^k({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) = H^k_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \oplus W_{k+a+b} H^k_{\mathrm{Eis}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}).$$ Moreover, the map ${H}^k(\overline {\mathcal{A}}_2,j_{!\ast}{\mathbb{V}}_{a,b}) \to W_{k+a+b}H^k({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is surjective and maps the cuspidal cohomology into the inner cohomology. Hence if $H^k_{\mathrm{res}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ and $W_{k+a+b} H^k_{\mathrm{Eis}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ have the same dimension, then $H^k_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \to H^k_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is an isomorphism.
Now we have seen that $H^k_{\mathrm{res}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is nonzero only for $k \in \{2,4\}$ and $a=b$ even, so these are the only cases where it is not automatic that $H^\bullet_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \to H^\bullet_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is an isomorphism. The dimension of $H^k_{\mathrm{res}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is $s_{a+b+4}-s_{a+b+4}'$ in these cases. From Harder’s paper we see that $W_{k+a+b}H^k_{\mathrm{Eis}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \neq 0$ only for $k =2$ and $a=b$ even, in which case its dimension, too, is $s_{a+b+4}-s_{a+b+4}'$. But then $H^2_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) \to H^2_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ is an isomorphism by the preceding paragraph, and then it is an isomorphism also in degree $4$ since both the cuspidal and the inner cohomology satisfy Poincaré duality.
The equality of dimensions above is not surprising, since Harder explicitly constructs these pure Eisenstein cohomology classes as residues of Eisenstein series associated to cusp forms for $\operatorname{SL}(2,{\ensuremath{{\ensuremath{\mathbf{Z}}}}})$ with nonvanishing central value. So in a sense the dimension argument in the preceding theorem is unnecessarily convoluted. See also [@residual] which describes in general all possible contributions from the residual spectrum to the Eisenstein cohomology of a Siegel threefold.
The main theorem of the paper follows from this result, as we now explain.
Up to semi-simplification we have $$H^\bullet_c({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) = H^\bullet_! ({\mathcal{A}}_2,{\mathbb{V}}_{a,b})\oplus H^\bullet_{c,{\mathrm{Eis}}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b}),$$ and $H^\bullet_{c,{\mathrm{Eis}}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$ was as already remarked determined in [@harder]. By the preceding proposition we have $H^\bullet_!({\mathcal{A}}_2,{\mathbb{V}}_{a,b}) = H^\bullet_{\mathrm{cusp}}({\mathcal{A}}_2,{\mathbb{V}}_{a,b})$, and the latter has been determined already in this section. Summing up the Eisenstein cohomology and the cuspidal contribution gives the result.
|
---
abstract: 'Using the $SU(2)$ gauge coupling, $g_{W^\pm} (M^2_{W^\pm})$, at the high-energy scale of $M_{W^\pm}$, defined by the (theoretical value of the) leptonic $W$-width, rather than using the low-energy value, defined via the Fermi coupling, $G_\mu$, in the Born approximation, and supplementing with Coulomb corrections and initial state radiation, errors with respect to the exact one-loop results for the differential cross section of $e^+e^-\to W^+W^-$ are below 1% at LEP 2 energies at all $W^+W^-$ production angles. A similar procedure is suggested to incorporate leading bosonic loop effects into four-fermion production in the fermion-loop scheme. The resulting accuracy below 1% is sufficient for LEP 2 experiments.'
author:
- |
[**M. Kuroda**]{}\
Institute of Physics, Meiji Gakuin University, Yokohama 244, Japan\
[**I. Kuss and D. Schildknecht**]{}\
Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany
date: |
hep-ph/9705294\
BI-TP 97/15\
May 1997
title: |
**A Simple Approximation of the One-Loop Corrected\
Cross Section for $e^+e^- \to W^+W^-$at LEP 2[^1]**
---
The process of $W$-pair production in $e^+e^-$ annihilation is presently studied experimentally at LEP 2. In the future, it will be one of the outstanding processes at a linear collider in the TeV energy range. It yields direct experimental information on the non-Abelian couplings characteristic for the $SU(2) \times U(1)$ electroweak theory, and it allows to put bounds on potential non-standard $Z_0W^+W^-$ and $\gamma W^+W^-$ couplings [@bkrs; @ks]. Within the $SU(2) \times U(1)$ electroweak theory, the calculation of the radiative corrections to this process has received much attention [@bohm]-[@been3].
The exact evaluation of the full one-loop electroweak corrections leads to complicated and lengthy expressions in terms of twelve $s$- and $t$-dependent form factors. Actually, only those (three) form factors which appear in amplitudes of the form of the Born approximation are numerically important [@bohm2; @fleisch]. Unfortunately, however, no simple analytic form for these form factors, valid at arbitrary $e^+e^-$ energies, has been given so far. For the LEP 2 energy range, however, a simple approximation has indeed been suggested [@bohm2]. In this approximation, the Born approximation is evaluated in terms of the appropriately introduced Fermi coupling, $G_\mu$, and the high-energy electromagnetic coupling, $\alpha (s)$, and it is supplemented by Coulomb corrections and by initial state radiation employing the structure function method. By comparing the improved Born approximation (IBA) with the full one-loop results, accuracies of 1.5% to 2% were found [@been; @ditt] in the angular distributions in the LEP 2 energy range. For energies above 500 GeV, a simplification of the exact ${\cal O}
(\alpha)$ corrections has been given in terms of an asymptotic high-energy expansion [@been2].
It is the purpose of the present note to point out that a slight modification of the previously suggested [@bohm2] improved Born approximation for the LEP 2 energy range improves its accuracy to values well below 1%. Accordingly, such an approximation will be sufficient for all practical purposes at LEP 2. Our results are obtained by replacing the low-energy value of the $SU(2)$ gauge coupling, $g^2_{W^\pm}(0) \equiv 4 \sqrt 2 M^2_{W^\pm} G_\mu$ previously employed in [@bohm2; @been; @ditt], by its high-energy value, defined by the leptonic $W^\pm$ width, $g^2_{W^\pm} (M^2_{W^\pm}) \equiv 48 \pi \Gamma_l^W/M_{W^\pm}$ [@ditt2] more appropriate for the LEP 2 energy scale[^2]. In essence, this approach amounts to employing a different renormalization scheme, defined [@ditt2] by using $\Gamma_l^W$ instead of $G_\mu$ as experimental input.
The Born amplitude for the process $e^+e^- \to W^+W^-$, in the notation of refs. [@been; @ditt], takes the form $${\cal M}_{Born} (\kappa, \lambda_+, \lambda_-, s,t) =
g^2_{W^\pm} {1\over 2} \delta_{\kappa -}
{\cal M}_I + e^2 {\cal M}_Q,\label{one}$$ where the dependence on energy and momentum transfer squared, $s$ and $t$, and on the electron and $W^\pm$ boson helicities, $\kappa = \pm 1$ and $\lambda_\pm
= 0, \pm 1$, is contained in ${\cal M}_I$ and ${\cal M}_Q$. The $SU(2)$ gauge coupling and the electromagnetic coupling in (\[one\]) have been denoted by $g_{W^\pm}$ and $e$, respectively. Even though (\[one\]) is easily obtained by starting from the Feynman rules for $t$-channel neutrino and $s$-channel $\gamma$ and $Z_0$ exchange, we prefer to gain (\[one\]) directly from the electroweak theory in the $BW_3$ base, i.e. before diagonalization of $BW_3$ mixing.
\[fig1\]
From the diagrams (a), (b) and (c) in Fig. 1, one immediately obtains $${\cal M}_I = {1 \over {s - M^2_Z}} {\cal M}_s + {1 \over t} {\cal M}_t.
\label{two}$$ One recognizes the correspondence of this expression to diagrams (a) and (b) in Figure 1, diagram (c) supplying the substitution $M^2_W \to M^2_Z = (1 + (g^\prime/
g_{W^\pm})^2) M^2_W$ in the $s$-channel term[^3]. For the explicit forms of the $s$-channel and $t$-channel quantities, ${\cal M}_s$ and ${\cal M}_t$, we refer to [@been].
The $B$-propagator, to all orders in $BW_3$ mixing, according to diagram (e) becomes
(470,40) (120,20)[1.5]{} (120,20)(150,20)[2]{}[3]{} (130,27)[$B$]{} (163,20)[13]{}[0.5]{} (176,20)(206,20)[2]{}[3]{} (206,20)[1.5]{} (186,27)[$B$]{} (216,17)[=]{} (235,15)[$\frac{\displaystyle s-M_W^2}{\displaystyle s(s-M_Z^2)}$,]{} (451,15)[(3)]{}
where $M^2_B \equiv (g^\prime/g_{W^\pm})^2 M^2_W$ for the square of the $B$-boson mass and $-(g^\prime /
g_{W^\pm})$ $\cdot M^2_W$ for the mixing strength were used. For right-handed electrons, only the $B$-coupling (to the weak hypercharge current) of diagram (d) is relevant, implying, with (3) and $g^{\prime 2} M^2_W = e^2M^2_Z$, that $${\cal M}_Q = -{M_Z^2 \over s(s-M_Z^2)} {\cal M}_s.
\label{four}$$ This expression holds equally well for left-handed electrons, where contributions from the diagrams (d) and (a)+(c) make up one half of ${\cal M}_Q$ each.
We feel that the above derivation of (\[one\]) illuminates in the most straight-forward manner the decomposition of (\[one\]) into a weak $SU(2)$ and an electromagnetic piece, where the latter one for right-handed electrons is entirely induced by the $B$-boson coupling to the hypercharge current. Moreover, the origin of the double-pole structure in (\[four\]) as a result of $BW_3$ mixing becomes immediately obvious. The double pole leads to a suppression of the amplitude (\[four\]) relative to (\[two\]) at high energies, which in the case of longitudinal $W^\pm$ bosons is compensated, however, by the longitudinal polarization vectors.
The calculation of the cross section for $e^+e^- \to W^+W^-$ from (\[one\]) requires the specification of a scale at which the $SU(2)$ gauge coupling $g^2_{W^\pm}$ and the electromagnetic coupling $e^2$ are to be defined. As $W$ pairs are produced at LEP 2 at energies of $2M_{W^\pm} \lsim \sqrt s \sim 200 GeV$, it is natural to choose a high-energy scale, such as $\sqrt s$. We expect that it is sufficiently accurate to use the scale $M_W \simeq M_Z$ instead of $\sqrt s$. Accordingly, we choose [@burk] $$\left({{e^2} \over {4 \pi}}\right)^{-1} =
\alpha^{-1} (M^2_Z) = 128.89 \pm 0.09,
\label{five}$$ for the electromagnetic coupling, and define $g^2_{W^\pm} (M^2_W)$ by the leptonic width of the $W^\pm$ [@ditt2][^4] $$g^2_{W^\pm} (M^2_{W^\pm}) = 48 \pi {{\Gamma_l^W}\over {M_{W^\pm}}}.
\label{six}$$ Expressing $\Gamma_l^W$ in terms of the Fermi coupling, $$\Gamma^W_l = {{G_\mu M^3_W} \over {6 \sqrt 2 \pi (1 +
{\Delta y}^{SC})}},
\label{seven}$$ including one-loop corrections, one finds that $g^2_{W^\pm}
(M_W^2)$ differs from the gauge coupling, $g^2_{W^\pm} (0)$, defined from $\mu$-decay, by the correction factor $(1 + {\Delta y}^{SC})^{-1}$, $$g^2_W (M^2_{W^\pm}) = {{g^2_{W^\pm} (0)} \over
{1 + {\Delta y}^{SC}}}.
\label{eight}$$ The “scale-change (SC)” part, ${\Delta y}^{SC}$, of the coupling parameter $\Delta y$ of ref. [@ditt2], takes care of the change in scale between $\mu$-decay and $W^\pm$-decay. It is given by $${\Delta y}^{SC} = \Delta y^{SC}_{ferm} + {\Delta y}^{SC}_{bos},
\label{nine}$$ where the fermionic part, $\Delta y^{SC}_{ferm}$, is essentially due to contributions arising from light fermion loops in the $W^\pm$ propagator. For $m_t \to \infty$, it is given by $$\Delta y^{SC}_{ferm} \vert_{m_t \to \infty} = - {{3 \alpha (M^2_Z)} \over
{4 \pi s^2_0}} \simeq - 8.01 \cdot 10^{-3},
\label{ten}$$ while for $m_t = 180$ GeV, $$\Delta y^{SC}_{ferm} \vert_{m_t = 180 GeV} = - 7.79 \times 10^{-3}.
\label{11}$$ This negative contribution to ${\Delta y}^{SC}$ is largely compensated by the bosonic one, ${\Delta y}^{SC}_{bos}$, which is practically independent[^5] of the Higgs boson mass and is given by[^6] $${\Delta y}^{SC}_{bos} = 11.1 \times 10^{-3}.
\label{12}$$ Accordingly, the SU(2) coupling, $g^2_{W^\pm} (M_{W^\pm}^2)$, to be used when evaluating the cross section for the process $e^+e^- \to W^+W^-$ in the Born approximation (\[one\]) is given by $$g^2_{W^\pm} (M^2_{W^\pm}) = {{4 \sqrt 2 G_\mu M^2_{W^\pm}} \over {1 +
{\Delta y}^{SC}}},
\label{13}$$ with a correction term, due to scale change, whose magnitude, $${\Delta y}^{SC} = 3.3 \times 10^{-3},
\label{14}$$ is practically independent of the precise values of $m_t$ and $M_H$. Even though there is such a strong cancellation between fermions and bosons in ${\Delta y}^{SC}$, thus implying the fairly small value of ${\Delta y}^{SC}$ in (\[14\]), the correction induced by ${\Delta y}^{SC}$ will be seen to be decisive for providing the announced accuracy, better than 1%, in the expression for the $W^\pm$ pair-production cross section.
Including the Coulomb correction and the initial state radiation (ISR) in soft photon approximation, the improved Born approximation for the differential cross section takes the form $$\begin{aligned}
\left({{d \sigma} \over {d \Omega}}\right)_{IBA} &=& {\beta \over {64 \pi^2 s}}
\left\vert {{2 \sqrt 2 G_\mu M^2_W} \over {1 + {\Delta y}^{SC}}}
{\cal M}^\kappa_I \delta_{\kappa^-} + 4 \pi\alpha (M^2_Z) {\cal M}^\kappa_Q
\right\vert^2\cr
&&+\left({{d\sigma} \over {d \Omega}}\right)_{Coul} (1 - \beta^2)^2 +
\left({{d\sigma} \over
{d\Omega}}\right)_{ISR}.\label{15}\end{aligned}$$ The only difference of the present work with respect to refs. [@bohm2; @been; @ditt] consists in the inclusion of ${\Delta y}^{SC}$ which introduces according to (\[13\]) the appropriate high-energy scale for the $SU(2)$-coupling strength. We note the dependence of $(d\sigma/d\Omega)_{ISR}$ in (\[15\]) on the choice of the photon splitting scale, $Q^2$, inherently connected with the structure function method. This method consists of evaluating the leading logarithmic QED corrections thus all contributions proportional to $(\alpha/\pi)\ln(m_e^2/Q^2)$.
The deviation of the differential cross section in the improved Born approximation (without the ${\Delta y}^{SC}$ correction) from the full one-loop result normalized by the Born cross section, $\Delta_{IBA}$, was worked out numerically in refs. [@bohm2; @been; @ditt]. The introduction of ${\Delta y}^{SC}$ in (\[15\]) simply amounts to an additive correction to $\Delta_{IBA}$. This additive correction, $\delta \Delta_{IBA}$, is calculated by evaluating $$\delta \Delta_{IBA} = {{({{d\sigma} \over {d\Omega}})_{IBA~({\Delta y}^
{SC} \not= 0)} - ({{d\sigma} \over {d\Omega}})_{IBA~({\Delta y}^
{SC} = 0)}}\over {({{d\sigma} \over {d\Omega}})_{Born}}},\label{16}$$ and the quality of the approximation (\[15\]) to the full one loop result is accordingly quantified by $$\Delta_{IBA}+\delta\Delta_{IBA}.\label{17}$$ We note that the magnitude of $\delta \Delta_{IBA}$ may easily be estimated due to the fact that the ${\cal M}_I$ part dominates the cross section (\[15\]). Indeed, neglecting ${\cal M}_Q$ in (\[15\]), one obtains from (\[16\]), $$\delta \Delta_{IBA} \simeq - 2 {\Delta y}^{SC} = -0.66\% \label{18}$$ as a rough estimate. This value will be somewhat enhanced or diminished, depending on whether the interference term of the ${\cal M}_I$ with the ${\cal M}_Q$ term in (\[15\]) is negative (as in the forward region) or positive (as in the backward region).
The results for $\delta\Delta_{IBA}$ are given in Table \[table1\] and the percentage quality of our improved Born approximation (\[15\]), $\Delta_{IBA}+\delta\Delta_{IBA}$, is compared with $\Delta_{IBA}$. The values for $\Delta_{IBA}$ are taken from Table 4 in ref. [@been]. They are based on the choice of $Q^2\equiv s$ for the photon splitting scale $Q^2$. One observes that indeed the deviation of the unpolarized cross section from the full one-loop results is improved to less than 1% as a consequence of introducing ${\Delta y}^{SC}$ in (\[15\]). As the scale $Q^2$ is by no means theoretically uniquely fixed, we also show, in Table \[table2\], the results for the different choice of $Q^2=M_W^2$. Even though the uncorrected quality of the approximation, $\Delta_{IBA}$[^7], is better in this case than for $Q^2=s$, the inclusion of $\delta\Delta_{IBA}$ again leads to an improvement of the quality of the approximation also in this case, and values below approximately 0.5% are reached. In other words, the qualitative improvement in the approximation (\[15\]), obtained by introducing $\Delta y^{SC}\neq 0$, is independent of the choice of $Q^2$.
For completeness, in Tables \[table1\] and \[table2\], we also present the results for the cross section for left-handed electrons, which obviously do not differ much from the results for the unpolarized cross section, since the right-handed cross section is suppressed by about two orders of magnitude compared with the left-handed one. The right-handed cross section by itself is obviously unaffected by introducing ${\Delta y}^{SC}$.
A final comment concerns the inclusion of the decay of the $W^\pm$’s into fermion pairs which has to be incorporated into a completely realistic description of the process of $W^\pm$ pair production. A gauge-invariant description of the process $e^+e^-\to 4$ fermions at one-loop order was recently given in the fermion-loop approximation [@loopscheme]. In this connection it seems worth while to come back to the decomposition of $\Delta y^{SC}$ into fermion-loop and bosonic contributions in (\[nine\]), (\[11\]) and (\[12\]). We note that taking into account fermion-loop contributions only, the estimate (\[18\]) becomes $$\delta\Delta_{IBA}\vert_{ferm}\simeq -2\Delta y^{SC}_{ferm}
\vert_{m_t = 180 GeV} \simeq +1.56\%,
\label{19}$$ and the total deviation $\Delta_{IBA}+\delta\Delta_{IBA}$ of (\[15\]) from the full one-loop results (using $\Delta_{IBA}\simeq 1.2\%$ from Table \[table1\]) rises to values of the order of 2.8% for the total cross section. Therefore, we expect that four-fermion production evaluated in the fermion-loop approximation [@loopscheme] is also enhanced by as much as approximately 2.8% relative to the (so far unknown) outcome of a full one-loop calculation incorporating bosonic loops as well. It is gratifying that a simple procedure of taking into account bosonic loops to improve the results of the fermion-loop calculations of four-fermion production immediately suggests itself. We suggest to approximate bosonic loop corrections by carrying out the substitution $$G_\mu\to G_\mu/(1+\Delta y^{SC}_{bos})
\label{20}$$ with $\Delta y^{SC}_{bos}=11.1\cdot 10^{-3}$ in the four-fermion production amplitudes evaluated in the fermion-loop approximation. Substitution (\[20\]) practically amounts to using $g_{W^\pm}(M_{W^\pm}^2)$ in four-fermion production as well. With substitution (\[20\]) it is indeed to be expected that the deviation of four-fermion production in the fermion-loop scheme will be diminished from the above estimated value of $\simeq 2.8\%$ to a value below 1%.
In summary, the simple procedure of introducing the $SU(2)$ gauge coupling $g_{W^\pm} (M^2_W)$ at the high-energy scale, approximated by $s \simeq M^2_W$, or in other words, by introducing a renormalization scheme, in which the $SU(2)$ coupling is defined by the (theoretical value of the radiatively corrected) leptonic width of the $W$-boson, allows one to incorporate most of the electroweak radiative corrections to the process of $e^+e^- \to W^+W^-$ in the LEP 2 energy range of $\sqrt s\, \lsim\, 200 GeV$. Adding the Coulomb corrections and the initial state radiation in the leading logarithmic approximation provides a scheme which approximates the full one-loop results with an accuracy better than 1%. Moreover, we suggest a simple recipe to approximately incorporate bosonic corrections into four-fermion production calculations, which so far are available in the fermion-loop approximation only. The overall accuracy thus obtained should be sufficient for the analysis of $W$ pair production at LEP 2.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank S. Dittmaier and H. Spiesberger for useful discussions.
[|r|r|r|c||r|r|c|]{} &&\
&&$\delta\Delta_{IBA}$& & $\Delta_{IBA}$&$\delta\Delta_{IBA}$&\
\
&1.45&-0.72&0.73&1.45&-0.72&0.73\
10&1.63&-0.73&0.90&1.63&-0.73&0.90\
90&1.44&-0.72&0.72&1.44&-0.72&0.72\
170&1.26&-0.70&0.56&1.26&-0.70&0.56\
\
&1.27&-0.71&0.56&1.28&-0.71&0.57\
10&1.67&-0.74&0.93&1.67&-0.74&0.93\
90&1.17&-0.71&0.46&1.18&-0.71&0.47\
170&0.75&-0.67&0.08&0.77&-0.67&0.10\
\
&1.26&-0.71&0.55&1.28&-0.71&0.57\
10&1.71&-0.75&0.96&1.71&-0.75&0.96\
90&1.03&-0.69&0.34&1.06&-0.70&0.36\
170&0.59&-0.62&-0.03&0.69&-0.63&0.06\
\
&1.02&-0.70&0.32&1.06&-0.71&0.35\
10&1.57&-0.75&0.82&1.57&-0.75&0.82\
90&0.67&-0.68&-0.01&0.72&-0.69&0.03\
170&0.10&-0.58&-0.48&0.32&-0.64&-0.32\
\
&1.24&-0.70&0.54&1.28&-0.71&0.57\
10&1.67&-0.74&0.93&1.67&-0.75&0.92\
90&0.95&-0.68&0.27&1.01&-0.69&0.32\
170&0.58&-0.57&0.01&0.83&-0.59&0.24\
\
&1.60&-0.70&0.90&1.65&-0.71&0.94\
10&1.77&-0.74&1.03&1.77&-0.74&1.03\
90&1.55&-0.66&0.89&1.64&-0.68&0.96\
170&1.61&-0.53&1.08&1.94&-0.56&1.38\
[|r|r|r|c||r|r|c|]{} &&\
&&$\delta\Delta_{IBA}$& & $\Delta_{IBA}$&$\delta\Delta_{IBA}$&\
\
&0.97&-0.72&0.25&0.97&-0.72&0.25\
10&1.14&-0.73&0.41&1.14&-0.73&0.41\
90&0.95&-0.72&0.23&0.96&-0.72&0.24\
170&0.78&-0.70&0.08&0.78&-0.70&0.08\
\
&0.77&-0.71&0.06&0.78&-0.71&0.07\
10&1.17&-0.74&0.43&1.17&-0.74&0.44\
90&0.67&-0.71&-0.04&0.68&-0.71&-0.03\
170&0.25&-0.67&-0.42&0.27&-0.67&-0.40\
\
&0.70&-0.71&-0.01&0.73&-0.71&0.02\
10&1.17&-0.75&0.42&1.17&-0.75&0.42\
90&0.48&-0.69&-0.21&0.51&-0.70&-0.19\
170&0.05&-0.62&-0.57&0.15&-0.63&-0.48\
\
&0.43&-0.70&-0.27&0.47&-0.71&-0.24\
10&0.99&-0.75&0.24&0.99&-0.75&0.24\
90&0.09&-0.68&-0.59&0.14&-0.69&-0.55\
170&-0.48&-0.58&-1.06&-0.26&-0.64&-0.90\
\
&0.63&-0.70&-0.07&0.67&-0.71&-0.04\
10&1.07&-0.74&0.33&1.07&-0.75&0.32\
90&0.35&-0.68&-0.33&0.41&-0.69&-0.28\
170&-0.02&-0.57&-0.59&0.23&-0.59&-0.36\
\
&0.94&-0.70&0.24&0.99&-0.71&0.28\
10&1.11&-0.74&0.37&1.12&-0.74&0.38\
90&0.90&-0.66&0.24&0.99&-0.68&0.31\
170&0.96&-0.53&0.43&1.28&-0.56&0.72\
[99]{} M. Bilenky, J.L. Kneur, F.M. Renard and D. Schildknecht, Nucl. Phys. [**B409**]{} (1993) 22; Nucl. Phys. [**B419**]{} (1994) 240. I. Kuss and D. Schildknecht,Phys. Lett. [**B383**]{} (1996) 470. M. Böhm, A. Denner, T. Sack, W. Beenakker, F. Berends and H. Kuijf,Nucl. Phys. [**B304**]{} (1988) 463. J. Fleischer, F. Jegerlehner and M. Zrałek,Z. Phys. [**C42**]{} (1989) 409. M. Böhm, A. Denner and S. Dittmaier,Nucl. Phys. [**B376**]{} (1992) 29; err. ibid. [**B391**]{} (1993) 483. J. Fleischer, J.L. Kneur, K. Kołodziej, M. Kuroda and D. Schildknecht,Nucl. Phys. [**B378**]{} (1992) 443; err. ibid. [**B426**]{} (1994) 246. W. Beenakker, A. Denner, S. Dittmaier and R. Mertig, Phys. Lett. [**B317**]{} (1993) 622;W. Beenakker, A. Denner, S. Dittmaier, R. Mertig and T. Sack, Nucl. Phys. [**B410**]{} (1993) 245. W. Beenakker and A. Denner, Int. J. Mod. Phys. [**A9**]{} (1994) 4837 W. Beenakker et. al.,in Physics at LEP2, eds. G. Altarelli, T. Sjöstrand, F. Zwirner, CERN 96-01 Vol. 1, p. 79, hep-ph/9602351. S. Dittmaier,Talk at the 3rd International Symposion on Radiative Corrections, Cracow, Poland, August 1996, hep-ph/9610529. S. Dittmaier, D. Schildknecht and G. Weiglein,Nucl. Phys. [**B465**]{} (1996) 3. H. Burkhardt and B. Pietrzyk,Phys. Lett. [**B356**]{} (1995) 398;S. Eidelman and F. Jegerlehner,Z. Phys. [**C67**]{} (1995) 585. W. Beenakker et. al., hep-ph/9612260, NIKHEF 96-031, PSI-PR-96-41
[^1]: Partially supported by the Ministry of Education and Culture, Japan, under Grant-in-Aid for basic research program C (no. 08640391), the EC-network contract CHRX-CT94-0579 and the Bundesministerium für Bildung und Forschung, Bonn, Germany
[^2]: We approximate $g^2_{W^\pm} (s \gsim 4M^2_W)$ by $g^2_{W^\pm} (M_W^2)$.
[^3]: In addition, diagram (c) yields a contribution proportional to $e^2$ (via the relation $(g')^2M_W^2=e^2M_Z^2$) which is recognized as a part of ${\cal M}_Q$.
[^4]: We note that $g^2_{W^\pm}(M_{W^\pm}^2)$ as defined by (\[six\]) differs from $g^2_{W^\pm}(M_{W^\pm}^2)$ as defined by (6) in ref. [@ditt2] by the factor $r\equiv 1+c_0^2\cdot 3\alpha/4\pi$, where $c_0^2\cdot 3\alpha/4\pi\simeq 1.34\cdot 10^{-3}$. The factor $r$ corresponds to the factor $1+3\alpha/4\pi$ in the $Z^0$ width, where it is conventionally introduced in order to explicitly separate photon radiation from all other electroweak one-loop corrections. The introduction of the factor $r$ in the $W^\pm$ width allowed [@ditt2] to correctly define the magnitude of isospin breaking by one-loop weak interactions when passing from the charged boson coupling $g_{W^\pm}(M_{W^\pm}^2)$ to the neutral boson coupling $g_{W^0}(M_Z^2)$, while keeping the usual convention of separating photon radiation from other loop corrections in the expression for the $Z^0$ width. For the purposes of the present paper we have removed the factor $r$. This amounts to including all one-loop radiative corrections in $g_{W^\pm}^2(M_{W^\pm}^2)$ as defined in (\[six\]). All qualitative conclusions of the present work remain the same if $\Delta y^{SC}\simeq 3.3\cdot 10^{-3}$ from (\[14\]), corresponding to (\[six\]), (\[seven\]) and (\[eight\]), is replaced by $\Delta
y^{SC}\simeq 4.6\cdot 10^{-3}$ from ref. [@ditt2].
[^5]: The parameter ${\Delta y}^{SC}_{bos} = 11.13 \times 10^{-3}$ for $M_H = 100$ GeV, while ${\Delta y}^{SC}_{bos} = 11.08 \times 10^{-3}$ for $M_H = 300$ GeV and $\Delta y^{SC}_{bos} = 11.07 \times 10^{-3}$ for $M_H = 1000$ GeV [@ditt2].
[^6]: Note that $\Delta
y^{SC}_{bos}$(this paper) $=\Delta y^{SC}_{bos}$(Table 1 in ref. [@ditt2]) $-1.34\cdot 10^{-3}$. Compare footnote 3.
[^7]: We thank S. Dittmaier for providing us with the values of $\Delta_{IBA}$ for the photon splitting scale $Q^2=M_W^2$.
|
---
abstract: 'We consider the cosmological models for the higher dimensional spacetime which includes the curvatures of our space as well as the curvatures of the internal space. We find that the condition for the integrability of the cosmological equations is that the total space-time dimensions are $D=10$ or $D=11$ which is exactly the conditions for superstrings or M-theory. We obtain analytic solutions with generic initial conditions in the four dimensional Einstein frame and study the accelerating universe when both our space and the internal space have negative curvatures.'
author:
- Masakazu Sano and Hisao Suzuki
title: Integrable Cosmological Models From Higher Dimensional Einstein Equations
---
Introduction
============
There has been much attention to the understanding cosmology from the superstring or M-theory. The recent observation of Type Ia supernovae and CMB measurement by WMAP indicates our universe is an accelerating universe. In String/M-Theory, there is the no-go theorem [@bio1] which is an obstacle to realize the accelerating universe. The no-go theorem indicates that the warped compactification with the static internal space does not give rise to the four-dimensional de Sitter spacetime under some assumptions. This implies that the warped compactification can not lead the accelerating universe with the static internal space.
One way to avoid the no-go theorem is to employ the time-dependent internal space [@bio2]. pace. If the curvature of the internal space is negative in the four-dimensional Einstein frame, it has been shown that the effective Lagrangian has positive potentials [@bio3]-[@bio7]. This positive potential gives rise to the acceleration of the four-dimensional spacetime in the four-dimensional Einstein frame. In the four-dimensional Einstein frame, the scale factor of the internal space appears as a scalar field with potentials which come from the curvature of the internal s S-brane solutions also lead to accelerating solutions [@bio8]-[@bio14]. The flux field has a role of a positive potential and contributes to the acceleration of the four-dimensional spacetime. Scalar perturbations of scale factors have been shown in [@bio4] that the eternally accelerating universe is realized in the eleven-dimensional spacetime by the external and internal spaces which possess a negative curvature. Recently, it has been also shown by scalar perturbations of scale factors that the eternally accelerating universe occurs in the ten-dimensional spacetime [@bio5]. The fixed point analysis [@bio29] also showed that the eternal acceleration is realized if two spaces possess a negative curvature.
In general, it is difficult to solve the Einstein equations exactly because of the non-linearity of the Einstein equations. Therefore, most of the analytic solutions are special solutions with particular initial conditions. However, it is desirable to find integrable models for the analysis of the initial conditions for our universe. Up to now, few integrable models have been found. In $p$-brane and cosmological solutions which were inspired with String/M-theory, it is known that there exist a few classes of models whose solutions can be reduced to the Liouville or Toda type [@bio15]-[@bio21]. The Toda equation is integrable and provides exact solutions for us. In [@bio15]-[@bio21], the metric has two spatial parts whose curvature is flat for all spaces or for one of two spaces. However, if both spaces have curvatures, it is very difficult to solve even vacuum Einstein equations exactly. When our universe starts from the quantum era, it is natural to expect that we have spatial curvatures whose values are comparable with the curvature to the direction of time. The curvatures with respect to the spatial directions can be regarded as potential energies whereas the time dependence can be regarded as the kinetic energy of the effective actions. If the universe started from quantum fluctuations, it is natural to expect that the order of the potential energies are the same as the order of the kinetic energy. Therefore, it is highly desirable to find integrable models with spatial curvatures both for our space and the internal space.
The accelerating universe occurs when two spatial parts have the negative curvature as shown in [@bio4]-[@bio5]. This analysis performed by the perturbation of scale factors. In [@bio29] the fixed point analysis was performed and also showed that the eternally accelerating universe occurs with two spatial parts whose curvature is negative. In this paper, we will try to solve $D$-dimensional vacuum Einstein equations with two homogeneous spaces exactly.
In order to find integrable cosmological models, we will adopt a method for solving Einstein equations analyzed in [@bio22]. In the method used in [@bio22], the Einstein equations can be reduced to an analytically mechanical problem with one gauge degree of freedom. The gauge degree of freedom originates from the choice of the time variable. By using time variables which include scalar fields, the condition of the integrability has been classified. We will review this method in the next section and find that we are able to obtain general solutions for those vacuum Einstein equations. Actually, it will turn out that the classification of [@bio22] was not complete and we will show that a new type of integrable models is useful for our analysis.
We will start with the generic space ansatz with total dimensions $D$ and our spatial dimensions $d$. Strangely enough, we will find that the condition for the integrability is that the total dimension is ten or eleven, which is exactly correspond to the consistency conditions of superstrings or M-theory. If the integrable condition is satisfied there is an additional conserved quantity aside from the Hamiltonian constraint. Therefore the system has two conserved quantities for two dynamical variables and then the system reduces to the integrable case.
In [@bio26]-[@bio28], the relation between the integrable condition and the conserved quantity was investigated from the Hamiltonian viewpoint. The same integrability condition was obtained and it was shown that there exist the conserved quantity under the integrability condition.
The integrable system does not necessarily have the simple analytic solutions [@bio26]-[@bio28]. In this paper, it will be shown that we are able to derive the analytic solutions by the particular choices of the time variable in $D=10$ and $D=11$. It is very important to note that the time variable used in this paper easily realizes the analytic solutions. The other choices of the time variable make it difficult to solve the equations of motion.
We investigated the cosmological behavior in $D=10$ because this case includes the four-dimensional spacetime and the six-dimensional internal space. It was found that the accelerating universe occurs if two spaces have a negative curvature.
This paper is organized as follows. In the next section, we will construct the effective action which has two potential terms arising from curvatures of the two spaces. We will also show that the system is integrable when the total dimensions of the space-time is ten or eleven. In section three, we will solve Einstein equations completely and general solutions will be given. In section four, we will consider the cosmological property of the case of ($D=10,\,d=3$) in detail and the analytic solutions representing the accelerating universe when two spatial parts have negative curvature. The asymptotic behavior of the metric is also discussed. Finally, we conclude with a discussion of our results and some implications.
Effective Action and conditions for integrability
=================================================
Let us consider the $D$-dimensional spacetime constructed with two spatial parts which have curvatures. For simplicity, we will consider the vacuum Einstein Equations. We consider that the $D$-dimensional space consists of two homogeneous spaces whose sizes depend on time. Namely, the metric ansatz for the $D$-dimensional spacetime is given by, $$\begin{aligned}
ds_{D}^{2}&=G_{MN}(X )\,dX^{M}dX^{N} \notag \\
&=-e^{2\,\widetilde{n}(t)}dt^{2}+e^{2\,\widetilde{A}(t)} g_{ij}(x)\,dx^{i}dx^{j} \notag \\
&\qquad \qquad \qquad \qquad +e^{2\, \alpha (t)}g_{AB}(y)\,dy^{A}dy^{B}, \label{a1}\end{aligned}$$ where $$\begin{cases}
i,\,j=1,\,2,\,\cdots ,\,d \\
A,\,B=d+1,\,d+2,\,\cdots ,\,D-1 \\
M,\,N=0,\,1,\,\cdots ,\,D-1. \\
\end{cases} \notag$$ $e^{2\,\widetilde{A}(t)}$ and $e^{2\,\alpha(t)}$ are scale factors of the $d$- and $(D-d-1)$-dimensional spaces whose metrics are $g_{ij}(x)\,dx^{i}dx^{j}$ and $g_{AB}(y)\,dy^{A}dy^{B}$. respectively. These scale factors depend on time variable $t$.
We assume that both two spatial manifolds are homogeneous spaces (Einstein spaces); $$\begin{aligned}
{}^{(d)} \widetilde{R}_{ij}(g(x))&=(d-1)\,k_{(d)}\,g_{ij}(x), \label{a2} \\
{}^{(D-d-1)} \widetilde{R}_{AB}(g(y))&=(D-d-2)\,k'_{(D-d-1)}\,g_{AB}(y), \label{a3}\end{aligned}$$ where $k_{(d)}$ and $k'_{(D-d-1)}$ represent the curvature of Einstein spaces. For our physical $d$-dimensional space, we assume homogeneous and isotropic space. However we do not need the explicit representation of $g_{ij}(x)$ and $g_{AB}(y)$ to derive the effective action.
The ($d+1$)-dimensional Einstein frame is realized by the following conformal transformations; $$\begin{split}
&\widetilde{A}(t) \longrightarrow A(t)-\frac{1}{d-1}\,(D-d-1)\,\alpha (t), \\
&\widetilde{n}(t)\longrightarrow n(t)-\frac{1}{d-1}\,(D-d-1)\,\alpha (t).
\end{split} \label{a8}$$ In fact, the $D$-dimensional Einstein$-$Hilbert Lagrangian can be written as $$\begin{split}
\sqrt{-G}\,R&=\sqrt{-g_{d+1}(x)\,g_{D-d-1}(y)}\,e^{d\,\widetilde{A}+\widetilde{n}+(D-d-1)\,\alpha}\,R \\
\longrightarrow & \sqrt{-g_{d+1}(x)\,g_{D-d-1}(y)}\,e^{d\,A+n}\,(\,{}^{(d+1)}R+\cdots \,).
\end{split} \notag$$ Under the conformal transformation (\[a8\]), we obtain an effective action in the ($d+1$)-dimensional Einstein frame [@bio3]; $$\begin{split}
&\frac{1}{d(d-1)\,V_{D-1}}\int d^{D}X\,\sqrt{-G}\,R \\
\longrightarrow &\int dt \,
\Biggl[ e^{dA-n}\,\Bigl\{ -\dot{A}^{2}+X^{2}\,\dot{\alpha}^{2} \Bigr\} \\
&-e^{dA+n}\Bigl\{ \, -k_{(d)}\,e^{-2A}-\widetilde{k}_{(D-d-1)}\,e^{-\frac{2(D-2)}{d-1}\,\alpha} \, \Bigr\} \Biggr],
\end{split} \label{a9}$$ where $V_{D-1}\equiv \int d^{D-1}X\,\sqrt{-\det g_{ij}(x)}\,\sqrt{\det g_{AB}(y)}$ and $$\begin{split}
X^{2}&=\frac{(D-d-1)(D-2)}{d(d-1)^{2}}, \\
\widetilde{k}_{(D-d-1)}&=\frac{(D-d-1)(D-d-2)}{d(d-1)}\,k'_{(D-d-1)}.
\end{split} \label{a10}$$ The effective action (\[a9\]) shows that the scale factor of the internal space appears as a scalar field with the potential terms. The two effective potentials are generated by the curvature of Einstein spaces which can be easily seen from the fact that the potential terms are directly proportional to the curvatures.
We would like to solve equations of motion derived from the ($d+1$)-dimensional effective action (\[a9\]). In general, Einstein equations are highly non-linear equations and difficult to be solved exactly. The effective action (\[a9\]) indeed results in non-linear equations. A way to analyze the system is to utilize a gauge degree of freedom [@bio24]. Let us recall that we can choose a time variable via the lapse function $e^{n(t)}$ which is a non-dynamical quantity [@bio24]. This gauge degree of freedom represents the invariance under the coordinate transformation of time. Because of this gauge degree of freedom, we can freely choose the gauge to solve the system. We will take the lapse function as $$e^{n(t)}=e^{p\,\alpha(t)+q\,A(t)}, \label{a11}$$ where $p$ and $q$ are any real numbers. We would like to look for the more convenient transformation of two dynamical variables.
We use the following transformation between ($A,\,\alpha$) and ($U_{+},\,U_{-}$), $$\begin{aligned}
e^{2A}&=U_{+}^{\frac{M_{1}+1}{d-1}}\,U_{-}^{\frac{M_{2}+1}{d-1}}, \notag \\
e^{2\alpha}&=U_{+}^{\frac{(M_{1}+1)}{(d-1)\,X}}\,U_{-}^{-\frac{(M_{2}+1)}{(d-1)\,X}}, \label{a14} \\
e^{2n}&=U_{+}^{-2+\frac{d\,(M_{1}+1)}{d-1}}\,U_{-}^{-2+\frac{d\,(M_{2}+1)}{d-1}}, \notag \end{aligned}$$ where we have defined $$\begin{split}
M_{1}=\frac{(d-2+q)\,X+p}{(d-q)\,X-p}, \\
M_{2}=\frac{(d-2+q)\,X-p}{(d-q)\,X+p}. \label{a13}
\end{split}$$ By using these variables, we can re-write the effective action (\[a9\]) as [@bio22] $$\begin{aligned}
&\mathcal{L}_{\text{eff}}=-\frac{(M_{1}+1)(M_{2}+1)}{(d-1)^{2}}\,\dot{U}_{+}\dot{U}_{-}
+k_{(d)}\,U_{+}^{M_{1}}U_{-}^{M_{2}} \notag \\
&+\widetilde{k}_{(D-d-1)}\,U_{+}^{-1+\frac{M_{1}+1}{d-1}\Bigl[ d-\frac{D-2}{
(d-1)X} \Bigr]}U_{-}^{-1+\frac{M_{2}+1}{d-1}\Bigl[d+\frac{D-2}{
(d-1)X} \Bigr]}. \label{a12}\end{aligned}$$ The effective action $\int dt \mathcal{L}_{\text{eff}}(M_{1},\,M_{2})$ can be transformed to the action of some other parameters $N_{1},N_{2}$, $\int d\xi \mathcal{L}_{\text{eff}}(N_{1},\,N_{2})$ by the change of time coordinates, $dt=N(\xi)\, d\xi$. This is realized by following transformations; $$\begin{split}
&U_{+}=V_{+}^{\frac{N_{1}+1}{M_{1}+1}}, \quad
U_{-}=V_{-}^{\frac{N_{2}+1}{M_{2}+1}}, \\
&N=V_{+}^{-1+\frac{N_{1}+1}{M_{1}+1}}V_{-}^{-1+\frac{N_{2}+1}{M_{2}+1}}.
\end{split} \label{a15}$$ These transformations preserve the form of the effective action (\[a12\]). It is possible to connect a solution in some parameters ($M_{1},\,M_{2}$) to many other solutions by above transformations (\[a15\]). Because of this gauge degree of freedom, we can solve the Einstein equations with a particular choice of the parameters.
In [@bio22], $M_{1}=M_{2}=0$ was considered. In our cases, this condition leads the following potential $$\begin{aligned}
W&=-k_{(d)} \notag \\
&-\widetilde{k}_{(D-d-1)}U_{+}^{-1+\frac{1}{d-1}\Bigl[ d-\frac{D-2}{
(d-1)X} \Bigr]}U_{-}^{-1+\frac{1}{d-1}\Bigl[ d+\frac{D-2}{
(d-1)X} \Bigr]}. \notag \end{aligned}$$ If the form of the potential becomes $W=-k_{(d)}
-\widetilde{k}_{(D-d-1)}\,U_{+}$ or $W=-k_{(d)}
-\widetilde{k}_{(D-d-1)}\,U_{-}$, Einstein equations are soluble as shown in [@bio22]. A model with this type of the potential also studied in [@bio25]. But the above potential cannot take such that potential because the total dimension has to be $D=1$ in order to satisfy $-1+\frac{1}{d-1}\,\Bigl[\, d\pm \frac{D-2}{
(d-1)X} \,\Bigr]=0$. Therefore our model is not correspond to the model considered in [@bio22].
A convenient choice of the parameters is $M_{1}=M_{2}=1$ which implies $$\begin{aligned}
&\mathcal{L}_{\text{eff}}=-\frac{4}{(d-1)^{2}}\,\dot{U}_{+}\dot{U}_{-}
+k_{(d)}\,U_{+}\,U_{-} \notag \\
&+\widetilde{k}_{(D-d-1)}\,U_{+}^{-1+\frac{2}{d-1}\,\Bigl[\, d-\frac{D-2}{
(d-1)X} \,\Bigr]}U_{-}^{-1+\frac{2}{d-1}\,\Bigl[\, d+\frac{D-2}{
(d-1)X} \,\Bigr]}. \label{a16}\end{aligned}$$ The above effective Lagrangian shows the second term is an interaction similar to a harmonic oscillator and third term represents a non-linear interaction which is an obstacle to solve equations of motion. If the power of $U_{+}$ or $U_{-}$ is simplified, it is possible to solve the equations analytically. To perform this procedure, we will impose a condition in which the non-linear term in the effective action does not depend on $U_+$; $$-1+\frac{2}{d-1}\,\Bigl[\, d-\frac{D-2}{(d-1)X}\Bigr]=0 ,\label{condition}$$ where $X$ was defined in (\[a10\]). We will see below that the system is integrable if this condition is satisfied. Before considering the integrability, let us solve the condition (\[condition\]). The condition (\[condition\]) can be rewritten as $$(d-1)(D-d-5)=4.$$ From this, we can immediately derive the condition of the integrability $D$ and $d$ as follows $$\begin{cases}
D=10,\qquad d=3 \\
D=11,\qquad d=2,\,5,
\end{cases} \label{a17}$$ where (\[a10\]) was used. Note that $D=10$ is the critical dimension of the superstring theories and $D=11$ is the dimension of the M-theory! Moreover, $d=3$ means our spacetime is four dimensions. Therefore, we have integrable cosmological models for a realistic setup.
We are going to show that the system is integrable if the condition (\[condition\]) is satisfied. By using the condition, we can rewrite the effective Lagrangian and the Hamiltonian constraint as $$\begin{aligned}
\mathcal{L}_{\text{eff}}&=-\frac{4}{(d-1)^{2}}\,\dot{U}_{+}\dot{U}_{-}
+k_{(d)}\,U_{+}U_{-} \notag \\
& \qquad \qquad \qquad \qquad +\widetilde{k}_{(D-d-1)}\,U_{-}^{2(d+1)/(d-1)}, \label{a18} \\
H&=0=-\frac{4}{(d-1)^{2}}\,\dot{U}_{+}\dot{U}_{-}
-k_{(d)}\,U_{+}U_{-} \notag \\
& \qquad \qquad \qquad \qquad -\widetilde{k}_{(D-d-1)}\,U_{-}^{2(d+1)/(d-1)}, \label{a19}\end{aligned}$$ where the second equation represents the total energy conservation which can be derived by the variation of $n(t)$ in the action (\[a9\]).
We get equations of motion by the variation with respect to $U_+$ and $U_-$ as follows $$\begin{aligned}
&-\frac{4}{(d-1)^{2}}\,\ddot{U}_{+}-k_{(d)}\,U_{+} \notag \\
& \qquad \qquad -\widetilde{k}_{(D-d-1)}\,\frac{2(d+1)}{d-1}\,U_{-}^{(d+3)/(d-1)}=0, \label{a20} \\
&-\frac{4}{(d-1)^{2}}\,\ddot{U}_{-}-k_{(d)}\,U_{-}=0. \label{a21}\end{aligned}$$ Because of the equation (\[a21\]), we can easily find the following conserved quantities; $$\epsilon =\frac{2}{(d-1)^2} \dot{U}_{-}^2+\frac{k_{(d)}}{2}{U}_{-}^2.\label{cons2}$$ Since the system has two conserved constants (\[a19\]) and (\[cons2\]) for two dynamical variables, the total system is classically integrable.
In [@bio26]-[@bio28], the integrability was discussed from the Hamiltonian viewpoint. It is essential idea that they looked not only for functions Poisson-commuting with the Hamiltonian $H$, but also for a function $F$ satisfying an equation of the form $$\{ F,\,H \}_{\text{P.B.}}=\phi H \notag$$ for some unknown function $\phi$. The Hamiltonian constraint $H=0$ indicates $\{ F,\,H \}_{\text{P.B.}}=\phi H=0$, therefore the function $F$ becomes a conserved quantity on this Hamiltonian constraint. Using this method, same consequence (\[a17\]) was obtained in [@bio26]-[@bio28]. In our model, $\epsilon$ satisfies $\{ \epsilon ,\, H\}_{\text{P.B.}}=0$ where we used canonical momenta $P_{+}=-(4/(d-1)^{2})\dot{U}_{-}$, $P_{-}=-(4/(d-1)^{2})\dot{U}_{+}$, the Hamiltonian (\[a19\]) and the Poisson bracket $$\{ q_{1},\,q_{2} \}_{\text{P.B.}}\equiv\sum_{i=+,-} \Biggl[ \frac{\partial q_{1}}{\partial U_{i}}\frac{\partial q_{2}}{\partial P_{i}}
-\frac{\partial q_{1}}{\partial P_{i}}\frac{\partial q_{2}}{\partial U_{i}} \Biggr]. \notag$$ $\epsilon$ commutes with the Hamiltonian and then is a conserved quantity. This means that the system becomes integrable, because the system has two conserved quantities for two dynamical variables. If the condition (\[condition\]) is not satisfied, $\epsilon$ dose not commute with the Hamiltonian derived from (\[a16\]) and $\{ \epsilon,\,H \}_{\text{P.B.}}\neq \phi H$. In the next section, we are going to show that it is quite easy to derive the analytic solutions by the choice of the time variable, ($M_{1}=M_{2}=1$).
We can impose the other type of requirement that the non-linear term in the effective action (\[a16\]) does not depend on $U_+$. It turns out that interchanging $U_+$ and $U_-$ can be achieved by the replacement $d \rightarrow D-d-1$ which means that the interchange of the internal and the external space. This symmetry should be present because we are just considering the evolution of two homogeneous spaces. At the level of the effective action this equivalence results from the re-parametrization of the time coordinate. For instance, we will take $-1+\frac{M_{1}+1}{d-1}\,\Bigl[\, d-\frac{D-2}{
(d-1)X} \,\Bigr]=-1+\frac{M_{2}+1}{d-1}\,\Bigl[\, d+\frac{D-2}{
(d-1)X} \,\Bigr]=1$ in (\[a12\]) and impose $-1+\frac{2(d-1)}{d-(D-2)/(d-1)X}=0$ which gives $$\begin{cases}
D=10,\qquad d=6 \\
D=11,\qquad d=5,\,8
\end{cases} \label{a22}$$ where we have used (\[a10\]). The effective Lagrangian is identical to (\[a18\]) just by interchanging the two spaces.
General Solutions
=================
In the previous section, we have seen that the Einstein equations for two homogeneous spaces are integrable if the total dimensions are ten or eleven. In this section, we will discuss the case $D=10,\,d=3$ which is most relevant for four-dimensional physics. General solutions of ($D=11,\, d=2$) and ($D=11,\, d=5$) are shown in Appendix \[Appendix\].
In this case, the equations of motion (\[a20\])-(\[a21\]) and the Hamiltonian constraint (\[a19\]) are written by $$\begin{aligned}
&\ddot{U}_{+}+k_{(3)}\,U_{+}
+4\,\widetilde{k}_{(6)}\,U_{-}^{3}=0, \label{b1} \\
&\ddot{U}_{-}+k_{(3)}\,U_{-}=0, \label{b2} \\
&\dot{U}_{+}\dot{U}_{-}
+k_{(3)}\,U_{+}U_{-}
+\widetilde{k}_{(6)}\,U_{-}^{4}=0, \label{b3}\end{aligned}$$ where $\widetilde{k}_{(6)}=5\,k'_{(6)}$. Let us first consider the equation of motion (\[b2\]). The solution of (\[b2\]) can be easily obtained as $$U_{-}=
\begin{cases}
A_{1}\,\cos [\,\sqrt{k_{(3)}}\,t+A_{2}\, ], \qquad ~~\,(k_{(3)}>0) \\
A_{1}\,t+A_{2}, \qquad \qquad \qquad ~~~~\,~\,(k_{(3)}=0) \\
A_{1}\,\cosh [\, \sqrt{-k_{(3)}}\,t+A_{2}\, ], \quad ~\,(k_{(3)}<0)
\end{cases} \label{b4}$$ where $A_{1}$ and $A_{2}$ are constants of integrations. These equations show that the behavior of $U_{-}$ is controlled by the curvature of the three-dimensional Einstein space. Substituting this $U_{-}$ into the equation of motion (\[b1\]), we obtain following equations of motion, $$\begin{split}
&k_{(3)}>0\,;\quad \ddot{U}_{+}+k_{(3)}\,U_{+} \\
&\qquad \qquad \quad +4\,\widetilde{k}_{(6)}\,\Bigl(\, A_{1}\,\cos [\,\sqrt{k_{(3)}}\,t+A_{2}\, ] \,\Bigr)^{3}=0, \\
&k_{(3)}=0\,;\quad \ddot{U}_{+}
+4\,\widetilde{k}_{(6)}\,\Bigl(\, A_{1}\,t+A_{2}\, \,\Bigr)^{3}=0, \\
&k_{(3)}<0\,;\quad \ddot{U}_{+}+k_{(3)}\,U_{+} \\
& \qquad \qquad +4\,\widetilde{k}_{(6)}\,\Bigl(\, A_{1}\,\cosh [\,\sqrt{-k_{(3)}}\,t+A_{2}\, ] \,\Bigr)^{3}=0.
\end{split} \label{b5}$$ These equations of motion concretely show that the $U_{+}$ have received the forced power from the $U_{-}$. The internal space gives the effect of the forced oscillation to the motion of $U_{+}$.
In the cases of $k_{(3)}<0$ or $k_{(3)}>0$, it is useful to adopt the following transformations; $$\begin{split}
&k_{(3)}>0\,;\quad U_{+}=f(t) \\
&\qquad\qquad\qquad\qquad-\frac{3\,\widetilde{k}_{(6)}A_{1}^{3}}{2\,\sqrt{k_{(3)}}}\,t\, \sin [\,\sqrt{k_{(3)}}\,t+A_{2}\,] \\
&\qquad\qquad\qquad\qquad
+\frac{\widetilde{k}_{(6)}A^{3}_{1}}{8\,k_{(3)}}\,\cos [\,3(\,\sqrt{k_{(3)}}\,\,t+A_{2}\,)\,] \\
&k_{(3)}<0\,;\quad U_{+}=f(t) \\
&\qquad\qquad\qquad\qquad-\frac{3\,\widetilde{k}_{(6)}A_{1}^{3}}{2\,\sqrt{-k_{(3)}}}\,t\, \sinh [\,\sqrt{-k_{(3)}}\,t+A_{2}\,] \\
&\qquad\qquad\qquad\qquad
+\frac{\widetilde{k}_{(6)}A^{3}_{1}}{8\,k_{(3)}}\,\cosh [\,3(\,\sqrt{-k_{(3)}}\,\,t+A_{2}\,)\,]
\end{split} \label{b6}$$ Using these relations, we can simplify the equation of motion (\[b5\]) as $$\begin{split}
& \ddot{f}(t)+ k_{(3)}\,f(t)=0, \qquad ~~~~\,\,(\,k_{(3)}>0\,)\\
& \ddot{f}(t)+ (\,-k_{(3)}\,)\,f(t)=0. \qquad (\,k_{(3)}<0\,)
\end{split} \notag$$ Then, answers for this equation are simply $$\begin{split}
&f(t)=B_{1}\,\cos[\, \sqrt{k_{(3)}}\,\,t+B_{2} \,], (k_{(3)}>0) \\
&f(t)=B_{1}\,\sinh [\, \sqrt{-k_{(3)}}\,\,t+B_{2} \,], (k_{(3)}<0)
\end{split} \notag$$ where $B_{1}$ and $B_{2}$ are the constants of integrations. Combining all these things, we finally obtain the solutions given by $$\begin{split}
&k_{(3)}>0\,;\quad U_{+}=B_{1}\,\cos [\, \sqrt{k_{(3)}}\,\,t+B_{2} \,] \\
&\qquad\qquad\qquad\qquad-\frac{3\,\widetilde{k}_{(6)}A_{1}^{3}}{2\,\sqrt{k_{(3)}}}\,t\, \sin [\,\sqrt{k_{(3)}}\,t+A_{2}\,] \\
&\qquad\qquad\qquad\qquad
+\frac{\widetilde{k}_{(6)}A^{3}_{1}}{8\,k_{(3)}}\,\cos [\,3(\,\sqrt{k_{(3)}}\,\,t+A_{2}\,)\,], \\[6pt]
&k_{(3)}=0\,;\quad U_{+}=B_{1}\,t+B_{2}-\frac{\widetilde{k}_{(6)}}{5A^{2}_{1}}\,(A_{1}\,t+A_{2})^{5}, \\[7pt]
&k_{(3)}<0\,;\quad U_{+}=B_{1}\,\sinh [\, \sqrt{-k_{(3)}}\,\,t+B_{2} \,] \\
&\qquad\qquad\qquad\quad-\frac{3\,\widetilde{k}_{(6)}A_{1}^{3}}{2\,\sqrt{-k_{(3)}}}\,t\, \sinh [\,\sqrt{-k_{(3)}}\,t+A_{2}\,] \\
&\qquad\qquad\qquad\quad
+\frac{\widetilde{k}_{(6)}A^{3}_{1}}{8\,k_{(3)}}\,\cosh [\,3(\,\sqrt{-k_{(3)}}\,\,t+A_{2}\,)\,].
\end{split} \label{b7}$$ The term proportional to $\widetilde{k}_{(6)}$ indicates the resonance, as the frequency of the harmonic and forced oscillation are identical.
The Hamiltonian constraint (\[b3\]) gives a constraint on four constants of integrations, $$\begin{split}
&k_{(3)}>0\,;\quad \frac{9\widetilde{k}_{(6)}A^{3}_{1}}{8}+k_{(3)}\,B_{1}\,\cos [A_{2}-B_{2}]=0, \\
&k_{(3)}=0\,;\quad B_{1}=0, \\
&k_{(3)}<0\,;\quad \frac{9\widetilde{k}_{(6)}A^{3}_{1}}{8}+(-k_{(3)})\,B_{1}\,\sinh [A_{2}-B_{2}]=0.
\end{split} \label{b8}$$ We shall consider the metric which is given by $$e^{2n}=e^{2A}=U_{+}U_{-},\qquad e^{2\alpha}=\sqrt{\frac{U_{+}}{U_{-}}}, \label{b9}$$ where we have used (\[a10\]), (\[a14\]) and $M_{1}=M_{2}=1$. These equations shows that the four-dimensional part is the conformal metric, $e^{2n}=e^{2A}$. Therefore, in the four-dimensional Einstein frame, ten-dimensional metric is $$\begin{aligned}
ds_{10}^{2}&=e^{-6\alpha}[\,e^{2A}(-dt^{2}+g_{ij}(x)\,dx^{i}dx^{j})\,] \notag \\
& \qquad \qquad \qquad \qquad +e^{2\alpha}\,g_{AB}(y)\,dy^{A}dy^{B} \notag \\
&= \Bigg( \frac{U_{+}}{U_{-}} \Biggr)^{-3/2}\,\Bigl[ U_{+}U_{-}\,(-dt^{2}+g_{ij}(x)\,dx^{i}dx^{j}) \Bigr] \notag \\
&\qquad \qquad \qquad +\Biggl( \frac{U_{+}}{U_{-}}\Biggr)^{1/2}\,g_{AB}(y)\,dy^{A}dy^{B}. \label{b10}\end{aligned}$$ For $\widetilde{k}_{(6)}\neq 0$, metric components are $$\begin{aligned}
&(\,k_{(3)}>0 \,) \notag \\
& e^{2A}=\frac{9\widetilde{k}_{(6)}A^{4}_{1}}{8\,k_{(3)}}\,\Biggl[
-\frac{\cos [\, \sqrt{k_{(3)}}\,\,t+B_{2} \,]}{\cos[A_{2}-B_{2}]} \notag \\
& -\frac{4\sqrt{k_{(3)}}}{3}\,\,t\, \sin [\,\sqrt{Y_{(3)}}\,t+A_{2}\,] \notag \\
& +\frac{1}{9}\,\cos [\,3(\,\sqrt{k_{(3)}}\,\,t+A_{2}\,)\,] \Biggr]
\,\cos [\,\sqrt{k_{(3)}}\,t+A_{2}\, ], \notag \\[10pt]
&e^{2\alpha}=\sqrt{\frac{9\widetilde{k}_{(6)}A^{2}_{1}}{8\,k_{(3)}}}\,\Biggl\{ \Biggl[
-\frac{\cos [\, \sqrt{k_{(3)}}\,\,t+B_{2} \,]}{\cos[A_{2}-B_{2}]} \notag \\
& -\frac{4\sqrt{k_{(3)}}}{3}\,\,t\, \sin [\,\sqrt{k_{(3)}}\,t+A_{2}\,] \notag \\
& +\frac{1}{9}\,\cos [\,3(\,\sqrt{k_{(3)}}\,\,t+A_{2}\,)\,] \Biggr]
\,\frac{1}{\cos [\,\sqrt{k_{(3)}}\,t+A_{2}\, ] } \Biggr\}^{\frac{1}{2}}, \notag \end{aligned}$$ $$\begin{aligned}
&(\,k_{(3)}=0\,) \notag \\
& e^{2A}=\Biggl[ -\frac{\widetilde{k}_{(6)}}{5A^{2}_{1}}\,(A_{1}\,t+A_{2})^{5}+B_{2} \Biggr]\,(A_{1}\,t+A_{2}) , \notag \\[8pt]
& e^{2\alpha}=\Biggl\{ \Biggl[ -\frac{\widetilde{k}_{(6)}}{5A^{2}_{1}}\,(A_{1}\,t+A_{2})^{5}
+B_{2} \Biggr] \frac{1}{A_{1}\,t+A_{2}} \Biggr\}^{\frac{1}{2}}, \notag \end{aligned}$$ $$\begin{aligned}
&(\,k_{(3)}<0 \,) \notag \\
& e^{2A}=\frac{-9\widetilde{k}_{(6)}A^{4}_{1}}{8\,(-k_{(3)})}\,\Biggl[
\frac{\sinh [\, \sqrt{-k_{(3)}}\,\,t+B_{2} \,]}{\sinh[A_{2}-B_{2}]} \notag \\
&+\frac{4\sqrt{-k_{(3)}}}{3}\,\,t\, \sinh [\,\sqrt{-k_{(3)}}\,t+A_{2}\,] \notag \\
&+\frac{1}{9}\cosh [\,3(\,\sqrt{-k_{(3)}}\,\,t+A_{2}\,)\,] \Biggr]
\cosh [\,\sqrt{-k_{(3)}}\,t+A_{2}\, ], \notag \\[10pt]
& e^{2\alpha}=\sqrt{\frac{-9\widetilde{k}_{(6)}A^{2}_{1}}{8\,(-k_{(3)})}}\,\Biggl\{ \Biggl[
\frac{\sinh [\, \sqrt{-k_{(3)}}\,\,t+B_{2} \,]}{\sinh[A_{2}-B_{2}]} \, \notag \\
& +\frac{4\sqrt{-k_{(3)}}}{3}\,\,t\, \sinh [\,\sqrt{-k_{(3)}}\,t+A_{2}\,] \notag \\
& +\frac{1}{9}\cosh [3(\,\sqrt{-k_{(3)}}\,\,t+A_{2}\,)\,] \Biggr]
\,\frac{1}{\cosh [\sqrt{-k_{(3)}}\,t+A_{2} ] } \Biggr\}^{\frac{1}{2}}.
\label{b11}\end{aligned}$$
Cosmological Characteristic for (D=10, d=3)
===========================================
The solutions that we have obtained in the previous section include a metric which has realistic dimensions $D=10,\,d=3$. The total dimension of this spacetime is ten dimension equal to the critical dimension of superstrings and the physical space-time has four dimensions. Therefore, it is very interesting how the universe evolve with time. In this section, we shall consider the behavior of the ten-dimensional spacetime. We will analytically show that the four-dimensional part of the ten-dimensional spacetime accelerates eternally, which has been analyzed in [@bio5] by the qualitative method and in [@bio29] by the fixed point analysis on the phase space.
We shall consider metric (\[b11\]). In $k_{(3)}>0$, $e^{2\,A}$ and $e^{2\,\alpha}$ take oscillatory behavior. $e^{2\,A}$ starts from zero and end up with zero because $U_{-}=A_{1}\,\cos[\sqrt{k_{(3)}}\,t+A_{1}]$ oscillates between two zeros. On the other hand, $e^{2\,\alpha}$ diverges when $U_{-}$ and $e^{2\,A}$ become zero, and then, the case of $k_{(3)}>0$ may not have a stable internal space.
Similarly, in $k_{(3)}=0$, the scale factor of the internal space diverges at $U_{-}=A_{1}\,t+A_{2}=0$ and $e^{2\,A}$ takes zero when $U_{-}=0$. For $k_{(3)}=0$ and $\widetilde{k}_{6}<0$, the asymptotic behavior becomes $e^{2\,A}\rightarrow t^{6}$ and $e^{2\,\alpha}\rightarrow t^{2}$ at $t\rightarrow \infty$ and the ten-dimensional metric (\[b10\]) has the behavior as follows $$ds_{10}^{2}\rightarrow (\,-dt^{2}+ds_{3}^{2}\,)+t^{2}\,ds_{6}^{2}. \label{c1}$$
This means that the four-dimensional part of the metric does not depend on $t$ in the ten-dimensional frame. If we take $ds^{2}_{3}$ to the three-dimensional Euclid space, the four-dimensional part becomes the Minkowski spacetime at $t \rightarrow \infty$. The internal space becomes large in this region.
If $k_{(3)}<0$, an interesting phenomenon occurs. The three-dimensional space expands with acceleration eternally. This acceleration is extracted from the negative curvature of the internal space, $\widetilde{k}_{(6)}<0$. The curvature of the internal space acts like a positive cosmological constant in four dimensions. We assume the internal space is the Einstein space with the negative curvature. This case, $k_{(3)}<0$ and $\widetilde{k}_{(6)}<0$, is equivalent to the situation suggested in [@bio5] in which it was shown by scalar perturbations of scale factors that the acceleration of the three-dimensional space occurs in the four-dimensional Einstein frame. In [@bio29] the fixed point analysis also indicated that the eternal acceleration occurs for $k_{(3)}<0$ and $\widetilde{k}_{(6)}<0$. We can confirm these facts by using the analytic solutions.
Defining a proper time $d\tau \equiv \sqrt{U_{+}\,U_{-}}\,dt$, the velocity and acceleration of $e^{A}$ are given by $de^{A}/d\tau$ and $d^{2}e^{A}/d\tau^{2}$. These velocity and acceleration are shown in fig.\[\[fig1\]\]-\[\[fig2\]\] where we neglect overall factor in (\[b11\]).
![The velocity of $e^{A}$. $k_{(3)}=-1$, $A_{2}=0.5$, $B_{2}=0$.[]{data-label="fig1"}](fig1.eps){width="54mm"}
(0,109) (-174,80)[$\frac{de^{A}}{d\tau}$]{} (0,40)[$t$]{}
![The acceleration of $e^{A}$. $k_{(3)}=-1$, $A_{2}=0.5$, $B_{2}=0$.[]{data-label="fig2"}](fig2.eps){width="55mm"}
(21,0) (-178,77)[$\frac{d^{2}e^{A}}{d\tau^{2}}$]{} (0,50)[$t$]{}
From these figures, we can extract following facts. The three-dimensional space evolves with the very large positive velocity and negative acceleration, then the three-dimensional space decelerates quickly at a first stage. The acceleration of the three-dimensional space turns to positive at some time, and then the acceleration decreases gradually. The three-dimensional space finally expands with a positive velocity and the zero acceleration at the infinite future. In particular, the three-dimensional space accelerates forever. This fact coincides with [@bio5] and [@bio29].
We shall consider the asymptotic behavior of $e^{2\,A}$ and $e^{2\,\alpha}$ in (\[b11\]). $U_{-}=A_{1}\,\cosh[\sqrt{-k_{(3)}}\,t+A_{2}]$ is not singular for appropriate constants of integrations. $A_{1}$ and $A_{2}$, which implies $e^{2\,\alpha}$ may not diverge when $e^{2\,A} \rightarrow 0$. For $t \rightarrow \infty$, $e^{2\,A}\rightarrow e^{4\,\sqrt{-k_{(3)}}\,t}$, $e^{2\,\alpha}\rightarrow e^{\,\sqrt{-k_{(3)}}\,t}$ and $$ds^{2}_{10}\rightarrow e^{\,\sqrt{-k_{(3)}}\,t}(\,-dt^{2}+ds^{2}_{3}+ds^{2}_{6}\,). \label{c2}$$ It is found that this ten-dimensional metric is conformally equivalent to $-dt^{2}+ds^{2}_{3}+ds^{2}_{6}$ at large $t$. In four-dimensional Einstein frame, this metric (\[c2\]) behaves as $$\begin{split}
ds^{2}_{10}\rightarrow &e^{\,-3\sqrt{-k_{(3)}}\,t}[e^{\,4\sqrt{-k_{(3)}}\,t}(-dt^{2}+ds^{2}_{3})] \\
& \qquad \qquad \qquad \qquad \qquad \quad +e^{\,\sqrt{-k_{(3)}}\,t}ds^{2}_{6}. \label{c3}
\end{split}$$ The above metric (\[c3\]) shows that the internal space also expands. The proper time, for the four-dimensional frame, is given by $\tau \sim e^{\,2\sqrt{-k_{(3)}}\,t}$ and then, the three-dimensional space expands with $e^{2\,A}\sim \tau^{2}$ which shows that the expansion has the uniform velocity at large $\tau$. The three-dimensional space has a negative curvature $k_{(3)}<0$ in this case. Therefore the four-dimensional part can be the Milne universe for the four-dimensional frame. The metric (\[c3\]) also indicates that the internal space becomes large with $e^{2\alpha} \sim e^{\,\sqrt{-k_{(3)}}\,t}$, and then it is intuitively expected that the curvature of the internal space to decrease. In fact, it is found in (\[a9\]) that the potential term $ -\widetilde{k}_{(6)}e^{-8\alpha} $ vanishes at $t \rightarrow \infty$.
If $t \rightarrow 0$, $ds^{2}_{10}\rightarrow t^{-1/2}(-dt^{2}+ds^{2}_{3})+t^{1/2}ds^{2}_{6}$ and in the four-dimensional frame, $$ds^{2}_{10}\rightarrow t^{-3/2}[\,t\,(-dt^{2}+ds^{2}_{3})\,]+t^{1/2}ds^{2}_{6}. \label{c4}$$ The proper time for the four-dimensional frame is given by $\tau \sim t^{3/2}$ and the three-dimensional space expands with $e^{2\,A}\sim \tau^{2/3}$. The acceleration of this scale factor is $d^{2} e^{A}/d\tau^{2} \sim -(2/9)\, \tau^{-5/3}$ and then the acceleration diverges at $\tau\rightarrow 0$.
As a final example, we shall consider $\widetilde{k}_{(6)}=0$ and $k_{(3)}<0$. In this case, we can find that the constants of integrations satisfy $A_{2}=B_{2}$ in (\[b8\]). Using (\[b4\]), (\[b7\]) and (\[b9\]), the asymptotic behavior of $e^{2\,A}$ and $e^{2\,\alpha}$ are $e^{2\,\alpha}\rightarrow \text{const.}$ and $e^{2\,A}\rightarrow e^{2\,\sqrt{-k_{(3)}}\,t}$ at $t \rightarrow \infty$. The ten-dimensional metric leads to $$ds^{2}_{10}\rightarrow e^{2\,\sqrt{-k_{(3)}}\, t} (-dt^{2}+ds^{2}_{3})+ds^{2}_{6}. \label{c5}$$ This metric shows that the internal space does not depend on $t$. The proper time is defined as $\tau \sim e^{\,\sqrt{-k_{(3)}}\,t}$ in ten dimension. Therefore, the ten-dimensional metric is represented as $$ds^{2}_{10}\rightarrow (-d\tau^{2}+\tau^{2}\, ds^{2}_{3})+ds^{2}_{6} \label{c6}$$ whose structure is the product space of the Milne universe and a flat six-dimensional space. It is possible to transform the Milne universe into the Minkowski spacetime by coordinate transformations. In this case, the above metric (\[c6\]) becomes the product spacetime with the four-dimensional Minkowski spacetime and the flat six-dimensional internal space at $t\rightarrow \infty$.
Conclusions
===========
We have considered the vacuum Einstein equations in the $D$-dimensional spacetime and obtained integrable cosmological models. Those Einstein equations have two potential terms arising from the curvature of the $d$- and $(D-d-1)$-dimensional Einstein spaces. It was thought that solving Einstein equations with two curved spaces are very difficult. However we have pointed out that the total dimension should be $D=10$ or $D=11$ to make those Einstein equations integrable as cosmological models. The integrability is guaranteed by the conserved quantity which commutes with the Hamiltonian. The integrable system does not necessarily have the analytic solutions. It is very important to note that the time variable used in this paper easily realizes the analytic solutions. It is interesting that models with superstrings or M-theory are more tractable as cosmological models than other dimensional models.
For ($D=10$, $d=3$), we have obtained the accelerating universe with two spatial parts whose curvature is negative. The three-dimensional space expands with the acceleration, but the six-dimensional internal space also expands. The external space finally approaches the expansion whose acceleration tends to zero. It may be difficult for this case to give an account of the realistic acceleration at late time. To obtain more realistic models, we need to construct a model whose internal space is fixed dynamically whereas our space is going to expand more drastically. In the context of pure gravity solutions we have treated in this paper, we cannot get such solutions. The flux field, dilaton and the world volume actions may have a role for the interesting behavior such as fixing the internal spaces [@bio23].
It would be more interesting to find solutions for more general setup.
We would like to thank N. D. Hari Dass for valuable discussions and Eiichi Takasugi and Tetsuyuki Yukawa for useful comments. We would like to thank Nobuyoshi Ohta for informing us papers [@bio26]-[@bio29].
General solutions in $D=11$ {#Appendix}
===========================
We can obtain other solutions corresponding to ($D=11,\,d=2$) and ($D=11,\,d=5$) from the equations (\[a18\]) and (\[a19\]).
In ($D=11,\,d=2$), the equations of motion and the Hamiltonian constraint are $$\begin{aligned}
&4\,\ddot{U}_{+}+k_{(2)}\,U_{+}
+6\,\widetilde{k}_{(8)}\,U_{-}^{5}=0, \label{b12} \\
&4\ddot{U}_{-}+k_{(2)}\,U_{-}=0, \label{b13} \\
&4\dot{U}_{+}\dot{U}_{-}
+k_{(2)}\,U_{+}U_{-}
+\widetilde{k}_{(8)}\,U_{-}^{6}=0, \label{b14}\end{aligned}$$ and for ($D=11,\,d=5$), $$\begin{aligned}
&\frac{1}{4}\,\ddot{U}_{+}+k_{(5)}\,U_{+}
+3\,\widetilde{k}_{(5)}\,U_{-}^{2}=0, \label{b15} \\
&\frac{1}{4}\,\ddot{U}_{-}+k_{(5)}\,U_{-}=0, \label{b16} \\
&\frac{1}{4}\,\dot{U}_{+}\dot{U}_{-}
+k_{(5)}\,U_{+}U_{-}
+\widetilde{k}_{(5)}\,U_{-}^{3}=0. \label{b17}\end{aligned}$$ For ($D=11,\,d=2$), solutions are given by $$U_{-}=
\begin{cases}
A_{1}\,\cos [\,\frac{\sqrt{k_{(2)}}}{2}\,t+A_{2}\, ], \qquad ~~\,(k_{(2)}>0) \\
A_{1}\,t+A_{2}, \qquad \qquad \qquad ~~~~\,~(k_{(2)}=0) \\
A_{1}\,\cosh [\, \frac{\sqrt{-k_{(2)}}}{2}\,t+A_{2}\, ], \quad ~\,\,(k_{(2)}<0)
\end{cases} \label{b18}$$ $$\begin{aligned}
&k_{(2)}>0\,;\quad U_{+}=B_{1}\,\cos [\, \frac{\sqrt{k_{(2)}}}{2}\,\,t+B_{2} \,] \notag \\
&\qquad\qquad\qquad -\frac{15\,\widetilde{k}_{(8)}A_{1}^{5}}{16\,\sqrt{k_{(2)}}}\,\,t\, \sin [\,\frac{\sqrt{k_{(2)}}}{2}\,t+A_{2}\,] \notag \\
&\qquad\qquad\qquad
+\frac{15\widetilde{k}_{(8)}A^{5}_{1}}{64\,k_{(2)}}\,\cos [\,3(\,\frac{\sqrt{k_{(2)}}}{2}\,\,t+A_{2}\,)\,] \notag \\
&\qquad\qquad\qquad+\frac{\widetilde{k}_{(8)}A^{5}_{1}}{64\,k_{(2)}}\,\cos [\,5(\,\frac{\sqrt{k_{(2)}}}{2}\,\,t+A_{2}\,)\,], \notag \end{aligned}$$ $$\begin{aligned}
&k_{(2)}=0\,;\quad U_{+}=B_{1}\,t+B_{2}-\frac{\widetilde{k}_{(8)}}{28A^{2}_{1}}\,(A_{1}\,t+A_{2})^{7}, \notag\end{aligned}$$ $$\begin{aligned}
&k_{(2)}<0\,;\quad U_{+}=B_{1}\,\sinh [\, \frac{\sqrt{-k_{(2)}}}{2}\,\,t+B_{2} \,] \notag \\
&\qquad\qquad\qquad-\frac{15\,\widetilde{k}_{(8)}A_{1}^{5}}{16\,\sqrt{-k_{(2)}}}\,\,t\, \sinh [\,\frac{\sqrt{-k_{(2)}}}{2}\,t+A_{1}\,] \notag \\
&\qquad\qquad\qquad
+\frac{15\widetilde{k}_{(8)}A^{5}_{1}}{64\,k_{(2)}}\,\cosh [\,3(\,\frac{\sqrt{-k_{(2)}}}{2}\,\,t+A_{2}\,)\,] \notag \\
&\qquad\qquad\qquad+\frac{\widetilde{k}_{(8)}A^{5}_{1}}{64\,k_{(2)}}\,\cosh [\,5(\,\frac{\sqrt{-k_{(2)}}}{2}\,\,t+A_{2}\,)\,],
\label{b19}\end{aligned}$$ and constraints are given by $$\begin{aligned}
&k_{(2)}>0\,;\quad \frac{5\widetilde{k}_{(8)}A^{5}_{1}}{4}+k_{(2)}\,B_{1}\,\cos [A_{2}-B_{2}]=0, \notag \\
&k_{(2)}=0\,;\quad B_{1}=0, \\
&k_{(2)}<0\,;\quad \frac{5\widetilde{k}_{(8)}A^{5}_{1}}{4}+(-k_{(2)})\,B_{1}\,\sinh [A_{2}-B_{2}]=0. \notag
\label{b20}\end{aligned}$$ For ($D=11,\,d=5$), solutions are $$U_{-}=
\begin{cases}
A_{1}\,\cos [\,2\sqrt{k_{(5)}}\,t+A_{2}\, ], \qquad ~~\,\,\,(k_{(5)}>0) \\
A_{1}\,t+A_{2}, \qquad \qquad \qquad ~~~~~~\,~\,\,(k_{(5)}=0) \\
A_{1}\,\cosh [\, 2\sqrt{-k_{(5)}}\,t+A_{2}\, ], \quad ~~\,(k_{(5}<0)
\end{cases} \label{b21}$$ $$\begin{aligned}
&k_{(5)}>0\,;\quad U_{+}=B_{1}\,\cos [\, 2\sqrt{k_{(5)}}\,\,t+B_{2} \,] \notag \\
&\qquad\qquad \qquad \quad -\frac{3\,\widetilde{k}_{(5)}A_{1}^{2}}{2\,k_{(5)}} \notag \\
&\qquad\qquad \qquad \quad+\frac{\widetilde{k}_{(5)}A^{2}_{1}}{2\,k_{(5)}}\,\cos [\,2(\,2\sqrt{k_{(5)}}\,\,t+A_{2}\,)\,], \notag \end{aligned}$$ $$\begin{aligned}
&k_{(5)}=0\,;\quad U_{+}=B_{1}\,t+B_{2}-\frac{\widetilde{k}_{(5)}}{A_{1}^{2}}\,(A_{1}\,t+A_{2})^{4}, \notag \end{aligned}$$ $$\begin{aligned}
&k_{(5)}<0\,;\quad U_{+}=B_{1}\,\sinh [\, 2\sqrt{-k_{(5)}}\,\,t+B_{2} \,] \notag \\
&\qquad\qquad \qquad \quad-\frac{3\,\widetilde{k}_{(5)}A_{1}^{2}}{2\,k_{(5)}} \notag \\
&\qquad\qquad\qquad\quad
+\frac{\widetilde{k}_{(5)}A^{2}_{1}}{2\,k_{(5)}}\,\cosh [\,2(\,2\sqrt{-k_{(5)}}\,\,t+A_{2}\,)\,].
\label{b22}\end{aligned}$$ The constraints are $$\begin{split}
&k_{(5)}>0\,;\quad k_{(5)}\,A_{1}\,B_{1}\,\cos [A_{2}-B_{2}]=0, \\
&k_{(5)}=0\,;\quad B_{1}=0, \\
&k_{(5)}<0\,;\quad (-k_{(5)})\,A_{1}\,B_{1}\,\sinh [A_{2}-B_{2}]=0.
\end{split} \label{b23}$$ For ($D=11,\,d=2$) and three dimensional Einstein frame, the metric is $$\begin{aligned}
ds_{11}^{2}&=\Bigg( \frac{U_{+}}{U_{-}} \Biggr)^{-8/3}\,\Bigl[ (U_{+}U_{-})^{2}\,(-dt^{2}+g_{ij}(x)\,dx^{i}dx^{j}) \Bigr] \notag \\
&\qquad \qquad \qquad \qquad +\Biggl( \frac{U_{+}}{U_{-}}\Biggr)^{1/3}\,g_{AB}(y)\,dy^{A}dy^{B}, \label{b24} \end{aligned}$$ $$\begin{aligned}
& e^{2\,A}=
e^{2\,n}=(U_{+}\,U_{-})^{2},\qquad e^{2\,\alpha}=\Biggl( \frac{U_{+}}{U_{-}}\Biggr)^{1/3}. \label{b25}\end{aligned}$$ For ($D=11,\,d=5$) and six dimensional Einstein frame, the metric is given by $$\begin{aligned}
ds_{11}^{2}&=\Bigg( \frac{U_{+}}{U_{-}} \Biggr)^{-5/6}\Bigl[ (U_{+}U_{-})^{1/2}(-dt^{2}+g_{ij}(x)\,dx^{i}dx^{j}) \Bigr] \notag \\
& \qquad \qquad \qquad \qquad +\Biggl( \frac{U_{+}}{U_{-}}\Biggr)^{2/3}g_{AB}(y)\,dy^{A}dy^{B}, \label{b26} \end{aligned}$$ $$\begin{aligned}
& e^{2\,A}
=e^{2\,n}=(U_{+}\,U_{-})^{1/2},\qquad e^{2\,\alpha}=\Biggl( \frac{U_{+}}{U_{-}}\Biggr)^{2/3}. \label{b27}\end{aligned}$$
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|
---
abstract: |
A method for determining quantum variance asymptotics on compact quotients attached to non-split quaternion algebras is developed in general and applied to “microlocal lifts” in the non-archimedean setting. The results obtained are in the spirit of recent work of Sarnak–Zhao.
The arguments involve a careful analytic study of the theta correspondence, the interplay between additive and multiplicative harmonic analysis on quaternion algebras, the equidistribution of translates of elementary theta functions, and the Rallis inner product formula.
address: 'ETH Z[ü]{}rich, Department of Mathematics, R[ä]{}mistrasse 101, CH-8092, Z[ü]{}rich, Switzerland'
author:
- 'Paul D. Nelson'
bibliography:
- 'refs.bib'
title: 'Quantum variance on quaternion algebras, II'
---
Introduction {#sec-1}
============
Overview {#sec-1-1}
--------
The quantum variance problem (see e.g. [@nelson-variance-73-2 §1], [@MR848319], [@2009arXiv0911.4312Z §15.6], [@MR3204186 §4.1.3], [@2013arXiv1303.6972S; @MR1361757; @MR1465794; @luo-sarnak-mass; @MR2103474; @MR2651907]) concerns sums of the shape $$\label{eq:qv-sums-classical}
\sum_{\varphi \in \mathcal{F}}
\langle \varphi, \Psi_1 \varphi \rangle
\langle \Psi_2 \varphi, \varphi \rangle.$$ Here $\Psi_1,\Psi_2$ are fixed mean zero functions on the unit cotangent bundle of a Riemannian manifold $M$ with ergodic geodesic flow, $\mathcal{F}$ traverses a sequence of families of microlocal lifts of Laplace eigenfunctions with eigenvalues in $[0, T^2]$, and $T \rightarrow \infty$. The problem is to determine the leading order asymptotic behavior of . The difficulty of the problem may be appreciated by comparing the expected magnitude $\asymp T$ for for typical $\Psi_1 = \Psi_2$ with the best known general upper bound $O(T^{\dim(M)} / \log T)$ (see e.g. [@nelson-variance-73-2 §1] and references for details).
Although a mathematically rigorous solution to the problem seems hopeless on general $M$, Sarnak–Zhao [@2013arXiv1303.6972S] (following Luo–Sarnak [@luo-sarnak-mass] and Zhao [@MR2651907]) managed to solve it completely on $M = \operatorname{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ for $\mathcal{F}$ consisting of Hecke eigenfunctions. It is natural to seek analogous results on other arithmetic quotients, such as the compact quotients attached to orders in quaternion division algebras. The method of Luo, Sarnak and Zhao demonstrates the tremendous power of *parabolic Fourier expansions*, such as the $q$-expansions $\sum a_n q^n$ enjoyed by classical holomorphic modular forms on $\operatorname{SL}_2(\mathbb{Z})$ at the cusp $\infty$, to establish results that are inaccessible by means of semiclassical analysis or trace formulas alone. Conversely, their technique is fundamentally limited to *split* quotients, such $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ and its congruence covers, on which such expansions are available.
In this article, we develop systematically a method for studying quantum variance on *non-split* arithmetic quotients arising from non-split quaternion algebras, in contrast to the split matrix algebra $M_2(\mathbb{Q})$ underlying the quotient $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ considered by Luo, Sarnak and Zhao. Our main result (Theorem \[thm:main-result-for-microlocal-stuff\], stated in §\[sec-1-5\]) concerns families of “microlocal lifts” on non-split $p$-adic arithmetic quotients attached to quaternion algebras. We focus on $p$-adic quotients to simplify the analysis; this passage does not affect the fundamental difficulty of the problem, and is orthogonal to the primary novelty that the quotient is non-split. Our core estimates (Theorem \[thm:main-estimate-general-variance\], stated in §\[sec-4\]) are developed in some generality.
Trace formulas and linear statistics {#sec:trace-formula-linear}
------------------------------------
Let $\mathbf{X} := \Gamma \backslash G$ be the quotient by an arithmetic lattice of the points of a semisimple $\mathbb{Q}$-group over a local field, such as the real numbers, and let $\mathcal{F}$ be a “large” collection of eigenfunctions $\varphi : \mathbf{X} \rightarrow \mathbb{C}$. It is natural to ask for the asymptotic statistics, as “$\mathcal{F} \rightarrow \infty$” in some sense, of the random measure on $\mathbf{X}$ sending a test function $\Psi$ to $\langle \varphi, \Psi \varphi \rangle$, where $\varphi \in
\mathcal{F}$ is sampled randomly with respect to (say) the normalized counting measure.
The *linear* statistics of this random measure are captured by the *mean* $\Psi \mapsto \mathbb{E}_{\varphi \in \mathcal{F}} \langle \varphi, \Psi \varphi \rangle$. When $\mathcal{F}$ admits a nice harmonic-analytic description, it can be (at least approximately) picked off by a convolution kernel $f \in C_c^\infty(G)$. The mean can then be studied using trace formula techniques: by integrating the pretrace formula $$\label{eq:intro-pretrace-fromula}
\sum_{\gamma \in \Gamma} f(x^{-1} \gamma y)
\approx
\sum_{\varphi \in \mathcal{F} }
\overline{\varphi(x)} \varphi(y) \quad (x,y \in \Gamma \backslash G)$$ over the diagonal against $\Psi$, one obtains an identity $$\label{eq:intro-pretrace-fromula-1}
\sum_{\varphi \in \mathcal{F} }
\langle \varphi, \Psi \varphi \rangle
\approx
\int_{x \in \Gamma \backslash G}
\Psi(x) \sum_{\gamma \in \Gamma} f(x^{-1} \gamma x)$$ whose RHS may be studied by methods for bounding orbital integrals much as in the “Weyl’s law” case $\Psi \equiv 1$. (For an example of such arguments, see §\[sec:mean-statistics\].)
Higher-order statistics such as the $n$-point correlations $$(\Psi_1,\dotsb,\Psi_n)
\mapsto
\mathbb{E}_{\varphi \in \mathcal{F}}
\langle \varphi, \Psi_1 \varphi \rangle
\dotsb
\langle \varphi, \Psi_n \varphi \rangle$$ are more mysterious. The quantum variance problem concerns the quadratic statistics about which trace formulas alone say little.
Hecke multiplicativity and variance statistics
----------------------------------------------
Until this work and its prequel, the only known asymptotic formulas for higher-order statistics in this setting of §\[sec:trace-formula-linear\] were those of Luo–Sarnak–Zhao concerning $\operatorname{SL}_2(\mathbb{Z}) \backslash \operatorname{SL}_2(\mathbb{R})$. The point of departure for their method is that when the eigenfunctions $\varphi$ admit Fourier expansions with coefficients $\lambda(n)$ enjoying a “doubling identity” of the shape $$\label{eqn:hecke-multiplicativity}
\lambda(m) \lambda(n) = \sum \lambda(\dotsb),$$ one can try to reduce variance statistics to linear ones and apply trace formulas such as . This method does not apply when such expansions are not available.
Theta functions and variance statistics {#sec:theta-funct-vari}
---------------------------------------
When the space $\mathbf{X}$ arises from a quaternion algebra $B$ (over $\mathbb{Q}$, say), the Eichler/Shimizu theta correspondence provides an analogue of the doubling identity that suggests a natural strategy for studying quantum variance. We pursue this strategy here. Let $\mathcal{F}$ be a family of eigenfunctions on $\mathbf{X}$. Oversimplifying for now, Shimizu’s theorem (see [@MR783511 II.1]) says that one can find
- a space $\mathbf{X} '$ (a congruence cover of $\PGL_2(\mathbb{Z})
\backslash \PGL_2(\mathbb{R})$),
- a function of three variables $\Theta : \mathbf{X} \times \mathbf{X} \times \mathbf{X} '
\rightarrow \mathbb{C}$ (a theta kernel), and
- for each $\varphi \in \mathcal{F}$, a function $\Phi_\varphi : \mathbf{X} ' \rightarrow
\mathbb{C}$ (an Eichler/Jacquet–Langlands lift)
with the property that $$\label{eq:shimizu-identity}
\overline{\varphi(x)} \varphi(y)
=
\int_{z}
\Phi_\varphi(z)
\Theta(x,y;z)
\quad \text{ for all $\varphi \in \mathcal{F}$
and $x,y \in \mathbf{X}$.
}$$ By integrating the diagonal case $x=y$ of against $\Psi$, one obtains $$\label{eq:identity-pre-parseval}
\langle \varphi, \Psi \varphi \rangle
=
\int_{z}
\Phi_\varphi(z)
\int_x \Psi(x)
\Theta(x,x;z).$$ If the functions $\Phi_\varphi$ are orthogonal to one another and the family $\mathcal{F}$ is sufficiently “complete,” then a cavalier application of Parseval’s formula to suggests that $$\label{eq:formula-seesaw}
\sum_{\varphi \in \mathcal{F}}
(\int_z |\Phi_\varphi|^2 )^{-1}
\langle \varphi, \Psi_1 \varphi \rangle
\langle \Psi_2 \varphi, \varphi \rangle
=
\int_{z}
(\int_x \Psi_1(x)
\Theta(x,x;z))
\overline{(\int_y \Psi_2(y)
\Theta(y,y;z))}.$$ The LHS may be understood as a reasonable proxy for the quantum variance of $\mathcal{F}$ provided that the weights $\int_z |\Phi_\varphi|^2$ are sufficiently uniform in $\varphi$. One aim of this article is to develop robust techniques for determining the asymptotics of the RHS of , which is not *a priori* any simpler to analyze than the LHS. A second aim is to apply the resulting machinery to an interesting family of automorphic forms.
We have oversimplified by neglecting that the theta kernel $\Theta$ produced by Shimizu’s theorem may (and generally does) depend upon the automorphic form $\varphi$. For the above argument to make sense, we need to choose one $\Theta$ that works for every element of the family $\mathcal{F}$. It is natural instead to interpret as *defining* a (weighted) family $\mathcal{F}$ in terms of $\Theta$. A third aim of this article is then to clarify in general how to invert the association $\Theta \mapsto \mathcal{F}$.
Microlocal lifts in the non-archimedean setting\[sec:setting-overview\] {#sec-1-2}
-----------------------------------------------------------------------
Let $k$ be a non-archimedean local field of characteristic zero. Denote by $\mathfrak{o}$ its maximal order, $\mathfrak{q}$ its maximal ideal, and $q := \# \mathfrak{o}/\mathfrak{q}$. For example, one can take $(k,\mathfrak{o},\mathfrak{q},q)
:= (\mathbb{Q}_p,\mathbb{Z}_p, p \mathbb{Z}_p, p)$ for some rational prime number $p$.
Fix a totally real number field $F$ having $k$ as its completion. Fix a discrete cocompact subgroup $\Gamma$ of $\PGL_2(k)$ arising from a maximal order in a (non-split) totally definite quaternion algebra over $F$ (see §\[sec:strong-approx\] for details). Set $$\mathbf{X} := \Gamma \backslash \PGL_2(k).$$ To simplify the present exposition, we assume that $F$ has odd narrow class number, so that $\mathbf{X}$ comes with a natural family of commuting Hecke operators (see §\[sec:hecke-ops\]), which also commute with the right translation action by $\PGL_2(k)$. Fix any $\PGL_2(k)$-invariant measure on $\mathbf{X}$; denote by $\operatorname{vol}(\mathbf{X})$ the total volume and define $L^2(\mathbf{X})$ and $\langle, \rangle$ by integrating.
\[defn:eigenfunctions-intro\] By an *eigenfunction*, we shall mean a nonzero function $\varphi : \mathbf{X} \rightarrow \mathbb{C}$ that is smooth (i.e., right-invariant under some open subgroup), is an eigenfunction under every Hecke operator, and generates an irreducible representation of $\PGL_2(k)$ under right translation.
Some of the Hecke operators are involutions; an eigenfunction will be called *even* if it has eigenvalue $+1$, rather than $-1$, under each such involution.
Let $N$ be a large integer and let $\omega : \mathfrak{o}^\times \rightarrow \mathbb{C}^\times$ be a unitary character of conductor $N$, thus $\omega$ is trivial on $1 + \mathfrak{q}^{N}$ but not on $1 + \mathfrak{q}^{N-1}$. An eigenfunction $\varphi$ will be called a *microlocal lift of orientation $\omega$* if it satisfies $\varphi(x g) = \omega(a^2/\det(g)) \varphi(x)$ for all $x \in \mathbf{X}$ and all $$\label{eq:microlocal-lift-subgroup-defn-0}
g = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in
\operatorname{GL}_2(\mathfrak{o}) \cap \begin{pmatrix}
\mathfrak{o} & \frac{1}{2} \mathfrak{q}^{\lfloor N/2 \rfloor} \\
\mathfrak{q}^{\lceil N/2 \rceil} & \mathfrak{o}
\end{pmatrix}.$$
There is a finite set $\mathcal{F}_\omega$ consisting of unit vector microlocal lifts of orientation $\omega$ so that $\{ \text{microlocal lifts of orientation $\omega$} \} =
\bigsqcup_{\varphi \in \mathcal{F}_\omega} \mathbb{C}^\times
\varphi$. (see §\[sec:deduction-main-thm-microlocal\] or [@nelson-padic-que]). Let $\mathcal{F}_N$ denote the union of $\mathcal{F}_\omega$ over all $\omega$ of conductor $N$.
There is an explicit $c > 0$ so that for $N$ large enough, $$\label{eq:family-size-booyah}
|\mathcal{F}_N| = c q^{2 N}.$$
By a trace formula computation (see §\[sec:mean-statistics\]).
From a representation-theoretic or microlocal-analytic perspective, $\mathcal{F}_N$ is analogous to the family of microlocal lifts of Hecke–Maass eigenforms on a compact arithmetic hyperbolic surface with Casimir eigenvalues $1/4 + t^2$, $t \in [T, 2 T]$, $T \approx q^{N}$. For example, their limit measures enjoy diagonal invariance for local reasons. We refer to [@nelson-padic-que Thm 25, Rmk 26] for a discussion of the sense in which the microlocal lifts considered here are actually “lifts.”
Context: worst case behavior and mean statistics {#sec-1-3}
------------------------------------------------
It was shown recently in [@nelson-padic-que] that microlocal lifts on $\mathbf{X}$ satisfy an analogue of the arithmetic quantum unique ergodicity theorem [@MR2680500; @MR2195133]: for each continuous $\Psi : \Gamma \backslash
\PGL_2(k) \rightarrow \mathbb{C}$, $$\label{eqn:padic-que-result-worst-case-booyah}
\lim_{N \rightarrow \infty}
\max_{\varphi \in \mathcal{F}_N}
|
\langle \varphi, \Psi \varphi \rangle
- \langle \Psi,1 \rangle / \langle 1, 1 \rangle
|
= 0.$$ The Lindel[ö]{}f hypothesis predicts more precisely that $$\label{eq:lindelof-prediction}
\langle \varphi, \Psi \varphi \rangle
= \langle \Psi,1 \rangle /\langle 1,1 \rangle
+ O(q^{-N/2 + o(N)})
\text{ for $\Psi$ fixed and $\varphi \in \mathcal{F}_N$},$$ and it is generally believed that this prediction is essentially optimal.
\[lem:mean-stats-bizooyah\] For each continuous $\Psi : \Gamma \backslash
\PGL_2(k) \rightarrow \mathbb{C}$, $$\label{eq:mean-stats-bizooyah}
\lim_{N \rightarrow \infty}
|\mathcal{F}_N|^{-1}
\sum_{\varphi \in \mathcal{F}_N}
\langle \varphi, \Psi \varphi \rangle
= \langle \Psi,1 \rangle /\langle 1,1 \rangle.$$
This follows by averaging ; alternatively, one can argue as in §\[sec:trace-formula-linear\] using the pretrace formula (see §\[sec:mean-statistics\]).
Definition of variance sums {#sec-1-4}
---------------------------
Consider for each large natural number $N$ the random measure on $\mathbf{X}$ sending a continuous function $\Psi : \mathbf{X} \rightarrow \mathbb{C}$ to $\langle \varphi, \Psi \varphi \rangle$, where $\varphi \in \mathcal{F}_N$ is sampled uniformly at random. The lemma of §\[lem:mean-stats-bizooyah\] says that this random measure has expectation tending as $N \rightarrow \infty$ to the invariant probability measure on $\mathbf{X}$. The most natural definition of the variance of that random measure would then be the bilinear form on smooth mean zero functions $\Psi_1,\Psi_2 : \mathbf{X} \rightarrow
\mathbb{C}$ given by $$|\mathcal{F}_N|^{-1}
\sum_{\varphi \in \mathcal{F}_N}
\langle
\varphi, \Psi_1 \varphi
\rangle
\langle \Psi_2 \varphi , \varphi
\rangle.$$ For technical reasons related to our method (which the reader may infer already from ), we consider instead the slightly modified (and unnormalized) sums $$\label{eq:variance-defn-for-refined-intro}
V_N(\Psi_1,\Psi_2)
:=
\sum_{\varphi \in \mathcal{F}_N}
\iota_\varphi
\langle
\varphi, \Psi_1 \varphi
\rangle
\langle \varphi \Psi_2, \varphi
\rangle,$$ where $\iota_\varphi
:=
L^{(S)}(\operatorname{ad}\varphi,1)$ is a standard “harmonic weight” (see §\[sec:standard-l-function\], §\[sec:variance-statistics\]). Such weights are positive and mild in that their size is $\iota_{\varphi} = q^{o(N)}$ (see ) and their mean essentially constant. Although there are well-developed techniques for removing them as in [@2013arXiv1303.6972S], we retain them here for simplicity.
It follows from the compactness of $\mathbf{X}$ that every smooth function on $\mathbf{X}$ is a finite linear combination of eigenfunctions, so in studying , we may assume by linearity that $\Psi_1,\Psi_2$ are eigenfunctions. It is natural to assume further that
- $\Psi_1, \Psi_2$ are *even*, as otherwise $V_N(\Psi_1,\Psi_2) = 0$ by parity considerations, and that
- $\Psi_1,\Psi_2$ are *strongly of mean zero* in that they are orthogonal to the finite collection of one-dimensional subrepresentations of $L^2(\mathbf{X})$, as otherwise $V_N(\Psi_1,\Psi_2) = 0$ by basic properties of the families $\mathcal{F}_N$ (see [@nelson-padic-que Lem 51]).
Statement of main result {#sec-1-5}
------------------------
By and , one expects $V_N(\Psi_1,\Psi_2)$ to have magnitude at most $O(q^{N})$. It should be possible to confirm this expected upper bound using the Cauchy–Schwarz inequality, the triple product formula, and well-developed techniques for averaging families of $L$-functions as in [@2013arXiv1303.6972S], but the true asymptotics of $V_N(\Psi_1,\Psi_2)$ are much subtler: in the off-diagonal case that $\Psi_1, \Psi_2$ generate distinct irreducible representations $\pi_1 \neq \pi_2 \subseteq L^2(\mathbf{X})$, one expects the *signs* of the quantities $\langle \varphi, \Psi_1 \varphi \rangle$ and $\langle \varphi, \Psi_2 \varphi \rangle$ to vary independently, suggesting additional cancellation in . The primary novelty in the following result is that we detect such cancellation in the off-diagonal case; a secondary novelty is that in the diagonal case, we determine the main term.
\[thm:main-result-for-microlocal-stuff\] For even eigenfunctions $\Psi_1, \Psi_2$ strongly of mean zero, the limit $$\label{eq:main-result-intro-1}
\lim_{N \rightarrow \infty}
q^{-N} V_N(\Psi_1, \Psi_2)$$ exists. The limit is zero unless $\Psi_1,\Psi_2$ generate the same irreducible representation $\pi \subseteq L^2(\mathbf{X})$. In that case, it is equal to $$\label{eq:main-result-intro-main-term}
c_0 L^{(S)}(\pi,\tfrac{1}{2})
\int_{h \in N(H)}
(\int_{x \in \mathbf{X}}
\Psi_1(x h)
\overline{\Psi_2(x)}),
$$ where (see §\[sec:deduction-main-thm-microlocal\] for details)
- $c_0$ is an explicit positive constant,
- $S$ denotes the set of “bad places,”
- $L^{(S)}(\pi,\tfrac{1}{2})$ denotes the central $L$-value without Euler factors in $S$, and
- $N(H)$ denotes the normalizer of the diagonal torus $H$ in $\PGL_2(k)$.
The integral converges as written (see §\[sec:local-Xi\], §\[sec:local-convergence-lemmas\]).
As in [@2013arXiv1303.6972S], the identity admits an intriguing semiclassical interpretation whereby the arithmetical values $L^{(S)}(\pi,\tfrac{1}{2})$ quantify the deviation of the asymptotic quantum variance from the (symmetrized) classical variance of the diagonal flow.
We establish the stronger assertion that $q^{-N} V_N(\Psi_1, \Psi_2)$ differs from its limit by $O(N q^{-N})$ (see §\[sec:deduction-main-thm-microlocal\]). We expect that our method is capable of refining that error term to $O(q^{-N})$ and proving that such an estimate is essentially best possible (see [@nelson-variance-73-2 §6.5]), but we do not pursue such refinements here.
We note as in [@MR2103474] that Theorem \[thm:main-result-for-microlocal-stuff\] confirms the Lindel[ö]{}f prediction on average and implies that it is essentially optimal if it is true.
The off-diagonal case is that in which the method of Luo–Sarnak–Zhao requires a cusp. It is conceivable that one could establish the diagonal case of Theorem \[thm:main-result-for-microlocal-stuff\] by averaging the triple product formula as in [@2013arXiv1303.6972S]. Our method does not use the triple product formula.
Comparison with the prequel
---------------------------
We highlight some differences between the results and aims of [@nelson-variance-73-2] and those of this article.
1. In [@nelson-variance-73-2], the observables $\Psi_1, \Psi_2 : \mathbf{X} \rightarrow \mathbb{C}$ were restricted to be right-invariant by the maximal compact subgroup $K := \PGL_2(\mathfrak{o})$. In this article, we study arbitrary fixed observables on the full “phase space” $\mathbf{X}$ rather than on the “configuration space” $\mathbf{X} / K$. The simplifying restriction of the prequel allowed us to get by with some *ad hoc* computations in places where a more systematic approach is required here.
This jump in complexity is analogous to that from [@MR2103474; @MR2651907] to [@2013arXiv1303.6972S], but the manner in which the new complexity is addressed differs completely (in a stronger sense than that the methods themselves differ completely): In [@2013arXiv1303.6972S], phase space observables were treated by an inductive technique involving weight isotypic vectors, while the methods developed here apply directly to general phase space observables.
2. In the prequel, we considered families $\mathcal{F}_N'$ of balanced newvectors. Here, we consider families $\mathcal{F}_N$ of microlocal lifts. The methods developed in Part 1 of this article apply robustly to a large class of families; given those methods, neither $\mathcal{F}_N$ nor $\mathcal{F}_N'$ is much more difficult to analyze than the other. We focused in [@nelson-variance-73-2] on newvector families because of their familiarity. We focus here on the families $\mathcal{F}_N$ because of their strong analogy with those in the motivating work of Luo–Sarnak–Zhao and because the formulas for the main term for $\mathcal{F}_N$ are more aesthetically appealing than those for $\mathcal{F}_N'$.
3. The aim of [@nelson-variance-73-2] was to introduce the method by application to the simplest non-trivial non-split case of the quantum variance problem and in the most elementary language possible.[^1] The aim here is instead to develop the technique as clearly as possible in its natural generality. We hope the two articles serve complementary purposes.
4. It may be worth recording that for the reasons indicated in the previous three points, the two articles have essentially no logical overlap.
Discussion of method {#sec-1-6}
--------------------
The general strategy of §\[sec:theta-funct-vari\] applies in the setting of Theorem \[thm:main-result-for-microlocal-stuff\]: modulo a preliminary technical partition of the family $\mathcal{F} = \mathcal{F}_N$, we construct $f = f_N$ for which holds and then $\Theta$ for which holds. (For a precise definition of $f$, see the end of §\[sec:appl-micro-local-prelims\].) The integral $\int_z \Psi_i(z) \Theta(x,x;z)$ does not define a theta lift of $\Psi_i$ in the traditional sense, but instead decomposes as a sum of products $\theta_i(z) h_i(z)$, where $\theta_i$ is a variant of the Jacobi theta function and $h_i$ is a theta lift of $\Psi_i$. The RHS of then decomposes as a sum of inner products $$\label{eq:ip-4-theta-before-rearr}
\langle \theta_1 h_1, \theta_2 h_2 \rangle$$ Suppose we can approximate each such inner product by $$\label{eq:ip-4-theta-after-rearr}
\langle \theta_1, \theta_2 \rangle \langle h_1, h_2 \rangle.$$ The Rallis inner product formula [@2012arXiv1207.4709T; @MR2837015] for theta lifts applies to ; summing it up, we obtain $$\label{eq:asymptotic-after-rallis-in-intro-sketch}
\sum_{\varphi \in \mathcal{F}}
(\int |\Phi_\varphi|^2 )^{-1}
\langle \varphi, \Psi_1 \varphi \rangle
\langle \Psi_2 \varphi, \varphi \rangle
\approx
(\ast)
\int_{h \in G}
I_f(h)
(\int_{x \in \mathbf{X}}
\Psi_1(x h)
\overline{\Psi_2(x)}),$$ where:
- $\approx$ means up to the error incurred by replacing each term with ;
- $(\ast)$ means “modify by a central $L$-value as in Theorem \[thm:main-result-for-microlocal-stuff\];” and
- $I_f(h) := \int_{g \in G} \mathfrak{S} f(h^{-1} g h)
\overline{\mathfrak{S} f(g)}$, where $\mathfrak{S} f(g)
:= (f(g) + f(g - \operatorname{tr}(g)))/2$.
To complete the proof of Theorem \[thm:main-result-for-microlocal-stuff\], it suffices now to show that
1. $I_f : G \rightarrow \mathbb{C}$ tends to the “delta distribution” on the normalizer $N(H)$ as the parameter $N$ tends to $\infty$, and that
2. the error hidden by $\approx$ is negligible.
Problem (i) is purely local. Problem (ii) involves both local and global difficulties; a critical global input to its solution was developed in [@nelson-theta-squared].
To indicate in more detail how this works, assume for simplicity that $k$ arises as a completion of $\mathbb{Q}$. The proof may then be summarized by the sequence $$\begin{aligned}
q^{-N} V_N(\Psi_1,\Psi_2)
&= \label{eqn:proof-outline-1}
\langle \theta(z) h_{1}(q^{2 N} z),
\theta(z) h_2(q^{2 N} z) \rangle
\\
&=\label{eqn:proof-outline-2}
\langle |\theta|^2(z), \overline{h_1} h_2(q^{2 N} z) \rangle
\\
&=\label{eqn:proof-outline-3}
\langle |\theta|^2, 1 \rangle \langle 1, \overline{h_1} h_2
\rangle
+ O(N q^{-N})
\\
&= \label{eqn:proof-outline-4}
c_0 L^{(S)}(\pi,1/2)
\int_{h \in N(H)}
(\int_{x \in \mathbf{X}}
\Psi_1(x h)
\overline{\Psi_2(x)})
+ O(N q^{-N}),\end{aligned}$$ where $\theta$ is essentially the weight $1/2$ Jacobi theta function, $z$ denotes an integration parameter in the upper half-plane, $h_1,h_2$ are *fixed* weight $3/2$ theta lifts of $\Psi_1,\Psi_2$ to some congruence cover of $\Gamma_0(4) \backslash \mathbb{H}$, and inner products are taken on congruence covers of $\Gamma_0(4) \backslash \mathbb{H}$ with respect to the natural probability measures. The first step was discussed in §\[sec:theta-funct-vari\] in high level terms. The obvious second step forms the cornerstone of the argument; see [@MR2373356] for related discussion. The third step is a variant of the equidistribution of Hecke operators; we have discussed it extensively in [@nelson-variance-73-2; @nelson-theta-squared]. The fourth step is an explication of the Rallis inner product formula followed by the asymptotic analysis of the integral $I_f(h)$ discussed above.
The shape of the key inner product
----------------------------------
The precise shape of the RHS of (namely, the “separation” of $\theta(z)$ and $h_i(q^{2 N} z)$ by the dilation $z \mapsto q^{2 N} z$) is evidently crucial to the success of the method, so we sketch how it arises. Let $\phi = \phi_N : M_2(k) \rightarrow \mathbb{C}$ denote the Schwartz–Bruhat function related to the kernel $f = f_N$ by $$\phi(x) :=
\begin{cases}
0 & \text{ if } x \notin \operatorname{GL}_2(k) \\
1_{\mathfrak{o}^\times}(\det(x)) f(\operatorname{pr}(x)) & \text{ if } x \in \operatorname{GL}_2(k),
\end{cases}$$ where $\operatorname{pr}: \operatorname{GL}_2(k) \rightarrow \PGL_2(k)$ denotes the canonical projection. Let $\mathcal{F}$ denote the Fourier transform on $M_2(k)$. (We hope the dual use of this symbol for Fourier transforms and families introduces no confusion.) Then $$\label{eq:key-approximation-for-fourier-transform-of-conv-kernel-sketch}
\mathcal{F} \phi
(
\begin{pmatrix}
d + a & b \\
c & d - a
\end{pmatrix}
)
\approx
\begin{cases}
1 & |d| = O(1); |b|, |c| = O(q^{N/2}); |a| \asymp q^N \\
0 & \text{otherwise}
\end{cases}$$ The detailed statement and proof of the computation may be found in §\[sec:fourier-transf-conv-kern\]. The key features are the shape of the support and the uniform smoothness under simultaneous dilation of the parameters $a,b,c$. Ignoring the less important variable $d$, one can think of $\mathcal{F} \phi$ as capturing roughly the Fourier transform of the pullback of $f$ to the Lie algebra of $\PGL_2(k)$. We note in passing that the subgroup $N(H)$ arises eventually from as the “normalizer of the limiting support” of $\mathcal{F} \phi$.
Consider now the Hecke twisted pretrace formula $$\label{eq:hecke-twisted-pretrace}
\sum_{\gamma \in M_n} f(x^{-1} \gamma y) =\sum_{\varphi \in
\mathcal{F}_N} \sqrt{n} \lambda_\varphi(n) \overline{\varphi
(x)} \varphi(y),$$ where $M_n$ is as in the classical definition of the Hecke operator $T_n$ and $\sqrt{n} \lambda_{\varphi}(n)$ denotes the Hecke eigenvalue. Let $R \subseteq B$ denote the maximal order underlying the construction of $\Gamma$. Taking $x=y=:g$ in and summing against $e(n z)$ over positive integers $n$ having only “good” prime divisors gives $$\sum_{\gamma \in R}
\phi(g^{-1} \gamma g)
e(\det(\gamma) z)
=
\sum_{\varphi \in \mathcal{F}_N}
|\varphi|^2(g)
\Phi_\varphi(z)$$ where $e(z) := e^{2 \pi i z}$ and $\Phi_\varphi(z) := \sum \sqrt{n} \lambda_\varphi(n) e(n z)$ denotes an Eichler/Jacquet–Langlands lift of $\varphi$. Suppose henceforth that $k = \mathbb{Q}_p$. By the inversion formula for theta functions, we obtain $$\begin{aligned}
\sum_{\varphi \in \mathcal{F}_N}
|\varphi|^2(g)
\Phi_\varphi(-1/z)
&\approx
\sum_{\gamma \in R[1/p]}
\mathcal{F} \phi(g^{-1} \gamma g)
e(\det(\gamma) z) \\
&\approx
\sum_{m,\beta}
\mathcal{F} \phi(m + g^{-1} \beta g)
e(m^2 z)
e(\det(\beta) z),\end{aligned}$$ where the sum is over $m \in \mathbb{Z}[1/p]$ and $\beta \in R[1/p]^0$. (Here $\approx$ means “up to unimportant inaccuracies.”) By integrating against $\Psi(g)$, we obtain $$\label{eq:key-sketch-identity-pre-parseval}
\sum_{\varphi \in \mathcal{F}_N}
\Phi_{\varphi}(-1/z)
\langle \varphi, \Psi \varphi \rangle
= \sum_{m,\gamma}
e(m^2 z) e(\det(\beta) z)
\int_{g \in \Gamma \backslash G}
\Psi(g) \mathcal{F} \phi(m + g^{-1} \beta g).$$ The multiplicity one theorem on $\mathbf{X}$ implies that for $\varphi, \varphi ' \in \mathcal{F}_N$, $$\label{eq:consequence-of-mult-one-in-sketch}
\langle \Phi_\varphi, \Phi_{\varphi '} \rangle
\approx \iota_{\varphi} \text{ if } \varphi = \varphi ',
\quad = 0 \text{ otherwise}.$$ By and the assumption that $\Psi$ is fixed, one deduces that $$\label{eqn:key-observation-for-proof-outline-1}
\int_{g \in \Gamma \backslash G}
\Psi(g) \mathcal{F} \phi(m + g^{-1} \beta g)
\approx 1_{\mathbb{Z}_p}(m)
\phi_{\Psi}''(p^N \beta)$$ where $\phi_{\Psi}''$ is independent of $N$. (A “toy version” of such reasoning: if a function on $\mathbb{R}^3$ is smooth with respect to polar coordinates and supported away from the origin, then it is smooth.) By Parseval applied to , and , one arrives at an identity of the form .
Organization of this paper {#sec-1-7}
--------------------------
The main result is Theorem \[thm:main-result-for-microlocal-stuff\], stated somewhat informally above and more precisely (and generally) as Theorem \[thm:main-variance-microlocal-adelic-formulation\] in the final section §\[sec:deduction-main-thm-microlocal\] of this paper, which also contains the proof.
To present the proof as clearly as possible, we separate the difficulties concerning general families from those specific to the $\mathcal{F}_N$ considered above. The former may be found in Part I, the latter in Part II. The two parts are weakly coupled.
The main result of Part I is Theorem \[thm:main-estimate-general-variance\], stated in §\[sec-4\]. Its proof involves local (§\[sec-2\]) and global (§\[sec-3\]) preliminaries. Its conclusion reduces the quantum variance problem to local problems of three sorts, which are treated in Part II for the families $\mathcal{F}_N$:
1. The comparatively mild difficulties associated with producing a convolution kernel $f$ that picks off a given family $\mathcal{F}$, as in .
2. Those associated with determining the main term (i.e., asymptotically evaluating distributions such as the $I_f$ considered in §\[sec-1-6\]).
3. Those concerned with bounding error terms (i.e., differences between inner products as in and ). An important point is that Theorem \[thm:main-estimate-general-variance\] reduces this global problem to a local one.
For the latter two problems, the key input is a careful analysis (§\[sec:fourier-transf-conv-kern\]) of the “Fourier transform” of the convolution kernel $f$.
The reader might first study carefully §\[sec:general-notation\], §\[sec-4-1\], §\[sec:main-general-estimate-key-defns\], §\[sec-4-3\], §\[sec:general-estimates-specialized-single-place\], §\[sec:element-attached-to-N-sigma\], §\[sec-7-1\], §\[sec-8-2\], §\[sec:9-setting\] and §\[sec:variance-statistics\], skipping or skimming any sections before or between; we have made some effort to keep those sections essentially self-contained.
General notation\[sec:general-notation\] {#sec-1-8}
----------------------------------------
For a quaternion algebra $B$ over a field or adele ring $A$, we denote by $\iota : B \rightarrow B$ the main involution, by $\operatorname{nr}: B \rightarrow A$ the reduced norm $\operatorname{nr}(x) := x x^{\iota}$, by $\operatorname{tr}: B \rightarrow A$ the reduced trace $\operatorname{tr}(x) := x + x^{\iota}$, and by $$B^0 := \{x \in B : \operatorname{tr}(x) = 0\}$$ the subspace of traceless quaternions. We employ the notations $$n(x) := \begin{pmatrix}
1 & x \\
0 & 1
\end{pmatrix},
\quad
a(y) :=
\begin{pmatrix}
y & 0 \\
0 & 1
\end{pmatrix},
\quad
t(y) :=
\begin{pmatrix}
y & 0 \\
0 & y^{-1}
\end{pmatrix},
\quad
n'(x) :=
\begin{pmatrix}
1 & 0 \\
x & 1
\end{pmatrix}$$ $$\operatorname{Ad}(g) x := g x g^{-1},
\quad
\operatorname{Ad}(g) f(x) := f(g^{-1} x g)$$ and $$\mathfrak{S} f(x) := \frac{f(x) + f(x - \operatorname{tr}(x))}{2}$$ whenever they make sense. For example, this is the case if $g$ belongs to the unit group $B^\times$ of a quaternion algebra $B$ over $A$ as above and $x$ belongs to (resp. $f$ is a function on) one of the sets $B^\times/A^\times, B, B^0$. (As we explain below, one may interpret $\mathfrak{S}$ as “projection onto $\O_1$-invariants.”)
We define the right regular representation $\rho_{\operatorname{reg}}(g) f(x) := f(x g)$ whenever it makes sense.
Given a local (resp. global) field $F$ and a nontrivial unitary character $\psi$ of $F$ (resp. of $\mathbb{A}/F$) and an element $a \in F^\times$, we denote by $\psi^a$ the nontrivial unitary character with the same domain as $\psi$ given by $\psi^a(x) := \psi(a x)$.
For a finite-dimensional vector space $V$ over a local field or adele ring, we denote by $\mathcal{S}(V)$ the space of Schwartz–Bruhat functions $\phi : V \rightarrow \mathbb{C}$, topologized as usual (see e.g. [@MR0165033 §11]).
Let $G$ be a group over an adele ring or a finite product of local fields. We let $C_c^\infty(G)$ denote the space of smooth compactly supported functions; as usual, smooth means infinitely differentiable (resp. locally constant) with respect to the archimedean (resp. non-archimedean) variables. Assume that we have equipped $G$ with a Haar measure. Let $\pi$ be a smooth representation of a group that contains $G$. Let $f \in C_c^\infty(G)$. We then define the operator $\pi(f) \in \operatorname{End}(\pi)$ by $\pi(f) v := \int_{g \in G} f(g) \pi(g) v$.
The use of Vinogradov notation is standard: $A = O(B)$, $A \ll B$ and $B \gg A$ each signify that $|A| {\leqslant}c |B|$ for some “constant” $c$, with dependencies indicated by subscripts; $A \asymp B$ signifies that $A \ll B \ll A$.
We write $1_E$ for the characteristic function of a subset $E$ of some set $X$. For an assertion $A$, we set $1_A := 1$ if $A$ is true and $1_A := 0$ if $A$ is false.
We set $\mathbb{C}^{(1)} := \{ z \in \mathbb{C}^{\times} : |z| =
1\}$.
We adopt the convention that main results (Theorems \[thm:main-result-for-microlocal-stuff\], \[thm:main-estimate-general-variance\], \[thm:main-variance-microlocal-adelic-formulation\] in §\[sec-1-5\], §\[sec-4-3\], §\[sec:variance-statistics\], respectively) are called theorems, the most important intermediary results original to this article (Propositions \[prop:main-error-estimate-global-adelic-general\], \[prop:after-extracting-main-term\], \[prop:harmonic-analytic-isolation-1\], \[prop:key-fourier-estimate-microlocal-kernel\], \[prop:desired-main-term-identity-do-it-up\], \[prop:local-error-estimates-stmt\] in §\[sec-3-5-5\], §\[sec:main-identity\], §\[sec:element-attached-to-N-sigma\], §\[sec-6-3\], §\[sec-7-1\], §\[sec-8-2\]) are called propositions, and everything else (including deep cited work) is called a lemma.
Local preliminaries {#sec-2}
===================
The purpose of this section is collect local definitions, notation and identities for later use. The notation introduced here should be self-descriptive with the exception of that for the similitude Weil representation $\Omega$ defined in §\[sec:defn-local-omega\].
Let $k$ be a local field of characteristic $\neq 2$, thus $k$ is either $\mathbb{R}$, $\mathbb{C}$ or a finite extension of $\mathbb{Q}_p$ or (for $p \neq 2$) of $\mathbb{F}_p(t)$. The assumption on the characteristic is relevant only for sections discussing the Weil representation.
Let $\psi : k \rightarrow \mathbb{C}^{(1)}$ be a nontrivial unitary character of $k$, and let $B$ be a quaternion algebra over $k$. Set $G := B^\times/ k^\times$. When $k$ is non-archimedean, let $R \subset B$ be a maximal order.
Generalities {#sec-2-1}
------------
### The number field {#sec-2-1-1}
We denote by $|.| := |.|_k : k \rightarrow \mathbb{R}_{{\geqslant}0}$ the normalized absolute value, so that $d(c x) = |c| \, d x$ for $c \in k$ and any Haar measure $d x$ on $k$.
Let $\zeta_k(s)$ denote the local zeta function, thus $\zeta_k(s) = \pi^{-s/2} \Gamma(s/2),
2 (2 \pi)^{-s} \Gamma(s)$ or $(1-q^{-s})^{-1}$ as $k = \mathbb{R}, \mathbb{C}$, or a non-archimedean local field with residue field of cardinality $q$.
Recall that $B$ is *split* if it is isomorphic to the algebra $M_2(k)$ of $2 \times 2$ matrices. Otherwise, $B$ is called *non-split* or *ramified*; in that case, it is the unique (up to isomorphism) quaternion division algebra over $k$, and the group $G$ is compact.
When $k$ is non-archimedean, we denote by $\mathfrak{o}$ the maximal order, by $\mathfrak{q}$ the maximal ideal, by $q := \# \mathfrak{o} / \mathfrak{q}$ the cardinality of the residue field, by $\varpi \in \mathfrak{q} = \varpi \mathfrak{o}$ a uniformizer (thus $|\varpi| = q^{-1}$), and by $\Delta_{\psi}$ the absolute conductor of $\psi : k \rightarrow \mathbb{C}^{(1)}$, thus $\Delta_{\psi} = q^d$ if $\psi$ is trivial on $\mathfrak{q}^{-d}$ but not on $\mathfrak{q}^{-d-1}$. Recall that $\psi$ is *unramified* if $\Delta_{\psi} = 1$.
### Measures\[sec:local-measures\] {#sec-2-1-2}
For $X \in \{k,B^0,B\}$, define the perfect pairing $\langle , \rangle : X \otimes X \rightarrow k$ by $\langle x,y \rangle := x y$ if $X = k$ and by $\langle x, y \rangle := \operatorname{tr}(x^{\iota} y)$ if $X = B^0,B$. Equip $X$ with the Haar measure $d x$ for which the Fourier transform $\mathcal{F} : \mathcal{S}(X) \rightarrow \mathcal{S}(X)$ defined by $\mathcal{F} f(\xi)
:=
\int_{x \in X}
f(x)
\psi(\langle x, \xi \rangle)
\, d x$ satisfies the inversion formula $$\label{eq:fourier-inversion-for-normalization}
\mathcal{F} \mathcal{F} f(x) = f(-x).$$ Equip $k^\times$ with the Haar measure $\int_{k^\times } f
:=
\int_{x \in k^\times}
f(x) \, \frac{d x}{|x|}$.
The quotient $k^\times / k^{\times 2}$ is finite. We equip it with the Haar measure compatible with the squaring map, so that for $f \in C_c(k^\times)$, $$\label{eqn:compatibility-squaring-map-local-measure}
\int_{x \in k^\times}
f(x)
\,
\frac{d x}{|x|}
=
\int_{y \in k^\times/k^{\times 2}}
(\int_{z \in k^\times}
f(y z^2) \, \frac{d z}{|z|}).$$ For $f : k^\times / k^{\times 2} \rightarrow
\mathbb{C}$, one has explicitly $\int_{x \in k^\times / k^{\times 2}}
f(x)
=
\frac{|2|_k}{2}
\sum_{x \in k^\times / k^{\times 2}}
f(x)$.
Equip $G$ with the Haar measure $d g$ for which the integral formula $$\label{eq:integarl-formula-for-integrating-over-B}
\int_{x \in B}
f(x) \, d x
= \int_{g \in G}
(\int_{z \in k^\times}
|\operatorname{nr}(z g)|^2 f(z g) \, \frac{d z}{|z|}) \, d g$$ holds for $f \in C_c(B)$. When $B = M_2(k)$ is split, so that $G = \PGL_2(k)$, a direct calculation with differential forms gives for $f \in C_c(G)$ that $$\label{eqn:explicit-M2-integral-formula}
\int_G f
=
\int_{x_1,x_2 \in k}
\int_{y \in k^\times}
f(
n'(x_1)
n(x_2)
a(y)
)
\, d x_1 \, d x_2 \, \frac{d y}{|y|}.$$
### Volume formulas\[sec:local-vol-formulas\] {#sec-2-1-3}
Assume (for all but the final assertion of §\[sec-2-1-3\]) that $k$ is non-archimedean. Write $\operatorname{vol}(E \subseteq X)$ to denote the volume of $E$ with respect to the measure that we have defined on $X$. Let $J {\leqslant}G$ denote the image of $R^\times$; if $B$ is split, then $J$ is a maximal compact subgroup of $G$, otherwise it has index $2$ in the compact group $G$. Abbreviate $\operatorname{vol}(\mathfrak{o}) := \operatorname{vol}(\mathfrak{o} \subseteq k)$, $\operatorname{vol}(\mathfrak{o}^\times ) := \operatorname{vol}(\mathfrak{o}^\times
\subseteq k^\times )$, $\operatorname{vol}(J) := \operatorname{vol}(J \subseteq G)$, $\operatorname{vol}(R) := \operatorname{vol}(R \subseteq B)$ and $\Delta := \Delta_{\psi}$. Let $\Delta_{B}$ denote the reduced discriminant, thus $\Delta_{B_\mathfrak{p}} = 1$ or $q$ according as $B$ splits or ramifies. Set $\zeta_B(s) := \zeta_k(2 s) \zeta_k(2 s - 1)$ if $B$ splits and $\zeta_B(s) := \zeta_k(2 s)$ otherwise.
\[lem:local-vol-formulas\]
1. $\operatorname{vol}(\mathfrak{o}) = \Delta^{-1/2}$, $\operatorname{vol}(\mathfrak{o}^\times)
= \zeta_k(1)^{-1} \Delta^{-1/2}$.
2. $\operatorname{vol}(R) = \Delta_B^{-1} \Delta^{-4/2}$, $\operatorname{vol}(J) = \zeta_k(1) \zeta_B(1)^{-1} \Delta_B^{-1} \Delta^{-3/2}$.
3. If $B$ is split, then $$\frac{\operatorname{vol}(R)}{\operatorname{vol}(J) \Delta^{-1/2}}
= \zeta_k(2).$$
4. If $k$ is real, $B$ is non-split and $\psi(x) = e^{2 \pi i x}$, then $\operatorname{vol}(G) = 4 \pi^2$.
For (i)—(iii), we may reduce by dimensional analysis to the case $\Delta = 1$. The required formulas then follow from applied to $f = 1_\mathfrak{o}$ or $f = 1_R$ and by applied to $f = 1_{1 + \varpi R}$ (see [@MR580949 Lem 2.4.3] for details). For (iv), set $f(x) := e^{- 2 \pi \operatorname{nr}(x)}$. Apply to see that $\int_B f = 1$. Apply and the substitution $z \mapsto z / (2 \pi \operatorname{nr}(g))^{1/2}$ to deduce that $(2 \pi)^2 = \operatorname{vol}(G) \int_{z \in \mathbb{R}^\times} |z|^4
e^{-|z|^2} \, \frac{d z}{|z|} = \operatorname{vol}(G)$.
### Cartan decomposition {#sec:cartan-decomposition}
Suppose $B = M_2(k)$, so that $G = \PGL_2(k)$. Let $K {\leqslant}G$ be the standard maximal compact subgroup. Then $G = \cup_{y \in k^\times : |y| {\leqslant}1} K a(y) K$. When $k$ is non-archimedean, one has for $f \in C_c(K \backslash G / K)$ the integral formula $$\label{eq:cartan-decomp-integral-formula}
\int_{G}
f
=
\operatorname{vol}(K)
\sum_{m {\geqslant}0}
q^m
(1 + 1_{m > 0} q^{-1})
f(a(\varpi^m)).$$
### The $\Xi$-function\[sec:local-Xi\] {#sec-2-1-4}
Given a maximal compact subgroup $K {\leqslant}G$, let $\Xi : G \rightarrow \mathbb{R}_{>0}$ denote the Harish–Chandra function relative to $K$:
- If $B$ is non-split, then $\Xi \equiv 1$.
- If $B$ is split, then $\Xi(g) = \langle g v, v \rangle$ where $v$ is a $K$-invariant unit vector in the unitary induction of the trivial character of a Borel subgroup of $G$ (see [@MR946351]).
The following properties of $\Xi$ are relevant for us:
1. It satisfies $\Xi(1) = 1$, and is bi-$K$-invariant.
2. If $B$ is split, then under any identification $G = \PGL_2(k)$, one has $\Xi(a(y)) \asymp \log(t)/t^{1/2}$ with $t := |y| + |y|^{-1}$.
3. Let $\pi$ be an irreducible unitary representation of $G$. If $B$ is split, assume that $\dim(\pi) > 1$. Then there exists $\delta > 0$ so that for $v_1, v_2 \in \pi$, one has $\langle g v_1, v_2 \rangle
\ll_{v_1,v_2} \Xi(g)^{\delta}$ for all $g \in G$. (See for instance [@michel-2009 §2.5.1] for a more precise assertion).
### Convergence lemmas\[sec:local-convergence-lemmas\]
The estimates collected here are standard. \[sec-2-1-5\]
\[lemma:cheap-matrix-coeff-schwartz-space-B-estimate-via-Xi\] Either let $X$ be one of the spaces $B^0, B$ and take $\phi_1, \phi_2 \in \mathcal{S}(X)$, or let $X = G$ and take $\phi_1,\phi_2 \in C_c^\infty(G)$. For $g \in G$, one then has $$\langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle_{L^2(X)}
\ll_{\phi_1,\phi_2} \Xi(g)^2.$$
We treat the case $X = B^0$; the other cases are similar. The estimate is trivial unless $B$ is split, so assume that $B = M_2(k)$. By the Cartan decomposition, we may assume that $g = a(y)$ with $|y| {\leqslant}1$. It suffices then (ignoring logarithmic factors) to show that $$\int_{a,b,c \in k}
\overline{\phi_1}
(
\begin{pmatrix}
a & y^{-1} b \\
y c & -a
\end{pmatrix}
)
{\phi_2}
(
\begin{pmatrix}
a & b \\
c & -a
\end{pmatrix}
)
\ll_{\phi_1,\phi_2} |y|.
$$ For this, we substitute $b \mapsto y b$ and appeal to the rapid decay of $\phi_1,\phi_2$.
\[lemma:convergence-Xi-along-G-and-H\] Let $\delta > 0$.
1. The integral $\int_{g \in G} \Xi^{2+\delta}(g)$ converges.
2. Let $E$ be a separable quadratic subalgebra of $B$. Let $H {\leqslant}G$ denote the image of $E^\times$. Equip $H$ with some Haar measure. Then the integral $\int_{h \in H} \Xi^\delta(h)$ converges.
We may assume that $B = M_2(k)$ is split and that $E$ is the split diagonal torus, as otherwise the groups involved are compact. The convergence of the second integral then follows from that of $\int_{y \in k^\times} \min(|y|,|y|^{-1})^\delta$. For the first integral, we integrate using the Cartan decomposition; the convergence follows similarly.
### Conventions
By a *representation* of a $k$-group, we shall always mean
- a smooth representation, if $k$ is non-archimedean, and otherwise
- the space of smooth vectors in a unitary representation.
Weil representations\[sec:local-weil-reps\] {#sec-2-2}
-------------------------------------------
### Quadratic spaces\[sec:local-quadratic-spaces\] {#sec-2-2-1}
Let $V$ be a quadratic space over $k$, thus $V$ is a finite-dimensional $k$-vector space equipped with a non-degenerate quadratic form $q_V : V \times V \rightarrow k$. We denote by $b_V : V \otimes V \rightarrow k$ the associated non-degenerate bilinear form given by $b_V(x,y)
:= q_V(x+y) - q_V(x) - q_V(y)$.
Recall that $\O(V) := \{g \in \operatorname{GL}(V) : q_V(g x) = q_V(x) \text{ for all } x
\in V\}$, $\operatorname{GO}(V) := \{g \in \operatorname{GL}(V) : \text{ there exists
$\lambda \in k^\times$ so that } q_V(g x) = \lambda q_V(x)
\text{ for all } x \in V\}$, and $\operatorname{SO}(V) := \operatorname{SL}(V) \cap \O(V)$. The group $\operatorname{GO}(V)$ contains the subgroup $k^\times$ of scalar operators, and we set $\operatorname{PGO}(V) := \operatorname{GO}(V) / k^\times$.
Let $\mu_V$ denote the measure on $V$ that is $(\psi,b_V)$-self dual, i.e., that for which $\mathcal{F} : \mathcal{S}(V) \rightarrow \mathcal{S}(V)$ defined by $\mathcal{F} \phi(\xi)
:=
\int_{x \in V}
\phi(x)
\psi(b_V(x,\xi))
\, d \mu_V(x)$ satisfies $\mathcal{F} \mathcal{F} \phi(x) = \phi(-x)$.
The following examples of quadratic spaces are relevant for us:
1. $V = B$, $q_V = \operatorname{nr}$, so that $b_V(x,\xi) = \operatorname{tr}(x^{\iota} \xi) = \langle x,\xi
\rangle$.
2. $V = B^0$, $q_V$ the restriction of $\operatorname{nr}$. The natural map $\operatorname{Ad}: G \rightarrow \operatorname{SO}(B^0)$ is an isomorphism.
3. $V = k$, regarded as a subspace of $B$, and $q_V$ the restriction of $\operatorname{nr}$, thus $q_V(x) = x^2$ and $b_V(x,y) = 2 x y$ for $x \in V$. In this case, we denote the orthogonal group by $\O_1(k) := \O(V) \cong \{\pm 1\}$.
For $V = B, B_0$, the measure $d \mu_V(x)$ coincides with $d x$ as defined in §\[sec:local-measures\].
### Metaplectic group {#sec-2-2-2}
Let $\operatorname{Mp}_2(k)$ denote the metaplectic double cover of $\operatorname{SL}_2(k)$. It is convenient to identify $\operatorname{Mp}_2(k)$ with $\operatorname{SL}_2(k) \times \mu_2$, where $\mu_2 := \{\pm 1\}$, with the group law given by $(s_1,\zeta_1) (s_2,\zeta_2) = (s_1 s_2, \zeta_1 \zeta_2
c(s_1,s_2))$ for a cocycle $c : \operatorname{SL}_2(k) \times \operatorname{SL}_2(k) \rightarrow \{\pm 1\}$ as in [@MR0424695 p.19] or [@nelson-theta-squared §4.4]. Thus $\mu_2$ is a central subgroup of $\operatorname{Mp}_2(k)$, and one has a short exact sequence $1 \rightarrow \mu_2 \rightarrow \operatorname{Mp}_2(k) \xrightarrow{\operatorname{pr}}
\operatorname{SL}_2(k) \rightarrow 1$.
### Weil representation\[sec:local-weil-repn\] {#sec-2-2-3}
For a quadratic space $V$, one has the Weil representation [@MR0165033] on the Schwartz–Bruhat space $\mathcal{S}(V)$: $$\rho_{\operatorname{Weil}}^{\psi,V} : \operatorname{Mp}_2(k)
\times \O(V) \rightarrow \operatorname{GL}(\mathcal{S}(V)).$$ This representation is continuous [@MR0165033 §39] for the standard topologies on all spaces involved and extends to a unitary representation on $L^2(V) := L^2(V,\mu_V)$.
For the remainder of §\[sec:local-weil-repn\], abbreviate $\rho := \rho_{\operatorname{Weil}}^{\psi,V}$. For $s \in \operatorname{Mp}_2(k)$ or $g \in \O(V)$ we abbreviate $\rho(s) := \rho(s,1)$ and $\rho(g) := \rho(1,g)$; one then has $\rho(s)
\rho(g)
=
\rho(g)
\rho(s)$.
Elements $\zeta$ of the central subgroup $\mu_2$ of $\operatorname{Mp}_2(k)$ act by the scalar operators $\rho(\zeta) = (-1)^{\dim(V)}$, so $\rho$ factors through $\operatorname{SL}_2(k)$ if and only if $\dim(V)$ is even.
There is a quartic character $\chi_{\psi,V} : k^\times \rightarrow \mathbb{C}^{(1)}$ and an eighth root of unity $\gamma_{\psi,V} \in \mathbb{C}^{(1)}$ so that, abbreviating $n(b) := (n(b),1),
t(a) := (t(a),1), w := (\begin{pmatrix}
& 1 \\
-1 &
\end{pmatrix}, 1) \in \operatorname{Mp}_2(k)$ for $b \in k$, $a \in k^\times$, one has for $\phi \in \mathcal{S}(V)$, $x \in V$ that $$\begin{aligned}
\rho(n(b))
\phi(x)
&=
\psi(b q_V(x)) \phi(x),
\\
\rho(t(a))
\phi(x)
&=
\chi_{\psi,V}(a) |a|^{\dim(V)/2} \phi(a x),
\\
\rho(w) \phi(x)
&=
\gamma_{\psi,V} \mathcal{F} \phi(x).\end{aligned}$$ If $V = M_2(k)$, then $\chi_{\psi,V}$ is trivial and $\gamma_{\psi,V} = 1$.
Elements $g$ of the orthogonal group $\O(V)$ act by $\rho(g) \phi(v) := \phi(g^{-1} v)$. Suppose that $V = B^0$, so that $\operatorname{Ad}: G \xrightarrow{\cong} \operatorname{SO}(B^0)$. For $g \in G$ and $\phi \in \mathcal{S}(B^0)$, the function $\operatorname{Ad}(g) \phi$ as defined in §\[sec:general-notation\] agrees with $\rho(\operatorname{Ad}(g)) \phi$: both send $x \in B^0$ to $\phi(g^{-1} x g)$.
### Factorization\[sec:factorization-weil-repn\] {#sec-2-2-4}
Let $V$ be a quadratic space that admits an orthogonal decomposition $V = V' \oplus V''$ as a sum of two quadratic spaces. (The relevant example is when $V = B, V' = k, V'' = B^0$.)
Recall the dense inclusion $\mathcal{S}(V') \otimes \mathcal{S}(V'') \rightarrow
\mathcal{S}(V)$ obtained by identifying $\phi ' \otimes \phi '' \in
\mathcal{S}(V') \otimes \mathcal{S}(V'')$ with the function $V' \oplus V'' \ni \alpha ' + \alpha '' \mapsto \phi '(\alpha ') \phi '' (\alpha
'')$.
Given continuous linear operators $T, T', T''$ on $\mathcal{S}(V), \mathcal{S}(V'), \mathcal{S}(V'')$, respectively, write $T = T' \otimes T''$ to denote that $T (\phi ' \otimes \phi '') = T' \phi ' \otimes T'' \phi ''$ for all $\phi ' \in \mathcal{S}(V'), \phi '' \in
\mathcal{S}(V'')$. In this sense, one has $\rho_{\operatorname{Weil}}^{V,\psi}(s) = \rho_{\operatorname{Weil}}^{V',\psi}(s) \otimes
\rho_{\operatorname{Weil}}^{V'',\psi}(s)$ for all $s \in
\operatorname{Mp}_2(k)$.
We denote by $1 \otimes \rho_{\operatorname{Weil}}^{V'',\psi}(s)$ the operator on $\mathcal{S}(V)$ sending $\phi ' \otimes \phi ''$ to $\phi ' \otimes \rho_{\operatorname{Weil}}^{V'',\psi}(s) \phi ''$.
### Extension to similitudes\[sec:defn-local-omega\] {#sec-2-2-5}
The following definitions were inspired by [@MR783511 I.3]. Let $\Omega$ denote the space of functions $\phi : k^\times \times
B \rightarrow \mathbb{C}$ satisfying the conditions:
- For each $t \in k^\times$, the function $\phi[t] : B
\rightarrow \mathbb{C}$ given by $\phi[t](x) := \phi(t,x)$ belongs to the Schwartz–Bruhat space $\mathcal{S}(B)$.
- One has $\phi(z^2 t, x) = \phi(t, z x)$ for all $t,z \in
k^\times$, $x \in B$.
Let $\rho_{\operatorname{Weil}} : \PGL_2(k) \times \operatorname{PGO}(B)
\rightarrow \operatorname{GL}(\Omega)$ denote the representation characterized by the identities: for $s \in \operatorname{SL}_2(k), y \in k^\times, g \in \operatorname{GO}(B)$, $$\begin{aligned}
(\rho_{\operatorname{Weil}}(s) \phi)[t]
&= \rho_{\operatorname{Weil}}^{\psi^t,B}(s) (\phi[t]),
\\
(\rho_{\operatorname{Weil}}(a(y)) \phi)[t]
&=
|y| \phi[t y],
\\
(\rho_{\operatorname{Weil}}(g) \phi)(t,x)
&=
\phi(\lambda(g) t, g^{-1} x)\end{aligned}$$ where $\lambda : \operatorname{GO}(B) \rightarrow k^\times$ denotes the similitude factor.
More generally, if $V$ is any even-dimensional quadratic space, then the representation $\rho_{\operatorname{Weil}}^{\psi,V}$ factors through $\operatorname{SL}_2(k) \times \O(V)$. One can induce it to a representation of $\operatorname{GL}_2(k) \times \operatorname{GO}(V)$ on $\mathcal{S}(k^\times \times V)$, whose isomorphism class is independent of $\psi$. By taking coinvariants for the action by the center, one arrives at a representation of $\PGL_2(k) \times \operatorname{PGO}(V)$. In the relevant case that $V = B$, the representation obtained in that way is realized by $\Omega$. Our global discussion concerns the restriction of $\Omega$ to $\operatorname{SL}_2(k) \times \O_1(k) \times \O(B^0)$, which embeds as the “even subspace” of $\oplus_{\tau \in k^{\times} / k^{\times 2}}
\rho_{\operatorname{Weil}}^{\psi^{\tau},F} \otimes
\rho_{\operatorname{Weil}}^{\psi^{\tau},B^0}$.
Equip $\Omega$ with the invariant hermitian norm $\|.\|_{\Omega}$ given by $$\label{eqn:inner-product-on-Omega-1}
\|\phi\|^2_{\Omega}
:=
\int_{t \in k^\times / k^{\times 2}}
|t|^2
\int_{x \in B}
|\phi|^2(t,x),$$ or equivalently (by , ), $$\label{eqn:inner-product-on-Omega-2}
\|\phi\|^2_{\Omega}
=
\int_{g \in G}
|\operatorname{nr}(g)|^2
\int_{t \in k^\times}
|t|^2
|\phi|^2(t,g)
\, \frac{d t}{|t|} \, d g.$$
Define $\mathfrak{S} : \Omega \rightarrow \Omega$ and $\operatorname{Ad}(g) : \Omega \rightarrow \Omega$ ($g \in G$) by applying the general definition (§\[sec:general-notation\]) to the second coordinate, so that for $\phi \in \Omega$ and $(t,x) \in k^\times \times B$, one has $(\mathfrak{S} \phi)[t] = \mathfrak{S} (\phi[t])$, $\mathfrak{S} \phi(t,x) =
(\phi(t,x) + \phi(t,\operatorname{tr}(x)- x))/2$, $\operatorname{Ad}(g) \phi = \rho_{\operatorname{Weil}}(\operatorname{Ad}(g)) \phi$, $(\operatorname{Ad}(g) \phi)[t] = \operatorname{Ad}(g) (\phi[t])$, $\operatorname{Ad}(g) \phi(t,x) =
\phi(t,g^{-1} x g)$.
### The distinguished element {#sec:dist-elem}
Suppose temporarily that $k$ is non-archimedean and that $B \cong M_2(k)$ is split; similar considerations apply to non-split $B$, but we do not need them. The *distinguished element* $\phi^0 \in \Omega$ (with respect to the chosen maximal order $R \subset B$) is then defined by $$\phi^0(t,x)
:=
\frac{
\int_{z \in k^\times}
1_{R}(z x)
1_{\mathfrak{o}^\times}(z^{-2} t)
\, \frac{d z}{|z|}
}
{
\int_{z \in k^\times}
1_{\mathfrak{o}^\times}(z)
\, \frac{d z}{|z|}
}.$$ Note that $\phi^0$ takes values in $\{0,1\}$. One verifies directly that $\phi^0$ is $\PGL_2(\mathfrak{o})
\times K'$-invariant, where $K' {\leqslant}\operatorname{PGO}(B)$ denotes the image of the $\O(B)$-stabilizer of $R$. Moreover, $\mathfrak{S} \phi^0 = \phi^0$.
\[lem:norm-of-distinguished-vector-in-Omega-local\] $\|\phi^0\|^2_{\Omega}
= \operatorname{vol}(R)$.
Since $\phi$ takes values in $\{0 ,1\}$, one has $\|\phi^0\|^2_{\Omega}
=
\int_{t \in k^\times / k^{\times 2}}
|t|^2
\int_{x \in B}
\phi^0(t,x)$. By expanding the definition of $\phi^0$ and using that $\int_{x \in B} 1_R(z x)
= |z|^{-4} \operatorname{vol}(R)$ and that $|t|^2 |z|^{-4} 1_{\mathfrak{o}^\times}(z^{-2} t)
= 1_{\mathfrak{o}^\times}(z^{-2} t)$, our task reduces to showing that $\int_{t \in k^\times / k^{\times 2}}
\int_{z \in k^\times}
1_{\mathfrak{o}^\times}(z^{-2} t)
\, \frac{d z}{|z|}
=
\int_{z \in k^\times}
1_{\mathfrak{o}^\times}(z^{-2} t)
\, \frac{d z}{|z|}$, as follows from .
Let $K {\leqslant}G$ denote the image of $R^\times$. We may then fix an identification $B = M_2(k)$ under which $G = \PGL_2(k)$, $R = M_2(\mathfrak{o})$, $K = \PGL_2(\mathfrak{o})$.
\[lem:formula-for-how-Ad-acts-on-distinguish-element-inner-products\] Let $\phi_1, \phi_2 \in \mathbb{C} \phi^0$. Let $g \in K a(\varpi^m) K$ for some $m \in \mathbb{Z}_{{\geqslant}0}$ (see §\[sec:cartan-decomposition\]). Then $\langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle_{\Omega}
= q^{-m} \langle \phi_1, \phi_2 \rangle_{\Omega}$.
We expand the definitions and use that $\operatorname{vol}(g R g^{-1} \cap R) = q^{-m} \operatorname{vol}(R)$.
Generic representations of $\operatorname{PGL}_2$ {#sec-2-3}
-------------------------------------------------
We refer to [@MR1431508 §4.4, §4.6] for details on and proofs of the facts collected here. Let $\pi$ be an irreducible representation of $\PGL_2(k)$. Recall that $\pi$ is *generic* if it admits a Whittaker model $\mathcal{W}(\pi,\psi)$, consisting of $W : \PGL_2(k) \rightarrow \mathbb{C}$ satisfying $W(n(x) g) = \psi(x) W(g)$. It then admits a Kirillov model $\mathcal{K}(\pi,\psi)$, consisting of $W : k^\times \rightarrow \mathbb{C}$ of the form $W(y) := W'(a(y))$ for some $W' \in \mathcal{W}(\pi,\psi)$. The vector space $\mathcal{K}(\pi,\psi)$ is independent of $\psi$ and contains $C_c^\infty(k^\times)$. Recall that $\pi$ is *unramified* if the space $\pi^{\PGL_2(\mathfrak{o})}$ of $\PGL_2(\mathfrak{o})$-invariant vectors in $\pi$ is nonzero, and that in that case, $\dim(\pi^{\PGL_2(\mathfrak{o})}) = 1$.
Suppose for the remainder of §\[sec-2-3\] that $\pi$ is generic and unramified. Let $\psi^0$ be an unramified unitary character of $k$. There is then a unique $\PGL_2(\mathfrak{o})$-invariant vector $W_{\pi}^0$ in the Kirillov model $\mathcal{K}(\pi,\psi^0)$ of $\pi$ for which $W_{\pi}^0(1) = 1$. There is a unique unordered pair $\{\alpha, \beta \}$ of complex numbers, the *Satake parameters* of $\pi$, so that for $y \in k^\times$ with $|y| = q^{-n}$, $$\label{eq:explicit-formula-W-pi-0}
W_{\pi}^0(y)
=
|y|^{1/2}
\sum _{\substack{
i, j \in \mathbb{Z}_{{\geqslant}0}:
i + j = n
}
}
\alpha^{i} \beta^j
=
1_{\mathfrak{o}^\times}(y) |y|^{1/2}
\frac{\alpha^{n+1} - \beta^{n+1}}{\alpha - \beta }.$$ One has in general $\alpha \beta = 1$; if moreover $\pi$ is unitary, then either $|\alpha| = |\beta| = 1$ or $\alpha,\beta \in (-q^{1/2}, q^{1/2}) \subseteq
\mathbb{R}$. The *adjoint $L$-factor* is defined for $s \in \mathbb{C}$ by $$L(\operatorname{ad}\pi,s)
:= (1 - \alpha \beta^{-1} q^{-s})^{-1}
(1 - q^{-s})^{-1}
(1 - \alpha^{-1} \beta q^{-s})^{-1}.$$
\[lem:local-computation-norm-of-whittaker-newvector\] If $\pi$ is unitary and ${\mathrm{Re}}(s) {\geqslant}0$, then $L(\operatorname{ad}\pi, s)$ is finite, and one has the identity $$\label{eq:local-computation-norm-of-whittaker-newvector}
\int_{y \in k^\times}
|W_{\pi}^0(y)|^2
|y|^s
\, \frac{d y}{|y|}
= \frac{L(\operatorname{ad}\pi,1 + s)}{\zeta_k(2 + 2 s)}
\Delta_{\psi}^{-1/2}
\frac{\zeta_k(1 + s)}{\zeta_k(1)}$$ in which the LHS converges absolutely.
This is a standard calculation.[^2]
Representations of $G$ {#sec-2-4}
----------------------
Let $\pi$ be an irreducible representation of $G$. Define a compact open subgroup $J {\leqslant}G$ in the following two cases:
- if $k$ is non-archimedean, take for $J {\leqslant}G$ the image of the unit group $R^\times$ of the chosen maximal order $R \subseteq B$;
- if $k$ is real and $B$ is non-split, set $J := G$.
In either case, set $\operatorname{vol}(J) := \int_{g \in G} 1_J(g)$ and $e_J := \operatorname{vol}(J)^{-1} 1_J \in C_c^\infty(G)$.
### Hecke kernels and theta kernels\[sec:hecke-kernels-local\] {#sec-2-4-1}
Assume that $k$ is non-archimedean. For $y \in k^\times$, the *normalized Hecke kernel* $T_y \in C_c^\infty(J \backslash G / J)$ is defined to be the element with the property that $|y|^{-1} \operatorname{vol}(J) T_y$ is the characteristic function of the image in $G$ of the subset $\{b \in R : |\operatorname{nr}(b)| = |y| \}$ of $B^\times$. For example, if $y \in \mathfrak{o}^\times$, then $T_y = e_J$.
\[lem:relation-distinguished-elt-hecke-kernel\] Let $y \in k^\times$, $g \in G$. Choose $\tilde{g} \in B^\times$ with image $g$. Then $$\rho_{\operatorname{Weil}}(a(y))
\phi^0(\operatorname{nr}(\tilde{g})^{-1}, \tilde{g})
=
|y| \phi^0(y \operatorname{nr}(\tilde{g})^{-1}, \tilde{g})
= \operatorname{vol}(J)
T_y(g)
$$ where $\phi^0 \in \Omega$ is the distinguished element (§\[sec:dist-elem\]).
We must verify that $$\label{eq:formula-for-hecke-kernel}
|y|^{-1} \operatorname{vol}(J) T_y(g) =
\frac{
\int_{z \in k^\times}
1_R(z \tilde{g}) 1_{\mathfrak{o}^\times}(y \operatorname{nr}(z \tilde{g})^{-1})
\, \frac{d z}{|z|}
}
{
\int_{z \in k^\times}
1_{\mathfrak{o}^\times}(z)
\, \frac{d z}{|z|}
}.$$ Let $g \in G$. The RHS of is independent of $\tilde{g}$, and both sides take values in $\{0,1\}$. The RHS of is nonzero iff its integrand is nonzero for some $z \in k^\times$, i.e., iff for some $z \in k^\times$ the element $b := z \tilde{g}$ lies in $R$ and $|\operatorname{nr}(b)| = |y|$, i.e., iff the LHS of is nonzero.
### Hecke functionals and standard $L$-factors {#sec-2-4-2}
Continue to assume that $k$ is non-archimedean. Recall that $\pi$ is *unramified* if the space $\pi^{J}$ of $J$-invariant vectors in $\pi$ is nonzero; it is known then that $\dim(\pi^J) = 1$.
Suppose for remainder of §\[sec-2-4-2\] that $\pi$ is unramified. There is then a unique functional $\lambda_\pi : C_c^\infty(J \backslash G / J) \rightarrow
\mathbb{C}$ so that $\pi(T) v = \lambda_\pi(T) v$ for all $T \in C_c^\infty(J \backslash G / J), v \in \pi^J$. We may evaluate this functional on the elements $T_y$ attached above to $y \in k^\times$:
- If $B$ is split, then there is a unique unordered pair $\{\alpha,\beta\}$ of complex numbers (the *Satake parameters*) satisfying $\alpha \beta = 1$ so that with $|y| = q^{-n}$, $$\label{eq:explicitf-romula-lambda-pi-tY}
\lambda_\pi(T_y)
=
|y|^{1/2}
\sum _{\substack{
i, j \in \mathbb{Z}_{{\geqslant}0}:
i + j = n
}
}
\alpha^{i} \beta^j
=
1_{\mathfrak{o}^\times}(y)
|y|^{1/2}
\frac{\alpha^{n+1} - \beta^{n+1}}{\alpha - \beta }$$ (see e.g. [@MR1431508 §4.6]).
- If $B$ is non-split, then there is a unique unramified quadratic character $\eta$ of $k^\times$ so that $\lambda_\pi(T_y)
= |y| \eta(y)$.
The *standard $L$-factor* is then the meromorphic function defined for $s \in \mathbb{C}$ by $$L(\pi,s) := \begin{cases}
(1 - \alpha q^{-s})^{-1} (1 - \beta q^{-s})^{-1} & \text{ if $B$ is split}, \\
(1 - \eta(\varpi) q^{-s-1/2})^{-1} & \text{ if $B$ is non-split}.
\end{cases}$$
### The local Jacquet–Langlands correspondence {#sec-2-4-3}
The Jacquet–Langlands lift $\pi_{\operatorname{JL}}$ of $\pi$ is an irreducible representation of $\PGL_2(k)$ attached to $\pi$. The following properties of the association $\pi \mapsto \pi_{\operatorname{JL}}$ are relevant for us:
- $\pi_{\operatorname{JL}}$ is generic if (and only if) either
- $B$ is non-split, or
- $B$ is split and $\dim(\pi) > 1$.
- If $B$ is split, then $\pi_{\operatorname{JL}}$ corresponds to $\pi$ under the isomorphism $G \cong \PGL_2(k)$. In particular, if $\pi$ is unramified, then so is $\pi_{\operatorname{JL}}$, and Satake parameters (see §\[sec-2-3\], §\[sec-2-4-2\]) are preserved.
Assume now that that $k$ is non-archimedean, that $B$ is split, that $\pi$ is unramified, and that $\dim(\pi) > 1$. Then $\pi_{\operatorname{JL}}$ is generic and unramified. Let $W^0_\pi : k^\times \rightarrow \mathbb{C}$ denote the function attached to $\pi_{\operatorname{JL}}$ in §\[sec-2-3\]. By and , $$\label{eqn:key-local-identity-relating-whittaker-and-hecke}
W_\pi^0(y) = \lambda_\pi(T_y).$$
### Local integrals {#sec:local-integrals-for-rallis-ipf}
Assume first that $k$ is non-archimedean, that $\pi$ is unramified, and that $\pi$ is unitary. Retain the notation of §\[sec-2-4-2\].
\[lem:local-rallis-ipf-unram-calc\] Suppose that $B$ is split and that $\dim(\pi) > 1$.
1. $|\alpha|, |\beta| < q^{1/2}$. In particular, $L(\pi,\tfrac{1}{2})$ is finite.
2. Let $\phi_1, \phi_2$ belong to the line $\mathbb{C} \phi^0$ spanned by the distinguished element $\phi^0 \in \Omega$. Let $v_1, v_2 \in \pi^J$. Then the identity $$\label{eqn:local-rallis-ipf-unram-calc}
\int_{g \in G}
\langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle
\langle g v_1, v_2 \rangle
=
\frac{L(\pi,\frac{1}{2})}{\zeta_k(2)}
\operatorname{vol}(J) \langle \phi_1, \phi_2 \rangle
\langle v_1, v_2 \rangle$$ holds, with the LHS converging absolutely.
For (i), see [@MR1431508 Thm 4.6.7]. For (ii), the convergence follows from §\[sec:local-convergence-lemmas\]. Let $\{\alpha,\beta\}$ denote the Satake parameters of $\pi$ and set $t_1 := \alpha q^{-1/2}, t_2 := \beta q^{-1/2}$, so that $L(\pi,\tfrac{1}{2})^{-1} = (1 - t_1) (1-t_2)$. The Macdonald formula [@MR1431508 Thm 4.6.6] says that $\langle g v_1, v_2 \rangle = (u_1 t_1^m + u_2 t_2^m)
\langle v_1, v_2 \rangle$, where $$u_1 :=
\frac{1}{1 + q^{-1}}\frac{1 - q^{-1} \beta/\alpha
}{1 - \beta / \alpha
},
\quad
u_2 :=
\frac{1}{1 + q^{-1}}\frac{1 - q^{-1} \alpha / \beta }{1 -
\alpha / \beta }.$$ By the Cartan decomposition and Lemma \[lem:formula-for-how-Ad-acts-on-distinguish-element-inner-products\], we obtain $$\int_{g \in G} \langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle
\langle g v_1, v_2 \rangle =
\operatorname{vol}(J) \langle \phi_1, \phi_2 \rangle \langle v_1, v_2 \rangle
\Sigma,$$ where $\Sigma := \sum_{i=1,2} \sum_{m {\geqslant}0} (1 + 1_{m>0} q^{-1})
t_i^m$. We compute that $\sum_{i=1,2} u_i (1 + q^{-1} t_i) ( 1 - t_i)^{-1} =
L(\pi,\tfrac{1}{2}) \Sigma '$ with $\Sigma ' := \sum_{i=1,2} u_i (1 + q^{-1} t_i) ( 1 -
t_{i'})^{-1}$, $\{i, i'\} = \{1,2\}$. Direct calculation gives $\Sigma ' = \zeta_k(2)^{-1}$, as required.
Suppose now that $B$ is non-split, so that $\pi$ is the one-dimensional representation corresponding to the character $G \ni g \mapsto \eta(\operatorname{nr}(g)) \in \{\pm 1\}$, as in §\[sec-2-4-2\]. Let $v_1,v_2 \in \pi$. Recalling that $[G:J] = 2$, we have $$\label{eqn:local-rallis-ipf-integral-nonsplit-finite}
\int_{g \in G} \langle \operatorname{Ad}(g) e_J, e_J \rangle_{L^2(G)}
\langle g v_1, v_2 \rangle
=
\langle v_1, v_2 \rangle
\cdot
\begin{cases}
0 & \text{ if $\eta$ is nontrivial,} \\
2 & \text{ if $\eta$ is trivial.}
\end{cases}$$
Suppose finally that $k \cong \mathbb{R}$, that $B$ is non-split, and that $\pi$ is trivial. For $v_1,v_2 \in \pi$, one then has $$\label{eqn:local-rallis-ipf-integral-nonsplit-real}
\int_{g \in G} \langle \operatorname{Ad}(g) e_J, e_J \rangle_{L^2(G)}
\langle g v_1, v_2 \rangle
=
\langle v_1, v_2 \rangle.$$
Global preliminaries {#sec-3}
====================
In this section we collect those preliminaries for the proof of Theorem \[thm:main-estimate-general-variance\] whose discussion makes sense independently of that proof.
Let $F$ be a number field with adele ring $\mathbb{A}$, let $B$ be a quaternion algebra over $F$, and let $\psi$ be a nontrivial unitary character of $\mathbb{A}/F$.
Generalities {#sec-3-1}
------------
### Notation {#sec-3-1-1}
We denote by $\mathcal{O}_F$ or simply $\mathcal{O}$ the ring of integers in $F$. We denote by $\mathfrak{p}$ a place of $F$, finite or infinite. A subscripted $\mathfrak{p}$ denotes completion; for example, $\mathcal{O}_\mathfrak{p}$ denotes the ring of integers of $F_{\mathfrak{p}}$ if $\mathfrak{p}$ is finite. For a finite set of places $S$, a subscripted $S$ denotes a product taken over $S$, such as in $F_S := \prod_{\mathfrak{p} \in S} F_\mathfrak{p}, B_S :=
\prod_{\mathfrak{p} \in S} B_\mathfrak{p}$.
The character $\psi$ factors as $\psi(x) = \prod \psi_\mathfrak{p}(x_\mathfrak{p})$, where $\psi_\mathfrak{p}$ is a nontrivial unitary character of $F_\mathfrak{p}$. For a place $\mathfrak{p}$, let $\zeta_\mathfrak{p} := \zeta_{F_\mathfrak{p}}$ denote the local Euler factor. Let $\xi_F(s) := \prod \zeta_\mathfrak{p}(s)$ denote the Dedekind zeta function (absolutely convergent for ${\mathrm{Re}}(s) > 1$) and $\xi_F^*(1) := \operatorname{res}_{s \rightarrow 1} \xi_F(s)$ its residue. For a finite set $S$ of places that contains the infinite places, let $\zeta_F^{(S)}(s) :=
\prod_{\mathfrak{p} \in S}
\zeta_\mathfrak{p}(s)$ denote the partial Dedekind zeta function.
### Groups
For an algebraic $F$-group $\mathbf{G}$ we write $G := \mathbf{G}(F), G_\mathfrak{p} :=
\mathbf{G}(F_\mathfrak{p})$, $G_\mathbb{A} := \mathbf{G}(\mathbb{A})$, $G_S := \mathbf{G}(F_S) = \prod_{\mathfrak{p} \in S} G_\mathfrak{p}$, $[G] := G \backslash G_\mathbb{A}$. This notation applies notably to the $F$-group $\operatorname{{\mathbf P}{\mathbf B}}^\times$ given by $\operatorname{{\mathbf P}{\mathbf B}}^\times(A) := (B \otimes_F A)^\times/A^\times$ and also to the $F$-groups ${{\mathbf P}{\mathbf G}{\mathbf L}}_2$, ${{\mathbf S}{\mathbf L}}_2$. We similarly abbreviate $[\operatorname{Mp}_2] := \operatorname{SL}_2(F) \backslash \operatorname{Mp}_2(\mathbb{A})$ (see §\[sec:global-metaplectic-gp\]).
### Measures\[sec:global-measures\] {#sec-3-1-2}
When $\mathbf{G}$ is semisimple, we equip $G_\mathbb{A}$ and $[G]$ with Tamagawa measures. Then $\operatorname{vol}([\operatorname{SL}_2]) = 1$ and $\operatorname{vol}([\PGL_2]) = \operatorname{vol}([\PB^\times]) = 2$. Denote by $\langle , \rangle_{G}$ the corresponding inner product on $L^2([G])$; we omit the subscripted $G$ if it is clear by context.
For each place $\mathfrak{p}$, the character $\psi_\mathfrak{p}$ induces (via the recipe of §\[sec:local-measures\]) a Haar measure on $F_\mathfrak{p}$, $B_\mathfrak{p}$, $F_\mathfrak{p}^{\times} / F_{\mathfrak{p}}^{\times
2}$, $\PB^\times_\mathfrak{p}$; we equip $\mathbb{A}, B_\mathbb{A}$, $\mathbb{A}^{\times} / \mathbb{A}^{\times 2}$ and $\PB^\times_\mathbb{A}$ with the corresponding restricted product measures. (This defines the Tamagawa measure on $\PB^\times_\mathbb{A}$.) The quotient measures on $\mathbb{A}/F$ and $B_\mathbb{A}/B$ are then probability measures. We likewise equip finite products such as $F_S$ or $\PB^\times_S$ with product measures.
We equip $\mathbb{A}^\times$ with the regularized product of the measures constructed in §\[sec:local-measures\]: for a factorizable function $f = \prod f_\mathfrak{p} \in C_c^\infty(\mathbb{A}^\times)$ for which $f_\mathfrak{p} = 1_{\mathcal{O}_\mathfrak{p}^\times}$ for almost all finite primes $\mathfrak{p}$, we set $$\int_{y \in \mathbb{A}^\times}
f(y) \, \frac{d y}{|y|}
:=
\frac{1}{\xi_F^*(1)}
\prod_{\mathfrak{p}}
\zeta_\mathfrak{p}(1)
\int_{y \in F_\mathfrak{p}^\times}
f_\mathfrak{p}(y)
\, \frac{d y}{|y|}.$$ We thereby obtain a quotient Haar $\frac{d y}{|y|}$ on $\mathbb{A}^\times / F^\times$ whose pushforward under $|.| : \mathbb{A}^\times / F^\times \rightarrow
\mathbb{R}^\times_+$ is the standard Haar measure $\frac{d t}{|t|}$ on $\mathbb{R}^\times_+$, where $d t$ denotes Lebesgue measure.
The quotient measure on the discrete group $F^{\times } / F^{\times 2}$ compatible with the squaring map is half the counting measure, i.e., for finitely-supported $f : F^\times \rightarrow \mathbb{C}$, one has $\sum_{x \in F^\times } f(x)
= \frac{1}{2} \sum_{y \in F^\times / F^{\times 2}}
(\sum_{z \in F^\times} f(y z^2) )$. On $\mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}$, we take the quotient measure induced by the exact sequence $1 \rightarrow F^\times / F^{\times 2}
\rightarrow
\mathbb{A}^\times / \mathbb{A}^{\times 2} \rightarrow
\mathbb{A}^\times / F^\times \mathbb{A}^{\times 2} \rightarrow
1$, where $F^{\times} / F^{\times 2}$ is equipped with half the counting measure. Thus for $f \in C_c(\mathbb{A}^\times / \mathbb{A}^{\times 2})$, $$\label{eq:integral-formula-involving-squares}
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}}
\frac{1}{2} \sum_{a \in F^\times / F^{\times 2}}
f(a y)
= \int_{\mathbb{A}^\times / \mathbb{A}^{\times 2}} f.$$ By decomposing the Haar on $\mathbb{A}^\times$ in two ways, one finds for $f \in C_c(\mathbb{A}^\times / F^\times)$ that $\int_{\mathbb{A}^\times / F^\times} f = \int_{x \in
\mathbb{A}^\times / F^{\times } \mathbb{A}^{\times 2}} \int_{y
\in \mathbb{A}^\times / F^\times} f(x y^2)$; moreover, $\operatorname{vol}(\mathbb{A}^{\times} / F^\times \mathbb{A}^{\times 2}) = 2$. Finally, for $f \in C_c^\infty([\PGL_2])$, $$\label{eq:sl2-vs-pgl2-integrals}
\int_{[\PGL_2]} f
=
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}}
\int_{s \in [\operatorname{SL}_2]}
f(s a(y)).$$
### The $\Xi$-function\[sec:Xi-global\] {#sec-3-1-3}
Fix a maximal compact subgroup $K = \prod K_\mathfrak{p} {\leqslant}\PB^\times_{\mathbb{A}}$. Let $\Xi : \PB^\times_{\mathbb{A}} \rightarrow \mathbb{C}$ be the product $\Xi(g) := \prod \Xi_\mathfrak{p}(g_\mathfrak{p})$ of the functions $\Xi_\mathfrak{p}$ on $\PB^\times_\mathfrak{p}$ attached in §\[sec:local-Xi\] to the factors $K_\mathfrak{p}$.
### Conventions {#sec:conventions-cusp-forms}
A *cusp form* is a smooth vector in the Hilbert space $L^2_{\operatorname{cusp}}([G])$ of square-integrable cuspidal functions. A *cuspidal automorphic representation* $\pi$ of $G_\mathbb{A}$ is the space of smooth vectors in an irreducible subrepresentation of $L^2_{\operatorname{cusp}}([G])$.
Automorphic forms on $\PGL_2$ {#sec-3-3}
-----------------------------
### Fourier expansions {#sec:aut-forms-fourier-exp}
\[sec-3-3-1\] Let $\varphi : [\PGL_2] \rightarrow \mathbb{C}$ be a smooth function. It admits the Fourier expansion $\varphi(n(x) a(y)) = c_{\varphi}(y) + \sum_{\tau \in F^\times} \psi(\tau x)
W_{\varphi}(\tau y)$, where $c_{\varphi}(y) := \int_{x \in \mathbb{A}/F} \varphi(n(x) a(y))$ denotes the constant term and $$W_{\varphi}(y) := \int_{x \in \mathbb{A}/F} \psi(- x) \varphi(n(x)
a(y))$$ denotes the (diagonal restriction of) the Whittaker function. The standard Borel subgroup of $\PGL_2(\mathbb{A})$ has dense image in $[\PGL_2]$, so $\varphi$ is determined by the values $\varphi(n(x) a(y))$ for $x \in \mathbb{A}, y \in \mathbb{A}^\times$. Recall that $\varphi$ is *cuspidal* if $c_{\varphi} = 0$; in that case, $\varphi$ is determined by $W_\varphi$.
### Kirilov model {#sec-3-3-2}
Let $\pi \subseteq L^2([\PGL_2])$ be a cuspidal automorphic representation; it is (the smooth completion of) a restricted tensor product $\otimes \pi_\mathfrak{p}$, where $\pi_\mathfrak{p}$ is a generic for every $\mathfrak{p}$ and unramified for almost all finite $\mathfrak{p}$. Let $\mathcal{K}(\pi,\psi) := \{W_\varphi : \varphi \in \pi \}$. The natural map $\pi \rightarrow \mathcal{K}(\pi,\psi)$ is a linear isomorphism under which the pure tensors in $\pi$ correspond to the factorizable functions $W(y) = \prod W_\mathfrak{p}(y_\mathfrak{p})$, where $W_\mathfrak{p}$ belongs to the local Kirillov model $\mathcal{K}(\pi_\mathfrak{p},\psi_\mathfrak{p})$ and satisfies $W_\mathfrak{p} = W_{\pi_\mathfrak{p}}^0$ (see §\[sec-2-4-3\]) for almost all finite $\mathfrak{p}$.
\[lem:inner-product-formula-petersson-vs-kirillov\] Let $S$ be a finite set of places of $F$ that contains all infinite places as well as any places $\mathfrak{p}$ at which $\pi$ ramifies. Let $\varphi \in \pi$. The integral $I(s) := \int_{y \in \mathbb{A}^\times}
|W_\varphi(y)|^2 |y|^s \, \frac{d y}{|y|}$ converges absolutely for complex numbers $s$ with positive real part, extends to a meromorphic function on the complex plane, and satisfies $$\label{eqn:rs-ipf}
2 \operatorname{res}_{s \rightarrow 0} I(s) = \|\varphi\|^2.$$
See for instance [@michel-2009 Lem 2.2.3].[^3]
### Adjoint $L$-function {#sec:adjoint-l-function}
For $\pi$, $S$ as in the lemma of §\[sec-3-3-2\], the *partial adjoint $L$-function* is defined for ${\mathrm{Re}}(s) > 1$ by the absolutely-convergent Euler product $L^{(S)}(\operatorname{ad}\pi,s) := \prod_{\mathfrak{p} \notin S} L(\operatorname{ad}\pi_\mathfrak{p},s)$; it continues meromorphically to the complex plane, and is holomorphic for (at least) ${\mathrm{Re}}(s) {\geqslant}1$ (see [@MR533066]).
Automorphic forms on $\PB^\times$ {#sec-3-4}
---------------------------------
### Jacquet–Langlands lifts {#sec-3-4-1}
Let $\pi = \otimes \pi_\mathfrak{p} \subseteq L^2([\PB^\times])$ be a cuspidal automorphic representation with $\dim(\pi) > 1$. By [@MR0333081 Prop 4], $$\label{eqn:generic-at-each-place-if-not-1-diml}
\text{$\dim(\pi_\mathfrak{p}) > 1$ for any prime $\mathfrak{p}$ at
which $B$ splits.}$$ The Jacquet–Langlands lift $\pi_{\operatorname{JL}} = \otimes \pi_{\operatorname{JL},\mathfrak{p}} \subseteq
L^2([\PGL_2])$ is the unique cuspidal automorphic automorphic representation for which $\pi_{\operatorname{JL},\mathfrak{p}} = (\pi_\mathfrak{p})_{\operatorname{JL}}$ for each place $\mathfrak{p}$. If $\mathfrak{p}$ is a finite prime at which $B$ splits and for which $\pi_\mathfrak{p}$ is unramified, then $(\pi_{\operatorname{JL}})_\mathfrak{p}$ is unramified. The association $\pi \mapsto \pi_{\operatorname{JL}}$ is injective.
### The pretrace formula\[sec:pretrace-formula\] {#sec-3-4-2}
Assume that $B$ is non-split, so that $[\PB^\times]$ is compact. Fix a maximal compact subgroup $K$ of $\PB^\times_\mathbb{A}$. The pretrace formula asserts that for $f \in C_c^\infty(\PB_\mathbb{A}^\times)$ and $x \in \PB^\times_\mathbb{A}$, $$\label{eq:pretrace-formula-general}
\sum_{\pi}
\sum_{\varphi}
\overline{\varphi (x)}
\pi(f) \varphi(x)
= \sum_{\gamma \in \PB^\times} f(x^{-1} \gamma x),$$ where $\pi$ traverses the irreducible subrepresentations of $L^2([\PB^\times])$ and $\varphi$ traverses an orthonormal basis $\mathcal{B}(\pi)$ of $\pi$ consisting of $K$-isotypic vectors. Only finitely many summands on the RHS of are nonzero, while the condition on $\mathcal{B}(\pi)$ implies that the LHS of converges absolutely, or indeed, rapidly: Let $C(\pi) := \prod_\mathfrak{p} C((\pi_\mathfrak{p})_{\operatorname{JL}}) \in
\mathbb{R}_{{\geqslant}1}$ denote the analytic conductor of $\pi$ (see e.g. [@michel-2009 §3.1.8, §4.1.4]). Then for each $A {\geqslant}0$, one has[^4] $$\label{eq:rapid-convergence-of-pretrace-formula}
\sum_{\pi} C(\pi)^A
\sum_{\varphi}
|\overline{\varphi}(x) \pi(f) \varphi(x)|
< \infty.$$ Note also that there exists $A_0 > 3$ so that (see e.g. [@michel-2009 (2.15)]). $$\label{eq:polynomial-growth-of-reps}
\sum_{\pi} C(\pi)^{-A_0}
< \infty$$
Let $S$ be a set of places containing the infinite ones. Let $R \subseteq B$ be a maximal order. For each $\mathfrak{p} \notin S$, let $J_\mathfrak{p} {\leqslant}\PB^\times_\mathfrak{p}$ denote the image of $R_\mathfrak{p}^\times$, as in §\[sec-2-4\], and set $J := \prod_{\mathfrak{p} \notin S} J_\mathfrak{p}$. Suppose that $f = f_S \otimes (\otimes_{\mathfrak{p} \notin S}
T_{y_\mathfrak{p}})$ for some $f_S \in C_c^\infty(\PB_S^\times)$ and $y \in \mathbb{A}^\times$, where $T_{y_\mathfrak{p}}$ is the Hecke kernel as defined in §\[sec:hecke-kernels-local\] relative to $J_\mathfrak{p}$. The formula then specializes to $$\sum_\pi
(\sum_{\varphi}
\overline{\varphi(x)}
\pi(f_S) \varphi(x))
\prod_{\mathfrak{p} \notin S}
\lambda_{\pi_\mathfrak{p}}(T_{y_\mathfrak{p}})
=
\sum_{\gamma \in \PB^\times}
f_S(x_S^{-1} \gamma x_S)
\prod_{\mathfrak{p} \notin S}
T_{y_\mathfrak{p}}(x_\mathfrak{p}^{-1} \gamma x_\mathfrak{p}),$$ where $\pi \subseteq L^2([\PB^\times])$ now traverses the subrepresentations that are unramified outside $S$ (i.e., that contain a nonzero $J$-fixed vector) and $\varphi$ traverses an orthonormal basis of $K$-isotypic vectors for the $J$-fixed subspace $\pi^J$ of $\pi$.
### $L$-functions {#sec:standard-l-function}
Let $\pi \subseteq L^2([\PB^\times])$ be a cuspidal automorphic representation with $\dim(\pi) > 1$. Let $S$ be a finite set of places containing all infinite places as well as any places at which either $B$ or $\pi$ ramifies.
The *partial standard $L$-function* is defined for ${\mathrm{Re}}(s) > 1$ by the absolutely-convergent Euler product $L^{(S)}(\pi,s) := \prod_{\mathfrak{p} \notin S}
L(\pi_\mathfrak{p},s)$; it continues meromorphically to the complex plane, and is holomorphic for (at least) ${\mathrm{Re}}(s) {\geqslant}1/2$ (see e.g. [@MR1431508 §3.5]).
Set $L^{(S)}(\operatorname{ad}\pi,s) := L^{(S)}(\operatorname{ad}\pi_{\operatorname{JL}}, s)$ (see §\[sec:adjoint-l-function\]). By [@HL94] (cf. [@2009arXiv0904.2429B §2.9]), one has $$\label{eq:HL}
C(\pi)^{-\eps} \ll_{\eps} L^{(S)}(\operatorname{ad}\pi,1) \ll_{\eps}
C(\pi)^{\eps}
\text{ for each } \eps > 0.$$
Theta functions {#sec-3-2}
---------------
### Metaplectic group\[sec:global-metaplectic-gp\] {#sec-3-2-1}
Let $\operatorname{Mp}_2(\mathbb{A})$ denote the metaplectic double cover of $\operatorname{SL}_2(\mathbb{A})$; it fits into a short exact sequence $1 \rightarrow \mu_2 \rightarrow \operatorname{Mp}_2(\mathbb{A})
\xrightarrow{\operatorname{pr}} \operatorname{SL}_2(\mathbb{A})$. We may identify it with $\operatorname{SL}_2(\mathbb{A}) \times \mu_2$ with the group law given by $(s_1,\zeta_1) (s_2,\zeta_2) = (s_1 s_2, \zeta_1 \zeta_2
c(s_1,s_2))$, where $c$ is the product of the cocycles from §\[sec:global-metaplectic-gp\]. We identify $\operatorname{SL}_2(F)$ with its image under the unique splitting $\operatorname{SL}_2(F)
\hookrightarrow \operatorname{Mp}_2(\mathbb{A})$.
We may similarly define $\operatorname{Mp}_2(F_S)$ as a double cover of $\operatorname{SL}_2(F_S)$ for any collection $S$ of places of $F$.
### Quadratic spaces {#sec-3-2-2}
One defines quadratic spaces $V$ over $F$ as in §\[sec:local-quadratic-spaces\]. The relevant examples are still $V = B, B^0, F$. We equip $V_\mathbb{A}$ with the $(\psi,b_V)$-self dual measure $\mu_V$. That measure is the product of the measures $\mu_{V_\mathfrak{p}}$ on the local spaces $V_{\mathfrak{p}}$ attached to $\psi_\mathfrak{p}$, and is independent of $\psi$: it assigns volume one to a fundamental domain for $V_\mathbb{A}/V$.
### Weil representation\[sec:weil-repn-global\] {#sec-3-2-3}
For a quadratic space $V$ over $F$, the Schwartz–Bruhat space $\mathcal{S}(V_\mathbb{A})$ factors as the (completed) restricted tensor product $\mathcal{S}(V_\mathbb{A}) = \otimes
\mathcal{S}(V_\mathfrak{p})$. The Weil representation $\rho_{\operatorname{Weil}}^{\psi,V} : \operatorname{Mp}_2(\mathbb{A})
\times \O(V_\mathbb{A}) \rightarrow
\operatorname{GL}(\mathcal{S}(V_\mathbb{A}))$ is given by $\rho_{\operatorname{Weil}}^{\psi,V} = \otimes
\rho_{\operatorname{Weil}}^{\psi_{\mathfrak{p}},V_{\mathfrak{p}}}$ in the evident sense.
We may similarly define a Weil representation $\rho_{\operatorname{Weil}}^{\psi,V} : \operatorname{Mp}_2(F_S)
\times \O(V_S)
\rightarrow \operatorname{GL}(\mathcal{S}(V_S))$ for a finite set $S$ of places of $F$.
### Theta kernels\[sec:theta-kernels\] {#sec-3-2-5}
Let $V$ be a quadratic space over $F$. For $\phi \in
\mathcal{S}(V_\mathbb{A})$, $s \in \operatorname{Mp}_2(\mathbb{A})$ and $g \in \O(V_\mathbb{A})$, set $\theta_{\psi}(\phi)(s,g)
:=
\sum_{x \in V}
\rho_{\operatorname{Weil}}^{\psi,V}(s,g) \phi(x)$. The sum converges absolutely and defines a smooth function $\theta_{\psi}(\phi) : [\operatorname{Mp}_2]
\times [\O(V)] \rightarrow \mathbb{C}$. We employ notation such as $\theta_{\psi}(\phi;s,g) := \theta_{\psi}(\phi)(s,g)$. Observe that $$\label{eq:equivariance-theta-kernel}
\theta_{\psi}(\phi;s s', g g')
= \theta_{\psi}(\rho_{\operatorname{Weil}}^{\psi,V}(s',g') \phi;s,g)$$
### Elementary theta functions\[sec:elem-theta-fns\] {#sec-3-2-6}
Let $V = F$, regarded as a quadratic subspace of $B$ as in §\[sec:local-quadratic-spaces\]. In that case, we abbreviate $\O_1(F) := \O(V) \cong \{\pm 1\}$. For $\phi \in \mathcal{S}(V_\mathbb{A}) = \mathcal{S}(\mathbb{A})$, we denote also by $\theta_{\psi}(\phi)$ the elementary theta function on $[\operatorname{Mp}_2]$ obtained by restricting to the first factor the theta kernel defined in §\[sec:theta-kernels\], thus $\theta_{\psi}(\phi)(s) := \theta_{\psi}(\phi)(s,1)
=
\sum_{x \in F}
\rho_{\operatorname{Weil}}^{\psi,F}(s) \phi(x)$. By , $$\label{eq:equivariance-etf}
\rho_{\operatorname{reg}}(s) \theta_{\psi}(\phi)
= \theta_{\psi}(\rho_{\operatorname{Weil}}^{\psi,F}(s) \phi)
\text{ for }
s \in \operatorname{Mp}_2(\mathbb{A}).$$ The $\O_1(F)$-invariance of the theta kernel says that for $\phi \in \mathcal{S}(\mathbb{A})$, $$\label{eqn:O1-invariance-elementary-theta-fn}
\text{$\theta_{\psi}(\phi) = \theta_{\psi}(\phi_-)$
with $\phi_-(x) := \phi(-x)$.}$$
### Ternary theta lifts {#sec-3-2-7}
Suppose $V = B^0$. Given $\Psi : [\PB^\times] \rightarrow \mathbb{C}$ and $\phi \in \mathcal{S}(B^0_\mathbb{A})$ and $s \in \operatorname{Mp}_2(\mathbb{A})$, set $\theta_{\psi}(\phi,\Psi;s)
:=
\int_{g \in [\PB^\times]}
\Psi(g)
\theta_{\psi}(\phi;s,\operatorname{Ad}(g))$ where $\operatorname{Ad}: \PB^\times_\mathbb{A} \xrightarrow{\cong}
\operatorname{SO}(B^0_\mathbb{A})$ is the isomorphism induced by the notation of §\[sec:general-notation\]. If $\Psi$ is a cusp form, then the integral converges absolutely and defines a cusp form $\theta_{\psi}(\phi,\Psi) : [\operatorname{Mp}_2] \rightarrow \mathbb{C}$. By , $$\label{eq:equivariance-ttf}
\rho_{\operatorname{reg}}(s) \theta_{\psi}(\phi,\Psi)
= \theta_{\psi}(\rho_{\operatorname{Weil}}^{\psi,B^0}(s) \phi, \Psi)
\text{ for }
s \in \operatorname{Mp}_2(\mathbb{A}),$$ $$\label{eq:equivariance-ttf-2}
\theta_{\psi}(\operatorname{Ad}(g)\phi, \rho_{\operatorname{reg}}(g) \Psi)
=
\theta_{\psi}(\phi, \Psi)
\text{ for }
g \in \PB^\times_\mathbb{A}.$$
### Factorization {#sec-3-2-8}
If the quadratic space $V$ decomposes as the direct sum $V' \oplus V''$ of quadratic subspaces, then the factorization of the Weil representation (§\[sec:factorization-weil-repn\]) implies the factorization of theta functions: for $g = g' \times
g''
\in \O(V'_\mathbb{A}) \times \O(V_\mathbb{A} '')
{\leqslant}\O(V_\mathbb{A})$ and $\phi = \phi ' \otimes \phi '' \in \mathcal{S}(V_\mathbb{A})$ with $\phi ' \in \mathcal{S}(V'_\mathbb{A}),
\phi '' \in \mathcal{S}(V_\mathbb{A} '')$ (see §\[sec:factorization-weil-repn\]), $$\label{eqn:Factorization-of-theta-fns}
\theta_{\psi}(\phi;s,g)
=
\theta_{\psi}(\phi ';s,g')
\theta_{\psi}(\phi '';s,g'').$$ With $\phi '_-$ as in and notation as in §\[sec:general-notation\], one has $$\label{eq:action-of-S-on-pure-tensors-for-B-0}
\operatorname{Ad}(g) \phi = \phi ' \otimes \operatorname{Ad}(g) \phi ''$$ $$\label{eq:action-of-S-on-pure-tensors-for-B}
\mathfrak{S} \phi = \phi '_- \otimes \phi '',$$
Equidistribution of products of pairs of elementary theta functions\[sec:equid-prod-pairs-theta\] {#sec-3-5}
-------------------------------------------------------------------------------------------------
The purpose of this section is to recall and apply some results from [@nelson-theta-squared]. Let $\tau_1,\tau_2 \in F^\times$. Throughout this section we regard $\psi,\tau_1,\tau_2,F,B$ as fixed: implied constants may depend upon them without explicit mention. We assume also (for technical convenience) that $B$ is non-split.
### Some asymptotic notation {#sec:some-asympt-notat}
Given a topological vector space $\mathcal{S}$, we adopt the convention (similar to “big O notation”) of denoting by $\mathcal{C}(\phi)$ any quantity depending continuously upon $\phi \in \mathcal{S}$; the continuity is assumed uniform in all auxiliary parameters except those explicitly labelled “fixed.” The space $\mathcal{S}$ itself is always regarded as fixed, of course. This convention applies in particular to Schwartz–Bruhat spaces of finite-dimensional vector spaces over local fields, over finite products of local fields, or over adele rings.
Similarly to the “$\eps$-convention” of analytic number theory, we allow the precise meaning of $\mathcal{C}(\phi)$ to change from one occurrence to the next. When we specifically wish to distinguish between several such quantities, we use the notation $\mathcal{C} '(\phi), \mathcal{C} ''(\phi)$, and so on.
For example, let $V$ be a vector space over $F$ (always assumed finite-dimensional). Let $\hat{F}$ denote the ring of finite adeles, so that $\mathbb{A} = F_\infty \times \hat{F}$ with $F_\infty := \prod_{\mathfrak{p} | \infty} F_\mathfrak{p}$. Similarly, write $V_\mathbb{A} = V_\infty \times \hat{V}$. The Schwartz–Bruhat space $\mathcal{S}(V_\mathbb{A})$ factors as the algebraic tensor product $\mathcal{S}(V_\infty) \otimes
\mathcal{S}(\hat{V})$. Suppose given some quantities $a(\phi;t_1,t_2)$ and $b(\phi;t_1,t_2)$ depending upon $\phi \in \mathcal{S}(V_\mathbb{A})$ and some auxiliary parameters $t_1,t_2$. The notation $$\label{eq:example-a-phi-bounded-by-C-phi}
a(\phi;t_1,t_2)
\ll b(\phi;t_1,t_2) \mathcal{C}(\phi) \text{ for fixed $t_2$ }
$$ means that for each $t_2$ and each $\phi_f \in \mathcal{S}(\hat{V})$ there is a finite collection $\mathcal{P}$ of polynomials on $V_\infty$ and a finite collection $\mathcal{D}$ of translation-invariant differential operators on $V_\infty$ (thus $\mathcal{D}$ consists of linear combinations of monomials $\frac{\partial }{\partial x_{i_1}}
\dotsb \frac{\partial }{\partial x_{i_n}}$ with respect to some coordinates $x_j : V_\infty \rightarrow \mathbb{R}$) so that for all $\phi_\infty \in \mathcal{S}(V_\infty)$ and all $t_1$, $$|
a(\phi_\infty \otimes \phi_f;t_1,t_2)
|
{\leqslant}|b(\phi_\infty \otimes \phi_f;t_1,t_2)|
\sum_{P \in \mathcal{P}}
\sum_{D \in \mathcal{D}}
\|P D \phi_\infty \|_{L^\infty(V^\infty)}.$$ One could just as well write “the functional $\mathcal{S}(V_\mathbb{A}) \ni \phi \mapsto
a(\phi;t_1,t_2)/b(\phi;t_1,t_2)$ is defined and continuous, uniformly in $t_1$,” but that would be stilted.
It would be “better” to work not with the notation $\mathcal{C}$ but instead with a system of adelic Sobolev norms on $\mathcal{S}(V_\mathbb{A})$, such as those attached by the recipe of [@michel-2009 §2] and [@nelson-theta-squared §4.6, §5.3] to the basic Weil representation of the metaplectic group of the symplectic space $V^*_\mathbb{A} \oplus V_\mathbb{A}$. The advantage of doing so would be to obtain polynomial dependence in Theorem \[thm:main-result-for-microlocal-stuff\] upon the levels of $\Psi_1, \Psi_2$. The disadvantages would be to lengthen the article and introduce technical overhead irrelevant to our primary aims.
### Simple estimates for lattice sums {#sec:simple-estimates-for-lattice-sums}
\[lem:cheap-lattice-sum-bound-Rn-Zn\] Let $n \in \mathbb{Z}_{{\geqslant}0}$. Let $A {\geqslant}0$ and $t_0 > 0$ be fixed. For each $\phi \in \mathcal{S}(\mathbb{R}^n)$ and $t > t_0$, one has $\sum_{v \in \mathbb{Z}^n - \{0\}} |\phi(t v)| \ll |t|^{-A} \mathcal{C}(\phi)$.
The LHS of the required estimate is bounded by $$C
|t|^{-A}
\sup_{x \in \mathbb{R}^n}
|x|^{n+1+A} |\phi(x)|$$ with $C := |t_0|^{-(n+1)} \sum_{v \in \mathbb{Z}^n - \{0\}} |v|^{-(n+1+A)} < \infty$.
\[lem:cheap-lattice-sum-bound-adelic\] Let $V$ be a vector space over $F$. Let $A {\geqslant}0$ and $t_0 > 0$ be fixed. For $\phi \in \mathcal{S}(V_\mathbb{A})$ and $y \in \mathbb{A}^\times$ with $|y| > t_0$, one has $\sum_{v \in V - \{0\}}
|\phi(y v)|
\ll |y|^{-A} \mathcal{C}(\phi)$.
Observe first that the action of $\mathbb{A}^\times$ on $\mathcal{S}(V_\mathbb{A})$ by dilation is continuous. Let $\mathbb{A}^{(1)} := \{y \in \mathbb{A}^\times : |y| = 1\}$ denote the subgroup of norm one ideles. By the compactness of $\mathbb{A}^{(1)} / F^\times$ and the previous observation, it suffices to consider the case that $y_\mathfrak{p} = 1$ for all finite $\mathfrak{p}$ and $y_\mathfrak{p} = t$ for all infinite $\mathfrak{p}$, where $t \in \mathbb{R}_+^\times$ satisfies $t > t_1 := t_0^{1/[F:\mathbb{Q}]}$. Each $\phi_f \in \mathcal{S}(\hat{V})$ is bounded and satisfies $\operatorname{supp}(\phi_f) \cap V \subseteq L$ for some lattice $L \subseteq V$, so it suffices to show for each fixed $A {\geqslant}0$ and fixed lattice $L \subseteq V$ that for all $\phi \in \mathcal{S}(V_\infty)$ and $t > t_1$, one has $\sum_{v \in L - \{0\}} |\phi(t v)| = O(|t|^{-A} \mathcal{C}(\phi))$. By choosing a $\mathbb{Z}$-basis of $L$, we reduce to Lemma \[lem:cheap-lattice-sum-bound-Rn-Zn\].
### Simple estimates for theta functions {#sec-3-5-4}
Recall that the *Iwasawa decomposition* asserts that each $s \in \operatorname{SL}_2(\mathbb{A})$ may be written in the form $s = n(x) t(y) k$, where $x \in \mathbb{A}, y \in \mathbb{A}^\times$ and $k$ belongs to the standard maximal compact subgroup of $\operatorname{SL}_2(\mathbb{A})$. The decomposition is not unique, but the quantities $x$ and $|y|$ depend only upon $s$.
Recall that the *height* function $\operatorname{ht}: [\operatorname{Mp}_2] \rightarrow \mathbb{R}_{>0}$ factors through $\operatorname{ht}: [\operatorname{SL}_2] \rightarrow \mathbb{R}_{>0}$ where it is given for $g \in [\operatorname{SL}_2]$ by $\operatorname{ht}(g) := \max_{\gamma \in \operatorname{SL}_2(F)} \operatorname{ht}_\mathbb{A}(\gamma
g)$, where $\operatorname{ht}_{\mathbb{A}} : \operatorname{SL}_2(\mathbb{A}) \rightarrow
\mathbb{R}_{>0}$ is defined with respect to the Iwasawa decomposition $s = n(x) t(y) k$ by $\operatorname{ht}_{\mathbb{A}}(s) := |y|^{1/2}$. One has $\int_{[\operatorname{SL}_2]} \operatorname{ht}^{1-\eps} < \infty$ for $\eps > 0$. *Reduction theory* says that the image of $\operatorname{ht}$ is bounded from below by some $c > 0$ depending only upon $F$.
Recall that the nontrivial unitary character $\psi$ of $\mathbb{A}/F$ is regarded as fixed.
\[lem:crude-bound-for-cuspidal-theta-functions\] Let $A {\geqslant}0$ be fixed. Let $\Psi \in L^1( [\PB^\times])$ with $\langle \Psi, 1 \rangle = 0$. Let $\phi \in \mathcal{S}(B_\mathbb{A}^0)$. For $s \in \operatorname{Mp}_2(\mathbb{A})$, one has $\theta_{\psi}(\phi,\Psi;s)
\ll \operatorname{ht}(s)^{-A}
\mathcal{C}(\phi)
\|\Psi \|_{L^1}$.
Since $\Psi$ has mean zero, $$\theta_{\psi}(\phi,\Psi;s)
= \int_{g \in [\PB^\times]}
\Psi(g)
\sum_{v \in V - \{0\}}
\rho_{\operatorname{Weil}}^{\psi,V}(s,\operatorname{Ad}(g))
\phi(v).$$ By the Iwasawa decomposition and reduction theory, we may assume that $s = n(x) t(y) k$ with $|y| \gg 1$. Since $B$ is non-split, we may fix a compact subset $U$ of $\PB^\times_\mathbb{A}$ containing a fundamental domain for $[\PB^\times]$. Then $$|\theta_{\psi}(\phi,\Psi;s)|
{\leqslant}\|\Psi \|_{L^1}
|y|^{3/2}
\sup_{g \in U}
\sum_{v \in B^0 - \{0\}}
|\rho_{\operatorname{Weil}}^{\psi,B^0}(k,\operatorname{Ad}(g))
\phi(y v)|.$$ Since the Weil representation is continuous ([@MR0165033 §39]), we may reduce to the case $k = 1$ and $g = 1$, in which the required estimate follows from Lemma \[lem:cheap-lattice-sum-bound-adelic\] of §\[sec:simple-estimates-for-lattice-sums\].
\[lem:crude-bound-for-elementary-theta-functions\] For $\phi \in \mathcal{S}(\mathbb{A})$ and $s \in
\operatorname{Mp}_2(\mathbb{A})$, one has $\theta_{\psi}(\phi;s)
\ll
\operatorname{ht}(s)^{1/4}
\mathcal{C}(\phi)$.
We argue as in the proof of Lemma \[lem:crude-bound-for-cuspidal-theta-functions\], but take into account the contribution from $0 \in F$ to the definition of $\theta_{\psi}(\phi)$.
### Main estimate: the case of pure tensors {#sec-3-5-3}
\[lem:main-result-for-theta-squared\] Let $\phi_1', \phi_2' \in \mathcal{S}(\mathbb{A})$ and $\phi_1'', \phi_2'' \in \mathcal{S}(B_\mathbb{A}^0)$. Let $\Psi_1, \Psi_2 : [\PB^\times] \rightarrow \mathbb{C}$ be integrable functions of mean zero. Let $\tau_1, \tau_2 \in F^\times$ be fixed. Abbreviate $\theta_i :=
\theta_{\psi^{\tau_i}}(\phi_i')$ and $h_i := \theta_{\psi^{\tau_i}}(\phi_i'',\Psi_i)$. Then for all $s \in \operatorname{Mp}_2(\mathbb{A})$, $$\langle
\theta_1 \cdot \rho_{\operatorname{reg}}(s) h_1,
\theta_2 \cdot \rho_{\operatorname{reg}}(s) h_2
\rangle
= \langle \theta_1, \theta_2 \rangle
\langle h_1, h_2 \rangle
+ O(\Xi(s) \prod_{i=1,2} \mathcal{C}(\phi_i')
\mathcal{C}(\phi_i'')
\|\Psi_i\|_{L^1}
).$$
The main result of [@nelson-theta-squared] gives an estimate nearly of the required shape, but instead with the error term $\Xi(s) \mathcal{S}(\phi_1') \mathcal{S}(\phi_2')
\mathcal{S}^{\mathbf{X}}(h_1 \overline{h_2})$, where $\mathcal{S}, \mathcal{S}^{\mathbf{X}}$ are adelic Sobolev norms whose relevant properties we recall shortly. By the cuspidality of $h_1, h_2$ and axioms (S3b) and (S4e) of [@michel-2009], we may replace the expression $\mathcal{S}^{\mathbf{X}}(h_1 \overline{h_2})$ first with $\mathcal{S}^{\mathbf{X}}(h_1) \mathcal{S}^{\mathbf{X}}(h_2)$ and then with $\mathcal{S}(h_1) \mathcal{S}(h_2)$. Our task thereby reduces to showing for $i=1,2$ that $\mathcal{S}(\phi_i') \ll \mathcal{C}(\phi_i')$ and $\mathcal{S}(h_i) \ll \mathcal{C}(\phi_i'') \|\Xi\|_{L^1}$.
To that end, we must recall something about the norms $\mathcal{S}$ (see [@michel-2009 §2] and [@nelson-theta-squared §4.6, §5.3] for details). They have the form $\mathcal{S}(v) = \|\Delta^d v\|$, where $d \in \mathbb{Z}_{{\geqslant}0}$ is fixed but large enough, and $\Delta$ acts linearly on the space of smooth vectors in any unitary representation $\pi$ of $\operatorname{Mp}_2(\mathbb{A})$. If $\pi$ factors as $\pi_\infty \otimes \pi_{\operatorname{fin}}$, then likewise $\Delta = \Delta_\infty \otimes \Delta_{\operatorname{fin}}$. If $\pi$ is the Weil representation on the Schwartz–Bruhat space $\mathcal{S}(V_\mathbb{A}) = \mathcal{S}(V_\infty) \otimes
\mathcal{S}(\hat{V})$ of a quadratic space $V$ over $F$, then $\Delta_{\infty}$ is a finite order differential operator with polynomial coefficients, hence is continuous for the Schwartz topology; since any linear operator on $\mathcal{S}(\hat{V})$ is continuous, it follows that $\Delta$ defines a continuous operator on $\mathcal{S}(V_\mathbb{A})$ for the Schwartz–Bruhat topology.
The continuity of $\Delta$ implies that $\mathcal{S}(\phi_i')
= \|\Delta^d \phi_i'\|
\ll \mathcal{C}(\phi_i)$, giving one of the two required estimates. The operator $\Delta$ is moreover natural in that for an equivariant morphism $f : \pi \rightarrow \pi '$ of $\operatorname{Mp}_2(\mathbb{A})$-representations, one has $\Delta \circ f = f \circ \Delta$. Thus $$\mathcal{S}(h_i)
= \|\Delta^d h_i\|
= \|
\theta_{\psi^{\tau_i}}(\Delta^d \phi_i'', \Psi_i)
\|.$$ By Lemma \[lem:crude-bound-for-cuspidal-theta-functions\] of §\[sec-3-5-4\], we have $\mathcal{S}(h_i) \ll \mathcal{C}(\Delta^d \phi_i'')
\|\Psi_i\|_{L^1}$. The continuity of $\Delta$ gives $\mathcal{C}(\Delta^d \phi_i'') \ll \mathcal{C} '(\phi_i'')$, and the required estimate follows.
### Factorization {#sec:quantitative-factorization}
Let $V', V''$ be vector spaces over $F$ and $V := V' \oplus V''$.
\[lem:quantitative-factorization\] Let $\ell : \mathcal{S}(V_\mathbb{A} ') \otimes
\mathcal{S}(V_\mathbb{A} '') \rightarrow \mathbb{C}$ be an algebraic linear functional on the algebraic tensor product of Schwartz–Bruhat spaces satisfying an estimate of the form $$\ell(\phi' \otimes \phi '')
\ll \mathcal{C}'(\phi ') \mathcal{C}''(\phi '').$$ Then $\ell$ extends to a continuous functional $\ell : \mathcal{S}(V_\mathbb{A}) \rightarrow \mathbb{C}$ satisfying $$\ell(\phi) \ll \mathcal{C}(\phi)$$ for all $\phi \in \mathcal{S}(V_\mathbb{A})$, where $\mathcal{C}$ depends only upon $\mathcal{C} '$ and $\mathcal{C} ''$.
This is essentially the Schwartz kernel theorem, as extended by Bruhat [@MR0140941 §5]. For concreteness, we sketch a proof. There exists $\nu = \prod \nu_\mathfrak{p} \in C_c^\infty(V'_\mathbb{A})$ such that $\sum_{\lambda \in V'} \nu(\lambda + x)^2 = 1$ for all $x \in V'_\mathbb{A}$, thus $\nu$ is a “square-root of a partition of unity.” Let $m \in \mathbb{A}^\times$ be large enough that the map $\operatorname{supp}(\nu) \rightarrow V'_\mathbb{A} / m V'$ is injective. Set $\Lambda_1 := V'$ and $\Lambda_2 := m^{-1} V'$. Fix non-degenerate bilinear forms $V'_\mathbb{A} \otimes V'_\mathbb{A}
\rightarrow \mathbb{A}$ and $V''_\mathbb{A} \otimes V''_\mathbb{A} \rightarrow
\mathbb{A}$; denote them by $(x,y) \mapsto x \cdot y \in \mathbb{A}$. By Fourier inversion on the compact group $V'_\mathbb{A} / m V'$, there exists $c > 0$ so that for all $f \in C^\infty(V'_\mathbb{A})$ and $x \in V'_\mathbb{A}$, $$\label{eq:fourier-expansion-of-half-of-partitino-of-unity}
\nu(x)^2 f(x) = c \nu(x) \sum_{\lambda_2 \in \Lambda_2}
\psi(\lambda_2 \cdot x) \int_{y \in V'_\mathbb{A}} \nu(y) f(y)
\psi(-\lambda_2 \cdot y).$$ For $t' \in V'_\mathbb{A}, t'' \in V''_\mathbb{A}$, we apply to $f(x) := \phi( (t'' - \lambda_1) + x)$ and set $x := t' + \lambda_1$, giving $$\begin{aligned}
\phi(t' + t'')
&=
\sum_{\lambda_1 \in \Lambda_1}
\nu(t' + \lambda_1)^2
\phi( (t'' - \lambda_1) + (t' + \lambda_1))
\\
&=
\sum_{\substack{
\lambda_1 \in \Lambda_1 \\
\lambda_2 \in \Lambda_2
}
}
\phi_{\lambda_1,\lambda_2}'(t')
\phi_{\lambda_1,\lambda_2}''(t''),
\end{aligned}$$ where $$\phi_{\lambda_1,\lambda_2}'(t')
:=
c \nu (t' + \lambda_1)
\psi(\lambda_2 \cdot (t' + \lambda_1)),$$ $$\label{eq:}
\phi_{\lambda_1,\lambda_2}''(t'')
= \int_{y \in \mathbb{A}}
\nu(y) \phi(t'' - \lambda_1 + y) \psi(- \lambda_2 \cdot y).$$ “Integration by parts” gives readily that $$\sum
|\ell(\phi'_{\lambda_1,\lambda_2} \otimes \phi''_{\lambda_1,\lambda_2})|
\ll
\sum
|\mathcal{C} '(\phi'_{\lambda_1,\lambda_2})
\mathcal{C} ''(\phi''_{\lambda_1,\lambda_2})|
\ll
\mathcal{C}(\phi)$$ for suitable $\mathcal{C}(\phi)$. It follows that $\ell$ admits the required extension and satisfies the required estimate.
### Main estimate: the general case\[sec:main-estimate-for-equidistribution-pairs-theta\] {#sec-3-5-5}
Temporarily denote by $\mathcal{A}_0$ denote the space of integrable functions $\Psi : [\PB^\times] \rightarrow \mathbb{C}$ of mean zero. Let $\mathcal{E}_{\tau_1,\tau_2} : \mathcal{S}(B_\mathbb{A}) \otimes
\mathcal{S}(B_\mathbb{A})
\otimes \mathcal{A}_0 \otimes \mathcal{A}_0 \rightarrow
\mathbb{C}$ denote the sesquilinear form given for $\phi_i = \phi_i' \otimes \phi_i'' \in
\mathcal{S}(B_\mathbb{A})$ with $\phi_1',\phi_2' \in \mathcal{S}(\mathbb{A}),
\phi_1'',\phi_2'' \in \mathcal{S}(B_\mathbb{A}^0)$ by $$\begin{aligned}
\mathcal{E}_{\tau_1,\tau_2}(\phi_1,
\phi_2,
\Psi_1, \Psi_2)
:=
\langle
\theta_1 h_1, \theta_2 h_2 \rangle
- \langle \theta_1, \theta_2 \rangle
\langle h_1, h_2 \rangle,\end{aligned}$$ where we abbreviate $\theta_i := \theta_{\tau_i}(\phi_i')$ and $h_i := \theta_{\tau_i}(\phi_i'',\Psi_i)$. (The definition makes sense: *a priori* estimates as in §\[sec-3-5-4\] and the density of $\mathcal{S}(\mathbb{A}) \otimes \mathcal{S}(B_\mathbb{A}^0)$ in $\mathcal{S}(B_\mathbb{A})$ allow us to extend $\mathcal{E}_{\tau_1,\tau_2}$ continuously from its initial domain.)
\[prop:main-error-estimate-global-adelic-general\] For $\phi_1, \phi_2 \in \mathcal{S}(B_\mathbb{A}),
\Psi_1,\Psi_2 \in \mathcal{A}_0$ and $s \in \operatorname{Mp}_2(\mathbb{A})$, one has with $\rho_0^{\tau}(s) := \rho_{\operatorname{Weil}}^{\psi^{\tau}, B^0}(s)$ the estimate $$\mathcal{E}_{\tau_1,\tau_2}((1 \otimes \rho_0^{\tau_1}(s))\phi_1,
(1 \otimes \rho_0^{\tau_2}(s))\phi_2,
\Psi_1,\Psi_2)
\ll \Xi(s)
\prod_{j=1,2}
\Sob(\phi_j)
\|\Psi_j\|_{L^1}.$$ The implied constant and the uniformity of the continuity of $\Sob(\phi_j)$ depend at most upon $\psi,\tau_1,\tau_2,F,B$. The operator $1 \otimes \rho^{\tau_1}(s)$ is defined as in §\[sec:factorization-weil-repn\].
The lemma of §\[sec:quantitative-factorization\] reduces the general case of Proposition \[prop:main-error-estimate-global-adelic-general\] to the special case in which $\phi_i = \phi_i' \otimes \phi_i''$ for $i=1,2$, which follows from the lemma of §\[sec-3-5-3\] upon recalling from that $\theta_{\psi_\tau}$ intertwines $\rho_{0}^{\tau}$ with $\rho_{\operatorname{reg}}$.
### Invariance properties {#sec:equivariance-summary-for-E-tau-tau}
We record these for later use.
\[lem:equivariance-summary-for-E-tau-tau\] For $g_1, g_2 \in \PB^\times_\mathbb{A}$ and $s \in \operatorname{Mp}_2(\mathbb{A})$, one has $$\begin{aligned}
\mathcal{E}_{\tau_1,\tau_2}(\phi_1, \phi_2, \Psi_1, \Psi_2)
&=
\mathcal{E}_{\tau_1,\tau_2}(\mathfrak{S} \phi_1, \phi_2,
\Psi_1, \Psi_2)
\\
&=
\mathcal{E}_{\tau_1,\tau_2}(\phi_1, \mathfrak{S} \phi_2,
\Psi_1, \Psi_2)
\\
&=
\mathcal{E}_{\tau_1,\tau_2}(\operatorname{Ad}(g_1) \phi_1, \operatorname{Ad}(g_2) \phi_2,
\rho_{\operatorname{reg}}(g_1) \Psi_1, \rho_{\operatorname{reg}}(g_2)\Psi_2)
\\
&=
\mathcal{E}_{\tau_1,\tau_2}(\rho_{\operatorname{Weil}}^{\psi^{\tau_1},B}(s) \phi_1, \rho_{\operatorname{Weil}}^{\psi^{\tau_2},B}(s) \phi_2,
\Psi_1, \Psi_2).
\end{aligned}$$
The first two identities follow from and , the remaining from , , , , and the translation invariance of the Petersson inner product.
Simillitude theta functions\[sec:non-traditional-theta-lifts\] {#sec-3-2-9}
--------------------------------------------------------------
### Weil representation\[sec:similitudes-global\] {#sec-3-2-4}
For each place $\mathfrak{p}$ of $F$, let $\Omega_{\mathfrak{p}}$ denote the representation of $\PGL_2(F_{\mathfrak{p}}) \times \operatorname{GO}(B_\mathfrak{p})$ attached as in §\[sec:defn-local-omega\] to the tuple $(F_\mathfrak{p},B_{\mathfrak{p}},\psi_{\mathfrak{p}})$. Let $\Omega$ denote the restricted tensor product of the spaces $\Omega_{\mathfrak{p}}$ with respect to the distinguished elements, which we denote now by $\phi_{\mathfrak{p}}^0 \in
\Omega_{\mathfrak{p}}$. We may and shall identify $\Omega$ with the space of functions $\phi : \mathbb{A}^\times \times B_\mathbb{A} \rightarrow
\mathbb{C}$ such that
- For each $t \in \mathbb{A}^\times$, the function $\phi[t] : B_\mathbb{A} \rightarrow \mathbb{C}$ given by $\phi[t](x) := \phi(t,x)$ belongs to the Schwartz–Bruhat space $\mathcal{S}(B_\mathbb{A})$;
- $\phi(z^2 t, x) = \phi(t, z x)$ for all $z,t \in \mathbb{A}^\times, x \in B_\mathbb{A}$.
- There is a compact subset $C$ of $\mathbb{A}^\times / \mathbb{A}^{\times 2}$ such that $\phi[t] = 0$ for all $t \notin C$ (i.e., for all $t \in \mathbb{A}^\times$ whose image in $\mathbb{A}^\times / \mathbb{A}^{\times 2}$ lies outside $C$);
- There is an open subgroup $U$ of $\mathbb{A}^\times / \mathbb{A}^{\times 2}$ such that $\phi[t u] = \phi[t]$ for all $t \in \mathbb{A}^\times, u \in U$.
We equip $\Omega$ with the invariant hermitian norm $\|.\|$ obtained by tensoring those on the factors $\Omega_\mathfrak{p}$, thus $$\label{eqn:inner-product-on-Omega-1-adelic}
\|\phi\|^2_{\Omega}
:=
\int_{t \in \mathbb{A}^\times / \mathbb{A}^{\times 2}}
|t|^2
\int_{x \in B_\mathbb{A}}
|\phi|^2(t,x).$$ The group $\PGL_2(\mathbb{A}) \times \operatorname{GO}(B_\mathbb{A})$ acts on $\Omega$ by the representation $\rho_{\operatorname{Weil}}$ obtained as the restricted tensor product of those defined in §\[sec:defn-local-omega\]. We define $\mathfrak{S} : \Omega \rightarrow \Omega$ and (for $g \in \PB^\times_\mathbb{A}$) $\operatorname{Ad}(g) : \Omega \rightarrow \Omega$ as in §\[sec:defn-local-omega\]. Note that $\mathfrak{S}$ does *not* preserve pure tensors: for $\phi
= \otimes \phi_\mathfrak{p} \in \Omega$, $$\begin{aligned}
\mathfrak{S} \phi (t,x)
&= (\phi (t,x) + \phi (t,x -
\operatorname{nr}(x)))/2,
\\
\otimes \mathfrak{S} \phi_\mathfrak{p}(x,t)
&= \prod (\phi (t_\mathfrak{p},x_\mathfrak{p}) + \phi (t_\mathfrak{p},x_\mathfrak{p} -
\operatorname{nr}(x_\mathfrak{p} )))/2,\end{aligned}$$ and these are not in general the same. They *do* coincide if $\# \{ \mathfrak{p} : \mathfrak{S} \phi_\mathfrak{p}
\neq \phi_\mathfrak{p} \}
{\leqslant}1$.
### Theta functions {#theta-functions}
For $\phi \in \Omega$, $s \in \PGL_2(\mathbb{A}), g \in \operatorname{GO}(B_\mathbb{A})$, set $$\label{defn:theta-knerle-for-Omega}
\Theta(\phi;s,g)
:=
\frac{1}{2} \sum_{\tau \in F^\times / F^{\times 2}}
\sum_{x \in B}
\rho_{\operatorname{Weil}}(s,g)
\phi(\tau,x).$$ The sum is well-defined, converges absolutely and defines a smooth function $\Theta(\phi)$ on $[\PGL_2] \times [\operatorname{GO}(B)]$. For a cusp form $\Psi : [\PB^\times] \rightarrow \mathbb{C}$ and $s \in \PGL_2(\mathbb{A})$, set $$\Theta(\phi,\Psi;s) := \int_{g \in [\PB^\times]}
\Psi(g) \Theta(\phi;s,\operatorname{Ad}(g)).$$ The integral (together with similar integrals below) converges absolutely and defines a cusp form $\Theta(\phi,\Psi) : [\PGL_2] \rightarrow \mathbb{C}$.
$\Theta(\phi,\Psi)$ is not a theta lift in the traditional sense: the integral in its definition is with respect to the orthogonal group of $B^0$ rather than that of $B$.
### Fourier expansion {#sec:four-expans}
Let $\phi \in \Omega$, and let $\Psi : [\PB^\times] \rightarrow
\mathbb{C}$ be a cusp form.
For $x \in \mathbb{A}$, $y \in \mathbb{A}^\times$, one has $$\Theta(\phi,\Psi;n(x) a(y))
=
\sum_{\tau \in F^\times}
\psi(\tau x)
W(\Theta(\phi,\Psi), \tau y)$$ where $W(\Theta(\phi,\Psi),y)
:=
\int_{g \in [\PB^\times]}
\Psi(g)
\sum_{\gamma \in \PB^\times}
|y|
\phi(y \operatorname{nr}(\gamma)^{-1}, g^{-1} \gamma g)$.
By direct unfolding as in [@MR0333081], one has for $g \in
\PB^\times_{\mathbb{A}}$ that $$\begin{split}
\Theta(\phi, n(x) a(y);\operatorname{Ad}(g))
&=
\frac{1}{2}
\sum_{\tau \in F^\times / F^{\times 2}}
|y| \phi(0,\tau y) \\
&\quad + \sum_{\tau \in F^\times}
\psi(\tau x)
W(\Theta(\phi),\operatorname{Ad}(g),\tau y),
\end{split}$$ where $W(\Theta(\phi),\operatorname{Ad}(g),y)
:=
\sum_{\gamma \in \PB^\times}
|y|
\phi(y \operatorname{nr}(\gamma)^{-1}, g^{-1} \gamma g)$. We conclude by integrating against $\Psi$.
### Restriction to $\operatorname{SL}_2$
Let $\phi, \Psi$ be as above.
Suppose that for each $y \in \mathbb{A}^\times$, one has $\phi [y] = \phi '[y] \otimes \phi ''[y]$ for some $\phi '[y] \in \mathcal{S}(\mathbb{A})$, $\phi ''[y] \in \mathcal{S}(B_\mathbb{A}^0)$. Then for $y \in \mathbb{A}^\times$ and $s \in \operatorname{SL}_2(\mathbb{A})$, one has $$\label{eqn:simliitude-theta-expand-as-sum-of-pure-tensors-and-over-a}
\Theta(\phi,\Psi;s a(y))
=
\frac{1}{2}
\sum_{\tau \in F^\times \backslash F^{\times 2}}
|y|
\theta_{\psi^\tau}(\phi'[\tau y];s)
\theta_{\psi^\tau}(\phi''[\tau y],\Psi;s).$$
We derive first using that for $g \in \O(B_{\mathbb{A}})$, $$\Theta(\phi;s a(y),g)
=
\frac{1}{2}
\sum_{\tau \in F^\times \backslash F^{\times 2}}
|y|
\theta_{\psi^\tau}(\phi[a y];s,g),$$ hence by that for $g \in \O(B_\mathbb{A}^0)$, $$\Theta(\phi;s a(y),g)
=
\frac{1}{2}
\sum_{\tau \in F^\times \backslash F^{\times 2}}
|y|
\theta_{\psi^\tau}(\phi'[\tau y];s)
\theta_{\psi^\tau}(\phi''[\tau y];s,g).$$ We integrate against $\Psi$ to conclude.
### Unfolding the inner product {#sec:unfold-inner-product-nontraditional-theta-over-sl2}
Let $\phi_1, \phi_2 \in \Omega$. Let $\Psi_1, \Psi_2 : [\PB^\times] \rightarrow \mathbb{C}$ be cusp forms.
Suppose that for each $y \in \mathbb{A}^\times$, one has $\phi_i [y] = \phi_i '[y] \otimes \phi_i ''[y]$ for some $\phi_i '[y] \in \mathcal{S}(\mathbb{A})$, $\phi_i ''[y] \in \mathcal{S}(B_\mathbb{A}^0)$. Abbreviate $\theta_i := \theta_{\psi^{\tau_i}}(\phi_i'[\tau_i y])$, $h_i := \theta_{\psi^{\tau_i}}(\phi_i''[\tau_i y],\Psi_i)$. Then the identity $$\label{eq:inner-product-pgl2-made-into-sl2}
\langle \Theta(\phi_1,\Psi_1),
\Theta(\phi_2,\Psi_2) \rangle_{\PGL_2}
=
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}}
|y|^2
\frac{1}{2^2}
\sum_{\tau_1,\tau_2 \in F^\times / F^{\times 2}}
\langle \theta_1 h_1, \theta_2 h_2 \rangle_{\operatorname{SL}_2}$$ holds, with both sides converging absolutely.
The LHS is an inner product of cusp forms, hence convergent. On the RHS, we may replace the $y$-integral by a finite sum, since the domain $\mathbb{A}^{\times} / F^{\times } \mathbb{A}^{\times 2}$ is compact and the integrand is invariant under an open subgroup. For individual $y$, the sum over $\tau_1, \tau_2$ has only finitely many nonzero summands, each of which consists of an inner product whose convergence is clear (see §\[sec-3-5-4\]). The expansion implies for $y \in \mathbb{A}^\times, s \in \operatorname{SL}_2(\mathbb{A})$ that $\Theta(\phi_i,\Psi_i,a(y) s) = \frac{1}{2} \sum _{\tau_i \in
F^\times / F^{\times 2} } |y| \theta_i(s) h_i(s)$, so the required identity follows from the formula relating integrals over $[\PGL_2]$ and $[\operatorname{SL}_2]$.
In this paper, we consider several expressions shaped like the RHS of . On a first (or perhaps on any) reading, one should focus on the contributions from $y = \tau_1 = \tau_2 = 1$; under some class number and unit group restrictions, these turns out to be the relevant ones for the proof of Theorem \[thm:main-result-for-microlocal-stuff\]. (We considered imposing such restrictions for the sake of presentation, but found that doing so obfuscated rather than simplified.)
Inner product formulas {#sec-3-7}
----------------------
### Elementary theta functions {#sec:elem-theta-ipf}
We recall part of [@nelson-theta-squared Thm 2].
Suppose $\phi_1, \phi_2 \in \mathcal{S}(\mathbb{A})$ satisfy $\phi_1(x) = \phi_1(-x), \phi_2(x) = \phi_2(-x)$. Let $\tau_1, \tau_2 \in F^\times$. Set $\theta_1 := \theta_{\psi^{\tau_1}}(\phi_1),
\theta_2 := \theta_{\psi^{\tau_1}}(\phi_2)$. Then $\langle \theta_1, \theta_2 \rangle_{\operatorname{SL}_2}= 0$ unless $\tau_1 = \tau_2$, in which case $\langle \theta_1, \theta_2 \rangle_{\operatorname{SL}_2}
=
2
\langle \phi_1, \phi_2 \rangle_{L^2(\mathbb{A})}$.
### Ternary theta lifts {#sec:ternary-theta-ipf}
As in [@nelson-variance-73-2 §12.3], we explicate Gan–Takeda [@MR2837015 Thm 6.6] (compare with [@MR3291638 Prop 2.8 (i)]).
Let $\pi_1, \pi_2 \subseteq L^2([\PB^\times])$ be cuspidal automorphic representations that are not one-dimensional. Let $\Psi_1 \in \pi_1, \Psi_2 \in \pi_2$ and $\phi_1, \phi_2 \in \mathcal{S}(B_\mathbb{A}^0)$. Let $\tau \in F^\times$. Set $h_i := \theta_{\psi^{\tau}}(\phi_i,\Psi_i)$ for $i=1,2$.
1. If $\pi_1 \neq \pi_2$, then $\langle h_1, h_2 \rangle_{\operatorname{SL}_2} = 0$.
2. Suppose $\pi_1 = \pi_2 =: \pi$. Let $S$ be a finite set of places of $F$ that containing all archimedean places, as well as any finite places at which $B$ ramifies, and that is sufficiently large in terms of $\Psi_i, \phi_i$. Then $\langle h_1, h_2 \rangle_{\operatorname{SL}_2}$ equals $$\frac{L^{(S)}(\pi,\tfrac{1}{2})}{\zeta_F^{(S)}(2)}
(\prod_{\mathfrak{p} \notin S}
\operatorname{vol}(K_\mathfrak{p}) )
\int_{g \in \PB^\times_S}
\langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle_{L^2(B_\mathbb{A}^0)}
\langle \pi(g) \Psi_1, \Psi_2 \rangle_{\PB^\times}$$ with $L^{(S)}(\pi,\tfrac{1}{2})$ as in §\[sec:standard-l-function\].
### Induction to $\Omega$ {#sec:induction-omega}
We now combine the previous two lemmas and sum them up. Temporarily denote by $\mathcal{A}_0$ the space of cusp forms $\Psi : [\PB^\times] \rightarrow \mathbb{C}$ that are orthogonal to all one-dimensional representations. For $\tau_1, \tau_2 \in F^\times$, let $\mathfrak{m} : \Omega \otimes \Omega \otimes \mathcal{A}_0
\otimes \mathcal{A}_0 \rightarrow \mathbb{C}$ denote the sesquilinear form given for $\phi_1, \phi_2 \in \Omega$ admitting factorizations $\phi_i[y] = \phi_i'[y] \otimes \phi_i''[y]$ by $$\mathfrak{m}(\phi_1,\phi_2,\Psi_1,\Psi_2)
:=
\int_{y \in \mathbb{A}^\times / F^{\times }
\mathbb{A}^{\times 2}}
|y|^2
\frac{1}{2^2}
\sum_{\tau_1,\tau_2 \in F^\times / F^{\times 2}}
\langle \theta_1, \theta_2 \rangle
\langle h_1, h_2 \rangle,$$ where we abbreviate $\theta_i := \theta_{\psi^{\tau_i}}(\phi_i'[\tau_i y])$ and $h_i := \theta_{\psi^{\tau_i}}(\phi_i''[\tau_i y],\Psi_i)$. The relevance of $\mathfrak{m}$ may be inferred from §\[sec:unfold-inner-product-nontraditional-theta-over-sl2\].
The definition makes sense: as in the proof of §\[sec:unfold-inner-product-nontraditional-theta-over-sl2\], the $y$-integral is really a finite sum, and the sum over $\tau_1, \tau_2$ has only finitely many nonzero summands. Each summand defines a sesquilinear form on $\mathcal{S}(\mathbb{A}) \otimes \mathcal{S}(B_\mathbb{A}^0)
\otimes \mathcal{A}_0$ that extends continuously to $\mathcal{S}(B_\mathbb{A}) \otimes \mathcal{A}_0$ by the *a priori* estimates of §\[sec-3-5-4\].
Let $\pi_1, \pi_2 \subseteq L^2([\PB^\times])$ be cuspidal automorphic representations that are not one-dimensional. Let $\Psi_1 \in \pi_1, \Psi_2 \in \pi_2$. Let $\phi_1, \phi_2 \in \Omega$.
1. If $\pi_1 \neq \pi_2$ then $\mathfrak{m}(\phi_1,\phi_2,\Psi_1,\Psi_2) = 0$.
2. Suppose $\pi_1 = \pi_2 =: \pi$. Let $S$ be a large enough finite set of places. Then $\mathfrak{m}(\phi_1,\phi_2,\Psi_1,\Psi_2)$ equals $$\frac{L^{(S)}(\pi,\tfrac{1}{2})}{\zeta_F^{(S)}(2)}
(\prod_{\mathfrak{p} \notin S}
\operatorname{vol}(K_\mathfrak{p}) )
\int_{g \in \PB^\times_S}
\langle \operatorname{Ad}(g) \mathfrak{S} \phi_1, \mathfrak{S} \phi_2 \rangle_{\Omega}
\langle \pi(g) \Psi_1, \Psi_2 \rangle_{\PB^\times}.$$
It suffices to consider the case that $\phi_1, \phi_2$ admit factorizations as in the definition of $\mathfrak{m}$. By and , we may assume that $\mathfrak{S} \phi_i = \phi_i$, or equivalently, that $\phi_i'(t,x) = \phi_i'(t,-x)$. By the lemmas of §\[sec:elem-theta-ipf\] and §\[sec:ternary-theta-ipf\], we have $\langle \theta_1, \theta_2 \rangle = 0$ unless $\tau_1 = \tau_2$ and then $\langle h_1, h_2 \rangle = 0$ unless $\pi_1 = \pi_2$; in that case, the formulas from those lemmas and the identities $$\langle \phi_1'[y \tau], \phi_2'[y \tau] \rangle_{L^2(\mathbb{A})}
\langle \operatorname{Ad}(g) \phi_1''[y \tau], \phi_2''[y \tau]
\rangle_{L^2(B_\mathbb{A}^0)}$$ $$=
\langle \operatorname{Ad}(g) \phi_1[y \tau], \phi_2[y \tau] \rangle_{L^2(B_\mathbb{A})}$$ and (see , ) $$\int_{y \in \mathbb{A}^{\times} / F^\times \mathbb{A}^{\times
2}
}
|y|^2
\frac{1}{2}
\sum_{\tau \in F^\times / F^{\times 2}}
\langle \operatorname{Ad}(g) \phi_1[y \tau], \phi_2[y \tau]
\rangle_{L^2(B_\mathbb{A})}$$ $$=
\int_{y \in \mathbb{A}^{\times} / \mathbb{A}^{\times
2}
}
|y|^2
\langle \operatorname{Ad}(g) \phi_1[y], \phi_2[y]
\rangle_{L^2(B_\mathbb{A})}
=
\langle \operatorname{Ad}(g) \phi_1, \phi_2 \rangle_{\Omega}$$ combine to give the required conclusion.
Estimates for general quantum variance sums\[sec:estimates-general-var\] {#sec-4}
========================================================================
In this section we introduce general families of quantum variance sums, propose a candidate for their leading asymptotics, and state a general “estimate” comparing the two.
Notation {#sec-4-1}
--------
Let $F$ be a number field with adele ring $\mathbb{A}$. Fix a nontrivial unitary character $\psi$ of $\mathbb{A}/F$. Let $B$ be a non-split quaternion algebra over $F$. Fix a maximal order $R \subseteq B$ and a finite set $S$ of places of $F$, containing all archimedean places as well as any finite places at which $B$ ramifies. Retain the (unsurprising) notation of §\[sec-3-1\].
Since $B$ is non-split, the quotient $[\PB^\times]=
\PB^\times \backslash \PB^\times_\mathbb{A}$ is compact, and so $L^2([\PB^\times])$ is completely reducible. Let $A^{\flat}$ denote the set of irreducible subrepresentations of the Hilbert space $L^2([\PB^\times])$. For each $\pi^{\flat} \in A^{\flat}$, let $\pi {\leqslant}\pi^{\flat}$ denote the subspace of smooth vectors. Set $A := \{\pi : \pi^\flat \in A^{\flat}\}$. Let $\mathcal{A}$ denote the algebraic direct sum $\oplus_{\pi \in A} \pi$, regarded as a pre-unitary representation of the group $\PB^\times_\mathbb{A}$. We introduce the following additional notation:
- $K = \prod K_\mathfrak{p}$: a maximal compact subgroup of $\PB^\times_\mathbb{A}$. For $\mathfrak{p} \notin S$, we assume that $K_\mathfrak{p} {\leqslant}\PB^\times_{\mathfrak{p}}$ is the image of $R_\mathfrak{p}^\times$.
- $\mathcal{A}_0 {\leqslant}\mathcal{A}$: the orthogonal complement of the one-dimensional subrepresentations. (We had earlier, in §\[sec-3-5\] and §\[sec:induction-omega\], used the same symbol to denote some *larger* spaces than what we call here $\mathcal{A}_0$. This abuse of notation should introduce no confusion.)
- $A_0 := \{\pi \in A : \pi \subseteq \mathcal{A}_0\} =
\{\pi \in A : \dim(\pi) > 1\}$, so that $\mathcal{A}_0 = \oplus_{\pi \in A_0} \pi$.
- $\mathcal{A}^S := \{\varphi \in \mathcal{A} :
\rho_{\operatorname{reg}}(k) \varphi = \varphi \text{ for all }
k \in K_\mathfrak{p}, \mathfrak{p} \notin S
\}, \mathcal{A}_0^S := \mathcal{A}_0 \cap \mathcal{A}^S$: the “unramified outside $S$” subspaces of $\mathcal{A}, \mathcal{A}_0$.
- $A^S := \{\pi \in A : \pi \cap \mathcal{A}^S \neq \{0\}\},
{A}_0^S := {A}^S \cap {A}_0$: the subsets consisting of those $\pi$ that are unramified outside $S$.
- $\mathcal{B}(V)$, for $V$ a $K$-invariant subspace of $\mathcal{A}$: an orthonormal basis for the closure of $V$ that consists of $K$-isotypic elements of $V$.
Fix Haar measures on $\PB^\times_S$ and $[\PB^\times]$; we do not require any compatibility between them. Because $B$ is non-split, each $\pi \in A_0$ is cuspidal. Let $L^{(S)}(\pi,s)$, $L^{(S)}(\operatorname{ad}\pi,s)$ be as in §\[sec:standard-l-function\].
Key definitions\[sec:main-general-estimate-key-defns\] {#sec-4-2}
------------------------------------------------------
By , and , the sums considered in the definitions to follow converge absolutely.
### The basic distributions {#sec:omega-pi}
For $\pi \in A^S$, define $\omega_\pi : C_c^\infty(\PB^\times_S) \otimes \mathcal{A}^S
\rightarrow \mathbb{C}$ by $\omega_\pi(f,\Psi)
:=
\sum_{\varphi \in \mathcal{B}(\pi \cap \mathcal{A}^S)}
\langle \varphi, \Psi \cdot \pi(f) \varphi \rangle$. The definition is independent of the choice of orthonormal basis.
\[example:harmonic-sum-attached-to-orth-proj\] If $\pi(f) = 0$, then $\omega_\pi(f,\Psi) = 0$. If $\pi(f)$ is the orthogonal projector onto a one-dimensional subspace $\mathbb{C} \varphi$ of $\pi$ with unit basis vector $\varphi$, then $\omega_\pi(f,\Psi) = \langle \varphi, \Psi \varphi \rangle$.
### Quantum variance sums
For $f \in C_c^\infty(\PB^\times_S)$, define the sesquilinear form $\mathcal{V}_f : \mathcal{A}_0^S
\otimes \mathcal{A}_0^S \rightarrow \mathbb{C}$ by $$\mathcal{V}_f(\Psi_1,\Psi_2)
:=
\sum_{\pi \in A_0^S}
L^{(S)}(\operatorname{ad}\pi,1)
\omega_\pi( f,\Psi_1)
\overline{
\omega_\pi( f,\Psi_2) }.$$
### Proposed limiting variance
For $f \in C_c^\infty(\PB^\times_S)$, define the sesquilinear form $\mathcal{M}_f : \mathcal{A}_0^S \otimes \mathcal{A}_0^S
\rightarrow \mathbb{C}$ by requiring for $\Psi_1 \in \pi_1 \in A_0^S, \Psi_2 \in \pi_2
\in A_0^S$ that $\mathcal{M}_f(\Psi_1,\Psi_2) := 0$ unless $\pi_1 = \pi_2 =: \pi$, in which case $$\mathcal{M}_f(\Psi_1,\Psi_2)
:=
c_3
L^{(S)}(\pi,\tfrac{1}{2})
\int_{g \in \PB^\times_S}
\langle \operatorname{Ad}(g) \mathfrak{S} f, \mathfrak{S} f \rangle_{L^2(\PB^\times_S)}
\langle \pi(g) \Psi_1, \Psi_2 \rangle_{\PB^\times}$$ where $$\label{eq:defn-of-c}
c_3 :=
\zeta_F^{(S)}(2) \operatorname{vol}([\PB^\times])^{-1}.$$ The integral converges absolutely (see §\[sec:local-convergence-lemmas\]).
### Thickening $\PB^\times$ inside $B$ {#sec:heartsuit}
Fix, once and for all, a nonzero element $W_S \in
C_c^\infty(F_S^\times)$. For $\tau \in F^\times$, define the linear map $\heartsuit^{\tau} : C_c^\infty(\PB^\times_S)
\rightarrow \mathcal{S}(B_S)$ by $$\heartsuit^{\tau} f(x)
:=
\frac{W_S(\tau \operatorname{nr}(x))}{|\tau \operatorname{nr}(x)|_S}
1_{B_S^\times}(x)
f(\operatorname{pr}(x)),$$ where $\operatorname{pr}: B_S^\times \rightarrow \PB^\times_S$ denotes the natural projection.
Statement of main result {#sec-4-3}
------------------------
The statement involves the metaplectic group (§\[sec:global-metaplectic-gp\]) and the Weil representation (§\[sec:weil-repn-global\]). For $s \in \operatorname{Mp}_2(F_S)$, we abbreviate $\rho^{\tau}(s) := \rho_{\operatorname{Weil}}^{\psi^{\tau},B}(s)$ and $\rho_0^\tau(s) := \rho_{\operatorname{Weil}}^{\psi^{\tau},B^0}(s)$; these operators act respectively on $\mathcal{S}(B_S)$ and $\mathcal{S}(B_S^0)$. The operators $1 \otimes \rho_0^{\tau_i}(s)$ on $\mathcal{S}(B_S)$ are defined using the decomposition $B_S = F_S \oplus B_S^0$, as in §\[sec:factorization-weil-repn\].
\[thm:main-estimate-general-variance\] There is a finite subset $X$ of $F^\times$ and a finite collection $(\eps_{\tau_1,\tau_2})_{\tau_1,\tau_2 \in X}$ of sesquilinear forms $\eps_{\tau_1,\tau_2} : \mathcal{S}(B_S) \otimes
\mathcal{S}(B_S) \otimes \mathcal{A}_0^S \otimes
\mathcal{A}_0^S \rightarrow \mathbb{C}$, depending only upon $F$, $\psi$, $S$ and $W_S$, with the following properties:
1. [**Relevance.**]{} For $f \in C_c^\infty(\PB^\times_S)$, one has the following identity of sesquilinear forms on $\mathcal{A}_0^S$: $$\label{eq:relevance}
\mathcal{V}_f
= \mathcal{M}_f
+ \sum_{\tau_1,\tau_2 \in X} \eps_{\tau_1,\tau_2}(\heartsuit^{\tau_1} f, \heartsuit^{\tau_2} f, \cdot, \cdot).$$
2. [**$\O_1(F)$-invariance.**]{} $$\eps_{\tau_1,\tau_2}(\mathfrak{S} \phi_1, \phi_2, \Psi_1,
\Psi_2)
=
\eps_{\tau_1,\tau_2}( \phi_1, \phi_2, \Psi_1,
\Psi_2),$$ $$\eps_{\tau_1,\tau_2}(\phi_1, \mathfrak{S}\phi_2, \Psi_1,
\Psi_2)
=
\eps_{\tau_1,\tau_2}(\phi_1, \phi_2, \Psi_1,
\Psi_2).$$
3. [**$\operatorname{SO}(B_S^0)$-invariance.**]{} For $g_1,g_2 \in \PB^\times_S$, $$\eps_{\tau_1,\tau_2}(\operatorname{Ad}(g_1) \phi_1, \operatorname{Ad}(g_2) \phi_2, \rho_{\operatorname{reg}}(g_1) \Psi_1,
\rho_{\operatorname{reg}}(g_2) \Psi_2)$$ $$= \eps_{\tau_1,\tau_2}(\phi_1, \phi_2, \Psi_1,
\Psi_2).$$
4. [**Metaplectic invariance.**]{} For $s \in \operatorname{Mp}_2(F_S)$, $$\eps_{\tau_1,\tau_2}(\rho^{\tau_1}(s) \phi_1, \rho^{\tau_2}(s) \phi_2, \Psi_1,
\Psi_2)
=
\eps_{\tau_1,\tau_2}(\phi_1, \phi_2, \Psi_1,
\Psi_2).$$
5. [**Main estimate.**]{} For $s \in \operatorname{Mp}_2(F_S)$, $$\eps_{\tau_1,\tau_2}((1 \otimes \rho_0^{\tau_1}(s)) \phi_1,
(1 \otimes \rho_0^{\tau_2}(s)) \phi_2, \Psi_1,
\Psi_2)$$ $$\ll
\Xi(s)
\prod_{i=1,2}
\Sob(\phi_i) \|\Psi\|_{L^1},$$ where $\Xi$ denotes the Harish–Chandra function (§\[sec:Xi-global\]) and $\Sob(\phi_i)$ denotes a quantity that varies continuously with $\phi_i$ (see §\[sec:some-asympt-notat\]). The implied constants and the uniformity in the continuity of $\Sob(.)$ depend at most upon $F,\psi,S,W_S$.
6. [**Construction.**]{} $\eps_{\tau_1,\tau_2}$ factors explicitly through the theta correspondence in the sense of §\[sec-4-5-5\] and the remark of §\[sec-8-6\].
For the application to Theorem \[thm:main-result-for-microlocal-stuff\], the crucial assertions are the relevance, the $\operatorname{SO}(B_S^0)$-invariance, and the main estimate. The $\O_1(F)$-invariance and metaplectic invariance are employed to simplify the presentation of the proof. The construction is not applied in this paper, but may be useful for further refinements and extensions.
One purpose of Part II is to give evidence that Theorem \[thm:main-estimate-general-variance\] is useful.
The formulation of Theorem \[thm:main-estimate-general-variance\] is independent of the choice of measures on $\PB_S^\times$ and on $[\PB^\times]$.
Theorem \[thm:main-estimate-general-variance\] minus the “main estimate” is like a trace formula: $\mathcal{V}_f$ is a sum over automorphic forms, $\mathcal{M}_f$ is like the “identity” contribution, and the $\eps_{\tau_1,\tau_2}$ are the “interesting” contributions which one would like in practice to show have negligible size. One difference is that $\mathcal{V}_f$ has a quadrilinear (rather than bilinear) dependence upon the automorphic forms $\varphi$.
Theorem \[thm:main-estimate-general-variance\] likely extends to the split case $B = M_2(F)$ after incorporating contributions from the continuous spectrum into the definitions of §\[sec:main-general-estimate-key-defns\] and replacing $\|\Psi_i\|_{L^1}$ with $\|\operatorname{ht}^A \Psi_i\|_{L^1}$ for some fixed large enough $A > 0$.
Proof of Theorem \[thm:main-estimate-general-variance\] {#sec:proof-theor-refthm:m}
-------------------------------------------------------
### Measures {#sec-4-5-1}
With a view to applications, we have formulated Theorem \[thm:main-estimate-general-variance\] in a measure-independent fashion. For the proof, it is convenient to take on $[\PB^\times]$ the Tamagawa measure, so that $$\label{eq:defn-of-c-2}
c_3 = \frac{1}{2} \zeta_F^{(S)}(2),$$ and to fix measures on $\PB^\times_{\mathbb{A}}, \PB^\times_\mathfrak{p}$ and hence on $\PB^\times_S = \prod_{\mathfrak{p} \in S}
\PB^\times_\mathfrak{p}$ as in §\[sec:global-measures\].
### The $\heartsuit$ operator: local {#sec-2-5}
Suppose temporarily (for §\[sec-2-5\] only) that $k$ is a local field, $\psi$ is a nontrivial unitary character of $k$, $B$ is a quaternion algebra over $k$, $G := B^\times / k^\times$, and $W \in C_c^\infty(k^\times)$. Recall from §\[sec:defn-local-omega\] the definition of $\Omega$. We define a linear map $\heartsuit : C_c^\infty(G) \rightarrow \Omega$ by $$\heartsuit f(t,x)
:=
\frac{W(t \operatorname{nr}(x))}{|t \operatorname{nr}(x)|}
1_{B^\times}(x)
f(x).$$ (By abuse of notation, we write $f(x)$ for the value taken by $f$ at the image of $x$ under the natural projection $B^\times \rightarrow G$.)
By inspecting the definitions, one has the identities of maps $C_c^\infty(G) \rightarrow \Omega$ $$\mathfrak{S} \heartsuit = \heartsuit \mathfrak{S},
\quad
\operatorname{Ad}(g) \heartsuit = \heartsuit \operatorname{Ad}(g)
\text{ (for $g \in G$)}.$$ By inspecting the definitions, one has for $y \in k^\times, b \in B^\times$ that $$\label{eqn:local-heartsuit-formula}
\rho_{\operatorname{Weil}}(a(y)) \heartsuit f(\operatorname{nr}(b)^{-1},b)
=
|y| \heartsuit f(y \operatorname{nr}(b)^{-1}, b)
=
\mathfrak{W}(y) f(b).$$ By the formula for $\|.\|_{\Omega}$, one obtains $$\label{eqn:local-norm-heartsuit-f}
\|\heartsuit f\|_{\Omega} =
\|f\|_{L^2(G)} \,
\|\mathfrak{W}\|_{L^2(k^\times, |x|^{-1} \, d x)}.$$
### The $\heartsuit$ operator: global\[sec:heartsuit-global\] {#sec-3-6}
We revert to the global setting of §\[sec:estimates-general-var\]. Let $\pi \in A_0^S$. Recall from §\[sec:similitudes-global\] the definition of $\Omega$. We define a linear map $\heartsuit : C_c^\infty(\PB^\times_S) \rightarrow \Omega$ by $$\heartsuit f(t,x) :=
\frac{W_S(t_S \operatorname{nr}(x_S))}{|t_S \operatorname{nr}(x_S)|}
1_{B_S^\times}(x_S)
f(x_S)
\prod_{\mathfrak{p} \notin S} \operatorname{vol}(K_{\mathfrak{p}})^{-1}
\phi_\mathfrak{p}^0(t_\mathfrak{p},x_\mathfrak{p}),$$ where $\phi_\mathfrak{p}^0 \in \Omega_\mathfrak{p}$ is defined with respect to $R_\mathfrak{p}$ (see §\[sec:dist-elem\]). This definition and that of §\[sec:heartsuit\] are obviously similar; we record their precise relationship below in §\[sec-4-5-5\].
By and the lemma of §\[sec:hecke-kernels-local\], one has for $y \in \mathbb{A}^\times$, $b \in B_\mathbb{A}^\times$ that $$\label{eqn:formulas-for-heartsuit-f-global}
|y| \heartsuit f(
y \operatorname{nr}(b)^{-1}, b)
=
W_S(y_S) f(b_S)
\prod_{\mathfrak{p} \notin S}
T_{y_\mathfrak{p}}(b_\mathfrak{p}),$$ with $T_{y_\mathfrak{p}}$ as in §\[sec:hecke-kernels-local\]. By combining with Lemma \[lem:norm-of-distinguished-vector-in-Omega-local\] of §\[sec:dist-elem\], one obtains $$\label{eqn:norm-of-heartsuit-f-global}
\|\heartsuit f\|^2_{\Omega}
=
\|f\|_{L^2(\PB^\times_S)}^2
\int_{t \in F_S^\times}
|\mathfrak{W}|^2(t)
\, \frac{d t}{|t|}
\prod_{\mathfrak{p} \notin S}
\frac{\operatorname{vol}(R_\mathfrak{p})}{\operatorname{vol}(K_\mathfrak{p})^2}.$$
\[lem:main-term-eval-for-heartsuit\] Let $\pi \in A_0^S$. Let $\Psi_1,\Psi_2 \in \pi$ be $\prod_{\mathfrak{p} \notin S} K_\mathfrak{p}$-invariant vectors. For $f \in C_c^\infty(\PB^\times_S)$, the quantity $\mathfrak{m}(\heartsuit f, \heartsuit f, \Psi_1,
\Psi_2)$ (see §\[sec:induction-omega\]) equals $$c_2
L^{(S)}(\pi,\tfrac{1}{2})
\int_{g \in \PB_S^\times}
\langle \operatorname{Ad}(g) \mathfrak{S} f, \mathfrak{S} f \rangle_{L^2(\PB_S^\times)}
\langle \pi(g) \Psi_1, \Psi_2 \rangle_{\PB^\times},$$ where $$\label{eq:defn-c2-frak-W}
c_2
:=
\frac{1}{\zeta^{(S)}_F(2)}
(\prod_{\mathfrak{p} \notin S}
\frac{\operatorname{vol}(R_\mathfrak{p})}{\operatorname{vol}(K_\mathfrak{p})})
\int_{y \in F_S^\times}
|W_S|^2(y)
\, \frac{d y}{|y|}.$$
By the lemma of §\[sec:induction-omega\], the polarization of and the commutativity $\heartsuit \operatorname{Ad}(g) = \operatorname{Ad}(g) \heartsuit$, the required identity holds if we replace $S$ with some possibly larger finite set of places $S' \supseteq S$. To deduce the identity as written, we apply (using to verify its hypotheses).
### A specific Eichler/Jacquet–Langlands lift {#sec:Phi-pi}
For $\pi \in A_0^S$, let $\Phi_\pi \in \pi_{\operatorname{JL}}$ denote the element of the Jacquet–Langlands lift of $\pi$ having the Fourier expansion $\Phi_\pi(n(x) a(y)) =
\sum_{\tau \in F^\times} \psi(\tau x) W_\pi(\tau y)$, where the Whittaker function $W_\pi : \mathbb{A}^\times \rightarrow \mathbb{C}$ is given by $W_\pi(y) := W_S(y_S) \prod_{\mathfrak{p} \notin S}
W_{\pi_\mathfrak{p}}^0(y_\mathfrak{p})$ (see §\[sec:aut-forms-fourier-exp\], §\[sec-3-3-2\]).
One has $\|\Phi_\pi\|^2 = c_1 L^{(S)}(\operatorname{ad}\pi,1)$, where $$\label{eq:c-of-frak-W-defn}
c_1
:=
\frac{2}{ \zeta_F^{(S)}(2)}
(\prod_{\mathfrak{p} \notin S}
\Delta_{\psi_{\mathfrak{p}}}^{-1/2})
\int_{y \in F_S^\times} |W_S|^2(y) \, \frac{d y}{|y|}.$$ If $\pi, \pi ' \in A_0^S$ are distinct, then $\langle \Phi_{\pi}, \Phi_{\pi '} \rangle = 0$.
The conclusion in the case $\pi \neq \pi '$ is the multiplicity one theorem for $\PB^\times$ combined with the injectivity of $\pi \mapsto \pi_{\operatorname{JL}}$. The formula is a consequence of the lemma of §\[sec-3-3-2\] and the corresponding local calculation (§\[sec-2-3\]).
### The normalizing scalar
Recall from , and the scalars $c_1,c_2,c_3$. By the local volume formulas of §\[sec:local-vol-formulas\], $$\label{eqn:relation-c1-c2-c}
c_1^{-1} c_2 = c_3.$$
### Application of the pretrace formula {#sec-4-5-2}
Recall the theta functions $\Theta(\phi,\Psi)$ attached in §\[sec:non-traditional-theta-lifts\] to each $\phi \in \Omega, \Psi \in \mathcal{A}_0$. Let $f \in C_c^\infty(\PB^\times_S)$, $\Psi \in \mathcal{A}_0^S$.
\[lem:abs-conv-of-silly-sum\] $\sum_{\pi \in A_0^S} |\omega_\pi( f,\Psi)|
\| \Phi_\pi \|_{L^p([\PGL_2])} < \infty$ for $p=2,\infty$.
By and , it suffices to show that $\| \Phi_\pi \|_{L^p([\PGL_2])} \ll C(\pi)^{O(1)}$. The case $p=2$ follows from the lemma of §\[sec:Phi-pi\] and . The case $p=\infty$ reduces to the case $p=2$ by axioms (S2a) and (S3b) of [@michel-2009 §2.4], wherein the quantities $\mathcal{S}_d(\Phi_\pi)$ may be estimated using [@michel-2009 §3.2.5]. A direct proof of this convergence also follows by a rearrangement of the arguments given below.
\[lem:fourier-coefficients-of-nontraditional-theta-lifts\] $\Theta(\heartsuit f,\Psi)
= \sum_{\pi \in A_0^S} \omega_\pi( f,\Psi)
\Phi_\pi$.
Set $\Phi_1 := \Theta(\heartsuit f,\Psi)$ and $\Phi_2 := \sum_{\pi \in A_0^S} \omega_\pi( f,\Psi)
\Phi_\pi$; we must show that $\Phi_1 = \Phi_2$. Since $\Phi_1, \Phi_2$ are cuspidal, it will suffice to demonstrate the equality of their Whittaker functions $W_1, W_2 : \mathbb{A}^\times \rightarrow \mathbb{C}$ as defined in §\[sec:aut-forms-fourier-exp\]. By the lemma of §\[sec:four-expans\] and , we have $$W_1(y)
= \sum_{\gamma \in \PB^\times}
\int_{g \in [\PB^\times]}
\Psi(g)
W_S(y_S)
f(g_S^{-1} \gamma g_S)
\prod_{\mathfrak{p} \notin S}
T_{y_\mathfrak{p}}(g_\mathfrak{p}^{-1} \gamma g_\mathfrak{p}).$$ The definition of $\Phi_\pi$ implies (using Lemma \[lem:abs-conv-of-silly-sum\] to justify the interchange of summation with the Fourier integral over the compact group $\mathbb{A}/F$) that $$W_2(y)
= \sum_{\pi \in A_0^S} \omega_\pi( f,\Psi) W_S(y_S)
\prod_{\mathfrak{p} \notin S}
W_{\pi_\mathfrak{p}}^0(y_\mathfrak{p}).$$ Since $\Psi \in \mathcal{A}_0^S$, we have $\omega_\pi(f,\Psi) = 0$ for all $\pi \in A^S$ with $\pi \notin A_0^S$, so it suffices to establish for all $y \in \mathbb{A}^\times$, $g \in \PB^\times_{\mathbb{A}}$ the pointwise identity $$\sum_{\gamma \in \PB^\times}
f(g_S^{-1} \gamma g_S)
\prod_{\mathfrak{p} \notin S}
T_{y_\mathfrak{p}}(g_\mathfrak{p}^{-1} \gamma g_\mathfrak{p})
=
\sum_{\pi \in A^S}
(\sum_{\varphi \in \mathcal{B}(\pi \cap \mathcal{A}^S)}
\overline{\varphi}(g)
\pi(f) \varphi(g))
\prod_{\mathfrak{p} \notin S}
W_{\pi_\mathfrak{p}}^0(y_\mathfrak{p}),$$ which follows from the pretrace formula (§\[sec:pretrace-formula\]) and the identity $W_{\pi_\mathfrak{p}}^0(y_\mathfrak{p})
= \lambda_{\pi_\mathfrak{p}}(T_{y_\mathfrak{p}})$ (see ).
Lemma \[lem:fourier-coefficients-of-nontraditional-theta-lifts\] and its proof are in the spirit of arguments of Shimizu [@MR0333081 §4], but we were unable to relate them precisely (e.g., by deducing one from the other).
### Some sesquilinear forms
Define $\mathcal{V}, \mathcal{M}, \mathcal{E} : \Omega \otimes \Omega \otimes \mathcal{A}_0
\otimes \mathcal{A}_0 \rightarrow \mathbb{C}$ by requiring that for $\phi_1, \phi_2 \in \Omega$ satisfying $\phi_i[y] = \phi_i'[y] \otimes \phi_i''[y]$ with $\phi_i'[y] \in \mathcal{S}(\mathbb{A})$ and $
\phi_i''[y] \in \mathcal{S}(B_\mathbb{A}^0)$ for all $y \in \mathbb{A}^\times$, one has with the abbreviations $\theta_i := \theta_{\psi^{\tau_i}}(\phi_i'[\tau_i y])$ and $h_i := \theta_{\psi^{\tau_i}}(\phi_i''[\tau_i y],\Psi_i)$ that $$\mathcal{V}(\phi_1,\phi_2,\Psi_1,\Psi_2)
:= c_1^{-1}
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}} |y|^2
\frac
{
1
}
{
2^2
}
\sum_{\tau_1,\tau_2 \in F^\times / F^{\times 2}}
\langle
\theta_1 h_1, \theta_2 h_2 \rangle,$$ $$\mathcal{M}(\phi_1,\phi_2,\Psi_1,\Psi_2)
:= c_1^{-1}
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}} |y|^2
\frac
{
1
}
{
2^2
}
\sum_{\tau_1,\tau_2 \in F^\times / F^{\times 2}}
\langle \theta_1, \theta_2 \rangle
\langle h_1, h_2 \rangle$$ and $\mathcal{E} := \mathcal{V} - \mathcal{M}$, or equivalently, $$\mathcal{E}(\phi_1,\phi_2,\cdot ,\cdot )
:= c_1^{-1}
\int_{y \in \mathbb{A}^\times / F^\times \mathbb{A}^{\times 2}} |y|^2
\frac
{
1
}
{
2^2
}
\sum_{\tau_1,\tau_2 \in F^\times / F^{\times 2}}
\mathcal{E}_{\tau_1,\tau_2}(\phi_1[\tau_1 y], \phi_2[\tau_2
y],\cdot,\cdot),$$ where $\mathcal{E}_{\tau_1,\tau_2} : \mathcal{S}(B_\mathbb{A}) \otimes
\mathcal{S}(B_\mathbb{A})
\otimes \mathcal{A}_0 \otimes \mathcal{A}_0 \rightarrow
\mathbb{C}$ is as in §\[sec:main-estimate-for-equidistribution-pairs-theta\]. The definitions makes sense for the same reasons as in §\[sec:induction-omega\]. The identity $$\label{eqn:c1-times-error-equals-bla}
\mathcal{V}(\phi_1,\phi_2,\Psi_1,\Psi_2)
= c_1^{-1} \langle \Theta(\phi_1, \Psi_1), \Theta(\phi_2, \Psi_2) \rangle.
$$ follows from §\[sec:unfold-inner-product-nontraditional-theta-over-sl2\] when $\phi$ is a pure tensor, hence in general by linearity.
### The main identities {#sec:main-identity}
\[prop:after-extracting-main-term\] Let $f \in C_c^\infty(\PB^\times_S)$ and $\Psi_1,\Psi_2 \in \mathcal{A}_0^S$. Then $$\label{eq:V-equals-Vf}
\mathcal{V}(\heartsuit f,\heartsuit f,\Psi_1,\Psi_2)
= \mathcal{V}_f(\Psi_1,\Psi_2),$$ $$\label{eq:M-equals-Mf}
\mathcal{M}(\heartsuit f,\heartsuit f,\Psi_1,\Psi_2)
= \mathcal{M}_f(\Psi_1,\Psi_2),$$ $$\label{eq:Vf-equals-Mf-plus-E}
\mathcal{V}_f(\Psi_1,\Psi_2)
=
\mathcal{M}_f(\Psi_1,\Psi_2)
+
\mathcal{E}(\heartsuit f, \heartsuit f,
\Psi_1, \Psi_2).$$
: By , Lemma \[lem:fourier-coefficients-of-nontraditional-theta-lifts\] of §\[sec-4-5-2\], and the lemma of §\[sec:Phi-pi\], $$\begin{aligned}
\mathcal{V}(\heartsuit f,\heartsuit f,\Psi_1,\Psi_2)
&=
c_1^{-1} \langle \Theta(\heartsuit f, \Psi_1), \Theta(\heartsuit f,
\Psi_2) \rangle
\\
&=
c_1^{-1} \sum_{\pi_1,\pi_2 \in A_0^S}
\left\langle
\omega_{\pi_1}(f, \Psi_1),
\omega_{\pi_2}(f, \Psi_2)
\right\rangle
\\
&=
\sum_{\pi \in A_0^S}
L^{(S)}(\operatorname{ad}\pi,1)
\omega_{\pi}( f, \Psi_1),
\overline{\omega_{\pi}( f, \Psi_2)}
\\
&=
V_f(\Psi_1,\Psi_2).
\end{aligned}$$
: by the lemma of §\[sec-3-6\] and .
: by , and the definition of $\mathcal{E}$.
### Completion of the proof {#sec-4-5-5}
We now apply Proposition \[prop:main-error-estimate-global-adelic-general\] (see §\[sec-3-5-5\]) and Proposition \[prop:after-extracting-main-term\] to prove Theorem \[thm:main-estimate-general-variance\]. This final part of the argument is of a purely technical nature and involves no major new ideas. Indeed, its main purpose is to recast the content of those propositions in terms of $\mathcal{S}(B_S)$ rather than the less “user-friendly” space $\Omega$.
By weak approximation, we may choose a compact fundamental domain $Y \subset
\mathbb{A}^\times / \mathbb{A}^{\times 2}$ for $\mathbb{A}^\times / F^{\times} \mathbb{A}^{\times 2}$ with the property that $y_\mathfrak{p} = 1$ for all $y \in Y$ and $\mathfrak{p} \in S$. Choose a finite set $X \subseteq F^\times$ of representatives for the finite set $$\label{eq:elements-such-that-multiplying-by-something-in-Y-gives-squares-mod-units-outside-S}
\{\tau \in F^\times / F^{\times 2} :
\text{
there exists }
y \in Y
\text{ so that }
y_\mathfrak{p} \tau \in F_\mathfrak{p}^{\times 2} \mathcal{O}_\mathfrak{p}^\times
\text{ for all }
\mathfrak{p} \notin S
\}.$$ For $y \in Y, \tau \in X$, let $\diamondsuit^{\tau y} : \mathcal{S}(B_S) \hookrightarrow
\mathcal{S}(B_\mathbb{A})$ denote the map $\diamondsuit^{y \tau} \Phi := \Phi \otimes
(\otimes_{\mathfrak{p} \notin S} \phi_\mathfrak{p}^0[\tau
y_\mathfrak{p} ])$, where $\phi_{\mathfrak{p}}^0 \in \Omega_{\mathfrak{p}}$ denotes as usual the distinguished element. For $f \in C_c^\infty(\PB_S^\times)$, one then has $\diamondsuit^{\tau y} \heartsuit^{\tau} f = \heartsuit f[\tau y]$. Observe that for $\mathfrak{p} \notin S$ and $t \in F_\mathfrak{p}^\times$, one has $\phi_\mathfrak{p}^0[t] = 0$ unless $t \in F_\mathfrak{p}^{\times 2}
\mathcal{O}_\mathfrak{p}^\times$. It follows that for $\tau \in F^\times$ and $y \in Y$, one has $\heartsuit f[\tau y] = 0$ unless $\tau$ belongs to the set , hence that $$\label{eq:penultimate-identity-before-proving-main-general-thm}
\mathcal{E}(\heartsuit f, \heartsuit f, \cdot, \cdot)
=
\frac{c_1^{-1}}{2^2}
\int_{y \in Y}
|y|^2
\sum_{\tau_1,\tau_2 \in X}
\mathcal{E}_{\tau_1,\tau_2}(\diamondsuit^{\tau_1 y}
\heartsuit^{\tau_1} f,
\diamondsuit^{\tau_2 y} \heartsuit^{\tau_2} f,
\cdot,\cdot).$$ Define $\eps_{\tau_1,\tau_2} : \mathcal{S}(B_S) \otimes \mathcal{S}(B_S)
\otimes \mathcal{A}_0^S
\otimes \mathcal{A}_0^S \rightarrow \mathbb{C}$ by $$\label{eq:defn-of-the-eps-guys-yay}
\eps_{\tau_1,\tau_2}(\Phi_1,\Phi_2,\Psi_1,\Psi_2)
:=
\frac{c_1^{-1}}{2^2}
\int_{y \in Y}
|y|^2
\mathcal{E}_{\tau_1,\tau_2}(\diamondsuit^{\tau_1 y}
\Phi_1,
\diamondsuit^{\tau_2 y} \Phi_2,
\Psi_1,\Psi_2).$$ We verify the assertions made in Theorem \[thm:main-estimate-general-variance\]:
1. The “relevance” follows from , and Proposition \[prop:after-extracting-main-term\].
2. Since $\mathfrak{S} \phi_\mathfrak{p}^0 = \phi_\mathfrak{p}^0$, one has $\mathfrak{S} \diamondsuit^{\tau_i y} = \diamondsuit^{\tau_i
y} \mathfrak{S}$. For $g \in \PB^\times_S$ and $s \in \operatorname{Mp}_2(F_S)$, one has $\operatorname{Ad}(g) \diamondsuit^{\tau_i y} = \diamondsuit^{\tau_i y}
\operatorname{Ad}(g)$ and $\rho^{\tau_i}(s) \diamondsuit^{\tau_i y} =
\diamondsuit^{\tau_i y} \rho^{\tau_i}(s)$. Thus the “$\O_1(F)$-invariance,” “$\operatorname{SO}(B_S^0)$-invariance” and “metaplectic invariance” follow from §\[sec:equivariance-summary-for-E-tau-tau\].
3. The “main estimate” is the content of Proposition \[prop:main-error-estimate-global-adelic-general\].
Classicalization {#sec-4-4}
----------------
We begin discussing how to relate the setting of Theorem \[thm:main-estimate-general-variance\] to that of Theorem \[sec-1\]. We complete this discussion in §\[sec:deduction-main-thm-microlocal\].
### Specialization to a single place\[sec:general-estimates-specialized-single-place\] {#sec-4-4-1}
We specialize the definitions of §\[sec:main-general-estimate-key-defns\] to the case that ramification is concentrated at a single place $\mathfrak{q}$ of $F$, finite or infinite. This is the case required for the proof of Theorem \[thm:main-result-for-microlocal-stuff\].
Assume that $S$ is the set of places $\mathfrak{p}$ for which either
- $\mathfrak{p}$ is infinite,
- $\mathfrak{p}$ is a finite place at which $B$ ramifies, or
- $\mathfrak{p} = \mathfrak{q}$.
Assume that for each $\mathfrak{p} \notin S - \{\mathfrak{q}\}$, the completion $B_\mathfrak{q}$ is non-split, or equivalently, that $\PB^\times_\mathfrak{q}$ is compact. There are the following possibilities:
1. $\mathfrak{q}$ is real, in which case $F$ is totally real and $B$ ramifies at every infinite place other than $\mathfrak{q}$.
2. $\mathfrak{q}$ is complex, in which case $F$ is real and $B$ ramifies at every infinite place other than $\mathfrak{q}$.
3. $\mathfrak{q}$ is finite, in which case $F$ is totally real and $B$ is totally definite.
For each place $\mathfrak{p}$, define the compact open subgroup $J_\mathfrak{p} {\leqslant}\PB^\times_\mathfrak{p}$ as in §\[sec-2-4\] by taking for $J_\mathfrak{p}$ the image of $R_\mathfrak{p}^\times$ if $\mathfrak{p}$ is finite and taking $J_\mathfrak{p} := \PB^\times_\mathfrak{p}$ if $\mathfrak{p}$ is infinite. Set $J := \prod_{\mathfrak{p} \neq \mathfrak{q}} J_\mathfrak{p}$. In addition to the notation of §\[sec-4-1\], we now introduce a superscripted $J$, as in $\mathcal{A}^J, \mathcal{A}_0^J, \pi^J$ to denote the $J$-fixed subspace. We denote by $\mathcal{A}^J_+ \subseteq \mathcal{A}^J$, $\mathcal{A}^J_{0+} \subseteq \mathcal{A}_0^J$ the “even” subspaces consisting of $\varphi$ that are $\PB^\times_{\mathfrak{p}}$-invariant for all $\mathfrak{p} \in S - \{\mathfrak{q}\}$. Thus, for instance, $\mathcal{A}_{0+}^J
\subseteq \mathcal{A}_0^J
\subseteq \mathcal{A}_0^S
\subseteq \mathcal{A}_0 \subseteq \mathcal{A}$. We denote by $A_0, A^J, A_0^J, A^J_+, A^J_{0+}$ the set of all $\pi \in A$ having nonzero intersection with the space having the corresponding scripted notation.
Set $G := \PB^\times_\mathfrak{q}$, and let $f \in C_c^\infty(G)$. For $\mathfrak{p} \in S - \{q\}$, set $e_{J_\mathfrak{p}} :=
\operatorname{vol}(J_\mathfrak{p})^{-1} 1_{J_\mathfrak{p}} \in
C_c^\infty(\PB^\times_\mathfrak{p})$. Define $\tilde{f} \in
C_c^\infty(\PB^\times_S)$ by the formula $$\tilde{f} (g)
:=
f(g_\mathfrak{q})
\prod_{\mathfrak{p} \in S - \{\mathfrak{q} \}}
e_{J_\mathfrak{p}}(g_\mathfrak{p}).$$ For $\pi \in A_0^J$ and $\Psi \in \mathcal{A}_0^J$, set $\omega_\pi(f,\Psi) :=
\sum_{\varphi \in \mathcal{B}(\pi \cap \mathcal{A}^J)}
\langle \varphi, \Psi \cdot \pi(f) \varphi \rangle$. Then $\omega_\pi(\tilde{f},\Psi) = \omega_\pi(f,\Psi)$ (see §\[sec:omega-pi\]). Since $\dim(\pi_\mathfrak{p}) = 1$ for all $\mathfrak{p} \in S -
\{\mathfrak{q} \}$, one has for $\Psi \in \pi' \in \mathcal{A}_0^J$ that $\omega_\pi(f,\Psi) = 0$ unless $\pi ' \in \mathcal{A}_{0+}^J$.
Let $V_{f}, M_{f} : \mathcal{A}_{0+}^J
\otimes \mathcal{A}_{0+}^J \rightarrow \mathbb{C}$ denote the sesquilinear forms obtained by restricting the forms $\mathcal{V}_{\tilde{f}}, \mathcal{M}_{\tilde{f}} : \mathcal{A}_0^S \otimes
\mathcal{A}_0^S \rightarrow \mathbb{C}$. Then $$V_{f}(\Psi_1,\Psi_2)
=
\sum_{\pi \in A_0^J}
L^{(S)}(\operatorname{ad}\pi,1)
\omega_\pi(f,\Psi_1)
\overline{\omega_\pi(f,\Psi_2)}.$$ By the observation that $\mathfrak{S} e_{J_\mathfrak{p} } = e_{J_\mathfrak{p}}$ for $\mathfrak{p} \in S - \{\mathfrak{q}\}$ and the local calculations and , we see for $\Psi_1 \in \pi_1 \in A_{0+}^J$ and $\Psi_2 \in \pi_2 \in A_{0+}^J$ that $M_{f}(\Psi_1,\Psi_2) = 0$ unless $\pi_1 = \pi_2 =: \pi$, in which case $$M_{f}(\Psi_1,\Psi_2)
=
c_4
L^{(S)}(\pi,\tfrac{1}{2})
\int_{g \in G}
\langle \operatorname{Ad}(g) \mathfrak{S} f, \mathfrak{S} f \rangle_{L^2(G)}
\langle \pi(g) \Psi_1, \Psi_2 \rangle_{\PB^\times}$$ where $$\label{eq:defn-of-c4}
c_4 := 2^t
\zeta_F^{(S)}(2) \operatorname{vol}([\PB^\times])^{-1}.$$ with $t$ the number of finite primes $\mathfrak{p} \in S - \{\mathfrak{q}\}$.
### Strong approximation\[sec:strong-approx\] {#sec-4-4-2}
Retaining the notation of §\[sec:general-estimates-specialized-single-place\], we record here how the quotient $[\PB^\times]/J$ unadelizes under some assumptions. Recall that $G := \PB^\times_{\mathfrak{q}}$. Let $\Gamma {\leqslant}G$ denote the image of $\PB^\times \cap J$ under the inclusion $\PB^\times \hookrightarrow G$. Then $\Gamma$ is a discrete cocompact subgroup of $G$, and the natural map $$\label{eq:map-defining-adelic-orbit}
\Gamma \backslash G
\xrightarrow{\iota}
[\PB^\times]/J$$ is injective.
\[lem:consequence-of-strong-approx\] Suppose that $F$ has odd narrow class number and either that
1. $B_\mathfrak{q}$ is split, or that
2. $\mathfrak{q}$ is infinite and $B$ has class number one.
Then $\iota$ is bijective.
The class number assumption on $F$ implies that $$\label{eq:class-number-odd-definite-case}
F_+^\times
F_{\infty+}^\times
F_{\mathfrak{q}}^\times
\prod_{\mathfrak{p} < \infty}
\mathcal{O}_\mathfrak{p}^\times
\mathbb{A}^{\times 2}
=
\mathbb{A}_+^\times :=
F_{\infty+}^\times
\mathbb{A}_f^\times$$ where $F_{\infty+}^\times$ denotes the connected component of the unit group of $F_{\infty} := \prod_{\mathfrak{p}|\infty} F_\mathfrak{p}$, $\mathbb{A}_f^\times := \prod_{\mathfrak{p} < \infty}
F_\mathfrak{p}^\times$ denotes the group of finite ideles, and $F_+^\times := F^\times \cap F_{\infty+}^\times$. Set $J' := \PB^\times_{\mathfrak{p}_0} J$; it is open in $\PB^\times_{\mathbb{A}}$. Under the reduced norm map $\operatorname{nr}: \PB^\times_\mathbb{A}
\rightarrow \mathbb{A}^\times_+/\mathbb{A}^{\times 2}$, we have $\operatorname{nr}(\PB^\times) = F_+^\times \mathbb{A}^{\times
2}/\mathbb{A}^{\times 2}$ and $\operatorname{nr}(J') = F_{\infty+}^\times F_{\mathfrak{p}_0}^\times
\prod_{\mathfrak{p} < \infty}
\mathcal{O}_{\mathfrak{p}}^\times
\mathbb{A}^{\times 2} / \mathbb{A}^{\times 2}$; from our assumption , it follows that $$\label{eq:norm-relation-relevant-for-strong-approx}
\operatorname{nr}(\PB^\times) \operatorname{nr}(J') = \operatorname{nr}(\PB_\mathbb{A}^\times).$$ If $B_\mathfrak{q}$ splits, then $\PB^\times_\mathfrak{q}$ is non-compact, so the strong approximation theorem and imply that $\PB^\times J' = \PB_{\mathbb{A}}^\times$, hence that $\iota$ is surjective. In the remaining case, the surjectivity of $\iota$ holds by the definition of “the class number of $B$.”
\[rmk:classical-volume-computation\] Suppose the conclusion of the lemma holds and that $\mathfrak{q}$ is finite. It is then natural to ask for the volume of $\Gamma \backslash G$ with respect to a Haar $\nu$ on $\Gamma \backslash G$ obtained as the quotient of some given Haar $\mu$ on $G$. Using that $\operatorname{vol}([\PB^\times]) = 2$ with respect to Tamagawa measure, one can show (as in §\[sec:mean-statistics\] below) that $$\nu(\Gamma \backslash G) = 2 \frac{
\zeta_F(2)
\Delta_B
\Delta_F^{3/2}
}{ (4 \pi^2)^{[F:\mathbb{Q}]} \prod_{\mathfrak{p} \in
\operatorname{ram}_f(B)} \zeta_\mathfrak{p}(1)} \mu(K_{\mathfrak{q}}),$$ where $K_{\mathfrak{q}} {\leqslant}G$ denotes a maximal compact subgroup, $\Delta_F$ the absolute discriminant, $\Delta_B$ the absolute reduced discriminant, and $\operatorname{ram}_f(B)$ the set of finite places at which $B$ ramifies.
### Hecke operators\[sec:hecke-ops\] {#sec-4-4-3}
For each finite place $\mathfrak{p} \neq \mathfrak{q}$, let $\mathcal{H}_{\mathfrak{p}}$ denote the Hecke algebra consisting of compactly-supported bi-$J_{\mathfrak{p}}$-invariant distributions on $\PB^\times_{\mathfrak{p}}$. It acts on $\mathcal{A}^J$. It admits a standard generator $T_{\mathfrak{p}}$ given by convolution against $|\varpi|^{-1} T_{\varpi} \in C_c^\infty(J_\mathfrak{p}^\times
\backslash \PB_\mathfrak{p}^\times / J_\mathfrak{p}^\times)$, where $\varpi$ is a generator of the maximal ideal in $\mathcal{O}_\mathfrak{p}$ and $T_{\varpi}$ is the normalized Hecke kernel defined in §\[sec:hecke-kernels-local\]. If $B$ ramifies at $\mathfrak{p}$, then $T_{\mathfrak{p}}$ is an involution; otherwise, it has degree $|\mathfrak{p}| + 1$, where $|\mathfrak{p}|$ denotes the absolute norm.
The following are equivalent for $\varphi \in \mathcal{A}^J$.
1. $\varphi$ generates an irreducible representation of $G$ and is an eigenfunction of $T_\mathfrak{p}$ for each finite $\mathfrak{p} \neq \mathfrak{q}$.
2. $\varphi$ generates an irreducible representation $\pi$ of $\PB^\times_\mathbb{A}$.
Assume that these conditions hold. Then $\pi \in A^J$, and the following are equivalent:
1. $T_\mathfrak{p} \varphi = \varphi$ for each finite prime $\mathfrak{p} \neq \mathfrak{q}$ at which $B$ ramifies.
2. $\pi \in A^J_+$.
In view of the lemma, the following definition faithfully extends and makes precise Definition \[defn:eigenfunctions-intro\] of §\[sec:setting-overview\]:
\[defn:eigenfunction-general\] An *eigenfunction* is a nonzero element $\varphi \in \mathcal{A}^J$ that belongs to some $\pi \in A^J$; it is *even* if $\pi \in A^J_+$ and *strongly of mean zero* if $\pi \in A^J_{0}$ (and both if $\pi \in A^J_{0+}$).
Our aim is now to prove Theorem \[thm:main-result-for-microlocal-stuff\] by application of Theorem \[thm:main-estimate-general-variance\]. We retain the general notation of §\[sec:general-notation\]. The following additional notation is in effect for §\[sec:appl-micro-local-prelims\]–§\[sec:local-estimates-error\]:
- $k$: a non-archimedean local field of characteristic $\neq 2$.
- $\mathfrak{o}, \mathfrak{q}, q, |.| = |.|_k, \operatorname{ord}= \operatorname{ord}_k$: as in §\[sec-2-1-1\].
- $B:= M_2(k)$: the algebra of $2 \times 2$ matrices, so that $\operatorname{nr}= \det : B \rightarrow k$.
- $G := B^\times / k^\times = \PGL_2(k)$.
Equip $G$ with any Haar measure; we choose it more precisely when we must.
Preliminaries\[sec:appl-micro-local-prelims\] {#sec-5}
=============================================
Conductors {#sec-5-1}
----------
Let $c(\omega)$ denote the (log-)conductor of a character $\omega : \mathfrak{o}^\times \rightarrow \mathbb{C}^\times$, that is, the smallest nonnegative integer $n$ for which $\omega$ has trivial restriction to $\mathfrak{o}^\times \cap (1 + \mathfrak{q}^n)$. Note that $c(\omega) = 0$ precisely when $\omega = 1$. Let $c(\chi) := c(\chi|_{\mathfrak{o}^\times})$ denote the conductor of a character $\chi : k^\times \rightarrow \mathbb{C}^\times$.
Principle series representations {#sec-5-2}
--------------------------------
For a character $\chi : k^\times \rightarrow \mathbb{C}^\times$, we denote by $\chi \boxplus \chi^{-1}$ the corresponding induced representation; it consists of smooth functions $v : G \rightarrow \mathbb{C}$ satisfying $v(n(x) a(y) g) = |y|^{1/2} \chi(y) v(g)$ for all $x,y,g \in k,k^\times,G$, with the group $G$ acting by right translation. If $c(\chi^2) \neq 0$, then $\chi \boxplus \chi^{-1}$ is irreducible and generic.
Microlocal lifts {#sec-5-3}
----------------
We recall the specialization to trivial central characters of the main definition from [@nelson-padic-que] and record some basic properties.
### Definition {#sec-5-3-1}
For each integer $N > \operatorname{ord}_k(2)$, fix a decomposition $N - \operatorname{ord}_k(2) = N_1 + N_2$ into nonnegative integers $N_1, N_2$ that tend to $\infty$ with $N$. For $N$ large enough, we assume for the sake of consistency with §\[sec:setting-overview\] that $N_1 = \lfloor N/2 \rfloor - \operatorname{ord}_k(2), N_2 = \lceil N/2 \rceil$.
Let $\mathcal{A}$ be a representation of $G$. We say that a vector $\varphi \in \mathcal{A}$ is a *microlocal lift* if
1. $\varphi \neq 0$,
2. $\varphi$ generates an irreducible subrepresentation $\pi \subseteq A$, and
3. there is a character $\omega$ of $\mathfrak{o}^\times$ whose square $\omega^2$ is nontrivial so that with $N := c(\omega)$ and for all $$\label{eq:microlocal-lift-subgroup-defn}
g = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in
\operatorname{GL}_2(\mathfrak{o}) \cap \begin{pmatrix}
\mathfrak{o} & \mathfrak{q}^{N_1} \\
\mathfrak{q}^{N_2} & \mathfrak{o}
\end{pmatrix},
$$ one has $g \varphi = \omega(a^2/\operatorname{nr}(g)) \varphi$.
We call $\omega$ the *orientation* of $\varphi$; it is determined by $\varphi$.
We refer again to [@nelson-padic-que Thm 25, Rmk 26] for a discussion of the sense in which the microlocal lifts considered here are actually “lifts.”
The quantity denoted $N$ in [@nelson-padic-que] has been renamed here to $N - \operatorname{ord}_k(2)$ in order to make the formulas for the variance statistics hold more uniformly when $k$ extends $\mathbb{Q}_2$. Amusingly, the convention chosen here is less natural from the perspective of linear statistics (see §\[sec:mean-statistics\]).
### Classification {#sec-5-3-2}
Let $\pi$ be an irreducible representation of $G$. By [@nelson-padic-que Lem 22], we have:
\[lem:determination-microlocal-lifts\]
1. Suppose that $\pi$ is isomorphic to $\chi \boxplus \chi^{-1}$ for some character $\chi$ of $k^\times$ for which $c(\chi^2) \neq 0$. Then the set of microlocal lifts in $\pi$ is a disjoint union $\mathbb{C}^\times \varphi_+ \bigsqcup \mathbb{C}^\times \varphi_-$, where $\varphi_{+}$ and $\varphi_-$ are microlocal lifts of orientations $\omega := \chi|_{\mathfrak{o}^\times}$ and $\omega^{-1}$, respectively.
2. Otherwise, $\pi$ contains no microlocal lifts.
\[cor:get-nice-subspaces-of-microlocal-lifts-assuming-no-inverses\] Let $X$ be a set consisting of characters $\omega$ of $\mathfrak{o}^\times$ for which $\omega^2 \neq 1$. Assume that $\omega \in X \implies \omega^{-1}
\notin X$. The set of microlocal lifts in $\pi$ with orientation in $X$ is then either empty or of the form $\mathbb{C}^\times \varphi$ for some $0 \neq \varphi \in \pi$.
### Projectors {#sec-5-3-3}
Let $\omega$ be a character of $\mathfrak{o}^\times$ for which $\omega^2$ is nontrivial. Set $N := c(\omega)$. Let $\widetilde{\mathfrak{J}}$ denote the subgroup of $\operatorname{GL}_2(\mathfrak{o})$ appearing on the RHS of . Let $\mathfrak{J} {\leqslant}G$ denote the image of $\widetilde{\mathfrak{J}}$ under the projection $\operatorname{GL}_2(k)
\rightarrow G$. By matrix multiplication and the inequalities $\min(N_1,N_2) {\geqslant}0$, $\max(N_1,N_2) {\geqslant}1$, one has for $x_1,x_2 \in k$ and $y \in k^\times$ that $$\label{eqn:explicit-description-of-J-bar}
n'(x_2) n(x_1) a(y) \in
\mathfrak{J}
\iff
x_1 \in \mathfrak{q}^{N_1},
x_2 \in \mathfrak{q}^{N_2},
y \in \mathfrak{o}^\times$$ Define $f_\omega \in C_c^\infty(G)$ as follows, for $g \in G$:
- If $g \notin \mathfrak{J}$, then $f_\omega(g) := 0$.
- If $g \in \mathfrak{J}$ is image of $\tilde{g} = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in \widetilde{\mathfrak{J}}$, then $f_\omega(g) := \operatorname{vol}(\mathfrak{J})^{-1}
\omega(\operatorname{nr}(\tilde{g})/a^2)$.
The definition is independent of the choice of $\tilde{g}$.
\[lem:harmonic-analytic-isolation-0\] Let $\pi$ be an irreducible unitary representation of $G$.
1. If $\pi$ contains a microlocal lift $\varphi$ of orientation $\omega$, then $\pi(f_\omega)$ is the rank one orthogonal projector onto the subspace $\mathbb{C} \varphi$ of $\pi$.
2. Otherwise, $\pi(f_\omega) = 0$.
$f_\omega$ is supported on $\mathfrak{J}$. By direct calculation, the restriction of $\operatorname{vol}(\mathfrak{J}) f_\omega$ to $\mathfrak{J}$ defines a unitary character $\xi$ of $\mathfrak{J}$ with the property that condition (iii) in §\[sec-5-3-1\] reads “$g \varphi = \xi^{-1}(g) \varphi$ for all $g \in \mathfrak{J}$.” The conclusion follows from elementary Fourier analysis on the compact group $\mathfrak{J}$.
The convolution kernel of interest
----------------------------------
### Taylor expansion of the logarithm {#sec:tayl-expans-logar}
Let $N,N_0$ be positive integers. Assume that $N_0$ is large enough in terms of $\operatorname{ord}_k(2)$ and that $N$ is large enough in terms of $N_0$ that $$\label{eq:basic-N-N0-ord-2-assumptions}
N > N_0 > \operatorname{ord}_k(2),$$ $$\label{eq:basic-N-N0-ord-2-assumptions-2}
N {\geqslant}2 N_0 + \operatorname{ord}_k(2).$$ In applications, we take $N_0$ large enough but fixed, and let $N \rightarrow \infty$.
Set $\mathfrak{o}^\times_0 := 1 + \mathfrak{q}^{N-N_0}$, regarded as a subgroup of $\mathfrak{o}^\times$. Then $$\label{eqn:index-of-O-0}
[\mathfrak{o}^\times:\mathfrak{o}_0^\times]
= q^{N-N_0} \zeta_k(1)^{-1}.$$ Regard $1 + \mathfrak{q}^N$ as a subgroup of $\mathfrak{o}^\times_0$. Fix a uniformizer $\varpi$, i.e., a generator of the $\mathfrak{o}$-ideal $\mathfrak{q}$.
\[lem:additive-vs-mult-over-local-field-local-taylor-sorta\] The map $$\iota :
\mathfrak{o}^\times_0 / (1 + \mathfrak{q}^{N})
\rightarrow
\mathfrak{q}^{-N_0} / \mathfrak{o}$$ $$\iota(u) := \varpi^{-N} (u-1)$$ is an isomorphism of groups.
By inspection, $\iota$ is well-defined and has a well-defined inverse $\iota^{-1}$, hence is bijective. For $x,y \in \mathfrak{q}^{-N_0}/\mathfrak{o}$, one has $\iota^{-1}(x) \iota^{-1}(y) = \iota^{-1}(x+y) + \eps$ with $\eps := \varpi^{2 N} x y$. By , one has $\eps \in \mathfrak{q}^{2 N - 2 N_0} \subseteq
\mathfrak{q}^N$, so $\iota^{-1}$ is a homomorphism.
### Partitioning the characters of given conductor {#sec-5-4}
Motivated by the corollary of §\[sec-5-3-2\], we record here a partition of the characters of $\mathfrak{o}^\times$ having given conductor into nice subsets that do not contain the inverses of any characters that they contain. Recall that a unitary character $\sigma : k \rightarrow
\mathbb{C}^{(1)}$ is *unramified* if it is trivial on $\mathfrak{o}$ but not on $\mathfrak{q}^{-1}$. Let $\Sigma$ denote the set of equivalence classes of unramified unitary characters $\sigma$ of $k$, with two such characters declared equivalent if they have the same restriction to $\mathfrak{q}^{-N_0}$. For any $\sigma \in \Sigma$, the map $$\mathfrak{o}^\times / (1 + \mathfrak{q}^{N_0}) \rightarrow
\Sigma$$ $$\xi \mapsto \text{ (the class of the character
$[x \mapsto \sigma(\xi x)]$)}$$ is bijective, so $\Sigma$ is a finite set of cardinality $$\label{eq:Sigma-cardinality}
|\Sigma| = \zeta_k(1)^{-1} q^{N_0}.$$
For each $\sigma \in \Sigma$, let $\omega_\sigma : \mathfrak{o}^\times_0 \rightarrow
\mathbb{C}^\times$ denote the function defined by $\omega_\sigma(u) := \sigma (\iota(u))$. Both $\omega_\sigma$ and the association $\Sigma \ni \sigma \mapsto \omega_{\sigma}$ are then well-defined. By the lemma of §\[sec:tayl-expans-logar\], we see that
- $\omega_\sigma$ is a character of $\mathfrak{o}_0^\times$ of conductor $N$ (in the sense that it has trivial restriction to $1 + \mathfrak{q}^{N}$ but not to $1 + \mathfrak{q}^{N-1}$), and that
- each character of $\mathfrak{o}^\times_0$ of conductor $N$ is of the form $\omega_\sigma$ for some unique $\sigma \in \Sigma$.
Let $\mathcal{X}_N$ denote the set of characters $\omega$ of $\mathfrak{o}^\times$ for which $c(\omega) = N$. Since each such $\omega$ restricts to a character of $\mathfrak{o}_0^\times$, we have a partition $$\label{eq:partition-of-chars-into-packets}
\mathcal{X}_N = \bigsqcup_{\sigma \in \Sigma}
\mathcal{X}_N^\sigma$$ where $\mathcal{X}_N^\sigma := \{\omega \in \mathcal{X}_N : \omega|_{\mathfrak{o}^\times_0} = \omega_{\sigma}\}$.
\[lem:properties-of-partition-of-characters\] $\omega \in \mathcal{X}_N^\sigma \implies \omega^{-1} \notin
\mathcal{X}_N^\sigma$.
If $\omega, \omega^{-1} \in \mathcal{X}_N^\sigma$ then $\omega_\sigma = \omega|_{\mathfrak{o}_0^\times} =
\omega_\sigma^{-1}$, hence $\omega_\sigma^2 = 1$, hence $\sigma(2 x) = 1$ for all $x \in \mathfrak{q}^{-N_0}$, contrary to our assumptions that $\sigma$ is unramified and $N_0 > \operatorname{ord}_k(2)$.
### Definition {#sec:element-attached-to-N-sigma}
By *the element $f \in C_c^\infty(G)$ attached to $(N,\sigma)$* we shall mean the function $$f := \sum_{\omega \in \mathcal{X}_N^{\sigma}}
f_\omega,$$ where $f_\omega$ is as in §\[sec-5-3-3\]. The element $f$ depends also upon $N_0$; we regard that element as fixed in applications, and so omit its dependence from the terminology.
\[prop:harmonic-analytic-isolation-1\] Let $f \in C_c^\infty(G)$ be attached to $(N,\sigma)$. Let $\pi$ be an irreducible unitary representation of $G$.
1. If $\pi$ contains a microlocal lift $\varphi$ of orientation $\omega$ for some $\omega \in \mathcal{X}_N^\sigma$, then $\pi(f)$ is the rank one orthogonal projector onto the subspace $\mathbb{C} \varphi$ of $\pi$.
2. Otherwise, $\pi(f) = 0$.
By combining the lemmas of §\[lem:determination-microlocal-lifts\], §\[sec-5-3-3\], §\[sec-5-4\].
The $\heartsuit^{\tau}$ operator {#sec:defn-heartsuit-local-again-without-Omega}
--------------------------------
For $\tau \in k^\times$, define $\heartsuit^{\tau} : C_c^\infty(G) \rightarrow
\mathcal{S}(B)$ by the formula $\heartsuit^{\tau} f (g) :=
1_{\mathfrak{o}^\times}(\tau \operatorname{nr}(g)) f(g)$. (This is a variant of the definition of §\[sec:heartsuit\], adapted to the local setting.)
Fourier transforms of convolution kernels\[sec:fourier-transf-conv-kern\] {#sec-6}
=========================================================================
We study here the “Fourier transform” of the $f \in C_c^\infty(G)$ attached to $(N,\sigma)$ (see §\[sec:element-attached-to-N-sigma\]).
This section may be safely skipped on a first reading: the results stated here are used in the proofs of the main results of §\[sec:local-estimates-main-statements\] and §\[sec:local-estimates-error\], but the details of those proofs are not needed to understand the overall structure of the proof given in §\[sec:deduction-main-thm-microlocal\] of the main result of the article.
Throughout this section, we fix $\sigma \in \Sigma$ and choose an arbitrary unramified unitary character $\psi : k
\rightarrow \mathbb{C}^{(1)}$ belonging to the class $\sigma$.
Measures\[sec:measures-for-fourier-computation\] {#sec-6-1}
------------------------------------------------
We use $\psi$ to define measures on $k, B, G$ as in §\[sec:local-measures\]. The measures so obtained on $k$ and $B$ assign volume one to maximal compact subrings. The measure on $G$ satisfies the integral formulas , .
\[lem:volume-of-J-bar\] $\operatorname{vol}(\mathfrak{J}) = |2|_k^{-1} q^{-N} \zeta_k(1)^{-1}$.
This follows from the description of $\mathfrak{J}$ given by , the integral formula , the local volume formulas (§\[sec:local-vol-formulas\]), and the consequence $q^{-N_1 - N_2} = |2|_k^{-1} q^{-N}$ of the definitions of $N_1, N_2$.
Coordinates {#sec-6-2}
-----------
On $B$, we employ the coordinates $$\label{eq:x-xi-in-fancy-notation-etc}
x = \begin{pmatrix}
d - a/2 & b \\
c & d + a/2
\end{pmatrix},
\quad
\xi = \begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix}.$$ By a simple change of variables, one verifies that the Haar measure on $B$ is given in either set of coordinates by integrating over the coordinate variables with respect to the chosen Haar measure on $k$. The main involution $\iota : B \rightarrow B$ is given by $(a,b,c,d) \mapsto (-a,-b,-c,d)$, $(\alpha,\beta,\gamma,\delta) \mapsto
(-\alpha,-\beta,-\gamma,\delta)$ and the Fourier transform $\mathcal{F} : \mathcal{S}(B) \rightarrow
\mathcal{S}(B)$, defined as in §\[sec:local-measures\] or §\[sec:local-quadratic-spaces\] by $\mathcal{F} \phi(\xi)
:=
\int_{x \in B}
\phi(x)
\psi(\langle x,\xi \rangle)
\, d x$ with $\langle x, \xi \rangle := \operatorname{tr}(x^{\iota} \xi)$, by $$\label{eq:fourier-transform-definition-spelled-out}
\mathcal{F} \phi(\xi )
=
\int_{a,b,c,d \in k}
\phi(x)
\psi(\alpha a + \delta d
- \beta c - \gamma b).$$
The general notation of §\[sec:general-notation\] gives us for each space $V \in \{C_c^\infty(G), \mathcal{S}(B)\}$ maps $\mathfrak{S} : V \rightarrow V$ and $\operatorname{Ad}(g) : V \rightarrow V$ for $g \in G$. These maps commute with each other and with $\mathcal{F}, \heartsuit^{\tau}$ in every conceivable sense: $\mathfrak{S} \heartsuit^{\tau} = \heartsuit^{\tau} \mathfrak{S}$, $\mathfrak{S} \operatorname{Ad}(g) = \operatorname{Ad}(g) \mathfrak{S}$, $\heartsuit^{\tau} \operatorname{Ad}(g) = \operatorname{Ad}(g) \heartsuit^{\tau}$, $\mathcal{F} \operatorname{Ad}(g) = \operatorname{Ad}(g) \mathcal{F}$, $\mathcal{F}
\mathfrak{S} = \mathfrak{S} \mathcal{F}$.
Statement of result {#sec-6-3}
-------------------
Asymptotic notation here refers to the $N \rightarrow \infty$ limit. Implied constants depend at most upon the field $k$ and the integer $N_0$.
\[prop:key-fourier-estimate-microlocal-kernel\] Let $f \in C_c^\infty(G)$ be attached to $(N,\sigma)$ (see §\[sec:element-attached-to-N-sigma\]). Set $$\label{eqn:phi-symmetrized-fourier-transform-of-f}
\phi := \mathcal{F} \mathfrak{S} \heartsuit^1 f =
\mathfrak{S} \mathcal{F} \heartsuit^1 f =
\mathcal{F} \heartsuit^1 \mathfrak{S} f
\in \mathcal{S}(B).$$ Define $\Phi : k^4 \rightarrow \mathbb{C}$ in terms of the coordinates by $\Phi(\alpha,\beta,\gamma,\delta) := \phi(\xi)$. One then has $\Phi(\alpha,\beta,\gamma,\delta) \neq 0$ only if $$\label{eq:asymptotic-properties-of-entries-of-support-of-fourier-transform}
|\alpha| \asymp q^N, \quad
|\beta|, |\gamma| = o(q^N), \quad
|\delta| = O(1).$$ One has $$\label{eq:some-invariance-of-fourier-transform-udner-add-and-multiply}
\Phi(u \alpha,u \beta,u \gamma,\delta)
=
\Phi(\alpha,\beta,\gamma,\delta) \text{ for all }
u \in 1 + \mathfrak{q}^{N_0}.$$ The function $I(\alpha,\delta) :=
\int_{\beta,\gamma \in k} \Phi(\alpha,\beta,\gamma,\delta)$ satisfies $$\label{eq:weyl-invariance-fourier-transform-explicated-coordinates}
I(\alpha,\delta) = I(-\alpha,\delta)$$ and $$\label{eq:I-rescaled-independent-of-N}
q^{-N} I(\varpi^{-N} \alpha,\delta) \text{ is independent of $N$}.$$ Moreover, $$\label{eq:explicit-normalizing-computation}
\int_{\alpha,\delta \in k}
|2 \alpha|^{-2}
\left\lvert
I(\alpha,\delta)
\right\rvert^2
=
C q^{N-N_0}$$ with $C := (2 \zeta_k(1))^{-1}$.
We shall subsequently refer only to the properties laid out in the statement of Proposition \[prop:key-fourier-estimate-microlocal-kernel\], but it may be instructive to record that if $q$ is odd, one can establish (extending the proof of Proposition \[prop:key-fourier-estimate-microlocal-kernel\]) the explicit formula $$\Phi(\alpha,\beta,\gamma,\delta) =
\frac{q^{- N_0}}{\zeta_k(1)}
1_{\varpi^{-N} \mathfrak{o}^\times}(\alpha)
1_{\mathfrak{q}^{-N_2}}(\beta)
1_{\mathfrak{q}^{-N_1}}(\gamma)
1_{\mathfrak{q}^{-N_0}}(\delta)
\operatorname{co}(\delta / (\varpi^N \alpha)),$$ where $\operatorname{co}(t) := (\psi(t) + \psi(-t))/2$. Otherwise, $k$ is a finite extension of $\mathbb{Q}_2$, and a similar but more complicated formula holds.
Proofs\[sec:main-comp-fourier-pfs\] {#sec-6-4}
-----------------------------------
The purpose of this section (which may be safely skipped on a first reading) is to prove Proposition \[prop:key-fourier-estimate-microlocal-kernel\].
\[lem:easy-formula-for-f-yay\] Let $f$ be attached to $(N,\sigma)$. Let $x \in B$. In the coordinates , $$\label{eq:easy-formula-for-f-yay}
\heartsuit^1 f(x)
=
C_0
1_{\mathfrak{o}^\times}(d)
1_{\mathfrak{q}^{N-N_0}}(a)
1_{\mathfrak{q}^{N_1}}(b)
1_{\mathfrak{q}^{N_2}}(c)
\psi (\frac{a d - b c}{\varpi^N d^2})$$ where $C_0 := q^{2 N - N_0} |2|_k$.
By and the lemma of §\[lem:volume-of-J-bar\], $$\label{eqn:compute-volumes-J-etc}
C_0 = [\mathfrak{o}^\times:\mathfrak{o}_0^\times]
\operatorname{vol}(\mathfrak{J})^{-1}.$$ By Fourier analysis on $\mathfrak{o}^\times$, we have for $u \in \mathfrak{o}^\times$ the expansion $$\label{eq:fourier-expansion-of-zeta-local}
[\mathfrak{o}^\times:\mathfrak{o}_0^\times]
1_{\mathfrak{o}_0^\times}(u) \omega_{\sigma}(u)
= \sum_{\omega \in \mathcal{X}_N^\sigma} \omega(u).$$ Applying and to the definition of $f$ gives $\heartsuit^1 f(x) = C_0 \kappa \psi(\zeta)$, where $$\kappa :=
1_{\mathfrak{q}^{N_1}}(b)
1_{\mathfrak{q}^{N_2}}(c)
1_{\mathfrak{o}^\times}(d - a/2)
1_{\mathfrak{o}^\times}(d + a/2)
1_{\mathfrak{q}^{N-N_0}}
(\frac{\operatorname{nr}(g)}{(d-a/2)^2} - 1),$$ $$\zeta
:=
\varpi^{-N}
(
\frac{\operatorname{nr}(x)}{(d-a/2)^2} - 1
).$$ By our assumption on the largeness of $N$ relative to $N_0$ and identifies such as $$\label{eq:first-determinant-expansion-in-annoying-f-initial-lemma}
\frac{\operatorname{nr}(x)}{(d-a/2)^2} - 1
= \frac{a}{d-a/2}
- \frac{b c}{(d - a/2)^2},$$ we verify directly that $$\label{eq:basic-congruence-equivalence-initial-study-of-f}
\kappa
=
1_{\mathfrak{o}^\times}(d)
1_{\mathfrak{q}^{N-N_0}}(a)
1_{\mathfrak{q}^{N_1}}(b)
1_{\mathfrak{q}^{N_2}}(c).
$$ Similarly, we verify using that for $a,b,c,d$ in the support of the RHS of , the congruences $$\frac{a}{d-a/2}
\equiv \frac{a d}{d^2} \pmod{\mathfrak{q}^N},
\quad
\frac{b c}{(d-a/2)^2}
\equiv \frac{b c}{d^2} \pmod{\mathfrak{q}^N}$$ hold. It follows from these and that $$\kappa \psi(\zeta)
= \kappa \psi(\frac{a d - b c}{2 \varpi^N d^2}).$$ This completes the proof of .
With notation as in the lemma, the quantity $f(x)$ depends only upon the congruence classes $$a \pmod{\mathfrak{q}^{N}},
\quad
b \pmod{2 \mathfrak{q}^{N_1}},
\quad
c \pmod{2 \mathfrak{q}^{N_2}},
\quad
d \pmod{\mathfrak{q}^{N_0}}.$$
Immediate; it may help to recall that $N = \operatorname{ord}_k(2) + N_1 + N_2$.
We now prove Proposition \[prop:key-fourier-estimate-microlocal-kernel\]. By the formula for $\heartsuit^1 f$ given in the lemma and the explication of the Fourier transform, we have $$\Phi(\alpha,\beta,\gamma,\delta)
=
\int_{a,b,c,d \in k}
F(a,b,c,d)
\psi(
\alpha a + \delta d
- \beta c - \gamma b),$$ where $$\label{eq:defn-F-explicated-f}
F(a,b,c,d)
:=
C_0
1_{\mathfrak{o}^\times}(d)
1_{\mathfrak{q}^{N-N_0}}(a)
1_{\mathfrak{q}^{N_1}}(b)
1_{\mathfrak{q}^{N_2}}(c)
\operatorname{co}(\frac{a d - b c}{\varpi^N d^2}),$$ with $\operatorname{co}(x) :=
( \psi (x)
+
\psi (-x))/
2$. The smoothness properties of $f$ (hence of $F$), as enunciated in the corollary, imply corresponding decay properties of its Fourier transform, namely that $\Phi(\alpha,\beta,\gamma,\delta)
\neq 0$ only if $\alpha \in \mathfrak{q}^{-N},
\beta \in 2^{-1} \mathfrak{q}^{-N_2},
\gamma \in 2^{-1} \mathfrak{q}^{-N_1},
\delta \in \mathfrak{q}^{-N_0}$. We thereby obtain all assertions in except for the lower bound on $|\alpha|$. To establish the latter, we compute for $d \in \mathfrak{o}^\times$ that $$\label{eq:computation-relevant-for-alpha-support-of-fourier-transform}
\int_{a \in k}
1_{\mathfrak{q}^{N-N_0}}(a)
\operatorname{co}(\frac{a d }{\varpi^N d^2})
\psi(\alpha a)
=
q^{N_0-N}
\rho(\alpha,d),$$ where $$\rho(\alpha,d) :=
\frac{1_{\mathfrak{q}^{N_0-N}}(\alpha + \frac{1}{
\varpi^N d}) +
1_{\mathfrak{q}^{N_0-N}}(\alpha - \frac{1}{\varpi^N d})
}{2
}.$$ One has $\rho(\alpha,d) \neq 0$ only if $\alpha \in \pm \frac{1}{\varpi^N d} +
\mathfrak{q}^{N_0 - N}
\subseteq \frac{1}{\varpi ^N} \mathfrak{o}^\times$, in which case $|\alpha| \asymp q^N$.
The invariance is equivalent to the identity $F(u a, u b, u c,d) = F(a,b,c,d)$ for $u \in 1 + \mathfrak{q}^{N_0}$, which follows from and the congruence $u a d - u^2 b c \equiv a d - b c \pmod{\mathfrak{q}^N}$ for $(a,b,c,d) \in \operatorname{supp}(F)$.
We now verify . By Fourier inversion, $$\label{eq:I-of-alpha-delta-after-Fourier-inversion}
I(\alpha,\delta)
=
\int_{a,d \in k}
F(a,0,0,d)
\psi(\alpha a + \delta d).$$ Thus reduces to $F(a,0,0,d) = F(-a,0,0,d)$, follows from .
To establish , we recall the definition of $C_0$ from the lemma and apply to and the change of variables $a \mapsto \varpi^N a$, giving $$I(\varpi^{-N} \alpha,\delta) =
C'
q^N
\int_{a,d \in k}
1_{\mathfrak{o}^\times}(d)
1_{\mathfrak{q}^{-N_0}}(a)
\operatorname{co}(a/d)
\psi(\alpha a + \delta d)$$ for some unimportant scalar $C' = |2| q^{-N_0}$ that does not depend upon $N$.
We turn finally to . Denote temporarily by $\mathcal{I}$ its LHS. By and Parseval, $$\label{eq:normalizing-computation-consequence-of-parseval}
\mathcal{I} =
\int_{\alpha,d \in k}
|2 \alpha|^{-2}
\left\lvert
\int_{a \in k}
F(a,0,0,d)
\psi(\alpha a)
\right\rvert^2.$$ We compute with the help of that $$\int_{a \in k}
F(a,0,0,d)
\psi(\alpha a)
=
q^{N_0-N} C_0
1_{\mathfrak{o}^\times}(d)
\rho(\alpha,d).$$ Substituting this and the identity $q^{N_0-N} C_0 = q^N |2|_k$ into gives $$\mathcal{I} =
q^{2 N}
\int_{\alpha,d \in k}
|\alpha |^{-2} 1_{\mathfrak{o}^\times}(d)
|\rho(\alpha,d)|^2.$$ We substitute $\alpha \mapsto \varpi^{-N} \alpha$; since $|\varpi^{-N}|^{-1} = q^{-N}$, we obtain $$\mathcal{I} =
q^{N}
\int_{\alpha,d \in k}
|\alpha|^{-2}
1_{\mathfrak{o}^\times}(d)
\left\lvert \frac{1_{\mathfrak{q}^{N_0}}(\alpha + 1/d)
+
1_{\mathfrak{q}^{N_0}}(\alpha - 1/d)
}{
2}
\right\rvert^2.$$ By the consequence $2 \notin \mathfrak{q}^{N_0}$ of the assumption , we have $$1_{\mathfrak{o}^\times}(d)
1_{\mathfrak{q}^{N_0}}(\alpha + 1/d)
1_{\mathfrak{q}^{N_0}}(\alpha - 1/d) = 0,$$ whence $$\begin{aligned}
\mathcal{I}
&= q^N
\int_{\alpha,d \in k}
|\alpha|^{-2}
1_{\mathfrak{o}^\times}(d)
\frac{1_{\mathfrak{q}^{N_0}}(\alpha + 1/d)
+
1_{\mathfrak{q}^{N_0}}(\alpha - 1/d)
}{
2^2}
\\
&=
\frac{2 q^{N-N_0} ( 1 - \tfrac{1}{q}) }{2^2}
=
\frac{q^{N-N_0} }{2 \zeta_k(1)}
\end{aligned}$$ We thereby arrive at with the constant $$C =
q^{N_0-N}
\cdot
\frac{q^{N-N_0} }{2 \zeta_k(1)}
=
\frac{1}{2 \zeta_k(1)},$$ as required.
Estimates for the main term\[sec:local-estimates-main-statements\] {#sec-7}
==================================================================
Statement of result {#sec-7-1}
-------------------
Let $E$ denote the diagonal subalgebra of $B$. Let $H {\leqslant}G$ denote the image of $E^\times$. Thus $H = \{a(y) : y \in k^\times \}$ is the diagonal split torus. Let $N(H)$ denote the normalizer in $G$ of $H$. Let $$W := \left\{ \begin{pmatrix}
1 & \\
& 1
\end{pmatrix},
\begin{pmatrix}
& 1 \\
1 &
\end{pmatrix}\right\}$$ denote the Weyl group $W \cong N(H) / H$. Equip $H$ and $N(H)$ the measures $$\int_H f
:=
\int_{y \in k^\times}
f(y) \, \frac{d y}{|y|},
\quad
\int_{N(H)}
f :=
\sum_{w \in W} \int_{h \in H}
f(w h),$$ where $d y$ denotes (as in §\[sec:measures-for-fourier-computation\]) the Haar measure on $k$ assigning volume one to $\mathfrak{o}$.
\[prop:desired-main-term-identity-do-it-up\] Let $\Psi : G \rightarrow \mathbb{C}$ be a function with the following properties:
1. There is an open subgroup $U$ of $G$ so that $$\label{eq:matrix-coeff-bi-invariance}
\Psi(u_1 g u_2) = \Psi(g) \text{ for all $u_1,u_2 \in U$, $g \in G$.}$$
2. One has (see §\[sec:local-Xi\] concerning $\Xi$) $$\label{eq:matrix-coeff-decay-weak-generic}
\Psi(g) \ll \Xi(g)^{\delta} \text{ for some } \delta > 0.$$
Let $N$ be a positive integer taken sufficiently large in terms of $\Psi$. Let $f \in C_c^\infty(G)$ be attached to $(N,\sigma)$ for some $\sigma \in \Sigma$ (see §\[sec:element-attached-to-N-sigma\]). Then $$\label{eqn:main-term-what-we-wanna-show-just-after-reducing-to-local-problem}
\int_{g \in G}
\langle \operatorname{Ad}(g) \mathfrak{S} f, \mathfrak{S} f
\rangle_{L^2(G)}
\Psi(g)
=
q^{N-N_0} \frac{1}{2}
\int_{N(H)} \Psi.$$
The condition implies the absolute convergence of both sides of (see §\[sec:local-convergence-lemmas\]).
\[rmk:main-term-thm-applies-to-mx-coefs\] Let $\pi$ be an irreducible unitary representation of $G$ with $\dim(\pi) > 1$, and let $v_1,v_2 \in \pi$. The hypotheses of Proposition \[prop:desired-main-term-identity-do-it-up\] are then satisfied by $\Psi(g) := \langle g v_1, v_2 \rangle$ (see §\[sec:local-Xi\]).
The LHS of is independent of the choice of Haar measure on $G$ (noting that the definition of $f$ involves the factor $\operatorname{vol}(\mathfrak{J})^{-1}$).
The proof of Proposition \[prop:desired-main-term-identity-do-it-up\] occupies the remainder of §\[sec:local-estimates-main-statements\].
Reduction to matrix calculus\[sec:determine-main-term\] {#sec-7-2}
-------------------------------------------------------
We fix measures on $G,B,k$ and define $\mathcal{F}$ as in §\[sec:measures-for-fourier-computation\].
### Application of Parseval {#sec:application-parseval}
Set $\phi := \mathcal{F} \mathfrak{S} \heartsuit^1 f \in \mathcal{S}(B)$. By Parseval, and the volume formulas of §\[sec:local-vol-formulas\], we have for $g \in G$ that $$\label{eq:parseval-application-to-main-term}
\langle \operatorname{Ad}(g) \mathfrak{S} f, \mathfrak{S} f \rangle_{L^2(G)}
= \zeta_k(1)
\langle \operatorname{Ad}(g) \phi, \phi \rangle_{L^2(B)}.$$
We may now informally explain as follows: The support properties say that $\phi$ is concentrated on the subspace of diagonal matrices, whose normalizer is $N(H)$. Thus $\langle \operatorname{Ad}(g) \phi, \phi \rangle$ should be small unless $g$ is close to $N(H)$. One has (morally) $\operatorname{Ad}(h) \phi \approx \phi$ for elements $h \in H$ of size $O(1)$. The identity says (morally) that $\operatorname{Ad}(w) \phi \approx \phi$ for all $w \in W$. Thus the distribution $g \mapsto \langle \operatorname{Ad}(g) \phi, \phi \rangle$ should conceivably approximate some multiple of the Haar measure on $N(H)$.
### Principal congruence subgroups {#sec:princ-congr-subgr}
For a positive integer $m$, we let $K[m]$ denote the $m$th principal congruence subgroup of $G$; we define this to mean the image in $G$ of the depth $m$ principal congruence subgroup $$\begin{pmatrix}
1 + \mathfrak{q}^m & \mathfrak{q}^m \\
\mathfrak{q}^m & 1 + \mathfrak{q}^m
\end{pmatrix}
{\leqslant}\operatorname{GL}_2(\mathfrak{o}).$$ One has a diffeomorphism $$\label{eqn:description-of-Km}
\begin{split}
\mathfrak{q}^m \times \mathfrak{q}^m \times (1 +
\mathfrak{q}^m)
&\xrightarrow{\cong} K[m] \\
(x_1,x_2,y)
&\mapsto n'(x_2) n(x_1) a(y).
\end{split}$$
### Properties of $\phi$
Let $m$ be any positive integer for which $m {\geqslant}N_0$. Proposition \[prop:key-fourier-estimate-microlocal-kernel\] implies that $$\label{eq:unit-invariance-assumed-of-phi}
\phi(\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix})
=
\phi(\begin{pmatrix}
\delta/2 + u \alpha & u \beta \\
u \gamma & \delta/2 - u \alpha
\end{pmatrix})
\text{ for all }
u \in 1 + \mathfrak{q}^m$$ and for $N$ large enough also that $$\label{eq:support-phi-contained-in-E-nought-of-m}
\operatorname{supp}(\phi) \subseteq E(m),$$ where $E(m)$ denotes the following set of “near-diagonal” matrices: $$\label{eq:definition-of-sort-of-neighborhood-of-diagonals}
E(m) := \left\{
\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix}
:
\delta \in k, \alpha \in k;
\beta,\gamma \in 2 \alpha \mathfrak{q}^m
\right\}.$$ Recall also from that $$\label{eq:weyl-invariance-after-integrating-out-upper-and-lower-variables}
\int_{\beta,\gamma \in k}
\phi (
\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix}
)
=
\int_{\beta,\gamma \in k}
\phi (
\begin{pmatrix}
\delta/2 - \alpha & \beta \\
\gamma & \delta/2 + \alpha
\end{pmatrix}
).$$
### The key computation {#sec:key-computation}
Proposition \[prop:desired-main-term-identity-do-it-up\] follows immediately from , the normalizing computation (giving the factor $(2 \zeta_k(1))^{-1}$) and the following:
Let $\Psi : G \rightarrow \mathbb{C}$ and $\phi \in \mathcal{S}(B)$ be arbitrary. Suppose there exists an open subgroup $U$ of $G$ and a positive integer $m$ so that $$\label{eq:concentration-of-phi-quantified-eps-U}
K[m] {\leqslant}U.$$ and so that , , , and hold. Then $$\int_{g \in G}
\langle \operatorname{Ad}(g) \phi, \phi \rangle
\Psi(g)
=$$ $$(\int_{N(H)} \Psi )
\int_{\alpha,\delta \in k}
|2 \alpha|^{-2}
\left\lvert
\int_{\beta,\gamma \in k}
\phi (
\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix}
)
\right\rvert^2.$$
The proof involves Hensel’s lemma, the Weyl integral formula, and related arguments; we complete it in §\[sec:appendix-proof-theorem\].
The lemma and its proof generalize readily to non-split quaternion algebras $B$ and/or non-split separable quadratic subalgebras $E \subseteq B$. We focus on the relevant split diagonal case for the sake of concreteness.
Some matrix calculus\[sec:some-mx-calc\] {#sec-7-3}
----------------------------------------
One purpose of this section is to prove the lemma of §\[sec:key-computation\]. Another is to develop preliminaries for the proof of Proposition \[prop:local-error-estimates-stmt\], below.
### Notation {#sec-7-3-1}
We introduce on $B^0$ the coordinates $$[\alpha,\beta,\gamma] := \begin{pmatrix}
\alpha & \beta \\
\gamma & - \alpha
\end{pmatrix}.$$ The Weyl group $W \cong N(H)/H$ has a natural right action on $G/H$, denoted by juxtaposition. Let $(G/H)[m]$ denote the image of $K[m]$ under the quotient map $G \rightarrow G/H$; by , the map $\mathfrak{q}^m \times \mathfrak{q}^m
\ni (x_1,x_2) \mapsto n'(x_1) n(x_2) H \in (G/H)[m]$ is a bijection. Let $E^0 := E \cap B^0 = \{[\alpha,0,0] : \alpha \in k\}$ denote the subspace of traceless diagonal matrices. Set $E^0(m) :=
\{[\alpha,\beta,\gamma] : \alpha \in k;
\beta,\gamma \in 2 \alpha \mathfrak{q}^m \}
\subseteq B^0$. The sets $E^0(m) - \{0\}$ form a shrinking system of open neighborhoods of $E^0 - \{0\}$. Their images $\mathbb{P} E^0(m)$ in the projective plane $\mathbb{P} B^0$ form a fundamental system of open neighborhoods of the point $\mathbb{P} E^0$.
### Measures {#sec-7-3-2}
We fix measures on $G,B,B^0,k$ as in §\[sec:measures-for-fourier-computation\]. One then has $$\int_{B^0}
f
= \int_{\alpha,\beta,\gamma \in k}
f ([\alpha,\beta,\gamma]),$$ $$\label{eq:formula-relating-B-and-B0}
\int_{B} f
= \int_{\alpha,\beta,\gamma,\delta \in k}
f (\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2-\alpha
\end{pmatrix})
=
\int_{\delta \in k}
(\int_{B_0} f_\delta),$$ where for $f \in C_c(B)$ we define $f_\delta \in C_c(B^0)$ by $f_\delta(\xi) := f(\delta/2 + \xi)$.
We use the notation $\mathbb{E}$ to denote an average over a compact group or orbit thereof with respect to the evident invariant probability measure. Averages over $K[m]$ may be computed by the formula (cf. ) $$\label{eq:formula-averages-over-Km}
\mathbb{E}_{g \in K[m]}
f(g)
=
\mathbb{E}_{\substack{
x,y \in \mathfrak{q}^m \\
z \in 1 + \mathfrak{q}^m
}
}
f(n'(x) n(y) a(z)).$$
Equip $G/H$ with the quotient Haar; by , $$\label{eq:integral-formula-G-mod-H}
\int_{g \in G/H} f(g)
= \int_{x_1,x_2 \in k}
f(n'(x_2) n(x_1)),$$ $$\label{eq:integral-over-G-mod-H-m}
\int_{x \in (G/H)[m]}
f(x)
:= \int_{x \in G/H} 1_{(G/H)[m]}(x)
f(x)
=
\int_{x_1,x_2 \in \mathfrak{q}^m}
f(n'(x_1) n(x_2)),$$ $$\label{eq:integral-over-G-mod-H-m-averaged}
\mathbb{E}_{x \in (G/H)[m]}
f(x)
=
\mathbb{E}_{x_1,x_2 \in \mathfrak{q}^m}
f(n'(x_1) n(x_2)).$$
### Basic observations
By inspection, $$\label{eq:weyl-invariance-of-cone-nbhd}
\operatorname{Ad}(N(H)) E^0 = E^0,
\quad
\operatorname{Ad}(W) E^0(m) = E^0(m).$$ By direct calculations such as $$\label{eq:basic-conjugation-action-on-diagonal}
\operatorname{Ad}(n'(x) n(y) a(z) [1,0,0])
=
[1 + 2 x y, - 2 y, 2 x(1 + 2 y)].$$ one verifies also that $$\label{eq:adjoint-action-small-elements-near-preserve-diagonal-neighborhood}
\operatorname{Ad}(K[m]) E^0(m) = E^0(m).$$
### Applications of Hensel’s lemma {#sec-7-3-3}
\[lem:hensel-lemma-statement\] Let $m,n \in \mathbb{Z}_{{\geqslant}1}$. Set $M := (\mathfrak{q}^m)^{\oplus n}$. Let $f_1,\dotsc,f_n \in \mathfrak{o}[X_1,\dotsc,X_n]$ be polynomials in the variables $X_1,\dotsc,X_n$ with coefficients in $\mathfrak{o}$ satisfying $f_1(0) = \dotsb = f_n(0) = 0$. Let $f : M \rightarrow k^{\oplus n}$ denote the function given by $x \mapsto (f_1(x),\dotsc,f_n(x))$. Set $J :=\det(\partial f_i / \partial x_j) \in
\mathfrak{o}[X_1,\dotsc,X_n]$. Assume that $J(x) \in \mathfrak{o}^\times$ for all $x \in M$. Then $f$ induces a diffeomorphism $f : M \rightarrow M$.
One argues as in the proof of Hensel’s lemma that $f$ is bijective. The conclusion then follows from the inverse function theorem.
\[key-application-hensel-lemma\] Let $\alpha_0 \in k - \{0\}$. Then the map $$(1 + \mathfrak{q}^m)
\times
(G/H)[m]
\rightarrow
[(1 + \mathfrak{q}^m) \alpha_0, 2 \mathfrak{q}^m \alpha_0, 2 \mathfrak{q}^m \alpha_0]$$ $$(\lambda,x)
\mapsto
\lambda \operatorname{Ad}(x) [\alpha_0,0,0]$$ is a well-defined diffeomorphism whose Jacobian has constant valuation.
We may assume that $\alpha_0 = 1$. For $x_1,x_2,x_3 \in \mathfrak{q}^m$, define $y_1,y_2,y_3 \in k$ by $(1 + x_1) \operatorname{Ad}(n'(x_2) n(x_3)) [1,0,0]
= [1 + y_1, 2 y_2, 2 y_3]$. By a calculation similar to , $$\begin{aligned}
y_1 &= x_1 + 2 x_2 x_3 + 2 x_1 x_2 x_3, \\
y_2 &= - x_3 ( 1 + x_1), \\
y_3 &= x_2 (1 + x_1) (1 + 2 x_3).
\end{aligned}$$ Since $m {\geqslant}1$, one sees that $y_1,y_2,y_3 \in \mathfrak{q}^m$ and that the Jacobian of the map $(x_1,x_2,x_3) \mapsto
(y_1,y_2,y_3)$ belongs to $\mathfrak{o}^\times$ at every point of its domain. The conclusion follows now from Lemma \[lem:hensel-lemma-statement\].
\[cor:we-can-write-stuff-near-diagonal-nicely\] For each $\xi = [\alpha,\beta,\gamma] \in E^0(m)$ there exists $\alpha_0 \in \alpha(1 + \mathfrak{q}^m)$ and $g \in K[m]$ so that $\xi = \operatorname{Ad}(g) [\alpha_0,0,0]$.
\[lem:basic-integral-formula-without-hypotheses\] Let $f \in C_c(B^0)$ and $\xi_0 = [\alpha_0,\beta_0,\gamma_0] \in E^0(m) - \{0\}$. Then $$\mathbb{E}_{
\substack{
\lambda \in 1 + \mathfrak{q}^m
\\
g \in K[m]
}
}
f(\lambda \operatorname{Ad}(g) \xi_0)
=
\mathbb{E} _{
\substack{
\alpha \in (1 + \mathfrak{q}^m) \alpha_0 \\
\beta, \gamma \in 2 \alpha_0 \mathfrak{q}^m
}
}
f ([\alpha,\beta,\gamma]).$$
By the corollary, we may write $\xi_0 = \operatorname{Ad}(g_0)[\alpha_0',0,0]$ with $\alpha_0' \in (1 + \mathfrak{q}^m) \alpha_0$ and $g_0 \in K[m]$. By the change of variables $g \mapsto g g_0^{-1}$, we reduce to proving the required identity in the special case $\beta_0 = \gamma_0 = 0$. In that case, we replace the average over $g \in K[m]$ with an average over $x \in (G/H)[m]$ and appeal to Lemma \[key-application-hensel-lemma\] and the change of variables formula.
\[lem:basic-integral-formula-for-diag-inv-f\] Suppose $f \in C_c(B^0)$ is supported on $E^0(m)$ and satisfies $f(\lambda x) = f(x)$ for all $x \in B^0$ and $\lambda \in 1 + \mathfrak{q}^m$. Let $\xi_0 \in B^0 - \{0\}$. Then $$\begin{aligned}
\label{eqn:basic-integral-formula-for-diag-inv-f}
\mathbb{E}_{g \in K[m]}
f(\operatorname{Ad}(g) \xi_0)
&=
1_{E^0(m)}(\xi_0)
\mathbb{E}_{\beta,\gamma \in 2 \mathfrak{q}^m \alpha_0} f
([\alpha_0,\beta,\gamma])
\\
\nonumber
&=
1_{\alpha_0 \neq 0}
e_{2 \mathfrak{q}^m \alpha_0}(\beta_0)
e_{2 \mathfrak{q}^m \alpha_0}(\gamma_0)
\int_{\beta,\gamma \in k}
f ([\alpha_0,\beta,\gamma,-\alpha_0])
\end{aligned}$$ where $e_\mathfrak{a} := \operatorname{vol}(\mathfrak{a})^{-1} 1_\mathfrak{a}$ denotes the normalized characteristic function of a compact open subgroup $\mathfrak{a}$ of $k$.
If $\xi_0 \notin E^0(m)$, then the vanishing of the LHS follows from , so suppose that $\xi_0 \in E^0(m)$. The $(1+\mathfrak{q}^m)$-invariance of $f$ allows us to rewrite the LHS of as $\mathbb{E}_{\lambda \in 1 + \mathfrak{q}^m, g \in K[m]}
f(\lambda \operatorname{Ad}(g) \xi_0)$. We apply Lemma \[lem:basic-integral-formula-without-hypotheses\] to the latter, invoking again the $(1+\mathfrak{q}^m)$-invariance of $f$ to simplify the conclusion.
\[lem:useful-for-main-term-evaluation-inverse-weyl-integral-formula\] Let $f$ be as in Lemma \[lem:basic-integral-formula-for-diag-inv-f\] and $\alpha_0 \in k - \{0\}$. Then $$\label{eqn:useful-for-main-term-evaluation-inverse-weyl-integral-formula}
\int_{x \in (G/H)[m]}
f(\operatorname{Ad}(x) [\alpha_0,0,0])
=
|2 \alpha_0|^{-2}
\int_{\beta,\gamma \in k}
f([\alpha_0,\beta,\gamma]).$$
Using the integral formulas and , the LHS of may be rewritten $\operatorname{vol}(\mathfrak{q}^m)^2
\mathbb{E}_{g \in K[m]} f(\operatorname{Ad}(g)[\alpha_0,0,0])$. The conclusion follows from Lemma \[lem:basic-integral-formula-for-diag-inv-f\] upon noting that $\operatorname{vol}(\mathfrak{q}^m)^2 e_{2 \alpha_0 \mathfrak{q}_m}(0)^2
= |2 \alpha_0|^{-2}$.
### Expanding near the singular points {#sec-7-3-4}
\[lem:we-know-when-conjugation-nearly-stabilizes-diagonal\] For $x \in G/H$, the following are equivalent:
1. There exists $w \in W$ so that $x w \in (G/H)[m]$.
2. There exists $\tau \in E^0 - \{0\}$ so that $\operatorname{Ad}(x) \tau \in E^0(m)$.
3. For all $\tau \in E^0$, one has $\operatorname{Ad}(x) \tau \in E^0(m)$.
\(i) implies (ii): immediate from and . (ii) implies (iii): immediate from the definitions and the fact that $\dim(E^0) = 1$. (iii) implies (i): By the corollary to Lemma \[key-application-hensel-lemma\] of §\[sec-7-3-3\], we have $\operatorname{Ad}(x) [1,0,0] \in \operatorname{Ad}(g) E^0$ for some $g \in K[m]$; then $\operatorname{Ad}(g^{-1} x) E^0 = E^0$, hence $g^{-1} x \in N(H) = W H$, hence $x \in K[m] W H$.
\[lem:expansion-integrals-near-singular-pts\]
1. Suppose $f \in C_c(G/H)$ is supported on $(G/H)[m] W$. Then $$\int_{G/H} f
= \sum_{w \in W}
\int_{x \in (G/H)[m]} f(x w).$$
2. Suppose $f_1 \in C_c(B^0)$ is supported on $E^0(m)$ and $f_2 \in C_c(G/H)$ is arbitrary. Let $\tau \in E^0$. Then $$\int_{x \in G/H}
f_1(\operatorname{Ad}(x) \tau) f_2(x)
=
\sum_{w \in W}
\int_{x \in (G/H)[m]}
f_1(\operatorname{Ad}(x w) \tau)
f_2(x w).$$
1. The nontrivial Weyl element $w \in W$ acts on $x = n'(x_1) n(x_2) H \in (G/H)[m]$ by the formula (for $x_2 \neq 0$) $x w = n'(x_1 + 1/x_2) n(-x_2) H$, hence $(G/H)[m] \cap (G/H)[m] w = \emptyset$. The conclusion follows.
2. By Lemma \[lem:we-know-when-conjugation-nearly-stabilizes-diagonal\], the function $f(x) := f_1(\operatorname{Ad}(x) \tau) f_2(x)$ is supported on $(G/H)[m]$, so we may apply (i).
### A variant of the Weyl integral formula {#sec-7-3-5}
\[lem:weyl-integral-formula-variant\] Let $f \in C_c(B^0)$ be supported on $\cup_{g \in G} g E^0 g^{-1}$. Then $$\int_{\alpha,\beta,\gamma \in k}
f([\alpha,\beta,\gamma])
=
\frac{1}{|W|}
\int_{\alpha \in k}
|2 \alpha|^{-2}
\int_{x \in G/H}
f(\operatorname{Ad}(x) [\alpha,0,0]).$$
We omit the proof. In the special case that $\operatorname{supp}(f) \subseteq (G/H)[m] W$, the conclusion follows from Lemma \[lem:useful-for-main-term-evaluation-inverse-weyl-integral-formula\] of §\[sec-7-3-3\] and Lemma \[lem:expansion-integrals-near-singular-pts\] of §\[sec-7-3-4\]; this special case suffices for our purposes and serves also to check the normalization.
### Proof of the lemma of §\[sec:key-computation\] {#sec-7-3-6}
\[sec:appendix-proof-theorem\] Let $\phi \in \mathcal{S}(B)$. By , $$\langle \operatorname{Ad}(g) \phi, \phi \rangle_{L^2(B)}
= \int_{\delta \in k}
\langle \operatorname{Ad}(g) \phi_\delta, \phi_\delta \rangle_{L^2(B^0)},$$ where $\phi_\delta \in \mathcal{S}(B^0)$ is given by $\phi_\delta(\xi) := \phi(\delta/2 + \xi)$ for $\xi \in B^0$. The proof of the lemma of §\[sec:key-computation\] thereby reduces to that of the following:
Let $\Psi : G \rightarrow \mathbb{C}$ and $\phi \in \mathcal{S}(B^0)$ be arbitrary. Define $I : E^0 \rightarrow \mathbb{C}$ by $I([\alpha,0,0])
:=
\int_{\beta,\gamma \in k}
\phi([\alpha,\beta,\gamma])$. Suppose there exists an open subgroup $U$ of $\PGL_2(k)$ and a positive integer $m$ so that $$\label{eq:Km-in-U-again-with-B0}
K[m] {\leqslant}U$$ $$\label{eq:unit-invariance-assumed-of-phi-2}
\text{
$\phi(\lambda x) = \phi(x)$
for all $x \in B^0$ and $\lambda \in 1 + \mathfrak{q}^m$.
}
$$ $$\label{eq:support-phi-contained-in-E-nought-of-m-2}
\operatorname{supp}(\phi) \subseteq E^0(m)$$ $$\label{eq:I-alpha-neg-alpha}
I([\alpha,0,0]) = I([-\alpha,0,0]),$$ and so that the decay and smoothness assumptions , concerning $\Psi$ hold. Then $$\label{eq:main-term-integral-so-annoying-without-cleverness-both-B0}
\int_{g \in G}
\Psi(g) \langle \operatorname{Ad}(g) \phi, \phi \rangle =
(\int_{N(H)}
\Psi)
\int_{\alpha \in k}
|2 \alpha|^{-2}
|I([\alpha,0,0])|^2.$$
Note first that the RHS of converge absolutely, thanks to , Lemma \[lemma:convergence-Xi-along-G-and-H\] of §\[sec-2-1-5\], and the compactness of the support of $\phi$. We thereby reduce to establishing the claimed identity in the special case that $\Psi$ is the characteristic function of some $U \times U$-orbit; in particular, we may assume that $\Psi$ is compactly-supported. For this reason, we may neglect convergence issues in the arguments to follow.
Equip $E^0$ with the measure $\int_{E^0} f := \int_{\alpha \in k} f([\alpha,0,0])$ and define $D : E^0 \rightarrow \mathbb{R}_{{\geqslant}0}$ by $D([\alpha,0,0]) := |2 \alpha|^2$, so that the RHS of reads $$\label{eq:desired-RHS-for-weyl-int-formula-appl}
\sum_{w \in W}
\int_{h \in H}
\Psi(w h)
\int_{\tau \in E^0}
D(\tau)^{-1} |I(\tau)|^2.$$ We now successively transform the LHS of . By expanding the definitions and applying the integral formula of §\[sec-7-3-5\], we obtain $$\frac{1}{|W|}
\int_{g \in G}
\Psi(g)
\int_{\tau \in E^0}
D(\tau)
\int_{x \in G/H}
\overline{\phi(\operatorname{Ad}(g^{-1} x) \tau)}
\phi(\operatorname{Ad}(x) \tau).$$ We execute the (cosmetic) change of variables $x \mapsto g x$, swap orders of integration, and apply the (crucial) substitution $g \mapsto g x^{-1}$ to arrive at $$\frac{1}{|W|}
\int_{\tau \in E^0}
D(\tau)
\int_{x \in G/H}
\overline{\phi(\operatorname{Ad}(x) \tau)}
\int_{g \in G}
\Psi(g x^{-1})
\phi(\operatorname{Ad}(g) \tau).$$ By factoring $g = y h$ with $y \in G/H$, $h \in H$, we obtain $$\frac{1}{|W|}
\int_{\tau \in E^0}
D(\tau)
\int_{x,y \in G/H}
\overline{\phi(\operatorname{Ad}(x) \tau)}
\phi(\operatorname{Ad}(y) \tau)
\int_{h \in H}
\Psi(y h x^{-1}).$$ We apply Lemma \[lem:expansion-integrals-near-singular-pts\] of §\[sec-7-3-4\] to the $x,y$ integrals, giving $$\begin{aligned}
\frac{1}{|W|}
\sum_{w_1,w_2 \in W}
\int_{\tau \in E^0}
\int_{\substack{
x,y \in (G/H)[m]
}
}
D(\tau)
\overline{\phi(\operatorname{Ad}(x w_1) \tau)}
\phi(\operatorname{Ad}(y w_2) \tau)
I'(x,y)
\end{aligned}$$ where $I'(x,y)
:=
\int_{h \in H}
\Psi(y w_2 h w_1^{-1} x^{-1})$. For each $\tau \in E^0 - \{0\}$, one has by our assumption that $I'(x,y) = \int_{h \in H} \Psi(w_2 h w_1^{-1})
= \int_{h \in H} \Psi(w_2 w_1^{-1} h)$, by Lemma \[lem:useful-for-main-term-evaluation-inverse-weyl-integral-formula\] of §\[sec-7-3-3\] that $\int_{x \in (G/H)[m]} \phi(\operatorname{Ad}(x) \tau) = D(\tau)^{-1} I(\tau)$, and by our assumption that $I(\operatorname{Ad}(w) \tau) = I(\tau)$ for all $w \in W$. We obtain $$\frac{1}{|W|}
\sum_{w_1,w_2 \in W}
\int_{h \in H}
\Psi(w_2 w_1^{-1} h)
\int_{\tau \in E^0}
D(\tau)^{-1} |I(\tau)|^2,$$ which simplifies to .
Estimates for the error terms\[sec:local-estimates-error\] {#sec-8}
==========================================================
Statement of result {#sec-8-2}
-------------------
Recall the definitions of the Harish–Chandra function $\Xi$ (§\[sec:local-Xi\]) and the Weil representation (§\[sec:local-weil-reps\]). Let $\psi : k \rightarrow
\mathbb{C}^{(1)}$ be an unramified unitary character. For $\tau \in k^\times$, set $$\rho^\tau := \rho_{\operatorname{Weil}}^{\psi^\tau,B},
\quad
\rho_0^\tau := \rho_{\operatorname{Weil}}^{\psi^\tau,B^0}.$$ Let $\tau_1, \tau_2 \in k^\times$. The relevance of the following definition may be inferred from the statement of Theorem \[thm:main-estimate-general-variance\].
\[defn:good-sesquilinear-form\] Let $\ell : \mathcal{S}(B) \otimes \mathcal{S}(B) \rightarrow
\mathbb{C}$ be a sesquilinear form. We say that $\ell$ is *good* (relative to $\psi,\tau_1,\tau_2$) if:
1. There is an open subgroup $U$ of $G$ so that for $g_1,g_2 \in U$, $\phi_1, \phi_2 \in \mathcal{S}(B)$ and $s \in \operatorname{Mp}_2(k)$, one has $$\begin{aligned}
\ell(\phi_1, \phi_2)
&=
\label{eqn:symmetrization-of-ell}
\ell(\mathfrak{S} \phi_1, \phi_2)
= \ell(\phi_1, \mathfrak{S} \phi_2)
\\
&=
\ell(\operatorname{Ad}(g_1) \phi_1, \operatorname{Ad}(g_2) \phi_2)
\label{eqn:orthogonal-smoothness-of-ell}
\\
&=
\ell(\rho^{\tau_1}(s) \phi_1, \rho^{\tau_2}(s) \phi_2)
\label{eqn:metaplectic-invariance-of-ell}.
\end{aligned}$$
2. For all $\phi_1,\phi_2 \in
\mathcal{S}(B)$ there exists $C {\geqslant}0$ so that for all $s \in \operatorname{Mp}_2(k)$, $$\label{eq:key-decay-for-ell}
|\ell((1 \otimes \rho_0^{\tau_1}(s)) \phi_1,
(1 \otimes \rho_0^{\tau_2}(s)) \phi_2)|
{\leqslant}C \Xi(s).$$
\[prop:local-error-estimates-stmt\] For each good sesquilinear form $\ell : \mathcal{S}(B) \otimes \mathcal{S}(B) \rightarrow
\mathbb{C}$ and $N_0 > \operatorname{ord}_k(2)$ there exists $C {\geqslant}0$ so that for large positive integers $N$ and all $\sigma \in \Sigma$, the element $f \in C_c^\infty(G)$ attached to $(N,\sigma)$ (see §\[sec:element-attached-to-N-sigma\]) satisfies $$\label{eqn:required-estimate-involving-ell}
|\ell(\heartsuit^{\tau_1} f, \heartsuit^{\tau_2} f)| {\leqslant}C N.$$
The proof of Proposition \[prop:local-error-estimates-stmt\] occupies §\[sec-8-3\]–§\[sec-8-6\].
Preliminary reduction {#sec-8-3}
---------------------
If Proposition \[prop:local-error-estimates-stmt\] holds for $(\psi,\tau_1,\tau_2)$, then it holds formally also for $(\psi^{1/u}, \tau_1 u, \tau_2 u)$ for each $u \in \mathfrak{o}^\times$. We may and shall thereby reduce (for notational convenience) to the case that $\psi \in \sigma$ (see §\[sec-5-4\]).
Smoothing with respect to the adjoint action\[sec:smoothing-wrt-adjoint\] {#sec-8-4}
-------------------------------------------------------------------------
Let $m$ be a positive integer. Assume the following: $$\label{eq:m-geq-N0}
m {\geqslant}N_0,$$ $$\label{eq:N-large-wrt-m}
\text{$N$ is large enough in terms of $m$.}$$ Let $U {\leqslant}G$ denote the $m$th principal congruence subgroup (see §\[sec:princ-congr-subgr\]). Let $f \in C_c^\infty(G)$ be attached to $(N,\sigma)$. We study the effect of smoothing $f$ under the adjoint action of $U$.
\[lem:explicit-smoothed-out-variant-of-fourier-kernel\] Set $\phi := \mathcal{F} \mathfrak{S} \heartsuit^1 f \in
\mathcal{S}(B)$, as in §\[sec-6-3\]. Define $\phi^U \in \mathcal{S}(B)$ by $$\label{eq:defn-pih-m}
\phi^U(x) := \mathbb{E}_{g \in U}
\phi(\operatorname{Ad}(g) x).$$ Then $$\label{eq:desired-formula-for-phi-m}
\phi^U (
\begin{pmatrix}
\delta/2 + \alpha & \beta \\
\gamma & \delta/2 - \alpha
\end{pmatrix}
)
=
I(\alpha,\delta)
e_{2 \alpha \mathfrak{q}^m}(\beta)
e_{2 \alpha \mathfrak{q}^m}(\gamma).$$ In particular, $$\label{eq:eventual-N-stability-of-phi-m}
q^{-N}
\phi^U (\begin{pmatrix}
\delta/2 + \varpi^{-N} \alpha & \varpi^{-N} \beta \\
\varpi^{-N} \gamma & \delta/2 - \varpi^{-N} \alpha
\end{pmatrix}
)
\text{ is independent of } N.$$
Proposition \[prop:key-fourier-estimate-microlocal-kernel\] implies that for $N$ large enough, $$\label{eq:support-condition-relevant-for-smoothing}
\operatorname{supp}(\phi) \subseteq E(m)$$ with $E(m)$ as in , and also that $\phi$ satisfies the smoothness property noted above. To deduce from and is a calculus problem; to solve it, we apply Lemma \[lem:basic-integral-formula-for-diag-inv-f\] of §\[sec-7-3-3\] to the functions $\phi_\delta \in \mathcal{S}(B^0)$ attached to $\phi \in \mathcal{S}(B)$ and $\delta \in k$ by $\phi_\delta(\xi) := \phi(\delta/2 + \xi)$ for $\xi \in
B^0$. The final assertion follows from .
Metaplectic interpretation\[sec:error-estimates-metaplectic\] {#sec-8-5}
-------------------------------------------------------------
Retain the notation and setting of §\[sec:smoothing-wrt-adjoint\]. We now interpret in terms of the Weil representation. Recall the double cover $\operatorname{pr}: \operatorname{Mp}_2(k) \rightarrow \operatorname{SL}_2(k)$ and the elements $n(b), t(a), w \in \operatorname{Mp}_2(k)$ as in §\[sec:local-weil-repn\]. To reduce clutter in the formulas to follow, we introduce some notation and terminology:
Let $a_1,a_2$ be quantities depending implicitly upon the large positive integer $N$ and a field element $\tau \in k^\times$ (as well as the field $k$ containing $\tau$, of course). Thus $a_i = a_i(\tau,N)$. Write $a_1 \approx a_2$ to denote that $a_1 = \gamma |\tau|^{c/4} b$, where
- $\gamma \in \mathbb{C}^{(1)}$ is an eighth root of unity that may depend upon $\tau$ and $N$, and
- $c \in \mathbb{Z}$ depends neither upon $\tau$ nor upon $N$.
We say that a quantity $a$ is *essentially independent of $N$* if $a \approx b$, where $b$ is independent of $N$.
\[lem:essential-independence-after-translation\] Let $\tau \in k^\times$. If $\tau \notin \mathfrak{o}^\times k^{\times 2}$, then $\heartsuit^{\tau} f = 0$. Otherwise there exists $\phi_0 \in \mathcal{S}(B)$ that is essentially independent of $N$ so that $$\label{eqn:essential-indepdence-0}
\operatorname{Ad}(e_U)
\rho^{\tau}(w)
\mathfrak{S}
\heartsuit^{\tau} f
=
q^{N/2}
(1 \otimes \rho_0^{\tau}(t(\varpi^{N})))
\phi_0.$$
The first assertion is immediate from the definitions of $\heartsuit^{\tau}$ and $f$. Abbreviate $f' := \mathfrak{S} f$. Since $\heartsuit^\tau f' = \mathfrak{S} \heartsuit^{\tau} f$, it suffices now to show for $\tau \in \mathfrak{o}^\times k^{\times 2}$ that the required conclusion holds in the inverted form $$\label{eqn:essential-indepdence-1}
q^{-N/2} (1 \otimes \rho_0^{\tau}(t(\varpi^{-N}))) \operatorname{Ad}(e_U)
\rho^{\tau}(w) \heartsuit^{\tau} f'
\text{
is essentially independent of } N.$$ The case $\tau = 1$ of follows from the lemma of §\[sec-8-4\] and the formulas describing of Weil representation. In general, we see by inspecting the definitions that $\heartsuit^{\tau} f'
=
\heartsuit^{\nu^2} f'
\approx
\rho^{\tau}(t(\nu)) \heartsuit^1 f'$ and that $\rho^\tau(w)
\approx
\rho^\tau(t(\nu))
\rho^1(w)$, hence that $$\label{eqn:essential-indepdence-2}
\rho^{\tau}(w)
\heartsuit^{\tau} f'
\approx
\rho^\tau(t(\nu))
\rho^1(w)
\rho^{\tau}(t(\nu)) \heartsuit^1 f'
\approx
\rho^1(w) \heartsuit^1 f'.$$ Moreover, $\rho_0^{\tau}(t(\varpi^{-N})) \approx
\rho_0^1(t(\varpi^{-N}))$. Thus the identity for general $\tau \in \mathfrak{o}^\times k^{\times 2}$ reduces to the $\tau = 1$ case already established.
Completion of the proof {#sec-8-6}
-----------------------
We now prove Proposition \[prop:local-error-estimates-stmt\]. For $i=1,2$, set $\phi_i := \mathfrak{S} \rho^{\tau_i}(w) \heartsuit^{\tau_i}
f$. By and , we have $$\ell(\heartsuit^{\tau_1} f, \heartsuit^{\tau_2} f) = \ell(\phi_1,\phi_2).$$ By averaging over $g_1,g_2 \in U$, we have $$\ell(\phi_1,\phi_2) = \ell(\phi_1^U,\phi_2^U),$$ with notation as in §\[sec:smoothing-wrt-adjoint\]. By shrinking $U {\leqslant}G$ as necessary, we may assume that $U$ is the $m$th principal congruence subgroup for some $m {\geqslant}N_0$; since $\ell$ and $N_0$ are independent of $N$, we may assume also that $N$ is sufficiently large in terms of $m$. By the lemma of §\[sec-8-5\], we may suppose that $\tau_1, \tau_2 \in \mathfrak{o}^\times k^{\times 2}$, in which case $$\phi_i^U \approx q^{N/2} (1 \otimes
\rho_0^{\tau_i}(t(\varpi^N)) \phi_0,$$ where $\phi_0 \in \mathcal{S}(B)$ is essentially independent of $N$ (see §\[sec:error-estimates-metaplectic\]). Our task thereby reduces to establishing the estimate $$\ell(
(1 \otimes \rho_0^{\tau_1}(t(\varpi^N))) \phi_0,
(1 \otimes \rho_0^{\tau_2}(t(\varpi^N))) \phi_0
) \ll N q^{-N},$$ which follows finally from the condition in the definition of “good.”
The sesquilinear forms $\ell$ to which we apply Proposition \[prop:local-error-estimates-stmt\] below may be assumed to have an additional property (beyond being “good”). That property is not directly relevant for the immediate purposes of this paper, but may be useful in future work, so we briefly record it: There are irreducible unitary representations $\pi_1, \pi_2$ of $G$ of dimension $> 1$ (arising as local components of cuspidal automorphic representations) so that $\ell$ factors as $\ell = \tilde{\ell} \circ ((1 \otimes \theta_1) \otimes (1 \otimes
\theta_2))$, where
1. $\sigma_i$ is the local $\psi^{\tau_i}$-theta lift of $\pi_i$ as in [@MR1103429], i.e., the irreducible representation of $\operatorname{Mp}_2(k)$ for which one has $\operatorname{Hom}_G(\rho_0^{\tau_i},\pi_i) =
\operatorname{Hom}_\mathbb{C}(\sigma_i,\mathbb{C})$,
2. $\tilde{\ell} :
\mathcal{S}(k) \otimes \mathcal{S}(k) \otimes
\sigma_1 \otimes \sigma_2
\rightarrow \mathbb{C}$ is a sesquilinear form invariant by the diagonal action of $\operatorname{Mp}_2(k)$,
3. $\theta_i : \mathcal{S}(B^0) \rightarrow \sigma_i$ is a basis element for the one-dimensional space of $G$-equivariant maps $\rho_0^{\tau_i} \rightarrow \sigma_i$, and
4. $1 \otimes \theta_i : \mathcal{S}(B) = \mathcal{S}(k) \otimes
\mathcal{S}(B^0) \rightarrow \mathcal{S}(k) \otimes \sigma_i$ and $(1 \otimes \theta_1) \otimes (1 \otimes \theta_2) :
\mathcal{S}(B) \otimes \mathcal{S}(B) \rightarrow
\mathcal{S}(k) \otimes \mathcal{S}(k) \otimes \sigma_1 \otimes
\sigma_2$ are the evident maps.
The $\phi = \heartsuit f$ of interest in this paper concentrate on semisimple elements. If one instead considers $\phi$ supported close to the nilcone (as arise naturally when studying classical newvectors), then one may exploit bounds towards temperedness of the $\sigma_i$ and the above factorization of $\ell$ to produce estimates sharper than those that follow from $\ell$ being good.
Bounds for partial orbital integrals {#sec:bounds-part-orbit}
------------------------------------
The contents of this short miscellaneous section are used below to deduce the assertions made in §\[sec-1\] concerning the family cardinality and mean statistics; they are directly related neither to the rest of §\[sec-8\] nor to the main new ideas of this paper.
The parameter $N_0$ as in §\[sec:element-attached-to-N-sigma\] is regarded here as fixed once and for all. Recall that a *regular semisimple* element $\gamma \in B^\times$ is one which is diagonalizable with distinct eigenvalues over an algebraic closure $\overline{k}$ of $k$, or equivalently, for which $\operatorname{tr}(\gamma)^2 \neq 4 \operatorname{nr}(\gamma)$.
\[lem:bounds-part-orbit\] Let $\gamma \in B^\times$ be regular semisimple. Let $U_2$ be a compact open subgroup of $G$ that is small enough in terms of $\gamma$. Let $N$ be large enough in terms of $(\gamma,U_2)$. Let $f \in C_c^\infty(G)$ be attached to $(N,\sigma)$ for some $\sigma \in \Sigma$. Then $\int_{u \in U_2} f(u^{-1} \gamma u) = 0$.
Let $M_1$ denote the semidirect product $k \rtimes k^\times$. Set $M := M_1 \times G$. The group $M$ consists of triples $(x,y,z) \in k \times k^\times \times G$ with the group law $(x_1,y_1,z_1)(x_2,y_2,z_2) = (x_1 + y_1 x_2, y_1 y_2, z_1
z_2)$. For $(x,y,z) \in M$ and $b \in B$, set $(x,y,z) \cdot b := z (y b + x) z^{-1}$. This formula defines an action of $M$ on $B$.
Set $\tau := \operatorname{nr}(\gamma)^{-1}$ and $\phi := \heartsuit^{\tau} f \in \mathcal{S}(B)$. For $g \in G$, one then has $\tau \operatorname{nr}( g^{-1} \gamma g) = 1$ and thus $\phi(g^{-1} \gamma g) = f(g^{-1} \gamma g)$. By the lemma of §\[sec-6-4\] (or directly from the definitions), there is an open subgroup $U_1 {\leqslant}M_1$, depending only upon $\tau$, so that $\phi(u_1 \cdot b) = \phi(b)$ for all $u_1 \in U_1, b \in B$. Setting $U := U_1 \times U_2 {\leqslant}M$, our task reduces to showing that $$\label{eq:goal-vanishing-local-orbital-integral-for-phi}
\mathbb{E}_{u \in U} \phi(u^{-1} \gamma u) = 0$$ for $N$ large enough in terms of $(\gamma,U)$. To that end, observe that the orbit map $M \rightarrow B$ given by $m \mapsto m \cdot \gamma$ is submersive at the identity: it suffices to check this claim over $\overline{k}$, where it follows by direct calculation in the diagonal case. By Hensel’s lemma as in §\[sec-7-3-3\], together with the assumption that $U {\leqslant}M$ is small enough in terms of $\gamma$, we see that
1. the orbit $O := U \cdot \gamma \subseteq B$ is open,
2. the orbit map $U \rightarrow O$ is a diffeomorphism, and
3. the pushforward of the probability Haar on $U$ under the orbit map is the probability measure on $O$ induced by the Haar on $B$.
Our goal is thus equivalent to showing that $\int_O \phi = \langle \phi, 1_O \rangle_{L^2(B)} = 0$. Since the Fourier transform of $1_O$ has compact support, we reduce by Parseval to showing that the support of $\mathcal{F} \phi$ tends to $\infty$ as $N \rightarrow \infty$, which follows from the condition $\alpha \asymp q^{-N}$ of Proposition \[prop:key-fourier-estimate-microlocal-kernel\] (arguing as in §\[sec:error-estimates-metaplectic\] to relate $\heartsuit^{\tau}, \heartsuit^{1}$).
Deduction of the main theorem\[sec:deduction-main-thm-microlocal\] {#sec-9}
==================================================================
Setting {#sec:9-setting}
-------
We adopt here the notation and setting of §\[sec:general-estimates-specialized-single-place\], but assume now that $\mathfrak{q}$ is a finite place and that $B_\mathfrak{q}$ is split. Our existing assumptions imply then that $F$ is totally real and that $B$ is totally definite.
Set $k := F_\mathfrak{q}$, fix an identification $\PB^\times_\mathfrak{q} = \PGL_2(k)$, and adopt the notation $\mathfrak{o}, \mathfrak{q}, q$ from §\[sec-2-1-1\]. Let $N$ be a large positive integer, and let $\omega : \mathfrak{o}^\times \rightarrow \mathbb{C}^\times$ be a character of conductor $N$. Let $\tilde{\mathcal{F}}_\omega$ denote the set of nonzero vectors $\varphi \in \mathcal{A}_0^J$ for which
- there exists $\pi \in A_0^J$ so that $\varphi \in \pi$, and
- $\varphi$ is a microlocal lift of orientation $\omega$ in the sense of §\[sec-5-3\].
For $\pi \in A_0^J$, the set $\pi \cap \tilde{\mathcal{F}}_\omega$ is either empty or of the form $\mathbb{C}^\times \varphi_1 \sqcup \mathbb{C}^\times \varphi_2$ for some $\varphi_1, \varphi_2 \in \pi^J$ (see §\[lem:determination-microlocal-lifts\]), hence $\tilde{\mathcal{F}}_\omega$ is a union of scaling classes $\mathbb{C}^\times \varphi$. Choose a set $\mathcal{F}_\omega$ consisting of one unit vector from each such scaling class, so that $\tilde{\mathcal{F}}_\omega = \bigsqcup_{\varphi \in
\mathcal{F}_\omega} \mathbb{C}^\times \varphi$. The discussion of §\[sec:setting-overview\] and §\[sec-1-4\] applies, giving us a sequence of sets $\mathcal{F}_N := \sqcup_{\omega \in \mathcal{X}_N}
\mathcal{F}_\omega \subset \mathcal{A}^J_0$ indexed by large positive integers $N$.
Define $\Sigma$ and $\mathcal{X}_N^{\sigma}$ as in §\[sec-5-4\] with respect to some fixed but large enough natural number $N_0$. The partition $\mathcal{X}_N = \bigsqcup_{\sigma \in \Sigma}
\mathcal{X}_N^\sigma$ of the group of characters of $\mathfrak{o}^\times$ of conductor $N$ induces a partition $\mathcal{F}_N = \bigsqcup_{\sigma \in \Sigma}
\mathcal{F}_N^\sigma$ of the family of microlocal lifts. We emphasize that $|\Sigma| = O(1)$.
Mean statistics {#sec:mean-statistics}
---------------
We have included this section to complete the discussion of §\[sec-1\]; it has nothing to do with the main new ideas of this paper.
Since smooth functions on $\mathbf{X}$ are uniformly dense in the space of continuous functions, the lemmas of §\[sec-1-2\] and §\[sec-1-3\] are consequences of the following:
Fix $\Psi \in \mathcal{A}$. Assume that $N$ is large enough in terms of $\Psi$. Then $$\label{eq:local-weyl-law-final-section}
\sum_{\varphi \in \mathcal{F}_N}
\langle \varphi, \Psi \varphi \rangle
=
c q^{2 N}
\frac{\langle 1, \Psi \rangle}{ \langle 1, 1 \rangle}$$ where with $\Delta_F, \Delta_B$ the absolute (reduced) discriminants and $\operatorname{ram}_f(B)$ the set of finite places at which $B$ ramifies, $$c :=
2
\frac
{
\zeta_F(2)
\Delta_B
\Delta_F^{3/2}
}
{
(4 \pi^2)^{[F:\mathbb{Q}]}
\prod_{\mathfrak{p} \in \operatorname{ram}_f(B)}
\zeta_\mathfrak{p}(1)
}
\frac{
|2|_k
}
{
\zeta_k(1) \zeta_k(2)
}.$$
Suppose that $F = \mathbb{Q}$, that $B$ is ramified precisely at $\{\infty,D\}$ for some prime $D \in \mathbb{Z}_{{\geqslant}1}$, and that $\mathfrak{q}$ corresponds to some prime $q \in \mathbb{Z}_{{\geqslant}1}$. By taking $\Psi = 1$ in the lemma and evaluating $c$, we obtain that for $N$ large enough, $$|\mathcal{F}_N|
=
q^{2 N}
\frac{D-1}{12}
(1 - \frac{1}{q})
(1 - \frac{1}{q^2})
\cdot \begin{cases}
1/2 & q = 2 \\
1 & q \text{ is odd.}
\end{cases}$$
It will suffice to show for each $\sigma \in \Sigma$ that $$\label{eq:local-weyl-law-final-section-lhs-rewritten}
|\Sigma| \sum_{\varphi \in \mathcal{F}_N^\sigma} \langle \varphi, \Psi \varphi \rangle$$ is eventually equal to the RHS of . For the remainder of the proof, set $\mathbf{G} := \operatorname{{\mathbf P}{\mathbf B}}^\times$. Let $f = \prod_\mathfrak{p} f_\mathfrak{p} \in
C_c^\infty(G_\mathbb{A})$ be given by $f_\mathfrak{p} := \operatorname{vol}(J_\mathfrak{p})^{-1} 1_{J_\mathfrak{p}}$ for $\mathfrak{p} \neq \mathfrak{q}$ (see §\[sec:general-estimates-specialized-single-place\]) and by taking for $f_\mathfrak{q} \in
C_c^\infty(G_\mathfrak{q}) = C_c^\infty(\PGL_2(k))$ the element attached to $(N,\sigma)$ as in §\[sec:element-attached-to-N-sigma\]. By Proposition \[prop:harmonic-analytic-isolation-1\], the example of §\[sec:omega-pi\], and the pretrace formula (§\[sec:pretrace-formula\]), we have $$\label{eqn:integrated-pretrace-formula}
\sum_{\varphi \in \mathcal{F}_N^\sigma} \langle \varphi, \Psi
\varphi \rangle
=
\int_{g \in [G]}
\Psi(g)
\sum_{\gamma \in G}
f(g^{-1} \gamma g).$$ As in the proof of the trace formula, the RHS of may be folded as $\sum_{\{\gamma \}} I(\gamma)$, where $\gamma$ traverses a set of representatives for the $G$-conjugacy classes in $G$ and $I(\gamma) := \int_{h \in [G_\gamma]} \int_{g \in
G_{\gamma,\mathbb{A}} \backslash \mathbf{G}_{\mathbb{A}}} \Psi(h g)
f(g^{-1} \gamma g)$; here $\mathbf{G}_\gamma$ denotes the centralizer. The function $f$ is supported in a fixed (i.e., independent of $N$) compact subset of $G_\mathbb{A}$, hence $I(\gamma) = 0$ for $\gamma$ outside some fixed finite collection of representatives.
Let $\mu$ denote the Tamagawa measure on $\PB^\times_\mathbb{A}$. Recall the subgroup $\mathfrak{J} {\leqslant}G_\mathfrak{q}$ arising in the definition of $f$. Set $\mathfrak{J} ' := \mathfrak{J} \times \prod_{\mathfrak{p} \neq \mathfrak{q}} J_\mathfrak{p}$. Since $|\Sigma| \cdot |\mathcal{X}_N^\sigma| =
|\mathcal{X}_N|$, one has $$\label{eq:Sigma-I-of-1-rewrite}
|\Sigma|
I(1)
=
\langle \Psi, 1 \rangle
|\Sigma| f(1)
=
\langle \Psi, 1 \rangle
|\mathcal{X}_N|
\mu(\mathfrak{J} ')^{-1}.$$ To compute the latter volume, it is convenient to factor $\mu = \prod \mu_\mathfrak{p}$ into the local measures $\mu_\mathfrak{p}$ as defined in §\[sec:local-measures\] relative to the *standard* nontrivial unitary character $\psi = \prod \psi_\mathfrak{p}$ of $\mathbb{A}/F$, i.e., that for which $\psi_\mathfrak{p}(x) = e^{2 \pi i x}$ for infinite places $\mathfrak{p}$. For a finite place $\mathfrak{p}$, $\Delta_{F_\mathfrak{p}} = \Delta_{\psi_\mathfrak{p}}$ is then the absolute discriminant of $F_\mathfrak{p}$. We record some consequences of the volume formulas of §\[sec:local-vol-formulas\] and §\[sec:measures-for-fourier-computation\]:
1. If $\mathfrak{p}$ is finite and $B_\mathfrak{p}$ splits, then $\mu_\mathfrak{p}(J_\mathfrak{p}) =
\zeta_\mathfrak{p}(2)^{-1}
\Delta_{B_\mathfrak{p}}^{-1}
\Delta_{F_\mathfrak{p}}^{-3/2}$.
2. If $\mathfrak{p}$ is finite and $B_\mathfrak{p}$ ramifies, then $\mu_\mathfrak{p}(J_\mathfrak{p}) =
\zeta_\mathfrak{p}(1)
\zeta_\mathfrak{p}(2)^{-1} \Delta_{B_\mathfrak{p}}^{-1}
\Delta_{F_\mathfrak{p}}^{-3/2}$.
3. If $\mathfrak{p}$ is infinite, then $\mu_\mathfrak{p}(J_\mathfrak{p}) =
\mu_\mathfrak{p}(G_\mathfrak{p})
= 4 \pi^2$.
4. $\mu_{\mathfrak{q}}(\mathfrak{J}) /
\mu_{\mathfrak{q}}(J_\mathfrak{p})
= |2|_k^{-1} q^{-N} \zeta_k(1)^{-1} \zeta_k(2)$.
Therefore $$\mu(\mathfrak{J} ')^{-1}
=
\frac{
\zeta_F(2)
\Delta_B
\Delta_F^{3/2}
}
{
(4 \pi^2)^{[F:\mathbb{Q}]}
\prod_{\mathfrak{p} \in \operatorname{ram}_f(B)}
\zeta_\mathfrak{p}(1)
}
\frac{
|2|_k
q^N
\zeta_k(1)}{\zeta_k(2)}.$$ Since $|\mathcal{X}_N| = q^N / \zeta_k(1)^2$ and $\langle 1,1 \rangle = \mu([\PB^\times]) = 2$, we obtain $$\label{eq:Sigma-I-of-1-rewrite-2}
|\mathcal{X}_N|
\mu(\mathfrak{J} ')^{-1}
= c q^{2 N}/ 2 = c q^{2 N} / \langle 1,1 \rangle.$$ By and , the contribution from $I(1)$ to gives the required RHS of .
To complete the proof, it suffices now to show for each fixed $1 \neq \gamma \in G$ that $I(\gamma) = 0$ for $N$ large enough. Let $U$ be a small enough but fixed compact open subgroup of $G_\mathfrak{q}$ under which $\Psi$ is invariant. By a change of variables in the definition of $I(\gamma)$, our task reduces to showing for all $g \in G_{\mathbb{A}}$ that $$\label{eq:penultimate-goal-for-linear-stats}
\int_{u \in U}
f(u^{-1} g^{-1} \gamma g u) = 0 \text{ for $N$ large enough.}$$ Fix a compact set $E$ (independent of $N$) containing the support of $f$, and let $j : G_{\mathbb{A}} \rightarrow G_{\mathbb{A}}$ denote the orbit map $j(g) := g^{-1} \gamma g$. Since $B$ is non-split, the group element $\gamma$ is regular semisimple (see §\[sec:bounds-part-orbit\]). The map $G_{\gamma,\mathbb{A}} \backslash G_{\mathbb{A}} \rightarrow
G_{\mathbb{A}}$ induced by $\gamma$ is thus proper, and so the set $j^{-1}(E)$ meets only a fixed finite collection of double cosets $G_{\gamma,\mathbb{A}} g U$. For this reason, it suffices to establish for each *fixed* $g \in G_{\mathbb{A}}$. The conclusion follows in that case from the lemma of §\[sec:bounds-part-orbit\].
Variance statistics {#sec:variance-statistics}
-------------------
### The sums
Define the sesquilinear forms $V_N : \mathcal{A}_{0+}^J \otimes \mathcal{A}_{0+}^J
\rightarrow \mathbb{C}$ by $$V_N(\Psi_1,\Psi_2)
=
\sum_{\varphi \in \mathcal{F}_N}
L^{(S)}(\operatorname{ad}\varphi,1)
\langle
\varphi, \Psi_1 \varphi
\rangle
\langle \Psi_2 \varphi, \varphi
\rangle.$$ We have written $L^{(S)}(\operatorname{ad}\varphi,1) := L^{(S)}(\operatorname{ad}\pi,1)$ for $\varphi \in \pi \in A_0^J$.
### The leading constant
Set $\mathbf{X} := [\PB^\times]/J$. Equip it with the quotient measure induced by any Haar measure on $[\PB^\times]$; the formulation of our results will not depend upon this choice. Set $$\label{eq:defn-of-c0}
c_0 := 2^{\# \operatorname{ram}_f(B)}
\zeta_F^{(S)}(2)
\operatorname{vol}(\mathbf{X})^{-1} \frac{1}{2 \zeta_k(1)}.$$
In the setting of the example of §\[sec:mean-statistics\], suppose that we identify $\mathbf{X}$ with $\Gamma \backslash \PGL_2(\mathbb{Q}_q)$ as in §\[sec:strong-approx\] and equip $\mathbf{X}$ with the quotient measure induced by the Haar on $\PGL_2(\mathbb{Q}_q)$ assigning volume one to $\PGL_2(\mathbb{Z}_q)$. Then $\operatorname{vol}(\mathbf{X})= (D-1)/12$ (see the remark of §\[sec:strong-approx\]). After some calculation, we obtain $$c_0
=
2 \pi^2
\frac{2 (1 + D^{-1})}{D}
\frac{(1 - q^{-1}) (1 - q^{-2})}{2}$$
### The proposed limit
Let $V_\infty : \mathcal{A}_{0+}^J \otimes \mathcal{A}_{0+}^J
\rightarrow \mathbb{C}$ denote the sesquilinear form given for $\Psi_i \in \pi_i \in A_{0+}^J$ ($i=1,2$) by $V_\infty(\Psi_1,\Psi_2) := 0$ unless $\pi_1 = \pi_2 =: \pi$, in which case $$V_\infty(\Psi_1,\Psi_2) :=
c_0 L^{(S)}(\pi,\tfrac{1}{2})
\int_{h \in N(H)}
\langle \pi(h) \Psi_1, \Psi_2 \rangle_{L^2(\mathbf{X})},$$ where the measure on $N(H)$ is as in §\[sec:local-estimates-main-statements\].
### Main result
In view of the discussion of §\[sec:strong-approx\] and §\[sec:hecke-ops\], the following result makes precise (and mildly generalizes) Theorem \[thm:main-result-for-microlocal-stuff\]:
\[thm:main-variance-microlocal-adelic-formulation\] Let $\Psi_1, \Psi_2 \in \mathcal{A}_{0+}^J$ be fixed (i.e., independent of $N$). Then $$q^{-N} V_N(\Psi_1,\Psi_2)
= V_\infty(\Psi_1,\Psi_2)
+ O(N q^{-N}).$$
The proof divides into five steps:
1. The partition $\mathcal{F}_N = \bigsqcup_{\sigma \in \Sigma}
\mathcal{F}_N^\sigma$ of the family induces a decomposition $V_N = \sum_{\sigma \in \Sigma} V_N^\sigma$ of the variance. Since $|\Sigma| \asymp 1$, it will suffice to show that $$q^{-N} V_N^{\sigma}(\Psi_1,\Psi_2)
= |\Sigma|^{-1} V_\infty(\Psi_1,\Psi_2)
+ O(N q^{-N}).$$
2. Let $f \in C_c^\infty(\PB^\times_\mathfrak{q})$ be attached to $(N,\sigma)$ (see §\[sec:element-attached-to-N-sigma\]). Recall from §\[sec:general-estimates-specialized-single-place\] the definitions of $V_f,M_f$. By Proposition \[prop:harmonic-analytic-isolation-1\] and the example of §\[sec:omega-pi\], we have $V_N^\sigma = V_f$.
3. By and , we have $c_4 q^{N-N_0} \frac{1}{2}
=
q^N |\Sigma|^{-1} c_0$. Feeding this calculation into Proposition \[prop:desired-main-term-identity-do-it-up\] gives $$M_f(\Psi_1,\Psi_2)
= q^N |\Sigma|^{-1} V_\infty(\Psi_1,\Psi_2)$$ for $N$ large enough in terms of $\Psi_1,\Psi_2$. We reduce to showing that $$\label{eqn:desired-error-estimate--do-it-1}
V_f(\Psi_1,\Psi_2) - M_f(\Psi_1,\Psi_2) \ll N.$$
4. Fix a nontrivial unitary character $\psi$ of $\mathbb{A}/F$ whose component $\psi_\mathfrak{q}$ is unramified. Fix a nonzero element $W_S = \prod_{\mathfrak{p} \in S}
W_\mathfrak{p} \in C_c^\infty(F_S^\times)$ for which $W_\mathfrak{q} := 1_{\mathfrak{o}^\times}$. We apply Theorem \[thm:main-estimate-general-variance\] with respect to $\psi$ and $W_S$; our task thereby reduces to showing for fixed $\tau_1,\tau_2 \in F^\times$ that $$\label{eq:penultimate-task-almost-there}
\eps_{\tau_1,\tau_2}(\heartsuit^{\tau_1} \tilde{f},
\heartsuit^{\tau_2} \tilde{f},
\Psi_1,\Psi_2)
\ll N,$$ where $\tilde{f} \in C_c^\infty(\PB^\times_S)$ is as in §\[sec:general-estimates-specialized-single-place\] and $\heartsuit^{\tau} \tilde{f} \in \mathcal{S}(B_S)$ as in §\[sec:heartsuit\].
5. One has $\heartsuit^{\tau} \tilde{f}
= (\heartsuit^{\tau} f) \otimes \phi^{\tau}$ where $\heartsuit^{\tau} f \in \mathcal{S}(B_\mathfrak{q})$ is as in §\[sec:defn-heartsuit-local-again-without-Omega\] and $\phi^{\tau} = \otimes_{\mathfrak{p} \in S - \{\mathfrak{q} \}}
\phi_\mathfrak{p}^{\tau}$ with $\phi_\mathfrak{p}^{\tau} \in
\mathcal{S} (B_\mathfrak{p} )$ independent of $N$. The LHS of is thus equal to $\ell(\heartsuit^{\tau_1} f,
\heartsuit^{\tau_2} f)$, where $\ell : \mathcal{S}(B_\mathfrak{q}) \otimes
\mathcal{S}(B_\mathfrak{q})
\rightarrow \mathbb{C}$ denotes the sesquilinear form given by $$\ell(\Phi_1,\Phi_2)
:=
\eps_{\tau_1,\tau_2}( \Phi_1
\otimes \phi^{\tau_1},
\Phi_2
\otimes \phi^{\tau_2},
\Psi_1,\Psi_2).$$ Observe that $\ell$ is independent of $N$. Theorem \[thm:main-estimate-general-variance\] says that $\ell$ is good in the sense of §\[sec-8-2\] with respect to $(\psi_\mathfrak{q},\tau_1,\tau_2)$ (taking for $U {\leqslant}\PB^\times_\mathfrak{q}$ any open subgroup under which $\Psi_1,\Psi_2$ are invariant). Proposition \[prop:local-error-estimates-stmt\] implies finally that $\ell(\heartsuit f, \heartsuit f) \ll N$, giving , as required.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We gratefully acknowledge the support of NSF grant OISE-1064866 and SNF grant SNF-137488 during the work leading to this paper.
[^1]: In the prequel, the base field is unimportantly restricted to be $\mathbb{Q}_2$ (rather than $\mathbb{Q}_p$ for an odd prime $p$) in order to reduce the set of bad primes from $\{2, p\}$ to $\{2\}$, noting that $2$ is always bad when studying half-integral weight forms.
[^2]: Since we lack a convenient reference sharing our measure normalizations, we record the proof. The LHS of expands to $\operatorname{vol}(\mathfrak{o}^\times) \sum_{n {\geqslant}0} |a_n|^2 x^n$, where $a_n := (\alpha^{n+1} - \beta^{n+1})/(\alpha-\beta)$ and $x := q^{-(1+s) n}$. One has $\operatorname{vol}(\mathfrak{o}^\times) = \zeta_k(1)^{-1}
\Delta_\psi^{-1/2}$. The identity $\sum a_n x^n = (1 - \alpha x)^{-1} (1 - \beta x)^{-1}$ and [@MR1431508 Lem 1.6.1] imply that $\sum_{n {\geqslant}0} |a_n|^2 x^n = (1 - x^2) (1 - \alpha
\overline{\alpha } x)^{-1} (1 - \alpha \overline{\beta }
x)^{-1} (1 - \beta \overline{\alpha} x)^{-1} (1 - \beta
\overline{\beta} x)^{-1}$. Invoking the consequence $\{\alpha \overline{\alpha }, \alpha \overline{\beta }, \beta
\overline{\alpha}, \beta \overline{\beta }\} = \{1, \alpha
\beta^{-1}, \beta \alpha^{-1}, 1\}$ of the assumed unitarity of $\pi$, we obtain $\sum_{n {\geqslant}0} |a_n|^2 x^n = \zeta_k(2 + 2 s)^{-1} L(\operatorname{ad}\pi,1 + s) \zeta_k(1 + s)$, as required.
[^3]: The cited reference normalizes measures differently than we do. For the convenience of the reader, we sketch here the proof that is normalized correctly, taking for granted the meromorphicity of $I(s)$. By unfolding, $I(s) = \int_{y \in \mathbb{A}^\times / F^\times} \int_{x
\in \mathbb{A}/F} |\varphi(n(x) a(y))|^2 |y|^s \,
\frac{d y}{|y|}$. Because the pushforward under $|.| : \mathbb{A}^\times / F^\times \rightarrow
\mathbb{R}_+^\times$ of $\frac{d y}{|y|}$ is the standard Haar measure $\frac{d t}{t}$ on $\mathbb{R}^\times_+$ (see §\[sec:global-measures\]), it follows that $\operatorname{res}_{s \rightarrow 1} I(s) = \lim_{t \rightarrow 0}
\mathbb{E}_{y : |y| = t} \int_{x \in \mathbb{A}/F}
|\varphi(n(x) a(y))|^2$ where $\mathbb{E}$ denotes an average with respect to the probability measure invariant by the norm one idele class group. By the equidistribution of the horocycle flow and the normalization $\operatorname{vol}(\mathbb{A}/F) = 1$, that limit is the integral of $|\varphi|^2$ with respect to the probability Haar on $[\PGL_2]$. By the consequence $\operatorname{vol}([\PGL_2]) = 2$ of our measure normalizations, we obtain .
[^4]: For instance, this follows (in overkill fashion) from the proof of [@MR1616155 Thm 9.1].
|
---
abstract: 'High quality single wall carbon nanotube quantum dots have been made showing both metallic and semiconducting behavior. Some of the devices are identified as small band gap semiconducting nanotubes with relatively high broad conductance oscillations for hole transport through the valence band and low conductance Coulomb blockade oscillations for electron transport through the conduction band. The transition between these regimes illustrates that transport evolves from being wave-like transmission known as Fabry-Perot interference to single particle-like tunneling of electrons or holes. In the intermediate regime four Coulomb blockade peaks appear in each Fabry-Perot resonance, which is interpreted as entering the SU(4) Kondo regime. A bias shift of opposite polarity for the Kondo resonances for one electron and one hole in a shell is in some cases observed.'
author:
- 'K. Grove-Rasmussen'
- 'H. I. Jørgensen'
- 'P. E. Lindelof'
bibliography:
- 'PhysicaE.bib'
title: 'Fabry-Perot interference, Kondo effect and Coulomb blockade in carbon nanotubes'
---
l
Introduction
============
Single wall carbon nanotubes (SWCNT) are interesting objects for the study of low dimensional mesoscopic systems, where the observed phenomena crucially depends on the coupling to the contacts. For good contact, the SWCNT acts as an electron wave guide creating resonances at certain energies. Such systems are regarded as open quantum dots with the resonances corresponding to the broad energy levels of the quantum dot [@Liang]. In the opposite limit of very low transparency, the electrons are forced to tunnel one by one due to Coulomb blockade and the energy levels sharpens due to their longer life time [@Tans1998; @BokrathQDropes1997]. In this so-called closed quantum dot regime the electron number on the SWCNT is well-defined except at charge-degeneracy points where single electron tunneling occurs. An intermediate regime also exists in which the electron number on the dot is still fixed but significant cotunneling is allowed. This leads to different kinds of Kondo effects related to the total excess spin [@GoldhaberGordon1998Nature; @NygaardKondo; @Paaske] and/or the orbital degree of freedom on the SWCNT quantum dot [@JarilloOrbitalKondo2005Nature; @Makarovski]. We will in this paper examine the transition between these regimes in small band gap semiconducting nanotubes, where the coupling to the SWCNT is different for transport through the valence and conduction band [@Cao2002NatureMaterials]. High quality measurement is presented from each transparency regime showing that each Fabry-Perot oscillation develops into four Coulomb blockade resonances with finite conductance in the valleys for intermediate transparency (Kondo effect) and well known Coulomb diamond patterns with four-fold degenerate shell structure at lowest transparency [@Sapmazprb].
Experimental methods
====================
SWCNTs are grown from predefined catalyst islands by chemical vapor deposition on a highly doped silicon substrate capped by a 500nm silicon dioxide layer. After growth, pairs of electrodes are defined by electron beam lithography some microns away from the catalyst islands in hope that one SWCNT bridges the gap between the two electrodes. In the case of the small band gap semiconducting SWCNT the electrodes consist of Au/Pd bilayers (40nm/10nm). Finally bonding pads of Au/Cr are made by optical lithography connecting the electron beam lithography defined electrodes. The devices are electrically probed at room temperature with typical resistances in the range of 20-200k$\Omega$. These are cooled to low temperatures where device characteristics clearly reveals if only one SWCNT is in the gap. More details on device fabrication can be found in Refs. [@MSS2006Proc; @hij]. The measurements presented in this article are mostly made at 4K in a DC-setup.
Metallic and Semiconducting nanotubes
=====================================
SWCNTs have the remarkable property that depending on the exact arrangement of the carbon atoms they can either be metals or semiconductors. Figure \[fig1\] shows the gate (and temperature) dependence of the linear conductance for two different types of SWCNTs, which defines whether the nanotube is semiconducting or metallic. Figure \[fig1\](a) displays the behavior of a metallic SWCNT characterized by its relatively constant linear conductance as a function of gate voltage close to room temperature. On the contrary the behavior shown in Fig. \[fig1\](b) corresponds to a semiconducting SWCNT due to its strongly gate dependent linear conductance [@TansTransistor]. The semiconducting SWCNT has clearly high conductance for negative gate voltages, which is due to hole transport through the valence band as indicated in the inset showing the band structure (p-type). In the range of 4V $<V_{gate}<10$V the chemical potential of the leads is aligned with the band gap, i.e., no conduction occurs. At higher temperatures signs of electron transport through the conduction band is seen at high positive gate voltages due to thermal excitation of electrons into the conduction band. The SWCNT shown in Fig.\[fig1\](b) has a relative large band gap, while conduction for small band gap semiconductors reappears for positive gate voltage in the gate range shown due to electron transport through the conduction band (see below). Note, that the resistance between the two devices differ by an order of magnitude due to different coupling to the leads. In general the coupling can to some extent be controlled by choice of electrode material [@Babic].
Figure \[fig1\](a-b) also shows the temperature dependence, where the gate dependence of the linear conductance for both devices evolves into regular oscillations at low temperature. These oscillations are a manifestation of Coulomb blockade, which as mentioned above happens for SWCNT weakly coupled to the leads. The regularity of the Coulomb oscillations indicates whether good quality of the SWCNT has been obtained or if more than one SWCNT is bridging the gap between the electrodes. For high quality SWCNTs regular oscillations should persist through a gate region of typically $|V_{gate}|<10$V as shown in (a). On the contrary if the SWCNT has defects only small regions of gate voltage might have well behaved oscillations.
![Two different types of SWCNTs. (a) A metallic SWCNT identified by its weak gate voltage dependence at high temperature ($T= 220, 120, 38, 10$K, black to blue). At low temperatures oscillations in the conductance versus gate voltage are seen due to Coulomb Blockade. Inset: Band diagram of a metallic SWCNT (armchair). (b) A semiconducting SWCNT identified by its strong gate dependence of the linear conductance. Oscillations of the conductance at low temperature in the p-type region is due to single hole transport (Coulomb blockade). Inset: Band diagram of a semiconducting SWCNT with the electrochemical potential in the valence band corresponding to the situation for $V_{gate} < 4$V. Curves are taken at temperatures ($T=150, 70, 40, 30, 15, 10,
3.3$K, black to violet).[]{data-label="fig1"}](Fig1){width="48.00000%"}
Small band gap semiconducting nanotubes
=======================================
We will now focus on small band gap semiconducting SWCNTs, where the band gap is so small that transport can be tuned from hole transport through the valence band to electron transport through the conduction band by the back gate [@Jarillo-Herreroe-hsym; @CaoSmall; @Cao2004PhRvL; @Cao2002NatureMaterials].
{width="70.00000%"}
Figure \[fig2\](a) shows the linear conductance versus gate voltage at source-drain voltages $V_{sd} \sim 1$ m$e$V for a small band gap SWCNT at $T=4$K (device 1). The nature of the SWCNT is identified by the relatively high conductance region for negative gate voltages (hole transport) in contrast to the low conductance region for positive gate voltages (electron transport). The broad oscillations for hole transport for gate voltages between $-3$V and 0.5V are a manifestation of the Fabry-Perot interference pattern, [*i.e.*]{}, an open quantum dot [@Liang]. In contrast for positive gate voltages regular low conductance oscillations are observed due to Coulomb blockade. Figure \[fig2\](b) shows the high positive gate region for electron transport where relatively high conductance Coulomb blockade resonances are seen. They are spaced into four reflecting the spin and orbital degree of freedom. When this device is cooled to lower temperature (50mK) Kondo resonances are seen within the four-fold shell structure (not shown) in contrast to the lower conducting region, e.g., gate voltages from 5V to 8V, where only single electron tunneling is possible.
The different transparency of the n- and p-type regions can be understood from Fig. \[fig2\](c). For negative gate-voltages (left figure) the bands are bending in such a way that holes can tunnel from source into the valence band and out to drain. The Schottky barrier for hole transport is relatively small because the workfunction of the Pd contacts is close to the valence band edge leading to a relatively high conductance [@Cao2002NatureMaterials]. Transport can be changed to electron transport through the conduction band (right figure) by applying positive voltage to the gate. The Schottky barrier is in this case significantly larger leading to a low coupling of the SWCNT to the electrodes. Between these two transport regions no transport is allowed because the chemical potential in the leads is within the band gap of the semiconducting SWCNT.
{width="80.00000%"}
More information on the transport properties can be revealed by bias spectroscopy, where the differential conductance is measured as a function of gate and bias voltages. A bias spectroscopy plot of part of the Fabry-Perot region is shown in Fig. \[fig3\](a). The low conductance regions (red areas) form a mesh due to interference of the hole waves reflected back and forth. Regarding the device as a quantum dot the white regions at zero bias correspond to being in resonance, while a red regions are off resonance. The level spacing can be extracted as indicated in Fig. \[fig3\](b) by the black arrow $\Delta E \sim 4 $m$e$V consistent with the device being around $L= \frac{\hbar v_F \pi}{\Delta E} \sim 400$nm. Here we have used a linear dispersion since we are far from the band gap. Furthermore, a four-fold degeneracy of the level is assumed, which is clearly revealed in the n-type region of the device (see below). The device has a high asymmetric resistive coupling, because the conductance at resonance is lower than $4e^2/h$. The asymmetry $\Gamma_L/\Gamma_R$ can be found from $G_{res}=\frac{4e^2}{h}\frac{4\Gamma_L
\Gamma_R}{(\Gamma_L+\Gamma_R)^2}$, where $G_{exp,res}\sim2e^2/h$ is the conductance at resonance yielding $\Gamma_L/\Gamma_R=0.17$. The capacitive couplings to the source and the drain electrodes are also slightly different or/and the capacitive coupling to the gate is comparable to the source and drain capacitance because of the different slopes of the low conductance lines (red). The capacitances will be examined more closely in the Coulomb blockade case below.
Figure \[fig3\](b) shows a bias spectroscopy plot in the p-type region for another small band gap semiconducting SWCNT with lower transparency (device 2). Signs of Fabry-Perot oscillations are still observed, but each Fabry-Perot resonance is split into four smaller peaks [@Cao2002NatureMaterials]. These peaks are due to Coulomb blockade and single hole tunneling, [*i.e.*]{}, the four-fold degenerate level becomes visible due to quantization of the charge. The finite conductance in the valleys between the peaks are indication of SU(4) Kondo effect, which will be treated in more detail below [@JarilloOrbitalKondo2005Nature].
Figure \[fig3\](c) shows a bias spectroscopy plot in the n-type region of device 1, where the transparency is reduced even further. Clear Coulomb blockade diamonds and excited states are observed. The numbers indicate the relative electron filling of the SWCNT quantum dot and a number dividable by four corresponds to a filled shell identified as three small diamonds followed by a bigger diamond. A shell thus consists of a four-fold degenerate level as expected due to orbital and spin degrees of freedom. The charging energy can be extracted from the three small diamonds as half the source-drain height $U_c \sim 11$m$e$V (left red arrow). The orbital splitting and exchange energy are very small since the three consecutive small diamonds in one shell are almost equal in height, e.g., diamonds 5, 6 and 7. Every fourth diamond is bigger (filled shell) since the addition of the first electron in a new shell requires both a charging energy and a level spacing energy. The additional diamond height of the larger diamonds thus yields the level spacing $\Delta
E \sim 4 $m$e$V (right black arrow). This is consistent with the level spacing identified from the excited state lines shown for electron filling 7 (left black arrow). The asymmetry of the diamonds are due to the capacitive coupling of the source $C_s$, drain $C_d$ and gate $C_{g}$ electrodes to the SWCNT. Estimating the slopes $\alpha_s=0.51$ and $\alpha_d=-1.02$ of the two lines constituting the diamond (dashed red lines) as well as the gate voltage distance $\Delta V_g=33.6$mV between two Coulomb diamonds (diamond 7) the capacitances can be found. We find $C_s=4.7$aF, $C_d=4.6$aF and $C_{g}=4.8$aF by $\alpha_s=C_g/(C_g+C_d)$, $\alpha_d=-C_g/C_s$ and $\Delta V_g = e/C_g$, where $C=C_s+C_d+C_{g}$ is the total capacitance to the surroundings, $\alpha_{s/d}$ corresponds to aligning the electrochemical potentials of the dot with the source/drain and an asymmetric biasing with the drain on ground is used [@Hanson]. The equal magnitude of the gate and source/drain capacitive coupling thus makes the diamond asymmetric.
 with no single hole charging effects, while the four-fold periodicity becomes visible due to Coulomb blockade for the black curve from Fig. \[fig3\](b). This structure is even more evident when the coupling decreases further as shown for the n-type region of device 1 (magenta and blue curves). All the curves are translated along the gate axis, but not scaled.[]{data-label="fig:FPtoCB"}](Fig4){width="48.00000%"}
The transition between the different transparency regimes is even more clearly revealed in the linear conductance versus gate voltage shown in Fig. \[fig:FPtoCB\]. All curves are extracted at zero bias from bias spectroscopy plots in the p- or n-type region of the two small band gap semiconducting SWCNTs presented above (device 1 and 2). The most conducting device (red curve) shows broad Fabry-Perot oscillations with no sign of single hole transport consistent with holes added continuously to the SWCNT due to the relatively good coupling. When the coupling to the SWCNT weakens, the holes become more localized on the SWCNT and single hole transport is observed (black curve). Four Coulomb blockade resonances emerge in each broad Fabry-Perot resonance consistent with each resonance (level) being four-fold degenerate. For even weaker coupling a closed quantum dot is formed (magenta and blue curves), where the measurement stems from the n-type region of device 1. The curves are not scaled but only shifted along the gate axis. Similar observations have been made by Cao [*et al.*]{} on very clean *suspended* small band gap semiconducting SWCNTs and also in our measurement the Coulomb energy seem to diminish as the transparency increases [@Cao2002NatureMaterials]. This is seen by the distance between the peaks within one shell for the magenta curve is smaller than in the case of the black curve.
Intermediate regime
-------------------
We now want to examine the transport behavior in the regime of intermediate transparency in more detail. Figure \[fig5\](a) shows a bias spectroscopy plot taken from Fig. \[fig3\](b) with the filling of only two shells each containing a four-fold degenerate level, i.e., addition of 8 holes as the gate voltage becomes more negative. The numbers 0,..,4 show the number of holes in one of the shells. The Coulomb diamonds are still faintly visible despite the relative high transparency, where the big diamond(s) correspond to filled shells. The charging energy and level spacing can be estimated as above yielding $U_c\sim\Delta E\sim 2.5$meV (see arrows). Similarly, to the case of the closed quantum dot we deduce the capacitances of the device from the slopes of the diamond sides ($\alpha_d=-0.185 and \alpha_s=0.178$) and the voltage distance between two consecutive Coulomb blockade resonances ($\Delta
V_g=27$mV) in the shell giving $C_s=32$aF, $C_d=27$aF and $C_g=6$aF. In the case of the closed quantum dot of device 1, the gate capacitance is almost identical ($\sim 5$aF) consistent with the two devices having the same geometry. In contrast to the more closed dot (device 1), the source and drain capacitances are increased. We speculate that increasing the transparency to the source and drain contacts leads to an increase of the source and drain capacitances because the wave function extends below the contact. Such behavior is also consistent with the charging energy diminishing with increasing transparency as noted above. Figure \[fig5\](c) shows the bias cuts in the center of the diamonds for filling 1, 2 and 3 (see colored arrows), where clear zero bias peaks are present. These zero bias peak are manifestation of the SU(4) Kondo effect due to the four-fold degeneracy of the shell [@JarilloOrbitalKondo2005Nature; @ChoiPRL]. The FWHM of the Kondo peak for filling of 2 holes yields 1.5mV by a Lorentzian fit corresponding to a Kondo temperature of 5.1K higher than the measurement temperature of 4K. The center positions of the Kondo and Coulomb blockade resonances within a shell are at zero bias as expected indicated by the red dashed line.
Figure \[fig5\](b) shows another gate region from Fig.\[fig3\](b) with SU(4) Kondo effect showing a different behavior of the Coulomb blockade and Kondo resonance positions. The Coulomb blockade resonances are clearly shifted to more negative source-drain voltage as the holes are added to the shell with a slope of $\Delta V_{sd}/\Delta V_{gate}\sim 0.008$ (red dashed line). Furthermore, the Coulomb blockade resonances for the hole transitions 0’-1’ (1’-2’) and 3’-4’ (2’-3’) are shifted equally in bias but with different polarity. Similarly, the Kondo peak for holes fillings 1’ and 3’ (i.e., one electron) are shifted oppositely in bias ($\sim \pm 0.5$meV) as shown in Fig. \[fig5\](d), while the Kondo peak for hole filling 2’ is centered at zero bias. We note that an opposite slope is also found in Fig. \[fig3\](b) ($V_{gate}\sim 1.1$V). This behavior has previously been observed in carbon nanotubes showing four-fold shell structure [@Makarovski; @BabicKondo] and also for spin half Kondo effect in a GaAs based quantum dot [@Simmel]. In Ref. [@Makarovski] the authors speculate that it is related to SU(4) Kondo effect, because SU(4) theory predicts a shifted density of states compared to the spin half Kondo effect for electron filling 1 in equilibrium [@ChoiPRB]. However, in Ref.[@BabicKondo; @Simmel] and theoretical work on spin half Kondo effect [@Kraviec] the behavior was attributed to the asymmetry of the coupling to the leads, which shifts the Kondo peaks to finite biases. The asymmetry can be extract from the linear conductance of the Kondo resonance in the unitary limit or from current plateau corresponding to the total current through one level [@BabicKondo]. However, at this temperature the Kondo resonances are not saturated and the current plateaus are too smeared to extract the couplings. More experimental work at lower temperature as well as precise theoretical predictions of these shifts are needed to elaborate more on this effect.
 showing SU(4) Kondo effect. Numbers indicate the number of holes in a shell. The Coulomb diamonds are shown by white dashed lines and the white double arrows reveal the level spacing and the charging energy. The red dashed lines show how the Coulomb and Kondo resonances are shifted in bias as holes are added to the shell, i.e., no and finite slope for (a) and (b), respectively. The green, blue and red arrows point to bias cut for hole filling one, two and three displayed in (c-d). Clear Kondo peaks are observed, while they are shifted to opposite bias for hole filling 1’ and 3’ in (d).[]{data-label="fig5"}](Fig5){width="48.00000%"}
Conclusion
==========
In conclusion measurements on very clean single wall carbon nanotube quantum dots have been presented. We focused on small band gap semiconducting SWCNTs and the transition from an open to a closed quantum dot (Fabry-Perot interference to Coulomb blockade). The appearance of four peaks is observed in each Fabry-Perot resonance as the transparency is decreased, which is interpreted as entering the SU(4) Kondo regime. The Kondo resonances for one hole and one electron in a shell are in some cases shifted to opposite biases. At even lower transparency clear Coulomb blockade shell structure with a four-fold degeneracy due to orbital and spin degrees of freedom is observed.
Acknowlegdement
===============
We would like to thank Jens Paaske for discussions and the support of the EU-STREP Ultra-1D program.
|
---
abstract: 'We consider a registration-based approach for localizing sensor networks from range measurements. This is based on the assumption that one can find overlapping *cliques* spanning the network. That is, for each sensor, one can identify geometric neighbors for which all inter-sensor ranges are known. Such cliques can be efficiently localized using multidimensional scaling. However, since each clique is localized in some local coordinate system, we are required to *register* them in a global coordinate system. In other words, our approach is based on transforming the localization problem into a problem of registration. In this context, the main contributions are as follows. First, we describe an efficient method for partitioning the network into overlapping cliques. Second, we study the problem of registering the localized cliques, and formulate a necessary *rigidity* condition for uniquely recovering the global sensor coordinates. In particular, we present a method for efficiently testing rigidity, and a proposal for augmenting the partitioned network to enforce rigidity. A recently proposed semidefinite relaxation of global registration is used for registering the cliques. We present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state-of-the-art methods in terms of run-time, accuracy, and scalability.'
author:
- 'R. Sanyal, M. Jaiswal, and K. N. Chaudhury [^1]'
bibliography:
- 'citations.bib'
title: 'On a Registration-Based Approach to Sensor Network Localization'
---
Sensor networks, localization, scalability, rigidity, clique, multidimensional scaling, semidefinite programming.
Introduction
============
Recent developments in wireless communication and micro-electro-mechanics have proliferated the deployment of wireless sensor networks (WSN) [@YMG2008]. A typical WSN may consist of few tens to thousands of nodes. Each node is a low-power device equipped with transducers, power supply, memory, processor, radio transmitter, and actuators. A global positioning system (GPS) is often installed on some of the nodes. Such nodes are referred to as *anchor* nodes. However, only a small fraction of the nodes are equipped with GPS to minimize weight and power consumption. In this paper, we will use *sensor* to specifically refer to a node that does not have a GPS, while the term *node* will be used for both sensors and anchors. WSNs are mostly deployed in remote locations, and nodes have limited memory capacity, so wireless transmitters are used to transfer the sensor data to base stations. Due to power constraints, two nodes can communicate if and only if the inter-node distance is within some *radio range*, which we will denote by $r$ [@YMG2008]. We would like to note that although GPS modules are getting cheaper, deploying them in large scale would still be costly. Moreover, GPS comes with its own limitations [@YMG2008]. To calculate the position of a sensor using GPS alone, at least four line-of-sights (with satellites) are required. This might not be viable in case of bad weather. Furthermore, for underwater surveys and mining applications, it is not even feasible to have line-of-sights. In fact, in applications where the position information is crucial, localization algorithms can be used to back up GPS positioning. To meaningfully interpret the sensor data, one requires the locations of the sensors. A central problem in this regard is to estimate the sensor locations from the inter-sensor distances and the anchor locations. This problem is referred to as sensor network localization (SNL) [@SY2007; @MFA2007]. To set up the mathematical description of SNL, we introduce some notations that will be follow throughout the paper. Assume that we have a total of $N$ sensors and $K$ anchors. We label the sensors using $\cS=\{1,\ldots,N\}$, the anchors using $\cA=\{N+1,\ldots,N+K\}$, and $\cN = \cS \cup \cA$ denotes the nodes in general. Let $$\label{nodes}
\cX_s=\{\bar{\x}_i : i \in \cS\} \quad \text{and} \quad \cX_a=\{\bar{\a}_k : k \in \cA\}$$ denote the sensor and anchor locations. We assume $\bar{\x}_i$ and $\bar{\a}_k$ to be in $\mathbb{R}^d$, where $d$ is typically $2$ or $3$ [@YMG2008; @SY2007].
The distance between two nodes $i$ and $j$ (that are within the radio range $r$) can be calculated using different techniques, such as the received signal strength or the time of arrival [@MFA2007]. A *measurement graph* $\cG$ is used to encode the distance information [@SXG2015; @CLS2012]. Particularly, $\cG = (\cV,\cE)$, where $\cV(\cG)=\cN$, and $(i,j) \in \cE(\cG)$ if and only if the distance between the $i$-th and the $j$-th node is known. The problem is to compute the unknown sensor locations $\cX_s$ from the measured distances and the anchor locations $\cX_a$. We make the standard assumption that the anchor locations are noise-free [@SXG2015; @SL2014].
Optimization Algorithms
-----------------------
The decision version of the SNL problem is known to be computationally intractable [@Saxe1979]. The presence of noise makes the problem even more challenging in practice. Nonetheless, several methods have been proposed that can compute approximate solutions. A survey of the literature on SNL is beyond the scope of this paper. Instead, we will focus on some of the recent optimization methods that are related to the present work. We refer the interested reader to [@MFA2007] for a survey of algorithms that are not based on optimization.
The simplest optimization framework for SNL is that of strain minimization [@BLTYW2006]. In this approach, the sensor locations $\x_1, \dots , \x_N$ are obtained by minimizing the *strain* function $$\label{SNL_strain}
\!\sum_{(i,j) \in \cE } \! \left( \lVert \x_i - \x_j \rVert^2 - d_{ij}^2 \right)^2\! +\! \sum_{(i,k) \in \cE} \! \left( \lVert \x_i - \a_k \rVert^2 - d_{ik}^2 \right)^2.$$ In , the indices $i,j$ are reserved for $\cS$, and the index $k$ for $\cA$. Unfortunately, it is difficult to compute the global minimum of since it is non-convex in the variables [@SY2007]. In this regard, several approximation algorithms based on convex programming have been proposed, which can provably compute the global minimum under certain conditions. Based on the computing paradigm, one can broadly classify these as centralized and distributed algorithms.
Centralized algorithms employ a server to store the transmitted range measurements, based on which the sensor locations are computed. It was observed in [@DPG2001] that the distance bounds in SNL can be posed as semidefinite constraints. Later, in the seminal paper [@BLTYW2006], the authors showed how can be approximated using a convex semidefinite program (SDP). The main advantage of posing SNL as a convex program is that we can find the global minimizer of the problem independent of the initialization. The flip side, however, is that standard SDP solvers (e.g., SeDuMi [@Sturm1999]) are memory and computation intensive, and hence cannot be scaled to large-sized problems. For example, the SDP-based algorithm in [@BLTYW2006] can scale only up to a few hundred nodes [@WZYB2008]. To improve the scalability, a further edge-based relaxation of [@BLTYW2006] was proposed in [@WZYB2008]. While the relaxation can indeed scale up to $8000$ nodes, its performance is nevertheless inferior to that of the original SDP for medium-sized problems.
On the other hand, distributed algorithms divide the processing over the nodes. As a result, they exhibit better scalability compared to centralized methods. The main drawback is that they suffer from error propagation [@MFA2007]. Moreover, distributed methods such as the ones in [@SXG2015; @SL2014; @ADPV2012; @GTSC2013] can operate only in the presence of anchors (which might not be available, e.g., in indoor WSN). The distributed algorithm in [@ADPV2012] that can handle million sensors without any significant communication overhead. However, the localization accuracy of this method is conditioned on a good initialization. More recently, distributed methods based on convex programming have been proposed in [@SXG2015; @SL2014; @GTSC2013]. In particular, a distributed algorithm based on the alternating direction method of multipliers was proposed in [@SL2014]. However, as reported in [@SXG2015], the approach is computationally demanding since each node is required to solve an SDP per iteration, and also the communication overhead is significant. A distributed algorithm that is cheaper and requires a smaller communication overhead was later proposed in [@SXG2015]. One shortcoming of the latter method is that it requires the sensors to be in the convex hull of the anchors, which is difficult to guarantee in practice.
The present work was motivated by a class of centralized algorithms that use divide-and-conquer approaches to improve scalability [@CLS2012; @CKS2015b; @LT2009; @ZLGG2010]. The general mechanism is to partition $\cG$ into overlapping subgraphs, localize each subgraph using the induced distances, and finally register the subgraphs. The idea is to construct subgraphs that are denser than the original graph. Moreover, the smaller graphs can be efficiently localized. The algorithms essentially differ on how each subproblem is solved. For example, a Cuthill-McKee-type permutation is used in [@LT2009] to partition $\cG$. In [@CLS2012; @ZLGG2010], $\cG$ is partitioned using neighborhood subgraphs. To improve the localization, rigid subgraphs [@CW2009] are extracted from each neighborhood subgraph in [@CLS2012]. Recently, recursive spectral clustering was used in [@CKS2015b]. We note that the subgraphs obtained using the graph partitioning in [@CKS2015b; @LT2009] are not guaranteed to be rigid [@Laman1970; @GHT2010], and hence can result in poor localization. In fact, a few poorly localized subgraphs can adversely affect the overall registration. The algorithms in [@LT2009; @ZLGG2010] register the subgraphs in a sequential fashion, and inevitably suffer from error propagation. Recently, a least-square method was proposed in [@CKS2015a] that can register the subgraphs in a globally-consistent manner. In particular, it was demonstrated in [@CKS2015a] that global registration can successfully operate in adversarial situations where sequential methods fail. In this regard, we note that a *lateration* condition was introduced in [@CKS2015a] that can guarantee exact recovery in the noise-free setting. However, there is no known efficient algorithm for testing lateration.
Contributions
-------------
We propose a divide-and-conquer algorithm building on the ideas in [@CKS2015b; @CKS2015a; @SC2016]. In particular, we address the following issues that emerged from this line of work: testing and ensuring that each subgraph is rigid, formulating a testable condition for recovering the sensor coordinates, and developing a scalable algorithm for registering the localized subgraphs. In this context, the contributions are as follows:
\(i) To bypass the rigidity issue associated with the localization of each subgraph, we propose to use cliques. Cliques are trivially rigid and can be efficiently localized using multidimensional scaling [@BG2003]. However, given that finding cliques in a large graph is challenging, we first partition $\cG$ into neighborhood graphs [@CLS2012; @ZLGG2010]. We then extract a clique from each neighborhood graph using the algorithm from [@BG2015]. Finally, we augment the vertices of a clique to expand it into a maximal clique. We experimentally demonstrate that the complete process is fast for both random and structured geometric graphs.
\(ii) We study the problem of registering a system of localized cliques. In particular, we establish a *rigidity* condition that is necessary for recovering the original sensor coordinates. The proposed condition can be efficiently tested simply by computing the maximum flow between the vertices of an appropriate graph. We present supporting examples to conjecture that the proposed rigidity condition is also sufficient for exact recovery. Moreover, we demonstrate using numerical examples that the registration performance can be improved in the noisy setting by enforcing the rigidity condition.
We note that a registration-based approach for anchorless SNL was earlier proposed in [@SC2016] that uses cliques and cMDS. In the present work, we focus on anchored SNL (though the method can also be used for anchorless SNL). Moreover, we consider a different clique exploration process. Importantly, we investigate the rigidity problem associated with registration which was not discussed in [@SC2016].
Organization
------------
The rest of the paper is organized as follows. In Section \[RP\], we propose a rigidity criteria for the registration problem, and explain how this can be tested efficiently. The proposed graph partitioning is described in Section \[GP\] keeping the registration problem in mind. Classical multidimensional scaling is reviewed in Section \[cMDS\], which is used to localize the cliques. The registration algorithm is described in Section \[ADMM\]. Experimental results and comparisons are provided in Section \[exp\].
The Rigidity Problem {#RP}
====================
{width="0.7\linewidth"}
We first study the fundamental problem of *rigidity* whose resolution will be useful during the graph partitioning phase in Section \[GP\]. This problem is also relevant for other divide-and-conquer approaches [@CLS2012; @CKS2015b; @ZLGG2010; @CKS2015a], where a system of point sets are required to be registered. More precisely, consider the sensors $\cX_s$ and the anchors $\cX_a$ in , and subsets $\cC_1,\ldots,\cC_M \subset \cN$. Following [@CLS2012; @CKS2015b], we will refer to each $\cC_i$ as a *patch*. Moreover, we create an additional patch $\cC_{M+1}$ consisting solely of the anchors $\cA$. Assume that the points in each patch have been derived from the respective points in $\cX_s \cup \cX_a$ via a rigid transform. Let $$\label{LocGlob1}
\bar{\x}_k =\cR_i(\x_{k,i}) = \bO_i \x_{k,i} + \t_i \qquad (k \in \cC_i \backslash \cA),$$ and $$\label{LocGlob2}
\bar{\a}_l =\cR_i(\bar{\a}_l) \qquad (l \in \cC_i \cap \cA),$$ where $\x_{k,i}$ is the coordinate of the $k$-th point in the $i$-th patch, and $\cR_i=(\bO_i,\t_i)$ is the rigid transform associated with the $i$-th patch, where the orthogonal matrix $\bO_i$ represents rotation (or reflection) and $\t_i$ is the translation component. We will refer to the $(\x_{k,i})$’s as the *patch coordinates*. The patches and the patch coordinates together form a *configuration*. The *registration* problem is one of determining the unknown $\cX_s$ from the given configuration.
\[prob1\] Find $\x_1, \dots , \x_N$ and rigid transforms $\cQ_1, \dots , \cQ_M$ such that, for $1 \leq i \leq M$, $$\x_k = \cQ_i (\x_{k,i}) \quad \text{and} \quad \bar{\a}_l = \cQ_i (\bar{\a}_l),$$ where $k \in \cC_i \setminus \cA$ and $l \in \cC_i \cap \cA$.
In the noiseless setting, the solution (points and transforms) sought above exists trivially, namely, the ground truth $\x_k = \bar{\x}_k$ and $\cQ_i=\cR_i$. The rigidity problem is to determine whether the solution is unique (upto a global rigid transformation).
\[prob2\] Determine whether Problem \[prob1\] have a unique solution up to a rigid transform. That is, if $\x_1, \dots , \x_N$ is a solution of Problem \[prob1\], then is it necessary that for some rigid transform $\cR$, $\x_k = \cR ( \bar{\x}_k)$ and $\bar{\a}_l = \cR (\bar{\a}_l)$, where $k \in \cS$ and $l \in \cA$?
A set of points in $\R^d$ is said to be *non-degenerate* if their affine span is $\R^d$. Clearly, the cardinality of such points must be $d+1$ or more. For example, three points are non-degenerate in two-dimensions if and only if they are not collinear. We note that the transforms $\cQ_1, \dots , \cQ_M$ in Problem \[prob1\] are latent variables and do not appear in the Problem \[prob2\].
\[def\_rigidity\] A configuration is said to be *rigid* in $\mathbb{R}^d$ (or simply *rigid*) if the solution is unique in the sense of Problem \[prob2\]; otherwise, the configuration is said to be *flexible*.
To provide geometric insights to the rigidity problem, we consider simple instances of rigid and flexible configurations in Figures \[PC1\] and \[PC3\]. In particular, we wish to highlight the importance of overlaps among patches in determining rigidity. In Figure \[PC1\], patches $A$ and $B$ share two sensors. The patches can be reflected along the line joining sensors $1$ and $2$, but due to the presence of anchors in $A$ and $B$, reflection is ruled out. The configuration is thus rigid in the sense of Definition \[def\_rigidity\]. On the other hand, the configuration in Figure \[PC3\] is flexible since patches $C$ and $D$ can be reflected along the red dotted line.
The above concepts and definitions were motivated by the rigidity aspects of the SNL problem [@SY2007], and more generally, the distance-geometry problem [@Laman1970; @GHT2010]. Here, the problem is to determine if the available distance measurements uniquely define the sensor locations (modulo a rigid transform which leaves the distances unchanged). A fundamental result in this regard is that, if the original sensor locations are *generic* [@GHT2010], then the uniqueness problem can be completely resolved using just the measurement graph [@CW2009; @Laman1970; @GHT2010]. Our present objective is to come up with similar results for Problem \[prob2\]. At this point, we wish to emphasize that rigidity theory is solely concerned with exact measurements [@GHT2010; @ASNF2010]. The point is that the combinatorial structure of the problem should, in principle, be able to guarantee exact recovery of the ground truth when the measurements are perfect. The design of an algorithm that can provably recover the ground truth is however a completely different topic. We will present some representative examples in Section \[exp\], which suggest that rigidity can also help improve the algorithmic performance in the noisy setting. We will assume the following in the rest of the discussion.
\[assumption\] There are at least $d+1$ non-degenerate points in each patch.
Under the above assumption, we provide a necessary condition for rigidity. Before doing so, we note that a *lateration* criteria was earlier proposed in [@CKS2015a] that can guarantee rigidity. However, it is not known if there exists an efficient test for lateration. Moreover, a path configuration can be rigid without being laterated, that is, lateration is not necessary for rigidity. This fact is demonstrated with an example in Figure \[PC2\]. This motivated us to look for a criteria that is both necessary and sufficient for rigidity. We propose a necessary condition for rigidity that can be tested efficiently. We present some examples where the condition is also sufficient, and conjecture that this is true in general.
Before stating the result, we set up a special bipartite graph that captures the overlap-pattern among patches. Recall that a graph is said to be bipartite if the vertex set $\cV$ can be divided into disjoint subsets $\cV_1$ and $\cV_2$ such that there are no edges between the vertices of a given $\cV_i$.
\[defT\] We define the bipartite correspondence graph to be $\Gamma=(\cV_1,\cV_2,\cE)$, where $\cV_1( \Gamma)$ are the nodes, $\cV_2( \Gamma)$ are the patches, and $(k,i) \in \cE( \Gamma)$ if and only if $k \in \cC_i$.
The correspondence graph for the configuration in Figure \[PC1\] is shown on the right. Finally, we introduce a special notion of connectivity. Recall that a *path* is an ordered sequence of vertices $v_1,v_2,\ldots,v_n$ such that $(v_t,v_{t+1})$ is an edge for $1 \leq t \leq n-1$. The path is said to be between vertices $\alpha$ and $\beta$ (or the path connects $\alpha$ and $\beta$) if $v_1=\alpha$ and $v_n=\beta$. Two paths in a graph are said to be *vertex-disjoint* over a set $\Theta$ if they do not share a common vertex from $\Theta$. A set of paths are said to be $\Theta$-*disjoint* if any two paths are vertex-disjoint over $\Theta$.
\[defConnected\] The correspondence graph $\Gamma$ is said to be quasi $k$-connected if any two vertices in $\cV_2 \left( \Gamma \right)$ have $k$ or more $\cV_1(\Gamma)$-disjoint paths between them. Moreover, there exist two vertices in $\cV_2 \left( \Gamma \right)$ that are connected by exactly $k$ paths that are $\cV_1(\Gamma)$-disjoint.
For latter reference, we record the following characterization of quasi $k$-connectivity. The equivalence can be derived by adapting the proof of Menger’s theorem [@Diestel2005 Theorem 3.3.1].
\[Menger\] The following are equivalent.\
(a) The correspondence graph $\Gamma$ is quasi $k$-connected.\
(b) $\cE(\Gamma)$ can be divided into two disjoint subsets $E_1$ and $E_2$ such that the edges from $E_1$ and that from $E_2$ are\
(i) incident on at least $k$ common vertices from $\cV_1(\Gamma)$, and\
(ii) not incident on any common vertex from $\cV_2(\Gamma)$.
In Figure \[PC3\], notice that there are $3$ paths between any pair of vertices in $\cV_2(\Gamma)$, but at most two paths are $\cV_1(\Gamma)$-disjoint. In this case, the configuration is not rigid. In fact, we have the following result (cf. supplementary material for the proof).
\[THEOREM\] Under Assumption \[assumption\], if a configuration is rigid in $\mathbb{R}^d$, then its correspondence graph must be quasi $(d\!+\!1)$-connected.
Moreover, we see from the example in Figure \[PC3\] that, if $\Gamma$ fails to be quasi $( d + 1)$-connected, then the configuration is not rigid. We are yet to find a counter-example where the configuration is flexible yet $\Gamma$ is quasi $( d + 1)$-connected. Based on empirical evidences, we make the following conjecture.
Suppose Assumption \[assumption\] holds, and that any $k \geq d +1$ points in $\cX_s \cup \cX_a$ are non-degenerate. Then a configuration is rigid in $\mathbb{R}^d$ if and only if its correspondence graph is quasi $\left( d \! + \! 1 \right)$-connected.
The second assumption appears somewhat stringent at first sight. The relevance of this assumption is somewhat clear from the example in Figure \[PC1\]. Namely, if sensors $1,2$ and anchor $5$ are concurrent, then one can reflect patch $B$ about the line joining these points. We note that the use of some form of non-degeneracy assumption is standard in rigidity theory [@GHT2010].
Based on Definitions \[defT\] and \[defConnected\], it is not difficult to establish a relation between quasi connectivity and the maximum flow between the vertices of $\cV_2(\Gamma)$ [@CLRS2001]. In this context, recall that a vertex is said to have *capacity* $\kappa$ if the incoming and outgoing flows for the vertex are at most $\kappa$ [@CLRS2001].
\[connect\_cut\] Assume that each vertex in $\cV_1(\Gamma)$ is assigned unit capacity while computing the flow. Then the maximum flow between the vertices of $\cV_2(\Gamma)$ is at least $k$ if and only if $\Gamma$ is quasi $k$-connected.
The key point is that one can efficiently check if, under the assumption that the vertices in $\cV_1(\Gamma)$ have unit capacity, the maximum flow between the vertices of $\cV_2(\Gamma)$ is at least $k$. This can be done using the Ford-Fulkerson algorithm [@CLRS2001]. Note that we do not need to check the maximum flow for all pairs of vertices in $\cV_2(\Gamma)$. We can simply fix a vertex and check the maximum flow between this vertex and the remaining vertices.
Partitioning {#GP}
============
We now describe a heuristic for partitioning $\cG$ into overlapping patches such that the corresponding $\Gamma$ is quasi $(d \! + \! 1)$-connected. In this relation, we note that divide-and-conquer approaches have been proposed in [@CKS2015b; @CLS2012; @LT2009; @ZLGG2010], where $\cG$ is partitioned into overlapping patches. The difficulty with the approaches in [@CKS2015b; @LT2009] is that the patches and the resulting patch configuration are not guaranteed to be rigid.
We propose to bypass the former rigidity issue by using *cliques*, that is, complete subgraphs of the measurement graph. In other words, each patch is a clique in our approach. This is precisely why we choose to denote the patches as $\cC_i$ in Section \[RP\]. Cliques are trivially rigid [@Laman1970; @GHT2010], and can be localized using multidimensional scaling [@BG2003]. In particular, for each $i \in \cV(\cG)$, we extract a maximal clique containing $i$. The system of cliques forms a clique-cover. We recall that a clique is said to be maximal if it is not contained in a strictly larger clique. By targeting maximal cliques, we wish to minimize the number of cliques that are required to be registered in the final phase.
Let $\cG_i$ denote the *neighborhood* graph of some $i \in \cV(\cG)$. Namely, $\cG_i$ is the subgraph of $\cG$ induced by $i$ and its one-hop neighbors. For each vertex $i$, we want to find a maximal clique $\cC \subset \cV(\cG)$ containing $i$. In this regard, we note that it suffices to restrict the search to $\cG_i$.
\[clq1\] Let $\cC$ be a maximal clique. Then $i \in \cC$ if and only if $\cC \subset \cV(\cG_i)$.
Let $\cC$ be a maximal clique containing $i$. Then, for any $j \in \cC$, we have $(i,j) \in \cE(\cG)$. Hence, $j \in \cV(\cG_i)$. In the other direction, suppose that $\cC \subset \cV(\cG_i)$, but $i$ does not belong to $\cC$. Then, by appending vertex $i$ to $\cC$, we obtain a clique that strictly contains $\cC$, which contradicts the maximality of $\cC$.
Unfortunately, finding cliques is generally intractable [@BG2015]. Based on Proposition \[clq1\], we first extract a clique from a given subgraph $\cG_i$ using the algorithm in [@BG2015]. In this work, the combinatorial problem of finding maximal cliques is relaxed into a continuous optimization problem. The stationary points of the latter are computed using projected gradient descent. The key result of the paper is that one can provably locate a clique by running the gradient-descent for sufficient number of iterations and rounding the output [@BG2015 Theorem 7, Corollary 3]. The authors empirically noticed that the clique retuned by the algorithm is often maximal. Since the subgraphs $\cG_i$ are typically small for practical values of $r$, we found the algorithm in [@BG2015] to be quite efficient for our purpose.
The clique located within a given $\cG_i$ using the above clique-finding algorithm may not contain vertex $i$. In this case, we can in fact obtain a larger clique simply by appending vertex $i$ to the found clique. Generally, since the subgraphs are small, one can efficiently test for maximality, and keep appending nodes until the maximal clique is found. In practice, we noticed that the cliques returned by the algorithm in [@BG2015] are often maximal or near-maximal. As a result, the combined process of appending vertices and testing for maximality is quite fast. We note that an extracted clique can belong to two or more subgraphs. We discard the redundant cliques during the clique-finding process. At the end, suppose that we have located, say, $m$ maximal cliques, $\cC_1, \dots ,$ $\cC_m$, that cover the vertices of $\cG$. We next test the rigidity of the patch configuration. To do so, we append to the existing cliques an additional clique $\cC_{m+1}$ containing the anchors, and test if $\Gamma$ is quasi $( d+1)$-connected. If so, we set $M=m$, and proceed to the localization phase in Section \[cMDS\].
If $\Gamma$ fails the test, we proceed as follows. Following Proposition \[connect\_cut\], we know that there exist $s,t \in \cV_2(\Gamma)$ for which the maximum $s\text{-}t$ flow is $k \leq d$, where recall that the vertices in $\cV_1(\Gamma)$ are assigned unit capacity. In fact, the Ford-Fulkerson algorithm returns the value $k$ and the corresponding minimum cut $C=(S,T)$, where $s \in S$ and $t \in T$ [@CLRS2001]. Let $I_S$ and $I_T$ be the indices of the cliques in $S$ and $T$, that is, $I_S = S \cap \cV_2(\Gamma)$ and $I_T = T \cap \cV_2(\Gamma)$.
\[cut1\] Let $A = \cup_{\alpha \in I_S} \cC_{\alpha}$ and $B = \cup_{\beta \in I_T} \cC_{\beta}$. Then $\lvert A \cap B \rvert = k$.
From the max-flow-min-cut theorem [@CLRS2001], $\lvert A \cap B \rvert \leq k$. If $\lvert A \cap B \rvert$ is less than $k$, then there would be a vertex in $\cV_1(\Gamma)$ with more than one edge in the cut set [@CLRS2001]. However, since each vertex in $\cV_1(\Gamma)$ has unit capacity, this is not possible.
We wish to increase the maximum flow by extracting a clique and appending it to the existing configuration. In particular, the appended clique must contain some $i \in A\setminus B$ and $j \in B \setminus A$. Our task is to find a maximal clique containing $i$ and $j$. Define $\cG_{ij}$ be the common subgraph of $\cG_i$ and $\cG_j$, that is, $\cG_{ij}$ is the subgraph of $\cG$ induced by the vertices $\cV(\cG_i) \cap \cV(\cG_j)$. Similar to Proposition \[clq1\], we note the following.
Let $\cC_0$ be a maximal clique. Then $i,j \in \cC_0$ if and only if $\cC_0 \subset \cV(\cG_{ij})$.
After appending $\cC_0$ to the existing cliques, we obtain a new configuration. Accordingly, we update $\cV_2(\Gamma)$ and $\cE(\Gamma)$, and increase $m$ by $1$. In particular, we reorder the indices of the cliques so that $\cC_{m+1}$ continues to be the anchor clique. For the updated $\Gamma$, we recompute $I_S$ or $I_T$, and note that $m$ belongs to either of these. As a result, we conclude the following.
\[NoNewMinCut\] For the updated $\Gamma$, $\lvert A \cap B \rvert > k$.
Moreover, if the maximum flow is uniquely achieved for the vertices identified by the Ford-Fulkerson algorithm, then appending $\cC_0$ actually increases the maximum flow. We continue this process, in which we alternately augment the configuration and test rigidity, until we attain the maximum flow of $d+1$. It is possible that the process is prematurely terminated if we are unable to find a clique of size at least $d +1$. This typically happens in adversarial settings where $r$ is small, making $\cG$ extremely sparse. At the end of the process, assume that we have $M+1$ cliques, $\cC_1\ldots,\cC_{M+1}$, where $\cC_{M+1}$ is the anchor clique. For the simulations in Section \[exp\], we found that $M$ is typically about $30\%$ of the total number of nodes.
Localization {#cMDS}
============
Having partitioned the measurement graph into a system of overlapping cliques, we now localize them in parallel. Since all the inter-node distances are available in a clique, we can efficiently localize a clique using multidimensional scaling [@BG2003]. However, there are two types of cliques, namely, cliques without anchors and those with anchors. For the former, we can directly use classical multidimensional scaling (cMDS). In particular, suppose that the clique has $n$ sensors, and the distances are $\{d_{ij} : 1 \leq i,j \leq n\}$. Consider the $n \times n$ matrices $\bD$ and $\bB$ given by $$\bD_{ij} = \begin{cases}
d_{ij}^2 & \text{ if } \ i \neq j, \\
0 & \text{ otherwise,} \\
\end{cases}$$ and $$\bB = -\frac{1}{2} \Big(\bI_n-\frac{1}{n}\textbf{uu}^{\top}\Big) \mathbf{D} \Big(\bI_n-\frac{1}{n}\textbf{uu}^{\top} \Big),$$ where $\textbf{u}$ is the all-ones vector of length $n$, and $\bI_n$ is the $n \times n$ identity matrix. Since $\bB$ is symmetric, it has real eigenvalues and a full set of orthonormal eigenvectors. Let $\lambda_1 \geq \dots \geq \lambda_n$ be the sorted eigenvalues, and $\q_1,\ldots,\q_n$ be the corresponding eigenvectors.
Suppose that the available distances are exact, that is, $d_{ij}=\lVert \x_i - \x_j \rVert$ for some $\x_1,\ldots,\x_n$. Then $\bB \succeq \mathbf{0}$ and $\mathrm{rank}(\bB) \leq d$. The sensor locations can be taken to be $$\label{embedding}
\x_i = \big( \sqrt{\lambda_1} \q_1(i), \dots, \sqrt{\lambda_d} \q_d(i) \big)^{\top} \quad (1\leq i \leq n).$$
If the distances are noisy, $\bB$ can have negative eigenvalues and its rank can be greater than $d$. In this case, it is customary to use the positive eigenvalues and the corresponding eigenvectors in . The resulting inter-sensor distances are an approximation to the available distances, where the approximation error is determined by the rank of $\bB$, and the number and magnitude of the negative eigenvalues [@BG2003]. Perturbation analysis of cMDS is a well-researched topic and the method is known to be stable under deformations [@Sibson1979].
If a clique has one or more anchors, we have to take into consideration the stipulated anchor locations. More precisely, we have a constrained problem, where we need to reconstruct the sensor locations keeping the anchor variables fixed. In such scenarios, we can use cMDS followed by an alignment. Assume that, of the $n$ nodes, the first $k$ are anchors and the remaining are sensors. We first localize the $n$ nodes using cMDS, regardless of the anchor locations. Then we align the reconstructed anchors with the original anchors via a rigid transformation. In particular, if there is just one anchor, and if the reconstructed and original locations are $\x$ and $\bar{\a}$, then we translate the reconstructed nodes by $\bar{\a}-\x$. If there are more than one anchor, then we perform an optimal alignment using least-square fitting: $$\label{Arun}
\min_{\bO \in \O\left(d\right), \t \in \R^d} \ \sum_{i=1}^{k} \lVert \bO \x_{i}+\t - \bar{\a}_i \rVert^2,$$ where $\x_i$ and $\bar{\a}_i$ are the reconstructed and the original anchor locations. As is well-known, the minimum of has a simple closed-form solution [@AHB1987]. In particular, the optimal transform is given by $\bO^{\star}=\bV \bU^{\top}$ and $\t^{\star} = \bm{\mu} -\bO^{\star}\bm{\nu}$, where $$\bm{\mu}= \frac{1}{k} \sum_{i=1}^{k} \x_i \quad \text{and} \quad\bm{\nu} = \frac{1}{k} \sum_{i=1}^k \bar{\a}_i,$$ and $\bC=\bU \Sigma \bV^{\top}$ is the SVD of $\bC = \sum_{i=1}^{k} (\x_i - \bm{\mu})(\bar{\a}_i -\bm{\nu})^{\top}$. We apply the transform $\x \mapsto \bO^{\star}\x+\t^{\star}$ on the reconstructed sensors, and place the anchors in their stipulated locations. Finally, we refine the localization by minimizing the stress function using a gradient-based method [@BLTYW2006]. The refinement is particularly effective when the distances are noisy.
Registration {#ADMM}
============
As a final step, we need to register the localized cliques in a global coordinate system. While the least-square formulation of the registration problem has a closed-form solution for two cliques [@AHB1987], the problem is generally intractable when there are three or more cliques [@CKS2015a]. Recently, it was demonstrated in [@CKS2015a] that the least-square optimization can be approximated using a semidefinite program (SDP). Later, a scalable ADMM-based solver for this SDP was proposed in [@SC2016]. For completeness, we review the SDP relaxation and the ADMM solver.
In the absence of noise, the relation between the local and global coordinates are given by and . Since these are not expected to hold exactly in the presence of noise, the authors in [@CKS2015b] proposed to minimize the least-square objective $$\label{LS}
\sum_{i=1}^{M} \Big(\sum_{k \in \cC_i \setminus \cA} \alpha_{k,i}^2 + \lambda \sum_{l \in \cC_i \cap \cA} \beta_{l,i}^2 \Big),$$ where $$\alpha_{k,i}= \lVert \x_k - \bO_i \x_{k,i} - \t_i\rVert \ \ \text{and} \ \ \beta_{l,i}=\lVert \bO_{M+1}\bar{\a}_l - \bO_i \bar{\a}_l - \t_i \rVert$$ are the registration errors for sensors and anchors. The scale $\lambda > 0$ is used to combine the gross errors. The variables are the sensor coordinates $\x_k$ and the rigid transformations $(\bO_i, \t_i)$. The dummy variable $\bO_{M+1}$ is introduced to make the objective homogenous [@CKS2015b]. In terms of the matrix variables $$\bZ = \left[ \x_1 \cdots \x_{N} \ \t_1 \dots \t_{M} \right] \quad \text{and} \quad \bO = \left[ \bO_1 \cdots \bO_{M+1} \right],$$ we can express as $$\label{LS2}
\text{Trace} \left( \begin{bmatrix} \bZ & \bO \end{bmatrix} \begin{bmatrix} \bJ & -\bB^{\top} \\ -\bB & \bD \end{bmatrix} \begin{bmatrix} \bZ^{\top} \\ \bO^{\top} \end{bmatrix} \right),$$ where $$\begin{aligned}
& \bJ = \sum_{i = 1}^{M} \Big[ \sum_{k \in \cC_i \setminus \cA} \!\! \e_{k,i} \e_{k,i}^{\top} + \lambda \sum_{l \in \cC_i \cap \cA} \!\! \del_{N + i}^{N + M} {\del_{N + i}^{N + M}}^{\top} \Big], \\
& \bB = \sum_{i = 1}^{M+1} \Big[ \sum_{k \in \cC_i \setminus \cA} \!\! \left( \del_{i}^{M+1} \otimes \bI_d \right)\x_{k,i} \e_{k,i}^{\top} \\
& \quad + \lambda \sum_{l \in \cC_i \cap \cA} \!\! \left( \f_{i} \otimes \bI_d \right)\bar{\a}_l {\del_{N+i}^{N+M}}^{\top} \Big],\\
& \bD = \sum_{i = 1}^{M+1} \Big[ \sum_{k \in \cC_i \setminus \cA} \!\!\left( \del_{i}^{M+1} \otimes \bI_d \right)\x_{k,i} \x_{k,i}^{\top} \left( \del_{i}^{M+1} \otimes \bI_d \right)^{\top}\\
& \quad + \lambda \sum_{l \in \cC_i \cap \cA} \!\!\left( \f_{i} \otimes \bI_d \right)\bar{\a}_l \bar{\a}_l^{\top} \left( \f_{i} \otimes \bI_d \right)^{\top} \Big].
\end{aligned}$$ Here, $\otimes$ is the Kronecker product, $\del_i^{L}$ is the all-zero vector of length $L$ with unity at the $i$-th position, $$\e_{k,i} = \del_{k}^{N+M}-\del_{N+i}^{N+M} \quad \text{and} \quad \f_i =\del_{M+1}^{M+1}-\del_{i}^{M+1}.$$ The minimum of over $\bZ$ is attained when $\bZ^{\star} = \bO \bB \bJ^{-1}$. On substituting $\bZ^{\star}$ in , we get the following problem: $$\label{GRET}
\begin{aligned}
& \underset{\bG \succeq 0}{\text{min}} \quad \text{Trace} \left( \bC \bG \right) \\
& \text{s.t.} \quad \left[\bG\right]_{ii} =\bI_d \ (i = 1, \ldots, M\!+\!1), \ \text{rank}\left( \bG \right)=d.
\end{aligned}$$ where $\bC = \bD - \bB \bJ^{-1} \bB^{\top}, \bG = \bO^{\top} \bO,$ and the $d \times d$ matrix $\left[\bG\right]_{ii}$ denotes the $i$-th diagonal block of $\bG$. It was observed in [@CKS2015a] that the objective and the constraints in are convex, except for the rank condition. By dropping the rank constraint, the authors arrived at the following SDP relaxation: $$\label{GRET-SDP}
\min_{\bG \succeq 0} \ \mathrm{Trace}\left(\bC \bG \right) \ \text{s.t.} \ \left[\bG \right]_{ii}=\bI_d \ \ (i = 1, \ldots, M\!+\!1).$$ The global minimum of can be computed for small or even medium-sized problems using an interior-point solver [@Sturm1999]. However, such solvers are both memory and computation intensive. In particular, the cost of approximating the global minimum of within a given accuracy is $\cO(\left(M d\right)^{4.5})$ [@CKS2015a]. Since $M$ is of the order $\cO(|\cN|)$ in our case, the size of the SDP variables can be few hundreds or thousands. Interior-point solvers run out of memory for such large problems. To achieve scalability, an iterative solver based on ADMM was proposed in [@SC2016]. The ADMM updates are summarized in Algorithm \[solver\], where $\S_+$ is the set of symmetric positive semidefinite matrices of size $L=(M+1)d$, $\Omega$ is the set of symmetric matrices of size $L$ whose $d \times d$ diagonal blocks are identity, and $\Pi_\cS (\bA)$ denotes the projection of $\bA$ onto a convex set $\cS$. Notice that the only non-trivial computation is determining $\Pi_{\S_+} (\bA)$. This is obtained by computing the eigendecomposition of $\bA$ and setting the negative eigenvalues to zero. The projection $\Pi_\Omega (\bA)$ amounts to setting the $d \times d$ diagonal blocks of $\bA$ to $\bI_d$, while keeping the non-diagonal blocks unchanged. We initialize $\bH$ using the spectral algorithm in [@CKS2015a]. The Lagrange multiplier $\Lam$ is initially set to be the zero matrix. We use a condition from [@BPCPE2011] to terminate the iterations.
\[solver\] Initialize $\bH$ and $\Lam.$
We can establish the convergence of Algorithm \[solver\] using the analysis in [@BPCPE2011]. In particular, we have the following result.
\[thmADMM\] Starting with $\bH^0$ and $\Lam^0$, let $(\bG^{k},\bH^{k},\Lam^{k})_{k \geq 1}$ be the variables generated by Algorithm \[solver\]. Then
- Objective convergence: If $F^*$ is the optimum of , then $$\label{conv1}
\lim_{k \rightarrow \infty} \ \mathrm{Trace}(\bC \bG^k) = F^*.$$
- Asymptotic feasibility: For $1 \leq i \leq M$, $$\label{conv2}
\lim_{k \rightarrow \infty} \ [\bG^k]_{ii} = \bI_d.$$ That is, $(\bG^k)$ approaches the feasible set in .
In fact, since updates \[Gupdate\] and \[Hupdate\] in Algorithm \[solver\] are convex projections, we can establish the Theorem \[thmADMM\] using elementary results from convex analysis. This and other technical results will be reported separately [@JSC2016].
Having approximated the optimal $\bG$ using Algorithm \[solver\], we compute $\bO =[ \bO_1 \cdots \bO_{M+1}]$ using the rounding in [@CKS2015a]. The first $N$ columns of $\bZ = \bO \bB \bJ^{-1}$ are taken to be the estimated sensor locations. As a final step, we refine the locations using stress minimization [@BLTYW2006] and denote the result as $\widehat{\x}_1, \dots , \widehat{\x}_N$.
Numerical Experiments {#exp}
=====================
In this section, we conduct numerical simulations to demonstrate the performance of the proposed method. In particular, we illustrate the impact of rigidity on the performance of the registration algorithm, and study the timing of different phases of the proposed method and the scaling of the localization error with the noise level. We then compare with some of the state-of-the-art algorithms [@SL2014; @BLTYW2006; @WZYB2008; @SXG2015] in terms of accuracy, run-time, and scalability. The comparisons are performed on planar networks, namely, the random geometric graph (RGG) [@BLTYW2006; @WZYB2008; @CKS2015b], and the structured PACM logo [@CLS2012]. The diameter (maximum distance between any two points) of the logo is $16.92$. We consider the following noise model that was used in [@BLTYW2006; @WZYB2008; @CKS2015b]: $$d_{ij} =\lvert 1+\epsilon_{ij} \rvert \cdot \lVert \bar{\x}_i-\bar{\x}_j \rVert \qquad (i,j \in \cS),$$ and $$d_{ik} = \lvert 1+\epsilon_{ik}\rvert \cdot \lVert \bar{\x}_i - \bar{\a}_k \rVert \qquad (i \in \cS, k \in \cA),$$ where $\epsilon_{ij}$ and $\epsilon_{ik}$ are i.i.d. Gaussians with mean zero and standard deviation $\eta$. As mentioned earlier, we enforce symmetry by replacing $d_{ij}$ and $d_{ji}$ with their average [@SHCJ2010]. For a quantitative comparison of the localization accuracies, we use the average normalized error (ANE) [@CLS2012] given by $$\text{ANE}=\left\{\frac{\sum_{i=1}^{N} \lVert \widehat{\x}_i - \bar{\x}_i \rVert^2}{\sum_{i=1}^{N} \lVert \bar{\x}_i - \bar{\x}_c \rVert^2} \right\}^{1/2},$$ where $\bar{\x}_c$ is the centroid of the original sensor locations. Of course, we assume that the reconstruction has been optimally aligned with the ground truth before computing the ANE [@AHB1987]. We also present visual comparison of the localizations obtained using different methods. For all experiments, we have used $\lambda=1$ in and $\rho=0.01$ for the augmented Lagrangian.
$N$ $K$ $r$ $\eta$ $t_1$ $t_2 $ $t_3$
----- ----- ----- -------- ------- -------- -------
0 0.79 0.04 0.02
0.1 0.73 0.06 0.92
0 3.9 0.07 0.21
0.1 3.9 0.1 12.8
0 7.2 0.1 0.8
0.1 7.5 0.2 70.9
0 9.2 0.1 1.7
0.1 8.1 0.2 1.5
: Run-times (in seconds) of different phases of the algorithm – partitioning $(t_1)$, localization $(t_2)$ and registration $(t_3)$.[]{data-label="SelfCompare"}
Performance Analysis
--------------------
-- -- -- -------- ----------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- ------------------
$\eta$ Proposed `ESDP` [@WZYB2008] `SNLSDP` [@BLTYW2006] Proposed `ESDP` [@WZYB2008] `SNLDR` [@SXG2015] `SNLSDP` [@BLTYW2006] `E-ML` [@SL2014]
$0$ $0.1$sec $0.2$sec $0.3$sec $3.9\mbox{e-}16$ $7.9\mbox{e-}8$ $1.3\mbox{e-}4$ $1.3\mbox{e-}9$ $1.4\mbox{e-}3$
$0.1$ $0.2$sec $0.3$sec $0.4$sec $9.6\mbox{e-}2$ $9.5\mbox{e-}2$ $2\mbox{e-}1$ $9.6\mbox{e-}2$ $1.3\mbox{e-}1$
$0$ $0.3$sec $0.6$sec $0.5$sec $1.3\mbox{e-}15$ $1.1\mbox{e-}8$ $1\mbox{e-}4$ $1.1\mbox{e-}8$ $2.5\mbox{e-}3$
$0.1$ $0.6$sec $0.6$sec $1$sec $6.4\mbox{e-}2$ $8.8\mbox{e-}2$ $1.6\mbox{e-}1$ $6.4\mbox{e-}2$ $9.2\mbox{e-}2$
$0$ $0.5$sec $3.3$sec $0.6$sec $2.3\mbox{e-}15$ $4.1\mbox{e-}8$ $1\mbox{e-}2$ $1.2\mbox{e-}9$ $2.3\mbox{e-}3$
$0.1$ $1.2$sec $1.2$sec $0.8$sec $4\mbox{e-}2$ $7.3\mbox{e-}2$ $1.5\mbox{e-}1$ $4\mbox{e-}2$ $9.6\mbox{e-}2$
$0$ $2$sec $30$sec $19$sec $4\mbox{e-}14$ $3.1\mbox{e-}7$ $1.4\mbox{e-}2$ $2.5\mbox{e-}9$ $-$
$0.1$ $4$sec $7$sec $4$sec $1.7\mbox{e-}2$ $3.1\mbox{e-}2$ $1.1\mbox{e-}1$ $1.7\mbox{e-}2$ $-$
$0$ $5$sec $1.5$min $5.8$min $4.7\mbox{e-}14$ $1.5\mbox{e-}7$ $1.6\mbox{e-}2$ $8.8\mbox{e-}7$ $-$
$0.1$ $1$min $25$sec $7.6$min $1\mbox{e-}2$ $2\mbox{e-}2$ $7.2\mbox{e-}2$ $1\mbox{e-}2$ $-$
$0$ $10$sec $2.6$min $26$min $1.3\mbox{e-}13$ $9.9\mbox{e-}7$ $5.9\mbox{e-}3$ $3.6\mbox{e-}2$ $-$
$0.1$ $5.4$min $1.2$min $26$min $7\mbox{e-}3$ $1.3\mbox{e-}2$ $4.6\mbox{e-}2$ $4.1\mbox{e-}2$ $-$
$0$ $3.2$min $32.8$min $\star$ $5.6\mbox{e-}13$ $2.1\mbox{e-}7$ $-$ $-$ $-$
$0.05$ $4.2$min $32.8$min $\star$ $1.7\mbox{e-}3$ $3.2\mbox{e-}3$ $-$ $-$ $-$
$0$ $8.5$min $1$hr $\star$ $7.6\mbox{e-}13$ $2.3\mbox{e-}3$ $-$ $-$ $-$
$0.05$ $6.8$min $42.2$min $\star$ $1.3\mbox{e-}3$ $3.4\mbox{e-}3$ $-$ $-$ $-$
$0$ $15.3$min $1.4$hr $\star$ $2\mbox{e-}12$ $1.1\mbox{e-}5$ $-$ $-$ $-$
$0.01$ $20$min $1.4$hr $\star$ $2.5\mbox{e-}4$ $4.3\mbox{e-}4$ $-$ $-$ $-$
-- -- -- -------- ----------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- ------------------
To assess the performance of our method, we present simulation results on RGGs over the unit square [@BLTYW2006; @WZYB2008; @CKS2015b; @CLS2012]. To generate a RGG, we uniformly sample $N$ points on the unit square $\left[-0.5,0.5\right]^2$ and fix them to be the sensors. We additionally pick $K$ points at random from the square (distinct from the sensors) and fix them to be the anchors. We assume that the distance between two sensors, or between a sensor and an anchor, is known if it is at most $r$.
**:** To understand the importance of rigidity, a network consisting of $500$ sensors and $10$ anchors is considered where $r$ is taken to be $0.17$. We generate several instances of random graphs with the above parameters until we have an instance where the corresponding $\Gamma$ fails to be quasi $3$-connected. We run our algorithm at noise levels $\eta = 0$ and $0.1$ on these instances and record the localization results. We then augment the existing clique system to ensure that $\Gamma$ is quasi $3$-connected. This is done using the heuristics proposed in Section \[GP\]. We again run our algorithm at noise levels $\eta = 0$ and $0.1$ and record the results. A particular instance is reported in Figure \[GammaRigid\]. We notice that the proposed algorithm performs poorly if $\Gamma$ is not quasi $3$-connected. In particular, notice that the registration mechanism fails in specific regions of the network. This can be attributed to the “fold-over” phenomena associated with patches that are loosely connected to the rest of the patch system [@ZLGG2010]. However, when $\Gamma$ is forced to be quasi $3$-connected, we notice that the registration output improves significantly for both the noiseless and noisy cases. In fact, we achieve almost machine-level precision for the noiseless case.
**:** In Table \[SelfCompare\], we report the run-times of the three phases of the algorithm for networks of different sizes (we round $K$ to $10\% $ of $N$ in each case). The experiments were performed using MATLAB $8.2$ on a $4$-core workstation with $3.4$ GHz processor and $32$ GB memory. Notice that the timing of the localization and the partitioning phase increases almost linearly with the number of sensors. Interestingly, the timing does not vary much with the noise level for a fixed $N$. However, the timing of the final registration phase appears to depend heavily on the noise level. An explanation for this is that we use the solution of the spectral relaxation of as an initialization for the ADMM solver [@CKS2015a]. If the spectral relaxation turns out to be a good approximation of the optimal solution, then the ADMM solver converges in few iterations. Else, a large number of ADMM iterations are required.
**:** As a final analysis, we study the scaling of ANE with the noise level, when $r = 0.12,0.15, \text{ and } 0.18$. A fixed network consisting of $500$ sensors and $50$ anchors was used. For a particular $\eta$ and $r$, we averaged the ANEs obtained over $10$ realizations of the random graph and the measured distances. The results from a typical experiment are plotted in Figure \[Self\_perfor\]. We notice that the ANE increases almost linearly with $\eta$ when $r = 0.12$ and $0.15$. However, when $r=0.12$, the ANE tends to increase abruptly at large noise levels. The reason for this is that the registration process can fail when $r$ is low and $\eta$ is large (low signal-to-noise ratio scenario).
Comparison
----------
We now compare the proposed method with the following optimization-based methods: Edge-based Maximum Likelihood (`E-ML`) relaxation [@SL2014], SNL using SDP (`SNLSDP`) [@BLTYW2006], Edge-based SDP (`ESDP`) relaxation [@WZYB2008], SNL using Disk Relaxation (`SNLDR`) [@SXG2015]. `E-ML` uses distributed optimization to minimize a surrogate of the ML estimator for the distance measurements in SNL. `SNLDR` uses distributed optimization for a novel convex relaxation of the SNL problem. On the other hand, `SNLSDP` is a centralized algorithm which is based on an SDP-based relaxation of the SNL problem. `ESDP` is a further relaxation of `SNLSDP` that can handle large networks.
**:** The proposed method is compared with `E-ML`, `SNLSDP`, `ESDP` and `SNLDR` on random geometric graphs. The results are reported in Table \[Compare\]. For a fair comparison with `SNLDR`, we additionally placed an anchor at each of the four corners of the unit square to ensure that the sensors are in the convex hull of the anchors [@SXG2015]. The localization obtained using our method is comparable with that obtained from `SNLSDP` for small networks ($N \leq 500$). The performance of `SNLSDP` starts degrading when $N > 500$, and it cannot handle large networks ($N>1000$). On the other hand, the proposed method is able to maintain its performance across networks of all sizes. Notice that, though `ESDP` can scale up to networks of size $8000$, its performance falls off abruptly when $N>4000$. The proposed method is generally faster than `ESDP` and `SNLSDP`.
[|c|m[3.8cm]{}|m[3.8cm]{}|m[3.8cm]{}|m[3.8cm]{}|]{} $\eta$ & Proposed & `SNLSDP` [@BLTYW2006] & `ESDP` [@WZYB2008] & `SNLDR` [@SXG2015]\
$0$ & & & &\
$0.1$ & & & &\
[|c|m[3cm]{}|m[3cm]{}|m[3cm]{}|m[3cm]{}|m[3cm]{}|]{} $\eta$ & Original & Proposed & `SNLSDP` [@BLTYW2006] & `ESDP` [@WZYB2008] & `SNLDR` [@SXG2015]\
$0$ & & & & &\
$0.5$ & & & & &\
**:** We provide some visual comparison in Figures \[my-labe1\] and \[my-labe2\] for RGGs and the PACM logo [@CLS2012]. The latter consists of $425$ points sampled from the logo. We randomly set $43$ points as anchors. Notice that the reconstruction from the proposed method is visibly superior to the competing methods in either case, which is also reflected by the ANE. The accuracy is competitive with `SNLSDP`, but consistently better than the other methods. In particular, notice the poor localizations obtained using `SNLDR` when $\eta=0.5$.
Additional comparisons with [@SXG2015], [@BLTYW2006], [@WZYB2008], and [@ADPV2012] are provided in the supplementary material. The MATLAB code of our algorithm is publicly available [@SJC2017].
Conclusion
==========
We demonstrated that by transforming the localization problem into a registration problem, one can achieve scalability without compromising the localization accuracy. In particular, the convex relaxation of the registration problem appears to be better behaved in terms of scalability and approximation quality compared to the convex relaxations of the localization problem. For example, the proposed algorithm can localize a network of $8000$ nodes in $15$ minutes with almost machine-precision accuracy of $1\mbox{e-}12$. In contrast, the convex relaxation in [@BLTYW2006] cannot be scaled beyond $1000$ nodes. An exception in this regard is `ESDP`, which can be scaled to networks with thousands of nodes. However, its localization accuracy starts falling off with the increase in network size. A key contribution of the paper is that we formulated and analysed the rigidity problem associated with multi-patch registration. An open question that emerged from this analysis is whether quasi-connectivity is sufficient for the patch configuration to be rigid. Another relevant question that remains unaddressed is the impact of rigidity on the performance of the registration algorithm, both in terms of tightness and stability. These will be investigated in future work.
Supplementary
=============
Proof of Theorem II.8 {#ProofTheorem}
---------------------
In this section, we prove Theorem II.8. First, we recall a basic assumption that was made in this regard.
\[assumption\] There are at least $d+1$ non-degenerate points in each patch.
We now restate Theorem II.8.
\[THEOREM\] Under Assumption \[assumption\], if a configuration is rigid in $\mathbb{R}^d$, then its correspondence graph must be quasi $(d\!+\!1)$-connected.
To prove Theorem \[THEOREM\], we will need the following proposition:
\[Menger\] The following are equivalent.\
(a) The correspondence graph $\Gamma$ is quasi $k$-connected.\
(b) $\cE(\Gamma)$ can be divided into two disjoint subsets $E_1$ and $E_2$ such that the edges from $E_1$ and that from $E_2$ are\
(i) incident on at least $k$ common vertices from $\cV_1(\Gamma)$, and\
(ii) not incident on any common vertex from $\cV_2(\Gamma)$.
For completeness, we recall Problems II.1 and II.2 from the main manuscript.
\[prob1\] Find $\x_1, \dots , \x_N$ and rigid transforms $\cQ_1, \dots , \cQ_M$ such that, for $1 \leq i \leq M$, $$\label{prob6}
\x_k = \cQ_i (\x_{k,i}) \quad \text{and} \quad \bar{\a}_l = \cQ_i (\bar{\a}_l),$$ where $k \in \cC_i \setminus \cA$ and $l \in \cC_i \cap \cA$.
\[prob2\] Determine whether Problem \[prob1\] have a unique solution up to a rigid transform. That is, if $\x_1, \dots , \x_N$ is a solution of Problem \[prob1\], then is it necessary that for some rigid transform $\cR$, $$\label{des_reg_out1}
\x_k = \cR ( \bar{\x}_k) \quad \text{ and } \quad \bar{\a}_l = \cR (\bar{\a}_l),$$ where $k \in \cS$ and $l \in \cA$?
Moreover, we assume that the points in each patch have been derived from the respective points in $\cX_s \cup \cX_a$ via a rigid transform. Let $$\label{LocGlob1}
\bar{\x}_k =\cR_i(\x_{k,i}) = \bO_i \x_{k,i} + \t_i \qquad (k \in \cC_i \backslash \cA),$$ and $$\label{LocGlob2}
\bar{\a}_l =\cR_i(\bar{\a}_l) \qquad (l \in \cC_i \cap \cA),$$ We consider a different registration problem where the patch coordinates are replaced by the original coordinates.
\[prob4\] Find $\x_1, \dots , \x_N$ and rigid transforms $\cT_1, \dots , \cT_M$ such that, for $1 \leq i \leq M$, $$\label{prob7}
\x_k = \cT_i (\bar{\x}_{k}) \quad \text{and} \quad \bar{\a}_l = \cT_i (\bar{\a}_l),$$ where $k \in \cC_i \setminus \cA$ and $l \in \cC_i \cap \cA$.
A trivial solution is $\x_k = \bar{\x}_{k}$ and $\cT_i= (\bI_d, \mathbf{0})$. As with Problem \[prob2\], we can ask whether this is the only solution. It turns out that the questions are related.
\[prob5\] Problem \[prob1\] has an unique solution if and only if Problem \[prob4\] has an unique solution.
Combining , and , we can write $$\label{prob8}
\x_k = (\cT_i \circ \cR_i) (\x_{k,i}) \quad \text{and} \quad \bar{\a}_l = (\cT_i \circ \cR_i) ( \bar{\a}_l ).$$ Comparing with , we have $\cQ_i=\cT_i \circ \cR_i$. It follows that the $\cT_i$’s are unique if and only if the $\cQ_i$’s are unique. Moreover, the uniqueness of the $\x_k$’s follows from the uniqueness of the transforms and relations and .
We also make the observation concerning Problem \[prob4\] that $\cT_1, \dots , \cT_M$ satisfying are unique, i.e., $\cT_i=\cR(\bI_d, \mathbf{0})$ for some rigid transform $\cR$, if and only if the corresponding $\x_1,\ldots,\x_N$ are related to $\bar{\x}_1,\ldots,\bar{\x}_N$ via a rigid transform. If $\cT_i=\cR(\bI_d, \mathbf{0})$, then it follows from that $\x_k =\cR (\bar{\x}_k)$. Conversely, if $\x_k =\cR (\bar{\x}_k)$ for some rigid transform $\cR$, then the corresponding $\cT_i$ should necessarily be of the form $\cT_i=\cR(\bI_d, \mathbf{0})$. Indeed, if some $\cT_i \neq \cR(\bI_d, \mathbf{0})$, then we can construct a solution that is not related to $\bar{\x}_1,\ldots,\bar{\x}_N$ via a rigid transform, and this would lead to a contradiction.
To complete the proof of Theorem \[THEOREM\], it remains to show that, if the solution of Problem \[prob4\] is unique, then $\Gamma$ must be quasi $(d+1)$-connected. We will prove this by contradiction. As a first step, we note that $\Gamma$ is at least quasi-$1$ connected.
\[conn\] If Problem \[prob4\] has a unique solution, then $\Gamma$ must be quasi-$k$ connected for some $k \geq 1$.
Indeed, suppose that there exist non-empty subsets $S$ and $T$ of $\cV_2(\Gamma)$ such that there is no path between any $i \in S$ and $j \in T$. Define $$A = \bigcup_{\alpha \in S} \cC_{\alpha} \quad \text{ and } \quad B= \bigcup_{\beta \in T} \cC_{\beta}.$$ Clearly, $A \cap B$ must be empty. Else, we can find a path between some $i \in S$ and $j \in T$, which would violate our assumption. However, on setting $\cT_i=(\bI_d,\mathbf{0})$ for $i \in S$, and $\cT_j=(-\bI_d, \mathbf{0})$ for $j \in T$, we obtain a solution to Problem \[prob4\] which is different from the trivial solution. Hence, our assumption about the existence of $S$ and $T$ must be wrong.
In fact, we can make the stronger claim that $\Gamma$ is quasi $k$-connected, where $k \geq d+1$. To establish the claim, we show that the rigidity assumption is violated if $k \leq d$.
First, we introduce few notations about paths. Suppose that there are one or more paths between two vertices of $\cV_2(\Gamma)$. We denote the $j$-th vertex on the $i$-th path using $\sigma^j_i$. In particular, $\sigma_i^1$ and $\sigma_i^{p_i}$ are the initial and final vertices, where $p_i$ is the number of vertices on the path. Since $\Gamma$ is bipartite, $p_i$ must be odd, and $$\sigma_i^j \in \begin{cases}
\cV_1 \left(\Gamma\right) & \text{for} \ j = 2,4,\dots,p_i-1, \\
\cV_2 \left(\Gamma\right) & \text{for} \ j = 1,3,\dots,p_i. \\
\end{cases}$$ For $1 \leq j \leq (p_i-1)/2$, consider the vertices $$\sigma_i^{2j},\ \sigma_i^{2j-1}, \text{ and } \sigma_i^{2j+1}.$$ The first vertex represents a node, while the latter two represent patches. Moreover, the node belongs to both the patches. Therefore, for $1 \leq i \leq k$ and $1 \leq j \leq (p_i-1)/2$, $$\label{Theo_prof_1}
\bO_{\sigma_i^{2j-1}}\bar{\x}_{\sigma_i^{2j}}+\t_{\sigma_i^{2j-1}} = \bO_{\sigma_i^{2j+1}}\bar{\x}_{\sigma_i^{2j}}+\t_{\sigma_i^{2j+1}}.$$ To arrive at a contradiction, we show that if $k \leq d$, then there exists at least some $\cT_i=(\bO_i,\t_i),1\leq i \leq M,$ different[^2] from $(\bI_d,\bm{0})$ for which the system of equations in hold. To do so, we divide $\cV_2(\Gamma)$ into two disjoint sets. Note that, from Proposition \[Menger\], we can identify disjoint subsets $E_1,E_2 \subset \cE(\Gamma)$ such that the edges from $E_1$ and that from $E_2$ are not incident on any common vertex of $\cV_2(\Gamma)$. In particular, define $S \subset \cV_2(\Gamma)$ to be the vertices on which the edges of $E_1$ are incident. Similarly, let $T \subset \cV_2(\Gamma)$ be the vertices on which the edges of $E_2$ are incident. Then, $S$ and $T$ are non-empty and disjoint. Without loss of generality, we assume that the vertex corresponding to the anchor patch belongs to $S$. Since $\Gamma$ is quasi $k$-connected, we can find a distinct vertex $t \in T$ which is connected with the anchor patch vertex by paths $\sigma_1,\ldots,\sigma_k$ that are $\cV_1(\Gamma)$-disjoint.
Note that, since the anchor patch is fixed, $\bO_{M+1}=\bI_d$ and $\t_{M+1}=\bm{0}$. Therefore, we set $$\cT_i= (\bO_i,\t_i) = \begin{cases}
(\bO,\t) , & \text{ if } \ i \in T, \\
(\bI_d,\mathbf{0}) & \text{ if } \ i \in S,
\end{cases}$$ and show that if $k \leq d$, then we can find $(\bO,\t) \neq (\bI_d,\bm{0})$ such that holds. Note that Proposition \[Menger\] also tells us that the edges from $E_1$ and that from $E_2$ are incident on exactly $k$ common vertices from $\cV_1(\Gamma)$; we denoted these vertices using $\Omega$. It is also be reasoned that each path contains exactly one vertex from $\Omega$. Assume that $\sigma_i^{2q_i} \in \Omega$ be the vertex on the path $\sigma_i$. Therefore, $$\label{Theo_prof_2}
\bar{\x}_{\sigma_i^{2q_i}} = \bO \bar{\x}_{\sigma_i^{2q_i}}+\t.$$ If $k=1$, then $\bO=-\bI_d$ and $\t= 2\bar{\x}_{\sigma_i^{2q_i}}$ satisfy , and hence the equations in . On the other hand, if $2\leq k \leq d$, then we have $k$ equations similar to , one for each path. We eliminate $\t$ by subtracting the equations corresponding to $2 \leq i \leq k$ from the equation corresponding to $i=1$. This gives us $$\label{Theo_prof_3}
\bO(\bar{\x}_{\sigma_i^{2q_i}} - \bar{\x}_{\sigma_1^{2q_1}}) = \bar{\x}_{\sigma_i^{2q_i}}-\bar{\x}_{\sigma_1^{2q_1}} \qquad (2 \leq i \leq k).$$ We collect this into the fixed-point equation $\bO \bX = \bX$, where $$\bX = \left[ \bar{\x}_{\sigma_2^{2q_2}} - \bar{\x}_{\sigma_1^{2q_1}} \ \cdots \ \bar{\x}_{\sigma_k^{2q_k}} - \bar{\x}_{\sigma_1^{2q_1}}\right] \in \mathbb{R}^{d \times (k-1)}.$$ Now, if we assume that $k \leq d$, then we can find $\bO \neq \bI_d$ such that $\bO \bX = \bX$. In particular, we can find $\bO$ that acts as an identity transform on the space spanned by the columns of $\bX$, and as a non-trivial rotation on the orthogonal complement of this space. We set $\t$ using for this choice of $\bO$. One can verify that the above choice of $(\bO,\t) \neq (\bI_d,\bm{0})$ satisfies the equations in . This concludes the proof of Theorem \[THEOREM\].
Experiments
-----------
In this section, we report some additional numerical results to demonstrate the performance of the proposed algorithm.
**:** To study the effect of the number of anchors on the performance, we consider a random geometric graph (RGG) on $[-0.5,0.5]^2$ consisting of $500$ sensors. The sensing radius $r$ is set as $0.17$. We plot the ANE as a function of the number of anchors $K$ for different noise levels $\eta = 0.01, 0.05$ and $0.1$. The ANE is averaged over $100$ noise realizations. The results are reported in Figure \[KvsANE\]. We notice that the ANE falls off with increase in $K$, and saturates beyond a certain $K$.
**:** We compare the localization accuracy of the proposed method with `PLACEMENT` [@ADPV2012] on RGGs. The results are reported in Table \[Compare3\]. We notice that for both clean and noisy measurements, the proposed method performs better than `PLACEMENT`.
**:** We consider a RGG with $200$ sensors. For a fair comparison with `SNLDR` [@SXG2015], we placed 4 anchors at $(\pm 0.5, \pm 0.5)$ (so that the sensors are guaranteed to be in the convex hull of the anchors). We set $r=0.28$ and $\eta = 0.1$. The reconstructions are compared in Figure \[ANE\] along with the corresponding ANEs.
**:** We repeat Experiment 3 with $500$ sensors and $10$ anchors (with $4$ of them placed at $(\pm 0.5, \pm 0.5)$). The localizations obtained using `ESDP` [@WZYB2008], `SNLDR` [@SXG2015], `SNLSDP` [@BLTYW2006] and the proposed method are shown in Figure \[ANE\_deg\]. For this instance of RGG, the average node degree is $13.2$ and the minimum node degree is $3$. We note that for both Experiments 3 and 4, the ANE for the proposed method is the least.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank the editor and the anonymous reviewers for their thoughtful comments and suggestions. The authors also wish to thank Nicolas Gillis, Andrea Simonetto, Claudia Soares, and Arvind Agarwal for useful discussions and for providing the MATLAB code of their algorithms.
[^1]: The authors were supported by a Startup Grant from IISc Bangalore and an EMR Grant SERB/F/6047/2016-2017 from Department of Science and Technology, Government of India. The second author was supported by a NBHM Postdoctoral Fellowship from Department of Atomic Energy, Government of India. Address: Department of Electrical Engineering, Indian Institute of Science, India. Correspondence: {rajatsanyal,monika,kunal}@ee.iisc.ernet.in.
[^2]: Without loss of generality, we omit the global rigid transform $\cR$.
|
---
abstract: 'Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of “theta functions” on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions.'
address: |
School of Mathematics\
University of Edinburgh\
Edinburgh EH9 3FD\
UK
author:
- Travis Mandel
bibliography:
- 'main.bib'
title: 'Theta bases and log Gromov-Witten invariants of cluster varieties'
---
Introduction {#Intro}
============
Let $Y$ be a smooth compact variety over an algebraically closed field ${\Bbbk}$ of characteristic $0$, or more generally, a smooth integral separated Deligne-Mumford stack which is proper and finite type over ${\Bbbk}$ (an “orbifold” for short). The quantum cohomology ring $\operatorname{QH}^*(Y)$ is obtained by using (virtual) counts of holomorphic curves in $Y$ to deform the cup product on the cohomology ring of $Y$. The degree $0$ subalgebra $\operatorname{QH}^0(Y)$ is very simple, generated by the fundamental class $[Y]=\operatorname{Id}\in \operatorname{QH}^*(Y)$. But now suppose we have a log variety or log orbifold $Y^{\dagger}$ obtained by equipping $Y$ with a log structure, e.g., the data of a reduced effective normal crossings divisor $D\subset Y$. We call such $(Y,D)$ a **log pair**. Then already in codimension $0$ the structure is far more complicated, e.g., logarithmic analogs of the degree $0$ cohomology are often infinite-dimensional. The question of whether degree $0$ log classes admit an analog of the quantum cohomology product is highly non-trivial, and the resulting algebra $\operatorname{QH}_{\log}^0(Y^{\dagger})$ is expected to be very rich. Indeed, if $Y^{\dagger}=(Y,D)$ is a **log Calabi-Yau orbifold with maximal boundary** (i.e., $D$ contains a $0$-stratum and is in $|-K_Y|$), then the Frobenius structure conjecture [@GHK1 arXiv v1, Conj. 0.8] predicts that $\operatorname{Spec}\operatorname{QH}_{\log}^0(Y^{\dagger})$ is the mirror to $Y$. Furthermore, $\operatorname{QH}_{\log}^0(Y^{\dagger})$ should be naturally equipped with a canonical basis of “theta functions” which should agree with the theta functions constructed combinatorially by Gross, Hacking, Keel, Kontsevich, and Siebert [@CPS; @GHK1; @GHKK; @GHS]. Our main result is a proof of this conjecture for cluster varieties.
Cluster varieties were defined in [@FG1], giving geometric meaning to the cluster algebras of [@FZ]. Some examples of cluster varieties include Grassmannians [@ScottGCA] and other partial flag varieties [@GLS], double Bruhat cells of reductive Lie groups [@BFZ], various moduli of local systems (higher-Teichmüller spaces) [@FG0], and all two-dimensional log Calabi-Yau varieties with maximal boundary [@GHK3 §5]. By [@GHK3 §3.2], compactifications of cluster varieties can always be viewed as log Calabi-Yau orbifolds with maximal boundary, obtained from toric varieties by preforming certain non-toric blowups of hypertori in the boundary, cf. §\[nontoric\]. [@GHKK] constructed canonical theta bases on cluster varieties, and in the process settled many long-standing conjectures about cluster algebras. We will focus on **cluster log pairs** $(Y,D)$ as in Definition \[clp\]. These are the log pairs obtained by compactifying “leaves” of cluster $\s{X}$-varieties, which in our construction includes the usual symplectic leaves and also entire cluster $\s{X}$-spaces, cf. Remark \[fullX\]. All theta functions of [@GHKK] can be recovered from these cases, cf. Remark \[AllTheta\].
For a log pair $(Y,D)$, a simple toric blowup $\pi:(\wt{Y},\wt{D})\rar (Y,D)$ is a blowup $\pi:\wt{Y}\rar Y$ of $Y$ along a stratum of $D$, with $\wt{D}$ the reduced inverse image of $D$. A **toric blowup** is then a sequence of simple toric blowups. For $\pi_i:(Y_i,D_i)\rar (Y,D)$, $i=1,2$ two toric blowups of $(Y,D)$, we say that an irreducible component $D'_1\subset D_1$ is equivalent to an irreducible component $D'_2\subset D_2$ if they correspond to the same valuation on the function field of $Y$. Let $H^0_{\log}(Y,D,\bb{Z})$ denote the free Abelian group generated by $[Y]$ and $[kD']$ for $k\in \bb{Z}_{>0}$ and $D'$ an irreducible component of $\wt{D}$ for some toric blowup $(\wt{Y},\wt{D})$, up to equivalence. We refer to these generators $[Y]$ and $[kD']$ as **prime fundamental classes**.[^1] Let $\operatorname{NE}(Y)$ be the cone of effective curve classes in $Y$ up to numerical equivalence (cf. §\[N1\]). Let $\operatorname{QH}_{\log}^0(Y,D)$ denote the completion of $\?{\operatorname{QH}}^0_{\log}(Y,D):=H^0_{\log}(Y,D,\bb{Z})\otimes {\Bbbk}[\operatorname{NE}(Y)]$ with respect to the unique maximal monomial ideal of ${\Bbbk}[\operatorname{NE}(Y)]$, so $\operatorname{QH}_{\log}^0(Y,D)$ has the structure of a ${\Bbbk}\llb \operatorname{NE}(Y)\rrb$-module. We say that a complete curve $C$ in $Y$ is an **interior curve** if it is disjoint from $D$, and we say $(Y,D)$ is **interior-curve free** if it contains no interior curves. For example, $(Y,D)$ is interior-curve free whenever $Y\setminus D$ is affine.[^2] We say $(Y,D)$ **supports an ample divisor on its boundary** if there is a toric blowup $(\wt{Y},\wt{D})$ such that $\wt{D}$ supports an effective ample divisor.
\[MainNaive\] Let $(Y,D)$ be an interior-curve free cluster log pair as in Def. \[clp\]. For $\vartheta_1,\ldots,\vartheta_s$ prime fundamental classes of $H^0_{\log}(Y,D,\bb{Z})$, let $\pi:(\wt{Y},\wt{D})\rar (Y,D)$ be a toric blowup in which each $\vartheta_i$ is either $[Y]$ or is represented by $[k_iD_i]$ for $D_i$ an irreducible component of $\wt{D}$. Let $\beta\in \operatorname{NE}(\wt{Y})$. Let $(C,x_1,\ldots,x_s,x_{s+1},x_{s+2})$ be a fixed generic irreducible genus $0$ curve with $s+2$ marked points. Let $y$ be a generically specified point of $Y\setminus D$. Define $N_{\beta}^{\operatorname{naive}}(\vartheta_1,\ldots,\vartheta_s)$ to be the number of isomorphism classes of maps $\varphi:C\rar Y$ such that $\varphi(x_{s+1})=y$ and $\varphi^* \s{O}_{\wt{Y}}(\wt{D})=\s{O}_C(\sum_{i=1}^s k_ix_i)$, where we take $k_i=0$ if $\vartheta_i=[Y]$.[^3] All such maps are torically transverse.[^4] Define a ${\Bbbk}\llb \operatorname{NE}(Y)\rrb$-multilinear $s$-point function $\langle \cdot \rangle^{\operatorname{naive}}:\operatorname{QH}_{\log}^0(Y)^{s} \rar{\Bbbk}\llb \operatorname{NE}(Y)\rrb$ via $$\begin{aligned}
\langle \vartheta_1,\ldots,\vartheta_s\rangle^{\operatorname{naive}}:=\sum_{\beta\in \operatorname{NE}(\wt{Y})} z^{\pi_*(\beta)}N^{\operatorname{naive}}_{\beta}(\vartheta_1,\ldots,\vartheta_s).
\end{aligned}$$ Then there is a unique associative product $*$ on $\operatorname{QH}_{\log}^0(Y,D)$ making it into an associative ${\Bbbk}\llb \operatorname{NE}(Y)\rrb$-algebra such that $$\begin{aligned}
\langle \vartheta_1,\ldots,\vartheta_s\rangle^{\operatorname{naive}} = \langle \vartheta_1*\cdots *\vartheta_s\rangle^{\operatorname{naive}}\end{aligned}$$ for all $s$-tuples $\vartheta_1, \ldots, \vartheta_s$, $s\geq 1$ (and in fact, the $s=2$ and $s=3$ cases are sufficient to determine $*$). The algebra $\operatorname{QH}_{\log}^0(Y,D)$ is commutative with identity $[Y]$. If $(Y,D)$ supports an ample divisor on its boundary, then $*$ restricts to give a ${\Bbbk}[\operatorname{NE}(Y)]$-algebra structure on $\?{\operatorname{QH}}^0_{\log}(Y,D)$.
The algebra $\operatorname{QH}^0_{\log}(Y,D)$ is naturally a subalgebra of $\Gamma(\wt{\s{X}}_{{{\bf S}}^{\vee}},\s{O}_{\wt{\s{X}}_{{{\bf S}}^{\vee}}})$, where $\wt{\s{X}}_{{{\bf S}}^{\vee}}$ denotes a formal version of the Langlands dual $\s{X}$-space[^5] (cf. Remark \[Lang\]), and the prime fundamental classes are identified with theta functions constructed as in [@GHKK]. If $(Y,D)$ supports an ample divisor on its boundary, then $\?{\operatorname{QH}}^0_{\log}(Y,D)=\Gamma(\?{\s{X}}_{{{{\bf S}}}^{\vee}},\s{O}_{\?{\s{X}}_{{{{\bf S}}}^{\vee}}})$, where $\?{\s{X}}_{{{{\bf S}}}^{\vee}}$ is a partial compactification of the Langlands dual $\s{X}$-space (cf. Remark \[Lang\]), and again, the prime fundamental classes are the [@GHKK] theta functions.
The Frobenius structure conjecture {#FSC}
----------------------------------
For log Calabi-Yau varieties with maximal boundary $(Y,D)$, the Frobenius structure conjecture [@GHK1 arXiv v1, Conj. 0.8] predicts the existence of an algebra structure on $\operatorname{QH}^0_{\log}(Y,D)$, defined essentially as in Theorem \[MainNaive\] but using descendant log Gromov-Witten invariants in place of the naive curve counts above.
Let $(Y,D)$ be a **smooth** log Calabi-Yau orbifold with maximal boundary. Here, in addition to $Y$ being smooth as a Deligne-Mumford stack, we require that there exists a toric blowup $(Y',D')$ of $(Y,D)$ such that $Y'$ is smooth along $D'$ as a variety (i.e., any orbifold points in the boundary are resolvable by toric blowups), and such that the components of $D'$ are smooth (i.e., $D'$ is snc). For such a $(Y',D')$, let $S$ denote the dual intersection complex of $D'$. That is, if $D'=D_1+\ldots+D_n$, then $S$ is the simplicial complex with vertices $v_1,\ldots,v_n$, and with one $(k-1)$-cell with vertices $\{v_{i_1},\ldots v_{i_k}\}$ for each non-empty stratum $D_{i_1}\cap \cdots \cap D_{i_k}$. Let $B$ be the cone over $S$, and let $\Sigma'$ be the induced simplicial fan in $B$.
In the cone $\sigma\in \Sigma'$ spanned by $v_{i_1},\ldots v_{i_k}$, we have a set of integer points $\sigma(\bb{Z})$ defined as the $\bb{Z}_{\geq 0}$-span of $v_{i_1},\ldots v_{i_k}$ in $\sigma$. Let $B(\bb{Z}):=\bigcup_{\sigma\in \wt{\Sigma}} \sigma(\bb{Z})$ be the integer points of $B$. Note that there is a bijection between points $p\in B(\bb{Z})$ and prime fundamental classes $\vartheta_p\in \operatorname{QH}^0_{\log}(Y,D)$. The apex $0\in B(\bb{Z})$ corresponds to $\vartheta_0:=[Y]\in \operatorname{QH}^0_{\log}(Y,\bb{Z})$. For nonzero $p\in B(\bb{Z})$, we write $\vartheta_p=[|p|D_p]$, i.e., the ray through $p$ determines an irreducible component $D_p$ in the boundary of some toric blowup (up to equivalence), and $|p|\in \bb{Z}_{>0}$ is the index of $p$ (cf. Notation \[LatticeNotation\]). For notational convenience, we write $D_0:=Y$.
By a **tropical degree**, we mean a map $\Delta:J\rar B(\bb{Z})$ for some finite index-set $J$. In particular, for ${{\bf p}}$ an $s$-tuple of points $p_1,\ldots,p_s\subset B(\bb{Z})$, we consider $$\begin{aligned}
\label{Deltapp}
\Delta_{{{\bf p}}}:\{1,\ldots,s,s+1,s+2\}\rar B(\bb{Z})
\end{aligned}$$ defined by $\Delta_{{{\bf p}}}(i)=p_i$ for $i=1,\ldots,s$ and $\Delta_{{{\bf p}}}(s+1)=\Delta_{{{\bf p}}}(s+2)=0$. Given a tropical degree $\Delta$, let $\wt{Y}^{\dagger}=(\wt{Y},\wt{D})$ be a toric blowup with $\wt{Y}$ projective and smooth along $\wt{D}$ as a variety, and such that each $[D_{\Delta(j)}]$ with $\Delta(j)\neq 0$ is represented by an irreducible component of $\wt{D}$. For $\beta\in \operatorname{NE}(\wt{Y})$, let $\s{M}_{0,\Delta}^{\log}(\wt{Y}^{\dagger},\beta)$ denote [@GSlog; @AC]’s moduli stack[^6] of basic/minimal stable log maps $\varphi^{\dagger}:C^{\dagger}\rar \wt{Y}^{\dagger}$ over $\operatorname{Spec}{\Bbbk}$ satisfying the following collection of conditions:
- $C$ has genus $0$,
- $\varphi_*[C] = \beta$,
- $C^{\dagger}$ has $\#J$ marked points $\{x_i\}_{i\in J}$,
- For each $i\in J$, $\varphi(x_i)\in D_{\Delta(i)}$. Furthermore, if $t_1$ is the generator for the ghost sheaf of $\wt{Y}^{\dagger}$ at a generic point of $D_{\Delta(i)}$, and $t_2$ is the generator for the ghost sheaf of $C^{\dagger}$ at $x_i$, then $\varphi^{\flat}:t_1\mapsto |\Delta(i)|t_2$. We view this condition as being satisfied automatically for $i$ such that $\Delta(i)=0$.
When the component of $C$ containing $x_i$ is not mapped entirely into $\wt{D}$, this last condition means that the intersection multiplicity of $\varphi(C)$ with $D_{\Delta(i)}$ at $x_i$ is equal to $|p_i|$. The algebraic stack $\s{M}_{0,\Delta}^{\log}(\wt{Y}^{\dagger},\beta)$ has a virtual fundamental class $[\s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(\wt{Y}^{\dagger},\beta)]^{\operatorname{vir}}$ of virtual dimension $\dim(\wt{Y})+|\Delta|-3$.
Let $\operatorname{ev}_{i}:\s{M}_{0,\Delta}^{\log}(\wt{Y}^{\dagger},\beta)\rar \wt{Y}$ be the evaluation map $[\varphi^{\dagger}:C^{\dagger}\rar \wt{Y}]\mapsto \varphi(x_i)$. Let $\pi:\s{C}\rar \s{M}^{\log}_{0,\Delta}(\wt{Y}^{\dagger},\beta)$ denote the universal curve over the moduli space. Let $\omega_{\pi}$ denote the relative cotangent bundle of $\pi$, and let $\sigma_i$ denote the section of $\pi$ corresponding to $x_i$. Define $$\begin{aligned}
\label{psic1}
\psi_{i}:=c_1(\sigma_i^* \omega_{\pi}),\end{aligned}$$ i.e., $\psi_i$ is the first Chern class of the line bundle whose fiber over a point $[\varphi^{\dagger}:C^{\dagger}\rar \wt{Y}^{\dagger}]$ is the cotangent space to $C$ at $x_i$. We can now define the relevant Gromov-Witten numbers: $$\begin{aligned}
\label{Nbeta}
N_{\beta}(p_1,\ldots,p_s) := \int_{[\s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(\wt{Y}^{\dagger},\beta)]^{\operatorname{vir}}} \operatorname{ev}_{s+1}^*[\operatorname{pt}] \cdot \psi_{s+1}^{s-1}.\end{aligned}$$ We next define a ${\Bbbk}\llb \operatorname{NE}(Y)\rrb$-multilinear $s$-point function $$\langle \cdot \rangle:\operatorname{QH}_{\log}^0(Y)^{s} \rar{\Bbbk}\llb \operatorname{NE}(Y)\rrb$$ via $$\begin{aligned}
\label{spointfunction}
\langle \vartheta_{p_1},\ldots,\vartheta_{p_s}\rangle:=\sum_{\beta\in \operatorname{NE}(\wt{Y})} z^{\eta_*(\beta)}N_{\beta}(\vartheta_{p_1},\ldots,\vartheta_{p_s}).
\end{aligned}$$
\[FrobConj\] For any smooth log Calabi-Yau orbifold with maximal boundary $(Y,D)$, there is a unique associative product $*$ on $\operatorname{QH}_{\log}^0(Y,D)$ making $\operatorname{QH}_{\log}^0(Y,D)$ into an associative ${\Bbbk}\llb\operatorname{NE}(Y)\rrb$-algebra such that $$\begin{aligned}
\label{spoint}
\langle \vartheta_{p_1},\ldots,\vartheta_{p_s}\rangle = \langle \vartheta_{p_1}*\cdots *\vartheta_{p_s}\rangle\end{aligned}$$ for all $s$-tuples $p_1, \ldots, p_s\in B(\bb{Z})$, $s\geq 1$. Furthermore, the $s=2$ and $s=3$ cases of are sufficient to determine $*$. The algebra $\operatorname{QH}_{\log}^0(Y,D)$ is commutative with identity $\vartheta_0$. If $(Y,D)$ supports an ample divisor on its boundary, then $*$ restricts to give a ${\Bbbk}[\operatorname{NE}(Y)]$-algebra structure on $\?{\operatorname{QH}}^0_{\log}(Y,D)$.
\[logFCA\] The statement of the Frobenius structure conjecture in [@GHK1 arXiv v1, Conj. 0.8] defines $N_{\beta}$ slightly differently. Their log curves do not include the marked point $x_{s+2}$ (they have only $s+1$ marked points), and their $\psi$-class $\psi_{s+1}$ in is only raised to the power of $s-2$, not $s-1$. That these two definitions are equivalent follows from the Fundamental Class Axiom (generalized to this log setting by the same argument as in the non-log setting). The advantage of our version of $N_{\beta}$ is that it makes sense for $s=1$ and thus makes the setup more elegant. We similarly have an extra marked point and $\psi$-class factor in Theorem \[QHthm\] to allow for the $s=1$ cases.
\[MainThm\] Conjecture \[FrobConj\] holds for all cluster log pairs $(Y,D)$ (as in Def. \[clp\]). Furthermore, as in Theorem \[MainNaive\], the resulting algebra $\operatorname{QH}^0_{\log}(Y,D)$ is naturally a subalgebra of $\Gamma(\wt{\s{X}}_{{{\bf S}}^{\vee}},\s{O}_{\wt{\s{X}}_{{{\bf S}}^{\vee}}})$, and the prime fundamental classes are identified with theta functions constructed as in [@GHKK]. If $(Y,D)$ supports an ample divisor on its boundary, then $\?{\operatorname{QH}}^0_{\log}(Y,D)=\Gamma(\?{\s{X}}_{{{{\bf S}}}^{\vee}},\s{O}_{\?{\s{X}}_{{{{\bf S}}}^{\vee}}})$, where $\?{\s{X}}_{{{{\bf S}}}^{\vee}}$ is a partial compactification of the Langlands dual $\s{X}$-space, and again, the prime fundamental classes are the [@GHKK] theta functions.
Theorem \[MainNaive\] actually follows from Theorem \[MainThm\] via Proposition \[naive\], which says that, under the interior-curve free assumption, the descendant log Gromov-Witten counts $N_{\beta}(p_1,\ldots,p_s)$ agree with the corresponding naive counts $N^{\operatorname{naive}}_{\beta}(\vartheta_{p_1},\ldots,\vartheta_{p_s})$.
As a sample application, recall from [@CCGGK] that mirror symmetry for Fano manifolds predicts the equality of the “quantum period” of the Fano and the “classical period” of a mirror Landau-Ginzburg potential. This equivalence is a key tool in the ongoing Fano classification program outlined in loc cit. We will show in a separate paper [@ManFano] that this mirror equivalence of quantum and classical periods follows from the Frobenius structure conjecture, at least whenever the log curves being counted are torically transverse. Our results then imply this equivalence of mirror periods for Fano cluster varieties, thus yielding an algebro-geometric analog of [@Tonk Thm. 1.1].
Outline of the paper
--------------------
In §\[nontoric\] we review [@GHK3]’s realization of (compactified) cluster varieties as blowups of toric varieties. Then in §\[N1\] we describe the lattice $N_1(Y_{{{\bf S}}})$ of curve classes of a cluster variety $Y_{{{\bf S}}}$, and in **Theorem \[kappa\]** we show how to identify $N_1(Y_{{{\bf S}}})$ with the kernel of the exchange matrix of the cluster data.
In §\[ScatterIntro\] we review the construction of scattering diagrams and theta functions in the context used by [@GHKK]. **Lemma \[nondegen\]** (taken from [@Man3 Thm. 2.17]) is essentially the statement that a certain multilinear $s$-point function $\operatorname{Tr}^s$ is sufficient to determine the multiplication rule for the theta functions. Much of the work of this paper is then to show that the $s$-point function $\operatorname{Tr}^s$ agrees with the $s$-point function $\langle \cdot \rangle$ of .
It follows from Theorem \[kappa\] that the ring of theta functions can be viewed as an algebra over a completion of ${\Bbbk}[N_1(Y_{{{\bf S}}})]$. In **Lemma \[NECoeff\]** we see that the multiplication in fact restricts to give an algebra over ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$. This algebra can be identified with a ring of functions on the Langlands dual cluster variety, cf. **Remark \[LangRmk\]**.
Next, §\[Review\] reviews past results relating theta functions to tropical curve counts (§\[ScatTheta\]) and tropical curve counts to descendant log Gromov-Witten invariants of toric varieties (§\[SectionLogGW\]). The combination of these results immediately yields **Lemma \[GWToric\]**. Along the way, we prove (**Corollary \[AmplePolynomial\]**) that if $(Y,D)$ supports an ample divisor on its boundary, then the multiplication is polynomial over ${\Bbbk}[\operatorname{NE}(Y_{{{\bf S}}})]$ (i.e., we can restrict to $\?{\operatorname{QH}}^0_{\log}$), thus proving another piece of Theorem \[MainThm\].
We prove Theorem \[MainThm\] in §\[DegenS\]. The idea (inspired by the proof of [@GPS Prop. 5.3]) is to use a degeneration formula to relate the toric descendant log GW invariants from Lemma \[GWToric\] to the desired descendant log GW invariants of the cluster variety, cf. **Proposition \[DegenProp\]**. The main technical hurdle here is showing that the relevant log stable maps are all torically transverse. This is achieved in §\[trans\] (cf. **Lemma \[DaggerCircLem\]**) by proving that the tropicalizations of such log stable maps must be supported on the one-skeleton of the fan for the toric variety.
In **Proposition \[naive\]** we show that $\langle \cdot \rangle$ and $\langle \cdot \rangle^{\operatorname{naive}}$ agree whenever the cluster variety is interior-curve free, thus completing the proof of Theorem \[MainNaive\].
As suggested at the start of the introduction and in the notation, we view $\operatorname{QH}^0_{\log}(Y,D)$ as the degree $0$ part of a conjectural log version of quantum cohomology. We explain this viewpoint in the appendix, where we give a new description of the usual small quantum cohomology ring (**Theorem \[QHthm\]**).
Relation to other works {#other}
-----------------------
There are two nearly-completed works which prove versions of Conjecture \[FrobConj\] in other contexts.
In [@KY], Keel and Yue Yu prove a naive-counting version of the Frobenius structure conjecture whenever $Y\setminus D$ is an affine variety and contains a maximal-dimensional algebraic torus. Their approach is based on Berkovich analytic disks rather than log Gromov-Witten theory, and they obtain descriptions of all the structure constants in terms of these disks. By the uniqueness of the product $*$ in Theorem \[MainNaive\], their theta functions agree with ours in all cases where both results apply, i.e., for all cluster log pairs $(Y,D)$ such that $Y\setminus D$ is an affine variety.
As announced in [@GSInt Thm. 2.2], Gross and Siebert [@GSInt2] prove another version of Conjecture \[FrobConj\] in great generality. Rather than proving that the $s$-point functions uniquely determine the multiplication, they explicitly define all the structure constants in terms of newly developed punctured invariants, an extension of log invariants which satisfy a powerful splitting lemma. With this approach, they define an associative algebra associated to log pairs $(Y,D)$ whenever either $K_Y+D$ or $-(K_Y+D)$ is nef, cf. [@GSInt Thm. 2.2]. In particular, this includes the log Calabi-Yau cases $K_Y+D=0$. We note that [@KY] and [@GSInt2] also have results for a more general version of log Calabi-Yau varieties, cf. [@GSInt §2.2].
We note that [@FWY] also seems to be closely related. There, one considers the case of smooth $D$ and works with relative invariants, possibly with negative orders of tangency (which also appear in the punctured invariant setting of [@GSInt2]). Fan-Wu-You construct from this a relative version of the quantum cohomology ring.
Acknowledgements
----------------
I am very grateful to Sean Keel and Tony Yue Yu, and to Mark Gross and Bernd Siebert, for keeping me informed about their progress on their closely related projects (cf. §\[other\]), as well as for discussions that helped with various technical obstacles which arose in the writing of this and related papers. In particular, Sean Keel and Mark Gross encouraged the author to pursue this project early on and provided helpful feedback on a draft of this paper. Collaborations and discussions with Helge Ruddat were also invaluable. I have also benefited from discussions with Lawrence Barrott, Francesca Carocci, and Y.P. Lee.
Cluster varieties {#clusterSection}
=================
Here we review the construction of cluster varieties from [@FG1 §1.2] as reinterpreted in [@GHK3 §3.2].
\[LatticeNotation\] For any lattice $L$, let $L_{\bb{Q}}:=L\otimes \bb{Q}$, $L_{\bb{R}}:=L\otimes \bb{R}$, and $T_L:=L\otimes {\Bbbk}^*$. We say a nonzero element $v\in L$ is **primitive** if it is not a positive multiple of any other element of $L$. Also, for any nonzero $v\in L$, we let $|v|$ denote the **index** of $v$, i.e., $|v|$ is the unique positive integer such that $v$ is equal to $|v|$ times a primitive vector $v'\in L$. We use angled brackets $\langle \cdot, \cdot\rangle$ to denote the pairing between a lattice and its dual.
Construction via blowups of toric varieties {#nontoric}
-------------------------------------------
A **seed** is a collection of data ${{\bf S}}$ of the form $$\begin{aligned}
\label{seed}
{{\bf S}}:=(N,I,E:=\{e_i\}_{i\in I},F\subset I,B),\end{aligned}$$ where $N$ is a lattice of finite rank, $I$ is an index set with $|I| = \operatorname{rank}(N)$, $E$ is a basis for $N$, $F$ is a subset of $I$, and $B$ is a $\bb{Z}$-valued bilinear pairing on $N$. If $i\in F$, we say $e_i$ is **frozen**. Let $I_{\operatorname{uf}}:=I\setminus F$, and let $N_{\operatorname{uf}}$ be the span of $\{e_i\}_{i\in I_{\operatorname{uf}}}$. The pairing $B$ is required to have a **skew-symmetrizable** unfrozen part, meaning that there exists a skew-symmetric pairing $\omega$ on $N_{\operatorname{uf}}$ and a collection of positive rational numbers $\{d_i\}_{i\in I_{\operatorname{uf}}}$ such that $$\begin{aligned}
\label{skewsym}
B(e_i,e_j)=d_i\omega(e_i,e_j)\end{aligned}$$ for all $i,j\in I_{\operatorname{uf}}$. The **Langlands dual** seed ${{\bf S}}^{\vee}$ is obtained by replacing $B$ with its negative transpose $-B^T$ while keeping the rest of the seed data the same.
Let $M:=N^*=\operatorname{Hom}(N,\bb{Z})$. We have two maps $\pi_1, \pi_2:N\rar M$ given by $n\mapsto B(n,\cdot)$ and $n\mapsto B(\cdot,n)$, respectively. For $i=1,2$, let $K_i:=\ker \pi_i$, and let $N_i:=N/K_i$, which we identify with $\pi_i(N) \subset M$. We make the following assumptions, although (4) will be relaxed later:[^7]
\[assume\]
1. $\pi_2(N)$ is saturated in $M$.
2. $\pi_2(e_i)$ is primitive in $M$ for each $i\in F$.
3. $\pi_2(e_i)\neq \pi_2(e_j)$ for distinct $i,j\in F$.
4. There exists a non-singular complete projective fan $\Sigma$ in $N_2 \otimes \bb{R}$ such that
1. The rays of $\Sigma$ are precisely the rays generated by the vectors $\pi_2(e_i)$ for $i\in F$.
2. For each $i\in I_{\operatorname{uf}}$, $\pi_2(e_i)$ is nonzero and is contained in a ray of $\Sigma$, i.e., there is some $j\in F$ such that $\pi_2(e_j)$ points in the same direction as $\pi_2(e_i)$.
3. For distinct $i,j\in I_{\operatorname{uf}}$, the rays generated by $\pi_2(e_i)$ and $\pi_2(e_j)$ are either the same or have no cones of $\Sigma$ in common.
4. Let $\Sigma_i$ denote the set of cones of $\Sigma$ which have codimension $1$ in $N_2\otimes \bb{R}$ and have supports contained in $\pi_1(e_i)^{\perp}$. Then $\pi_2(e_i)$ is contained in the interior of $\bigcup_{\sigma\in \Sigma_i}\sigma$.
Note that Assumption \[assume\](1) implies that $N_1$ and $N_2$ are dual to each other via $\langle \pi_1(n_1),\pi_2(n_2)\rangle = B(n_1,n_2)$. We will write $N_1$ as $\?{M}$ and $N_2$ as $\?{N}$, so $\?{M}=\operatorname{Hom}(\?{N},\bb{Z})$. Let $r=\operatorname{rank}(\?{N})=\operatorname{rank}(\?{M})$.
Let us fix a fan $\Sigma$ as in Assumption \[assume\](4). Let $\operatorname{TV}_M(\Sigma)$ and $\operatorname{TV}_{\?{N}}(\Sigma)$ denote the toric varieties associated to $\Sigma$ when viewed as a fan in $M$ or $\?{N}$, respectively. For each $i\in I$, let $D_{\pi_2(e_i)}$ or simply $D_i$ denote the boundary divisor of $\operatorname{TV}_M(\Sigma)$ or $\operatorname{TV}_{\?{N}}(\Sigma)$ corresponding to the ray through $\pi_2(e_i)$ (whether we mean $D_i\subset \operatorname{TV}_M(\Sigma)$ or $D_i\subset \operatorname{TV}_{\?{N}}(\Sigma)$ should always be clear from context).
For each $i\in I_{\operatorname{uf}}$, let $H_i$ denote the scheme-theoretic intersection of $D_{\pi_2(e_i)}\subset \operatorname{TV}_M(\Sigma)$ with the scheme cut out by $(1+z^{e_i})^{|\pi_2(e_i)|}$. Let $\s{X}_{{{\bf S}},\Sigma}$ denote the scheme obtained by blowing up $\operatorname{TV}_M(\Sigma)$ along $H_i$ for each $i$, let $D^{\s{X}}_{{{\bf S}},\Sigma}$ denote the proper transform of the toric boundary of $\operatorname{TV}_M(\Sigma)$, and let $E^{\s{X}}_i$ denote the exceptional divisor resulting from blowing up $H_i$. Here, the $H_i$’s may intersect in codimension $2$, so we must choose an order in which to perform the blowups, taking proper transforms of the $H_i$’s and $E^{\s{X}}_i$’s at each step. Note that centers of blowups associated to distinct $i,j\in I_{\operatorname{uf}}$ are disjoint if $\pi_2(e_i)$ and $\pi_2(e_j)$ are in different rays, but otherwise the choice of ordering may have a codimension-two effect on the resulting space $\s{X}_{{{\bf S}},\Sigma}$, cf. Remark \[BlowupRmk\].
Consider the exact sequence $$\begin{aligned}
0 \rar K_2 \rar N \stackrel{\pi_2}{\rar} M \stackrel{\lambda}{\rar} K_1^* \rar 0.\end{aligned}$$ The surjection $\lambda$ induces a map $\lambda:\operatorname{TV}_M(\Sigma)\rar T_{K_1^*}$, and this lifts to a map $\lambda_{\s{X}}:\s{X}_{{{\bf S}},\Sigma}\rar T_{K_1^*}$. Let $Y_{{{\bf S}},\Sigma}$, or simply $Y_{{\bf S}}$, denote a general fiber[^8] of $\lambda_{\s{X}}$, and let $D_{{{\bf S}},\Sigma}$ or simply $D_{{\bf S}}$ denote the intersection of this fiber with $D^{\s{X}}_{{{\bf S}},\Sigma}$. Then $(Y_{{{\bf S}}},D_{{{\bf S}}})$ is a smooth log Calabi-Yau orbifold with maximal boundary. Let $E_i:=E^{\s{X}}_{i}\cap Y_{{{\bf S}}}$.
\[BlowupRmk\] We note that a fiber $(Y_{{\bf S}},D_{{\bf S}})$ could alternatively be constructed directly by essentially the same construction used to produce $\s{X}_{{{\bf S}},\Sigma}$. One simply replaces $\operatorname{TV}_M(\Sigma)$ with $\operatorname{TV}_{\?{N}}(\Sigma)$, and replaces and each $H_i$ with $\?{H}_i$, defined to be the scheme-theoretic intersection of $D_{\pi_2(e_i)}\subset \operatorname{TV}_{\?{N}}(\Sigma)$ with the scheme cut out by $(a_i+z^{\pi_1(e_i)})^{|\pi_2(e_i)|}$ for some general $a_i\in {\Bbbk}^*$. Then $E_i$ is the exceptional divisor associated to blowing up $\?{H}_i$. Note that for $\pi_2(e_i)$ and $\pi_2(e_j)$ parallel, the corresponding loci $\?{H}_i$ and $\?{H}_j$ might intersect. But thanks to Assumption \[assume\](4)(d), for general fibers of $\lambda_{\s{X}}$, they do not intersect in codimension-two strata of the boundary. We can therefore apply [@GHK3 Lem. 3.5(1)] to say that the ordering of the blowups only matters up to codimension at most two. This codimension-two ambiguity will not be important for us.
\[fullX\] Since only the unfrozen part of $B$ is required to be skew-symmetrizable, one can always add frozen vectors to obtain a seed ${{\bf S}}'$ such that $Y_{{{\bf S}}'}$ is a compactification of the full cluster $\s{X}$-variety $\s{X}_{{{\bf S}}}$, as opposed to just a compactification of a fiber of $\lambda_{\s{X}}:\s{X}_{{{\bf S}}}\rar T_{K_1^*}$.
These pairs $(Y_{{{\bf S}}},D_{{{\bf S}}})$ are examples of what we call cluster log pairs. In fact, our definition of cluster log pairs allows for more general boundary:
\[clp\] Let ${{\bf S}}=(N,I,E,F,B)$ be a seed ${{\bf S}}$ satisfying Assumptions \[assume\](1)-(3). Suppose there exists another seed $\wt{{{\bf S}}}=(\wt{N},\wt{I},\wt{E},\wt{F},\wt{B})$ satisfying all of Assumptions \[assume\](1)-(4) for some fan $\wt{\Sigma}$, and such the following hold: $N\subset \wt{N}$, $E\subset \wt{E}$, $I\subset \wt{I}$, $F\subset \wt{F}$ with $I_{\operatorname{uf}}=\wt{I}_{\operatorname{uf}}$, and $B=\wt{B}|_N$. Let $\Sigma$ be a complete sub cone-complex of $\wt{\Sigma}$ whose rays are precisely the rays generated by the vectors $\pi_2(e_i)$ for $i\in F$. Furthermore, suppose that the boundary strata of $(Y_{\wt{{{\bf S}}},\wt{\Sigma}},D_{\wt{{{\bf S}}},\wt{\Sigma}})$ associated to cones of $\Sigma\setminus \wt{\Sigma}$ can be blown down to obtain another smooth log Calabi-Yau orbifold with maximal boundary, which we denote $(Y_{{{\bf S}},\Sigma},D_{{{\bf S}},\Sigma})$. A **cluster log pair** is a pair $(Y_{{{\bf S}},\Sigma},D_{{{\bf S}},\Sigma})$ obtained in this way.
In particular, if ${{\bf S}}$ and $\Sigma$ do satisfy Assumptions \[assume\], then we can take $\wt{{{\bf S}}}={{\bf S}}$ and $\wt{\Sigma}=\Sigma$.
\[DMstack\] Note that the blowup loci $H_i$ and $\?{H}_i$ are possibly non-reduced, so even if $\Sigma$ is non-singular, the space $Y_{{{\bf S}},\Sigma}$ may still have orbifold singularities. This is why we work in the generality of Deligne-Mumford stacks.
\[AllTheta\] The constructions of [@GHKK] involve first constructing theta functions on the cluster variety with principle coefficients $\s{A}^{\operatorname{prin}}$ and then specializing to obtain theta functions on[^9] $\s{A}$ or $\s{X}$, cf. [@GHKK §7.2]. Theorem \[MainThm\] applies to determine the theta functions on $\s{X}^{\operatorname{prin}}$ (possibly with some benign modifications to the frozen parts of the seed data), and since $\s{X}^{\operatorname{prin}}$ is isomorphic to $\s{A}^{\operatorname{prin}}$, one can recover all the theta functions of [@GHKK].
Curve classes {#N1}
-------------
Letting $b_i$ denote the map blowing up $\?{H}_i$ as in Remark \[BlowupRmk\], let $C_i=b_i^{-1}(p)$ for a single generic point $p\in \?{H}_i$, so $C_i$ is a curve contained in $E_i$. Taking the proper transform of $C_i$ under any remaining blowups, and then the image under blowdowns of extra boundary components, we get a curve $C_i\subset E_i$ in $Y_{{\bf S}}$ satisfying $$\begin{aligned}
\label{CiEi}
[C_i].[E_i]=-\frac{1}{|\pi_2(e_i)|}.\end{aligned}$$
Let $A_*(Y_{{{\bf S}}})$ denote the integral Chow lattice of the smooth Deligne-Mumford stack $Y_{{{\bf S}}}$, cf. [@EG; @Kr]. Consider $N_1(Y_{{{\bf S}}})=A_1(Y_{{{\bf S}}})$. We see that $N_1(Y_{{{\bf S}}})$ is generated by classes pulled back from $\operatorname{TV}(\wt{\Sigma})$ and then pushed forward from $Y_{\wt{{{\bf S}}}}$ to $Y_{{{\bf S}}}$, together with the classes $|\pi_2(e_i)|[C_i]$ for $i\in I_{\operatorname{uf}}$. Let $\operatorname{NE}(Y_{{{\bf S}}})$ denote the cone in $N_1(Y_{{{\bf S}}})$ generated by classes of effective curves.
The following useful theorem[^10] identifies the lattice $K_2\subset N$ with the lattice $N_1(Y_{{{\bf S}},\Sigma})$.
\[kappa\] There is a unique isomorphism $$\begin{aligned}
\kappa:K_2\risom N_1(Y_{{{\bf S}},\Sigma})\end{aligned}$$ taking a vector $k=\sum_{i\in I} a_ie_i$ to a curve class $[C]_k$ such that $[C]_k.[E_i]=a_i$ for each $i\in I_{\operatorname{uf}}$, and $[C]_k.[D_i]=a_i$ for each $i\in F$.
Let us first assume that ${{\bf S}}=\wt{{{\bf S}}}$ and $\Sigma=\wt{\Sigma}$ in the construction of $(Y_{{{\bf S}},\Sigma},D_{{{\bf S}},\Sigma})$ as in Definition \[clp\]. Let $$\pi_{{{\bf S}}}:Y_{{{\bf S}}}\rar \operatorname{TV}(\Sigma):=\operatorname{TV}_{\?{N}}(\Sigma)$$ denote the blowdown map. By standard toric geometry (cf. [@Fult §3.4]), $A_{n-1}(\operatorname{TV}(\Sigma))$ is spanned by the classes of its boundary divisors. Hence, $A_{n-1}(Y_{{{\bf S}},\Sigma})$ is spanned by the boundary divisor classes $[D_i]$ with $i\in F$, together with the exceptional divisor classes $[E_i]$ with $i\in I_{\operatorname{uf}}$. So a class $[C]\in N_1(Y_{{{\bf S}},\Sigma})$ is indeed uniquely determined by its intersections with the classes $[D_i]$, $i\in F$ and $[E_i]$, $i\in I_{\operatorname{uf}}$.
We now check that such a $[C]_k$ exists for each $k\in K_2$. By definition, $\sum_{i\in I} a_ie_i\in K_2$ gives a relation $\sum_{i\in I} a_i\pi_2(e_i) = 0$, and such a relation corresponds to a class $[\?{C}]_k\in N_1(\operatorname{TV}(\Sigma))$ which, for each ray $\rho \in \Sigma$, satisfies $$\begin{aligned}
\label{Ckbar}
[\?{C}]_k.[D_{\rho}]=\sum_{\{i\in I:\pi_2(e_i)\in \rho\}} a_i|\pi_2(e_i)|.
\end{aligned}$$ Now let $$\begin{aligned}
[C]_k:=\pi_{{{\bf S}}}^*[{\?{C}}]_k-\sum_{i\in I_{\operatorname{uf}}} a_i|\pi_2(e_i)|[C_i]. \end{aligned}$$ Using , it is straightforward to check that $[C]_k$ has the desired intersection multiplicities with $[E_i]$ or $[D_i]$ for each $i\in I$.
Next, we want to check that $\kappa$ is surjective. It is clear that the image includes the pullback of any class from $N_1(\operatorname{TV}(\Sigma))$, so we just have to check that the image includes the classes $|\pi_2(e_i)|[C_i]$ for each $i\in I_{\operatorname{uf}}$. By Assumption \[assume\](2) and (4)(b), there is some $j\in F$ and $a\in \bb{Z}_{>0}$ such that $\pi_2(e_i)=a \pi_2(e_j)$. For $k=-e_i+ae_j$, the class $[\?{C}]_k$ is just $0\in N_1(\operatorname{TV}(\Sigma))$, so we have $$\begin{aligned}
\kappa(-e_i+ a e_j)=|\pi_2(e_i)|[C_i],\end{aligned}$$ as desired.
We have thus proved the claim when ${{\bf S}}=\wt{{{\bf S}}}$, $\Sigma=\wt{\Sigma}$. For the more general situation, let $\wt{\kappa}$ and $\wt{K_2}$ denote the appropriate data associated to $\wt{{{\bf S}}}$ and $\wt{\Sigma}$. Consider the blowdown $$\pi_{\Sigma}:Y_{\wt{{{\bf S}}},\wt{\Sigma}}\rar Y_{{{\bf S}},\Sigma}.$$ Note that the inclusion $N\subset \wt{N}$ identifies $K_2$ with $\wt{K}_2\cap N$. It follows from the projection formula for Chow rings that $\pi_{\Sigma}^* N_1(Y_{{{\bf S}},\Sigma})$ is a sublattice of $N_1(Y_{\wt{{{\bf S}}},\wt{\Sigma}})$ and consists of those classes which have $0$ intersection with the boundary divisors $D_i$ for each $i\in \wt{F}\setminus F$. Hence, $$(\wt{\kappa})^{-1}(\pi_{\Sigma}^* N_1(Y_{{{\bf S}},\Sigma})) = K_2.$$ We now see (using the projection formula again) that the desired map $\kappa$ is $$\begin{aligned}
\kappa:=(\pi_{\Sigma})_* \circ \wt{\kappa}|_{K_2}.\end{aligned}$$
Note that for each $i\in I_{\operatorname{uf}}$, the class $|\pi_2(e_i)|[C_i]\in N_1(Y_{{{\bf S}}})$ generates an extremal ray of the Mori cone $\operatorname{NE}(Y_{{{\bf S}}})$. Let $\operatorname{NE}(Y_{{\bf S}})_{{\bf S}}$ denote the localization of $\operatorname{NE}(Y_{{\bf S}})$ obtained by adjoining $-|\pi_2(e_i)|[C_i]$ for each $i\in I_{\operatorname{uf}}$. Let $N^{\oplus}$ denote the submonoid of $N$ spanned by the elements $e_i$, $i\in I$, i.e., $$\begin{aligned}
\label{Noplus}
N^{\oplus} = \left\{\sum_{i\in I_{\operatorname{uf}}} a_ie_i| a_i\in \bb{Z}_{\geq 0} \mbox{ for each } i\in I\right\}.\end{aligned}$$ Let $K_2^{\oplus}:=K_2\cap N^{\oplus}$.
\[NEloc\] $\kappa(K_2^{\oplus})\subset \operatorname{NE}(Y_{{\bf S}})_{{\bf S}}$.
This follows from the proof of Theorem \[kappa\], noting that when each $a_i\geq 0$, the class $[\?{C}]_k$ of can be represented by an effective curve in $\operatorname{TV}(\Sigma)$.
\[Lang\] The **Langlands dual $\s{X}$-space** is the space $\s{X}_{{{\bf S}}^{\vee}}$ associated to the Langlands dual seed ${{\bf S}}^{\vee}$ (with the boundary removed). It comes with a map $\lambda_{\s{X}}:\s{X}_{{{\bf S}}^{\vee}}\rar T_{K_2^*}=\operatorname{Spec}{\Bbbk}[K_2]$, which by Theorem \[kappa\] can be viewed as $$\begin{aligned}
\lambda_{\s{X}}:\s{X}_{{{\bf S}}^{\vee}} \rar \operatorname{Spec}{\Bbbk}[N_1(Y_{{{\bf S}},\Sigma})].\end{aligned}$$ By adding appropriate boundary strata, this can be extended to a family $$\begin{aligned}
\lambda_{\s{X}}:\?{\s{X}}_{{{\bf S}}^{\vee}} \rar \operatorname{Spec}{\Bbbk}[\operatorname{NE}(Y_{{{\bf S}},\Sigma})],\end{aligned}$$ and then one can define the completion $\wt{\s{X}}_{{{\bf S}}^{\vee}}$ of $\?{\s{X}}_{{{\bf S}}^{\vee}}$ at the boundary.
Scattering diagrams and theta functions
=======================================
Review of scattering diagrams and theta functions {#ScatterIntro}
-------------------------------------------------
We next recall the notion of a scattering diagram and the construction of theta functions. We continue to assume we have the data of a seed ${{\bf S}}$ and fan $\Sigma$ as in §\[nontoric\].
Recall $N^{\oplus}\subset N$ as in , and let $N^+:=N^{\oplus}\setminus \{0\}$. Consider ${\Bbbk}[N^{\oplus}]$. Let $\f{m}$ denote the unique maximal monomial ideal of ${\Bbbk}[N^{\oplus}]$, that is, the ideal generated by all $z^n$ with $n\in N^+$. For each $k\in \bb{Z}_{>0}$, we can take the quotient ${\Bbbk}[N^{\oplus}]/\f{m}^k$, and we thus define the inverse limit ${\Bbbk}\llb N^{\oplus}\rrb:=\varprojlim_k {\Bbbk}[N^{\oplus}]/\f{m}^k$.
Now let $$\begin{aligned}
\label{P}
P:=N^{\oplus}+\kappa^{-1}(\operatorname{NE}(Y_{{{\bf S}}}))\subset N.\end{aligned}$$ Let $A:={\Bbbk}[P]$ with its obvious $P$-grading. Let $\wh{A}$ denote the $N^+$-adic completion of ${\Bbbk}[P]$. I.e., for each $k\in \bb{Z}_{\geq 1}$, denote $$\begin{aligned}
kN^+:=\{n_1+\ldots,n_k\in P|n_i\in N^+ \mbox{ for each $i=1,\ldots,k$}\}.\end{aligned}$$ Then $\wh{A}$ consists of Laurent series $\sum_{p\in P} a_p z^p$ such that, for each $k\in \bb{Z}_{\geq 1}$, $a_p=0$ for all but finitely many $p\in P\setminus kN^+$. Equivalently, $\wh{A}={\Bbbk}\llb N^{\oplus}\rrb \otimes_{{\Bbbk}[N^{\oplus}]} {\Bbbk}[P]$, or if we define $A_k:={\Bbbk}[N^{\oplus}]/\f{m}^k \otimes_{{\Bbbk}[N^{\oplus}]} {\Bbbk}[P]$, then $\wh{A}=\varprojlim_k A_k$.
Similarly, let $K_2^+:=K_2^{\oplus}\setminus \{0\}$, where we recall $K_2^{\oplus}=K_2\cap N^{\oplus}$. Let $R:={\Bbbk}[K_2\cap P]$, and let $\wh{R}={\Bbbk}\llb K_2^{\oplus}\rrb \otimes_{{\Bbbk}[K_2^{\oplus}]} R$ be the $K_2^+$-adic completion of $R$. We sometimes view $A$ and $\wh{A}$ as algebras over $R$ and $\wh{R}$, respectively.
A **wall** $(\f{d},f)$ in $\?{N}_{\bb{R}}$ is the data of a function $$f\in {\Bbbk}\llb z^n\rrb\subset {\Bbbk}\llb N^{\oplus}\rrb$$ for some $n\in N^+$, and a convex (but not necessarily strictly convex) rational polyhedral cone $$\f{d}\subset \?{N}_{\bb{R}}$$ such that $f\equiv 1$ modulo $\f{m}$ and such that the linear span of $\f{d}$ contains $\pi_2(n)$. The vector $-\pi_2(n)\in \?{N}$ is called the **direction** of the wall. The wall is called **incoming** if $\f{d}$ contains $\pi_2(n)$ and **outgoing** otherwise.
A **scattering diagram** $\f{D}$ is a set of walls in $\?{N}_{\bb{R}}$ such that for each $k >0$, there are only finitely many walls $(\f{d},f)$ with $f$ not equivalent to $1$ modulo $\f{m}^k$. Given $\f{D}$, we let $\f{D}^k$ denote the finite scattering diagram consisting of walls $(\f{d},f) \in \f{D}$ for which $f\not\equiv 1$ modulo $\f{m}^k$.
Denote $\operatorname{Supp}(\f{D}):= \bigcup_{(\f{d},f)\in \f{D}} \f{d}$, and $$\begin{aligned}
\operatorname{Joints}(\f{D}):= \bigcup_{(\f{d},f)\in \f{D}} \partial \f{d} \cup \bigcup_{\substack{(\f{d}_1,f_1),(\f{d}_2,f_2)\in \f{D}\\
\operatorname{codim}(\f{d}_1\cap \f{d}_2\subset \?{N}_{\bb{R}}) = 2}} \f{d}_1\cap \f{d}_2.\end{aligned}$$ We will sometimes denote a wall $(\f{d},f)$ by just $\f{d}$. On the other hand, we may write $(\f{d},f\in {\Bbbk}\llb z^n\rrb)$ if we want to explicitly indicate the data of $n$.
Consider a smooth immersion $\gamma:[0,1]\rar \?{N}_{\bb{R}}\setminus \operatorname{Joints}(\f{D})$ with endpoints not in $\operatorname{Supp}(\f{D})$ which is transverse to each wall of $\f{D}$ it crosses. Let $(\f{d}_i,f_i\in {\Bbbk}\llb z^{n_i}\rrb)$, $i=1,\ldots, s$, denote the walls of $\f{D}^{k}$ crossed by $\gamma$, and say they are crossed at times $0<t_1\leq \ldots \leq t_s<1$, respectively (the ambiguity in the labelling when $t_i=t_{i+1}$ is unimportant). Define a ring automorphism $\theta_{\f{d}_i}$ of ${\Bbbk}[N^{\oplus}]/\f{m}^k$ which, for each $n\in N^{\oplus}$, acts via $$\begin{aligned}
\label{WallCross}
\theta_{\f{d}_i}(z^p):=z^pf_i^{\langle u_i, \pi_2(p)\rangle},\end{aligned}$$ where $u_i$ is the primitive element of $\f{d}_i^{\perp}\subset \?{M}$ which is positive on $-\gamma'(t_i)$. Let $\theta_{\gamma,\f{D}}^k:=\theta_{\f{d}_s} \circ \cdots \circ \theta_{\f{d}_1}\in \operatorname{Aut}(A_k)$. Finally, define the path-ordered product $$\begin{aligned}
\theta_{\gamma,\f{D}}:= \varprojlim_k \theta_{\gamma,\f{D}}^k \in \operatorname{Aut}(\wh{A}).\end{aligned}$$
We say two scattering diagrams $\f{D}$ and $\f{D}'$ are **equivalent** if $\theta_{\gamma,\f{D}} = \theta_{\gamma,\f{D}'}$ for each smooth immersion $\gamma$ as above. One says $\f{D}$ is **consistent** if each $\theta_{\gamma,\f{D}}$ depends only on the endpoints of $\gamma$.
The following theorem of Gross-Siebert and Kontsevich-Soibelman is fundamental to the theory of scattering diagrams.
\[KSGS\] Let $\f{D}_{\operatorname{in}}$ be a finite scattering diagram in $\?{N}_{\bb{R}}$ whose only walls have full hyperplanes as their supports. Then there is a unique-up-to-equivalence scattering diagram $\f{D}$, also denoted $\operatorname{Scat}(\f{D}_{\operatorname{in}})$, such that $\f{D}$ is consistent, $\f{D} \supset \f{D}_{\operatorname{in}}$, and $\f{D}\setminus \f{D}_{\operatorname{in}}$ consists only of outgoing walls.
Let us now fix a consistent scattering diagram $\f{D}$ in $\?{N}_{\bb{R}}=N_2\otimes \bb{R}$. Let $\varphi:\?{N}_{\bb{R}}\rar N_{\bb{R}}$ denote the integral $\Sigma$-piecewise-linear section of $\pi_2$ determined by setting $$\begin{aligned}
\label{varphi}
\varphi(\pi_2(e_i)):=e_i\end{aligned}$$ for each $i\in F$ and then extending linearly over the cones of $\Sigma$.
\[broken line\] Let $p \in \?{N}$, $Q\in \?{N}_{\bb{R}}\setminus \operatorname{Supp}(\f{D})$. A broken line $\gamma$ with ends $(p,Q)$ is the data of a continuous map $\gamma:(-\infty,0]\rar \?{N}_{\bb{R}}\setminus \operatorname{Joints}(\f{D})$, values $t_0 \leq t_1 \leq \ldots \leq t_{\ell-1} < t_{\ell} = 0$, and for each $i=0,\ldots,\ell$, an associated element $c_iz^{v_i}\in \wh{A}$, such that:
- $\gamma(0)=Q$.
- For $i=1\ldots, \ell$, $\gamma'(t)=-\pi_2(v_i)$ for all $t\in (t_{i-1},t_{i})$. Similarly, $\gamma'(t)=-\pi_2(v_0)$ for all $t\in (-\infty,t_0)$.
- $c_0=1$ and $v_0=\varphi(p)$.
- For $i=0,\ldots,\ell-1$, $\gamma(t_i)\in \f{d}_i$ for some wall $(\f{d}_i,f_i)\in \f{D}$, and $c_{i+1}z^{v_{i+1}}\neq c_iz^{v_i}$ is a monomial term in the power series expansion of $c_iz^{v_i}f_i^{\langle u_i,v_i\rangle}$, where $u_i$ is the primitive element of $\f{d}_i^{\perp}$ which is positive on $v_i$ (i.e., $c_iz^{v_i}f_i^{\langle u_i,v_i\rangle}$ is $\theta_{\f{d}_i}(c_iz^{v_i})$ as defined in for a smoothing of $\gamma$).
Fix a generic point $Q \in \?{N}_{\bb{R}}\setminus \operatorname{Supp}(\f{D})$. For any $p\in \?{N}$, we define a theta function $$\begin{aligned}
\label{vartheta-dfn}
\vartheta_{p,Q}:=\sum_{\operatorname{Ends}(\gamma)=(p,Q)} c_{\gamma}z^{n_{\gamma}} \in \wh{A}.\end{aligned}$$ Here, the sum is over all broken lines $\gamma$ with ends $(p,Q)$, and $c_{\gamma}z^{n_{\gamma}}$ denotes the monomial attached to the final straight segment of $\gamma$. In particular, we define $\vartheta_{0,Q}:=1$. One can prove that these functions $\vartheta_{p,Q}$ form a well-defined topological $\wh{R}$-module basis for $\wh{A}$, hence also for the $\wh{R}$-subalgebra $\wh{A}_{\Theta,Q}\subset \wh{A}$ which they generate, cf. [@Man3 Prop 2.14].
Furthermore, if $Q$ and $Q'$ are two generic points in $\?{N}_{\bb{R}}\setminus \operatorname{Supp}(\f{D})$, and if $\gamma$ is a smooth path from $Q$ to $Q'$ avoiding $\operatorname{Joints}(\f{D})$, then an important result of [@CPS] says that $\vartheta_{p,Q'}=\theta_{\gamma,\f{D}}(\vartheta_{p,Q})$. Hence, as an abstract algebra, $\wh{A}_{\Theta,Q}$ is independent of $Q$, and so we denote it by just $\wh{A}_{\Theta}$. Similarly, we denote $\vartheta_{p,Q}\in \wh{A}_{\Theta,Q}=\wh{A}_{\Theta}$ by simply $\vartheta_p$.
Given $f=\sum_{p\in \?{N}} c_p\vartheta_p\in \wh{A}_{\Theta}$, define $\operatorname{Tr}(f):=c_0\in \wh{R}$. This determines a symmetric multilinear $s$-point function $\operatorname{Tr}^s:\wh{A}_{\Theta}^s \rar \wh{R}$, $$\begin{aligned}
\label{Trs}
\operatorname{Tr}^s(f_1,\ldots,f_s):=\operatorname{Tr}(f_1\cdots f_s).\end{aligned}$$ The following lemma says that these $\operatorname{Tr}^s$ are sufficient to determine the entire multiplication structure.
\[nondegen\] $\operatorname{Tr}^2$ is non-degenerate as a symmetric $\wh{R}$-bilinear pairing on the $\wh{R}$-module $\wh{A}_{\Theta}$. Thus, given $\wh{A}_{\Theta}$ as a module over $\wh{R}$ topologically generated by $\{\vartheta_p\}_{p\in \?{N}}$, the multiplication rule giving the $\wh{R}$-algebra structure is uniquely determined by $\operatorname{Tr}^2$ and $\operatorname{Tr}^3$.
We will prove, for a certain $\f{D}_{\operatorname{in}}$, that $\operatorname{Tr}^s$ as defined here is given by the $s$-point function of . Specifically, the initial scattering diagram $\f{D}^{{\bf S}}_{\operatorname{in}}$ with which we shall work is $$\begin{aligned}
\label{Din}
\f{D}^{{\bf S}}_{\operatorname{in}}:=\left\{\left(\pi_1(e_i)^{\perp},(1+z^{e_i})^{|\pi_1(e_i)|}\right) : i\in I_{\operatorname{uf}}\right\},\end{aligned}$$ and we denote $$\begin{aligned}
\f{D}^{{\bf S}}:=\operatorname{Scat}(\f{D}^{{\bf S}}_{\operatorname{in}})\end{aligned}$$
Effectiveness of curve classes {#EffCurve}
------------------------------
From the statement that the theta functions form a topological $\wh{R}$-module basis for $\wh{A}_{\Theta}$, we know that for any $p,q\in \?{N}$, we can write $$\begin{aligned}
\vartheta_p\vartheta_q=\sum_r c_{pqr} \vartheta_r\end{aligned}$$ for some collection of “structure constants” $c_{pqr}\in \wh{R}$. Recall that $\wh{R}={\Bbbk}\llb K_2^{\oplus}\rrb \otimes_{{\Bbbk}[K_2^{\oplus}]} {\Bbbk}[K_2\cap P]$. By Lemma \[NEloc\], $\kappa(K_2^{\oplus}) \subset \operatorname{NE}(Y_{{{\bf S}}})_{{{\bf S}}}$, hence $\kappa(K_2\cap P) \subset \operatorname{NE}(Y_{{{\bf S}}})_{{{\bf S}}}$ as well. Hence, each $c_{pqr}$ is a formal sum of monomials of the form $c_{pqrs}z^s$ with $c_{pqrs}\in {\Bbbk}$ and $s\in \operatorname{NE}(Y_{{{\bf S}}})_{{{\bf S}}}$.
We would like to show that the localization of the Mori cone here is in fact not necessary, i.e., we want to show that $c_{pqrs}=0$ unless $s\in \operatorname{NE}(Y_{{{\bf S}}})\subset \operatorname{NE}(Y_{{{\bf S}}})_{{{\bf S}}}$. It then follows that the ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$-submodule $A_{\Theta}$ of $\wh{A}_{\Theta}$ spanned topologically by the theta functions is in fact a ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$-subalgebra.
For this, we take advantage of the operation of mutation. Namely, for the element $e_i\in E$ from our seed ${{\bf S}}$, we consider the birational map $\mu_i:T_{\?{N}} \dashrightarrow T_{\?{N}}$, $z^{\pi_1(n)}\mapsto z^{\pi_1(n)}(1+z^{\pi_1(e_i)})^{B(e_i,n)}$. [@GHK3] shows that this map can be interpreted geometrically (up to codimension $2$) as taking the blowup of $\?{H}_i$ as in Remark \[BlowupRmk\], followed by taking the blowdown of a certain locus $\wt{F}_i\rar \?{H}_i'$ as in Figure \[mutation-fig\]. Let $C_i'$ denote a generic fiber of the blowdown $\wt{F}_i\rar \?{H}_i'$, so $|\pi_2(e_i)|[C_i']$ generates another extremal ray of $\operatorname{NE}(Y_{{{\bf S}},\Sigma})$. Let $\operatorname{NE}(Y_{{{\bf S}}})_{\mu_i({{\bf S}})}$ denote the localization of $\operatorname{NE}(Y_{{{\bf S}}})$ obtained by adjoining $-|\pi_2(e_j)|[C_j]$ for $j\in I_{\operatorname{uf}}\setminus \{i\}$, along with $-|\pi_2(e_i)|[C'_i]$.
There is a well-known seed-mutation which associates a new seed ${{\bf S}}_i$ to each $i\in I$ above. We can use this to define a different scattering diagram $\f{D}^{\mu_i({{\bf S}})}$ and an associated algebra of theta functions. It follows from [@GHKK Thm 1.24] that the resulting theta functions have the same multiplication rule as before. Thus, for a coefficient $c_{pqrs}$ of theta the function multiplication to be nonzero, we must have $$\begin{aligned}
\label{sNE}
s\in \operatorname{NE}(Y_{{{\bf S}}})_{{\bf S}}\cap \bigcap_{i\in I_{\operatorname{uf}}} \operatorname{NE}(Y_{{{\bf S}}})_{\mu_i({{\bf S}})}.\end{aligned}$$ Since the classes $|\pi_2(e_i)|[C_i]$ and $|\pi_2(e_i)|[C'_i]$ for $i\in I_{\operatorname{uf}}$ are all distinct and extremal in $\operatorname{NE}(Y_{{{\bf S}}})$, the intersection is just $\operatorname{NE}(Y_{{{\bf S}}})$, as desired. We have thus proven the following:
\[NECoeff\] The theta functions $\{\vartheta_p\}_{p\in \?{N}}$ form a topological ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$-module basis for a ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$-algebra $A_{\Theta}$.
\[LangRmk\] By [@GHK3 §3.2], $\s{X}_{{{\bf S}}^{\vee}}$ is covered up to codimension $2$ by the initial cluster and the mutation-adjacent clusters. The same then follows for $\?{\s{X}}_{{{\bf S}}^{\vee}}$ and $\wt{\s{X}}_{{{\bf S}}^{\vee}}$. The birational automorphisms gluing these clusters in $\wt{\s{X}}_{\wh{{{\bf S}}}}$ are precisely the automorphisms associated to crossing the walls of $\f{D}^{{{\bf S}}}_{\operatorname{in}}$. Thus, the theta functions are realized as global regular functions on $\wt{\s{X}}_{{{\bf S}}^{\vee}}$.
Now suppose that the theta functions in fact generate a polynomial algebra over ${\Bbbk}[\operatorname{NE}(Y_{{{\bf S}}})]$. The identification of this algebra with $\Gamma(\?{\s{X}}_{{{{\bf S}}}^{\vee}},\s{O}_{\?{\s{X}}_{{{{\bf S}}}^{\vee}}})$ then follows from [@GHKK Thm. 0.3].
Theta functions, tropical curves, and log GW invariants of toric varieties {#Review}
==========================================================================
Tropical description of theta functions {#ScatTheta}
---------------------------------------
In this subsection we summarize some results from [@Man3] which express the theta functions in terms of certain counts of tropical curves. We work with a fixed a seed ${{\bf S}}$.
### The module of log derivations {#logDerivations}
Our setup for describing scattering diagrams in §\[ScatterIntro\] is a simplified version of the more general setup used in [@Man3]. We briefly explain how to translate between the two setups so that we can apply the results of [@Man3] to the setup here. See also [@Man3 Examples 2.1(i), 2.7(i), and 3.4(i)] for specializations of the general setup there to our situation, as well as [@Man3 Rmk. 2.3(i)] for an explanation of an additional difference between the setups (roughly, supports of walls here are equal to $\pi_2$ of the supports of walls there).
Recall that $\?{M}:=\pi_1(N)\subset M$ and $R:={\Bbbk}[K_2\cap P]$ where $K_2=\ker(\pi_2)$. As in [@GPS §1.1], consider the module of log derivations $\Theta_K(N^{\oplus})$ defined by $$\begin{aligned}
\Theta_K(N^{\oplus}):={\Bbbk}[N^{\oplus}]\otimes_{\bb{Z}}\?{M}\end{aligned}$$ with action on ${\Bbbk}[N^{\oplus}]$ via $R$-derivations defined by $$\begin{aligned}
f\otimes m(z^n):=f\langle n,m\rangle z^n.\end{aligned}$$ We will write $f\otimes m$ as $f\partial_m$. $\Theta_K(N^{\oplus})$ forms a Lie algebra with bracket $[a,b]:=ab-ba$, where multiplication means composition of derivations. In particular, $$\begin{aligned}
[z^{n_1}\partial_{m_1},z^{n_2}\partial_{m_2}]=z^{n_1+n_2}\partial_{\langle n_2,m_1\rangle m_2-\langle n_1,m_2\rangle m_1}.\end{aligned}$$ For each $n\in N^+$, let $\f{h}_n$ be the submodule of $\Theta_K(N^{\oplus})$ spanned by the element $z^n\partial_{\pi_1(n)}$. One easily checks that $\f{h}:=\bigoplus_{n\in N^+} \f{h}_n$ is a Lie subalgebra of $\Theta_K(N^{\oplus})$. Let $\wh{\f{h}}$ denote the completion of $\f{h}$ associated to the $N^+$-grading. For $n\in N^{+}$ primitive, let $\f{h}_n^{\parallel}$ denote the Lie subalgebra of $\wh{\f{h}}$ spanned topologically by elements of $\f{h}_{kn}$ for $k\in \bb{Z}_{>0}$.
Now, in the setup of [@Man3], our walls would be written as $(m,\f{d},g)$, where $m$ is an element of $\?{M}$ up to positive scaling such that the support $\f{d}$ is contained in $m^{\perp}\subset \?{N}_{\bb{R}}$, and $g$ is an element of $\f{h}_n^{\parallel}$ for some primitive $n\in m^{\perp}\subset N$ (recall that $m\in \?{M}\subset M$, so we can view $m^{\perp}$ as living in $\?{N}_{\bb{R}}$ or $N$). More precisely, consider a wall given in our setup by $(\f{d},f\in {\Bbbk}\llb n\rrb)$, and let $u$ be a primitive element of $\f{d}^{\perp}\cap \?{M}$. Then in the setup of [@Man3], this wall would be expressed as $(u,\f{d},\log(f)\partial_{u})$, cf. [@Man3 Ex. 2.8(i)]. The wall-crossing automorphism $\theta_{\f{d}}$ as in is then realized as the action of $\exp \left[\operatorname{sgn}\langle u,-\gamma'(t)\rangle \operatorname{ad}_{\log(f)\partial_u}\right]$.
In particular, recall the scattering diagram $\f{D}_{\operatorname{in}}:=\left\{\left(\pi_1(e_i)^{\perp},(1+z^{e_i})^{|\pi_1(e_i)|}\right) : i\in I_{\operatorname{uf}}\right\}$ from . In the setup of [@Man3], this could be written as $$\begin{aligned}
\label{DinMan3}
\f{D}_{\operatorname{in}}=\left\{\left( \pi_1(e_i),\pi_1(e_i)^{\perp},g_i:=\sum_{w=1}^{\infty} w \frac{(-1)^{w+1}}{w^2} z^{we_i} \partial_{\pi_1(e_i)}\right) : i \in I_{\operatorname{uf}} \right\}.\end{aligned}$$ Note that $\pi_1(e_i)=d_i\omega(e_i,\cdot)$, so $g_i$ is in fact in $\wh{\f{h}}$. Denoting $a_{iw}:=w\frac{(-1)^{w+1}}{w^2}$ (pulling a $w$-factor in front like this will be convenient in §\[DegenS\]), we can rewrite $g_i$ as $g_i=\sum_{w=1}^{\infty} g_{iw}$ where $$\begin{aligned}
\label{gi}
g_{iw}:= a_{iw} z^{we_i} \partial_{\pi_1(e_i)}.\end{aligned}$$
### Tropical Gromov-Witten invariants {#TropGW}
We now recall some background on tropical curves. It is convenient to work in greater generality than we will need.
Let $\?{\Gamma}$ be the topological realization of a connected finite tree without bivalent vertices, and denote the complement of the $1$-valent vertices by $\Gamma$. Let $\Gamma^{[0]}$, $\Gamma^{[1]}$, and $\Gamma^{[1]}_{\infty}$ denote the vertices, edges, and unbounded edges of $\Gamma$, respectively. We equip $\Gamma$ with a weight function $w:\Gamma^{[1]}\rar \bb{Z}_{\geq 0}$ such that if $w(E)=0$, then $E\in \Gamma^{[1]}_{\infty}$. A marking of $\Gamma$ is a bijection $\epsilon:J\rar \Gamma^{[1]}_{\infty}$ for some index set $J$. We denote $E_j:=\epsilon(j)$. Let $J^{\circ}\subset J$ denote the set of $j\in J$ for which $w(E_j)=0$, and let $J':=J\setminus J^{\circ}$.
\[TropCurveDfn\] A (genus $0$) parameterized tropical curve $(\Gamma,h)$ (in $\?{N}_{\bb{R}}$) is data $\Gamma$, $w$, $\epsilon$ as above (the weight and marking are suppressed in the notation), along with a continuous map $h:\Gamma\rar \?{N}_{\bb{R}}$ such that
- For each $E\in \Gamma^{[1]}$ with $w(E)>0$, $h|_E$ is a proper embedding into an affine line with rational slope. If $w(E)=0$, then $h(E)$ is a point.
- **The balancing condition**: For any edge $E\ni V$ with $w(E)>0$, denote by $u_{(V,E)}$ the primitive integral vector emanating from $h(V)$ into $h(E)$. Then $$\begin{aligned}
\label{balance}
\sum_{E\ni V} w(E) u_{(V,E)} =0.\end{aligned}$$
For non-compact edges $E_i\ni V$, we may denote $u_{(V,E_i)}$ as simply $u_{E_i}$. Similarly, we may write $u_E$ in place of $u_{(V,E)}$ if $V$ is clear or if the direction only matters up to scaling.
An isomorphism of marked parameterized tropical curves $(\Gamma,h)$ and $(\Gamma',h')$ is a homeomorphism $\Phi:\Gamma\rar \Gamma'$ respecting the weights and markings such that $h=h'\circ \Phi$. A [**tropical curve**]{} is then defined to be an isomorphism class of parameterized marked tropical curves. We will let $(\Gamma,h)$ denote the isomorphism class it represents, and we will often abbreviate this as simply $\Gamma$ or $h$.
A **tropical disk** is defined in the same way except that there is an edge $E_0\in \Gamma^{[1]}_{\infty}$ of possibly positive weight which is contracted by $h$. The associated label in $J$ is viewed as being in $J^{\circ}$, not $J'$. Let $V_0$ denote the vertex of $E_0$. The balancing condition is still required to hold at the vertex $V_0$ of $E_0$ for some uniquely determined primitive vector $u_{(V_0,E_0)}\in \?{N}$ (or for $u_{(V_0,E_0)}=0$ if $w(E_0)=0$, in which case the tropical disk can be viewed as a tropical curve).
Let $\operatorname{val}(V)$ denote the number of edges containing $V$ (the valence of $V$). Let $\operatorname{Flags}(\Gamma)$ denote the set of flags $(V,E)$, $V\in E$, of $\Gamma$. The [**type**]{} of a tropical curve or disk is the data of $\Gamma$, $w$, and $\epsilon$, along with the data of the map $u:\operatorname{Flags}(\Gamma)\rar \?{N}$, $(V,E)\mapsto u_{(V,E)}$. The **degree** $\Delta$ of a tropical curve or disk is the data of $J$ along with the corresponding map $\Delta:J\rar \?{N}$, $\Delta(j)=w(E_j)u_{E_j}$. In the case of a tropical disk, we say that $\Delta$ also remembers which edge is the special one $E_0$ which is contracted despite possibly having positive weight.
\[Constraints\] An affine constraint ${{\bf A}}$ is a tuple $(A_j)_{j\in J}$ of affine subspaces of $\?{N}_{\bb{R}}$ with rational slope, each equipped with a weight[^11] $w(A_j)\in \bb{Z}_{>0}$. A tropical curve or disk $(\Gamma,h)$ matches the constraint ${{\bf A}}$ if $h(E_j)\subset A_j$ for all $j\in J$. Now consider a map $\Psi:J^{\circ}\rar \bb{Z}_{\geq 0}$, denoting $s_j:=\Psi(j)$. We say $(\Gamma,h)$ satisfies the $\psi$-class conditions $\Psi$ if $$\begin{aligned}
\label{Psidfn}
\operatorname{val}(V)-3\geq \sum_{\substack{j\in J^{\circ}\\ E_j\ni V}} s_j\end{aligned}$$ for each vertex $V\in \Gamma^{[0]}$. Let $\langle V\rangle$ denote the multinomial coefficient $$\begin{aligned}
\label{V}
\langle V \rangle:= \frac{(\operatorname{val}(V)-3)!}{\prod_{\substack{j\in J^{\circ}\\ E_j\ni V}} s_j!}\end{aligned}$$ and denote $\langle \Gamma \rangle:=\prod_{V\in \Gamma^{[0]}} \langle V\rangle$.
Let $\f{T}_{0,\Delta}({{\bf A}},\Psi)$ denote the space of genus $0$ degree $\Delta$ tropical curves or disks which match the constraint ${{\bf A}}$ and satisfy a $\psi$-class condition $\Psi$. By [@MRud Lem 2.14] (which is stated for tropical curves but easily extends to allow for tropical disks), this space is finite (and is an equality for each $V$) for generic translates of the $A_j$’s whenever $$\begin{aligned}
\label{RigidDim}
\sum_{j\in J} \operatorname{codim}(A_j) + \sum_{j\in J^{\circ}} s_j= \#J+r-3. \end{aligned}$$ Under these conditions (i.e., when is satisfied and the $A_i$’s are chosen generically among their translations), $\f{T}_{0,\Delta}({{\bf A}},\Psi)$ and the tropical curves/disks it contains are called **rigid**. So assuming rigidity of $\f{T}_{0,\Delta}({{\bf A}},\Psi)$, after describing how to assign a “multiplicity” $\operatorname{Mult}(\Gamma)\in \bb{Z}_{\geq 1}$ to each $\Gamma\in\f{T}_{0,\Delta}({{\bf A}},\Psi)$, one can define $$\begin{aligned}
\label{GWtropDfn}
\operatorname{GW}_{0,\Delta}^{\operatorname{trop}}({{\bf A}},\Psi):=\sum_{(\Gamma,h)\in \f{T}_{0,\Delta}({{\bf A}},\Psi)} \langle \Gamma \rangle \operatorname{Mult}(\Gamma).\end{aligned}$$ For the cases we care about, $s_j$ will be nonzero for only one $j$, and so since rigidity ensures that is always an equality, $\langle \Gamma \rangle$ will always be $1$. We note that is independent of the generic translates of the $A_j$’s for tropical curves, but for tropical disks we will have to specify $A_0$ more precisely.
### Multiplicities of tropical curves and disks {#MultGWtrop}
The definition of $\operatorname{Mult}(\Gamma)$ used in [@MRud Lem/Def 2.16] (due to [@NS] when there are no $\psi$-classes) is given in a form which is impractical for the applications we consider here. This motivated the paper [@MRudMult], which shows that the same multiplicities can alternatively be computed as follows:
Consider a rigid $(\Gamma,h)\in \f{T}_{0,\Delta}({{\bf A}},\Psi)$. For each $j\in J$, let $\alpha_j$ denote an index-$w(A_j)$ element (unique up to sign) of $\Lambda^{\operatorname{codim}A_j} \?{M} \subset \Lambda^* \?{M}$ whose kernel is parallel to $A_j$, i.e., the contraction $\iota_n(\alpha_j)=0$ if and only if $n$ is parallel to $A_j$. We pick a flow on $\Gamma$ by choosing one vertex $V_0$ to serve as the sink. Using this flow, we will recursively associate an element $\omega_E$ of $\bb{Z}[\?{N}]\otimes \Lambda^* \?{M}$ (determined up to sign) to every edge $E$ of $\Gamma$. For each $j\in J$, we associate the element $$\omega_{E_j}:=z^{w(E_j)u_{E_j}}\otimes \alpha_j$$ to the edge $E_j$. Now consider a vertex $V\neq V_0$ with $E_1,\ldots,E_k$ flowing into $V$ and $E_V$ the unique edge flowing out of $V$. If $\omega_{E_i} = z^{n_i}\otimes \alpha_i$ is the element associated to $E_i$, $i=1,\ldots,k$, we define the element associated to $E_V$ to be $$\begin{aligned}
\label{lk}
\omega_{E_V} := l_k(z^{n_1}\otimes \alpha_1,\ldots,z^{n_k} \otimes \alpha_k) := z^{n_V} \otimes \iota_{n_V}(\alpha_1 \wedge \cdots \wedge \alpha_k),\end{aligned}$$ where $n_V:=n_1+\ldots+n_k$.
Finally, if $E_1,\ldots,E_s$ are the edges containing $V_0$, define $$\begin{aligned}
\omega_0:=\omega_{E_1} \cdots \omega_{E_s} \in 1\otimes \Lambda^{r} \?{M} \subset \bb{Z}[\?{N}]\otimes \Lambda^* \?{M}.\end{aligned}$$ Here, balancing ensures that the first factor is $1$ (the exponents of the monomial terms cancel out), and rigidity ensures that the wedge product is of top degree. Then [@MRudMult Thm. 1.2] says that $$\begin{aligned}
\label{MultDfn}
\operatorname{Mult}(\Gamma)=|\omega_0|\end{aligned}$$ by which we mean the index of $\omega_0$ in $\Lambda^r \?{M}$.
Finally, while [@MRudMult] focused on tropical curves, we note that the above recipe yields a well-defined number $\operatorname{Mult}(\Gamma)$ when $\Gamma$ is a rigid tropical disk as well.
### Theta functions from tropical curve counts {#ThetaFromTrop}
We next describe the specific tropical degrees and conditions used in the main theorems of [@Man3], applied to the scattering diagram $\f{D}:=\operatorname{Scat}(\f{D}_{\operatorname{in}})$ for $\f{D}_{\operatorname{in}}$ the initial scattering diagram associated to a seed ${{\bf S}}=(N,I,E:=\{e_i\}_{i\in I},F,B)$.
Let ${{\bf w}}:=({{\bf w}}_i)_{i\in I_{\operatorname{uf}}}$ be a tuple of weight vectors ${{\bf w}}_i:=(w_{i1},\ldots,w_{il_i})$ with $w_{i1} \leq \ldots \leq w_{il_i}$, $w_{ij}\in \bb{Z}_{>0}$. For $\Sigma_{l_i}$ denoting the group of permutations of $\{1,\ldots,l_i\}$, let $$\operatorname{Aut}({{\bf w}})\subset \prod_{i\in I_{\operatorname{uf}}} \Sigma_{l_i}$$ be the group of automorphisms of the second indices of the weights ${{\bf w}}_i$ which act trivially on ${{\bf w}}$.
Let ${{\bf p}}$ be an $s$-tuple $(p_1,\ldots,p_s)$ of elements of $\?{N}$. For $n\in N$ and $\varphi:\?{N}_{\bb{R}}\rar N_{\bb{R}}$ as in , let $\s{W}_{{{\bf p}}}(n)$ denote the set of weight vectors ${{\bf w}}$ such that $$\begin{aligned}
\sum_{i\in I_{\operatorname{uf}}} \sum_{j=1}^{l_i} w_{ij}e_i +\sum_{k=1}^s \varphi(p_k)= n.\end{aligned}$$ Let $$\begin{aligned}
J_{{{\bf w}},{{\bf p}}}:=\{(i,j)|i\in I_{\operatorname{uf}},j=1,\ldots,l_i\} \cup \{1,\ldots,s\}\cup \{\operatorname{out},\infty\}.\end{aligned}$$ We will often write the pairs $(i,j)$ as simply $ij$.
For ${{\bf w}}\in \s{W}_{{{\bf p}}}(n)$, let $\Delta_{{{\bf w}},{{\bf p}}}$ denote the degree $$\Delta_{{{\bf w}},{{\bf p}}}:J_{{{\bf w}},{{\bf p}}}\rar \?{N}$$ with $\Delta_{{{\bf w}},{{\bf p}}}((i,j)):=w_{ij}\pi_2(e_i)$, $\Delta_{{{\bf w}},{{\bf p}}}(k)=p_k$ for $k=1,\ldots,s$, $\Delta_{{{\bf w}},{{\bf p}}}(\operatorname{out})=-\pi_2(n)$, and $\Delta(\infty)=0$. We view $\Delta_{{{\bf w}},{{\bf p}}}$ as the degree of a tropical disk, with $E_{\operatorname{out}}$ being the special edge $E_0$.
Given a generic point $Q\in \?{N}_{\bb{R}}$ and ${{\bf w}}\in \s{W}_{{{\bf p}}}(n)$, we define the incidence conditions ${{\bf A}}_{{{\bf w}},{{\bf p}},Q}$ as follows: Each $A_{ij}$ is taken to be a generic translate of $\f{d}_i=\pi_1(e_i)^{\perp}\subset \?{N}_{\bb{R}}$ with $w(A_{ij})=|\pi_1(e_i)|$. We take $A_k:=\?{N}_{\bb{R}}$ for each $k=1,\ldots,s$ and for $k=\infty$ (i.e., the incidence conditions on the $E_k$’s are trivial), and we take $A_{\operatorname{out}}=Q$. For our $\psi$-class conditions, we define $\Psi_{{{\bf w}},{{\bf p}}}:J_{{{\bf w}},{{\bf p}}}^{\circ}\rar \bb{Z}_{\geq 0}$ by $\Psi_{{{\bf w}},{{\bf p}}}(\operatorname{out})=s-1$ and $\Psi_{{{\bf w}},{{\bf p}}}(k)=0$ for any other $k\in J_{{{\bf w}},{{\bf p}}}^{\circ}$.
In the setup of [@Man3], the marked point $\infty$ is not included, but $\Psi_{{{\bf w}},{{\bf p}}}(\operatorname{out})$ is taken to be $s-2$ instead of $s-1$. This change in the $\psi$-class condition forces the valence of the vertex $V_0\in E_{\operatorname{out}}$ to be higher by $1$, thus forcing the extra contracted edge $E_{\infty}$ to contain $V_0$, so there is an obvious bijection between the tropical curves/disks in the two setups. This modification of the tropical data corresponds to the geometric modification discussed in Remark \[logFCA\], and it allows us to avoid treating the $s=1$ case separately.
With these conditions and ${{\bf w}}\in \s{W}_{{{\bf p}}}(n)$, $\f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})$ is finite, so we can count its elements using the multiplicities $\operatorname{Mult}$ introduced above, yielding numbers $\operatorname{GW}^{\operatorname{trop}}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})$. Alternatively, [@Man3 §3.1.2] defines slightly different multiplicities $\operatorname{\wt{Mult}}$ (explained below), and with these one defines $$\begin{aligned}
\operatorname{N}^{\operatorname{trop}}_{{{\bf w}},{{\bf p}}}(Q):=\sum_{\Gamma \in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})} \operatorname{\wt{Mult}}(\Gamma) \in \wh{A}.\end{aligned}$$
For $\Gamma \in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})$, we define the multiplicity $\operatorname{\wt{Mult}}(\Gamma)$ by modifying the definition of $\operatorname{Mult}(\Gamma)$ as follows: For the sink of our flow, we use the vertex $V_0$ contained in $E_{\operatorname{out}}$. Then in the recursive construction, we associate to each edge $E$ an element $\omega_E$ of the Lie algebra $A\oplus \f{h}$ for $A$ as in §\[ScatterIntro\] and $\f{h}$ as in §\[logDerivations\]. The bracket is given by $$\begin{aligned}
\label{AhBracket}
[a_1 + h_1,a_2 +h_2]=[a_1,a_2]+h_1\cdot a_2 - h_2\cdot a_2 + [h_1,h_2],\end{aligned}$$ where $\cdot$ is the derivation action of $\f{h}$ on $A$.
To $E_{\operatorname{out}}$ and $E_{\infty}$ we associate the element $1\in A$. To $E_{ij}$ we associate the element $g_{iw_{ij}}$ as defined in . To $E_k$ for $k=1,\ldots,p$ we associate $z^{\varphi(p_k)}\in A$. Then at a vertex $V\neq V_0$ (necessarily trivalent, so with exactly two edges flowing in), instead of applying $l_2$ as in , we simply take the Lie bracket of the two incoming elements (for a certain choice of ordering) in order to produce the outgoing $\omega_{E_V}$. Finally, for each $E\ni V_0$, $\omega_E$ is in fact an element of $A$, and the product $\prod_{E\ni V} \omega_E$ in $A$ is, up to sign, equal to $\operatorname{\wt{Mult}}(\Gamma)$. This element $\operatorname{\wt{Mult}}(\Gamma)\in A$ has the form $ka_{{{\bf w}}}z^{n_{\operatorname{out}}}$ for some nonzero integer $k$, $$\begin{aligned}
a_{{{\bf w}}}:=\prod_{ij} a_{iw_{ij}},\end{aligned}$$ and $n_{\operatorname{out}}:=\sum_{ij} w_{ij}e_i + \sum_{k=1}^s \varphi(p_k)$, i.e., $n_{\operatorname{out}}$ is the element of $N$ such that ${{\bf w}}\in \s{W}_{{{\bf p}}}(n_{\operatorname{out}})$. As explained in [@Man3 Ex. 3.4(i)], the correct sign of $\operatorname{\wt{Mult}}(\Gamma)$ is the one for which $k$ is positive.
Now, in the definition of $\operatorname{Mult}(\Gamma)$, the elements $g_{iw_{ij}}=a_{iw_{ij}}z^{w_{ij}e_i}\partial_{\pi_1(e_i)}$ above would have instead been $z^{w_{ij}\pi_2(e_i)}\partial_{\pi_1(e_i)}$, while the elements associated to $E_k$ for $k=1,\ldots,s$ would have been $z^{p_k}$ instead of $z^{\varphi(p_k)}$. Then in the computation of $\operatorname{Mult}$, the element $\omega_{E_{\operatorname{out}}}$ is $z^{-\pi_2(n_{\operatorname{out}})}$ times a primitive element of $\Lambda^{\operatorname{top}} M$. The element $\omega_{E_{\infty}}$ is still $1$. Note that $\pi_2:N\rar \?{N}$ induces a map $\pi_2:A\oplus \f{h}\rar \bb{Z}[\?{N}]\otimes \Lambda^* \?{M}$ (i.e., applying $\pi_2$ to the exponents), and for $l_2$ as in and any $a,b\in A\oplus \f{h}$, we have $l_2(\pi_2(a),\pi_2(b))=\pi_2([a,b])$ for $[\cdot,\cdot]$ as in . One now checks that $$\begin{aligned}
\label{mM1}
\operatorname{\wt{Mult}}(\Gamma)=a_{{{\bf w}}}\operatorname{Mult}(\Gamma)z^{n_{\operatorname{out}}}.\end{aligned}$$
### Theta functions and scattering diagrams in terms of tropical invariants
We are now ready to state [@Man3]’s result expressing theta functions in terms of tropical disk counts.
\[TropFrob\] For ${{\bf p}}:=(p_i)_{i=1,\ldots,s}$ an $s$-tuple of elements of $\?{N}$, let $\alpha({{\bf p}};p)$ denote the $\vartheta_p$-coefficient of $\prod_{i=1}^s \vartheta_{p_i}$. Then $$\begin{aligned}
\label{TropFrobEqnQuot}
\alpha({{\bf p}};p)z^{\varphi(p)}=\sum_{r\in K_2^+} \sum_{{{\bf w}}\in \s{W}_{{{\bf p}}}(\varphi(p)+r)} \frac{\operatorname{N}_{{{\bf w}},{{\bf p}}}^{\operatorname{trop}}(Q)}{|\operatorname{Aut}({{\bf w}})|}\end{aligned}$$ for $Q$ chosen to be sufficiently close to the ray through $p$ (or for $Q$ anywhere if $p=0$).
\[Dints\] Using , note that we can rewrite as $$\begin{aligned}
\label{TropFrobEqnQuot2}
\alpha({{\bf p}};p)=\sum_{r\in K_2^+} \sum_{{{\bf w}}\in \s{W}_{{{\bf p}}}(\varphi(p)+r)} a_{{{\bf w}}}\frac{\operatorname{GW}^{\operatorname{trop}}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})}{|\operatorname{Aut}({{\bf w}})|}z^{r}.\end{aligned}$$ Given data ${{\bf p}}$, $p$, $r$, and ${{\bf w}}\in \s{W}_{{{\bf p}}}(\varphi(p)+r)$ as in , it follows immediately from the definition of $\Delta_{{{\bf w}},{{\bf p}}}$ and the description of $\kappa$ in Theorem \[kappa\] that $\kappa(r)\in N_1(Y_S,\bb{Z})$ is the class of a curve having intersection number $\sum_{j=1}^{l_i} w_{ij}$ with $[E_i]$ for each $i\in I_{\operatorname{uf}}$, plus for each ray $\rho$ of $\Sigma$, having intersecting number with $D_{\rho}$ equal to $\sum_{p_i\in \rho} |p_i|,$ plus $-|p|$ if $p\in \rho$.
\[AmplePolynomial\] If $(Y_{{{\bf S}},\Sigma}, D_{{{\bf S}},\Sigma})$ supports an ample divisor on its boundary, then the theta functions in fact generate an algebra over ${\Bbbk}[\operatorname{NE}(Y_{{{\bf S}},\Sigma})]$ (as opposed to over some formal completion of this).
By possibly replacing $\Sigma$ by a refinement, we have that $D_{{{\bf S}},\Sigma}$ supports an effective ample divisor $H$. After possibly replacing $H$ with $D_{{{\bf S}},\Sigma}+kH$ for $k$ a sufficiently large integer, we can always assume that $H$ is of the form $\sum_{\rho\in \Sigma^{[1]}} a_{\rho}D_{\rho}$ with each $a_{\rho}$ in $\bb{Z}_{\geq 1}$. By Lemma \[NECoeff\], the coefficient of $\vartheta_p$ in the theta function expansion of the product $\prod_{i=1}^s \vartheta_{p_i}$ is an (a priori formal) sum of monomials whose exponents correspond to elements of the Mori cone $\operatorname{NE}(Y_{{{\bf S}},\Sigma})$, hence have positive intersection with $H$. Furthermore, Remark \[Dints\] ensures that the intersections of these classes with $H$ are determined by ${{\bf p}}$ and $p$, and for fixed ${{\bf p}}$ we see that there are only finitely many possibilities for $p$ for which this intersection number with $H$ will in fact be positive. Thus, there are only finitely many $p$’s for which the $\vartheta_p$-coefficient can be nonzero.
So now it suffices to show that for fixed ${{\bf p}}$ and $p$, there are only finitely many curve classes with the associated intersection number $d$ with $H$. Note that there always exists a rational polyhedral cone $\Xi\subset N_1(Y_{{{\bf S}},\Sigma})$ such that $\operatorname{NE}(Y_{{{\bf S}},\Sigma})\subset \Xi$ and such that $H$ is positive on $\Xi\setminus \{0\}$.[^12] Since $\Xi$ is finitely generated, it clearly contains only finitely many elements whose pairing with $H$ is $d$, and so the same holds for $\operatorname{NE}(Y_{{{\bf S}},\Sigma})\subset \Xi$, as desired.
Log Gromov-Witten invariants {#SectionLogGW}
----------------------------
Let us recall and extend the setup from §\[FSC\]. Given a smooth log pair $Y^{\dagger}=(Y,D)$, we can consider the integral points $B(\bb{Z})$ of the cone over the dual intersection complex of some $\wt{D}$. In particular, when $(Y,D)$ is $(Y_{{{\bf S}},\Sigma},D_{{{\bf S}},\Sigma})$ or $(\operatorname{TV}(\wt{\Sigma}),\partial \operatorname{TV}(\wt{\Sigma}))$ for $\wt{\Sigma}$ a complete fan in $\?{N}$, the set $B(\bb{Z})$ is identified with $\?{N}$.
Recall that a tropical degree is a map $\Delta:J\rar B(\bb{Z})$ for some finite index-set $J$. Let $J^{\circ}:=\Delta^{-1}(0)$. Let $\Sigma$ be the fan in $B$ for $Y^{\dagger}$. Let $\eta:(\wt{Y},\wt{D})\rar (Y,D)$ denote a toric blowup corresponding to a refinement $\wt{\Sigma}$ of $\Sigma$. We can assume that $\wt{Y}$ is projective and that $\Delta$ is a [**torically transverse degree**]{} for $\wt{\Sigma}$, meaning that $\Delta(j)$ is contained in a ray $\rho\in \wt{\Sigma}$ for each $j\in J'$. Recall that $D_{\Delta(j)}\subset \wt{D}$ denotes the corresponding boundary component.
For $\beta\in \operatorname{NE}(\wt{Y})$, we consider the moduli stack $\s{M}_{0,\Delta}^{\log}(\wt{Y}^{\dagger},\beta)$ as in §\[FSC\]. We have evaluation maps $\operatorname{ev}_i$ for each $i\in J$ and $\psi$-classes $\psi_i$ as in .
Define a map $Z:J\rar A^*(\wt{Y})$, denoting $Z_j:=Z(j)$. Let $\Psi$ be a map from $J^{\circ}$ to $\bb{Z}_{\geq 0}$, and denote $s_j:=\Psi(j)$. We define log Gromov-Witten invariants by $$\begin{aligned}
\label{GWdef}
\operatorname{GW}^{\log}_{0,Y^{\dagger},\Delta, \beta}(Z,\Psi) := \int_{[\s{M}^{\log}_{0,\Delta}(\wt{Y}^{\dagger},\beta)]^{\operatorname{vir}}} \left(\bigcup_{j\in J^{\circ}} \psi_j^{s_j}\right) \cup \left(\bigcup_{j\in J} \operatorname{ev}_j^*(Z_j)\right).\end{aligned}$$ The main result of [@AW] ensures that this is independent of the choice of toric blowup $\wt{Y}$ of $Y$.
Suppose $(\wt{Y},\wt{D})$ above is a nonsingular complete toric variety with cocharacter lattice $\?{N}$ and fan $\wt{\Sigma}$, so $B(\bb{Z})=\?{N}$. Consider an affine linear subspace $A\subseteq \?{N}_{\bb{R}}$ with rational slope. Given a point $x$ in the big torus orbit of $Y$, we obtain a subvariety $Z_{A,x}$ as follows: Let $A^{\perp}$ denote the $m\in \?{M}$ which pair to $0$ with the tangent directions to $A$. Let $z^m(x)$ denote $z^m$ evaluated at $x$. Then $Z_{A,x}$ is the subvariety corresponding to the ideal sheaf $\langle z^m-z^m(x)|m\in A^{\perp}\rangle$. In particular, when $A\subset\?{N}_{\bb{R}}$ is just a point, $Z_{A,x}=x$. Note that $\dim_{{\Bbbk}}(Z_{A,x})=\dim_{\bb{R}}(A)$. We denote the corresponding Chow class (independent of $x$) by $[Z_A]$.
Now suppose we have data $\?{N},\Delta,{{\bf A}},\Psi$ as in §\[TropGW\]. Define $Z_{{{\bf A}}}:J\rar A_*(\wt{Y})$ by $Z_{{{\bf A}}}(j)=[Z_{A_j}]$. Let $[\Delta]\in \operatorname{NE}(\wt{Y},\bb{Z})$ be the unique curve class such that $$[\Delta].[D_{\rho}]=\sum_{\substack{j\in J \\ \Delta(j)\in \rho}} |\Delta(j)|$$ for each ray $\rho$ of $\wt{\Sigma}$. We define $$\begin{aligned}
\label{GWtoric}
\operatorname{GW}_{0,\Delta}^{\log}({{\bf A}},\Psi):=\operatorname{GW}^{\log}_{0,\operatorname{TV}(\wt{\Sigma})^{\dagger},\Delta,[\Delta]}(Z_{{{\bf A}}},\Psi).\end{aligned}$$
The genus $0$ case of [@MRud Thm 4.15], or alternatively [@AGr Cor 5.2], states the following ($\Delta$ here is assumed to be a tropical curve degree, not a tropical disk degree):
\[MRudThm\] $$\begin{aligned}
\operatorname{GW}_{0,\Delta}^{\log}({{\bf A}},\Psi) = \operatorname{GW}_{0,\Delta}^{\operatorname{trop}}({{\bf A}},\Psi).\end{aligned}$$
In particular, the tropical invariants of when $p=0$ can be replaced with the corresponding log invariants, yielding:
\[GWToric\] $$\begin{aligned}
\alpha({{\bf p}};0)=\sum_{r\in K_2^+} \sum_{{{\bf w}}\in \s{W}_{{{\bf p}}}(r)} a_{{{\bf w}}}\frac{ \operatorname{GW}_{0,\Delta_{{{\bf w}},{{\bf p}}}}^{\log}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})}{|\operatorname{Aut}({{\bf w}})|}z^{r}.\end{aligned}$$
Degeneration
============
In this section, we will use a degeneration of our cluster varieties to relate the toric log Gromov-Witten invariants from Lemma \[GWToric\] to log Gromov-Witten invariants of the cluster variety. First though, we will need an important new technical result which says that curves satisfying “somewhat generic” conditions are torically transverse.
Toric transversality lemma {#trans}
--------------------------
We will write ${{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ}$ to indicate the tropical incidence conditions ${{\bf A}}_{{{\bf w}},{{\bf p}},Q}$ as in §\[ThetaFromTrop\], except with each $A_{ij}$ chosen to contain the origin in $\?{N}_{\bb{R}}$ (rather than being chosen to be a generic translate). We write $A_{{{\bf w}},{{\bf p}},0}^{\circ}$ to indicate that $Q=0$ as well.
Consider the invariants $\operatorname{GW}_{0,\Delta_{{{\bf w}},{{\bf p}}}}^{\log}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})$ of Lemma \[GWToric\], and consider a curve $\varphi^{\dagger}=[\varphi^{\dagger}:C^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}(\operatorname{TV}(\wt{\Sigma})^{\dagger},[\Delta_{{{\bf w}},{{\bf p}}}])$. We say that $\varphi^{\dagger}$ satisfies **somewhat generic** incidence and $\psi$-class conditions if it satisfies generically chosen representatives of the conditions $Z_{{{\bf A}}_{{{\bf w}},{{\bf p}},Q}}(\operatorname{out})=:y^{\operatorname{out}}$ and $\Psi_{{{\bf w}},{{\bf p}}}$, along with representatives of $Z_{{{\bf A}}_{{{\bf w}},{{\bf p}},Q}}((i,j))$ for each $(i,j)\in J_{{{\bf w}},{{\bf p}}}$ which are not necessarily generic, but which at least intersect the interior of $D_i$. The tropicalization of $\varphi^{\dagger}$ (in the sense of [@GSlog])[^13] is in $\f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},0}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$. We wish to prove the following:[^14]
\[DaggerCircLem\] Suppose $[\varphi^{\dagger}:C^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}(\operatorname{TV}(\wt{\Sigma})^{\dagger},[\Delta_{{{\bf w}},{{\bf p}}}])$ satisfies somewhat generic incidence and $\psi$-class conditions representing the classes $Z_{{{\bf A}}_{{{\bf w}},{{\bf p}},Q}}$ and $\Psi_{{{\bf w}},{{\bf p}}}$ as in the invariants of Lemma \[GWToric\]. Assume $\Delta_{{{\bf w}},{{\bf p}}}$ is a torically transverse degree for $\wt{\Sigma}$. Then $\varphi(C)$ is torically transverse in $\operatorname{TV}(\wt{\Sigma})$.
First, we need the following simple observation:
\[StarTransverse\] A basic stable log curve in $\operatorname{TV}(\wt{\Sigma})^{\dagger}$ is torically transverse if and only if all of its tropicalizations are supported on the $1$-skeleton of $\wt{\Sigma}$.
Let $\star(0)\subset \?{N}_{\bb{R}}$ denote the union of all rational-slope lines through the origin. Let $[\varphi^{\dagger}:C^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}(\operatorname{TV}(\wt{\Sigma})^{\dagger},[\Delta_{{{\bf w}},{{\bf p}}}])$ be an arbitrary basic stable log map which satisfies our somewhat generic incidence and $\psi$-class conditions. By the assumption that $\Delta_{{{\bf w}},{{\bf p}}}$ is a torically transverse degree for $\wt{\Sigma}$, the tropicalization $(\Gamma,h)$ of $\varphi^{\dagger}$ (for any pullback of the log structure to the standard log point) is supported on the $1$-skeleton of $\wt{\Sigma}$ if and only if it is supported on $\star(0)$. Thus, to prove Lemma \[DaggerCircLem\], it suffices to prove that any $(\Gamma,h)$ obtained as a tropicalization of $\varphi^{\dagger}$ must be supported on $\star(0)$.
We will need some new definitions regarding tropical curves. By a [**contractible**]{} tropical curve we will mean a tropical curve as before, but now we allow compact positive-weight edges to be contracted by $h$. Each flag $E\ni V$ is still assigned a designated primitive direction $u_{(V,E)}$ (i.e., as part of the data of the contractible tropical curve) such that the balancing condition still holds and such that $u_{(V,E)}=-u_{(V',E)}$ for $V,V'$ the two vertices of $E$. One can define the type of a contractible tropical curve just as for the tropical curves of §\[TropGW\], keeping in mind that contracted edges have directions. Tropical curves as in §\[TropGW\] (i.e., without contracted compact positive-weight edges) will sometimes be referred to as **contracted** tropical curves.
Given a contractible tropical curve $(\Gamma',h')$, we can obtain a contracted tropical curve as follows: for each compact positive-weight edge of $\Gamma'$ contracted by $h'$, we simply contract the edge in the domain before applying $h'$ to get a new domain $\Gamma$. Then $h$ is the map $\Gamma\rar N_{\bb{R}}$ such that the contraction $\Gamma'\rar \Gamma$ composed with $h$ is equal to $h'$. We call this new tropical curve $(\Gamma,h)$ the [**contraction**]{} of $(\Gamma',h')$, and we say that $(\Gamma',h')$ is an [**expansion**]{} of $\Gamma$. We say that a contractible tropical curve is in some $\f{T}_{0,\Delta}({{\bf A}},\Psi)$ if its contraction is.
The point of this terminology is that the moduli space of basic stable log maps is stratified by tropical types (cf. [@ACGSdecomp]), with the type of a stratum $\sigma$ being a contraction of the type associated to any stratum of $\partial \sigma$.
Consider $\Gamma\in \f{T}_{0,\Delta}({{\bf A}},\Psi)$, possibly contractible. As in the computation of $\operatorname{Mult}(\Gamma)$ in , we choose a flow on $\Gamma$ by specifying a vertex $V_0$ to serve as the sink. We recursively define affine linear spaces $A_E$ associated to each edge $E\in \Gamma^{[1]}$ as follows. For each $j\in J$, we take $A_{E_j}:=A_j$. For each vertex $V\neq V_0$, if $E_1,\ldots,E_k$ are the edges flowing into $V$ and $E_{V}$ is the unique edge flowing out of $V$, then we define $$\begin{aligned}
A_E:=\bb{R}u_{E_V}+\bigcap_{i=1}^k A_{E_i}.\end{aligned}$$ Then for each $E\ni V_0$, we define $A_{V_0,E}:=A_E$. Of course, we could take $V_0$ to be any vertex of $\Gamma^{[0]}$, and in this way we obtain linear spaces $A_{V,E}$ for all flags of $\Gamma$. One sees by induction that each vertex $V$ must be contained in $A_{V,E}$ for each $E\ni V$.
Let us now specialize to the case of $\f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$. Recall from §\[nontoric\] that the seed data ${{\bf S}}$ included a form $B$ on $N$ which is skew-symmetrizable in the sense that there exists a skew-symmetric form $\omega$ on $N$ and positive integers $\{d_i\}_{i\in I_{\operatorname{uf}}}$ such that $B(e_i,e_j)=d_i\omega(e_i,e_j)$ for all $i,j\in I_{\operatorname{uf}}$. In particular, this implies that $\ker(\omega|_{N_{\operatorname{uf}}})=\ker(\pi_2|_{N_{\operatorname{uf}}})$, so $\omega$ induces a non-degenerate skew-symmetric form $\?{\omega}$ on $\?{N}_{\operatorname{uf}}:=\pi_2(N_{\operatorname{uf}})$. Furthermore, we see that $\pi_1(e_i)|_{\?{N}_{\operatorname{uf}}} = d_i\omega(e_i,\cdot)|_{\?{N}_{\operatorname{uf}}}$ for each $i\in I_{\operatorname{uf}}$. Hence, the conditions $A_{ij}$ of ${{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ}$ satisfy $A_{ij}\cap \?{N}_{\operatorname{uf},\bb{R}} = u_{E_{ij}}^{\?{\omega}\perp}$, where for $u\in \?{N}_{\operatorname{uf}}$, $u^{\?{\omega}\perp}:=\{n\in \?{N}_{\operatorname{uf},\bb{R}}:\?{\omega}(u,n)=0\}$. This motivates the following:
\[AE\] Let $\Gamma\in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$. For $V\in \Gamma^{[0]}$ and $E\ni V$, Let $\Gamma_{(V,E)}$ denote the closure in $\Gamma$ of the connected component of $\Gamma\setminus V$ which contains the interior of $E$. Suppose all unbounded edges of $\Gamma_{(V,E)}$ are labelled by pairs $(i,j)\in J_{{{\bf w}},{{\bf p}}}$. Then $u_E\in \?{N}_{\operatorname{uf}}$ and $A_{V,E}\cap \?{N}_{\operatorname{uf},\bb{R}}\subset u_E^{\?{\omega}\perp}$.
This follows from induction: The claim holds for the edges $E_{ij}$ by the observations preceding the Lemma. If $E_1,\ldots,E_k$ are the edges flowing into a vertex $V'$ and $E'$ is the edge flowing out, then $u_{E_i}\in \?{N}_{\operatorname{uf}}$ for each $i=1,\ldots,k$ implies the same for $u_{E'}$ by the balancing condition. Similarly, $A_{E_i}\subset u_{E_i}^{\?{\omega}\perp}$ for each $i$ implies that $$\bigcap_{i=1}^k A_{E_i} \subset \left(\sum_{i=1}^k w(E_i)u_{E_i}\right)^{\?{\omega}\perp}=u_{E'}^{\?{\omega}\perp},$$ hence $$\begin{aligned}
\bb{R}u_{E'}+\bigcap_{i=1}^k A_{E_i} \subset u_{E'}^{\?{\omega}\perp}, \end{aligned}$$ as desired.
The following Lemma says that $Q$ being generic is enough to ensure that tropical curves in $\f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$ resemble $s$-tuples of broken lines meeting at a point.
\[Vout\] Fix a generic $Q\in N_{\bb{Q}}$. Let $(\Gamma,h)\in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$. Let $V_{\operatorname{out}}$ be the vertex of $\Gamma$ contained in $E_{\operatorname{out}}$. Then each component of $\Gamma \setminus V_{\operatorname{out}}$ other than (the interior of) $E_{\operatorname{out}}$ contains precisely one unbounded edge of the form $E_k$ for $k=1,\ldots,s,\infty\in J_{{{\bf w}},{{\bf p}}}$.
Suppose a component $\Gamma'$ does not contain any such edge. Let $E'$ be the edge of $\Gamma'$ whose closure in $\Gamma$ contains $V_{\operatorname{out}}$. Then by Lemma \[AE\], ${{\bf A}}_{V_{\operatorname{out}},E'}$ has codimension at least one, and so the generic point $Q$ (and thus $V_{\operatorname{out}}$) cannot be contained in ${{\bf A}}_{V_{\operatorname{out}},E'}$. This gives a contradiction.
On the other hand, $\Psi_{{{\bf w}},{{\bf p}}}$ ensures that the valence of $V_{\operatorname{out}}$ is at least $(s+2)$, so no component of $\Gamma\setminus V_{\operatorname{out}}$ can contain more than one of the edges from $\{E_1,\ldots,E_s,E_{\infty},E_{\operatorname{out}}\}$.
A choice of generic $Q\in \?{N}_{\bb{Q}}$ determines (after a finite base change) a deformation $y_t^{\operatorname{out}}$ of the point $y^{\operatorname{out}}=:y_0^{\operatorname{out}}$ into the boundary of $\operatorname{TV}(\wt{\Sigma})$, cf. [@MRud §3.2.1]. Denote the limit in the boundary by $y_1^{\operatorname{out}}$. Since $y^{\operatorname{out}}$ was chosen generically, any curve $[\varphi^{\dagger}:C^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}]$ satisfying the somewhat generic conditions with $y^{\operatorname{out}}=y_0^{\operatorname{out}}$ will deform to a log curve $\varphi^{\dagger}_t$ satisfying the conditions with $y^{\operatorname{out}}$ replaced by $y_t^{\operatorname{out}}$. The curves $\varphi_0^{\dagger}$ and $\varphi_1^{\dagger}$ admit tropicalizations $(\Gamma,h)\in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},0}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$ and $(\Gamma_1,h_1)\in \f{T}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q}^{\circ},\Psi_{{{\bf w}},{{\bf p}}})$, respectively. Furthermore, since $\varphi_1^{\dagger}$ is a degeneration of $\varphi_0^{\dagger}$, $(\Gamma_1,h_1)$ must be have the same type as some expansion $(\Gamma',h')$ of $(\Gamma,h)$.
Suppose $\Gamma$ is not supported on $\star(0)$. The incidence conditions at least force $h(V_{\operatorname{out}})=0$, where $V_{\operatorname{out}}$ denotes the vertex contained in $E_{\operatorname{out}}$. Let $V'$ be a vertex of $\Gamma'$ of minimal distance from $V_{\operatorname{out}}$ which is not at $0$ and whose adjacent edges have directions not all parallel to the ray through $h(V')$. For $E'\ni V'$ the edge on the component of $\Gamma'\setminus V'$ containing $V_{\operatorname{out}}$, we necessarily have $u_{E'}$ parallel to the ray through $h(V')$, and so $A_{V',E'}$ contains the line $\ell_{V'}$ through $0$ and $h(V')$. By Lemma \[Vout\], the component of $\Gamma'\setminus V_{\operatorname{out}}$ containing $V'$ includes exactly one of the edges $E_k$, $k=1,\ldots,s$. It follows that $A_{V',E'}$ has dimension at most $2$.
If the dimension is $2$, then Lemma \[AE\] applies to the other edges $E_i$, $i=1,\ldots,l$ containing $V'$, so they satisfy $A_{V',E_i}\cap \?{N}_{\operatorname{uf},\bb{R}}\subset u_{E_i}^{\?{\omega}\perp}$. Since balancing forces these edges to point in different directions and $\?{\omega}$ is non-degenerate, these codimension $1$ spaces are non-equal, hence intersect to give a space of codimension at least $2$. But since each $A_{V',E_i}$ contains $V'$ and $0$, they must contain $\ell_{V'}$, so the intersection with $A_{V',E'}$ is $1$-dimensional, hence non-transverse. However (giving the edges of $\Gamma_1$ the same names as the corresponding edges of $\Gamma'$), translating $Q$ moves $h_1(E')$ independently from the other edges $E_i$ containing $V'$. So $h_1(V')$ is contained in a translate of the space $A_{V',E'}$ which has dimension at most $2$, but also in the space $\bigcap_{i=1}^{l} A_{V',E_i}$, which has codimension at least $2$ and intersects $A_{V',E'}$ non-transversely, thus giving a contradiction.
Similarly, if $A_{V',E'}$ is $1$-dimensional, then still one of the $A_{V',E_i}$ is contained in $u_{E_i}^{\omega\perp}$, which again must contain $\ell_{V'}$, hence have $1$-dimensional intersection with $A_{V',E'}$. This again is impossible for generic translates of $Q$ by the same argument. The claim follows.
Relating the invariants of $\operatorname{TV}(\wt{\Sigma})^{\dagger}$ and $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}})$. {#DegenS}
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Recall from Remark \[BlowupRmk\] that the pair $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}})$ can be constructed by, for each $i\in I_{\operatorname{uf}}$, blowing up the scheme-theoretic intersection $$\begin{aligned}
\?{H}_i:=D_{\pi_2(e_i)}\cap Z((a_i+z^{\pi_1(e_i)})^{|\pi_2(e_i)|}) \subset \operatorname{TV}_{\?{N}}(\wt{\Sigma}) \end{aligned}$$ for some $a_i\in {\Bbbk}^*$, possibly followed by some toric blowdowns which (by [@AW]) do not affect log invariants. Lemma \[GWToric\] allows us to express the theta functions associated to $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}})$ in terms of certain log Gromov-Witten numbers of $\operatorname{TV}_{\?{N}}(\wt{\Sigma})$. We wish to relate these to log Gromov-Witten numbers of $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}})$. More precisely, we wish to prove the following:
\[DegenProp\] Given a tuple ${{\bf p}}=(p_1,\ldots,p_s)$ of vectors in $\?{N}$, consider the tropical degree $\Delta_{{{\bf p}}}$ as in . Given $\beta\in \operatorname{NE}(Y_{{{\bf S}},\wt{\Sigma}})$, let $\s{W}(\beta)$ denote the set of weight vectors ${{\bf w}}$ such that $\sum_{j=1}^{l_i} w_{ij} = \beta.[E_i]$. Let $Z(s+1)=[\operatorname{pt}]$ and $Z(i)=[Y_{{{\bf S}},\wt{\Sigma}}]$ for all other $i$. Let $\Psi(s+1)=s-1$ and $\Psi(i)=0$ for all other $i$. Let $\eta$ denote the toric blowdown $Y^{\dagger}_{{{\bf S}},\wt{\Sigma}}\rar Y^{\dagger}_{{{\bf S}},\Sigma}$. Then $$\begin{aligned}
\label{DegenFormula}
\operatorname{GW}^{\log}_{0,Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\Delta_{{{\bf p}}},\beta}(Z,\Psi)z^{\eta_*\beta} = \sum_{r\in K_2^+} \sum_{{{\bf w}}\in \s{W}_{{{\bf p}}}(\varphi(p)+r)} a_{{{\bf w}}}\frac{\operatorname{GW}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}})}{|\operatorname{Aut}({{\bf w}})|}z^{r}.\end{aligned}$$
Here, $z^r$ is viewed as an element of ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}},\wt{\Sigma}})\rrb$ by using $\kappa$ to identify $K_2$ with $N_1(Y_{{{\bf S}},\Sigma})$. Remark \[Dints\] tells us already that $\kappa(r)$ is indeed equal to $\eta_*(\beta)$.
The strategy is to take a degeneration of $Y_{{{\bf S}},\wt{\Sigma}}$, pictured in Figure \[CDfig\], as is done for two-dimensional cases in [@GPS §5]. To do this, let $\?{N}':=\?{N}\oplus \bb{Z}$, and let $\wt{\Sigma}^{\times}$ denote the fan in $N'_{\bb{R}}$ equal to the product of $\wt{\Sigma}$ with the fan for $\bb{A}^1$. That is, for each cone $\sigma\in \wt{\Sigma}$, there are two cones in $\wt{\Sigma}^{\times}$ given by $\sigma\times \{0\}$ and $\wt{\sigma}:=\sigma \times \bb{R}_{\geq 0}$. Then $\operatorname{TV}(\wt{\Sigma}^{\times})=\operatorname{TV}(\wt{\Sigma})\times \bb{A}^1$, and the projection $t:\?{N}'\rar \bb{Z}$ induces a map of fans giving a projection $t^{\times}:\operatorname{TV}(\wt{\Sigma}^{\times}) \rar \bb{A}^1$.
For any $n\in \?{N}$, let $\rho_n$ denote the ray generated by $n$ in $\?{N}$ (or the origin of $n=0$), and recall that for $n\neq 0$, $n'$ denotes the primitive vector in $\?{N}$ with direction $n$. Now for each $i\in I_{\operatorname{uf}}$, we refine the cone $\wt{\rho}_{\pi_2(e_i)}\subset \wt{\Sigma}^{\times}$ by adding the ray $\bb{R}_{\geq 0}(\pi_2(e_i)',1)$. We then further refine the cones of $\wt{\Sigma}^{\times}$ until we achieve a non-singular fan $\wt{\Sigma}'$ such that the projection $t':\operatorname{TV}(\wt{\Sigma}')\rar \bb{A}^1$ induced by $t$ is projective.
Consider the cones in $\wt{\Sigma}'$ which are contained in $\wt{\rho}_{\pi_2(e_i)}$ for some $i\in I_{\operatorname{uf}}$, or in $\wt{\rho}_{p_i}$ for some $p_i$ from ${{\bf p}}$. Let $\wt{\Sigma}^{\circ}$ denote the set of all such cones except for the ones which are entirely contained in $(\?{N}_{\bb{R}},0)\subset \?{N}'_{\bb{R}}$. Let $\?{Y}_0^{\circ}\subset \operatorname{TV}(\wt{\Sigma}')$ denote the union of all toric strata of $\operatorname{TV}(\wt{\Sigma}')$ which correspond to cones in $\wt{\Sigma}^{\circ}$. Let $\?{D}_0^{\circ}\subset \?{Y}_0^{\circ}$ denote the union of all codimension-$2$ toric strata of $\operatorname{TV}(\wt{\Sigma}')$ which correspond to $2$-dimensional cones in $\wt{\Sigma}^{\circ}$. By assuming that $\wt{\Sigma}$ was sufficiently refined, we can assume that $\?{D}_0^{\circ}$ is non-singular (in particular, the strata of $\?{D}_0^{\circ}$ are disjoint).
Finally, we blow up $\operatorname{TV}(\wt{\Sigma}')$ along the subvariety cut out by the loci $\?{H}_i':=Z(a_i+z^{(\pi_1(e_i),0)})\cap D_{(\pi_2(e_i),0)}$ for each $i\in I_{\operatorname{uf}}$. Let $\wt{E}_i$, $i\in I_{\operatorname{uf}}$, denote the respective exceptional divisors. We denote the resulting projective log smooth family by $$\pi^{\dagger}:\wt{Y}_{{{\bf S}},\wt{\Sigma}}^{\dagger} \rar (\bb{A}^1)^{\dagger},$$ where the log structure of $\wt{Y}_{{{\bf S}},\wt{\Sigma}}^{\dagger}$ is the divisorial log structures with respect to the proper transform of the toric boundary of $\operatorname{TV}(\wt{\Sigma}')$, and for $(\bb{A}^1)^{\dagger}$ we use the divisorial log structure with respect to $0\in \bb{A}^1$.
Let $Y_0^{\circ}$ denote the the preimage of $\?{Y}_0^{\circ}$ under the blowups of the loci $\?{H}_i$. Similarly, let $D_0^{\circ}$ denote the proper transform of $\?{D}_0^{\circ}$ under these blowups. Let $\wt{Y}_{{{\bf S}},\wt{\Sigma}}^{\circ}$ denote the space $\wt{Y}_{{{\bf S}},\wt{\Sigma}}$ equipped with the divisorial log structure associated to the divisor $Y_0^{\circ}$. We then equip $Y_0^{\circ}$ with the log structure pulled back via the inclusion of $Y_0^{\circ}$ into $\wt{Y}_{{{\bf S}},\wt{\Sigma}}^{\circ}$. For each $i\in I_{\operatorname{uf}}$, let $\operatorname{Bl}_i$ denote the strata of $Y_0^{\circ}$ corresponding to cones in $\wt{\rho}_{\pi_2(e_i)}$. Similarly, for each $p_i$, $i=1,\ldots,s$, let $\operatorname{Bl}_{p_i}$ denote the strata of $Y_0^{\circ}$ corresponding to cones in $\wt{\rho}_{p_i}$.
Let $Y_t^{\dagger}$ denote $\pi^{-1}(t)$ with the log structure induced by the inclusion into $\wt{Y}_{{{\bf S}},\wt{\Sigma}}^{\dagger}$. Note that for $t\neq 0$, $Y_t^{\dagger}$ is simply the cluster variety $Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}$ whose log GW invariants we are interested in, while $Y_{0}^{\dagger}$ includes $$\begin{aligned}
\label{Y0}
Y_0^{\circ}=\operatorname{TV}(\wt{\Sigma})\cup \bigcup_{i\in I_{\operatorname{uf}}} \operatorname{Bl}_i\end{aligned}$$ (with slightly different log structure) along with some additional strata. Let $E_{i,t}:= \wt{E}_i \cap Y_{t}^{\dagger}$. Let $\operatorname{Bl}'_i$ denote the component of $\operatorname{Bl}_i$ containing $E_{i,0}$, and let $\?{\operatorname{Bl}}'_i$ denote the image of $\operatorname{Bl}'_i$ under blowing down the $E_{j,0}$’s which it contains. Let $\?{F}_i:=Z(a_i+z^{(\pi_1(e_i),0)})\cap \?{\operatorname{Bl}}'_i$, and let $F_i$ be the proper transform of $\?{F}_i$ under the blowups of the loci $\?{H}'_j$. There is a fibration of $\?{\operatorname{Bl}}'_i$ with generic fibers (those not contained in the boundary of $\?{\operatorname{Bl}}'_i$) being $\bb{P}^1$, and with the two components of $D_0^{\circ}\cap \?{\operatorname{Bl}}_{E_{i,0}}$ being sections. $\?{F}_i$ can be viewed as the union of the fibers which intersect $\?{H}'_i$. Let $C_{F_i}\in \operatorname{NE}(\operatorname{Bl}'_i)_{\bb{Q}}$ be the class of the proper transform of one of these $\bb{P}^1$-fibers in $\?{F}_i$. Let $D^{\circ}_{0,i}$ be the component of $D^{\circ}_0$ which intersects $F_i$.
Since $\pi^{\dagger}$ is log smooth, [@MRud Thm. A.3] says that the log Gromov-Witten invariants do not depend on $t$. Hence, when proving Proposition \[DegenProp\], we can replace $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}}) \cong Y_{t}^{\dagger}$ ($t\neq 0$) with $Y_{0}^{\dagger}$.
Consider the blowdown map $b:Y_0^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})$. Suppose $[\varphi^{\dagger}:C^{\dagger}\rar Y_0^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(Y^{\dagger}_0,\beta)$ satisfies generic representatives of the incidence condition $\operatorname{ev}_{\operatorname{out}}^*([\operatorname{pt}])$ and the $\psi$-class conditions $\psi_{\operatorname{out}}^{s-1}$. Then $b\circ \varphi:C\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}$ can be equipped with a log structure making it into a curve in some $\s{M}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}(\operatorname{TV}(\wt{\Sigma})^{\dagger},[\Delta_{{{\bf w}},{{\bf p}}}])$ which satisfies somewhat generic representatives of the conditions $Z_{{{\bf A}}_{{{\bf w}},{{\bf p}},Q}}$ and $\Psi_{{{\bf w}},{{\bf p}}}$. In particular, by Lemma \[DaggerCircLem\], $b\circ \varphi(C)$ must be torically transverse. Hence $\varphi(C)$ must be torically transverse, and furthermore, any components of $\varphi(C)$ mapping to any $\operatorname{Bl}_i$ must be supported on fibers of the blowdown map $b$.
We illustrate this setup in Figure \[CDfig\].
Consider the locus of $\varphi^{\dagger}\in \s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(Y_0^{\dagger},\beta)$ satisfying generic representatives of the conditions $Z$ and $\Psi$. Then the obstruction theory on this locus is unchanged if we view the maps $\varphi^{\dagger}$ as basic stable log maps to $Y_0^{\circ}$ instead of to $Y_0^{\dagger}$.
We saw above that the image of such $\varphi^{\dagger}$ is necessarily torically transverse, so the only boundary divisors such curves can intersect are those in $D_0^{\circ}$. Thus, forgetting the log structure along the other boundary divisors does not affect the obstruction theory.
The upshot is that since $D_0^{\circ}$ is smooth, we can now use the log degeneration formula[^15] of [@KLR Thm. 1.4]. Let $[\varphi_{\operatorname{TV}}^{\dagger}:C^{\dagger}\rar \operatorname{TV}(\wt{\Sigma})^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}(\operatorname{TV}(\wt{\Sigma})^{\dagger},[\Delta_{{{\bf w}},{{\bf p}}}])$ be a curve satisfying somewhat generic incidence and $\psi$-class conditions representing ${{\bf A}}_{{{\bf w}},{{\bf p}},Q}$ and $\Psi_{{{\bf w}},{{\bf p}}}$. The Gromov-Witten count of these curves is $$\begin{aligned}
\label{TVGW}
\operatorname{GW}^{\log}_{0,\Delta_{{{\bf w}},{{\bf p}}}}({{\bf A}}_{{{\bf w}},{{\bf p}},Q},\Psi_{{{\bf w}},{{\bf p}}}).\end{aligned}$$ Furthermore, any curve $\wt{\varphi}^{\dagger}$ contributing to $\operatorname{GW}^{\log}_{0,Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\Delta_{{{\bf p}}},\beta}(Z,\Psi)$ is obtained by taking one of these curves $\varphi_{\operatorname{TV}}^{\dagger}$ and gluing chains of $\bb{P}^1$’s as follows:
For each $(i,j)$, we glue to the marked point $x_{ij}$ a chain of $b_i$ $\bb{P}^1$’s where $b_i\geq 1$ is the number of components of $\operatorname{Bl}_i$. The first $(b_i-1)$ copies of $\bb{P}^1$ are just $w_{ij}|\pi_2(e_i)|$-fold covers of fibers of successive components of $\operatorname{Bl}_i$ with maximal tangency at $0$ and $\infty$ (where it intersects $D_0^{\circ}$). The final $\bb{P}^1$, denoted $C_{ij}$, maps to $\operatorname{Bl}'_i$, satisfies $$(\wt{\varphi}|_{C_{ij}})_*[C_{ij}]=w_{ij}|\pi_2(e_i)|[C_{F_i}],$$ and has maximal tangency with $D_{0,i}^{\circ}$ at a point $p_{ij}$ ($p_{ij}$ is a node in $\wt{C}^{\dagger}$ but can be viewed as a marked point in $C_{ij}$ using the degeneration formula). The nodes of this chain all have weight $w_{ij}|\pi_2(e_i)|$, and so the number of choices of log structures at these nodes, *modulo automorphisms*, is $$\begin{aligned}
\label{ijweight}
w_{ij}|\pi_2(e_i)|.\end{aligned}$$ Since $Z_{{{\bf A}}_{{{\bf w}},{{\bf p}},Q}}$ imposes the condition that $\varphi_{\operatorname{TV}}$ maps $x_{ij}$ to $H_i$, we should (via a Künneth decomposition of the diagonal class) impose on $C_{ij}$ a condition that $p_{ij}$ maps to some curve $F_i^{\vee}\subset D^{\circ}_{0,i}$ which has intersection multiplicity $1$ with $F_i$ in $\operatorname{Bl}'_i$. The resulting Gromov-Witten contribution of $C_{ij}$ is then reduced to the computation from [@GPS Prop. 5.2] (which was based on [@BP Thm. 5.1]), yielding $$\begin{aligned}
\label{ijcover}
\frac{(-1)^{w_{ij}-1}}{|\pi_2(e_i)|w_{ij}^2}.\end{aligned}$$ Also, for each $p_i$, $i=1,\ldots,s$, we must glue a chain of copies of $\bb{P}^1$ in $\operatorname{Bl}_{p_i}$, each being a $|p_i|$-fold cover of a $\bb{P}^1$ in a fiber of $\operatorname{Bl}_{p_i}$ with maximal tangency at each intersection with $D_0^{\circ}$. Such chains contribute a factor of $1$ to the Gromov-Witten count.
Finally, multiplying the toric Gromov-Witten count from , the node-weights $\prod_{ij} w_{ij}|\pi_2(e_i)|$ from , and the multiple-cover contributions $\prod_{ij} \frac{(-1)^{w_{ij}-1}}{|\pi_2(e_i)|w_{ij}^2}$ from , and then dividing by $|\operatorname{Aut}({{\bf w}})|$ to correct for over-counting caused by labellings of the $x_{ij}$’s that are no longer remembered, we obtain by the degeneration formula that the contribution to of the curves coming from degree $\Delta_{{{\bf w}},{{\bf p}}}$ is precisely the corresponding term from the right-hand side of , and then summing over all ${{\bf w}}$ yields the desired result.
It is immediate from Proposition \[DegenProp\] that $\operatorname{Tr}^s(\vartheta_{p_1},\ldots,\vartheta_{p_s})$, as defined in , is indeed given as in . The theta functions were constructed as elements of a commutative associative algebra with $\vartheta_0=1$, so we already know that these properties are satisfied. The fact that $\operatorname{Tr}^2$ and $\operatorname{Tr}^3$ uniquely determine the multiplication was Lemma \[nondegen\]. The identification of the base ring with ${\Bbbk}\llb \operatorname{NE}(Y_{{{\bf S}}})\rrb$ is Lemma \[NECoeff\], and the finiteness statement for cases where the boundary supports an ample divisor was Corollary \[AmplePolynomial\]. The relation to the the [@GHKK] theta functions on the Langlands dual cluster variety was Remark \[LangRmk\].
The Gromov-Witten numbers are naive counts
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Here we show that the log Gromov-Witten numbers $\operatorname{GW}^{\log}_{0,Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\Delta_{{{\bf p}}},\beta}(Z,\Psi)$ of Proposition \[DegenProp\] are in fact naive counts of rational curves, not just virtual counts (assuming interior-curve freeness). We denote by $$\begin{aligned}
\operatorname{Forget}:\s{M}^{\log}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)\rar \?{\s{M}}_{0,s+2}\end{aligned}$$ the forgetful/stabilization morphism taking $[\varphi^{\dagger}:C^{\dagger}\rar Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}]$ to the stabilization of the marked curve $C^{\dagger}$. We allow any $s\geq 1$.
\[naive\] Suppose $[\varphi^{\dagger}:C^{\dagger}\rar Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}]\in \s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)$ satisfies generically chosen representatives for the incidence and $\psi$-class conditions $Z$ and $\Psi$ as in Proposition \[DegenProp\]. Then $\varphi(C)$ is torically transverse. Let $[\operatorname{pt}]_Y$ denote the class of a point in $Y_{{{\bf S}},\wt{\Sigma}}$ and let $[\operatorname{pt}]_{\s{M}}$ denote the class of a point in $\?{\s{M}}_{0,s+2}$. Then the condition $\psi_{\operatorname{out}}^{s-1}$ can be replaced by $\operatorname{Forget}^*[\operatorname{pt}]_{\s{M}}$, i.e., $$\begin{aligned}
\label{psiforget}
\operatorname{GW}^{\log}_{0,Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\Delta_{{{\bf p}}},\beta}(Z,\Psi) = \int_{[\s{M}^{\log}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)]^{\operatorname{vir}}} \operatorname{Forget}^*[\operatorname{pt}]_{\s{M}} \cup \operatorname{ev}_{\operatorname{out}}^*[\operatorname{pt}]_Y.\end{aligned}$$ Furthermore, if $(Y_{{{\bf S}},\wt{\Sigma}},D_{{{\bf S}},\wt{\Sigma}})$ is interior-curve free, then is given by the naive count of irreducible torically transverse genus $0$ algebraic curves of tropical degree $\Delta_{{{\bf p}}}$ and class $\beta$, with generically specified image under $\operatorname{Forget}$, and with marked point $x_{\operatorname{out}}$ mapping to a generically specified point $y\in Y_{{{\bf S}},\wt{\Sigma}}$. These are the only basic stable log maps satisfying the generically specified representatives of the point ans $\psi$-class conditions.
For the toric transversality statement, we recall that by Lemma \[DaggerCircLem\], we had toric transversality in the central fiber of the degeneration of §\[DegenS\], so the claim here follows from the fact that toric transversality is an open condition.
To prove , let $\?{\psi}_{\operatorname{out}}$ denote the corresponding $\psi$-class on $\?{\s{M}}_{0,s+2}$. It is standard that $\?{\psi}_{\operatorname{out}}^{s-1}$ is the class of a point in $\?{\s{M}}_{0,s+2}$. Furthermore, $\psi_{\operatorname{out}}-\operatorname{Forget}^*\?{\psi}_{\operatorname{out}}$ is supported on the locus where the forgetful map destabilizes the curve-component $C_{\operatorname{out}}$ containing $x_{\operatorname{out}}$. If $C_{\operatorname{out}}$ is destabilized when forgetting the map, then it intersects $D_{{{\bf S}},\wt{\Sigma}}$ in at most one point because all such intersections must be marked points or nodes by [@GSlog Rmk 1.9]. But Lemma \[ExpectedDim\] below ensures that an irreducible genus $0$ curve hitting $D_{{{\bf S}},\wt{\Sigma}}$ in at most one point will not hit the generically specified point $y\in Y_{{{\bf S}},\wt{\Sigma}}$, contradicting the requirement that $x_{\operatorname{out}}$ maps to $y$.
For the claim about naive counts, we will first show that any $C$ satisfying the described generically specified conditions is irreducible. Suppose $C$ were reducible, and let $C'$ be a component not containing $x_{\operatorname{out}}$. We proceed by induction on $s$.
Using the condition $\operatorname{Forget}^* [\operatorname{pt}]_{\s{M}}$, we can assume $C$ contains no contracted components. Combined with the fact that $C$ is torically transverse and the interior-curve free assumption, it follows that no components of $C$ can map entirely into the boundary. Let $C_0$ denote the closure in $C$ of the component of $C\setminus C'$ which contains $x_{\operatorname{out}}$, and let $C_1$ denote the closure in $C$ of $C\setminus C_0$. By the interior-curve free assumption, $C'$ intersects the boundary, and as before, [@GSlog Rmk 1.9] tells us that any intersection of $C$ with $D_{{{\bf S}},\wt{\Sigma}}$ is either a node or a marked point. It follows that $C_1$ contains at least one of the marked points $x_i$, $i=1,\ldots,s$.
By the interior-curve free assumption and the fact that $C$ has no contracted components, $C_0$ must also intersect the boundary. This already provides a contradiction when $s=1$, thus proving our base-case. Furthermore, it now follows from the $\operatorname{Forget}^* [\operatorname{pt}]_{\s{M}}$ condition that $C_1$ cannot contain more than one marked point. Hence, $C_1$ must be irreducible, and now Lemma \[ExpectedDim\] applies as before to say that the image of $C_1$ must lie in a fixed locus $E$ of codimension at least one which we can assume does not contain $y$. For convenience, let us assume that the unique $x_i$ in $C_1$ is $x_s$.
Next, note that applying $\operatorname{Forget}$ destabilizes $C_1$, and upon stabilization, $x_s$ is identified with the point $x_s':=C_0\cap C_1$. Treating $x_s'$ as an interior marked point on $C_0$, one obtains a basic stable log map $\varphi^{\dagger}_0:C^{\dagger}_0\rar Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}$. Let $\varphi'_0:C'_0\rar Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}$ be the basic stable log map obtained by forgetting the marking at $x_s'$. By the inductive assumption, $C_0$ must be irreducible. We can therefore apply Lemma \[ExpectedDim\] to say that generically specifying $y$ and the underlying marked curve $(C_0,x_1,\ldots,x_{s-1},x_{\operatorname{out}},x_{\infty})$ is sufficient to determine $\varphi_0'$ up to finitely many choices. But then the condition that $x_s'$ maps to $E$ further determines the location of $x_s'$ up to finitely many choices, even though this point should independently be generically specified by the $\operatorname{Forget}^*[\operatorname{pt}]_{\s{M}}$ condition. This contradiction completes the proof of the claim that $C$ is irreducible.
We have thus shown that all curves satisfying the imposed conditions must lie in the open substack of irreducible curves $\s{M}^{\log,\operatorname{irr}}_{0,\Delta_{{{\bf p}}}}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)\subset \s{M}^{\log}_{0,\Delta_{{{\bf p}}}}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta).$ The result now follows from another application of Lemma \[ExpectedDim\].
The author learned of the following result and proof from Sean Keel and Tony Yue Yu, who are using a similar argument in [@KY]. The argument has previously appeared in [@Yu2 Proof of Prop. 5.1].
\[ExpectedDim\] Let $\s{M}^{\log,\operatorname{irr}}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)\subset \s{M}^{\log}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)$ denote the open substack parametrizing basic stable log maps with irreducible domain curve $C$. Label one of the markings as $x_{\operatorname{out}}$, with $\operatorname{ev}_{\operatorname{out}}$ being the corresponding evaluation map. Then there exists a codimension $1$ subscheme $V\subset Y_{{{\bf S}},\wt{\Sigma}}$ such that $\s{M}_V^{\circ}:=\operatorname{ev}_{\operatorname{out}}^{-1}(Y_{{{\bf S}},\wt{\Sigma}}\setminus V) \cap \s{M}^{\log,\operatorname{irr}}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)$ is smooth and has the expected dimension, i.e., $\dim(\s{M}_V^{\circ})=\dim(Y_{{{\bf S}},\wt{\Sigma}})+|\Delta|-3$.
By [@Hart Ch. III, Prop. 10.6], the derivative $d\operatorname{ev}_{\operatorname{out}}$ is surjective except on some codimension $1$ subscheme of $Y_{{{\bf S}},\wt{\Sigma}}$, and we take this subscheme to be $V$. To prove the claim then, we wish to show that deformations of basic stable log maps in $\s{M}_V^{\circ}$ are unobstructed. Let $T^{\log}Y^{\dagger}_{{{\bf S}},\wt{\Sigma}}=TY_{{{\bf S}},\wt{\Sigma}}(-D_{{{\bf S}},\wt{\Sigma}})$ denote the log tangent bundle of $Y^{\dagger}_{{{\bf S}},\wt{\Sigma}}$. Note that $d\operatorname{ev}_{\operatorname{out}}$ is given at a point $[\varphi^{\dagger}:C^{\dagger}\rar Y_{{{\bf S}},\wt{\Sigma}}^{\dagger}]\in \s{M}^{\log,\operatorname{irr}}_{0,\Delta}(Y_{{{\bf S}},\wt{\Sigma}}^{\dagger},\beta)$ by the restriction $$\begin{aligned}
d\operatorname{ev}_{\operatorname{out}}:H^0(C,\varphi^*T^{\log}Y^{\dagger}_{{{\bf S}},\wt{\Sigma}})\rar (\varphi^*T^{\log}Y^{\dagger}_{{{\bf S}},\wt{\Sigma}})_{x_{\operatorname{out}}}.\end{aligned}$$ Since $C\cong \bb{P}^1$, we can apply [@Kollar Ch. II, Def.-Prop. 3.8] to say that $\varphi^*T^{\log}Y^{\dagger}_{{{\bf S}},\wt{\Sigma}}$ is semi positive, i.e., is isomorphic to a direct sum of line bundles of the form $\s{O}(a_i)$ for various $a_i\geq 0$. In follows that $H^1(C,\varphi^* T^{\log}Y^{\dagger}_{{{\bf S}},\wt{\Sigma}})=0$, and so $\s{M}_V^{\circ}$ is unobstructed, as desired.
Proposition \[naive\] says that we can indeed interpret the Gromov-Witten counts of Theorem \[MainThm\] as the naive counts described in Theorem \[MainNaive\]. The result follows.
Relation to quantum cohomology {#appendix}
==============================
Here we explain how to view the structure from Conjecture \[FrobConj\] (the Frobenius structure conjecture) as part of a natural extension of (small) quantum cohomology to the log setting. We begin by describing quantum cohomology in a somewhat new way.
Let $Y$ be a smooth projective variety over our algebraically closed field ${\Bbbk}$ of characteristic $0$. Let $\operatorname{NE}(Y)$ be the cone of effective curve classes in $Y$ up to numerical equivalence, and define $$\begin{aligned}
\operatorname{QH}^*(Y):=H^*(Y,\bb{Q}\llb \operatorname{NE}(Y)\rrb),\end{aligned}$$ viewed for now as a $\bb{Q}\llb \operatorname{NE}(Y)\rrb$-module. We define a $\bb{Q}\llb \operatorname{NE}(Y)\rrb$-multilinear $s$-point function $$\langle \cdot \rangle:\operatorname{QH}^*(Y)^{\otimes s} \rar \operatorname{QH}^*(Y)$$ as follows. Given $\alpha_1,\ldots,\alpha_s\in H^*(Y,\bb{Q})$, and letting $[Y]$ denote the Poincaré dual to the fundamental class of $Y$, define $$\langle \alpha_1,\ldots,\alpha_s\rangle\in H^*(Y,\bb{Q}\llb\operatorname{NE}(Y)\rrb)$$ in terms of Gromov-Witten invariants via: $$\begin{aligned}
\label{spointQH}
\langle \alpha_1,\ldots,\alpha_s\rangle := \sum_{\beta\in \operatorname{NE}(Y)} z^{\beta} \int_{[\s{M}_{0,s+2}(Y,\beta)]^{\operatorname{vir}}} \operatorname{ev}_{y_1}^*(\alpha_1) \wedge \cdots \wedge \operatorname{ev}_{y_s}^*(\alpha_s) \wedge \psi_{y_{s+1}}^{s-1}\operatorname{ev}_{y_{s+1}}^*[Y] \wedge \operatorname{ev}_{s+2}^*[Y].\end{aligned}$$ This is then extended $\bb{Q}\llb \operatorname{NE}(Y)\rrb$-multilinearly to all of $\operatorname{QH}^*(Y)^{\otimes s}$.
\[QHthm\] There exists a unique associative product $*$ on $\operatorname{QH}^*(Y)$ which makes this $\bb{Q}\llb \operatorname{NE}(Y)\rrb$-module into a $\bb{Q}\llb \operatorname{NE}(Y)\rrb$-algebra and which satisfies $$\begin{aligned}
\label{qprod}
\langle \alpha_1,\ldots,\alpha_s\rangle = \langle \alpha_1 \cdots \alpha_s\rangle
\end{aligned}$$ for all $s$ and all $\alpha_1,\ldots,\alpha_s\in \operatorname{QH}^*(Y)$. Furthermore, $\operatorname{QH}^*(Y)$ with this product $*$ is the small quantum cohomology ring of $Y$.
The proof is based on the axioms of Gromov-Witten theory and the Topological Recursion Relation. We suggest [@Gr pg. 39 and Prop. 2.12] as a reference for these properties.
For $s\geq 2$, we can apply the Fundamental Class Axiom of Gromov-Witten theory to rewrite the Gromov-Witten invariants from as $$\begin{aligned}
\int_{[\s{M}_{0,s+1}(Y,\beta)]^{\operatorname{vir}}} \operatorname{ev}_{y_1}^*(\alpha_1) \wedge \cdots \wedge \operatorname{ev}_{y_s}^*(\alpha_s) \wedge \psi_{y_{s+1}}^{s-2}\operatorname{ev}_{s+1}^*[Y].\end{aligned}$$ Now for $s=2$, the Point Mapping Axiom implies that $\langle \alpha_1,\alpha_2\rangle$ equals the usual Poincaré pairing of $\alpha_1$ and $\alpha_2$. For $s=3$, the Dilation Axiom implies that $$\begin{aligned}
\langle \alpha_1,\alpha_2,\alpha_3\rangle = \sum_{\beta\in \operatorname{NE}(Y)} z^{\beta} \int_{[\s{M}_{0,3}(Y,\beta)]^{\operatorname{vir}}} \operatorname{ev}_{y_1}^*(\alpha_1) \wedge \operatorname{ev}_{y_2}^*(\alpha_2) \wedge \operatorname{ev}_{y_3}^*(\alpha_3),\end{aligned}$$ i.e., the usual $3$-point function. So the requirement that $\langle \alpha_1*\alpha_2,\alpha_3\rangle = \langle \alpha_1,\alpha_2,\alpha_3\rangle$ is the usual defining property of the quantum cohomology product $*$.
It remains to check that the usual quantum cohomology product satisfies for all $s$. The $s=1$ case is trivial, and the $s=2$ case follows from the Point Mapping Axiom as above. The general case then follows by inductively applying the Topological Recursion Relation.
It now seems natural to wonder whether there is an analog $H_{\log}^*(Y^{\dagger},\bb{Z})$ in the log setting which yields a log quantum cohomology ring $\operatorname{QH}_{\log}^*(Y^{\dagger})$ via the same recipe as above. Indeed, the prime fundamental classes of our §\[Intro\] are precisely the degree $0$ log Chow classes of [@Bar; @Herr], and we expect that the invariants of could be defined as in , i.e.,[^16] $$\begin{aligned}
\langle p_1,\ldots,p_s\rangle := \sum_{\beta\in \operatorname{NE}(Y)} z^{\beta} \int_{[\s{M}_{0,s+2}(Y,\beta)]^{\operatorname{vir}}} \operatorname{ev}_{y_1}^*(\alpha_{p_1}) \wedge \cdots \wedge \operatorname{ev}_{y_s}^*(\alpha_{p_s}) \wedge \psi_{y_{s+1}}^{s-1}\operatorname{ev}_{y_{s+1}}^*[\operatorname{pt}] \wedge \operatorname{ev}_{s+2}^*[Y].\end{aligned}$$ Here, $\alpha_{p_i}$ denotes the prime fundamental class associated to $p_i\in B(\bb{Z})$. Also, note that restricting to degree $0$ here forces us to replace $\operatorname{ev}_{y_{s+1}}^*[Y]$ by $\operatorname{ev}_{y_{s+1}}^*[\operatorname{pt}]$ (the trace is only nonzero on top-degree elements, so we are using multiplication by the class of a point to pull the trace back to degree $0$ elements). So this is the sense in which Conjecture \[FrobConj\] is an extension of degree $0$ quantum cohomology to the log setting.
In higher-degree, progress is obstructed by the fact that it is not even clear how to define $H_{\log}^*(Y^{\dagger},\bb{Z})$. One idea is to use an extension of [@Bar; @Herr]’s log Chow groups which includes “punctured classes” (conditions one can impose on punctured invariants), and the author hopes this will yield an appropriate definition of log quantum cohomology.
Alternatively, Ganatra and Pomerleano [@GP; @GP2] suggest taking $$\begin{aligned}
H_{\log}^*(Y^{\dagger},\bb{Z}):=\bigoplus_{p\in B(\bb{Z})} H^*(D_p^{\operatorname{KN}},\bb{Z}), \end{aligned}$$ where $D_{p}^{\operatorname{KN}}$ denotes the Kato-Nakayama space [@KN] associated to $D_p^{\dagger}$ (i.e., $D_p$ with the log structure pulled back from inclusion into $Y^{\dagger}$). Gross-Pomerleano-Siebert [@GPomS] seek to prove that working with punctured invariants and this choice of $H_{\log}^*(Y^{\dagger},\bb{Z})$ yields the symplectic cohomolgy ring $\operatorname{SH}_{\log}^*(Y^{\dagger})$ of $(Y,D)$. Closed string mirror symmetry predicts that $\operatorname{SH}^*_{\log}(Y^{\dagger})$ is isomorphic to the ring of polyvector fields on the mirror, so in particular, $\operatorname{SH}^0_{\log}(Y^{\dagger})$ should be isomorphic to the coordinate ring of the mirror (cf. [@Pas §1]). In this sense, the Frobenius structure conjecture should be the degree $0$ part of the closed string mirror symmetry conjecture, as noted in the original statement [@GHK1 arXiv v1, Conj. 0.8], and also in [@GSInt pg. 5]. The observation that this construction is analogous to that of quantum cohomology was also previously noted by Gross and Siebert [@GSInt Rmk 2.3].
[^1]: We note that our prime fundamental classes do indeed form a basis for the degree $0$ classes in the log Chow group $A^0_{\log}(Y^{\dagger})$ as in [@Bar; @Herr].
[^2]: In addition to the affine cases, generic two-dimensional log Calabi-Yau varieties with maximal boundary are interior-curve free by [@GHK_MLP Prop. 4.1], so Theorem \[MainNaive\] applies to all the theta functions constructed in [@GHK1]. Interior-curve freeness is not needed for the Gromov-Witten version of our main result, Theorem \[MainThm\].
[^3]: This condition means that, for $i=1,\ldots,s$, if $\vartheta_i=[k_iD_i]$, then $\varphi(C)$ intersects $D_i$ at $x_i$ with order $k_i$, and furthermore, these account for all intersections of $\varphi(C)$ with $\wt{D}$.
[^4]: A curve $\varphi:C\rar Y$ is torically transverse if $\varphi(C)$ is disjoint from all codimension-two strata of $D$.
[^5]: This is similar to the spaces from the formal Fock-Goncharov conjecture of [@GHKK §6], but with a different formal completion.
[^6]: Since any orbifold points of $\wt{Y}$ are away from the boundary, one can use [@AGV] to extend the construction of the relevant moduli stacks and their intersection theory to our orbifold setting. See [@GPS §5.5] for similar considerations from the viewpoint of relative stable maps.
[^7]: We note that most these assumptions can always be achieved by modifying only the frozen parts of the data, i.e., without modifying $I_{\operatorname{uf}}$, $N_{\operatorname{uf}}$, or $B|_{N_{\operatorname{uf}}}$. The only exception is the assumption in 4(b) that $\pi_2(e_i)\neq 0$ for $i\in I_{\operatorname{uf}}$, but vectors failing this assumption can be forgotten from the seed data without affecting the spaces we consider.
[^8]: If all of $B$ is skew-symmetrizable, i.e., determined as in by a skew-symmetric form $\wt{\omega}$ on $N$ and rational numbers $\{d_i\}_{i\in I}$, then the interior of $\s{X}_{{{\bf S}},\Sigma}$ admits a Poisson structure given by $\{z^{e_i},z^{e_j}\}=d_id_j\wt{\omega}(e_i,e_j)z^{e_i+e_j}$, and the interior of $Y_{{\bf S}}$ is a symplectic leaf of this Poisson structure.
[^9]: We will not further discuss cluster $\s{A}$-varieties, but these spaces (with general coefficients) can be similarly obtained by blowing up partial compactifications of $T_N$ along loci of the form $D_{e_i}\cap Z(a_i+z^{\pi_1(e_i)})$ for $i\in I_{\operatorname{uf}}$, cf. [@GHK3 §3.2]. The map $\pi_2$ lifts to realize these as the universal torsors over the fibers of $\lambda_{\s{X}}$, cf. [@GHK3 §4] and [@ManCox].
[^10]: A version of Theorem \[kappa\] in dimension $2$ without frozen vectors or the corresponding boundary divisors was proven in [@GHK3 Thm. 5.5]. The argument here is inspired by the proof in loc. cit.
[^11]: Most of [@MRud] treats these weights as being $1$, but a generalization allowing for other weights appears in [@MRud Def. 4.17]. Multiplying $w(A_j)$ by $k$ just has the effect of multiplying the multiplicities of our tropical curves by $k$.
[^12]: In fact, as the author has learned from Sean Keel, $D_{{{\bf S}},\Sigma}$ supporting an ample divisor implies that $(Y_{{{\bf S}},\Sigma}, D_{{{\bf S}},\Sigma})$ is log Fano, hence that $\operatorname{NE}(Y_{{{\bf S}},\Sigma})$ itself is rational polyhedral by [@BCHM Cor. 1.3.2].
[^13]: In this version of tropicalization from [@GSlog §1.4], when a component of the basic log curve maps to the toric stratum corresponding to a cone $\sigma\in \wt{\Sigma}$, there is corresponding vertex of the tropical curve which can live anywhere in $\sigma$, with the precise location in $\sigma$ depending on a choice of pullback to the standard log point. Nodes (respectively, marked points) of the log curve then correspond to compact edges (respectively, non-compact edges) of the tropicalization. The somewhat-genericness of the incidence conditions ensures that their tropicalizations pass through the origin.\[TropFoot\]
[^14]: Note that Lemma \[DaggerCircLem\] makes sense (and will hold) even in the $s=1$ case thanks to our convention of including the extra marked point $x_{\infty}$.
[^15]: Alternatively, we may use the main result of [@AMW] to say that the log Gromov-Witten invariants agree with the corresponding relative Gromov-Witten invariants, and then we may apply the relative degeneration formula of [@Li]. A more general degeneration formula allowing for non-smooth boundary divisors is also in progress [@ACGS].
[^16]: The formalism needed to actually define the log Gromov-Witten invariants in this way is still being developed by Barrott and Herr.
|
---
abstract: 'We discuss simulations of finite temperature nuclear matter on the lattice. We introduce a new approximation to nucleon matrix determinants that is physically motivated by chiral effective theory. The method involves breaking the lattice into spatial zones and expanding the determinant in powers of the boundary hopping parameter.'
author:
- 'Dean J. Lee'
- 'Ilse C.F. Ipsen'
bibliography:
- 'NuclearMatter.bib'
title: Zone Determinant Expansions for Nuclear Lattice Simulations
---
Introduction
============
We consider quantum simulations of nuclear matter on the lattice. In particular we address the problem of calculating the contribution of nucleon/nucleon-hole loops at nonzero nucleon density. With the help of auxiliary boson fields, all nucleon interactions can be written in terms of one-body interactions in a fluctuating background. In the grand canonical ensemble, the contribution of nucleon/nucleon-hole loops to the partition function equals the determinant of the one-body interaction matrix. Since the determinant of the interaction matrix for a general boson field configuration is not positive, stochastic methods such as hybrid Monte Carlo [@Scalettar:1986uy][@Gottlieb:1987mq][@Duane:1987de] do not give the sign or phase of the determinant. Instead one must rely on much slower and more memory intensive algorithms based on LU factorization, which decomposes matrices in terms of a product of upper and lower triangular matrices.
The number of required operations in LU factorization for an $n\times n$ matrix scales as $n^{3}$. It has been shown in the literature that repeated calculations of matrix determinants with only localized changes can be streamlined in various ways [@Scalapino:1981qs][@Gubernatis:1992a]. However it is difficult to avoid the poor scaling inherent in the method. If $V$ is the spatial volume and $\beta$ is the inverse temperature measured in lattice units, a simulation that includes nucleon/nucleon-hole loops requires $(V\beta)^{3}$ times more operations than the corresponding quenched simulation without loops. This slowdown should not be confused with the infamous fermion sign or phase problem [^1] which becomes significant at temperatures $T\leq1$ MeV. The computational bottleneck we are considering is due to the inefficiencies of the algorithm and persists at all temperatures. It is this numerical challenge which sets current limits on nuclear lattice simulations.
In this paper we introduce a new approach to approximating nucleon matrix determinants. We begin with a review of the current status of nuclear matter simulations on the lattice and look to chiral effective theory to determine the relative importance of various interactions. We then introduce the concept of spatial zones and suggest a new expansion of the nucleon determinant in powers of the hopping parameter connecting neighboring zones. Rigorous bounds on the convergence of this expansion are given as well as an estimate of the required size of the spatial zones as a function of temperature. We apply the expansion to a realistic lattice simulation of the interactions of neutrons and neutral pions.
Nuclear lattice simulations
===========================
Recently Müller, Koonin, Seki, and van Kolck* *[@Muller:1999cp], pioneered the study of quantum many body effects in infinite nuclear matter at finite density and temperature. In their work they considered only nucleon degrees of freedom and used an effective Hamiltonian of the form$$H=K+V_{c}+V_{\sigma}\text{,}$$ where $K$ is the kinetic energy, $V_{c}$ is the two-body scalar potential, and $V_{\sigma}$ is the two-body tensor potential. While the simulation was done on the lattice we will write their Hamiltonian in the more familiar continuum language. In the continuum $K$, $V_{c}$, and $V_{\sigma}$ take the form$$K=-\tfrac{1}{2m_{N}}\int d^{3}\vec{x}\psi_{\sigma\tau}^{\dagger}\vec{\nabla
}^{2}\psi_{\sigma\tau},$$$$V_{c}=\tfrac{1}{2}\int d^{3}\vec{x}d^{3}\vec{x}^{\prime}\psi_{\sigma\tau
}^{\dagger}(\vec{x})\psi_{\sigma^{\prime}\tau^{\prime}}^{\dagger}(\vec
{x}^{\prime})V_{c}(\vec{x}-\vec{x}^{\prime})\psi_{\sigma^{\prime}\tau^{\prime
}}(\vec{x}^{\prime})\psi_{\sigma\tau}(\vec{x}),$$$$V_{\sigma}=\tfrac{1}{2}\int d^{3}\vec{x}d^{3}\vec{x}^{\prime}\psi_{\xi\tau
}^{\dagger}(\vec{x})\psi_{\xi^{\prime}\tau^{\prime}}^{\dagger}(\vec{x}^{\prime})V_{\sigma}(\vec{x}-\vec{x}^{\prime})\vec{\sigma}_{\xi\tau
\kappa\lambda}\cdot\vec{\sigma}_{\xi^{\prime}\tau^{\prime}\kappa^{\prime
}\lambda^{\prime}}\psi_{\kappa^{\prime}\lambda^{\prime}}(\vec{x}^{\prime})\psi_{\kappa\lambda}(\vec{x}).$$ In our notation summations are implied over repeated indices. $\ m_{N}$ is the nucleon mass. $\ \psi_{\sigma\tau}^{\dagger}(\psi_{\sigma\tau})$ creates(annihilates) a nucleon of spin $\sigma$ and isospin $\tau,$ and $\vec{\sigma}_{\xi\tau\kappa\lambda}$ are the elements of a generalized Pauli spin-isospin matrix. Both potentials are assumed to have Skyrme-like on-site and next-nearest-neighbor interactions, $$V_{c}(\vec{x}-\vec{x}^{\prime})=V_{c}^{(0)}\delta(\vec{x}-\vec{x}^{\prime
})+V_{c}^{(2)}\vec{\nabla}^{2}\delta(\vec{x}-\vec{x}^{\prime}),$$ $$V_{\sigma}(\vec{x}-\vec{x}^{\prime})=V_{\sigma}^{(0)}\delta(\vec{x}-\vec
{x}^{\prime})+V_{\sigma}^{(2)}\vec{\nabla}^{2}\delta(\vec{x}-\vec{x}^{\prime
}).$$
The grand canonical partition function is given by$$Z=Tr\left[ \exp\left[ -\beta\left( H-\mu_{\tau}n_{\tau}\right) \right]
\right] ,$$ where $\mu_{\tau}$ is the isospin-dependent chemical potential and $n_{\tau}$ is the nucleon number operator for isospin index $\tau$. We can rewrite the quartic interactions in $V_{c}$ and $V_{\sigma}$ using the Hubbard-Stratonovich transformation [@Stratonovich:1958][@Hubbard:1959ub]. The Hubbard-Stratonovich transformation uses the identity,$$\exp(\tfrac{1}{2}A^{2})=\sqrt{2\pi}\int_{-\infty}^{\infty}d\varphi\exp
(-\tfrac{1}{2}\varphi^{2}-\varphi A),$$ where $A$ is any quantum operator. This allows the mapping of the interacting nucleon problem to a system of noninteracting nucleons coupled to a fluctuating background field. With this transformation the expectation value of any observable $O$ can be written as$$\left\langle O\right\rangle =\frac{\int D\phi G(\phi)\det(M(\phi))O(\phi
)}{\int D\phi G(\phi)\det(M(\phi))},$$ where $\phi$ collectively represents the Hubbard-Stratonovich fields (as well as any other bosonic fields), $M(\phi)$ is the one-body nucleon interaction matrix, and $G(\phi)$ is a function of the $\phi$’s.
Using this formalism Müller, *et. al.*, were able to measure the thermodynamic properties of nuclear matter and find signs of a first order phase transition from an uncorrelated Fermi gas to a clustered phase. They examined temperatures from 3.0 MeV to 100 MeV with up to 30 time steps and a spatial volume of $4^{3}$ with lattice spacing $a=1.842$ fm$.$ Unfortunately the LU factorization algorithm used to compute determinants in the simulation scales as $(V\beta)^{3}$ and thus going beyond lattice systems of size $4^{3}$ is problematic.
Chiral effective theory and nuclear forces
==========================================
There have been several recent efforts to describe nuclear forces starting from chiral effective theory. This line of study was initiated by Weinberg [@Weinberg:1990rz][@Weinberg:1991um][@Weinberg:1992yk]. The idea is to expand the nuclear interactions in powers of $q/\Lambda_{\chi}$, where $q$ is the typical external momentum of the nucleons and $\Lambda_{\chi}$ is the chiral symmetry breaking scale or equivalently the hadronic mass scale ($\sim1$ GeV). The momentum cutoff scale $\Lambda$ for the effective theory is set below $\Lambda_{\chi}$, and the renormalization group flow of operator coefficients from scale $\Lambda_{\chi}$ to $\Lambda$ suppress the effects of higher dimensional operators. Chiral symmetry in the limit of zero quark mass imposes additional constraints on the possible momentum and spin dependence of the interaction terms. Assuming naturalness for the renormalized coupling constants in the Lagrangian at the scale $\Lambda_{\chi
}$, one expects in the low energy effective theory that contributions to nucleon forces from operators at higher order in the chiral expansion are negligible.
Weinberg’s work was followed by applications of chiral effective theory to the nucleon potential [@Ordonez:1996rz] and alternative approaches to power counting without apparent fine tuning in the presence of long scattering lengths [@Kaplan:1998tg][@Kaplan:1998we]. Recent low energy studies [@Epelbaum:1998na][@Epelbaum:1998hg][@Epelbaum:1998ka][@Epelbaum:2002vt] have also integrated out pion fields to produce energy independent two- and three- nucleon potentials, and the effective theory approach has been used to calculate nuclear spectra as well as phase shifts and scattering lengths which compare favorably with potential model calculations.
In Weinberg’s power counting scheme one deals with infrared singularities in bound state problems by distinguishing between reducible and irreducible diagrams. Reducible diagrams are those that can be disconnected by cutting internal lines that correspond with particles in the initial or final state. In the notation of [@Ordonez:1996rz], the power of $q/\Lambda_{\chi}$ for any irreducible or non-reducible diagram is given by$$\nu=4-\tfrac{E_{f}}{2}+2L-2C+\sum_{i}V_{i}\delta_{i},$$ where $E_{f}$ is the number of external nucleon lines, $L$ is the number of loops, $C$ is the number of connected pieces, $V_{i}$ is the number of vertices of type $i$, and $\delta_{i}$ is the index of vertex $i$. The index $\delta_{i}$ is given by$$\delta_{i}=d_{i}+\tfrac{f_{i}}{2}-2\text{,}$$ where $d_{i}$ is the number of derivatives and $f_{i}$ is the number of nucleon fields in the vertex.
We let $N$ represent the nucleon fields,$$N=\left[
\begin{array}
[c]{c}p\\
n
\end{array}
\right] \otimes\left[
\begin{array}
[c]{c}\uparrow\\
\downarrow
\end{array}
\right] .$$ We use $\tau_{i}$ to represent Pauli matrices acting in isospin space, and we use $\vec{\sigma}$ to represent Pauli matrices acting in spin space. Pion fields are notated as $\pi_{i}$, and $\mu$ is the nucleon chemical potential. We denote the pion decay constant as $F_{\pi}\approx183$ MeV and let $$D=1+\pi_{i}^{2}/F_{\pi}^{2}.$$ The lowest order Lagrange density for low energy pions and nucleons is given by terms with $\delta_{i}=0$, $$\begin{aligned}
\mathcal{L}^{(0)} & =-\tfrac{1}{2}D^{-2}\left[ (\vec{\nabla}\pi_{i})^{2}-\dot{\pi}_{i}^{2}\right] -\tfrac{1}{2}D^{-1}m_{\pi}^{2}\pi_{i}^{2}+\bar{N}[i\partial_{0}-(m_{N}-\mu)]N\nonumber\\
& -D^{-1}F_{\pi}^{-1}g_{A}\bar{N}\left[ \tau_{i}\vec{\sigma}\cdot\vec
{\nabla}\pi_{i}\right] N-D^{-1}F_{\pi}^{-2}\bar{N}[\epsilon_{ijk}\tau_{i}\pi_{j}\dot{\pi}_{k}]N\nonumber\\
& -\tfrac{1}{2}C_{S}\bar{N}N\bar{N}N-\tfrac{1}{2}C_{T}\bar{N}\vec{\sigma
}N\cdot\bar{N}\vec{\sigma}N.\end{aligned}$$ $g_{A}$ is the nucleon axial coupling constant, and $\epsilon_{ijk}$ is the Levi-Civita symbol. For the purposes of power counting, the pion mass $m_{\pi
}$ is equivalent to one power of the momentum $q$. The nucleon mass term $\bar{N}N$ actually has index $\delta_{i}=-1$. However the coefficient of this term is fine-tuned using $\mu$ to set the nucleon density, and so the $\bar{N}N$ term is reduced to the same size as other terms with index $\delta_{i}=0$. At next order we have terms with $\delta_{i}=1$,$$\mathcal{L}^{(1)}=\tfrac{1}{2m_{N}}\bar{N}\vec{\nabla}^{2}N+...$$ The important point for our discussion is that the lowest order Lagrange density $\mathcal{L}^{(0)}$ describes static nucleons. Spatial hopping of nucleons first appears at subleading index $\delta_{i}=1$. This suggests that in some cases one could compute the determinant of the nucleon interaction matrix as an expansion in powers of the spatial hopping parameter. We should note one point of caution though. The $\mathcal{L}^{(1)}$ kinetic energy term cannot be ignored if the infrared singularities of $\mathcal{L}^{(0)}$ are not properly dealt with. Since diagrammatic methods are not used in nuclear matter Monte Carlo simulations, we cannot separate reducible and irreducible diagrams. However if the simulation is done at non-zero temperature $T$ then that will serve to regulate the infrared singularities. We will explicitly see the effect of temperature on the convergence of the expansion later in our discussion.
Spatial zones
=============
In [@Lee:2003a] spatial zones were used to calculate the chiral condensate for massless QED in three dimensions. In that paper the main problem was dealing with sign and phase problems that arise in the Hamiltonian worldline formalism with explicit fermions. The idea of the zone method is that fermions at inverse temperature $\beta$ with spatial hopping parameter $h$ have a localization length of$$l\sim\sqrt{\beta h}.$$ We now apply the zone idea to our determinant problem. Let $M$ be the nucleon matrix, in general an $n\times n$ complex matrix. We partition the lattice spatially into separate zones such that the length of each zone is much larger than the localization length $l$. Since most nucleon worldlines do not cross the zone boundaries, they would not be affected if we set the zone boundary hopping terms to zero. Hence we anticipate that the determinant of $M$ can be approximated by the product of the submatrix determinants for each spatial zone.
Let us partition the lattice into spatial zones labelled by index $j.$ Let $\{P_{j}\}$ be a complete set of matrix projection operators that project onto the lattice sites within spatial zone $j$. We can write$$M=\sum_{i,j}P_{j}MP_{i}=M_{0}+M_{E},$$ where$$\begin{aligned}
M_{0} & =\sum_{i}P_{i}MP_{i},\\
M_{E} & =\sum_{i\neq j}P_{j}MP_{i}.\end{aligned}$$ If the zones can be sorted into even and odd sets so that$$P_{j}MP_{i}=0$$ whenever $i$ is even and $j$ is odd or vice-versa, then we say that the zone partitioning is bipartite. We now have$$\begin{aligned}
\det(M) & =\det(M_{0})\det(1+M_{0}^{-1}M_{E})\nonumber\\
& =\det(M_{0})\exp(\text{trace}(\log(1+M_{0}^{-1}M_{E})))\text{.}$$ Using an expansion for the logarithm, we have$$\det(M)=\det(M_{0})\exp\left( \sum_{p=1}^{\infty}\frac{(-1)^{p-1}}{p}\text{trace}((M_{0}^{-1}M_{E})^{p})\right) .$$
Let us define $$\Delta_{m}=\det(M_{0})\exp\left( \sum_{p=1}^{m}\frac{(-1)^{p-1}}{p}\text{trace}((M_{0}^{-1}M_{E})^{p})\right) .$$ Let $\lambda_{k}(M_{0}^{-1}M_{E})$ be the eigenvalues of $M_{0}^{-1}M_{E}$ and $R$ be the spectral radius,$$R=\max_{k=1,...,n}(\left\vert \lambda_{k}(M_{0}^{-1}M_{E})\right\vert ).$$ It has been shown [@Ipsen:2003] that for $R<1$$$\frac{\left\vert \det(M)-\Delta_{m}\right\vert }{\left\vert \Delta
_{m}\right\vert }\leq cR^{m}e^{cR^{m}} \label{bound}$$ where$$c=-n\log(1-R). \label{prefactor}$$ The spectral radius $R$ determines the convergence of our expansion. $R$ can be reduced by increasing the size of the spatial zone relative to the localization length $l$. In the special case where the zone partitioning is bipartite, we note that for any odd $p,$$$\text{trace}((M_{0}^{-1}M_{E})^{p})=0.$$ In that case $$\Delta_{2m+1}=\Delta_{2m},$$ and so we gain an extra order of accuracy without additional work.
Application to neutron matter simulations
=========================================
We now illustrate the zone determinant expansion for a simple but realistic lattice simulation of neutron matter. The formalism we use is a merger of chiral effective theory and Euclidean lattice methods. In our analysis we focus on the convergence and accuracy of the zone determinant expansion method. In order to provide a detailed quantitative assessment we compare the zone determinant approximations with exact determinant results. Given the severe limitations of the exact determinant method, we are not able to probe large volumes nor comment on finite volume scaling. However a full account of results for large volume simulations as well as an introduction to the new approach combining chiral effective theory and lattice methods is forthcoming in another paper [@Borasoy:2003].
Our starting point is the same as that of Weinberg [@Weinberg:1990rz]. We start with the most general local Lagrange density involving pions and low-energy nucleons consistent with Lorentz and translational invariance, isospin symmetry, and spontaneously broken chiral symmetry. This will produce an infinite set of interaction terms with increasing numbers of derivatives and/or nucleon fields. The dependence of each term on the pion field is governed by the rules of spontaneously broken chiral symmetry. Degrees of freedom associated with heavier mesons such as the $\rho$, heavier baryons such as the $\Delta$’s as well as antinucleons are all integrated out. We also integrate out nucleons with momenta greater than the cutoff scale $\Lambda$. The contribution of these effects appear as coefficients of local terms in our pion-nucleon Lagrangian.
For simplicity we consider only neutrons and neutral pions. We let $\psi$ represent the neutron spin states,$$\psi=\left[
\begin{array}
[c]{c}\uparrow\\
\downarrow
\end{array}
\right] .$$ The terms in our effective pion-nucleon Euclidean action are $$S=S_{\pi\pi}+S_{\bar{N}N}+S_{\pi\bar{N}N}+S_{\bar{N}N\bar{N}N},$$ where$$S_{\pi\pi}={\textstyle\int}
d^{3}\vec{r}dr_{4}\left[ \tfrac{1}{2}(\tfrac{\partial\pi_{0}}{\partial r_{4}})^{2}+\tfrac{1}{2}(\vec{\nabla}\pi_{0})^{2}+\tfrac{1}{2}m_{\pi}^{2}\pi
_{0}^{2}\right] ,$$$$S_{\bar{N}N}={\textstyle\int}
d^{3}\vec{r}dr_{4}\left[ \psi^{\dagger}\tfrac{\partial\psi}{\partial r_{4}}-\psi^{\dagger}\tfrac{\vec{\nabla}^{2}\psi}{2m_{N}}+(m_{N}-\mu)\psi^{\dagger
}\psi\right] ,$$$$S_{\pi\bar{N}N}={\textstyle\int}
d^{3}\vec{r}dr_{4}\left[ -g\psi^{\dagger}\left( \vec{\sigma}\cdot\vec
{\nabla}\pi_{0}\right) \psi\right] ,$$$$S_{\bar{N}N\bar{N}N}={\textstyle\int}
d^{3}\vec{r}dr_{4}\left[ \tfrac{1}{2}C\text{:}\psi^{\dagger}\psi\psi
^{\dagger}\psi\text{:}\right] . \label{fourfermi}$$ We have kept terms in the lowest order chiral Lagrange density $\mathcal{L}^{(0)}$ containing neutrons and neutral pions. We have dropped the factors of $D^{-1}$ which at lowest order contribute to multipion processes. We have also chosen to include the neutron kinetic energy term even though it appears in $\mathcal{L}^{(1)}$. This is useful if we wish to recover the exact free neutron Fermi gas in the weak coupling limit.
On the lattice we let $a$ be the spatial lattice spacing and $a_{t}$ be the temporal lattice spacing. We use the notation $\hat{1},\hat{2},\hat{3},$ or $\hat{4}$ to represent vectors that extend exactly one lattice unit (either $a$ or $a_{t}$) in the respective direction. We use dimensionless lattice fields and dimensionless masses and couplings by multiplying by the corresponding power of $a$. For example, $\hat{m}_{\pi}=m_{\pi}\cdot a$, $\hat{m}_{N}=m_{N}\cdot a$, $\hat{g}=g\cdot a^{-1}$, $\hat{\mu}=\mu\cdot a$, and $\hat{C}=C\cdot a^{-2}$. We use standard fermion path integral conventions at finite time step [@Soper:1978dp][@Rothe:1997kp] and define$$\psi^{\prime}(\vec{n})=\psi(\vec{n}-\hat{4})$$ in order to write the lattice path integral in standard form. The lattice actions have the form$$\begin{aligned}
S_{\pi\pi} & =-\tfrac{a}{a_{t}}\sum_{\vec{n}}\pi(\vec{n})\pi(\vec{n}+\hat
{4})-\tfrac{a_{t}}{a}\sum_{\vec{n},l=1,2,3}\pi(\vec{n})\pi(\vec{n}+\hat
{l})\nonumber\\
& +((\tfrac{\hat{m}_{\pi}^{2}}{2}+3)\tfrac{a_{t}}{a}+\tfrac{a}{a_{t}})\sum_{\vec{n}}(\pi(\vec{n}))^{2},\end{aligned}$$$$\begin{aligned}
S_{\bar{N}N} & =\sum_{\vec{n}}\psi^{\dagger}(\vec{n})\psi^{\prime}(\vec
{n}+\hat{4})-\tfrac{a_{t}}{2\hat{m}_{N}a}\sum_{\vec{n},l=1,2,3}(\psi^{\dagger
}(\vec{n})\psi^{\prime}(\vec{n}+\hat{l})+\psi^{\dagger}(\vec{n})\psi^{\prime
}(\vec{n}-\hat{l}))\nonumber\\
& +(-1+(\hat{m}_{N}+\tfrac{3}{\hat{m}_{N}})\tfrac{a_{t}}{a}-\hat{\mu}\tfrac{a_{t}}{a})\sum_{\vec{n}}\psi^{\dagger}(\vec{n})\psi^{\prime}(\vec{n}),\end{aligned}$$$$\begin{aligned}
S_{\pi\bar{N}N} & =-\tfrac{\hat{g}a_{t}}{2a}{\displaystyle\sum_{\vec{n}}}
\left[ \left[ \psi_{\uparrow}^{\ast}(\vec{n})\psi_{\uparrow}^{\prime}(\vec{n})-\psi_{\downarrow}^{\ast}(\vec{n})\psi_{\downarrow}^{\prime}(\vec
{n})\right] \left[ \pi(\vec{n}+\hat{3})-\pi(\vec{n}-\hat{3})\right] \right]
\nonumber\\
& -\tfrac{\hat{g}a_{t}}{2a}{\displaystyle\sum_{\vec{n}}}
\left[ \psi_{\uparrow}^{\ast}(\vec{n})\psi_{\downarrow}^{\prime}(\vec
{n})\left[ \pi(\vec{n}+\hat{1})-\pi(\vec{n}-\hat{1})-i\pi(\vec{n}+\hat
{2})+i\pi(\vec{n}-\hat{2})\right] \right] \nonumber\\
& -\tfrac{\hat{g}a_{t}}{2a}{\displaystyle\sum_{\vec{n}}}
\left[ \psi_{\downarrow}^{\ast}(\vec{n})\psi_{\uparrow}^{\prime}(\vec
{n})\left[ \pi(\vec{n}+\hat{1})-\pi(\vec{n}-\hat{1})+i\pi(\vec{n}+\hat
{2})-i\pi(\vec{n}-\hat{2})\right] \right] ,\end{aligned}$$$$S_{\bar{N}N\bar{N}N}=\tfrac{\hat{C}a_{t}}{a}\sum_{\vec{n}}\psi_{\uparrow
}^{\ast}(\vec{n})\psi_{\uparrow}^{\prime}(\vec{n})\psi_{\downarrow}^{\ast
}(\vec{n})\psi_{\downarrow}^{\prime}(\vec{n}).$$ We can reexpress $S_{\bar{N}N\bar{N}N}$ using a discrete Hubbard-Stratonovich transformation [@Hirsch:1983] for $\hat{C}\leq0$, $$\begin{aligned}
& \exp(-\tfrac{\hat{C}a_{t}}{a}\psi_{\uparrow}^{\ast}(\vec{n})\psi_{\uparrow
}^{\prime}(\vec{n})\psi_{\downarrow}^{\ast}(\vec{n})\psi_{\downarrow}^{\prime
}(\vec{n}))\nonumber\\
& =\tfrac{1}{2}\sum_{s=\pm1}\exp\left[ -(\tfrac{\hat{C}a_{t}}{2a}+\lambda
s)(\psi_{\uparrow}^{\ast}(\vec{n})\psi_{\uparrow}^{\prime}(\vec{n})+\psi_{\downarrow}^{\ast}(\vec{n})\psi_{\downarrow}^{\prime}(\vec
{n})-1)\right] ,\end{aligned}$$ where $\lambda$ is defined by$$\cosh\lambda=\exp(-\tfrac{\hat{C}a_{t}}{2a}).$$
Results
=======
For the simulation results presented in this section we use the values $a^{-1}=150$ MeV, $a_{t}^{-1}=225$ MeV, $C=-4.0\cdot10^{-5}\ $MeV$^{-2}$, and $g=6.8\cdot10^{-3}$ MeV$^{-1}$. The value of $g$ is set according to the tree level Goldberger-Treiman relation [@Goldberger:1958vp]$$g=F_{\pi}^{-1}g_{A}=6.8\cdot10^{-3}\text{ MeV}^{-1}.$$ The value of $C$ is tuned to match the $s$-wave phase shifts on the lattice for nucleon scattering at lattice spacing $(150$ MeV$)^{-1}$. The calculation of phase shifts on the lattice is discussed in a forthcoming article which details the entire formalism [@Borasoy:2003].
We present data for three independent pion and Hubbard-Stratonovich field configurations on a $4^{3}\times6$ lattice at temperature $T=37.5$ MeV and neutron density $\rho=$ 0.57$\rho_{\text{nucl}}$ where nuclear density is $$\rho_{\text{nucl}}=2.8\cdot10^{14}\text{ g/cm}^{3}=1.2\cdot10^{9}\text{
MeV}^{4}\text{.}$$ We refer to spatial zones according to their $x,y,z$ lattice dimensions $[n_{x},n_{y},n_{z}].$ At this lattice spacing and temperature our localization length estimate is$$l\sim\sqrt{\beta h}=0.57,$$ and so we expect the zone determinant expansion to converge for even the smallest zones, $[n_{x},n_{y},n_{z}]=[1,1,1],$ consisting of groups of lattice points with the same spatial coordinate. In Table 1 we show the spectral radius $R$ of $M_{0}^{-1}M_{E}$ and the determinant expansion for $[1,1,1]$ zones for three independent pion and Hubbard-Stratonovich configurations at equilibrium. $$\begin{tabular}
[c]{||l|l|l|l||}\hline\hline
configuration & $\#1$ & $\#2$ & $\#3$\\\hline
$R$ & $0.538$ & $0.523$ & $0.534$\\\hline
$\log(\det(M))$ & $16.4727+0.3666i$ & $18.1193+0.4479i$ & $18.2612-0.1811i$\\\hline
$\log(\Delta_{0})$ & $12.1700+0.3807i$ & $13.7933+0.4900i$ & $14.0060-0.2075i$\\\hline
$\log(\Delta_{2})$ & $16.4964+0.3686i$ & $18.1792+0.4477i$ & $18.2841-0.1801i$\\\hline
$\log(\Delta_{4})$ & $16.4959+0.3659i$ & $18.1329+0.4468i$ & $18.2856-0.1805i$\\\hline
$\log(\Delta_{6})$ & $16.4664+0.3667i$ & $18.1147+0.4482i$ & $18.2548-0.1813i$\\\hline
$\log(\Delta_{8})$ & $16.4739+0.3666i$ & $18.1203+0.4478i$ & $18.2622-0.1810i$\\\hline\hline
\end{tabular}$$
Table 1: Convergence of determinant series for $[1,1,1]$ zones
We see that the spectral radius $R$ is less than $1$ as expected from the localization length estimate. We also find that the rigorous bounds in (\[bound\]) are satisfied. Since most of the eigenvalues of $M_{0}^{-1}M_{E}$ are much smaller in magnitude than the spectral radius $R$, we observe that the prefactor $c$ in (\[bound\]) is actually much larger than needed for these examples. From the data in Table 1 we find empirically$$\frac{\left\vert \det(M)-\Delta_{2m}\right\vert }{\left\vert \Delta
_{2m}\right\vert }=\frac{\left\vert \det(M)-\Delta_{2m+1}\right\vert
}{\left\vert \Delta_{2m+1}\right\vert }\sim R^{2m+1}.$$
We now repeat the same analysis with the same lattice dimensions and temperature but at much higher density, $\rho=$ 1.67$\rho_{\text{nucl}}$. The results for three independent pion and Hubbard-Stratonovich field configurations are shown in Table 2.$$\genfrac{}{}{0pt}{}{\begin{tabular}
[c]{||l|l|l|l||}\hline\hline
configuration & $\#1$ & $\#2$ & $\#3$\\\hline
$R$ & $0.520$ & $0.478$ & $0.536$\\\hline
$\log(\det(M))$ & $70.6349+0.4121i$ & $76.1762+0.7098i$ & $73.4831-0.1576i$\\\hline
$\log(\Delta_{0})$ & $64.8000+0.4406i$ & $71.0009+0.7989i$ & $67.8503-0.1809i$\\\hline
$\log(\Delta_{2})$ & $70.9530+0.4120i$ & $76.3927+0.7005i$ & $73.7923-0.1538i$\\\hline
$\log(\Delta_{4})$ & $70.6032+0.4114i$ & $76.1605+0.7106i$ & $73.4483-0.1583i$\\\hline
$\log(\Delta_{6})$ & $70.6390+0.4124i$ & $76.1778+0.7097i$ & $73.4888-0.1574i$\\\hline
$\log(\Delta_{8})$ & $70.6343+0.4120i$ & $76.1760+0.7098i$ & $73.4819-0.1576i$\\\hline\hline
\end{tabular}
}{\text{Table 2:\ \ Determinant series for [1,1,1] zones at }\rho
=1.67\rho_{\text{nucl}}}$$ Comparing Tables 1 and 2, we conclude that the zone determinant expansion appears to be unaffected by the increase in nucleon density.
On the other hand as the temperature decreases, the localization length increases and therefore the convergence of the zone determinant expansion should slow down. In Table 3 we show the expansion for a $6^{3}\times6$ lattice at $T=37.5$ MeV, $6^{3}\times9$ lattice at $T=25.0$ MeV, $6^{3}\times12$ lattice at $T=18.8$ MeV, $6^{3}\times18$ lattice at $T=12.5$ MeV. The chemical potential is kept at the same value used for the results in Table 1 ($\mu=0.8m_{N})$. $$\begin{tabular}
[c]{||l|l|l|l|l||}\hline\hline
$T$ & $37.5$ MeV & $25.0$ MeV & $18.8$ MeV & $12.5$ MeV\\\hline
$R$ & $0.5122$ & $0.6742$ & $0.7631$ & $0.8638$\\\hline
$\log(\det(M))$ & $65.8009-0.7250i$ & $37.6912-1.5930i$ & $16.0023+0.0541i$ &
$5.7618+0.0452i$\\\hline
$\log(\Delta_{0})$ & $51.3988-0.7888i$ & $20.7653-1.5752i$ & $4.1999+0.2033i$
& $0.1796-0.0019i$\\\hline
$\log(\Delta_{2})$ & $66.0028-0.7218i$ & $36.2570-1.6535i$ & $11.7590+0.0861i$
& $1.5071+0.0222i$\\\hline
$\log(\Delta_{4})$ & $65.8289-0.7243i$ & $38.0872-1.5800i$ & $15.9390+0.0337i$
& $3.8579+0.0399i$\\\hline
$\log(\Delta_{6})$ & $65.7926-0.7253i$ & $37.6528-1.5924i$ & $16.3121+0.0478i$
& $5.5534+0.0475i$\\\hline
$\log(\Delta_{8})$ & $65.8026-0.7249i$ & $37.6780-1.5943i$ & $16.0111+0.0575i$
& $6.0067+0.0473i$\\\hline\hline
\end{tabular}
$$
Table 3: Determinant series for $[1,1,1]$ zones for varying temperatures
The increase in $R$ and the slowdown in the series convergence is consistent with our intuition based on the localization length. Although we are looking at neutron matter in this example, we comment that this is the appropriate temperature range for observing the liquid-gas transition in symmetric nuclear matter ($T\sim20$ MeV) [@Elliott:2001hn][@Ma:2003dc][@Karnaukhov:2003vp]. The localization length becomes greater than $1$ for $T\sim10$ MeV, and so we expect the series to break down for $[1,1,1]$ zones at colder temperatures. This also happens to be the temperature at which long range interactions due to pairing become important. In order to see pairing correlations, one should therefore use larger zone sizes. However we note that at temperatures much less than $10$ MeV there is also a significant sign problem, and the numerical difficulties will be substantial independent of the method used to calculate determinants.
We now look at how convergence of the expansion is improved by using larger spatial zones. In Table 4 we show the determinant expansion for a $6^{3}\times6$ lattice at $T=37.5$ MeV for $[1,1,1]$, $[2,2,2],$ and $[3,3,3]$ zones [^2].$$\begin{tabular}
[c]{||l|l|l|l||}\hline\hline
zone & $[1,1,1]$ & $[2,2,2]$ & $[3,3,3]$\\\hline
$R$ & $0.5122$ & $0.3013$ & $0.3014$\\\hline
$\log(\det(M))$ & $65.8009-0.7250i$ & $65.8009-0.7250i$ & $65.8009-0.7250i$\\\hline
$\log(\Delta_{0})$ & $51.3988-0.7888i$ & $58.6585-0.7543i$ & $61.0471-0.7722i$\\\hline
$\log(\Delta_{2})$ & $66.0028-0.7218i$ & $65.8570-0.7231i$ & $65.8317-0.7233i$\\\hline
$\log(\Delta_{4})$ & $65.8289-0.7243i$ & $65.8015-0.7251i$ & $65.8007-0.7250i$\\\hline
$\log(\Delta_{6})$ & $65.7926-0.7253i$ & $65.8009-0.7250i$ & $65.8009-0.7250i$\\\hline
$\log(\Delta_{8})$ & $65.8026-0.7249i$ & $65.8009-0.7250i$ & $65.8009-0.7250i$\\\hline\hline
\end{tabular}
\ $$
Table 4: Determinant series for various zone sizes
It is somewhat unusual that the spectral radius $R$ is about the same for $[2,2,2]$ and $[3,3,3]$, however the convergence of the series clearly improves as we increase the zone size.
We now investigate the convergence of the determinant zone expansion for physical observables. Let us return to the data in Table 1 for a moment. We observe that at any given order the error appears to be about the same for each of the three independent configurations. Since the measurement of a physical observable does not dependent on the overall normalization of the partition function, this suggests that the zone determinant expansion could be more accurate in approximating physical observables. We now check to see if this is so for a particular example.
Let us define the neutron occupation number at site $\vec{r}$, $$\rho_{\uparrow(\downarrow)}(\vec{r})=\psi_{\uparrow(\downarrow)}^{\ast}(\vec{r})\psi_{\uparrow(\downarrow)}(\vec{r})=\psi_{\uparrow(\downarrow
)}^{\ast}(\vec{r})\psi_{\uparrow(\downarrow)}^{\prime}(\vec{r}+\hat{4}).$$ We measure the opposite spin radial distribution function,$$<\rho_{\uparrow}(\vec{r})\rho_{\downarrow}(0)>,$$ by sampling $20$ independent pion and Hubbard-Stratonovich field configurations. For $r_{x}=0$ we plot the results for the opposite spin radial distribution function in Fig. 1 using exact matrix determinants for a $4^{3}\times6$ lattice at $T=37.5$ MeV. In Figs. 2-6 we show the error in the radial distribution function if the estimate $\Delta_{m}$ is used in place of $\det(M)$ for $m=0,2,4,6,8$.
\[ptb\]
[exact.eps]{}
\[ptbptb\]
[order0.eps]{}
\[ptbptbptb\]
[order2.eps]{}
\[ptbptbptbptb\]
[order4.eps]{}
\[ptbptbptbptbptb\]
[order6.eps]{}
\[ptbptbptbptbptbptb\]
[order8.eps]{}
The approximation at order $0$ is a lot better than expected given the error of $\Delta_{0}$ in approximating $\det(M)$. Overall we find that the zone expansion is significantly more accurate for the radial distribution function than the expansion of the determinants. It is premature to say if this is typical of all physical observable measurements. Nevertheless if most of the error of $\Delta_{m}$ is in fact independent of the pion and Hubbard-Stratonovich configurations at equilibrium, then we expect an improvement in accuracy for most physical observables.
Summary and conclusions
=======================
We have discussed lattice simulations of finite temperature nuclear matter and a new approximation method called the zone determinant expansion for nucleon matrix determinants. The expansion is made possible by the small size of the spatial hopping parameter. We know from power counting in chiral effective theory that the spatial hopping parameter is suppressed relative to the leading order interactions at low energies. The zone determinant expansion is given by$$\det(M)=\det(M_{0})\exp\left( \sum_{p=1}^{\infty}\frac{(-1)^{p-1}}{p}\text{trace}((M_{0}^{-1}M_{E})^{p})\right) ,$$ where $M$ is the nucleon one-body interaction matrix, $M_{E}$ is the submatrix consisting of zone boundary hopping terms, and $M_{0}$ is the submatrix without boundary hopping terms. The convergence of the expansion is controlled by the spectral radius of $M_{0}^{-1}M_{E}$. Physically we expect the convergence to be rapid if the localization length of the nucleons$$l\sim\sqrt{\beta h}$$ is small compared to the size of the spatial zones.
We tested the zone determinant expansion using lattice simulations of neutron matter with self interactions and neutral pion exchange. The convergence of the expansion was measured for several configurations at temperature $T=37.5$ MeV and using $[1,1,1]$ spatial zones. By decreasing the temperature from $T=37.5$ MeV to $12.5$ MeV we found that the convergence of the expansion becomes slower, as predicted by the increase in the localization length. But we then showed that convergence could be accelerated by increasing the size of the zones from $[1,1,1]$ to $[3,3,3]$. Finally we looked at the convergence of the expansion for the opposite spin radial distribution function$$<\rho_{\uparrow}(\vec{r})\rho_{\downarrow}(0)>.$$ We found that the accuracy of the expansion for this physical observable was significantly better at each order than that for the expansion of the determinants.
The number of required operations for calculating the nucleon determinant using LU factorization for an $n\times n$ matrix scales as $n^{3}$. Therefore a nuclear lattice simulation that includes nucleon/nucleon-hole loops requires $(V\beta)^{3}$ times more operations than the quenched simulation without loops. This numerical challenge has been the most pressing limitation on finite temperature nuclear lattice simulations to date.
For the zone determinant expansion method at fixed zone size, the computation cost scales only as $f(m)\beta^{3}$ where $m$ is the order of the expansion. For a simulation on a lattice with spatial dimensions $8^{3}$, one can accelerate the simulation by a factor of about $10^{5}$ to $10^{7}$, depending on the expansion order and size of the spatial zone. The savings are greater on larger lattices and should facilitate future work in the area of finite temperature nuclear lattice simulations.
D.L. is grateful to B. Borasoy, T. Schaefer, R. Seki, U. van Kolck, and participants at the 2003 CECAM Sign Problem Workshop for discussions and comments. This work was supported in part by NSF Grants DMS-0209931 and DMS-0209695.
[^1]: We should mention recent work [@Chandrasekharan:2003wy] that shows in the chiral limit with static nucleons the sign problem does not occur. Thus there is some indirect relationship between the two problems.
[^2]: For $[2,2,2]$ the zone breakup is no longer biparitite since the $6\div2$ is odd. However this is only a boundary effect and we find that $\Delta_{2m+1}$ is extremely close to $\Delta_{2m}$.
|
---
abstract: |
Let $f$ be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain $D\subset\overline{\mathbb{C}}$ to which the function $f$ can be extended in a meromorphic and single-valued manner. ’Large’ means here that the complement $\overline{\mathbb{C}}\setminus D$ is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Padé approximation.
In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given.
Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev’s Problem.
author:
- Herbert R Stahl
title: |
Sets of Minimal Capacity and\
Extremal Domains
---
[^1]
\[s1\]Introduction
==================
We assume that $f$ is a function meromorphic in a neighborhood of infinity, and consider domains $D\subset\overline{\mathbb{C}}$ to which the function $f$ can be extended in a meromorphic and single-valued manner. The basic problem of our investigation is to find the domain with a complement of minimal (logarithmic) capacity. It will be shown that for any function $f$ that is meromorphic at infinity such a domain exists and is essentially unique. The domain is called extremal, and its complement is called the minimal set (or the set of minimal capacity). Formal definitions are given in the Sections \[s2\] and \[s3\].
Extremal domains play an important role in rational approximation, and there especially in the convergence theory of Padé approximants (cf. [@GoncharRakhmanov87], [@Nuttall80], [@NuttallSingh77], [@Nuttall90], [@Stahl86a], [@Stahl86b], [@Stahl87a], [@Stahl89], [@Stahl96], [@Stahl97], [@BakerGravesMorris] Chapter 6). Variants of the concept will also be useful in other areas of rational approximation and the theory of orthogonal polynomials.
Several elements of the material in the present article have already been studied in [@Stahl85a], [@Stahl85b], and [@Stahl85c]. Results from there will be revisited, proofs will be redone, and the whole concept will be extended and reformulated.
\[ptb\]
[Fig5a.eps]{}
\[s11\]A Concrete Example
-------------------------
As an illustration of the role played by extremal domains in the theory of Padé approximation, we consider a concrete example. Let $f$ be the algebraic function defined by $$f(z):=\sqrt[4]{\prod\nolimits_{j=1}^{4}(1-z_{j}/z)}+\sqrt[3]{\prod
\nolimits_{j=5}^{7}(1-z_{j}/z)}\label{f1a}$$ with $7$ branch points $z_{1},\ldots,z_{7}$ that have been chosen rather arbitrarily, but with the intention to get an evenly spread out configuration. The seven values are given in (\[f65c\]), further below, but their location can readily be read from Figure \[fig0\].
The rather simple construction of the function $f$ makes it easy to understand all possible meromorphic and single-valued continuations of $f$. Indeed, $f$ possesses a single-valued continuation throughout a domain $D\subset
\overline{\mathbb{C}}$ if, and only if, $\infty\in D$ and if each of the two sets $\{z_{1},\ldots,z_{4}\}$ and $\{z_{5},z_{6},z_{7}\}$ of branch points is connected in the complement $\overline{\mathbb{C}}\setminus D$.
The union of the $8$ arcs in Figure \[fig0\] form the set of minimal capacity for the function $f$, which we denote by $K_{0}(f,\infty)$, and by $D_{0}(f,\infty):=\overline{\mathbb{C}}\setminus K_{0}(f,\infty)$ we denote the extremal domain. Their definition and details about the calculation of the minimal set $K_{0}(f,\infty)$ will be given in Section \[s2\] and in the discussion of Example \[e65\] in Section \[s6\], further below.
Let $[63/62]_{f}$ be the Padé approximant of numerator and denominator degree $63$ and $62$, respectively, to the function $f$ developed at infinity. In Figure \[fig0\], the poles of this approximant are represented by stars. For any $n\in\mathbb{N}$ the Padé approximant $[n+1/n]_{f}=p/q$ is defined by the relation $$f(z)q(\frac{1}{z})-p(\frac{1}{z})=\text{O}(z^{-2n-2})\text{ \ \ as
\ \ }z\rightarrow\infty\label{f1b}$$ with $p$ and $q$ polynomials of degree at most $n+1$ and $n$, respectively. An comprehensive introduction to Padé approximation can be found in [@BakerGravesMorris].
The connection between Padé approximation and the minimal set $K_{0}(f,\infty)$ will be established in the next theorem, which covers functions of type (\[f1a\]). It has been proved in [@Stahl97] (cf. also [@BakerGravesMorris] Theorem 6.6.9), and is given here in a somewhat shortened and specialized form.
\[t1a\] For $n\rightarrow\infty$, the Padé approximants $[n+1/n]_{f}$ converge to the function (\[f1a\]) in capacity in the extremal domain $D_{0}(f,\infty)\subset\overline{\mathbb{C}}$ associated with $f$, and this convergence is optimal in the sense that it does not hold throughout any domain $\widetilde{D}\subset\overline{\mathbb{C}}$ with $\operatorname*{cap}(\widetilde{D}\setminus D_{0}(f,\infty))>0$.
Theorem \[t1a\] shows that extremal domains are convergence domains for Padé approximants, and this is also the case for our concrete example. In Figure \[fig0\] we observe that $61$ out of $63$ poles of the Padé approximant $[63/62]_{f}$ are distributed very nicely along the $8$ arcs that form the minimal set $K_{0}(f,\infty)=\overline{\mathbb{C}}\setminus
D_{0}(f,\infty)$. They are asymptotically distributed in accordance to the equilibrium distribution on the minimal set $K_{0}(f,\infty)$ (cf. [@Stahl97], Theorem 1.8), and they mark the places, where we don’t have convergence.
There are two poles that step out of line, and each one by a different reason: One of them lies close to the origin, where it approximates the simple pole of the function $f$ at the origin. Because of its correspondence to a pole of $f$, it is called systematic.
The other one, which lies at $z=-3.35+2.66\,i$, does not correspond to a singularity of the function $f$, and does obviously also not belong to any of the chains of poles along the arcs in $K_{0}(f,\infty)$. Such poles are called spurious in the theory of Padé approximation. Spurious poles always appear in combination with a nearby zero of the approximant. These pairs of poles and zeros are close to cancellation. They are a phenomenon that unfortunately cannot be ignored in Padé approximation (cf. [@Stahl98], [@Stahl97a], or [@BakerGravesMorris] Chapter 6). Convergence in capacity is compatible with the possibility of such spurious poles.
The convergence in capacity in Theorem \[t1a\] implies that almost all poles of the Padé approximants $[n+1/n]_{f}$ have to leave the extremal domain $D_{0}(f,\infty)$; they cluster on the minimal set $K_{0}(f,\infty)$. That they do this in a rather regular way is shown in Figure \[fig0\]. The picture does not change much for other values of $n$ only that the location, and possibly also the number of spurious poles may be different in each case.
If one wants to summarize the somewhat complicated convergence theory for diagonal Padé approximants in a short sentence one can say that extremal domains are for Padé approximants what discs are for power series.
\[s12\]The Outline of the Manuscript
------------------------------------
In the next two Sections \[s2\] and \[s3\], two alternative formal definitions are given for the extremality problem under investigation. In the second approach, the role of the function $f$ is taken over by a concrete Riemann surface $\mathcal{R}$ over $\overline{\mathbb{C}}$. Both formulations are equivalent.
Illustrative examples are discussed in Section \[s6\], but before that in the two Sections \[s4\] and \[s5\], general results about minimal sets and extremal domains are formulated and discussed. All proofs are postponed to later sections.
In Section \[s7\], a local version of the extremality problem is formulated and discussed. After that in Section \[s8\], the extremality problem is compared with some classical problems from geometric functions theory. For such problems there exists a broad range of tools and techniques, as for instance, boundary and inner variational methods, methods of extremal length, and techniques connected with quadratic differentials (cf. [@Pommerenke75], [@Goluzin], [@Kuzmina82], [@Kuzmina97]). Some of these ideas will play a role in our investigation. We shall use a solution of one of these problems as building block in one of our proofs.
Practically, no proofs are given in the Sections \[s2\] - \[s5\] and \[s7\]; they all are all postponed to the Sections \[s9\] and \[s10\]. In Section \[s110\], several auxiliary results from potential theory and geometric function theory are assembled, of which some have been modified quite substantially in order to fit their purpose in the present paper.
\[s13\]Some Special Aspects
---------------------------
It is a typical feature of the approach chosen in the present article that a general existence and uniqueness proof is put at the beginning of the analysis. This strategy has the advantage of giving great methodological liberty in later proofs of special properties. At these later stages, the knowledge of unique existence offers a free choice between different methods and techniques from the tool boxes of geometric function theory; and because of the uniqueness it is always clear that one is dealing with the same well defined object. The prize to be paid for this strategy is a rather abstract and somewhat heavy machinery for the uniqueness proof. The main tools there are potential-theoretic in nature.
It has been mentioned, and hopefully also illustrated by the introductory example (\[f1a\]), that extremality with respect to the logarithmic capacity arises in a very natural way in connection with diagonal Padé approximants. In rational approximation also other types of capacity are of interest, as for instance, condenser capacity or capacities in external fields, which become relevant in connection with rational interpolants (cf. [@Stahl96a]) or with essentially non-diagonal Padé approximants. The specific form of tools and methods in the present analysis should be helpful for such potential generalizations.
\[s2\]Basic Definitions and Unique Existence
============================================
In the present section we introduce basic definitions and formulate a theorem about the unique existence of a solution of the extremality problem.
Throughout the whole paper, we assume that $f$ is a function meromorphic in a neighborhood of infinity, and denote its meromorphic extensions by the same symbol $f$. By $\operatorname*{cap}(\cdot)$ we denote the (logarithmic) capacity.
\[s21\]The Definition of Problem $(f,\infty)$
---------------------------------------------
\[d21a\]A domain $D\subset\overline{\mathbb{C}}$ is called admissible for Problem $(f,\infty)$ if
- $\infty\in D$, and if
- $f$ has a single-valued meromorphic continuation throughout $D$.
By $\mathcal{D}(f,\infty)$ we denote the set of all admissible domains $D$ for Problem $(f,\infty)$. A compact set $K\subset\mathbb{C}$ is called admissible for Problem $(f,\infty)$ if it is the complement $\overline{\mathbb{C}}\setminus D$ of an admissible domain $D\in\mathcal{D}(f,\infty)$. By $\mathcal{K}(f,\infty)$ we denote the set of all admissible compact sets $K$ for Problem $(f,\infty)$.
Instead of meromorphic continuations, one could also consider analytic continuations in condition (ii) of Definition \[d21a\] without essentially changing the whole concept. This later option has been taken in [@Stahl85a], [@Stahl85b], and [@Stahl85c]. Meromorphic continuations have been chosen here because of their natural affiliation with rational approximation.
\[d21b\]A compact set $K_{0}=K_{0}(f,\infty)\subset\mathbb{C}$ is called minimal (or more lengthy: a set of minimal capacity with respect to Problem $(f,\infty)$) if the following three conditions are satisfied:
- $K_{0}\in\mathcal{K}(f,\infty)$.
- We have $$\operatorname*{cap}(K_{0})=\inf_{K\in\mathcal{K}(f,\infty)}\operatorname*{cap}(K).\label{f21a}$$
- We have $K_{0}\subset K_{1}$ for all $K_{1}\in\mathcal{K}(f,\infty)$ that satisfy condition (ii) with $K_{0}$ replaced by $K_{1}$.
The domain $D_{0}(f,\infty):=\overline{\mathbb{C}}\setminus K_{0}(f,\infty) $ is called extremal with respect to Problem $(f,\infty)$ (or short: extremal domain).
By $\mathcal{K}_{0}(f,\infty)$ we denote the set of all admissible compact sets $K$ of minimal capacity, i.e., all sets $K\in\mathcal{K}(f,\infty)$ that satisfy condition (ii), but not necessarily condition (iii), and by $\mathcal{D}_{0}(f,\infty)$ the set of all admissible domains $D\in
\mathcal{D}(f,\infty)$ such that $\overline{\mathbb{C}}\setminus
D\in\mathcal{K}_{0}(f,\infty)$.
With the introduction of the set of admissible domains $\mathcal{D}(f,\infty)$ and the definition of the extremal domain $D_{0}(f,\infty)$ together with its complementary minimal set $K_{0}(f,\infty)$, Problem $(f,\infty)$ is fully defined. The problem depends solely on the function $f$ given in neighborhood of infinity.
The point infinity plays a very special role for the function $f$ and also in the definition of the (logarithmic) capacity, which is reflected in condition (i) of Definition \[d21a\]. This special role is the reason why the symbol $\infty$ has been used besides of $f$ for the designation of Problem $(f,\infty)$.
\[s22\]Unique Existence
-----------------------
One of the central results in the present paper is the following existence and uniqueness theorem.
\[Unique Existence Theorem\]\[t22a\] For any function $f$, which is meromorphic in a neighborhood of infinity, there uniquely exists a minimal set $K_{0}(f,\infty)$ and correspondingly a unique extremal Domain $D_{0}(f,\infty)$ with respect to Problem $(f,\infty)$.
Among the three conditions in Definition \[d21b\], condition (ii) is most important, and condition (iii) plays only an auxiliary role. The situation becomes evident by the next proposition.
\[p22a\] Elements of the set $\mathcal{K}_{0}(f,\infty)$ differ at most in a set of capacity zero, and we have $$K_{0}(f,\infty)=\bigcap_{K\in\mathcal{K}_{0}(f,\infty)}K.\label{f22a}$$
The concept of extremal domains is most interesting if the function $f$ has branch points. In the absence of branch points, the concept becomes in a certain sense trivial, as the next proposition shows.
\[p22b\] If the function $f$ of Problem $(f,\infty)$ possesses no branch points, then the extremal domain $D_{0}(f,\infty)$ coincides with the Weierstrass domain $W_{f}\allowbreak\subset\overline{\mathbb{C}}$ for meromorphic continuation of the function $f$ starting at $\infty$.
In Section \[s6\] we shall discuss several concrete examples of functions $f$ together with their extremal domains $D_{0}(f,\infty)$ and minimal sets $K_{0}(f,\infty)$. These examples should give more substance to the formal definitions in the present section.
Several classical extremality problems from geometric function theory that are defined by purely geometric constraints are reviewed in Section \[s8\]. There exist similarities with Problem $(f,\infty)$, but there are also essential differences. The intention of the selection of examples in Section \[s6\] has been to illustrate these differences.
\[s3\]An Alternative Definition
===============================
In the present section a definition of the extremality problem is given that is equivalent to Problem $(f,\infty)$, but the role of the function $f$ is taken over by a Riemann surface $\mathcal{R}$. Of course, single-valuedness, or better its absence, lies at the heart of the idea of a Riemann surface, and so the alternative approach may shed light on the geometric background of Problem $(f,\infty)$. Since in all later sections, with the only exception of Subsection \[s42\], only Problem $(f,\infty)$ will be used as reference point, the alternative definition in the present section can be skipped in a first reading.
Let $\mathcal{R}$ be a Riemann surface over $\overline{\mathbb{C}}$, not necessarily unbounded, and let $\pi:\mathcal{R}\longrightarrow\overline
{\mathbb{C}}$ be its canonical projection. We assume that $\infty\in
\pi(\mathcal{R})$.
\[s31\]The Definition of Problem $(\mathcal{R},\infty^{(0)})
$
------------------------------------------------------------
\[d31a\] Let $\infty^{(0)}\ $be a point on the Riemann surface$\ \mathcal{R}$ with $\pi(\infty^{(0)})=\infty$. Then a domain $D\subset\mathcal{R}$ is called admissible for Problem $(\mathcal{R},\infty^{(0)})$ if the following two conditions are satisfied:
- $\infty^{(0)}\in D$.
- The domain $D$ is planar (also called schlicht), i.e., $\pi
\mid_{D}$ is univalent, or in other words, we have $\operatorname*{card}\left( (\pi^{-1}\circ\pi)(\{\zeta\})\cap D\right) =1$ for all $\zeta\in D$.
By $\mathcal{D}(\mathcal{R},\infty^{(0)})$ we denote the set of all admissible domains $D\subset\mathcal{R}$ for Problem $(\mathcal{R},\infty^{(0)})$.
\[d31b\] A compact set $K\subset\mathbb{C}$ is admissible for Problem $(\mathcal{R},\infty^{(0)})$ if it is of the form $K:=\overline{\mathbb{C}}\setminus\pi(D)$ with $D\in\mathcal{D}(\mathcal{R},\infty^{(0)})$.
By $\mathcal{K}(\mathcal{R},\infty^{(0)})$ we denote the set of all admissible compact sets $K\subset\mathbb{C}$ for Problem $(\mathcal{R},\infty^{(0)})$.
Notice that in contrast to admissible domains $D\in\mathcal{D}(f,\infty)$, now admissible domains $D\in\mathcal{D}(\mathcal{R},\infty^{(0)})$ are subdomains of the Riemann surface $\mathcal{R}$, while the admissible compact sets $K\in\mathcal{K}(\mathcal{R},\infty^{(0)})$ remain to be subsets of $\mathbb{C}$ like it has been the case in Definition \[d21a\].
Analogously to Definition \[d21b\], we define the minimal set and the extremal domain for Problem $(\mathcal{R},\infty^{(0)})$ as follows.
\[d31c\] A compact set $K_{0}=K_{0}(\mathcal{R},\infty^{(0)})\subset\overline{\mathbb{C}}$ is called minimal with respect to Problem $(\mathcal{R},\infty^{(0)})$ if the following three conditions are satisfied:
- $K_{0}\in\mathcal{K}(\mathcal{R},\infty^{(0)})$.
- We have $$\begin{aligned}
\operatorname*{cap}(K_{0}) & =\inf_{K\in\mathcal{K}(\mathcal{R},\infty
^{(0)})}\operatorname*{cap}(K)\label{f31a1}\\
& =\inf_{D\in\mathcal{D}(\mathcal{R},\infty^{(0)})}\operatorname*{cap}(\overline{\mathbb{C}}\setminus\pi(D)).\label{f31a2}$$
- We have $K_{0}\subset K_{1}$ for all $K_{1}\in\mathcal{K}(\mathcal{R},\infty^{(0)})$ that satisfy assertion (ii) with $K_{0}$ replaced by $K_{1}$.
A domain $D_{0}\in\mathcal{D}(\mathcal{R},\infty^{(0)})$ that satisfies $\overline{\mathbb{C}}\setminus\pi(D_{0})=K_{0}(\mathcal{R},\infty^{(0)})$ is called extremal with respect to Problem $(\mathcal{R},\allowbreak\infty
^{(0)})$, and it is denoted by $D_{0}(\mathcal{R},\infty^{(0)})$.
By $\mathcal{K}_{0}(\mathcal{R},\infty^{(0)})$ we denote the set of all compact sets $K$ that satisfy the two conditions (i) and (ii), but not necessarily condition (iii).
\[s32\]Unique Existence and Equivalence
---------------------------------------
For any Riemann surface $\mathcal{R}$ over $\overline{\mathbb{C}}$, there exists a meromorphic function $f$ such that $\mathcal{R}=\mathcal{R}_{f}$ is the natural domain of definition of $f$. On the other hand, the meromorphic continuation of a given function $f$, which is meromorphic in a neighborhood of infinity, defines a Riemann surface $\mathcal{R}_{f}$ over $\overline{\mathbb{C}}$ that contains a point $\infty^{(0)}\in\mathcal{R}_{f}$ with $\pi(\infty^{(0)})=\infty$, and this surface $\mathcal{R}_{f}$ is the natural domain of definition for the function $f$. From these observations we can conclude that the two Problems $(f,\infty)$ and $(\mathcal{R}_{f},\infty^{(0)})$ are equivalent.
It is an immediate consequence of the equivalence of both problems that the existence and uniqueness of a solution to Problem $(f,\infty)$ formulated in Theorem \[t22a\] carries over to Problem $(\mathcal{R},\infty^{(0)})$. Details are formulated in the next theorem.
\[t32a\] (i) For any Riemann surface $\mathcal{R}$ over $\overline
{\mathbb{C}}$ with $\infty^{(0)}\in\mathcal{R}$ and $\pi(\infty^{(0)})=\infty$, there uniquely exists a minimal set $K_{0}=K_{0}(\mathcal{R},\allowbreak\infty^{(0)})\subset\mathbb{C}$ for Problem $(\mathcal{R},\infty^{(0)})$, and correspondingly, there also uniquely exists an extremal domain $D_{0}=D_{0}(\mathcal{R},\allowbreak\infty^{(0)})\allowbreak
\subset\mathcal{R}$.
\(ii) Let the Riemann surface $\mathcal{R}=\mathcal{R}_{f}$ be the natural domain of definition for the function $f$, and let $f$ be assumed to be meromorphic in a neighborhood of infinity. Then the two extremal domains $D_{0}(f,\infty)$ and $D_{0}(\mathcal{R}_{f},\infty^{(0)})$ of the Definitions \[d21b\] and \[d31c\], respectively, are identical up to the canonical projection $\pi:\mathcal{R}_{f}\longrightarrow\overline{\mathbb{C}}$, i.e., we have $$D_{0}(f,\infty)=\pi\left( D_{0}(\mathcal{R}_{f},\infty^{(0)})\right)
.\label{f32a}$$ Further, we have $$K_{0}(f,\infty)=K_{0}(\mathcal{R}_{f},\infty^{(0)}).\label{f32b}$$
We assume that the function $f$ has the Riemann surface $\mathcal{R}=\mathcal{R}_{f}$ as its natural domain of definition and that the function element of $f$ at the point $\infty^{(0)}\in\mathcal{R}_{f}$, $\pi
(\infty^{(0)})=\infty$, is identical with the function $f$ at $\infty
\in\overline{\mathbb{C}}$.
It immediately follows from the two Definitions \[d21a\] and \[d31a\] that for each domain $\widetilde{D}\in\mathcal{D}(\mathcal{R}_{f},\infty^{(0)})$ we have $\pi(\widetilde{D})\in\mathcal{D}(f,\infty)$, and conversely, for each domain $D\in\mathcal{D}(f,\infty)$ there exists an admissible domain $\widetilde{D}\in\mathcal{D}(\mathcal{R}_{f},\infty^{(0)})$ with $\pi(\widetilde{D})=D$.
After these preparations, the theorem is an immediate consequence of the correspondence between the two sets $\mathcal{D}(\mathcal{R}_{f},\infty
^{(0)})$ and $\mathcal{D}(f,\infty)$ together with the two Definitions \[d21b\], \[d31c\], and Theorem \[t22a\].
The equivalence of the two Problems $(f,\infty)$ and $(\mathcal{R}_{f},\infty^{(0)})$ allows us to opt freely for one of the two approaches. In the present investigation we carry out the analysis in the framework of Problem $(f,\infty)$. However, in applications it is sometimes favorable to start from a Riemann surface $\mathcal{R}$. This approach will also give the intuitive background for the discussion of concrete examples in Section \[s6\].
\[s4\]Topological Properties
============================
Extremal problems in geometric function theory often lead to topologically simply structured and smooth solutions. In the next two sections it will be shown that a similar situation can be observed in our present investigations.
In Subsection \[s41\] we address topological properties of the minimal set $K_{0}(f,\allowbreak\infty)$, and corresponding results for the minimal set $K_{0}(\mathcal{R},\allowbreak\infty^{(0)})$ associated with Problem $(\mathcal{R},\infty^{(0)})$ are given in Subsection \[s42\].
\[s41\]Topological Properties of the Set $K_{0}(f,\infty)$
----------------------------------------------------------
The main result in the present section is a structure theorem for the minimal set $K_{0}(f,\infty)$. As usual, the function $f$ is assumed to be meromorphic in a neighborhood of infinity.
\[Structure Theorem\]\[t41a\] Let the function $f$ be meromorphic in a neighborhood of infinity, and let $K_{0}=K_{0}(f,\infty)$ be the minimal set for Problem $(f,\infty)$. There exist two sets $E_{0},E_{1}\subset\mathbb{C}$ and a family $\left\{ J_{j}\right\} _{j\in I}$ of open and analytic Jordan arcs such that $$K_{0}(f,\infty)=E_{0}\cup E_{1}\cup\bigcup_{j\in I}J_{j},\label{f41a}$$ and the components in (\[f41a\]) have the following properties:
- We have $\partial E_{0}\subset\partial D_{0}(f,\infty)$, and at each point $z\in\partial E_{0}$ the meromorphic continuation of the function $f$ has a non-polar singularity for at least one approach out of $D_{0}=D_{0}(f,\infty)$. The set $E_{0}\subset K_{0}$ is compact and polynomial-convex, i.e., $\overline{\mathbb{C}}\setminus E_{0}$ is connected.
- At each point $z\in E_{1}$ the function $f$ has meromorphic continuations out of $D_{0}$ from all possible sides, and these continuations lead to more than $2$ different function elements at the point $z$. The set $E_{1}$ is discrete in $\overline{\mathbb{C}}\setminus E_{0}$.
- All Jordan arcs $J_{j}$, $j\in I$, are contained in $\overline{\mathbb{C}}\setminus(E_{0}\cup E_{1})$, they are pair-wise disjoint, the function $f$ has meromorphic continuations to each point $z\in
J_{j}$, $j\in I$, from both sides of $J_{j}$ out of $D_{0}$, and these continuations lead to $2$ different function elements at each point $z\in
J_{j}$, $j\in I$.
The properties (i), (ii), and (iii) fully characterize all components on the right-hand side of (\[f41a\]).
\[r41a\]The family of Jordan arcs $\left\{ J_{j}\right\} _{j\in I}$ and also the set $E_{1}$ in (\[f41a\]) is empty if, and only if, all possible meromorphic continuations of the function $f$ are single-valued, i.e., if the function $f$ has no branch points. This situation has already been addressed in Proposition \[p22b\].
It follows from Theorem \[t41a\] that the boundary $\partial D_{0}(f,\infty)$ is smooth everywhere on $\partial D_{0}(f,\infty)\setminus
(\partial E_{0}\cup E_{1})$. More information about this aspect is given in the next theorem.
\[t41b\] The set $K_{0}(f,\infty)\setminus E_{0}$ is locally connected, and only a finite number ($>2$) of arcs $J_{j}$, $j\in I$, meets at each point of the set $E_{1}$.
In the next section (cf. Remark \[r52b\]), we shall see that the arcs $J_{j}$ that meet at a point $z\in E_{1}$ form a regular star at $z$.
Before we close the present subsection, we will discuss the two influences that determine the structure of the minimal set $K_{0}(f,\infty)$ in an informal way.
The principle of minimal capacity of the set $K_{0}(f,\infty)$ implies that the extremal domain $D_{0}(f,\infty)$ is as large as possible, and consequently it extends up to the natural boundary of the function $f$ (see also Definition \[d71a0\] in Subsection \[s71\], further below). On the other hand, the requirement of single-valuedness of the function $f$ in $D_{0}(f,\infty)$ can in general only be avoided by cuts in the complex plane $\overline{\mathbb{C}}$; these cuts separate different branches of the function $f$.
Both aspects, maximal extension and the principle of single-valuedness, find a specific balance in the topological structure of the minimal set $K_{0}(f,\infty)$. On one hand, there is the compact subset $E_{0}\subset
K_{0}(f,\infty)$, where on $\partial E_{0}$ meromorphic extensions of the function $f$ find a natural boundary. On the other hand, there is the part $K_{0}(f,\infty)\setminus E_{0}$ of $K_{0}(f,\infty)$, which essentially consists of analytic Jordan arcs $J_{j}$, $j\in I$, which cut $\overline
{\mathbb{C}}\diagdown E_{0}$ in such a way that different branches of the function $f$ are separated. They can be chosen with much liberty, and therefore optimization is possible. This optimization is done according to the principle of minimal capacity. We shall see in Section \[s5\], and more specifically in Section \[s7\], how a balance between forces leads to a state of equilibrium that determines the Jordan arcs $J_{j}$, $j\in
I$.
\[s42\]Topological Properties of the Set $K_{0}(\mathcal{R},\infty^{(0)})$
--------------------------------------------------------------------------
From Theorem \[t32a\] we know that the two Problems $(f,\infty)$ and $(\mathcal{R},\infty^{(0)})$ have equivalent solutions if there exists an appropriate relationship between the Riemann surface $\mathcal{R}$ and the function $f$. As a consequence of this equivalence, there exists a description of the topological properties of the set $K_{0}(\mathcal{R},\infty^{(0)})$ that corresponds to that given in Theorem \[t41a\]. However, now the function $f$ is no longer available, and its role has to be taken over by properties of the Riemann surface $\mathcal{R}$.
Let $\mathcal{R}$ be a Riemann surface over $\overline{\mathbb{C}}$. By $\partial D$ and $\overline{D}$ we denote the boundary and the closure of a domain $D\subset\mathcal{R}$ in $\mathcal{R}$. Further, we denote the set of all branch points of $\mathcal{R}$ by $Br(\mathcal{R})\subset\mathcal{R}$, and the relative boundary of the Riemann surface $\mathcal{R}$ over $\overline
{\mathbb{C}}$ by $\partial\mathcal{R}$. We set $\widetilde{\mathcal{R}}:=\mathcal{R}\cup\partial\mathcal{R}$. If the Riemann surface $\mathcal{R}$ is compact, then we have $\partial\mathcal{R=\emptyset}$.
The canonical projection $\pi:\mathcal{R}\longrightarrow\overline{\mathbb{C}}$ can be extended continuously to a projection $\widetilde{\pi}:\widetilde
{\mathcal{R}}\longrightarrow\overline{\mathbb{C}}$. We continue to denote the boundary of a domain $D$ in$\ \widetilde{\mathcal{R}}$ by the same symbol $\partial D$ as has been done in $\mathcal{R}$.
After this preparations, we are ready to formulate the analog of Theorem \[t41a\] for Problem $(\mathcal{R},\infty^{(0)})$.
\[t42a\] Let $D_{0}=D_{0}(\mathcal{R},\infty^{(0)})\subset\mathcal{R}$ and $K_{0}=K_{0}(\mathcal{R},\infty^{(0)})\subset\overline{\mathbb{C}}$ be the uniquely existing extremal domain and minimal set, respectively, for Problem $(\mathcal{R},\infty^{(0)})$. Like in Theorem \[t41a\], there exist two sets $E_{0},E_{1}\subset\mathbb{C}$ and a family $\left\{ J_{j}\right\} _{j\in
I}$ of analytic, open Jordan arcs in $\mathbb{C}$ such that representation (\[f41a\]) holds true with $K_{0}(f,\infty)$ replaced by $K_{0}(\mathcal{R},\infty^{(0)})$, i.e., we have $$K_{0}(\mathcal{R},\infty^{(0)})=E_{0}\cup E_{1}\cup\bigcup_{j\in I}J_{j}.\label{f42a0}$$
In the new situation, the components $E_{0},E_{1}$, and $\left\{
J_{j}\right\} _{j\in I}$ in (\[f42a0\]) can be characterized by the following properties:
- The boundary $\partial E_{0}$ of the compact set $E_{0}\subset
K_{0}$ is equal to $$\widetilde{\pi}((\partial D_{0}\cap\partial\mathcal{R})\cup(Br(\mathcal{R})\cap\overline{D_{0}})),\label{f42a}$$ and the set $E_{0}$ is the polynomial-convex hull of $\partial E_{0}$. (For a definition, see Definition \[d111b\] in Subsection \[s1101\], further below).
- The set $E_{1}\subset K_{0}$ is equal to $$E_{1}:=\left\{ \text{ }z\in K_{0}\setminus E_{0}\text{ }\right| \left.
\text{ }\operatorname*{card}(\pi^{-1}(\{z\})\cap\partial D_{0})>2\text{
}\right\} \label{f42b}$$ with $\pi$ being the canonical projection of $\mathcal{R}$ and not that of $\widetilde{\mathcal{R}}$. The set $E_{1}\subset K_{0}$ is discrete in $\overline{\mathbb{C}}\setminus E_{0}$.
- If $I\neq\emptyset$, then $K_{0}\setminus(E_{0}\cup E_{1})$ is the disjoint union of the analytic Jordan arcs $J_{j}$, $j\in I$. For each point $z\in J_{j}$, $j\in I$, we have $$\operatorname*{card}(\pi^{-1}(\{z\})\cap\partial D_{0})=2.\label{f42c}$$
\[s5\]Analytic Characterizations
================================
We now come to analytic characterizations of the Jordan arcs $J_{j}$, $j\in I$, in the minimal set $K_{0}(f,\infty)$ for Problem $(f,\infty)$. One method is based on quadratic differentials, and a related one involves the $S-$property (symmetry-property) of the extremal domain $D_{0}(f,\infty) $. In the last subsection we consider the special case that the set $E_{0}$ in Theorem \[t41a\] is finite, which leads to the interesting special case of rational quadratic differentials.
All results in the present section are formulated in the framework of Problem $(f,\infty)$. Their transfer to Problem $(\mathcal{R},\infty^{(0)}) $ is easily possible with the tools presented in Section \[s3\] and Subsection \[s42\].
\[s51\]The $S-$Property
-----------------------
A characteristic property of the extremal domain $D_{0}=D_{0}(f,\infty)$ for Problem $(f,\infty)$ is a specific behavior of the Green function $g_{D_{0}}(\cdot,\infty)$ on the Jordan arcs $J_{j}$, $j\in I$, in $K_{0}(f,\infty)$ that have been introduced in (\[f41a\]) of Theorem \[t41a\]. For a definition of the Green function we refer to Subsection \[s1103\], further below.
\[t51a\] Under the assumptions made in Theorem \[t41a\], we have $$\frac{\partial}{\partial n_{+}}g_{D_{0}}(z,\infty)=\frac{\partial}{\partial
n_{-}}g_{D_{0}}(z,\infty)\text{ \ for all \ }z\in J_{j}\text{, }j\in
I\text{,}\label{f51a}$$ with $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ denoting the normal derivatives to both sides of the arcs $J_{j}$, $j\in I$, that have been introduced in (\[f41a\]) of Theorem \[t41a\].
The symmetric boundary behavior (\[f51a\]) of the Green function $g_{D_{0}}(\cdot,\infty)$ is called the $S-$property of the extremal domain $D_{0}(f,\allowbreak\infty)$. In Section \[s7\], below, it will be shown that the $S-$property can be interpreted as a local condition for the minimality (\[f21a\]) in Definition \[d21b\].
While in Theorem \[t51a\] we get the $S-$property as a consequence of the minimality (\[f21a\]) in Definition \[d21b\], it will be proved in Theorem \[t73a\] in Subsection \[s73\] that the $S-$property is even equivalent to the minimality (\[f21a\]). As a consequence of this further going result it follows that the $S-$property can also be used as an alternative characterization of the extremal domain $D_{0}(f,\infty)$.
Notice that $I\neq\emptyset$ in (\[f51a\]) implies $\operatorname*{cap}(K_{0})>0$, and consequently, in this case, the Green function $g_{D_{0}}(\cdot,\infty)$ in (\[f51a\]) exists in a proper sense (cf. Subsection \[s1103\], further below). If on the other hand, we have $I=\emptyset$, then relation (\[f51a\]) is void.
From Theorem \[t41a\] we now know that the arcs $J_{j}$, $j\in I$, are analytic. Hence, the Green function $g_{D_{0}}(\cdot,\infty)$ has harmonic continuations across each arc $J_{j}$ from both sides (cf. Subsection \[s1103\]), and consequently the normal derivatives in (\[f51a\]) exist for each $z\in J_{j}$, $j\in I$.
\[s52\]Quadratic Differentials
------------------------------
The $S-$property can be described in an equivalent way by quadratic differentials. We say that a smooth arc $\gamma$ with parametrization $z:[0,1]\longrightarrow\overline{\mathbb{C}}$ is a trajectory of the quadratic differential $q(z)dz^{2}$ if we have $$q(z(t))\overset{\bullet}{z}(t)^{2}<0\text{ \ \ \ for all \ \ }t\in
(0,1).\label{f52a}$$
We note that there exists an associated family of orthogonal trajectories, which are defined by the same relation (\[f52a\]), but with an inequality showing in the other direction. As general reference to quadratic differentials and their trajectories we use [@Strebel84] or [@Jensen75]. Some of its local properties are assembled in Subsection \[s1105\], further below.
\[t52a\]Let $D_{0}=D_{0}(f,\infty)$, $E_{0},E_{1}\subset\overline
{\mathbb{C}}$, and $\left\{ J_{j}\right\} _{j\in I}$ be the objects introduced in Theorem \[t41a\], and let $g_{D_{0}}(\cdot,\infty)$ be the Green function in $D_{0}$. Then the Jordan arcs $J_{j}$, $j\in I$, are trajectories of the quadratic differential $q(z)dz^{2}$ with $q$ defined by $$q(z):=\left( 2\frac{\partial}{\partial z}g_{D_{0}}(z,\infty)\right)
^{2},\label{f52b}$$ where $\partial/\partial z=\frac{1}{2}\left( \partial/\partial x-i\,\partial
/\partial y\right) $ is the usual complex differentiation. The function $q$ has a meromorphic (single-valued) continuation throughout the domain $\overline{\mathbb{C}}\setminus E_{0}^{\prime}$ with $E_{0}^{\prime} $ denoting the sets of cluster points of $E_{0}$. Near infinity we have $$q(z)=\frac{1}{z^{2}}+\text{O}(z^{-3})\text{ \ \ as \ \ }z\rightarrow
\infty.\label{f52c}$$ The function $q$ has at most simple poles in isolated points of $E_{0}$, and it is analytic throughout $\overline{\mathbb{C}}\setminus E_{0}$.
It is not difficult to verify that the meromorphy of the function $q$ in $\overline{\mathbb{C}}\setminus E_{0}^{\prime}$ is equivalent to the $S-$property (\[f51a\]).
The local structure of the trajectories of quadratic differentials can rather easily be understood and described (for more details see Subsection \[s1105\], further below). Of special interest are neighborhoods of poles and zeros of the function $q$ in (\[f52b\]).
\[r52b\] Since we know from Theorem \[t52a\] that all Jordan arcs $J_{j}$, $j\in I$, are trajectories of a quadratic differential $q(z)dz^{2}$ that is meromorphic in $\overline{\mathbb{C}}\setminus E_{0}^{\prime}$, it follows from the local structure of the trajectories that all Jordan arcs $J_{j}$, $j\in I$, that end at an isolated point $z$ of$\ E_{0}\cup E_{1}$ form a regular star at this point.
\[s53\]Rational Quadratic Differentials
---------------------------------------
The description of the Jordan arcs $J_{j}$, $j\in I$, as trajectories of a quadratic differential $q(z)dz^{2}$ is especially constructive if the function $q$ in (\[f52b\]) is rational. This is the case if the set $E_{0}$ from Theorem \[t41a\] is finite. Algebraic functions $f$ are prototypical examples for this situation.
For the formulation of the main result in this direction, we need the notion of bifurcation points in $K_{0}(f,\infty)$, the associated bifurcation index, and the notion of critical points of the Green function $g_{D_{0}}(\cdot,\infty)$.
\[d53a\] Let the objects $K_{0}=K_{0}(f,\infty)$, $E_{0},E_{1}\subset\overline{\mathbb{C}}$, and $\left\{ J_{j}\right\} _{j\in I}$ be and ones as in the Theorems \[t41a\] or \[t52a\]. For each isolated point $z\in E_{1}\cup E_{0}$, the bifurcating index $i(z)$ is the number of different Jordan arcs $J_{j}$, $j\in I$, that end at this point $z$.
If $z$ is an isolated point of $K_{0}=K_{0}(f,\infty)$, then $z$ lies necessarily in $E_{0}$, and by definition we have $i(z)=0$ since $z$ has no contact to any arc in $K_{0}$. Such isolated points can exist; they are generated by isolated, essential singularities of the function $f$ that are no branch points.
\[d53b\] Let $D_{0}=D_{0}(f,\infty)$ be the extremal domain, and assume that $\operatorname*{cap}(K_{0}(f,\infty))>0$. By $E_{2}\subset D_{0}$ we denote the set of all critical points of the Green function $g_{D_{0}}(z,\infty)$, and for each $z\in E_{2}$ we denote the order of the critical point $z$ by $j(z),$ i.e., for $z\in E_{2}$, we have $$\frac{\partial^{l}}{\partial z^{l}}g_{D_{0}}(z,\infty)\left\{
\begin{array}
[c]{lll}=0\smallskip & \text{ \ for \ } & l=1,\ldots,j(z)\\
\neq0 & \text{ \ for \ } & l=j(z)+1.
\end{array}
\right. \label{f53a}$$ If $\operatorname*{cap}(K_{0}(f,\infty))=0$, then we set $E_{2}=\emptyset
$.
The sets $E_{1}$ and $E_{2}$ are always discrete in $\overline{\mathbb{C}}\setminus E_{0}$, while the set $E_{0}$ can be a mixture of isolated and cluster points. Because of this later possibility, it was necessary to distinguish the set $E_{0}^{\prime}$ of cluster points from the original set $E_{0}$ in Theorem \[t52a\]. The set $E_{1}\cup E_{0}\setminus E_{0}^{\prime}$ contains all isolated points of $E_{1}\cup E_{0}$. We have $E_{0}^{\prime}=\emptyset$ if and only if $E_{0}$ is finite.
\[p53a\] If $E_{0}$ is a finite set, then the sets $E_{1}$ and $E_{2}$ are necessarily also finite.
After these preliminaries, we are ready to formulate the central results of the present subsection.
\[t53a\] We use the same notations as in the Theorems \[t41a\] and \[t52a\], and assume that the set $E_{0}$ is finite. Then the function $q$ in (\[f52b\]) is rational, and we have the explicit representation $$q(z)=\prod_{v\in E_{0}\cup E_{1},\text{ }i(v)>0}(z-v)^{i(v)-2}\prod_{v\in
E_{2}}(z-v)^{2\,j(v)}.\smallskip\label{f53b}$$
Notice that there always exist points $z\in E_{0}$ with $i(z)=1$, which implies that $q$ always is a broken rational function. Actually, this assertion follows already from (\[f52c\]) in Theorem \[t52a\], and further we deduce from (\[f52c\]) that the denominator degree of $q$ is exactly $2$ degrees larger than its numerator degree.
The explicit formula (\[f53b\]) for $q$ can be very helpful for the numerical calculation of the analytic Jordan arcs $J_{j}$, $j\in I$, in $K_{0}(f,\infty)$. If the points of the sets $E_{0}$, $E_{1}$, and $E_{2}$ have been determined, then most of the work is done, and one can calculate the Jordan arcs $J_{j}$, $j\in I$, by solving a differential equation that is based on (\[f52a\]), (\[f52b\]), and (\[f53b\]). This procedure has, for instance, also been used for the calculation of the arcs in the minimal sets $K_{0}(f_{j},\infty)$, $j=1,\ldots,5$, in the Examples \[e61\] - \[e65\] that follow next. The critical part of the job is the calculation of the zeros of the function $q$ in (\[f53b\]). More information about this topic can be found at the end of the discussion of Example $f_{3}$ in Subsection \[e63\].
\[s6\]Examples
==============
In the present section we consider five specially chosen algebraic functions $f=f_{1},\ldots,f_{5}$, and discuss for each of them the solution of Problem $(f,\infty)$. Typically, we calculate and plot the minimal set $K_{0}(f,\infty)$, discuss particular features of its shape, and identify the sets $E_{0}$, $E_{1}$, $E_{2}$, and the family of Jordan arcs $J_{j}$, $j\in
I$, that have been introduced in Theorem \[t41a\] and in Definition \[d53b\]. Also the quadratic differential $q(z)dz^{2}$ from Theorem \[t53a\] is identified for each case.
Some of the examples depend on one or two parameters; and variations of these parameters will be done in order to understand the mechanisms that lead to special features of the minimal set $K_{0}(f,\infty)$. Of special interest are:
- The connectivity of the minimal set $K_{0}(f,\infty)$ together with the question of how it changes under variations of the function $f$.
- The identification of active versus inactive branch points of $f$. It turns out that in general not all branch points of the function $f$ play an active role in the determination of the minimal set $K_{0}(f,\infty)$, and for the calculation of $K_{0}(f,\infty)$ it is important to know already in advance which of them are active and which ones remain passive.
The presentation and discussion of the five examples demands comparatively much space, and there has been some hesitation to include all the material. But it is hoped that the expenses on space and efforts are counterbalanced by an improved understanding of the definitions and results presented in the last four sections.
\[e61\]Example $f_{1}$
----------------------
As a first, and in most aspects rather trivial example, we consider the function $$f_{1}(z):=\frac{1}{\sqrt{z^{2}-1}},\label{f61a}$$ which often appears in approximation theory, and has been included here as a warm-up exercise.
Clearly, the function has branch points at $-1$ and $1$. Therefore, the set $\mathcal{D}(f_{1},\infty)$ of admissible domains for Problem $(f_{1},\infty)$ from Definition \[d21a\] consists of all domains $D\subset\overline
{\mathbb{C}}$ such that $\infty\in D$ and that the two points $-1$ and $1$ are connected in the complement $K=\overline{\mathbb{C}}\setminus D$. The uniquely existing extremal domain of Theorem \[t22a\] is given by $$D_{0}(f_{1},\infty)=\overline{\mathbb{C}}\setminus\lbrack-1,1],\label{f61b}$$ and the minimal set by $K_{0}(f_{1},\infty)=[-1,1]$. As sets $E_{0}$, $E_{1}
$, $E_{2}$, and arcs $J_{j}$, $j\in I$, introduced in Theorem \[t41a\] and in Definition \[d53b\], we have $E_{0}=\{-1,1\}$, $E_{1}=E_{2}=\emptyset$, $I=\{1\}$, and $J_{1}=(-1,1)$. Solution (\[f61b\]) is a consequence of the monotonicity of $\operatorname*{cap}(\cdot)$ under projections onto straight lines (cf., Lemma \[l111c\] in Subsection \[s1101\], further below). The single arc $J_{1}=(-1,1)$ in $K_{0}(f_{1},\infty)$ is a trajectory of the quadratic differential $$\frac{1}{z^{2}-1}dz^{2},\label{f61c}$$ i.e., it satisfies the relation $$\frac{1}{z^{2}-1}dz^{2}<0,\label{f61d}$$ and (\[f61c\]) corresponds to Theorem \[t53a\].
\[e62\]Example $f_{2}$
----------------------
Next, we consider a function $f_{2}$ that depends on a parameter $\varphi$. For $\varphi\in\left( 0,\pi/2\right) $, we define $\left(
\varphi_{j}\right) _{j=1,\ldots,4}:=(\varphi,\pi-\varphi,\pi+\varphi
,2\pi-\varphi)$, $z_{j}:=\exp(i\,\varphi_{j})$, $j=1,\ldots,4$, $P_{4}(z):=\prod_{j=1}^{4}(1-z_{j}/z)$, and then we define the function $f_{2} $ as $$f_{2}(z):=\sqrt[2]{P_{4}(z)}\label{f62a}$$ with a choice of the sign of the square root in (\[f62a\]) so that $f_{2}(\infty)=1$. The function $f_{2}$ has the four branch points $z_{1},\ldots,z_{4}$, and it is symmetric with respect to the real and the imaginary axis. The symmetries lead to corresponding symmetries of the minimal set $K_{0}(f_{2},\infty)$ and the extremal domain $D_{0}(f_{2},\infty)$ for each $\varphi\in\left( 0,\pi/2\right) $.
\[ptb\]
[Fig1.eps]{}
For each $\varphi\in\left( 0,\pi/2\right) $, the set $\mathcal{D}(f_{2},\infty)$ of admissible domains introduced in Definition \[d21a\] consists of all domains $D\subset\overline{\mathbb{C}}$ such that $\infty\in
D$ and that at least two disjoint pairs of the four branch points $z_{1},\ldots,z_{4}$ are connected in $K=\overline{\mathbb{C}}\setminus D$. It is not necessary that all four points $z_{1},\ldots,z_{4}$ are connected, nor that a specific combination of pairs has to be connected in $K=\overline
{\mathbb{C}}\setminus D$.
From the uniqueness of the minimal set $K_{0}(f_{2},\infty),$ which has been proved in Theorem \[t22a\], it follows that from the variety of connectivities that are possible for the set $K\in\mathcal{K}(f_{2},\infty) $ and a given fixed parameter value $\varphi$, a specific one is selected as the minimal set $K_{0}(f_{2},\infty)$.
The shape and the connectivity of the minimal set $K_{0}(f_{2},\infty)$ depends on the parameter $\varphi$, and we distinguish the three cases $0<\varphi<\pi/4$, $\varphi=\pi/4$, and $\pi/4<\varphi<\pi/2$, which we will label as cases $a$, $b$, and $c$, respectively. In the three windows of Figure \[fig1\], the three cases are represented by the minimal sets $K_{0}(f_{2},\infty)$ for the parameter values $\varphi=\pi/6,$ $\varphi=\pi/4$, and $\varphi=101\pi/400$, respectively. The value $\varphi=101\pi/400$ has been chosen to be close to the critical value $\varphi=\pi/4$. The picture in the third window gives an impression of the metamorphosis of the set $K_{0}(f_{2},\infty)$ when $\varphi$ approaches and then crosses the critical value $\varphi_{0}=\pi/4$.
In the two cases $a$ and $c$, the minimal set $K_{0}(f_{2},\infty)$ consists of two components. We have $E_{0}=\{z_{1},\ldots,z_{4}\}$, $E_{1}=\emptyset$, $E_{2}=\{0\}$, $I=\{1,2\}$, and the two analytic Jordan arcs $J_{1}$ and $J_{2}$ in $K_{0}(f_{2},\infty)$ which connect the two pairs of branch points $\{z_{1},z_{4}\}$ and $\{z_{2},z_{3}\}$ in case $a$ and the two pairs $\{z_{1},z_{2}\}$ and $\{z_{3},z_{4}\}$ in case $c$.
The case $b$ corresponds to the single parameter value $\varphi=\pi/4$. Here, all four branch points $z_{1},\ldots,z_{4}$ are connected in $K_{0}(f_{2},\infty)$; the set is a continuum. We have $E_{0}=\{z_{1},\ldots
,z_{4}\}$, $E_{1}=\{0\}$, $E_{2}=\emptyset$, $I=\{1,\ldots,4\}$, and the four Jordan arcs $J_{1},\ldots,J_{4}$ in $K_{0}(f_{2},\infty)$ are the four segments $(0,z_{j})$, $j=1,\ldots,4$.
The two Jordan arcs $J_{1}$ and $J_{2}$ in the two cases $a$ and $c$, and also the $4$ Jordan arcs $J_{1},\ldots,\allowbreak J_{4}$ in case $b,$ are trajectories of the quadratic differential $$\frac{z^{2}}{\prod_{j=1}^{4}(z-z_{j})}dz^{2}.\label{f62b}$$
Taking advantage of the symmetry of the function $f_{2}$, one can show that for each $\varphi\in\left( 0,\pi/2\right) \setminus\{\pi/4\}$ the two arcs $J_{1}$ and $J_{2}$ are sections of an hyperbole. Indeed, it is not difficult to verify that the mapping $z\mapsto z^{2}$ maps the two arcs $J_{1}$ and $J_{2}$ onto one straight segment, which proves this last assertion.
\[e63\]Example $f_{3}$
----------------------
The third example is very similar to the second one, only that now the forth root is taken instead of the square root in (\[f62a\]). We use the same definitions for $\varphi$,$\ \varphi_{j}$, $z_{j}$, $j=1,\ldots,4 $, and $P_{4}$ as in Example \[e62\], and define function $f_{3}$ as $$f_{3}(z):=\sqrt[4]{P_{4}(z)}.\label{f63a}$$ The branch of the root $\sqrt[4]{\cdot}$ is chosen so that $f_{3}(\infty)=1 $. Although the basic structure of the two functions $f_{3}$ and $f_{2}$ is very similar, there exist decisive differences with respect to their meromorphic continuability. For each parameter value $\varphi\in\left(
0,\pi/2\right) $, the set $\mathcal{D}(f_{3},\infty)$ of admissible domains for Problem $(f_{3},\infty)$ consists of all domains $D\subset\overline
{\mathbb{C}}$ such that $\infty\in D$ and all four branch points $z_{1},\ldots,z_{4}$ are connected in the complementary set $K=\overline{\mathbb{C}}\setminus D$.
\[ptb\]
[Fig2.eps]{}
As in Example \[e62\], we distinguish three cases $a$, $b$, and $c$, which are again defined by $0<\varphi<\pi/4$, $\varphi=\pi/4$, and $\pi
/4<\varphi<\pi/2$, respectively. In all three cases, the minimal set $K_{0}(f_{3},\infty)$ is connected, and the extremal domain $D_{0}(f_{3},\infty)$ is simply connected. However, the minimal set $K_{0}(f_{3}\allowbreak\infty)$ is of a somewhat different structure in each of the three cases.
In case $b,$ the two functions $f_{2}$ and $f_{3}$ have an identical extremal domain $D_{0}(f_{3},\infty)$ and an identical minimal set $K_{0}(f_{3},\infty)=K_{0}(f_{2},\infty)$. The minimal set has already been shown in the middle window of Figure \[fig1\].
For the two other cases $a$ and $c$, two representatives of the minimal sets $K_{0}(f_{3},\allowbreak\infty)$ are shown in Figure \[fig2\]. The two cases are represented by the same two parameter values $\varphi=\pi/6$ and $\varphi=101\,\pi/400$ as already used before in Figure \[fig1\]. A new phenomenon now is the appearance of two bifurcation points in $K_{0}(f_{3},\infty)$, which are denoted by $z_{5}$ and $z_{6}$ in Figure \[fig2\].
In the two cases $a$ and $c$, we have $E_{0}=\{z_{1},\ldots,z_{4}\}$, $E_{1}=\{z_{5},z_{6}\}$, $E_{2}=\emptyset$, $I=\{1,\ldots,5\}$, and the five open analytic Jordan arcs $J_{1},\ldots,J_{5}$ in $K_{0}(f_{3},\infty)
$ connect the six points $z_{1},\ldots,z_{6}$ as shown in Figure \[fig2\]. These five Jordan arcs $J_{1},\ldots,\allowbreak J_{5}$, and also the four arcs in case $b$, are trajectories of the quadratic differential $$\frac{(z-z_{5})(z-z_{6})}{\prod_{j=1}^{4}(z-z_{j})}dz^{2}.\label{f63b}$$
Notice that in case $b$, we have $z_{5}=z_{6}=0$. In the two other cases, we always have $z_{5}=-z_{6}\neq0$. From a practical point of view the calculation of the two bifurcation points $z_{5}$ and $z_{6}$ is the main work and causes the main difficulties for the calculation of the arcs $J_{1},\ldots,\allowbreak J_{5}$. We want to take a closer look on this problem.
The form of the quadratic differential (\[f63b\]) already suggests that elliptic integrals should play a role in the analytic determination of the bifurcation points $z_{5}$ and $z_{6}$. Indeed, with the machinery presented in [@Lowien73], [@Blerch82], or [@Pirl69], it is not too difficult to formulate conditions that allow to determine the points $z_{5}$ and $z_{6}$. We reproduce the main elements of the procedure for case $a$, i.e., for the case $\varphi\in\left( 0,\pi/4\right) $, and define the function $$g(a,x):=\left\vert \sqrt{\frac{x-a}{x\,(x^{2}-2x\cos(2\varphi)+1)}}\right\vert
\text{ \ for \ }a\in(0,1)\text{, \ }x\in\mathbb{R.}\label{f63c}$$ The improper elliptical integral $$I(a):=\lim_{c\rightarrow+\infty}\left[ \int_{a}^{c}g(a,x)dx-\int_{-c}^{0}g(a,x)dx\right] \label{f63d}$$ is strictly monotonic for $a\in(0,1)$, and we have $I(0)>0$ and $I(1)<0$. Consequently, there uniquely exists $a_{0}\in(0,1)$ with $I(a_{0})=0$. The two bifurcation points $z_{5}$ and $z_{6}$ are then given by $$z_{5}=z_{5}(\varphi)=+\sqrt{a_{0}},\text{ \ \ }z_{6}=-z_{5}.\label{f63e}$$ For the special parameter value $\varphi=\pi/6$, for which the corresponding minimal set $K_{0}(f_{3},\infty)$ is shown in the first window of Figure \[fig2\], we get $$a_{0}=0.231584\text{, \ \ }z_{5}=0.481232\text{, \ and \ }z_{6}=-0.481232.\label{f63f}$$
In a derivation of the expressions (\[f63c\]) and (\[f63d\]), one has in a first step to transform the minimal set $K_{0}(f_{3},\infty)$ by the mapping $z\mapsto z^{2}$ into a continuum that connects the three points $0$, $e^{i2\varphi}$, and $e^{-i2\varphi}$.
After the reduction to a three-point problem, one can apply results that have been proved in [@Kuzmina68] (see also [@Kuzmina82], Theorem 1.5). In [@Kuzmina82], Theorem 1.5, the value $a_{0}$ is expressed as the solution of a system of four equations that involve Jacobi elliptical functions and theta functions. We have not investigated whether the approach is numerically easier to handle than the equation $I(a_{0})\overset{!}{=}0$, which is based on (\[f63d\]). In any case, the level of difficulties that arise already in this rather simply structured case of function $f_{3}$ gives an idea of the type of difficulties that arise if one has to determine the points in the set $E_{1}$ (and $E_{2}$) in a more general situation. In the next two examples these points have been calculated by a numerical method that has been developed by the author on an ad-hoc basis. It is based on a geometrical approach. The method will be published in a separate paper. Further comments about the numerical side of the problem will be made in Subsection \[s83\], further below.
\[e64\]Example $f_{4}$
----------------------
In the fourth example, we consider a modification of the function $f_{3}$, which itself has already been a modification of function $f_{2}$. We use again the definitions $\varphi$,$\ \varphi_{j}$, $z_{j}$, $j=1,\ldots,4$, and $P_{4}$ from Example \[e62\], and define the new function $f_{4}$ as $$f_{4}(z):=\sqrt[2]{\sqrt[2]{P_{4}(z)}-c}.\label{f64a}$$ In addition to the former parameter $\varphi$, there is now a second parameter $c$, which may assume arbitrary complex values $c\in\mathbb{C}$, but we shall consider only special situations. We discuss complex values of $c$ that lie near the origin, and in addition real values of $c$ in the interval $(0,1)$. The signs of the inner and outer square root in (\[f64a\]) are assumed to be chosen in such a way that both roots are positive for $z=\infty$ and $c=0$. In case of $c=0$, the two functions $f_{4}$ and $f_{3}$ are identical.
The study of the function $f_{4}$ and its associated minimal set $K_{0}(f_{4},\infty)$ will be more complex and involved than that of the last two examples, which in some sense have been preparations of the present example. Our main interest will be concentrated on the following three questions:
- It is not difficult to see that for almost all parameter constellations the function $f_{4}$ has $8$ branch points. But not all of them will always play an active role in the determination of the minimal set $K_{0}(f_{4},\infty)$, some of them are hidden away from $K_{0}(f_{4},\infty)$ somewhere on a ’lower’ sheet of the Riemann surface $\mathcal{R}_{f_{4}}$ that is defined by $f_{4}$. In the terminology of Section \[s3\], we can say that these inactive branch points on $\mathcal{R}_{f_{4}}$ stay away from the extremal domain $D_{0}(\mathcal{R}_{f_{4}},\infty^{(0)})\subset\mathcal{R}_{f_{4}}$. The first question in our discussion is therefore: Which of the branch points of $f_{4}$ are ’active’ and which ones are ’inactive’ for a given parameter constellation?
- We have already seen in Example \[e62\] that the connectivity of the minimal set $K_{0}(f_{4},\infty)$ can change. Motivated by this experience, the second question will be: What is the connectivity of the minimal set $K_{0}(f_{4},\infty)$ for a given parameter constellation, and how does it change with variations of the parameter values?
- At the end of the last example we have discussed in some detail the difficulties to find the points of the set $E_{1}$. In general these points are bifurcation points of the minimal set, and these points are crucial for the quadratic differential (\[f53b\]) in Theorem \[t53a\]. The third question is therefore: How do the bifurcation points of the minimal set $K_{0}(f_{4},\infty)$ depend on the parameter values, and at which parameter constellations do these points merge or split up?
The function $f_{4}$ has in general eight branch points; four of them are identical with those of the two functions $f_{2}$ and $f_{3}$, and they will be denoted again by $z_{1},\ldots,z_{4}$. These four branch points do not depend on the parameter $c$.
For every parameter $\varphi\in\lbrack0,\pi/2)$ there exists a whole region of parameter values $c$ such that only these four ’old’ branch points $z_{1},\ldots,z_{4}$ of $f_{4}$ appear in the minimal set $K_{0}(f_{4},\infty)$, and in these cases they are the only branch points that play an active role in the determination of $K_{0}(f_{4},\infty)$. All other branch points will be called ’inactive’.
\[ptb\]
[Fig3.eps]{}
Throughout the discussion, we keep the parameter $\varphi=\pi/6$ fixed, which implies that all minimal sets $K_{0}(f_{4},\infty)$ that will be considered during our discussion should be compared with the set $K_{0}(f_{3},\infty)$ in the first window of Figures \[fig2\].
In a first step we choose $$c=r\,e^{it}\text{ \ \ with \ \ }t\in\lbrack0,2\pi)\text{ \ \ and
\ \ }r>0\text{ \ small,}\label{f64b}$$ and see what happens. If $|c|>0$ is small, then the four new branch points $z_{5},\ldots,z_{8}$ of the function $f_{4}$ lie close to the four old branch points $z_{1},\ldots,z_{4}$. In Figure \[fig3\] the situation is shown for the parameter values $\varphi=\pi/6$ and $c=\sqrt{0.4}$. Of course, $\sqrt{0.4}$ is not very small, however, smaller values of $|c|$ lead to configurations that are difficult to plot.
\[t\]
[Fig4.eps]{}
While in (\[f64b\]) the parameter $t$ runs through $[0,2\pi)$, each one of the four new branch points $z_{5},\ldots,z_{8}$ encircles two times the corresponding old branch point $z_{1},\ldots,z_{4}$.
The interesting point is now that the four new branch points $z_{5},\ldots,z_{8}$ are elements of the minimal set $K_{0}(f_{4},\infty)$ only on one half of their twofold circular path. On the other half, they become ’inactive’, i.e., they are hidden away on another sheet of the Riemann surface $\mathcal{R}_{f_{4}}$. In this later case, the set $K_{0}(f_{4},\infty)$ contains only the four branch points $z_{1},\ldots,\allowbreak z_{4}$, and consequently, it is identical with the minimal set $K_{0}(f_{3},\infty)$, which has been shown in the first window of Figure \[fig2\].
It has already been said that in Figure \[fig3\], the minimal set $K_{0}(f_{4},\infty)$ is shown for the parameter values $\varphi=\pi/6$ and $c=\sqrt{0.4}$. This is a parameter constellation in which all eight branch points $z_{1},\ldots,z_{8}$ are active. In contrast to this, the parameter constellation $\varphi=\pi/6$ and $c=-\sqrt{0.4}$, which corresponds to $t=\pi$ in (\[f64b\]), leads to a minimal set $K_{0}(f_{4},\infty)$ that contains only the four old branch points $z_{1},\ldots,z_{4}$, and it is therefore identical with the minimal set $K_{0}(f_{3},\infty)$ shown in the first window of Figure \[fig2\].
Studying the minimal set $K_{0}(f_{4},\infty)$ for $|c|$ small, gives a good illustration of the phenomenon of active and inactive branch points. Of course, an extension of such a discussion to arbitrary values of $c\in\mathbb{C}$ would be possible, but it become rather complicated.
Next, we consider Problem $(f_{4},\infty)$ for the six specially chosen real parameter values $c=\sqrt{0.4},\allowbreak\sqrt{0.7},\allowbreak\sqrt
{0.705},\allowbreak\sqrt{0.715},\sqrt{0.74},\sqrt{0.76}$ and keep again $\varphi=\pi/6$ fixed. The selected values should be seen as representatives for the general situation of $c\in(0,1)$. The discussion will show why the specific selection is interesting.
There exists numerical evidence (but no analytic proof, so far) that at the critical parameter value $c_{0}=\sqrt[4]{1/2}$, the minimal set $K_{0}(f_{4},\infty)$ changes its connectivity. It is obvious that there exists $c_{0}\in(0,1)$ which is equal to, or lies close to $\sqrt[4]{1/2}$ such that for $0\leq c\leq c_{0}$ the set $K_{0}(f_{4},\infty)$ is connected, and for $c_{0}<c<1$ it is disconnected. In the disconnected case, it consists of three components. For $c\rightarrow c_{0}-0$, in each of the two half-planes $\{\operatorname*{Re}(z)\lessgtr0\}$ three bifurcation points of $K_{0}(f_{4},\infty)$ merge and form a new bifurcation point of order five in each of the two half-planes.
In Figure \[fig4\], the sequence of four minimal sets $K_{0}(f_{4},\infty) $ is shown for the parameter values we have $c=\sqrt{0.4},\allowbreak\sqrt
{0.7},\allowbreak\sqrt{0.705},\allowbreak\sqrt{0.715}$. The sequence shows the metamorphosis of the set $K_{0}(f_{4},\infty)$ while the parameter $c$ crosses the critical value $c_{0}=\sqrt[4]{1/2}=\sqrt{0.707...} $. In the four windows the set $K_{0}(f_{4},\infty)$ is shown only for the right half-plane.
In the first three windows of Figure \[fig4\], the minimal set $K_{0}(f_{4},\infty)$ is connected, and there are three bifurcation points $z_{9},z_{12},$ and $z_{13}$, each of order $3$, which then merge to a single bifurcation point when $c$ reaches the critical value $c_{0}$. At that moment, the new bifurcation point is of order $5$.
When the critical value $c_{0}$ has been passed, then the minimal set $K_{0}(f_{4},\infty)$ is disconnected, as shown in the fourth window of Figure \[fig4\]. There remains a bifurcation point $z_{9}$ of order $3$, and as a new phenomenon, we have a critical point of the Green function $g_{D_{0}}(\cdot,\infty)$, $D_{0}=D_{0}(f_{4},\infty)$, at $z_{11}$.
\[ptb\]
[Fig4b.eps]{}
Another interesting parameter value is $c_{1}=\sqrt{3/4},$ since at the parameter constellation $\varphi=\pi/6$ and $c_{1}=\sqrt{3/4}$ two pairs of branch points of the function $f_{4}$ collapse to simple zeros of $f_{4}$. These two simple zeros are located at $\pm\sqrt{2}$.
In Figure \[fig4b\], the transition process at the critical value $c_{1}=\sqrt{3/4}$ is represented by the two parameter values $c=\sqrt{0.74}$ and $c=\sqrt{0.76}$. One can see how the concerned components of $K_{0}(f_{4},\infty)$ change their shape from a type of vertical arcs to horizontal slits.
We conclude the discussion of Example \[e64\] by assembling informations about the sets $E_{0}$, $E_{1}$, $E_{2}$, and the arcs $J_{j}$, $j\in I$, introduced in Theorem \[t41a\] and in Definition \[d53b\]. This is done for the six parameter constellations of the two Figures \[fig4\] and \[fig4b\]. In addition we also give the quadratic differential $q(z)dz^{2}$ from Theorem \[t53a\]. This information corresponds to the whole set $K_{0}(f_{4},\infty)$, while in the Figures \[fig4\] and \[fig4b\] only restrictions to the right half-plane have been plotted.
For the three parameter values $c=\sqrt{0.4},\allowbreak\sqrt{0.7},\allowbreak\sqrt{0.705}$ the minimal set $K_{0}(f_{4},\allowbreak\infty)$ is connected, and with respect to $E_{0}$, $E_{1}$, $E_{2}$, $J_{j}$, $j\in I $, and the quadratic differential $q(z)dz^{2}$ we have identical structures.
We have $E_{0}=\{z_{1},\ldots,z_{8}\}$, $E_{1}=\{z_{9},\ldots,z_{14}\}$, and $E_{2}=\emptyset$. All eight branch points $z_{1},\ldots,z_{8}$ of $f_{4}$ are active, there are six bifurcation points $z_{9},\ldots,\allowbreak z_{14}$ and $13$ Jordan arcs $J_{j}$, $j\in I=\{1,\ldots,13\}$. In accordance to Theorem \[t53a\], all $13$ arcs $J_{j}$, $j\in I$, are trajectories of the quadratic differential $$\frac{\prod_{j=9}^{14}(z-z_{j})}{\prod_{j=1}^{8}(z-z_{j})}dz^{2}.\label{f64c}$$
For the three parameter values $c=\sqrt{0.715},\allowbreak\sqrt{0.74},\allowbreak\sqrt{0.76}$, which correspond to the fourth window in Figure \[fig4\] and the two windows in Figure \[fig4b\], the minimal set $K_{0}(f_{4},\infty)$ consists of three components. The sets $E_{0}$, $E_{1}$, $E_{2}$, $J_{j}$, $j\in I$, and the quadratic differential $q(z)dz^{2}$ are of the same structure in all three cases. There are two bifurcation points $z_{9}$, $z_{10}$, and the Green function $g_{D_{0}}(\cdot,\infty),$ $D_{0}=D_{0}(f_{4},\infty),$ has two critical points $z_{11}$ and $z_{12}$.
Thus, we have $E_{0}=\{z_{1},\ldots,z_{8}\}$, $E_{1}=\{z_{9},z_{10}\}$, and $E_{2}=\{z_{11},z_{12}\}$. There are $7$ Jordan arcs $J_{j}$, $j\in
I=\{1,\ldots,7\}$, and these arcs are trajectories of the quadratic differential $$\frac{\prod_{j=9}^{10}(z-z_{j})\prod_{j=11}^{12}(z-z_{j})^{2}}{\prod_{j=1}^{8}(z-z_{j})}dz^{2}.\label{f64d}$$
\[e65\]Example $f_{5}$
----------------------
As a last example, we come back to the algebraic function (\[f1a\]), which has already been used in the Introduction for a demonstration of the connection between Padé approximation and sets of minimal capacity. This function is now denoted as $f_{5}$, and it has been defined in (\[f1a\]) as $$f_{5}(z):=\sqrt[4]{\prod\nolimits_{j=1}^{4}(1-z_{j}/z)}+\sqrt[3]{\prod\nolimits_{j=5}^{7}(1-z_{j}/z)}\label{f65a}$$ with the $7$ branch points that have been chosen as $$\begin{aligned}
z_{1} & =1+3\,i,\text{ \ \ \ }z_{2}=-4+2\,i,\text{ \ \ \ }z_{3}=-4+i,\text{
\ \ \ }z_{4}=0+2\,i,\nonumber\\
z_{5} & =2+2\,i,\text{ \ \ \ }z_{6}=3+4\,i,\text{ \ \ \ \ \ }z_{7}=1+4\,i.\label{f65b}$$
The choice of the branch points was in principle arbitrary, but it reflects the intension to avoid symmetries in the minimal set $K_{0}(f_{5},\infty)$ of a sort that has been very dominant in the $3$ Examples \[e62\] - \[e64\].
From the structure of function $f_{5}$, we conclude that the set $\mathcal{D}(f_{5},\infty)$ of admissible domains for Problem $(f_{5},\infty)$ introduced in Definition \[d21a\] consists of all domains $D\subset
\overline{\mathbb{C}}$ such that $\infty\in D$ and that the elements of each of the two subsets of branch points $\{z_{1},\ldots,z_{4}\}$ and $\{z_{5},z_{6},z_{7}\}$ are connected in the complement $K=\overline
{\mathbb{C}}\setminus D$.
\[ptb\]
[Fig5b.eps]{}
It turns out that the minimal set $K_{0}(f_{5},\infty)$ consists of two components, and that indeed each of them connects one of the two sets $\{z_{1},\ldots,z_{4}\}$ and $\{z_{5},z_{6},z_{7}\}$. The set $K_{0}(f_{5},\infty)$ is shown in Figure \[fig5\]. It has three bifurcation points, which are denoted by $z_{8},z_{9},z_{10}$, and the Green function $g_{D_{0}}(\cdot,\infty)$ in the extremal domain $D_{0}=D_{0}(f_{5},\infty)$ possesses exactly one critical point, which is denoted by $z_{11}$ in Figure \[fig5\].
While the $7$ branch points $z_{1},\ldots,z_{7}$ in (\[f65b\]) can be considered as input to the problem, the location of the four other points $z_{8},\ldots,z_{11}$ has to be determined by the criterion of minimality of the set $K_{0}(f_{5},\infty)$. The calculation of these four points has been done numerically, and their values are $$\begin{aligned}
z_{8} & =-3.57021+1.50570\,i,\nonumber\\
z_{9} & =-1.28112+1.30991\,i,\nonumber\\
z_{10} & =\;\text{\ \ }1.54341+3.19816\,i,\label{f65c}\\
z_{11} & =\;\text{\ \ }0.64231+2.79311\,i.\nonumber\end{aligned}$$
The $8$ Jordan arcs $J_{j}$, $j\in I=\{1,\ldots,8\}$, in $K_{0}(f_{5},\infty)$ are trajectories of the quadratic differential $$\frac{(z-z_{11})^{2}\prod_{j=8}^{10}(z-z_{j})}{\prod_{j=1}^{7}(z-z_{j})}dz^{2}.\label{f65d}$$ The sets $E_{0}$, $E_{1}$, $E_{2}$ introduced in Theorem \[t41a\] and in Definition \[d53b\] are now $E_{0}=\{z_{1},\ldots,z_{7}\}$, $E_{1}=\{z_{8},z_{9},z_{10}\}$, and $E_{2}=\{z_{11}\}$.
\[s66\]Some General Remarks
---------------------------
The main motivation for the selection and presentation of the $5$ Examples \[e61\] - \[e65\] was to illustrate the variety of topological structures that are possible for the minimal set $K_{0}(f,\infty)$. Naturally, such examples should be kept simple, but even for the comparatively simply structured functions $f_{1},\ldots,f_{5}$ in the Examples \[e61\] - \[e65\], the shape and the connectivity of the minimal set $K_{0}(f,\infty)$ has not always been clear at the outset of the analysis.
Naturally, the situation becomes more technical and much more difficult to handle if the function $f$ becomes more complex, and especially, if it is no longer algebraic. As a consequence, the set $E_{0}$ may no longer be finite. For general functions $f$ it is very difficult to predict shape and connectivity of the minimal set $K_{0}(f,\infty)$. One way to get some information and a rough idea in this respect is to calculate poles of Padé approximants to the function $f$. This, by the way, has been done in the study of the function (\[f1a\]) in the Introduction, and the result in Figure \[fig0\] should be compared with Figure \[fig5\].
A critical task for the numerical calculation of the Jordan arcs $J_{j}$, $j\in I$, in the minimal set $K_{0}(f,\infty)$ is the calculation of the zeros in the quadratic differential (\[f53b\]) in Theorem \[t53a\]. For this purpose we have developed a numerical procedure, which has been used in the analysis of the Examples \[e63\] - \[e65\]. More details about this topic will be given in Subsection \[s83\], further blow.
\[s7\]A Local Criterion and Geometric Estimates
===============================================
The $S-$property (symmetry property), which has already been introduced and considered in Subsections \[s51\], will again take central stage in the first three subsections. We start with a definition of this property that characterises the whole domain, and will then show that it is a local condition for the minimality (\[f21a\]) in Definition \[d21b\]. As a somewhat surprising result in Subsection \[s73\], we shall formulate a theorem in which it is proved that the $S-$property is also sufficient for the global minimality (\[f21a\]) in Definition \[d21b\]. In the fourth subsections several inclusion relations for the minimal set $K_{0}(f,\infty) $ are presented that can be helpful in many practical situations.
\[s71\]A General Definition of the $S-$Property
-----------------------------------------------
In Theorem \[t51a\] of Subsection \[s51\] the $S-$property (\[f51a\]) appears as an important characteristic of the extremal domain $D_{0}(f,\infty)$ and its complementary minimal set $K_{0}(f,\infty)$. In the present subsection we define the $S-$property for arbitrary admissible domains $D\in\mathcal{D}(f,\infty)$. We start with an auxiliary definition.
\[d71a0\]An admissible domain $D\in\mathcal{D}(f,\infty)$ for Problem $(f,\infty)$ is called elementarily maximal if for every point $z\in\partial
D$ one of the following two assertions holds true.
- There exists at least one meromorphic continuation of the function $f$ out of the domain $D$ that has a non-polar singularity at $z$.
- There exist at least two meromorphic continuations of the function $f$ out of $D$ that lead to two non-identical function elements at $z$.
It is immediate that if an admissible domain $D\in\mathcal{D}(f,\infty)$ is not elementarily maximal, then the domain $D$ can be enlarged in a straight forward way without leaving the class $\mathcal{D}(f,\infty)$ of admissible domains. Hence, the elementarily maximal domains are the maximal elements in $\mathcal{D}(f,\infty)$ with respect to ordering by inclusion. We formulate this statement as a proposition.
\[p71a\]The elementarily maximal domains of Definition \[d71a0\] are the maximal elements in $\mathcal{D}(f,\infty)$ with respect to ordering by inclusion.
From the Structure Theorem \[t41a\] in Subsection \[s41\], we easily deduce that the extremal domain $D_{0}(f,\infty)$ is elementarily maximal, but of course, there exist many other maximal elements in $\mathcal{D}(f,\infty
)$. Often it is helpful, and in most situations also possible, to assume without loss of generality that an arbitrarily chosen admissible domain $D\in\mathcal{D}(f,\infty)$ is elementarily maximal.
After these preliminaries we come to the definition of the $S-$property of a domain.
\[d71a\]We say that an admissible domain $D\in\mathcal{D}(f,\infty)$ possesses the $S-$property (symmetry property) with respect to Problem $(f,\infty)$ if its complement $K=\overline{\mathbb{C}}\setminus D$ is of the form $$K=E_{0}\cup E_{1}\cup\bigcup_{j\in I}J_{j}\label{f71a2}$$ and
- assertion (i) of Definition \[d71a0\] holds true for every $z\in\partial E_{0}$,
- assertion (ii) of Definition \[d71a0\] holds true for every $z\in K\setminus E_{0}$,
- all $J_{j}$, $j\in I$, are open, analytic Jordan arcs,
- the set $E_{1}\subset K\setminus E_{0}$ is discrete in $\mathbb{C}\setminus E_{0}$, each point $z\in E_{1}$ is the end point of at least three different arcs of $\{J_{j}\}_{j\in I}$, and
- if $I\neq\emptyset$, then we have $$\frac{\partial}{\partial n_{+}}g_{D}(z,\infty)=\frac{\partial}{\partial n_{-}}g_{D}(z,\infty)\text{ \ for all \ }z\in J_{j},j\in I\label{f71a1}$$ with $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ denoting the normal derivatives to both sides of the arcs $J_{j}$, $j\in I$. By $g_{D}(\cdot,\infty)$ we denote the Green function in $D$.
If $I\neq\emptyset$, then it is immediate that $\operatorname*{cap}\left(
\partial D\right) >0$, and consequently the Green function $g_{D}(z,\infty)$ exists in this case in a proper way (see Subsection \[s1103\], further below). From identity (\[f71a1\]) one can deduce that the Jordan arcs $J_{j}$, $j\in I$, are analytic. Hence, the analyticity assumed in assertion (iii) of Definition \[d71a\] is implicitly also contained in assertion (v).
Because of the two assertions (i) and (ii) in Definition \[d71a\], a domain $D\in\mathcal{D}(f,\infty)$ with the $S-$property is also elementarily maximal in the sense of Definition \[d71a0\].
With Definition \[d71a\] and the Structure Theorem \[t41a\], we can rephrase Theorem \[t51a\] in Subsection \[s51\] as follows: The extremal domain $D_{0}(f,\infty)$ possesses the $S-$property. In Subsection \[s73\], below, we shall see that also the reversed conclusion holds true, i.e., if an admissible domain $D\in\mathcal{D}(f,\infty)$ possesses the $S-$property, then it is identical with the extremal domain $D_{0}(f,\infty)$ of Definition \[d21b\].
\[s72\]A Local Extremality Condition
------------------------------------
In the present subsection we show that Hadarmard’s boundary variation formula for the Green function implies that the $S-$property of Definition \[d71a\] is a local condition for the minimality of $\operatorname*{cap}\left( \overline{\mathbb{C}}\setminus D\right) $, i.e., $\operatorname*{cap}\left( \overline{\mathbb{C}}\setminus D\right) $ assumes a (local) minimum under local variations of the boundary of an admissible domain $D\in
\mathcal{D}(f,\infty)$ that $D$ possesses the $S-$property.
We start with the introduction of some notations that are needed for the setup of the boundary variation for Hadamard’s variation formula (for a very readable introduction to this topic we recommend the appendix of [@Courant50]). Let $D\subset\overline{\mathbb{C}}$ be a domain with $\infty\in D$, assume that in $\partial D$ there exists a smooth, open Jordan arc $\gamma\subset\partial D$, and assume further that the domain $D$ lies only on one side of $\gamma$. By $n(v)\in\mathbb{T}$, we denote the normal vector on $\gamma$ at the point $v\in\gamma$ that shows into $D$. Let $v_{0}\in\gamma$ be fixed, $\varphi\geq0$ a smooth function defined on $\gamma$ with compact support. We assume that the support of $\varphi$ is small and that it contains the point $v_{0}\in\gamma$ in its interior, and choose $\varepsilon\in\mathbb{R}$ with $|\varepsilon|>0$ small.
With these definitions we introduce a variation $\widetilde{D}$ of the domain $D$ by moving each boundary point $v\in\gamma$ along the vector $\varepsilon
\varphi(v)\,n(v)$. If $|\varepsilon|>0$ is sufficiently small, then the new domain $\widetilde{D}$ is well defined. Hadamard’s variation formula for the Green function $g_{D}(z,w)$ under this type of variation of the domain $D$ says that $$\begin{aligned}
& g_{\widetilde{D}}(z,\infty)-g_{D}(z,\infty)=\label{f72a1}\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\varepsilon}{2\pi}\int_{\gamma}\frac{\partial}{\partial n}g_{D}(v,\infty)\frac{\partial}{\partial n}g_{D}(v,z)\varphi(v)ds_{v}+\text{O}(\varepsilon^{2})\nonumber\end{aligned}$$ for $|\varepsilon|\rightarrow0$, where $\partial/\partial n$ denotes the normal derivative and $ds$ the line element on $\gamma$. The Landau symbol O$(\cdot)$ holds uniformly for $z$ varying on a compact subset of $\widetilde{D}\cap D$.
From (\[f72a1\]) and the connection between the logarithmic capacity and the Green function (cf. Lemma \[l113b\] in Subsection \[s1103\], further below), we then get $$\frac{\operatorname*{cap}(\overline{\mathbb{C}}\setminus\widetilde{D})}{\operatorname*{cap}(\overline{\mathbb{C}}\setminus D)}-1=\frac{\varepsilon
}{2\pi}\int_{\gamma}\left( \frac{\partial}{\partial n}g_{D}(v,\infty)\right)
^{2}\varphi(v)ds_{v}+\text{O}(\varepsilon^{2})\label{f72b1}$$ for $|\varepsilon|\rightarrow0$, which shows that Hadamard’s variation formula (\[f72a1\]) gives us an explicit expression for the first order variation of $\operatorname*{cap}(\overline{\mathbb{C}}\setminus D)$ under local variations of a smooth piece $\gamma$ of the boundary $\partial D$.
Let us now assume that the domain $D\subset\overline{\mathbb{C}}$ contains a smooth, open Jordan arc $J\subset\partial D$ with the property that on both sides of $J$ there are only points of $D$, i.e., $J$ is a cut in some larger domain. As before, by $n(v)\in\mathbb{T}$ we denote the normal vector to $J$ at a point $v\in J$, and assume that all normal vectors $n(v)\in\mathbb{T}$, $v\in J$, show towards the same side of $J$. Again, by $\varphi\geq0$ we denote a smooth function on $J$ with compact support, and assume that the support is contained in a neighborhood of $v_{0}\in J$. The parameter $\varepsilon\in\mathbb{R}$ with $|\varepsilon|>0$ plays the same role as before.
With the definitions just introduced, we first define a variation $J_{\varepsilon}$ of the arc $J$. The new arc $J_{\varepsilon}$ results from moving each point $v\in J$ along the vector $\varepsilon\varphi(v)\,n(v)$. For $|\varepsilon|>0$ sufficiently small, $J_{\varepsilon}$ is well defined and again a smooth Jordan arc. The variation $D_{\varepsilon}$ of the domain $D$ is then defined by replacing the arc $J$ by $J_{\varepsilon}$. This type of variation changes the boundary $\partial D$ only locally, but the changes take place in two subregions of $D$. The two pieces of the boundary $\partial
D$ that correspond to the arc $J$ are moved in opposite directions. Because of this variation at two places in $D$ in opposite directions, we deduce from (\[f72b1\]) that $$\begin{aligned}
& \frac{\operatorname*{cap}(\overline{\mathbb{C}}\setminus D_{\varepsilon})}{\operatorname*{cap}(\overline{\mathbb{C}}\setminus D)}-1=\label{f72c1}\\
& \;\;\;\;\;\;\frac{\varepsilon}{2\pi}\int_{J}\left[ \left( \frac{\partial
}{\partial n_{+}}g_{D}(v,\infty)\right) ^{2}-\left( \frac{\partial}{\partial
n_{-}}g_{D}(v,\infty)\right) ^{2}\right] \varphi(v)ds_{v}+\text{O}(\varepsilon^{2})\nonumber\end{aligned}$$ for $|\varepsilon|\rightarrow0$, where $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ denote the normal derivatives to both sides of $J$. From (\[f72c1\]) and the fact that the support of the function $\varphi
$ can be chosen as small as we want, we can conclude rather immediately that the symmetry (\[f71a1\]) in Definition \[d71a\] of the $S-$property is equivalent to the vanishing of the first order variation of $\operatorname*{cap}(\overline{\mathbb{C}}\setminus D)$. It follows immediately from assertion (ii) in Definition \[d71a\] and the local character of the variation of the arc $J\subset\partial D$ that the resulting variational domain $D_{\varepsilon}$ of the original domain $D$ belongs again to $\mathcal{D}(f,\infty)$ if $|\varepsilon|>0$ is small. The conclusion of our discussion is formulated in the next theorem.
\[t72a\] Let the complement $K=\overline{\mathbb{C}}\setminus D$ of an admissible domain $D\in\mathcal{D}(f,\infty)$ be of the form (\[f71a2\]) with two sets $E_{0}$, $E_{1}$, and the family of arcs $\{J_{j}\}$, $j\in I$, that satisfy the assertions (i) - (iv) in Definition \[d71a\]. Then the symmetry condition (\[f71a1\]) holds for every $z\in J_{j}$, $j\in I$, if, and only if, the first order variations of $\operatorname*{cap}(\overline
{\mathbb{C}}\setminus D)$ vanish for all local variations of these arcs $J_{j}$ done as just described, i.e., if we have $$\lim_{|\varepsilon|\rightarrow\infty}\frac{1}{\varepsilon}(\operatorname*{cap}(\overline{\mathbb{C}}\setminus D_{\varepsilon})-\operatorname*{cap}(\overline{\mathbb{C}}\setminus D))=0\label{f72d1}$$ for all such variations.$\smallskip$
\[s73\]$S-$Property and Uniqueness
----------------------------------
In the light of Theorem \[t72a\], the result of Theorem \[t51a\] can no longer surprise since we now know that the symmetry$\ $property (\[f51a\]) in Theorem \[t51a\] is a necessary condition for the minimality (\[f21a\]) in Definition \[d21b\]. The interesting and perhaps somewhat surprising result in the present section is the next theorem, in which the last conclusion is reversed; it is shown that the $S-$property is also sufficient for the minimality (\[f21a\]) in Definition \[d21b\].
\[t73a\] If an admissible domain $D\in\mathcal{D}(f,\infty)$ possesses the $S-$property in the sense of Definition \[d71a\], then $D$ is identical with the extremal domain $D_{0}(f,\infty)$ of Definition \[d21b\].
Since we know from Theorem \[t22a\] that the extremal domain $D_{0}(f,\infty)$ is unique, we can deduce a uniqueness result for the extremal domain $D_{0}(f,\infty)$ from the $S-$property as a corollary to Theorem \[t73a\].
\[c73a\]The $S-$property of an admissible domain $D\in\mathcal{D}(f,\infty)$ determines uniquely the extremal domain $D_{0}(f,\infty)$ of Problem $(f,\infty)$.
The interpretation of the $S-$property as a local condition for the minimality (\[f21a\]) in Definition \[d21b\] is interesting in itself, but it is also interesting for several applications in rational approximation. Hadarmard’s variation formula (\[f72b1\]), on the other hand, is not very helpful as a tool for proofs of the two important Theorems \[t22a\] and \[t41a\] since it requires the knowledge of smoothness of the arcs $J_{j}$, $j\in I$, in the boundary $\partial D$. However, this property is known only when most of the groundwork for the proofs has already been done.
\[s74\]Geometric Estimates
--------------------------
The minimal set $K_{0}(f,\infty)$ of Problem $(f,\allowbreak\infty)
$ is in general not convex. The rather trivial Example \[e61\] is perhaps the only case, where we have convexity. However, convexity can give rough, and sometimes also quite helpful, geometric estimates for the minimal set $K_{0}(f,\infty)$. Some results in this direction are contained in the next theorem.
\[t74a\] Let $K_{0}(f,\infty)$ be the minimal set for Problem $(f,\infty
)$, and let further $E_{0}\subset K_{0}(f,\infty)$ be the compact set that has been introduced in the Structure Theorem \[t41a\], i.e., $\partial E_{0}$ contains all non-polar singularities of the function $f$ that can be reached by meromorphic continuations of the function $f$ out of the extremal domain $D_{0}(f,\infty)$.
\(i) For any convex compact set $K\subset\mathbb{C}$ with the property that the function $f$ has a single-valued meromorphic continuation throughout $\overline{\mathbb{C}}\setminus K$, we have $$K_{0}(f,\infty)\subset K.\label{f74a1}$$
\(ii) Let $\operatorname*{Co}(E_{0})$ denote the convex hull of $E_{0}$. Then we have $$K_{0}(f,\infty)\subset\operatorname*{Co}(E_{0}).\label{f74a2}$$
\(iii) Let $K\subset\mathbb{C}$ be a convex compact set, $E\subset
\mathbb{C}\setminus K$ a set of capacity zero that is closed in $\mathbb{C}\setminus K$, and assume that the function $f$ has a single-valued meromorphic continuation throughout $\overline{\mathbb{C}}\setminus(K\cup E)$ . Then we have $$K_{0}(f,\infty)\subset K\cup E.\label{f74a3}$$
\(iv) There uniquely exist two sets $K_{\min}\subset\mathbb{C}$ and $E_{\min
}\subset\mathbb{C}\setminus K_{\min}$ with the same properties as assumed in assertion (iii) for the pairs of sets $\{K,E\}$ such that these sets are minimal with respect to inclusion among all pairs $\{K,E\}$ that satisfy the assumptions of assertion (iii), and we have $$K_{0}(f,\infty)\subset K_{\min}\cup E_{\min}.\label{f74a4}$$
\(v) Let $Ex(K_{\min})$ denote the set of extreme points of the convex set $K_{\min}$ from assertion (iv). Then we have $$Ex(K_{\min})\cup E_{\min}\subset E_{0}.\label{f74a5}$$
\[s8\]Geometrically Defined Extremality Problems
================================================
Extremality problems are a classical topic in geometric function theory, and among the different versions that are studied there we also find the kind of problems that are concerned with sets of minimal capacity. In the present section our interest concentrates on extremality problems that are defined purely by geometrical conditions since these problems have strong similarities with Problem $(f,\infty)$. But there also exist significant differences, which, of course, are the interesting aspects for our discussion.
In order to make this discussion more concrete, and also for later use in proofs, further below, we formulate two classical problems of the geometrical type. The first one is presented in two versions.
\[s81\]Two Classical Problems
-----------------------------
\[p81a\]*(Chebotarev’s Problem) Let finitely many points* $a_{1},\ldots,\allowbreak a_{n}\in\mathbb{C}$* be given. Find a continuum* $K\subset\mathbb{C}$* with* the property that $$a_{j}\in K\text{ \ \ \ \ \textit{for} \ \ }j=1,\ldots,n,\label{f81a}$$ *and further that the logarithmic capacity* $\operatorname*{cap}\left(
K\right) $* is minimal among all continua* $K\subset\mathbb{C}$* that satisfy (\[f81a\]).*
Problem \[p81a\] can be refined in a way that brings it closer to situations that could be observed in the Examples \[e63\], \[e64\], and \[e65\] in Section \[s6\].
\[p81b\]*Let* $m$* sets* $A_{i}\subset\mathbb{C}$*,* $i=1,\ldots,m$*, of finitely many points* $a_{ij}\in A_{i}$*,* $j=1,\ldots,n_{i}$*,* $i=1,\ldots,m$,* be given. Find* $m$* continua* $K_{1},\ldots,K_{m}\subset\mathbb{C}$* with* the property that $$a_{ij}\in K_{i}\text{ \ \ \ \ \textit{for} \ \ }i=1,\ldots,m,\text{\ }j=1,\ldots,n_{i},\label{f81b}$$ *and further that the logarithmic capacity* $\operatorname*{cap}\left(
K_{1}\cup\ldots\cup K_{m}\right) $* is minimal among all continua* $K_{1},\ldots,K_{m}\subset\mathbb{C}$* that satisfy (\[f81b\]).*
It is evident that Problem \[p81b\] has many similarities to the Problems $(f,\infty)$ in the Examples \[e61\] - \[e65\] in Section \[s6\]. However, these examples also illustrate some of the essential differences. Especially, there is the question about ’active’ versus ’inactive’ branch points and also the question about the connectivity of the minimal set $K_{0}(f,\infty)$. Such questions don’t exist for the classical problems, since there they are part of the setup of the problem. In Problem $(f,\infty)$ it is in general not possible to have answers to such questions in advance; the answers are part of the solution and not part of the definition as in Problem \[p81a\] and \[p81b\].
The functions $f$ in the Examples \[e61\] - \[e65\] are rather simple and transparent representatives for the functions possible in Problem $(f,\infty
)$. In the case of a more complex analytic function $f$, the minimal set $K_{0}(f,\infty)$ can be very complicated.
From a certain point of view, the two Problems \[p81a\] and \[p81b\] can be seen as special cases of Problems $(f,\infty)$, one has only to choose the function $f$ in an appropriate way. We exemplify this argument for Problem \[p81b\]. Let $f_{1}$ be defined as $$f_{1}(z):=\sum_{i=1}^{m}\prod_{j=1}^{n_{i}}\left[ 1-\frac{a_{ij}}{z}\right]
^{1/n_{i}},\label{f82a}$$ then it is immediate that the minimal set $K_{0}(f_{1},\infty)$ from Theorem \[t22a\] is the unique solution of Problem \[p81b\].
As a second example for a purely geometrically defined extremality problem we consider the following one:
\[p81c\]*Let two disjoint, finite sets of points* $a_{1},\ldots,\allowbreak a_{n}\in\overline{\mathbb{C}}$* and* $b_{1},\ldots,\allowbreak b_{m}\in\overline{\mathbb{C}}$* be given. Find two continua* $K,V\subset\overline{\mathbb{C}}$* with the property that* $$a_{j}\in K\text{ \ \ \textit{for}\ \ }j=1,\ldots,n,\text{ \ \textit{\ \ \ }}b_{i}\in V\text{ \ \ \textit{for}\ \ }i=1,\ldots,m,\label{f81c}$$ *and further that the condenser capacity* $\operatorname*{cap}\left(
K,V\right) $* is minimal among all pairs of continua* $K,V\subset
\overline{\mathbb{C}}$* that satisfy (\[f81c\]).*
For a definition of the condenser capacity we refer to [@SaTo] Chapter II.5. or [@Bagby]. Problem \[p81c\] has been included here because of two reasons: its solution will be used as an important element in one of the proofs further below, and secondly, it is perhaps the simplest example of its kind with non-unique solutions. In this respect, it underlines the importance and relevance of the uniqueness part in Theorem \[t22a\]. More about this second aspect follows in the next subsection.
\[s82\]Some Methodological and Historic Remarks
-----------------------------------------------
Problem \[p81a\] has apparently been mention for the first time in a letter by Chebotarev to G. Pólya (see [@Polya29]). The existence and uniqueness of a solution for this problem has been proved already shortly afterwards in 1930 by H. Grötzsch [@Groetzsch30] with his famous strip method. In [@Groetzsch30] one can also find a description of the analytic arcs in the minimal set by quadratic differentials, although the presentation has been done in a different language. In about the same time of [@Groetzsch30], M.A. Lavrentiev has formulated and studied Problem \[p81a\] in [@Lavrentiev30] and [@Lavrentiev34] in an equivalent but somewhat different setting.
A comprehensive review of methods and results relevant for the Problems \[p81a\], \[p81b\], and \[p81c\] can be found in the two long survey papers [@Kuzmina82], [@Kuzmina97]. We also mention in this respect the textbooks [@Goluzin] and [@Pommerenke75].
In the introduction to the present section it has been mentioned that a wide range of extremality problems has been studied in geometric function theory. There exists a correspondingly broad variety of methods (different types of variational methods, the methods of extremal length, quadratic differentials, etc.) for the analysis of such problems. For our purpose the survey papers [@Kuzmina82] and [@Kuzmina97] have provided a good coverage of the relevant literature.
In our proofs we shall need only properties of the solution of a special case of Problem \[p81c\] (see Definition \[d101a\] in Subsection \[s1011\], further below). In this problem the two sets $A=\{a_{1},\ldots,a_{n}\}$ and $B=\{b_{1},\ldots,b_{n}\}$ consist of points which are reflections of each other on the unit circle $\partial\mathbb{D}$, i.e., we assume that $b_{j}=1/\overline{a}_{j}$ for $j=1_{1},\ldots,n$. Under this assumption, Problem \[p81c\] can be seen as a hyperbolic version of Problem \[p81a\]. Indeed, the set $A$ of the given $n$ points is assumed to be contained in $\mathbb{D}$ and the logarithmic capacity $\operatorname{cap}(K)$ in Problem \[p81a\] is replaced by the hyperbolic capacity of $K\subset\mathbb{D}$ (see Subsection \[s1011\], further below).
Our last topic in the present subsection is concerned with the possibility of non-unique solutions to Problem \[p81c\]. We start with some remarks about Teichmüller’s problem, which practically is the most special situation of Problem \[p81c\]. If in Problem \[p81c\] both sets $A$ and $B$ consist of only $2$ points, then with the help of a Moebius transformation one can show that without loss of generality $3$ of the $4$ points can be chosen in a standardized way, which usually is done so that $A=\{-1,1\}$ and $B=\{b,\infty\}$ with $b$ being an arbitrary point in $\mathbb{C}\setminus\{-1,1\}$. Under these assumptions, the minimal condenser capacity $\operatorname*{cap}\left( K,V\right) $ of Problem \[p81c\] depends only on single complex variable $b$. The minimality problem in this special form has been suggested by O. Teichmüller in [@Teichmueller38], and it carries today his name. Its solution and the study of its properties has attracted some research interest (cf., [@Kuzmina82], Chapter 5.2, for a survey); we mention here only the very recent publication [@Heikkala06], where a numerical method for an efficient calculation of $\operatorname*{cap}\left( K,V\right) $ in dependence of $b\in\mathbb{C}\setminus\{-1,1\}$ has been developed and studied.
For our discussion, the cases with $b\in\left( -1,1\right) $ are of special interest, since Teichmüller’s problem has non-unique solutions exactly for the parameter values $b\in\left( -1,1\right) $. We consider the symmetric case $b=0$.
If in Problem \[p81c\], we choose $n=m=2$, $\left\{ a_{1},a_{2}\right\}
=\left\{ -1,1\right\} $, and $\left\{ b_{1},b_{2}\right\} =\left\{
0,\infty\right\} $, then it is not too difficult to verify by symmetry considerations that there exist at least two different solutions $(K,V)$. The first one is given by $K:=\{$ $e^{it}$ $|$ $\pi\leq t\leq2\pi$ $\}$ and $V:=\{$ $z$ $|$ $0\leq\operatorname*{Im}(z)\leq\infty$, $\operatorname*{Re}(z)=0$ $\}$, while the second one is its symmetric counterpart $\widetilde
{K}:=\{$ $e^{it}$ $|$ $0\leq t\leq\pi$ $\}$ and $\widetilde{V}:=\{$ $z$ $|$ $-\infty\leq\operatorname*{Im}(z)\leq0$, $\operatorname*{Im}(z)=0$ $\}$. This counterexample to uniqueness underlines that the uniqueness part of Theorems \[t22a\] cannot be trivial.
The proof of uniqueness of the solution to Problem $(f,\infty)$ is contained in Subsection \[s93\], and it has demanded some new ideas and concepts. A review of the uniqueness question for the general case of Problem \[p81c\] is contained in Chapter 5.4 of [@Kuzmina82].
\[s83\]The Numerical Calculation of the Set $K_{0}(f,\infty)
$
------------------------------------------------------------
From Theorem \[t41a\] we have a general knowledge of the structure of the minimal set $K_{0}(f,\infty)$, and we know that there uniquely exist two compact sets $E_{0}$, $E_{1}$, and a family of Jordan arcs $J_{j}$, $j\in I$, which are trajectories of a certain quadratic differential, and the union of these objects forms the set $K_{0}(f,\infty)$ in (\[f41a\]) of Theorem \[t41a\]. In each concrete case of a function $f$ that is not as simple as that in Example \[e61\], the determination of $E_{0}$, $E_{1},$ and $J_{j}$, $j\in I$, is a difficult and tricky task, and there is no general method at hand that can be applied in all situations.
The situation is different in the more special case of Theorem \[t53a\], where we have a rational quadratic differential $q(z)dz^{2}$, which can be used for the calculation of the Jordan arcs $J_{j}$, $j\in I$. In this more special situation, only two critical tasks have to be done: The first one consists in finding the set of branch points of the function $f$ in Problem $(f,\infty)$ that play an active role in the determination of the set $K_{0}(f,\infty)$; part of this first task is also the determination of the connectivity of the set $K_{0}(f,\infty)$. The second critical task is the calculation of the zeros in the quadratic differential (\[f53b\]) in Theorem \[t53a\]. This second task appears in a similar form if one wants to solve Problem \[p81b\], and therefore it has found already earlier attention in the literature. Some results in this direction have been reviewed in the discussion at the end of Example \[e63\].
In the analysis of the Examples \[e62\] - \[e65\] in Section \[s6\], the second task has been solved with the help of a numerical procedure that has been developed in an ad-hoc manner by the author. Details of the procedure will be published elsewhere.
\[s9\]Proofs I
==============
In the present section we prove Theorem \[t22a\] together with the accompanying Propositions \[p22a\], \[p22b\], and Theorem \[t32a\]. Thus, we are primarily dealing with a proof of the unique existence of a solution to the Problems $(f,\infty)$. Like in Theorem \[t22a\], we assume throughout the section that the function $f$ is meromorphic in the neighborhood of infinity.
\[s91\]Meromorphic Continuations Along Arcs
-------------------------------------------
The continuation of a function element along a given arc $\gamma$ is basic for any technique of meromorphic continuations. In the present subsection we introduce special sets of arcs and curves, and define on them a homotopy relation that is adapted to our special needs in later proofs. Toward the end of the subsection in Proposition \[p91a\], we prove a characterization of the domains in $\mathcal{D}(f,\infty)$ in terms of these newly introduced tools, i.e., a characterization of admissible domains for Problem $(f,\infty)$.
As a general notational convention, we denote the impression of a curve or an arc by the same symbol as use for the curve or the arc itself.
\[d91a\] By $\Gamma=\Gamma(f,\infty)$ we denote the set of all Jordan curves $\gamma$ with the following two properties:
- We have $\infty\in\gamma$.
- There exists a point $z\in\gamma\setminus\{\infty\}$, called separation point of $\gamma$, such that the curve $\gamma$ is broken down into the two closed partial arcs $\gamma^{-}$ and $\gamma^{+}$ connecting the two points $z$ and $\infty$. The function $f$ is assumed to possess meromorphic continuations along each of the two arcs, and these two arcs are not identical, i.e., we have $\gamma=\gamma^{+}-\gamma^{-}$ and $\gamma^{+}\cap\gamma^{-}=\{z,\infty\}$. (’Closed’ means here the arc contains its end points).
We assume that each Jordan curve $\gamma\in\Gamma$ has a parametrization of the form $$\gamma:\left[ -1,1\right] \longrightarrow\overline{\mathbb{C}}\label{f91a}$$ with $\gamma(-1)=\gamma(1)=\infty$ and $\gamma(0)=z$.
From (\[f91a\]), we have the parametrization $$\gamma^{+}:\left[ 1,0\right] \longrightarrow\overline{\mathbb{C}}\text{,\ \ \ }\gamma^{-}:\left[ -1,0\right] \longrightarrow\overline
{\mathbb{C}}\text{ }\label{f91b}$$ for the two partial arcs $\gamma^{-}$ and $\gamma^{+}$.
Whether a Jordan curves $\gamma$ with $\infty\in\gamma$ belongs to $\Gamma$ depends on the function $f$. A necessary and sufficient condition can be formulated as follows: We have $\gamma\in\Gamma$ if, and only if, the two meromorphic continuations of $f$ that start at $\infty$ and follow $\gamma$ in the two different directions cover the whole curve $\gamma$. We emphasize that the two continuations may hit non-polar singularities somewhere on the curve $\gamma$, but this is only allowed to happen after the separation point has already been passed.
Throughout the present section we assume that the separation point $z=z_{\gamma}\in\gamma\in\Gamma$ is chosen in an appropriate way, and we give details only if necessary.
In the next definition the set $\Gamma$ is divided into two subclasses.
\[d91b\] A Jordan curve $\gamma\in\Gamma=\Gamma(f,\infty)$ with partial arcs $\gamma^{-}$ and $\gamma^{+}$ belongs to the subclass $\Gamma_{0}=\Gamma_{0}(f,\infty)\subset\Gamma$ if the meromorphic continuations of the function $f$ along the two arcs $\gamma^{-}$ and $\gamma^{+}$ lead to the same function element at the separation point $z$ of $\gamma$. If, on the other hand, these continuations lead to two different function elements at $z$, then the curve $\gamma$ belongs to the subclass $\Gamma_{1}=\Gamma_{1}(f,\infty)\subset\Gamma$.
It is immediate that the two subsets $\Gamma_{0}$ and $\Gamma_{1}$ are disjoint, and we have $\Gamma=\Gamma_{0}\cup\Gamma_{1}$.
On the set $\Gamma$ we define a homotopy relation that fits our special needs. Two elements $\gamma_{0},\gamma_{1}\in\Gamma$ are considered to be homotopic if the two pairs $\{\gamma_{0}^{-},\gamma_{1}^{-}\}$ and $\{\gamma_{0}^{+},\gamma_{1}^{+}\}$ of partial arcs are homotopic in the usual sense, and if in addition property (ii) in Definition \[d91a\] is carried over from one to the other Jordan curve $\gamma_{0}$ and $\gamma_{1}$ in a continuous manner. More formally, we have the next definition.
\[d91c\] Two Jordan curves $\gamma_{0},\gamma_{1}\in\Gamma$ with partial arcs $\gamma_{j}^{\pm}$, $j=0,1$, and separation points $z_{j}$, $j=0,1$, are called homotopic (written $\sim$) if there exists a continuous function $h:\left[ -1,1\right] \times\left[ 0,1\right] \longrightarrow
\overline{\mathbb{C}}$ with the two following two properties:
- For $j=0,1$, we have $$\gamma_{j}(t)=h(t,j),\text{ \ \ \ \ \ }t\in\left[ -1,1\right] .\label{f31c}$$
- For each $s\in\left( 0,1\right) $ a Jordan curve $\gamma_{s}$ is defined by $$\gamma_{s}:=h(\cdot,s):\left[ -1,1\right] \longrightarrow\overline
{\mathbb{C}},\label{f31d}$$ and each $\gamma_{s}$ belongs to $\Gamma$ with separation point $\gamma
_{s}(0)$.
The equivalence class of $\gamma\in\Gamma$ with respect to the homotopy relation $\sim$ is denoted by $\left[ \gamma\right] $.
\[l91a\] The splitting of the set $\Gamma$ into the two subclasses $\Gamma_{0}$ and $\Gamma_{1}$ of Definition \[d91b\] is compatible with the homotopy relation of Definition \[d91c\].
The conclusion of the lemma is immediate.
The ring domain $R\subset\overline{\mathbb{C}}$ and the continuum $V\subset\mathbb{C}$ in the next lemma will be used at several places in the sequel. We say that $R$ is a ring domain in $\overline{\mathbb{C}}$ if $\overline{\mathbb{C}}\setminus R$ consists of two components.
\[l91b\] For any $\gamma_{0}\in\Gamma=\Gamma(f,\infty)$ there exists a ring domain $R\subset\overline{\mathbb{C}}$ with $\gamma_{0}\subset R $, for which the following five assertions hold true:
- The Jordan curve $\gamma_{0}$ separates the two components $A_{1} $ and $A_{2}$ of $\overline{\mathbb{C}}\setminus R$.
- Any Jordan curve $\gamma\subset R$ with $\infty\in\gamma$ that separates the two components $A_{1}$ and $A_{2}$ of $\overline{\mathbb{C}}\setminus R$ belongs to $\Gamma$.
- Any $\gamma\in\Gamma$ with $\gamma\subset R$ belong to $\gamma\in\lbrack\gamma_{0}]$, i.e., we have $\gamma\sim\gamma_{0}$ in the sense of Definition \[d91c\].
- If a Jordan curve $\gamma\subset R$ separates the two components $A_{1}$ and $A_{2}$ of $\overline{\mathbb{C}}\setminus R$, then any Jordan curve $\widetilde{\gamma}\subset R$ with $\infty\in\widetilde{\gamma}$, which is homotopic to $\gamma$ in $R$ (in the usual sense), belongs to $\Gamma$.
- If $\gamma_{0}\in\Gamma_{1}=\Gamma_{1}(f,\infty)$, then every admissible compact set $K\in\mathcal{K}(f,\infty)$ contains a continuum $V\subset\mathbb{C}$ that cross-sects $R$, i.e., we have $$V\cap A_{j}\neq\emptyset\text{ \ for\ \ }j=1,2\label{f91c1}$$ with $A_{1}$ and $A_{2}$ the two components of $\overline{\mathbb{C}}\setminus
R $. The set $\mathcal{K}(f,\infty)$ has been introduced in Definition \[d21a\].
Let $U^{-}$ and $U^{+}$ be two open and simply connected neighborhoods of the partial arcs $\gamma_{0}^{-}$ and $\gamma_{0}^{+}$ of $\gamma_{0}$, and let the function $f$ possess meromorphic continuations throughout $U^{-} $ and $U^{+}$. Let further $z_{0}=\gamma_{0}(0)$ denote the separation point of $\gamma_{0}$. By using $\varepsilon-$neighborhoods of $\gamma_{0}^{-}$ and $\gamma_{0}^{+}$, one can easily show that there exists a ring domain $R\subset\overline{\mathbb{C}}$ and an open disk $U_{0} $ with $z_{0}$ as its centre such that $$\begin{aligned}
& \overline{U}_{0}\subset U^{-}\cap U^{+}\text{, \ \ }\gamma_{0}\subset
R\subset U^{-}\cup U^{+},\label{f91d1}\\
& R\cap\partial U_{0}\text{ \ has exactly two components, and}\label{f91d2}\\
& R\setminus\overline{U}_{0}\text{ \ is a simply connected domain.}\label{f91d3}$$ Assertion (i) immediately follows from the construction of the ring domain $R
$ if the $\varepsilon-$neighborhoods of $\gamma_{0}^{-}$ and $\gamma_{0}^{+}$ are chosen sufficiently narrow.
Assertion (ii) follows from the following two facts: (a) any Jordan curve $\gamma$ in $R$ that separates the two components $A_{1}$ and $A_{2}$ is homotopic to $\gamma_{0}$ in the usual sense, and (b) $\gamma$ will intersect with $U_{0}$ because of (\[f91d2\]) and (\[f91d3\]). From the last assertion, it follows that we can choose a separation point $z$ for $\gamma$ anywhere in $\gamma\cap U_{0}$.
The assertions (iii) and (iv) are obvious completions of assertion (ii), and they follow rather immediately from the construction of $R$ in (\[f91d1\]), (\[f91d2\]), and (\[f91d3\]).
For the proof of assertion (v) we assume that $K$ is an arbitrary element of $\mathcal{K}(f,\infty)$, i.e., $\overline{\mathbb{C}}\setminus K$ is an admissible domain for Problem $(f,\infty)$ as introduced in Definition \[d21a\], and further we assume that $\gamma_{0}\in\Gamma_{1}$.
We considered the open set $R\setminus K$. From $\gamma_{0}\in\Gamma_{1}$ and assertion (iv) it follows that $$\gamma\cap K\neq\emptyset\label{f91d4}$$ for all Jordan curves $\gamma\subset R$ that separate $A_{1}$ from $A_{2}$. Indeed, if (\[f91e1\]) were false for some Jordan curve $\gamma$, then this curve could be modified near infinity in $R\setminus K$ into a Jordan curve $\widetilde{\gamma}\subset R\setminus K$ that is homotopic to $\gamma$ in $R$ and $\infty\in\widetilde{\gamma}$. From assertion (iv) we then know that $\widetilde{\gamma}\in\Gamma$. Since $R\setminus K\subset\overline{\mathbb{C}}\setminus K\in\mathcal{D}(f,\infty)$, we know from Definition \[d21a\] that the function $f$ has a single-valued meromorphic continuation along the whole curve $\widetilde{\gamma}$, which implies that $\widetilde{\gamma}\in
\Gamma_{0}$. On the other hand, from the assumption $\gamma_{0}\in\Gamma_{1}$ we deduce with assertion (iii) that also $\gamma_{1}\in\Gamma_{1}$. This last contradiction proves (\[f91d4\]).
Assertion (v) then follows from (\[f91d4\]) and the next Lemma \[l91c\]. The lemma is of independent interest for several applications at other places in the sequel.
\[l91c\] Let $R\subset\overline{\mathbb{C}}$ be a ring domain, $A_{1}$ and $A_{2}$ the two components of $\overline{\mathbb{C}}\setminus R$, and let $K\subset\mathbb{C}$ be a compact set. There exists a continuum $V\subset K$ with $$V\cap A_{j}\neq\emptyset\text{ \ for\ \ }j=1,2\label{f91e1}$$ if and only if $$\gamma\cap K\neq\emptyset\label{f91e2}$$ for every Jordan curve $\gamma\subset R$ that separates $A_{1}$ from $A_{2}$.
Let us first assume that there exists a Jordan curve $\gamma$ with the given properties for which (\[f91e2\]) is false, and let $O_{1}$ and $O_{2} $ be the interior and the exterior domain of $\gamma$. Then for any continuum $V\subset K$ satisfying (\[f91e1\]) we would have the contradiction that $V\subset O_{1}\cup O_{2}$ and $V\cap O_{j}\neq\emptyset$ for $j=1,2$.
Next, we assume that (\[f91e2\]) holds true, and set $B_{0}:=\overline
{R}\cap K$. Let $C_{j}\subset B_{0}$, $j\in I$, be the family of all components of $B_{0}$ that are disjoint from at least one of the two sets $A_{1}$ or $A_{2}$. The set $I$ is denumerable, we assume $I\subset\mathbb{N}$, and define $$B_{n}:=\overline{B_{0}\setminus\bigcup\nolimits_{j\in I,\text{ }j\leq n}C_{j}}\text{ \ \ for \ \ }n=1,2,\ldots\label{f91e3}$$ The assumption of (\[f91e2\]) implies that also $$\gamma\cap B_{n}\neq\emptyset\text{ \ \ for \ \ }n>0\label{f91e4}$$ and for every Jordan curve $\gamma\subset R$ that separates $A_{1}$ from $A_{2}$. Indeed, if there would exist an exceptional Jordan curve $\gamma$, then $\gamma$ could be modified into a Jordan curve $\widetilde{\gamma}\subset
R\setminus B_{0}$ that is homotopic to $\gamma$ in $R$, which then would contradict (\[f91e2\]).
From (\[f91e4\]) we deduce that $$B_{\infty}:=\bigcap\nolimits_{n\in\mathbb{N}}B_{n}\neq\emptyset.\label{f91e5}$$
The set $B_{\infty}$ contains only components that intersect simultaneously both sets $A_{1}$ and $A_{2}$, which proves the existence of a continuum $V\subset B_{\infty}\subset K$ satisfying (\[f91e1\]).
The following proposition has been the main reason and motivation for the introduction of the sets $\Gamma$, $\Gamma_{0}$, and $\Gamma_{1}$ of Jordan curves in the Definitions \[d91a\] and \[d91b\].
\[p91a\] Let $\Gamma=\Gamma(f,\infty)$, $\Gamma_{0}$, $\Gamma_{1}\subset\Gamma$ be the sets of Jordan curves introduced in the two Definitions \[d91a\] and \[d91b\], and let $\mathcal{D}(f,\infty)$ be the set of admissible domains for Problem $(f,\infty)$ introduced in Definition \[d21a\].
A domain $D\subset\overline{\mathbb{C}}$ with $\infty\in D$ belongs to $\mathcal{D}(f,\infty)$ if, and only if, the following two assertions hold true:
- The function $f$ has a meromorphic continuation along each closed Jordan arc $\gamma$ in $D$ that starts at $\infty$.
- For each Jordan curve $\gamma\in\Gamma_{1}$ we have $\gamma
\cap(\overline{\mathbb{C}}\setminus D)\neq\emptyset$.
Assertion (i) ensures that the function $f$ has a meromorphic continuation to each point of the domain $D$, and assertion (ii) guarantees that these continuations are single-valued. Hence, the two assertions (i) and (ii) imply $D\in\mathcal{D}(f,\infty)$.
The other direction of the proof follows also rather immediately from the two Definitions \[d21a\] and \[d91b\]. If $D\in\mathcal{D}(f,\infty)$, then clearly assertions (i) holds true; and if there would exist $\gamma_{1}\in\Gamma_{1}$ with $\gamma_{1}\subset D$, then this would contradict the assumption in Definitions \[d21a\] that the meromorphic continuation of the function $f$ in $D$ is single-valued.
\[s92\]The Existence of a Domain in $\mathcal{D}_{0}(f,\infty)$
---------------------------------------------------------------
In Definition \[d21b\], the set of all admissible domains with a complement of minimal capacity has been denoted by $\mathcal{D}_{0}(f,\infty
)$. In the present subsection we prove that $\mathcal{D}_{0}(f,\infty)$ is not empty.
\[p92a\]We have $\mathcal{D}_{0}(f,\infty)\neq\emptyset$.
The basic structure of the proof of Proposition \[p92a\] is simple and straightforward: A minimizing sequence of admissible compact sets $K_{n}\in\mathcal{K}(f,\infty)$, $n\in\mathbb{N}$, is chosen in such a way that in the limit the minimality condition (\[f21a\]) in Definition \[d21b\] is satisfied. The transition to the limit situation is done in the frame work of potential theory. It is shown that after some plausible corrections the resulting domain is admissible for Problem $(f,\infty)$. However, in the practical realization a number of technical hurdles have to be overcome; the whole proof is broken down in several consecutive steps, which are presented as lemmas.
In a first step, we deal with the very special situation that we have the value zero in the minimality (\[f21a\]) of Definition \[d21b\].
\[l92a\]If in (\[f21a\]) of Definition \[d21b\] we have $$\inf_{K\in\mathcal{K}(f,\infty)}\operatorname*{cap}(K)=0,\label{f92a}$$ then the subclass $\Gamma_{1}(f,\infty)$ of Jordan curves introduced in Definition \[d91b\] is empty.
For an indirect proof we assume $\Gamma_{1}=\Gamma_{1}(f,\infty)\neq\emptyset
$. Let $\gamma_{0}$ be an element of $\Gamma_{1}$, and let further $R\subset\overline{\mathbb{C}}$ be a ring domain with $\gamma_{0}\subset R$ as introduced in Lemma \[l91b\]. From assertion (iv) of Lemma \[l91b\] it follows that for every admissible compact set $K\in\mathcal{K}(f,\infty)$ there exists a continuum $V\subset K$ that intersects $R$, i.e., we have $$V\cap A_{j}\neq\emptyset\text{ \ for\ \ }j=1,2\label{f92a1}$$ and $A_{1}$, $A_{2}$ the two components of $\overline{\mathbb{C}}\setminus R$. From the lower estimate (\[f111b2\]) for the capacity given in Lemma \[l111a\], further below, we then conclude that $$\operatorname*{cap}(K)\geq\operatorname*{diam}(V)/4\geq\operatorname*{dist}(A_{1},A_{2})/4\text{.}\label{f92a2}$$ Since the right-hand side of (\[f92a2\]) is independent of $V$ and the choice of $K$, the estimate (\[f92a2\]) contradicts (\[f92a\]). Thus, we have proved that $\Gamma_{1}=\emptyset$.
In Lemma \[l92a\] a special case of Proposition \[p22b\] has been addressed, and we have the following corollary.
\[c92a\]If condition (\[f92a\]) is satisfied, then all meromorphic continuations of the function $f$ are single-valued, and consequently, the extremal domain $D_{0}=D_{0}(f,\infty)$ of Definition \[d21b\] is the Weierstrass domain $W_{f}\subset\overline{\mathbb{C}}$ for meromorphic continuation of the function $f$ starting at $\infty$.
It follows immediately from Definition \[d91b\] that $\Gamma_{1}=\emptyset$ is equivalent to the single-valuedness of all meromorphic continuations of $f$ in $\overline{\mathbb{C}}$, and consequently we have $D_{0}(f,\infty)=W_{f}$.
Thanks to Lemma \[l92a\], we can now assume without loss of generality for the remainder of the present subsection that $$\inf_{K\in\mathcal{K}(f,\infty)}\operatorname*{cap}(K)=:c_{0}>0.\label{f92b}$$
We select a sequence of admissible compact sets $K_{n}\in\mathcal{K}(f,\infty)$, $n\in\mathbb{N}$, such that $$\lim_{n\rightarrow\infty}\operatorname*{cap}(K_{n})=c_{0}.\label{f92c}$$
\[l92b\] There exists $r>0$ such that we can assume without loss of generality that the sequence $\{K_{n}\}$ in (\[f92c\]) satisfies $$K_{n}\subset\left\{ \text{\thinspace}|z|\leq r\text{\thinspace}\right\}
\text{ \ \ \ for all \ \ }n\in\mathbb{N}\text{.}\label{f92d}$$
Let $r>1$ be such that $f$ is meromorphic in $\left\{ \text{\thinspace
}|z|>r-1\text{\thinspace}\right\} $. For any admissible compact set $K\in\mathcal{K}(f,\infty)$, we denote by $\widetilde{K}$ the radial projection of $K$ onto the disk $\left\{ \text{\thinspace}|z|\leq
r\text{\thinspace}\right\} $ as defined in (\[f111d1\]) of Subsection \[s1101\], further below. It is not difficult to verify that because of $\overline{\mathbb{C}}\setminus K\in\mathcal{D}(f,\infty)$ we also have$\ \widetilde{D}:=\overline{\mathbb{C}}\setminus\widetilde{K}\in\mathcal{D}(f,\infty)$. One has only to check the conditions in Definition \[d21a\].
From Lemma \[l111c\] in Subsection \[s1101\], it then follows that $\operatorname*{cap}(\widetilde{K})\leq\operatorname*{cap}(K)$. Hence, any compact set $K_{n}$ in (\[f92c\]), which does not satisfy (\[f92d\]), can be replaced by $\widetilde{K}_{n}$, and because of (\[f92b\]), the limit (\[f92c\]) remains unchanged under such modifications.
In the sequel we assume that the inclusions (\[f92d\]) hold true for all compact sets $K_{n}\in\mathcal{K}(f,\infty)$, $n\in\mathbb{N}$, in (\[f92c\]).
Let $\omega_{n}$ be the equilibrium distribution of the compact set $K_{n}$, $n\in\mathbb{N}$, and let further $g_{n}=g_{D_{n}}(\cdot,\infty)$ be the Green function in the domain $D_{n}$ (for definitions of $\omega_{n}$ and $g_{n}$ see Section \[s110\], further below). As explained in Subsection \[s1104\], there always exists an infinite subsequence $N\subset\mathbb{N}$ such that the weak$^{\ast}$ limit $$\omega_{n}\overset{\ast}{\longrightarrow}\omega_{0}\text{ \ \ as
\ \ }n\rightarrow\infty\text{, }n\in N.\label{f92e1}$$ exists. Since inclusion (\[f92d\]) has been assumed to hold true for the sequence $\{K_{n}\}$, we have $$\operatorname*{supp}(\omega_{0})\subset\left\{ \text{\thinspace}|z|\leq
r\text{\thinspace}\right\} ,\label{f92e2}$$ and $\omega_{0}$ is a probability measures.
Using representation (\[f113b1\]) of Lemma \[l113b\] for the Green function $g_{n}$ we have $$g_{n}=-p(\omega_{n};\cdot)-\log\operatorname*{cap}\left( K_{n}\right)
\label{f92e21}$$ with $p(\omega_{n};\cdot)$ denotes the logarithmic potential of $\omega_{n}$, which has formerly been defined in Subsection \[s1102\], further below. From limit (\[f92e1\]) and the Lower Envelope Theorem \[t112a\] of potential theory (cf. Subsection \[s1102\], further below) we then conclude that $$\limsup_{N}g_{n}\,\leq\,g_{0}:=-p(\omega_{0};\cdot)-\log\left( c_{0}\right)
\label{f92e3}$$ with the constant $c_{0}$ introduced in (\[f92b\]). In (\[f92e3\]), equality holds quasi everywhere in $\mathbb{C}$ (for a definition of quasi everywhere see Definition \[d111a\], further below). It follows from (\[f92e1\]) and (\[f92e2\]) that outside of $\left\{ \text{\thinspace}|z|\leq r\text{\thinspace}\right\}
$ we have a proper limit in (\[f92e3\]) instead of the limes superior, and equality holds there instead of the inequality stated in (\[f92e3\]). In $\left\{ \text{\thinspace}|z|>r\text{\thinspace}\right\} $, the limit in (\[f92e3\]) holds locally uniformly.
\[d92a\]We define $$\widetilde{K}_{0}:=\overline{\left\{ \text{ }z\in\mathbb{C}\text{ }|\text{\ }g_{0}(z)=0\text{\ }\right\} },\label{f92e4}$$ and $\widetilde{D}_{0}:=\overline{\mathbb{C}}\setminus\widetilde{K}_{0}$.
The two sets $\widetilde{D}_{0}$ and $\widetilde{K}_{0}$ will become building blocks for extremal domains and minimal sets of Problem $(f,\infty) $, but several modifications and special considerations have to be made before the construction can be finished.
We note that the two sets $\widetilde{D}_{0}$ and $\widetilde{K}_{0}$, like the measure $\omega_{0}$ and the function $g_{0}$, depend on the subsequence $N\subset\mathbb{N}$ used in the limit (\[f92e1\]).
\[l92c\]We have $$\operatorname*{cap}(\widetilde{K}_{0})\,\leq\,c_{0}=\inf_{D\in\mathcal{D}(f,\infty)}\operatorname*{cap}(\overline{\mathbb{C}}\setminus D),\label{f92e6}$$ $\widetilde{K}_{0}\subset\left\{ \text{\thinspace}|z|\leq r\text{\thinspace
}\right\} ,$ and $\widetilde{D}_{0}$ is a domain.
The function $g_{0}$ introduced in (\[f92e3\]) is subharmonic in $\mathbb{C} $, which implies that the set $\widetilde{D}_{0}$ introduced in Definition \[d92a\] is a domain.
The inclusion $\widetilde{K}_{0}\subset\left\{ \text{\thinspace}|z|\leq
r\text{\thinspace}\right\} $ is an immediate consequence of (\[f92e2\]).
It remains to prove (\[f92e6\]). From (\[f92e3\]) it follows that $g_{0}\geq0$ everywhere in $\mathbb{C}$. Since the logarithmic potential of a finite measure is continuous quasi everywhere in $\mathbb{C}$ (cf. the introductory paragraphs of Subsection \[s1102\], further below), we conclude from (\[f92e4\]) that $$g_{0}(z)=0\text{ \ \ for quasi every \ }z\in\widetilde{K}_{0}.\label{f92f1}$$
We can assume without loss of generality that $\operatorname*{cap}(\widetilde{K}_{0})>0$ since otherwise (\[f92e6\]) is trivially true. From Lemma \[l112a\] in Subsection \[s1102\] we then know that the equilibrium distribution $\widetilde{\omega}_{0}$ on $\widetilde{K}_{0}$ is of finite energy. Hence, we can use the Principle of Domination from Theorem \[t112b\] in Subsection \[s1102\] for a comparison of the function $g_{0}=-p(\omega_{0};\cdot)-\log\left( c_{0}\right) $ from (\[f92e3\]) with the Green function $g_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}(\cdot,\infty)=-p(\widetilde{\omega}_{0};\cdot)-\log\operatorname*{cap}(\widetilde{K}_{0})$. In the last equation, we have used representation (\[f113b1\]) from Lemma \[l113b\] in Subsection \[s1103\]. With the Principle of Domination we deduce from (\[f92f1\]) that $$g_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}(z,\infty)\geq
\,g_{0}(z)\text{ \ \ for all \ \ }z\in\overline{\mathbb{C}}.\label{f92f2}$$ Comparing both side in (\[f92f2\]) near infinity yields the inequality $$\log\operatorname*{cap}(\widetilde{K}_{0})\leq\log c_{0},\label{f92f3}$$ which proves (\[f92e6\]).
It will turn out in (\[f92i2\]) and (\[f92i3\]), further below, that in (\[f92e6\]) we always have equality. Because of (\[f92f2\]), this means that we have $\omega_{0}=\widetilde{\omega}_{0}$, and therefore the measure $\omega_{0}$ has no mass points outside of $\widetilde{K}_{0}$.
In the next lemma, we see that $\overline{\mathbb{C}}\setminus\widetilde
{K}_{0} $ is indeed an important building block for an admissible domain with a minimal capacity, i.e., an element of $\mathcal{D}_{0}(f,\infty)$. The result should be seen in relation to Proposition \[p91a\].
\[l92d\]We have $\gamma\cap\widetilde{K}_{0}\neq\emptyset$ for every Jordan curve $\gamma\in\Gamma_{1}$.
Let $\gamma_{0}$ be an arbitrary element of $\Gamma_{1}$ with $\Gamma
_{1}=\Gamma_{1}(f,\infty)$ introduced in Definition \[d91b\], and let further $R\subset\overline{\mathbb{C}}$ be a ring domain with $\gamma
_{0}\subset R$ as introduced in Lemma \[l91b\]. Since $\gamma_{0}\in
\Gamma_{1}$, we know from assertion (iv) in Lemma \[l91b\] that for every $n\in\mathbb{N}$ there exists a continuum $V_{n}\subset K_{n}$ which cross-sects the ring domain $R$, i.e., we have $$V_{n}\cap A_{j}\neq\emptyset\text{ \ \ for \ }j=1,2\label{f92f4}$$ with $A_{1}$ and $A_{2}$ the two components of $\overline{\mathbb{C}}\setminus
R $.
Using Lemma \[l114b\] from Subsection \[s1104\] together with the assumptions (\[f92c\]), (\[f92d\]), and (\[f92e1\]), we conclude from (\[f92f4\]) that there exists a continuum $V\subset\left\{ \text{\thinspace
}|z|\leq r\text{\thinspace}\right\} $ that satisfy (\[f114f12\]) and (\[f114c13\]) in Lemma \[l114b\], i.e., we have $$V\cap A_{j}\neq\emptyset\text{ \ \ for \ }j=1,2,\text{ \ and}\label{f92f6}$$$$g_{0}(z)=0\text{ \ \ for \ \ }z\in V.\label{f92f5}$$ From (\[f92e4\]) and (\[f92f5\]), we then conclude that $V\subset
\widetilde{K}_{0}$. Because of (\[f92f6\]), we also know from Lemma \[l91c\] that $\gamma_{0}\cap V\neq\emptyset$, and consequently, we have shown that $\gamma_{0}\cap\widetilde{K}_{0}\neq\emptyset$.
In the proof of Lemma \[l92d\], Lemma \[l114b\] from Subsection \[s1104\], further below, has played a key role. The lemma will be essential at several other places in the sequel, and since its proof in Subsection \[s1104\] is based on Carathéodory’s Theorem about kernel convergence, we can say that the proof of the last lemma and also that of the other results is essentially based on Carathéodory’s fundamental Theorem. This theorem gives also importance to the use of the continua $V$ that have already appeared in the two Lemmas \[l91b\] and \[l91c\].
As a corollary of Lemma \[l92d\], we deduce that any meromorphic continuation of the function $f$ in the domain $\widetilde{D}_{0}$ is single-valued. Thus, one of the two conditions in Proposition \[p91a\] for a characterization of an admissible domain is satisfied by $\widetilde{D}_{0} $. What we still have not investigated is the question whether the function $f$ can be meromorphically continued to every point of $\widetilde{D}_{0}$, or how large the set of exceptional points in $\widetilde{D}_{0}$ can be if such a continuation is not possible. We start the investigation with the following definition.
\[d92b\]Let $\widetilde{E}_{0}\subset\widetilde{D}_{0}$ be the set of all points $z\in\widetilde{D}_{0}$ that satisfy the following two conditions:
- There exists a Jordan arc $\gamma$ in $\widetilde{D}_{0}$ with initial point $\infty$ and end point $z$, and the function $f$ has a meromorphic continuation along $\gamma\setminus\{z\}$.
- At the point $z$, the meromorphic continuation of $f$ along $\gamma$ has a non-polar singularity.
The next lemma is an immediate consequence of Lemma \[l92d\].
\[l92e\]Let $z_{0}\in\widetilde{D}_{0}$, and let $\gamma_{1},$ $\gamma
_{2}$ be two Jordan arcs that both satisfy condition (i) in Definition \[d92b\] with $z_{0}$ as end point. Then condition (ii) of Definition \[d92b\] is either simultaneously satisfied, or simultaneously not satisfied by the two arcs $\gamma_{1}$ and $\gamma_{2}$.
Let us assume that $\gamma_{1}$ and $\gamma_{2}$ are two Jordan arcs, which both satisfy condition (i) in Definition \[d92b\] with $z_{0}$ as end point, let $\gamma_{1}$ satisfy also condition (ii), but $\gamma_{2}$ not. Then after some modifications, if necessary, the composition $\gamma
:=\gamma_{1}-\gamma_{2}$ is a Jordan curve in $\widetilde{D}_{0}$. A separation point in the sense of Definition \[d91a\] can be chosen on $\gamma_{1}$ in the neighborhood of $z_{0}$, and we then have $\gamma\in
\Gamma$ with $\Gamma=\Gamma(f,\infty)$ introduced in Definition \[d91a\]. It follows from the assumptions made with respect to $\gamma_{1}$ and $\gamma
_{2}$ together with Definition \[d91a\] that $\gamma\in\Gamma_{1}$, but this would contradict Lemma \[l92d\].
The next lemma is a preparation of a proof of the result that $\operatorname*{cap}(\widetilde{E}_{0})=0$ for the set $\widetilde{E}_{0}$ that has been introduced in Definition \[d92b\]. Not only the formulation but also the proof this lemma is rather technical.
\[l92f\]Let $D_{1}$ be a simply connected and bounded domain with $\overline{D}_{1}\subset\widetilde{D}_{0}$. Then there exists $n_{1}\in\mathbb{N}$ such that $$\widetilde{E}_{0}\cap\overline{D}_{1}\subset K_{n}\text{ \ for all\ \ }n\geq
n_{1},\text{ }n\in N\label{f92g}$$ with $N\subset\mathbb{N}$ the subsequence used in the limit (\[f92e1\]).
With the assumptions made with respect to the domain $D_{1}$ it is rather immediate that there exists a ring domain $R\subset\widetilde{D}_{0}$ such that for one of the two components $A_{1}$ and $A_{2}$ of $\overline
{\mathbb{C}}\setminus R$, say $A_{1}$, we have $$\overline{D}_{1}\subset A_{1}\subset\widetilde{D}_{0}\text{ \ and \ \ }\infty\in R.\label{f92g1}$$ In addition to the domain $D_{1}$, we define the domain $D_{2}:=A_{1}\cup
R\subset\widetilde{D}_{0}$.
In a first step of the proof we show that there exists $n_{1}\in\mathbb{N}$ such that for each $n\geq n_{1}$, $n\in N$, there exists at least one Jordan curve $$\gamma\in\Gamma_{0}=\Gamma_{0}(f,\infty)\text{ \ \ with \ }\gamma\subset
R\setminus K_{n}.\label{f92g2}$$
The proof of (\[f92g2\]) will be given indirectly. For a negation of (\[f92g2\]) we consider the following two cases a and b:
Case a: There exists an infinite subsequence $N_{1}\subset N$ such that for each $n\in N_{1}$ and each Jordan curve $\gamma\subset R$ that separates $A_{1}$ from $A_{2}$ we have $$\gamma\cap K_{n}\neq\emptyset.\label{f92g3}$$
Case b: There exists an infinite subsequence $N_{2}\subset N$ such that for each $n\in N_{2}$ there exists at least one Jordan curve $$\gamma\in\Gamma_{1}=\Gamma_{1}(f,\infty)\text{ \ \ with \ }\gamma\subset
R\setminus K_{n}.\label{f92g4}$$ The sets $\Gamma_{0}$ and $\Gamma_{1}$ have been introduced in Definition \[d91b\].
If we have disproved Case a, then it follows from assertion (ii) of Lemma \[l91b\] that there exists $\gamma\in\Gamma$ with $\gamma\subset R\setminus
K_{n}$ for every $n\in N$ sufficiently large, and consequently, either (\[f92g2\]) or (\[f92g4\]) holds true for the particular Jordan curve $\gamma$. If also Case b is disproved, then it follows from assertion (iii) in Lemma \[l91b\] that for every $n\in N$ sufficiently large there exists a Jordan curve $\gamma$ which satisfies (\[f92g2\]), and we have accomplished the first step of the proof.
In order to disprove Case a, we observe that from Lemma \[l91c\] together with (\[f92g3\]), it follows that for each $n\in N_{1}$ there exists a continuum $V_{n}\subset K_{n}$ with $$V_{n}\cap A_{j}\neq\emptyset\text{ \ \ for \ \ }j=1,2\text{.}\label{f92g5}$$ As in the relations (\[f92f6\]) and (\[f92f5\]) in the proof of Lemma \[l92d\], we deduce from (\[f92g5\]) with the help of Lemma \[l114b\] in Subsection \[s1104\], further below, that there exists a continuum $V\subset\widetilde{K}_{0}$ with $V\cap R\neq\emptyset$, but the existence of $V$ contradicts the assumption $R\subset\widetilde{D}_{0}$. Hence, Case a is disproved.
In order to disprove Case b, we observe that from the existence of the Jordan curve $\gamma$ in (\[f92g4\]) and assertion (iv) in Lemma \[l91b\] it follows that for each $n\in N_{2}$ there exists a continuum $V_{n}\subset
K_{n}$ that satisfies (\[f92g5\]). With the same arguments as just used after (\[f92g5\]), we come to the same conclusion that there exists $V\subset\widetilde{K}_{0}$ with $V\cap R\neq\emptyset$, which again contradicts $R\subset\widetilde{D}_{0}$, and thus, also Case b is disproved.
As already said before, with a disproof of the two Cases a and b, we have shown that there exists $n_{1}\in\mathbb{N}$ such that for every $n\geq n_{1}$, $n\in N$ there exists a Jordan curve $\gamma$ which satisfies (\[f92g2\]).
In a second step, we prove indirectly the relations (\[f92g\]) for $n\geq
n_{1}$, $n\in N$. Let us assume that $(\widetilde{E}_{0}\cap\overline{D}_{1})\setminus K_{n_{0}}\neq\emptyset$ for a certain $n_{0}\geq n_{1}$, $n_{0}\in N$. Then there exists a point $$z_{0}\in(\widetilde{E}_{0}\cap\overline{D}_{1})\setminus K_{n_{0}}.\label{f92g6}$$
Let in accordance with Definition \[d92b\] $\gamma_{1}\subset\widetilde
{D}_{0}$ be a Jordan arc with initial point $\infty$ and end point $z_{0}$ such that the two conditions (i) and (ii) of Definition \[d92b\] are satisfied.
Since the function $f$ is single-valued and meromorphic in $D_{n_{0}}=\overline{\mathbb{C}}\setminus K_{n_{0}}\in\mathcal{D}(f,\infty)$, there exists a Jordan arc $\widetilde{\gamma}_{2}\subset D_{n_{0}}$ with initial point $\infty$ and end point $z_{0}$ and $f$ is meromorphic along $\widetilde{\gamma}_{2}$.
We know from (\[f92g2\]) that in $R\setminus K_{n_{0}}=R\cap D_{n_{0}}$ there exists a Jordan curve $\gamma_{0}$ with $\infty\in\gamma_{0}$, and along $\gamma_{0}$ the function $f$ has a single-valued meromorphic continuation. Hence, we can modify the arc $\widetilde{\gamma}_{2}$ in such a way that the modified Jordan arc $\gamma_{2}$ coincides with $\widetilde{\gamma}_{2}$ after its last contact with $\gamma_{0}$, but the whole Jordan arc $\gamma_{2}$ is contained in $D_{n_{0}}\cap D_{2}\subset D_{n_{0}}\cap\widetilde{D}_{0}$, and it connects $\infty$ with $z_{0}$. Clearly, the new Jordan arc $\gamma_{2}$ satisfies condition (i) of Definition \[d92b\], but it does not satisfy condition (ii). Hence, the two Jordan arcs $\gamma_{1}$ and $\gamma_{2}$ contradict Lemma \[l92e\]. This contradiction disproves the existence of $z_{0}$ in (\[f92g6\]), and completes the proof of lemma.
The proof of Lemma \[l92f\] has been very technical since in its background logic we were confronted with the possibility that in each admissible compact set $K_{n}$, $n\in N$, different non-polar singularities of the function $f$ may be ’active’ or ’inactive’. Illustrations for this phenomenon have been given in the examples of Section \[s6\]. In the situation of Lemma \[l92f\], it has turned out that with the selection of the subsequence $N\subset\mathbb{N}$ in (\[f92e1\]) all relevant choices in this respect have been fixed by the set $\widetilde{K}_{0}$.
\[l92g\]We have $$\operatorname*{cap}(\widetilde{E}_{0})=0\label{f92h1}$$ for the set $\widetilde{E}_{0}$ introduced in Definition \[d92b\].
For an indirect proof we assume that $$\operatorname*{cap}(\widetilde{E}_{0})>0.\label{f92h2}$$ Using an exhaustion of the domain $\widetilde{D}_{0}\cap\left\{
\text{\thinspace}|z|\leq r\text{\thinspace}\right\} $ by overlapping closed and simply connected domains, one can show because of (\[f92h2\]) that there exists a simply connected domain $D_{1}$ with $\overline{D}_{1}\subset\widetilde{D}_{0}\cap\left\{ \text{\thinspace}|z|\leq
r\text{\thinspace}\right\} $ such that $$\operatorname*{cap}(\widetilde{E}_{0}\cap\overline{D}_{1})>0.\label{f92h3}$$ The constant $r$ is the same as that in Lemma \[l92b\]. From (\[f92h3\]) and Lemma \[l92f\], we know that there exists $n_{1}\in\mathbb{N}$ such that (\[f92g1\]) holds true. Using Lemma \[l114a\] from Subsection \[s1104\], further below, we deduce from (\[f92g1\]) together with the assumptions (\[f92c\]) and (\[f92e1\]) that we have $$g_{0}(z)=0\text{ \ \ for quasi every \ }z\in\widetilde{E}_{0}\cap\overline
{D}_{1}\label{f92h4}$$ with $g_{0}$ the function introduced in (\[f92e3\]). From (\[f92h3\]), (\[f92h4\]), and (\[f92e4\]) in Definition \[d92a\], it then follows that $\widetilde{E}_{0}\cap\widetilde{K}_{0}\neq\emptyset$, but this contradicts Definition \[d92b\], and thus, the lemma is proved.
With the two Definitions \[d92a\] and \[d92b\], the Lemmas \[l92c\], \[l92d\], \[l92g\], and Proposition \[p91a\] we are prepared to prove Proposition \[p92a\] and close the present subsection.
**Proof of Proposition \[p92a\].** In a first step, we deal with the special case that (\[f92a\]) holds true. We then know from Lemma \[l92a\] and its Corollary \[c92a\] that the extremal domain $D_{0}=D_{0}(f,\infty)\in\mathcal{D}_{0}(f,\infty)$ of Definition \[d21b\] exists and is identical with the Weierstrass domain $W_{f}\subset\overline{\mathbb{C}}$ for meromorphic continuations of the function $f$ starting at $\infty$.
In the second step, we assume that the inequality in (\[f92b\]) is satisfied, and define $$D_{0}:=\widetilde{D}_{0}\setminus\widetilde{E}_{0},\label{f92i1}$$ and show that $D_{0}\in\mathcal{D}_{0}(f,\infty)$.
The set $\widetilde{E}_{0}$ is identical to its polynomial-convex hull $\widehat{E}_{0}$. Indeed, from Lemma \[l92d\] and from Lemma \[l111d\] in Subsection \[s1101\], further below, we deduce that $$\operatorname*{cap}(\widehat{E}_{0})=\operatorname*{cap}(\widetilde{E}_{0})=0.\label{f92i4}$$ From (\[f92i4\]) it follows that $\widehat{E}_{0}$ can have no inner points, and consequently, we have $\widehat{E}_{0}=\widetilde{E}_{0}$. This identity together with (\[f92i1\]) and Lemma \[l92c\] implies that $D_{0} $ is a domain.
From Lemma \[l92d\] we know that condition (ii) in Proposition \[p91a\] is satisfied. From (\[f92i1\]) and Definition \[d92b\] it follows that also condition (i) in Proposition \[p91a\] is satisfied. It therefore follows from Proposition \[p91a\] that $D_{0}$ is an admissible domain for Problem $(f,\infty)$, i.e., $D_{0}\in\mathcal{D}(f,\infty)$.
Since the capacity of a capacitable set does not change its value if a set of capacity zero is added or subtracted (cf. Lemma \[l111b\] in Subsection \[s1101\], further below), we deduce from the two Lemmas \[l92c\] and \[l92g\] that $$\operatorname*{cap}(\overline{\mathbb{C}}\setminus D_{0})=\operatorname*{cap}(\overline{\mathbb{C}}\setminus\widetilde{D}_{0})=\operatorname*{cap}(\widetilde{K}_{0})\leq c_{0}\text{ }\label{f92i2}$$ with the constant $c_{0}$ introduced in (\[f92b\]). From the minimality (\[f92b\]) and the fact that $D_{0}\in\mathcal{D}(f,\infty)$, we conclude that in (\[f92i2\]) a proper inequality is not possible. Hence, we have proved that $$\operatorname*{cap}(\overline{\mathbb{C}}\setminus D_{0})=\inf_{D\in
\mathcal{D}(f,\infty)}\operatorname*{cap}(\overline{\mathbb{C}}\setminus
D),\label{f92i3}$$ which implies that $D_{0}\in\mathcal{D}_{0}(f,\infty)$, and the proof of Proposition \[p92a\] completed. $\blacksquare$
\[s93\]Uniqueness up to a Set of Capacity Zero
----------------------------------------------
In the present subsection we prove that all admissible domains in $\mathcal{D}_{0}(f,\infty)$ differ only in a set of capacity zero. In Section \[s2\], this result has already been stated as the first part of Proposition \[p22a\], and there the sets $\mathcal{D}_{0}(f,\infty)$ and $\mathcal{K}_{0}(f,\infty)$ have also been introduced in Definition \[d21b\]. We formulate the result here as a proposition, which then will be proved at the end of the subsection after several auxiliary results have been formulated and proved.
\[p93a\]Sets in $\mathcal{K}_{0}(f,\infty)$ differ at most in a subset of capacity zero.
A key role in the proof of Proposition \[p93a\] is played by a number of special sets that are introduced in Definition \[d93a\] below. Especially the construction of the compact set $K_{\acute{}0}$ in (\[f93b6\]) can be seen as a type of convex combination, which will become more clear in Subsection \[s95\], further below.
We start with the formal set-up for an indirect proof of Proposition \[p93a\] and assume contrary to the assertion of the proposition that there exist at least two minimal compact sets $K_{1},K_{2}\in\mathcal{K}_{0}(f,\infty)$ that differ in a set of positive capacity, i.e., we assume $$\operatorname*{cap}\left( \left( K_{1}\setminus K_{2}\right) \cup\left(
K_{2}\setminus K_{1}\right) \right) >0\text{ \ \ for \ \ }K_{1},K_{2}\in\mathcal{K}_{0}(f,\infty)\text{.}\label{f93a1}$$ The corresponding admissible domains are defined as $$D_{j}:=\overline{\mathbb{C}}\setminus K_{j}\in\mathcal{D}_{0}(f,\infty),\text{
\ \ }j=1,2.\label{f93a2}$$ Since we have assumed $K_{1},K_{2}\in\mathcal{K}_{0}(f,\infty),$ we know that the minimality (\[f21a\]) in Definition \[d21b\] holds for both sets, i.e., we have $$\operatorname*{cap}(K_{1})=\operatorname*{cap}(K_{2})=\inf_{K\in
\mathcal{K}(f,\infty)}\operatorname*{cap}(K)=c_{0}\label{f93a3}$$ with $c_{0}$ the same constant as that introduced in (\[f92b\]). The two Green functions $g_{D_{j}}(\cdot,\infty)$ in the two domains $D_{j},$ $j=1,2,$ are denoted by $$g_{j}:=g_{D_{j}}(\cdot,\infty),\text{ \ \ }j=1,2.\label{f93a4}$$ From Lemma \[l113c\] in Subsection \[s1103\], further below, we know that assumption (\[f93a1\]) is equivalent to the assertion that the two Green functions $g_{1}$ and $g_{2}$ are not identical. From the harmonicity of $g_{1}-g_{2}$ in $D_{1}\cap D_{2}$, it then follows that $$g_{1}(z)\neq g_{2}(z)\text{ \ \ for almost all \ }z\in D_{1}\cap
D_{2},\label{f93a5}$$ and equality holds in $D_{1}\cap D_{2}$ on piece-wise analytic arcs. These arcs are part of the set $S_{0}$ that is formally defined in (\[f93b1\]) in the next definition. All sets introduced in Definition \[d93a\] will appear in subsequent lemmas that lead to the proof of Proposition \[p93a\] at the end of the present subsection.
\[d93a\]Under the assumptions (\[f93a1\]) and (\[f93a3\]) and with the same notations as introduced in (\[f93a2\]) and (\[f93a4\]), we define the sets $S_{0},K_{3},\widetilde{K}_{0},K_{10},K_{20},\allowbreak K_{0},\allowbreak D_{0}\subset\overline{\mathbb{C}}$ in the following way: $$\begin{aligned}
& S_{0}:=\overline{\left\{ \,z\in\overline{\mathbb{C}}\,\right. \left\vert
\,g_{1}(z)=g_{2}(z)\,\right\} },\medskip\label{f93b1}\\
& K_{3}:=\widehat{K_{1}\cup K_{2}},\medskip\label{f93b2}\\
& \widetilde{K}_{0}:=\widehat{S_{0}\cap K_{3}},\medskip\label{f93b3}\\
& K_{10}:=\left\{ \,z\in K_{1}\setminus\widetilde{K}_{0}\,\right. \left\vert
\,g_{1}(z)>g_{2}(z)\,\right\} ,\medskip\label{f93b4}\\
& K_{20}:=\left\{ \,z\in K_{2}\setminus\widetilde{K}_{0}\,\right. \left\vert
\,g_{2}(z)>g_{1}(z)\,\right\} ,\medskip\label{f93b5}\\
& K_{0}:=\widetilde{K}_{0}\cup K_{10}\cup K_{20},\medskip\label{f93b6}\\
& D_{0}:=\overline{\mathbb{C}}\setminus K_{0}.\label{f93b7}$$ The polynomial-convex hull of a bounded set $S\subset\mathbb{C}$ is denoted by $\widehat{S}$ (cf. Definition \[d111b\] in Subsection \[s1101\], further below).
For the proof of Proposition \[p93a\] the following strategy will be applied: First, it is proved that the set $D_{0}$ introduced in (\[f93b7\]) is a domain. Then it is shown that $D_{0}$ is an admissible domain, i.e., $D_{0}\in\mathcal{D}(f,\infty)$. After that in the final step, it is proved that the assumptions (\[f93a1\]) and (\[f93a3\]) imply that $\operatorname*{cap}(K_{0})<c_{0}$ for the compact set introduced in (\[f93b6\]). But such an estimate contradicts the minimality assumed in (\[f93a3\]). From a methodological point of view the last step is the most interesting and also the most challenging one.
We start with two lemmas in which topological and some potential-theoretic properties of sets from Definition \[d93a\] are investigated. The first lemma is of a more preparatory character.
\[l93a\]We set $$d:=g_{1}-g_{2},\label{f93c1}$$ and define the two sets $$\begin{aligned}
& B_{+}:=\{\,z\in\overline{\mathbb{C}}\setminus S_{0}\,|\,g_{1}(z)>g_{2}(z)\,\}=\{\,z\in\overline{\mathbb{C}}\setminus S_{0}\,|\,d(z)>0\,\},\medskip
\label{f93c2}\\
& B_{-}:=\{\,z\in\overline{\mathbb{C}}\setminus S_{0}\,|\,g_{1}(z)<g_{2}(z)\,\}=\{\,z\in\overline{\mathbb{C}}\setminus S_{0}\,|\,d(z)<0\,\}.\label{f93c3}$$ Both sets are open. The function $d$ is superharmonic in $B_{+}$ and subharmonic in $B_{-}$.
Let $C\subset\overline{\mathbb{C}}$ be an arbitrary component of $\overline{\mathbb{C}}\setminus S_{0}$. This component is broken down into the two sets $$C_{1}:=C\cap B_{+}\text{ \ and \ \ }C_{2}:=C\cap B_{-}\text{.}\label{f93c4}$$ Since the function $d$ is the difference of two Green functions, we know from Lemma \[l113a1\] in Subsection \[s1103\], further below, that $d$ is continuous outside of a measurable set $A\subset\overline{\mathbb{C}}$ with $\operatorname*{cap}(A)=0$. The definitions (\[f93c4\]) together with the continuity of $d$ then imply that $$\partial C_{j}\cap C\subset A\text{ \ \ for \ }j=1,2\text{.}\label{f93c5}$$ Indeed, if we assume that $z\in\partial C_{1}\cap C$ and $z\notin A$, then it follows from the continuity of $d$ in $C\setminus A$ that $d(z)=0$, and therefore $z\in S_{0}$, which contradicts the definition of the component $C$. For $j=2$ the same conclusion holds true.
Next, we show that we can have $$C_{j}\setminus A\neq\emptyset\label{f93c6}$$ at most for one of the two possibilities $j=1,2$. Indeed, it follows from $\operatorname*{cap}(A)=0$ and from Lemma \[l111e\] in Subsection \[s1101\], further below, that $C\setminus A$ is connected. If we assume that (\[f93c6\]) holds for both $j=1,2$, then it follows from the continuity of $d$ in $C\setminus A$ that there exists $z\in C\setminus A$ with $d(z)=0$. But this would imply that $z\in S_{0}$, which again contradicts the definition of the component $C$. We assume without loss of generality that $$C_{2}\subset A\text{ \ \ and \ }C_{1}\supset C\setminus A.\label{f93c7}$$ Let, as in Definition \[d113a2\] of Subsection \[s1103\], further below, $Rg(K_{1})$ denote the set $K_{1}$ minus all irregular points of $K_{1}$. From (\[f93c7\]) it follows that $$C\cap\overline{Rg(K_{1})}=\emptyset.\label{f93c8}$$ Indeed, if there exists $z\in C\cap\overline{Rg(K_{1})}$, then we know from part (iv) of Lemma \[l113a2\] in Subsection \[s1103\], further below, that $\operatorname*{cap}(C\cap K_{1})>0$, and further with Lemma \[l113a1\] in Subsection \[s1103\] we conclude that also $\operatorname*{cap}(C\cap
Rg(K_{1}))>0$.
Since $g_{1}(z)=0$ for all $z\in Rg(K_{1})$, it follows that $d(z)\leq0$ for $z\in Rg(K_{1})$ and therefore that $d(z)<0$ for $z\in C\cap Rg(K_{1})$. With (\[f93c7\]) this implies that $C\cap Rg(K_{1})\subset A$, and consequently, we have $\operatorname*{cap}(C\cap Rg(K_{1}))=0$. Since the last conclusion has led to a contradiction, (\[f93c8\]) is proved.
From (\[f93c8\]) and part (iii) of Lemma \[l113a2\] in Subsection \[s1103\], further below, we conclude that the function $d$ is superharmonic in the component $C$. Indeed, from the Lemma \[l113a2\] we know that $g_{1}
$ is harmonic in $C$, and on the other hand, $-g_{2}$ is superharmonic in $C$ (cf. Lemma \[l113e\] in Subsection \[s1103\]).
Since the minimum principle is valid for superharmonic functions, we conclude from (\[f93c7\]) and (\[f93c4\]) that $d(z)>0$ for all $z\in C$, and consequently, we have proved $$C\subset B_{+}.\label{f93c9}$$
The conclusion (\[f93c9\]) is conditional on the assumption made in (\[f93c5\]). The alternative choice in (\[f93c5\]) would have led to a reversed role for the two subsets $C_{+}$ and $C_{-}$, and we would have proved that $C\subset B_{-}$, and further that the function $d$ is subharmonic in $C$.
Putting both possibilities together, we have proved that each component of the open set $\overline{\mathbb{C}}\setminus S_{0}$ is completely contained in one of the two subsets $B_{+}$ or $B_{-}$. Hence, we have shown that these two sets are open. Further, it has been shown that the function $d$ is superharmonic (resp. subharmonic) in $B_{+}$ (resp. in $B_{-}$).
\[l93b\](i) The two sets $$B_{1}:=(K_{3}\setminus\widetilde{K}_{0})\cap B_{+}\text{ \ \ and \ \ }B_{2}:=(K_{3}\setminus\widetilde{K}_{0})\cap B_{-}\label{f93d1}$$ are disjoint, and they are open in $K_{3}$.
\(ii) We have $$\begin{aligned}
& B_{j}\setminus K_{j}=B_{j}\setminus K_{j0}\text{ \ \ for \ \ }j=1,2\text{,
\ and}\medskip\label{f93d2}\\
& \operatorname*{cap}(K_{10})=\operatorname*{cap}(K_{20})=0\text{.}\label{f93d3}$$
\(iii) Set $D_{3}:=\overline{\mathbb{C}}\setminus K_{3}$, then both sets $$D_{3}\cup(B_{j}\setminus K_{j})\text{, \ \ }j=1,2\text{,}\label{f93d4}$$ are domains.
\(vi) The set $D_{0}$ from (\[f93b7\]) is a domain, and we the decomposition $$D_{0}=D_{3}\cup(B_{1}\setminus K_{1})\cup(B_{2}\setminus K_{2}).\label{f93d5}$$
Assertion (i) is an immediate consequence of Lemma \[l93a\].
In order to prove assertion (ii), we observe that it follows from the definition of $K_{10}$ in (\[f93b4\]) together with (\[f93c2\]) and (\[f93d1\]) that $K_{10}=B_{1}\cap K_{1}$, which implies (\[f93d2\]) for $j=1$.
From the introduction of $K_{10}$ in (\[f93b4\]) and the definition of irregular points in the Definitions \[d113a\] and \[d113a2\] in Subsection \[s1103\], further below, it further follows that $K_{10}\subset Ir(K_{1})$, and the first part of (\[f93d3\]) therefore is a consequence of Lemma \[l113a1\] in Subsection \[s1103\]. Assertion (ii) follows analogously for $j=2$.
Since $\widetilde{K}_{0}$ is polynomial-convex by definition, each component of $K_{3}\setminus\widetilde{K}_{0}$ is a subset of a component of $\overline{\mathbb{C}}\setminus S_{0}$, which implies that $D_{3}\cup B_{j}$ is a domain for each $j=1,2$. Assertion (iii) then follows immediately from the second part of Lemma \[l111e\] in Subsection \[s1101\], further below, together with (\[f93d2\]) and (\[f93d3\]).
For a proof of assertion (iv), we observe that $$\begin{aligned}
D_{0} & =\overline{\mathbb{C}}\setminus(\widetilde{K}_{0}\cup K_{10}\cup
K_{20})=(\overline{\mathbb{C}}\setminus K_{3})\cup(K_{3}\setminus\widetilde
{K}_{0})\setminus(K_{10}\cup K_{20})\nonumber\\
& =D_{3}\cup(B_{1}\setminus K_{1})\cup(B_{2}\setminus K_{2}).\label{f93d6}$$ Indeed, the identities follow from the defining relations of the sets in Definition \[d93a\] together with (\[f93d2\]). From (\[f93d6\]) and assertion (iii) of the present lemma, it follows that $D_{0}$ is a domain.
The main result in Lemma \[l93b\] is the assertion that the set $D_{0}$ is a domain.
\[l93c\]The domain $D_{0}$ introduced in (\[f93b7\]) of Definition \[d93a\] is admissible for Problem $(f,\infty)$, i.e., we have $D_{0}\in\mathcal{D}(f,\infty)$ with $\mathcal{D}(f,\infty)$ introduced in Definition \[d21a\].
The basis of the proof is Proposition \[p91a\].
In a first step we prove that assertion (i) of Proposition \[p91a\] holds true. Let $f_{j}$, $j=1,2$, be the two single-valued meromorphic continuations of the function $f$ in Problem $(f,\infty)$ in the two admissible domains $D_{j}$, $j=1,2$, in (\[f93a2\]). In $D_{3}$ both functions are identical; but beyond this domain, the situation is different since we have assumed that the two domains $D_{1}$ and $D_{2}$ are not identical; the set $K_{3}$ has been defined in (\[f93b2\]). Using the two sets from (\[f93d1\]), we defined a new function $f_{0}$ as $$f_{0}(z):=\left\{
\begin{array}
[c]{lcc}f_{1}(z)=f_{2}(z) & \text{ \ for \ }\smallskip & z\in D_{3},\\
f_{1}(z) & \text{ \ for \ }\smallskip & z\in B_{1}\setminus K_{1},\\
f_{2}(z) & \text{ \ for \ } & z\in B_{2}\setminus K_{2}.
\end{array}
\right. \label{f93e1}$$ It follows from the assertions (iii) and (iv) of Lemma \[l93b\] that $f_{0}
$ is a meromorphic continuation of the function $f$ into $D_{0}$, which shows that assertion (i) of Proposition \[p91a\] holds true.
Assertion (ii) in Proposition \[p91a\] is a direct consequence of assertions (i) in Lemma \[l93b\]. Hence, it follows from Proposition \[p91a\] that $D_{0}\in\mathcal{D}(f,\infty)$.
We come now to the main part of the analysis in the present subsection, the proof of the inequality $\operatorname*{cap}(K_{0})<c_{0}$ for the compact set $K_{0}=\overline{\mathbb{C}}\setminus D_{0}$ introduced in (\[f93b6\]) in Definition \[d93a\]. In this proof two auxiliary functions $h_{0}$ and $h_{1}$ will play an important role; they are introduced in the next definition.
\[d93b\]With the sets introduced in Definition \[d93a\] and the notation $g_{1}$ and $g_{2}$ for the Green functions (\[f93a4\]), we define two functions $h_{0}$ and $h_{1}$ by $$h_{0}(z):=\left\{
\begin{array}
[c]{lcl}\frac{1}{2}(g_{1}(z)+g_{2}(z))\smallskip & \text{ \ for \ }\smallskip &
z\in\overline{\mathbb{C}}\setminus K_{3},\\
\frac{1}{2}\left| g_{1}(z)-g_{2}(z)\right| \smallskip & \text{ \ for
\ }\smallskip & z\in K_{3}\setminus\operatorname*{Int}(\widetilde{K}_{0}),\\
0 & \text{ \ for \ } & z\in\operatorname*{Int}(\widetilde{K}_{0}),
\end{array}
\right. \label{f93f1a}$$ $$h_{1}(z):=\left\{
\begin{array}
[c]{lcl}\frac{1}{2}\left| g_{1}(z)-g_{2}(z)\right| \smallskip & \text{ \ for
\ }\smallskip & z\in\overline{\mathbb{C}}\setminus K_{3},\\
\frac{1}{2}(g_{1}(z)+g_{2}(z))\smallskip & \text{ \ for \ }\smallskip & z\in
K_{3}\setminus\operatorname*{Int}(\widetilde{K}_{0}),\\
\frac{1}{2}(\widehat{g_{1}(z)+g_{2}(z)}) & \text{ \ for \ } & z\in
\operatorname*{Int}(\widetilde{K}_{0}).
\end{array}
\right. \label{f93f1b}$$ In (\[f93f1b\]), $\widehat{g_{1}+g_{2}}$ is the solution of the Dirichlet problem in each component $C$ of the interior $\operatorname*{Int}(\widetilde{K}_{0})$ of $\widetilde{K}_{0}$ with $(g_{1}+g_{2})|_{\partial
\widetilde{K}_{0}}$ as boundary function.$\smallskip$
In the next lemma a number of technical details are proved; they will be needed in the subsequent lemma.$\smallskip$
\[l93d\]If the assumptions (\[f93a1\]) and (\[f93a3\]) are satisfied, then with the notations from the Definitions \[d93a\], \[d93b\], and Lemma \[l93b\], the following assertions hold true:
\(i) There exists a signed measure $\sigma_{0}$ of finite energy with $$\operatorname*{supp}(\sigma_{0})\subset(K_{1}\cup K_{2})\setminus
\operatorname*{Int}(\widetilde{K}_{0})\label{f93f2a}$$ such that the function $h_{0}$ from (\[f93f1b\]) has the representation $$h_{0}=g_{0}(\cdot,\infty)+\int g_{0}(\cdot,v)d\sigma_{0}(v),\label{f93f2b}$$ where $g_{0}(\cdot,\cdot)$ is the Green function in the domain $D_{0}$. The measure $\sigma_{0}$ is carried by the set $$\Sigma_{0}:=(K_{1}\cup K_{2})\setminus\widetilde{K}_{0},\label{f93f2d}$$ and we have $$\sigma_{0}\neq0.\label{f93f2c}$$
\(ii) There exists a signed measure $\sigma_{1}$ of finite energy with $$\operatorname*{supp}(\sigma_{1})\subset S_{0}\cup K_{3}\label{f93f3a}$$ such that the function $h_{1}$ from (\[f93f1b\]) has the representation $$h_{1}=p(\sigma_{1};\cdot),\label{f93f3b}$$ where $p(\sigma_{1};\cdot)$ denotes the logarithmic potential of the measure $\sigma_{1}$ as introduced in (\[f112a1\]) in Subsection \[s1102\], further below. We have $$\sigma_{1}(\mathbb{C})=0.\label{f93f3c}$$
\(iii) With the notation (\[f93f2d\]), we have $$\begin{aligned}
& h_{0}(z)=h_{1}(z)\text{ \ \ for quasi every}\smallskip\text{ \ }z\in
\Sigma_{0},\label{f93f4a}\\
& h_{1}(z)=0\text{ \ \ \ \ \ \ \ for quasi every}\smallskip\text{ \ }z\in
S_{0}\setminus\operatorname*{Int}(K_{3}),\label{f93f4b}\\
& \sigma_{0}=-\sigma_{1}|_{K_{3}\setminus\widetilde{K}_{0}},\smallskip
\label{f93f4c}\\
& \sigma_{1}|_{\operatorname*{Int}(K_{3})}\leq0.\label{f93f4d}$$
In the first part of the proof we show that the two functions $h_{0}$ and $h_{1}$ introduced in (\[f93f1a\]) and (\[f93f1b\]) can be represented by potentials with measures of finite energy. This is done by an investigation of a sequence of auxiliary functions.
By $h_{2}$ we denote the function $$h_{2}:=\frac{1}{2}(g_{1}+g_{2})=r_{2}+p(\sigma_{2};\cdot),\label{f93f5a}$$ where the last equality is a consequence of (\[f93a4\]) together with representation (\[f113b1\]) for Green functions in Lemma \[l113b\] in Subsection \[s1103\], further below. It follows from (\[f93a3\]) and (\[f93a4\]) together with Lemma \[l113b\] that we have $$r_{2}=-\log(c_{0})\text{ \ \ and \ \ }\sigma_{2}=-\frac{1}{2}(\omega
_{1}+\omega_{2})\leq0\label{f93f5b}$$ with $\omega_{j}$ the equilibrium distribution on $K_{j}$, $j=1,2$.
From Lemma \[l112d\] together with Lemma \[l113b\] in the two Subsections \[s1102\] and \[s1103\], further below, we know that the function $\frac{1}{2}\left\vert g_{1}-g_{2}\right\vert $ can also be represented as a potential. We denote this function by $h_{3}$; and we have $$h_{3}:=\frac{1}{2}\left\vert g_{1}-g_{2}\right\vert =p(\sigma_{3};\cdot)\label{f93f6a}$$ with $\sigma_{3}$ a signed measure of finite energy and $$\operatorname*{supp}(\sigma_{3})\subset S_{0}\cup K_{1}\cup K_{2}.\label{f93f6b}$$ In (\[f93f6a\]), there is no constant because of (\[f93a3\]).
From Lemma \[l113a1\] in Subsection \[s1103\], further below, we know that the two Green functions $g_{1}$ and $g_{2}$, and consequently also the two functions $h_{2}$ and $h_{3}$ are continuous quasi everywhere in $\mathbb{C}$. Hence, it follows from (\[f93b1\]) in Definition \[d93a\] and (\[f93f6a\]) that $$h_{3}(z)=0\text{ \ \ for quasi every \ \ }z\in S_{0}.\label{f93f6c}$$ Because of (\[f93b2\]) in Definition \[d93a\], the two Green functions $g_{1}$ and $g_{2}$ are harmonic outside of $K_{3}$, and therefore we have equality for every $z\in S_{0}\setminus K_{3}$ in (\[f93f6c\]) without any exception.
Next, we use the balayage technique (cf. Definition \[d112a\] in Section \[s1102\], further below) for sweeping the masses of the two measures $\sigma_{2}$ and $\sigma_{3}$ out of the open set $\operatorname*{Int}(\widetilde{K}_{0})$. The two resulting balayage measures are denoted by $\sigma_{4}$ and $\sigma_{5}$, respectively. From part (i) of Definition \[d112a\] of the balayage applied to the measure $\sigma_{2}$, we get as consequence that the new function $h_{4}$ is of the form $$h_{4}:=r_{2}+p(\sigma_{4};\cdot)=\left\{
\begin{array}
[c]{lcl}\frac{1}{2}(\widehat{g_{1}+g_{2})} & \text{\ in }\smallskip &
\operatorname*{Int}(\widetilde{K}_{0}),\\
h_{2} & \text{\ quasi everywhere on }\smallskip & \partial\widetilde{K}_{0},\\
h_{2} & \ \text{in } & \overline{\mathbb{C}}\setminus\widetilde{K}_{0}.
\end{array}
\right. \label{f93f7a}$$ In (\[f93f7a\]) we have used (\[f93f5a\]). About the measure $\sigma_{4} $ we know that $$\begin{aligned}
& \operatorname*{supp}(\sigma_{4})\subset\operatorname*{supp}(\sigma
_{2})\setminus\operatorname*{Int}(\widetilde{K}_{0})\text{,}\smallskip
\label{f93f7e}\\
& \sigma_{4}|_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}=\sigma
_{2}|_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}=-\frac{1}{2}(\omega_{1}+\omega_{2})|_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}.\label{f93f7b}$$ On the other hand, from the balayage of the measure $\sigma_{3}$, it follows that the new function $h_{5}$ is of the form $$h_{5}:=p(\sigma_{5};\cdot)=\left\{
\begin{array}
[c]{lcl}0\text{ \ \ }\smallskip & \text{\ in }\smallskip & \operatorname*{Int}(\widetilde{K}_{0}),\\
0\text{ \ \ }\smallskip & \text{\ quasi everywhere on }\smallskip &
\partial\widetilde{K}_{0},\\
h_{3} & \ \text{in } & \overline{\mathbb{C}}\setminus\widetilde{K}_{0}.
\end{array}
\right. \label{f93f7c}$$ For the derivation of (\[f93f7c\]) identity (\[f93f6a\]) has been used. The balayage measure $\sigma_{5}$ satisfies $$\begin{aligned}
& \operatorname*{supp}(\sigma_{5})\subset\operatorname*{supp}(\sigma
_{3})\setminus\operatorname*{Int}(\widetilde{K}_{0}),\smallskip\label{f93f7f}\\
& \sigma_{5}|_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}=\sigma
_{3}|_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}\text{.}\label{f93f7d}$$
The function $\frac{1}{2}(\widehat{g_{1}+g_{2})}$ in the first line of (\[f93f7a\]) is the solution of the Dirichlet problem in each component of $\operatorname*{Int}(\widetilde{K}_{0})$ with boundary function $h_{2}|_{\partial\widetilde{K}_{0}}$ (cf. (\[f112c2\]) in Definition \[d112a\] in Section \[s1102\], further below). Analogously, in the first line of (\[f93f7c\]) the function $h_{5}=0$ is the solution of the Dirichlet problem in each component of $\operatorname*{Int}(\widetilde{K}_{0})$ with boundary function $h_{3}|_{\partial\widetilde{K}_{0}}$ since from (\[f93f6c\]) we know that $h_{3}(z)=0$ for quasi every $z\in\partial\widetilde{K}_{0}$.$\medskip$
The functions $h_{4}$ and $h_{5}$ and their associated measures $\sigma_{4}$ and $\sigma_{5}$ are the building blocks for the presentations by potentials for the two functions $h_{0}$ and $h_{1}$ introduced in Definition \[d93b\]. The proof of existence of such potentials is the main objectives in the present analysis. In the next step this aim will be achieved by using a method what pasting potentials together, which is described in Lemma \[l112e\] of Subsection \[s1102\], further below.$\smallskip$
Because of (\[f93b2\]) in Definition \[d93a\] we have $$\partial K_{3}\subset K_{1}\cup K_{2}\subset K_{3},\label{f93f8a}$$ and consequently $$h_{4}=h_{5}\text{ \ quasi everywhere on\ \ }\partial K_{3},\label{f93f8g}$$ which together with the representations (\[f93f7a\]) and (\[f93f7c\]) shows that the assumptions of Lemma \[l112e\] in Subsection \[s1102\] are satisfied. The domain $D$ in Lemma \[l112e\] is now $\operatorname*{Int}(K_{3})$.
From (\[f93f1a\]), (\[f93f1b\]), and the technique described in Lemma \[l112e\], we deduce that there exists two signed measures $\widetilde
{\sigma}_{0}$ and $\sigma_{1}$ of finite energy such that $$\begin{aligned}
& h_{0}(z)=r_{2}+p(\widetilde{\sigma}_{0};z)\text{ \ \ for quasi
every\smallskip\ \ }z\in\mathbb{C},\label{f93f8b}\\
& h_{1}(z)=p(\sigma_{1};z)\text{ \ \ \ \ \ \ \ for quasi every\smallskip
\ \ }z\in\mathbb{C}.\label{f93f8c}$$ Thus, in (\[f93f8b\]) and (\[f93f8c\]) we have representations by potentials for the piecewise defined functions $h_{0}$ and $h_{1}$, respectively.
Because of (\[f93f8a\]) and the properties of the functions $h_{4}$ and $h_{5}$ in the two sets $\operatorname*{Int}(K_{3})$ and $\overline
{\mathbb{C}}\setminus\operatorname*{Int}(K_{3})$, we have $$\begin{aligned}
& \operatorname*{supp}(\widetilde{\sigma}_{0})\subset(K_{1}\cup K_{2}\cup\widetilde{K}_{0})\setminus\operatorname*{Int}(\widetilde{K}_{0}),\smallskip\label{f93f8d}\\
& \operatorname*{supp}(\sigma_{1})\subset(S_{0}\setminus\operatorname*{Int}(K_{3}))\cup(K_{1}\cup K_{2}\cup\widetilde{K}_{0})\setminus\operatorname*{Int}(\widetilde{K}_{0}).\label{f93f8e}$$ From (\[f112h4\]) in Lemma \[l112e\] in Subsection \[s1102\] together with (\[f93f1b\]), (\[f93f5a\]), and (\[f93f7a\]), it further follows that $$\sigma_{1}|_{\operatorname*{Int}(K_{3})}=\sigma_{4}|_{\operatorname*{Int}(K_{3})}\leq0.\label{f93f8f}$$ The inequality in (\[f93f8f\]) is a consequence of (\[f93f7b\]).$\smallskip$
In order to prove a relationship between the two measures $\widetilde{\sigma
}_{0}$ and $\sigma_{1}$, we observe that from (\[f93f1a\]) and (\[f93f1b\]) in Definition \[d93b\] it follows that $$h_{0}(z)+h_{1}(z)=\max(g_{1}(z),g_{2}(z))\text{ \ \ for all \ \ }z\in
\overline{\mathbb{C}}\setminus\widetilde{K}_{0}.\label{f93f9a}$$ From (\[f93f9a\]) and Lemma \[l93a\] we deduce that $h_{0}+h_{1}$ is harmonic in $\mathbb{C}\setminus S_{0}$. Indeed, on the set $B_{+}$ introduced in Lemma \[l93a\], we have $h_{0}+h_{1}=g_{1}$. Since we know from Lemma \[l93a\] that the function $d=g_{1}-g_{2}$ is superharmonic in $B_{+}$, we deduce that $g_{1}$ is harmonic in $B_{+}$. On the set $B_{-}$ in Lemma \[l93a\], analogous considerations hold true.
From (\[f93f1a\]) and (\[f93f1b\]) in Definition \[d93b\] together with the two representations (\[f93f8b\]), (\[f93f8c\]), and the harmonicity of $h_{0}+h_{1}$ in $\mathbb{C}\setminus S_{0}$, it follows that $$\widetilde{\sigma}_{0}|_{\mathbb{C}\setminus S_{0}}=-\sigma_{1}|_{\mathbb{C}\setminus S_{0}},\label{f93f9b}$$ which is the relation between $\widetilde{\sigma}_{0}$ and $\sigma_{1}$ we were looking for.$\smallskip$
From (\[f93b6\]) in Definition \[d93a\] and (\[f93d3\]) in Lemma \[l93b\] we know that $\ K_{0}\ $ and $\widetilde{K}_{0})$ differ only in a set of capacity zero. Hence, from the defining property (\[f113a1\]) for Green functions, which has been stated at the beginning of Subsection \[s1103\], we then conclude that $$g_{0}(\cdot,v):=g_{\overline{\mathbb{C}}\setminus K_{0}}(\cdot,v)\equiv
g_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}(\cdot,v)\text{ \ \ for all
\ }v\in D_{0}.\label{f93f10a}$$
In the next step, we investigate the relation between Green function $g_{0}=g_{0}(\cdot,\infty)$ and the function $h_{0}$. From (\[f93f1a\]) in Definition \[d93b\] together with (\[f93f7a\]), (\[f93f6c\]), and (\[f93b3\]), we conclude that $$h_{0}(z)=0\text{ \ \ for quasi every \ \ }z\in\widetilde{K}_{0}.\label{f93f10b}$$
For the function $h_{0}$ we have representation (\[f93f8b\]). We will now show that if we sweep the measure $\widetilde{\sigma}_{0}$ out of the domain $\overline{\mathbb{C}}\setminus\widetilde{K}_{0}$ by balayage, we arrive at the Green function $g_{0}(\cdot,v)$. Let $\widehat{\sigma}_{0}$ be the balayage measure on $\partial\widetilde{K}_{0}$ resulting from sweeping $\widetilde{\sigma}_{0}$ out of $\overline{\mathbb{C}}\setminus\widetilde
{K}_{0}$, then it follows from Definition \[d112a\], part (ii), in Subsection \[s1102\] together with formula (\[f113e2\]) in Lemma \[l113e\] in Subsection \[s1103\] that $$\begin{aligned}
& r_{2}+p(\widehat{\sigma}_{0};z)-\int_{\overline{\mathbb{C}}\setminus
\widetilde{K}_{0}}g_{0}(v,\infty)d\widetilde{\sigma}_{0}(v)\nonumber\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=r_{2}+p(\widetilde{\sigma
}_{0};z)-\int_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}g_{0}(z,v)d\widetilde{\sigma}_{0}(v)\label{f93f10c}\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=h_{0}(z)-\int
_{\overline{\mathbb{C}}\setminus\widetilde{K}_{0}}g_{0}(z,v)d\widetilde
{\sigma}_{0}(v)=0\nonumber\end{aligned}$$ for quasi every $z\in\widetilde{K}_{0}$. In (\[f93f10c\]) the last equality follows from (\[f93f10b\] and the fact that $g_{0}(\cdot,v)=0$ quasi everywhere on $\widetilde{K}_{0}$ for all $v\in\overline{\mathbb{C}}\setminus\widetilde{K}_{0}$.
From (\[f93f10c\]) we deduce that $$g_{0}(\cdot,\infty)=h_{0}-\int_{\overline{\mathbb{C}}\setminus\widetilde
{K}_{0}}g_{0}(\cdot,v)d\widetilde{\sigma}_{0}(v).\label{f93f10d}$$ Indeed, since $\operatorname*{supp}(\widehat{\sigma}_{0})\subset\widetilde
{K}_{0} $, the right-hand side of (\[f93f10d\]) is harmonic in $\mathbb{C}\setminus\widetilde{K}_{0}$, has an appropriate behavior at infinity, and is equal to zero quasi everywhere on $\widetilde{K}_{0}$. Hence, identity (\[f93f10d\]) holds true since the right-hand side of (\[f93f10d\]) satisfies the defining property (\[f113a1\]) in Subsection \[s1103\] for the Green function $g_{0}(\cdot,\infty)=g_{D_{0}}(\cdot
,\infty)$.$\smallskip$
After this somewhat lengthy preparations we are ready to verify the individual statements of the lemma. We define $$\sigma_{0}:=\widetilde{\sigma}_{0}|_{K_{3}\setminus\widetilde{K}_{0}}=\widetilde{\sigma}_{0}|_{\mathbb{C}\setminus S_{0}}=-\sigma_{1}|_{\mathbb{C}\setminus S_{0}}.\label{f93f10e}$$ The second equality in (\[f93f10e\]) is a consequence of (\[f93f8d\]), and the last one is identical with (\[f93f9b\]).$\smallskip$
Representation (\[f93f2b\]) in the lemma follows directly from (\[f93f10d\]) and the introduction of the measure $\sigma_{0}$ in (\[f93f10e\]). That the set $\Sigma_{0}$ in (\[f93f2d\]) is a carrier of the measure $\sigma_{0}$ is a consequence of (\[f93f10e\]) and (\[f93f8d\]).$\smallskip$
Assertion (ii) is proved by (\[f93f8c\]) and (\[f93f8e\]).$\smallskip$
Identity (\[f93f4a\]) follows directly from (\[f93f1a\]) and (\[f93f1b\]) in Definition \[d93b\] and the fact that for the Green functions we have $g_{j}=0$ quasi everywhere on $K_{j}$, $j=1,2$, (cf. the defining property (\[f113a1\]) in Subsection \[s1103\], further below).$\smallskip$
Identity (\[f93f4b\]) is a consequence of (\[f93f6c\]) and the fact that $h_{1}=h_{3}=\frac{1}{2}\left| g_{1}-g_{2}\right| $ quasi everywhere on $\overline{\mathbb{C}}\setminus\operatorname*{Int}(K_{3})$.$\smallskip$
Identity (\[f93f4c\]) follows from (\[f93f10e\]) and (\[f93f8d\]), and at last identity (\[f93f4d\]) is practically identical with (\[f93f8f\]).$\smallskip$
Thus, only inequality (\[f93f2c\]) remains to be verified, and this will be done indirectly. Let us assume that $\sigma_{0}=0$. From (\[f93f10e\]), we then know that $\sigma_{1}|_{\mathbb{C}\setminus S_{0}}=0$. It then is a consequence of (\[f93f8f\]) that the potential $h_{1}=p(\sigma_{1};\cdot) $ is subharmonic in the domain $(\mathbb{C}\setminus S_{0})\cup
\operatorname*{Int}(K_{3})$. From (\[f93f1b\]), (\[f93f6c\]), and (\[f93f7c\]) we know that $$p(\sigma_{1};z)=0\text{ \ \ \ for quasi every \ \ }z\in S_{0}\setminus
\operatorname*{Int}(K_{3}).\label{f93f11a}$$ Since $\sigma_{1}$ is of finite energy, it follows from (\[f93f11a\]) and the subharmonicity of $h_{1}=p(\sigma_{1};\cdot)$ in $(\mathbb{C}\setminus
S_{0})\cup\operatorname*{Int}(K_{3})$ that $h_{1}(z)\leq0$ for all $z\in\mathbb{C}$. But the function $h_{1}$ is non-negative by definition, and so we have shown that $h_{1}\equiv0$. But the last identity contradicts the assumption (\[f93a5\]), and therefore assertion (\[f93f2c\]) is proved.
With the proof of Lemma \[l93d\] all preparations are done for beginning the last step in the indirect proof of Proposition \[p93a\] which is the next lemma.
\[l93e\]Under the assumptions (\[f93a1\]) and (\[f93a3\]), we have $$\operatorname*{cap}(K_{0})<c_{0}\label{f93g}$$ with $K_{0}$ the compact set introduced in (\[f93b6\]) of Definition \[d93a\], and $c_{0}$ the constant introduced in (\[f93a3\]).
From (\[f93f1a\]) in Definition \[d93b\] and the assumption made in (\[f93a3\]) we know that $$h_{0}(z)=-\log(c_{0})+\text{O}(z^{-1})\text{ \ \ as \ \ \ }z\rightarrow
\infty.\label{f93g1a}$$ As in Lemma \[l93d\], we abbreviate the Green function $g_{D_{0}}(\cdot,\cdot)$ by $g_{0}(\cdot,\cdot)$, and the special case $g_{0}(\cdot,\infty)$ by $g_{0}$. From the representation of Green functions in Lemma \[l113b\] in Subsection \[s1103\], further below, we know that $$g_{0}(z)=-\log(\operatorname*{cap}(K_{0}))+\text{O}(z^{-1})\text{ \ \ as
\ \ \ }z\rightarrow\infty.\label{f93g1b}$$ Hence, we have $$\begin{aligned}
\log\frac{\operatorname*{cap}(K_{0})}{c_{0}} & =\left. \left(
h_{0}(z)-g_{0}(z)\right) \right\vert _{z=\infty}\nonumber\\
& =\int g_{0}(v,\infty)d\sigma_{0}(v),\label{f93g1c}$$ where the last equation follows from (\[f93f2b\]) in Lemma \[l93d\].
If we knew that $\sigma_{0}$ were a purely negative measure, then we could get the desired estimate (\[f93g\]) very easily from (\[f93g1c\]). However, we cannot exclude that the measure $\sigma_{0}$ contains a positive part. Therefore, we have to go a more complicated way for getting an estimation for the integral $$I_{0}:=\int g_{0}(v,\infty)d\sigma_{0}(v).\label{f93g1d}$$ The technical results in Lemma \[l93d\] will provides the basis for the analysis.
Using in (\[f93g1d\]) representation (\[f93f2b\]) from Lemma \[l93d\] leads us to the expression $$I_{0}=\int h_{0}d\sigma_{0}-\int\int g_{0}(v,w)d\sigma_{0}(v)d\sigma
_{0}(w)=:I_{1}-I_{2}.\label{f93g2a}$$
From the positive definiteness of the Green function as a kernel in an energy formula, which has been stated in Lemma \[l113d\] in Subsection \[s1103\], further below, it follows together with (\[f93f2c\]) in Lemma \[l93d\] that $$I_{2}>0.\label{f93g2b}$$
In (\[f93g2a\]), there only remains the integral $I_{1}=\int h_{0}d\sigma_{0}$ to be estimated. This will be done after some transformations. First, we make the following general remark: From Lemma \[l93d\] we know that the two measures $\sigma_{0}$ and $\sigma_{1}$ are both of finite energy. Because of Lemma \[l112a\] in Subsection \[s1102\], further below, we have $\sigma_{0}(S)=\sigma_{1}(S)=0$ for every measurable set $S\subset\mathbb{C}$ of capacity zero. Consequently, integrals with respect to the measure $\sigma_{0}$ or $\sigma_{1}$ are equal if their integrands coincide quasi everywhere on a carrier of $\sigma_{0}$ or $\sigma_{1}$, respectively.
As in (\[f93f2d\]) in Lemma \[l93d\], we denote by $\Sigma_{0}$ the set $(K_{1}\cup K_{2})\setminus\widetilde{K}_{0}$. Since $\Sigma_{0}$ is a carrier of $\sigma_{0}$, we have $$\begin{aligned}
I_{1} & =\int h_{0}d\sigma_{0}\,=\,-\int_{\Sigma_{0}}h_{0}d\sigma
_{1}\label{f93g3a}\\
& =-\int_{\Sigma_{0}}h_{1}d\sigma_{1}\label{f93g3b}\\
& =-\int h_{1}d\sigma_{1}+\int_{\widetilde{K}_{0}\cap\operatorname*{Int}(K_{3})}h_{1}d\sigma_{1}.\label{f93g3c}$$ Indeed, the second equality in (\[f93g3a\]) follows from (\[f93f2d\]) and (\[f93f4c\]) in Lemma \[l93d\], the equality in (\[f93g3b\]) is a consequence of (\[f93f4a\]) in Lemma \[l93d\], and the equality in (\[f93g3c\]) follows from (\[f93f4b\]) in Lemma \[l93d\] and the fact that $\Sigma_{0}\cup(\widetilde{K}_{0}\cap\operatorname*{Int}(K_{3}))$ is a carrier of $\sigma_{1}|_{\operatorname*{Int}(K_{3})}$.
From Lemma \[l112b\], part (ii), in Subsection \[s1102\], together with (\[f93f4c\]) and (\[f93f2c\]) from Lemma \[l93d\], we conclude that $$\int h_{1}d\sigma_{1}=\int\int\log\frac{1}{|v-w|}d\sigma_{1}(v)d\sigma
_{1}(w)>0,\label{f93g4a}$$ i.e., we have used the positive definiteness of the logarithmic kernel.
Since $h_{1}\geq0$ by definition, it follows from (\[f93f4d\]) in Lemma \[l93d\] that $$\int_{\widetilde{K}_{0}\cap\operatorname*{Int}(K_{3})}h_{1}d\sigma_{1}\leq0.\label{f93g4b}$$
Putting (\[f93g1c\]), (\[f93g2a\]), (\[f93g2b\]), (\[f93g3c\]), (\[f93g4a\]), and (\[f93g4b\]) together, we conclude that $$\log\frac{\operatorname*{cap}(K_{0})}{c_{0}}<0,\label{f93g5a}$$ which proves (\[f93g\]).
With the proof of Lemma \[l93e\], the preparations of the proof of Proposition \[p93a\] are completed. Despite of the complexity of some of the preparatory lemmas, the basic structure of the approach is straight forward. It starts with assumption (\[f93a2\]), i.e., the assumption that there exist two essentially different admissible domains $D_{1}$ and $D_{2}$ with complements $K_{1}$ and $K_{2}$ of minimal capacity. Based on this assumption, a new admissible domain $D_{0}$ with a complement $K_{0}$ has been constructed in Definition \[d93a\], and it has then been shown in the last lemma that $\operatorname*{cap}(K_{0})$ is smaller than possible.
**Proof of Proposition \[p93a\].** The indirect proof of the proposition has been prepared by assumption (\[f93a1\]). The introduction of the two sets $K_{0}$ and $D_{0}=\overline{\mathbb{C}}\setminus K_{0}$ in (\[f93b6\]) and (\[f93b7\]) of Definition \[d93a\] provide the basis for the falsification of assumption (\[f93a1\]).
Indeed, in Lemma \[l93c\], it has been shown that for the domain $D_{0}$ is admissible for Problem $(f,\infty)$, i.e., $D_{0}\in\mathcal{D}(f,\infty)$, and in Lemma \[l93e\], it then is proved that the newly constructed set $K_{0}$ satisfies the inequality $\operatorname*{cap}(K_{0})<\operatorname*{cap}(K)$ for all $K\in\mathcal{K}_{0}(f,\infty)$, which contradicts the minimality (\[f21a\]) in Definition \[d21b\]. Hence, assumption (\[f93a1\]) is falsified, and Proposition \[p93a\] is proved. $\blacksquare$
The construction of the two sets $K_{0}$ and $D_{0}$ in Definition \[d93a\] can be seen as a special case of a general type of set-theoretical convex combination, which will be elaborated further in Definition \[d95a\] in Subsection \[s95\], below.
\[s94\]The Unique Existence of an Extremal Domain
-------------------------------------------------
In the present subsection we prove Theorem \[t22a\] and the two Propositions \[p22a\] and \[p22b\], which are all three concerned with the unique existence of an extremal domain for Problem $(f,\infty)$. With the two Propositions \[p92a\] and \[p93a\] in the last two Subsections \[s92\] and \[s93\], the main work for these proofs has already been done, we have only to put the different pieces together. We start with a technical lemma.
\[l94a\]The two sets $$\begin{aligned}
K_{0} & :=\bigcap_{K\in\mathcal{K}_{0}(f,\infty)}K\text{ \ \ \ and}\label{f94a1}\\
D_{0} & :=\bigcup_{D\in\mathcal{D}_{0}(f,\infty)}D\ \label{f94a2}$$ are well defined, and we have $$K_{0}\in\mathcal{K}_{0}(f,\infty)\text{ \ and \ \ }D_{0}\in\mathcal{D}_{0}(f,\infty)\label{f94a3}$$ with the two sets $\mathcal{K}_{0}(f,\infty)$ and $\mathcal{D}_{0}(f,\infty
)$ introduced in Definition \[d21b\].
From Proposition \[p92a\] we know that $\mathcal{K}_{0}(f,\infty
)\neq\emptyset$, hence, the sets $K_{0}$ and $D_{0}$ in (\[f94a1\]) and (\[f94a2\]) are well defined, and $D_{0}$ is a domain with $\infty\in
D_{0}$.
In order to prove (\[f94a3\]), we have only to show that $$D_{0}\in\mathcal{D}(f,\infty),\label{f94b1}$$ since if we know that $K_{0}\in\mathcal{K}(f,\infty)$, then the minimality condition (\[f21a\]) in Definition \[d21b\] follows immediately from (\[f94a1\]) together with the monotonicity of the capacity (cf. Subsection \[s1101\]). Relation (\[f94b1\]) will be proved with the help of Proposition \[p91a\] of Subsection \[s91\]; for this purpose we have to show that the two assertions (i) and (ii) in Proposition \[p91a\] hold true for the domain $D_{0}$ and the compact set $K_{0}$, respectively.
We start with assertion (i) in Proposition \[p91a\]. For every $z\in D_{0}$ there exists an admissible domain $D_{1}\in\mathcal{D}_{0}(f,\infty
)\subset\mathcal{D}(f,\infty)$ with $z\in D_{1}$. Since assertion (i) holds true for $D_{1},$ it holds true also for the larger domain $D_{0}$.
Next, we prove that also assertion (ii) in Proposition \[p91a\] holds true for the set $K_{0}$. Let $\gamma_{0}$ be an arbitrary Jordan curve of $\Gamma_{1}=\Gamma_{1}(f,\infty)$. From assertion (ii) in Proposition \[p91a\] we know that $$\gamma_{0}\cap K\neq\emptyset\label{f94b2}$$ for all $K\in\mathcal{K}_{0}(f,\infty)$. In order to prove that (\[f94b2\]) holds true also for the set $K_{0}$, we show in a first step that (\[f94b2\]) holds true for the intersection $K_{12}:=K_{1}\cap K_{2}$ of any two sets $K_{1},K_{2}\in\mathcal{K}_{0}(f,\infty)$. Indeed, let us assume that $$\gamma_{0}\cap K_{12}=\emptyset.\label{f94b3}$$ Let further $R\subset\overline{\mathbb{C}}\setminus K_{12}$ be a ring domain as introduced in Lemma \[l91b\] with $\gamma_{0}\subset R$. From Proposition \[p92a\] we know that $$\operatorname*{cap}(K_{1}\setminus K_{2})=0.\label{f94b4}$$ Hence, the set $K_{1}\setminus K_{2}$ cannot intersect the whole ring $R$, and consequently there exists a Jordan curve $\gamma_{1}\in\Gamma$ with $$\gamma_{1}\subset R\setminus(K_{1}\setminus K_{2})=R\setminus K_{1}\label{f94b5}$$ that separates the two components $A_{1}$ and $A_{2}$ of $\overline
{\mathbb{C}}\setminus R$. The equality in (\[f94b5\]) is a consequence of $K_{1}=K_{12}\cup K_{1}\setminus K_{2}$ and $R\cap K_{12}=\emptyset$.
From Lemma \[l91b\], part (ii), we know that $\gamma_{1}\thicksim\gamma_{0}$. Hence, from the assumption $\gamma_{0}\in\Gamma_{1}$ we conclude also $\gamma_{1}\in\Gamma_{1}$. On the other hand, it follows from (\[f94b5\]) that $\gamma_{1}\cap K_{1}=\emptyset$, which then contradicts assertion (ii) in Proposition \[p91a\], and with this falsification of (\[f94b3\]) we have proved that (\[f94b2\]) holds true for $K_{12}$.
With the same argumentation as that applied to the intersection $K_{12}$ of two elements from $\mathcal{K}_{0}(f,\infty)$, one can prove that relation (\[f94b2\]) holds true also for an intersection of finitely many elements from $\mathcal{K}_{0}(f,\infty)$, i.e., we can prove that $$\gamma_{0}\cap(K_{1}\cap\ldots\cap K_{m})\neq\emptyset\label{f94b6}$$ for an arbitrary $m\in\mathbb{N}$ and arbitrarily chosen sets $K_{j}\in\mathcal{K}_{0}(f,\infty)$, $j=1,\ldots,m$.
Let us now assume that relation (\[f94b2\]) does not hold true for the set $K_{0}$ of (\[f94a1\]), i.e., we assume $$\gamma_{0}\cap\bigcap_{K\in\mathcal{K}_{0}(f,\infty)}K=\emptyset.\label{f94b7}$$ An infinite intersection of compact sets can be empty only if already a finite intersection is empty. However, such a possibility has been excluded in (\[f94b6\]). Hence, we have proved that (\[f94b2\]) holds true for the set $K_{0}$, and as a consequence, we have shown that assertion (ii) in Proposition \[p91a\] holds true for the set $K_{0}$.
Having verified the two assertions (i) and (ii) in Proposition \[p91a\] for $D_{0}$ and $K_{0}$, it follows from the proposition that the domain $D_{0}$ is admissible, i.e., (\[f94b1\]) is proved, and the proof of the lemma is completed.
We now come to the three proofs of the central results from Section \[s2\].
**Proof of Theorem \[t22a\].** As minimal set $K_{0}=K_{0}(f,\infty)$ and as extremal domain $D_{0}=D_{0}(f,\infty)$ we choose the two sets introduced in (\[f94a1\]) and (\[f94a2\]). It follows immediately from Lemma \[l94a\] that $K_{0}$ satisfies the three conditions (i), (ii), and (iii) in Definition \[d21b\]. Hence, the existence side of Theorem \[t22a\] is established.
Uniqueness then follows immediately from (\[f94a1\]) in Lemma \[l94a\]. $\blacksquare$
**Proof of Proposition \[p22a\].** The proof is a combination of Proposition \[p93a\] and Lemma \[l94a\]. The first half-sentence in Proposition \[p22a\] has been proved in Proposition \[p93a\], and the second one is identical with (\[f94a1\]) in Lemma \[l94a\]. $\blacksquare
$
**Proof of Proposition \[p22b\].** If the function $f$ has no branch points, then we have $\Gamma_{0}=\Gamma$ and $\Gamma_{1}=\emptyset$ in Definition \[d91b\]. Hence, assertion (i) in Proposition \[p91a\] is trivially true, and therefore $D_{0}=D_{0}(f,\infty)$ is the largest domain to which the function $f$ can be meromorphically extended. Such a domain can be denoted as the Weierstrass domain for meromorphically continuation of $f$ if it is well-defined.
The domain $D_{0}$ is identical with the Weierstrass domain $W_{f}$ for analytic continuation of the function $f$ plus all polar singularities of $f$ that can be reached from within $W_{f}$, and which can be added without destroying the property that the resulting set is a domain. This completed the proof of Proposition \[p22b\]. $\blacksquare$
\[s95\]A Convexity Property
---------------------------
With the proof of Proposition \[p93a\] the main task of Subsection \[s93\] had been done. However, in the present subsection, we will add an extension to Definition \[d93a\]. It has already been mentioned after the proof Proposition \[p93a\] at the end of Subsection \[s93\] that the construction of the set $K_{0}$ in Definition \[d93a\] can be seen as a special case of a whole family of set-theoretic convex-combinations of the two sets $K_{1}$ and $K_{2}$ in Definition \[d93a\]. The extended construction leads to a whole continuum of sets $K_{h}$ with $h\in\left[ 0,1\right] $, and for the capacity of these sets $K_{h}$ we get an interesting inequality that generalizes inequality (\[f93g\]) in Lemma \[l93e\]. These extended results are certainly of independent interest, but they are also needed in Subsection \[s101\], below. The main properties of the new sets $K_{h}$, $h\in\left[ 0,1\right] $, are proved in Theorem \[t95a\].
\[d95a\]For two arbitrarily chosen admissible domains $D_{0},D_{1}\in\mathcal{D}(f,\allowbreak\infty)$ with corresponding compact sets $K_{j}=\overline{\mathbb{C}}\setminus D_{j}\in\mathcal{K}(f,\infty)$, $j=0,1$, that satisfy $$\operatorname*{cap}(K_{j})>0\text{,\ \ \ \ }j=0,1,\label{f95a}$$ we define a family of domains $D_{h}\subset\overline{\mathbb{C}}$, $0\leq
h\leq1$, (which will turn out to be admissible domains) together with a family of corresponding compact sets $K_{h}=\overline{\mathbb{C}}\setminus D_{h} $, $0\leq h\leq1$, in a way that generalizes Definition \[d93a\]. For $0\leq
h\leq1$ we define: $$\begin{aligned}
& S_{h}:=\overline{\left\{ \,z\in\overline{\mathbb{C}}\,\right. \left|
\,(1-h)\,g_{0}(z)=h\,g_{1}(z)\,\right\} },\medskip\label{f95b1}\\
& K_{3}:=\widehat{K_{0}\cup K_{1}},\medskip\label{f95b2}\\
& \widetilde{K}_{h}:=\widehat{S_{h}\cap K_{3}},\medskip\label{f95b3}\\
& K_{0,h}:=\left\{ \,z\in K_{0}\,\right. \left| \,(1-h)\,g_{0}(z)>h\,g_{1}(z)\,\right\} ,\medskip\label{f95b4}\\
& K_{1,h}:=\left\{ \,z\in K_{1}\,\right. \left| \,(1-h)\,g_{0}(z)<h\,g_{1}(z)\,\right\} ,\medskip\label{f95b5}\\
& K_{h}:=\widetilde{K}_{h}\cup K_{0,h}\cup K_{1,h},\medskip\label{f95b6}\\
& D_{h}:=\overline{\mathbb{C}}\setminus K_{h}\label{f95b7}$$ with $g_{j}=g_{D_{j}}(\cdot,\infty)$ the Green function in the domain $D_{j}$, $j=0,1$.
It is immediate that Definition \[d95a\] is a generalization of Definition \[d93a\]. The role of the two input sets $K_{1}$and $K_{2}$ in Definition \[d93a\] is now played by the two sets $K_{0}$ and $K_{1}$, respectively. The two set $K_{0}$ and $D_{0}$ in (\[f93b6\]) and (\[f93b7\]) of Definition \[d93a\] now correspond to the two sets $K_{1/2}$ and $D_{1/2}$, respectively, in the new terminology of Definition \[d95a\].
Another generalization in Definition \[d95a\] concerns the assumptions made with respect to the two compact input sets $K_{0}$ and $K_{1}$. While in Definition \[d93a\] the input sets have been assumed to be of minimal capacity, this assumption has been dropped without replacement in the extended definition.
The change of notation with respect to the input sets $K_{1}$and $K_{2}$ in Definition \[d93a\] into the sets $K_{1}$and $K_{2}$ in Definition \[d95a\] was necessary, and has the advantage that in the family of the newly defined sets $K_{h}$, $h\in\left[ 0,1\right] $, the two special sets $K_{0}$ and $K_{1}$ coincide with the two input sets $K_{0}$ and $K_{1}$ in Definition \[d95a\], which can easily be verified.
In the next theorem we prove that the newly defined domains $D_{h}$, $h\in\left[ 0,1\right] $, are all admissible for Problem $(f,\infty)$, and most importantly, we prove that the functional $\log\operatorname*{cap}(K_{h})$ depends on the index $h\in\left[ 0,1\right] $ in a strictly convex manner.
\[t95a\](i) Under the assumptions of Definition \[d95a\] we have $$D_{h}\in\mathcal{D}(f,\infty)\text{ \ \ and \ \ }K_{h}\in\mathcal{K}(f,\infty)\text{ \ \ for all \ \ }h\in\left[ 0,1\right] \label{f95c1}$$ with $\mathcal{K}(f,\infty)$ and $\mathcal{D}(f,\infty)$ the sets introduced in Definition \[d21a\].
\(ii) If in addition to the assumptions of Definition \[d95a\] we assume that $$\operatorname*{cap}\left( \left( K_{1}\setminus K_{0}\right) \cup\left(
K_{0}\setminus K_{1}\right) \right) >0,\label{f95c4}$$ then we have $$\log\operatorname*{cap}(K_{h})<(1-h)\,\log\operatorname*{cap}(K_{0})+h\,\log\operatorname*{cap}(K_{1})\text{ \ \ for \ \ }0<h<1.\label{f95c2}$$ (iii) Under the assumptions of Definition \[d95a\] we have the following continuity property: For any $h_{0}\in\left[ 0,1\right] $ and for any open set $U\subset\overline{\mathbb{C}}$ with $K_{h_{0}}\subset U$, there exists a neighborhood $V_{0}\subset\mathbb{R}$ of $h_{0}$ such that $$K_{h}\subset U\text{ \ \ \ for all \ \ }h\in V_{0}\cap\left[ 0,1\right]
.\label{f95c3}$$
Assertion (iii) in Theorem \[t95a\] means that in the Hausdorff metric the compact sets $K_{h}$ depend continuously on the parameter $h\in\left[
0,1\right] $.
Definition \[d93a\] has been the backbone of the proof of the essential uniqueness of minimal sets in Proposition \[p93a\] in Subsection \[s93\]. The important Lemma \[l93e\] in Subsection \[s93\] can be seen as a special case of the convexity relation (\[f95c2\]). It turns out that the proof of Theorem \[t95a\] is based on almost the same argumentations and techniques as those applied in the proof of Proposition \[p93a\], therefore we will now very closely follow the different stages of argumentations used in Subsection \[s93\]. As a consequence, we can shorten the proof of Theorem \[t95a\] considerably.
As a general policy, we will reformulate the content of lemmas and definitions from Subsection \[s93\] in such a way that it satisfies the needs of our new situation, but we will use shortcuts and will not repeat all details. Often it is only necessary to replace the difference $g_{1}-g_{2}$ of the two Green functions $g_{1}$ and $g_{2}$ from Definition \[d93a\] by the convex combination $(1-h)\,g_{0}+h\,g_{1}$ of the two Green functions $g_{0}$ and $g_{1}$ from Definition \[d95a\]. This change is evidently suggested by (\[f95b1\]) in Definition \[d95a\].
Thus, for instance, the difference $d:=g_{1}-g_{2}$ in (\[f93c1\]) will now be replaced by $$d(z):=((1-h)\,g_{0}+h\,g)(z)\text{ \ \ for \ \ }h\in\left[ 0,1\right] \text{
\ and \ }z\in\overline{\mathbb{C}}\text{.}\label{f95d1}$$
By using the same replacement repeatedly, one can transform all elements of Definition \[d93a\] into those of Definition \[d95a\]. In the same way the auxiliary definitions in the two Lemmas \[l93a\] and \[l93b\] can be adapted to the new situation, and like in Lemma \[l93c\], one can prove that $$K_{h}\in\mathcal{K}(f,\infty)\text{ \ \ and \ \ }D_{h}\in\mathcal{D}(f,\infty)\text{ \ \ for all \ \ }h\in\left[ 0,1\right] \label{f95d2}$$ with $K_{h}$ and $D_{h}$, $h\in\left[ 0,1\right] $, the sets introduced in (\[f95b6\]) and (\[f95b7\]), respectively. The last conclusion proves assertion (i) of Theorem \[t95a\].$\smallskip$
Analogously to (\[f93f1a\]) and (\[f93f1b\]) in Definition \[d93b\], we now introduce the two auxiliary functions $h_{0}$ and $h_{1}$ by defining $$h_{0}(z):=\left\{
\begin{array}
[c]{lcl}((1-h)\,g_{0}+h\,g_{1})(z)\smallskip & \text{ \ for \ }\smallskip &
z\in\overline{\mathbb{C}}\setminus K_{3},\\
\left| (1-h)\,g_{0}-h\,g_{1}\right| (z)\smallskip & \text{ \ for
\ }\smallskip & z\in K_{3}\setminus\operatorname*{Int}(\widetilde{K}_{h}),\\
0 & \text{ \ for \ } & z\in\operatorname*{Int}(\widetilde{K}_{h}),
\end{array}
\right. \label{f95d3a}$$ $$h_{1}(z):=\left\{
\begin{array}
[c]{lcl}\left| (1-h)\,g_{0}-h\,g_{1}\right| (z)\smallskip & \text{ \ for
\ }\smallskip & z\in\overline{\mathbb{C}}\setminus K_{3},\\
((1-h)\,g_{0}+h\,g_{1})(z)\smallskip & \text{ \ for \ }\smallskip & z\in
K_{3}\setminus\operatorname*{Int}(\widetilde{K}_{h}),\\
(\widehat{(1-h)\,g_{0}+h\,g_{1}})(z) & \text{ \ for \ } & z\in
\operatorname*{Int}(\widetilde{K}_{h})
\end{array}
\right. \label{f95d3b}$$ for $h\in\left[ 0,1\right] $ with sets $K_{3}$ and $\widetilde{K}_{h}$ that have been defined in (\[f95b2\]) and (\[f95b3\]). In (\[f95d3b\]), the expression $\widehat{(1-h)\,g_{0}+h\,g_{1}}$ denotes the solution of the Dirichlet problem in each component $C$ of the interior $\operatorname*{Int}(\widetilde{K}_{h})$ of $\widetilde{K}_{h}$ with $((1-h)\,g_{0}+h\,g_{1}))|_{\partial\widetilde{K}_{0}}$ as boundary function. Both functions $h_{0}$ and $h_{1}$ depend on the parameter $h$.
Representations for the functions $h_{0}$ and $h_{1}$ can be proved in the same way as has been done in Lemma \[l93d\]. Like in (\[f93f2b\]), we have a representation for $h_{0}$ that now takes the form $$h_{0}=g_{h}(\cdot,\infty)+\int g_{h}(\cdot,v)d\sigma_{0}(v)\label{f95d4}$$ with $g_{h}(\cdot,\cdot)$ the Green function of the domain $D_{h}$, $h\in\left[ 0,1\right] $. For the measure $\sigma_{0}$ in (\[f95d4\]) we have $$\begin{aligned}
& \text{ \ \ \ }\operatorname*{supp}(\sigma_{0})\subset(K_{0}\cup
K_{1})\setminus\operatorname*{Int}(\widetilde{K}_{h}),\label{f95d5}\\
& \sigma_{0}(\overline{\mathbb{C}}\setminus\Sigma_{0})=0\text{ \ \ for
\ \ }\Sigma_{0}:=(K_{1}\cup K_{2})\setminus\widetilde{K}_{0},\label{f95d6}$$ and if $0<h<1$, then we have $$\sigma_{0}\neq0\label{f95d7}$$ as a consequence of assumption (\[f95c4\]). These conclusions can be proved in exactly the same way as the corresponding assertions have been proved in the proof of Lemma \[l93d\].
The assertions (ii) and (iii) of Lemma \[l93d\] hold true also in the new situation if one substitutes the sets $S_{0}$, $K_{1}$, $K_{2}$, $\widetilde{K}_{0}$ by the sets $S_{h}$, $K_{0}$, $K_{1}$, $\widetilde{K}_{h}$. The sets $S_{h}$ and $\widetilde{K}_{h}$ have been introduced in Definition \[d95a\]. The proof of Lemma \[l93d\] is quite long and involved, and the same is true in the new situation if all details are taken into consideration. Since everything can be done in practically the same way as before, we will skip all details here.
In the new situation of Definition \[d95a\] the convexity relation (\[f95c2\]) is the analog of the inequality (\[f93g\]) in Lemma \[l93e\], and its proof can be done in quite the same way as that of Lemma \[l93d\]. Like in (\[f93g1b\]) and (\[f93g1c\]), from (\[f95d4\]) and (\[f95d3a\]) together with representation (\[f113b1\]) for Green functions in Lemma \[l113b\] of Subsection \[s1103\], further below, it follows that we have $$\begin{aligned}
& \log\operatorname{cap}(K_{h})-\left[ (1-h)\,\log\operatorname{cap}(K_{0})+h\,\log\operatorname{cap}(K_{1})\right] \medskip=\nonumber\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=\left(
h_{0}(z)-g_{h}(z,\infty)\right) |_{z=\infty}\label{f95d8}\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=\int
g_{h}(v,\infty)d\sigma_{0}(v).\nonumber\end{aligned}$$ In the verification of (\[f95d8\]), representation (\[f113b1\]) has been applied to the Green functions $g_{0}$, $g_{1}$, and $g_{h}$. By $g_{h}=g_{h}(\cdot,\cdot)$ we denote the Green function of the domain $D_{h} $ for $h\in\left[ 0,1\right] $.
In the same way as has been done in the proof of Lemma \[l93d\], it is then shown that $$\int g_{h}(v,\infty)d\sigma_{0}(v)<0\label{f95d9}$$ if, and only if, $0<h<1$. Together with (\[f95d8\]), the last conclusion proves assertion (ii) of Theorem \[t95a\].
It remains to prove assertion (iii), which will be done indirectly. We assume that there exist $h_{0}\in\left[ 0,1\right] $ and an open set $U\subset
\mathbb{C}$ with $K_{h_{0}}\subset U$ such that there exist $h_{n}\in\left[
0,1\right] $, $n\in\mathbb{N}$, with $$h_{n}\rightarrow h_{0}\text{ \ \ as \ \ }n\rightarrow\infty\text{ \ \ and
\ \ }K_{h_{n}}\setminus U\neq\emptyset\text{ \ \ for all \ }n\in
\mathbb{N}\text{.}\label{f95e1}$$ Without loss of generality, we can then select $x_{n}\in K_{h_{n}}\setminus U
$, $n\in\mathbb{N}$, such that $$x_{n}\rightarrow x_{0}\text{ \ \ as \ \ }n\rightarrow\infty\text{.}\label{f95e2}$$ From (\[f95b3\]) - (\[f95b6\]) it follows that $$x_{0}\in K_{3}\setminus U.\label{f95e3}$$ We shall prove that assertion (\[f95e3\]) is contradictory, which then implies that assertion (iii) of the theorem holds true. For disproving assertion (\[f95e3\]) we distinguish three different cases.
**Case 1:** We assume that $x_{0}\notin K_{0}\cup K_{1}$. Then both Green functions $g_{0}$ and $g_{1}$ are harmonic and continuous in a neighborhood of $x_{0}$. As a consequence, it follows from $x_{n}\in K_{h_{n}}$ for $n\in\mathbb{N}$ that also $x_{0}\in K_{h_{0}}$, which then disproves assertion (\[f95e3\]) since $K_{h_{0}}\subset U$.
We will give some more details of these last conclusions. From (\[f95b2\]) - (\[f95b6\]) together with $x_{n}\in K_{h_{n}}$ and $x_{0}\notin K_{0}\cup
K_{1}$, we deduce that $x_{n}\in\widetilde{K}_{h_{n}}$ for $n\in\mathbb{N}$ sufficiently large. From (\[f95b1\]) and (\[f95b3\]) we then know that $$(1-h_{n})\,g_{0}(x_{n})=h_{n}\,g_{1}(x_{n})\text{ \ \ for \ }n\in
\mathbb{N}\text{\ \ sufficiently large.}\label{f95e4}$$ From the continuity of $g_{0}$ and $g_{1}$ together with the limits in (\[f95e1\]) and (\[f95e2\]) we deduce from (\[f95e4\]) that $$(1-h_{0})\,g_{0}(x_{0})=h_{0}\,g_{1}(x_{0}),\label{f95e5}$$ which implies that $x_{0}\in K_{h_{0}}$, as stated above.
**Case 2:** Let us now assume that $x_{0}\in K_{0}\cap K_{1}$. From (\[f95b1\]) - (\[f95b6\]) it follows that $K_{0}\cap K_{1}\subset K_{h}$ for all $h\in\left[ 0,1\right] $. Consequently, we have $x_{0}\in K_{h_{0}}$, and because of $K_{h_{0}}\subset U$ this disproves (\[f95e3\]).
**Case 3:** We assume that $x_{0}\in\left( K_{0}\cup K_{1}\right)
\setminus\left( K_{0}\cap K_{1}\right) $. Because of symmetry, we can assume without loss of generality that $$x_{0}\in K_{0}\setminus K_{1}.\label{f95e6}$$ From (\[f95e6\]) it follows that the Green function $g_{1}$ is positive and harmonic in a neighborhood of $x_{0}$. Further, we can assume without loss of generality that $$x_{n}\notin K_{1}\text{ \ \ \ for all \ \ \ }n\in\mathbb{N}\text{.}\label{f95e7}$$ Because of (\[f95b1\]) - (\[f95b6\]), we can conclude from (\[f95e7\]) and $x_{n}\in K_{h_{n}}$ that $$(1-h_{n})\,g_{0}(x_{n})\geq h_{n}\,g_{1}(x_{n}),\label{f95e8}$$ which implies that $$\liminf_{n\rightarrow\infty}d_{0}(x_{n})\geq0\label{f95e9}$$ for $d_{0}:=(1-h_{0})\,g_{0}-h_{0}\,g_{1}$. In the last conclusion we have used the fact that the difference $g_{0}-g_{1}$ is bounded in a neighborhood of $x_{0}$.
The function $d_{0}$ is subharmonic and non-constant in the neighborhood of $x_{0}$.
If $d_{0}(x_{0})>0$, then it follows from (\[f95e6\]) and (\[f95b4\]) that $x_{0}\in K_{0,h_{0}}\subset K_{h_{0}}$. If, on the other hand, $d_{0}(x_{0})=0$, then it follows from (\[f95b1\]) and (\[f95b3\]) that $x_{0}\in\widetilde{K}_{h_{0}}\subset K_{h_{0}}$. If $d_{0}(x_{0})<0$, then it follows from (\[f95e9\]) that in each neighborhood of $x_{0}$ there exists a point $\widetilde{x}_{0}$ with $d_{0}(\widetilde{x}_{0})=0$, which, because of (\[f95b1\]) and (\[f95b3\]), again implies that $x_{0}\in\widetilde
{K}_{h_{0}}\subset K_{h_{0}}$. Hence, we have proved that $x_{0}\in K_{h_{0}}$, which disproves assertion (\[f95e3\]) because of $K_{h_{0}}\subset U$.
Since (\[f95e3\]) has been disproved for all three cases, the proof of assertion (iii) of Theorem \[t95a\] is completed.
\[s10\]Proofs II
================
In the present section we prove all results that have been stated in the Sections \[s4\], \[s5\], and \[s7\]. In the first subsection we deal with the special case of an algebraic function $f$. The results proved there are of independent interest, and this is especially true for Theorem \[t101b\] towards the end of the subsection. But besides of that they are also an essential preparation for the proofs of the main results from Sections \[s4\] and \[s5\], which will be given in Subsection \[s102\]. Results from Section \[s7\] are proved in the last two subsections.
\[s101\]Algebraic Functions
---------------------------
The particularity of algebraic functions $f$ with respect to our investigation is the fact that they possess only finitely many branch points and no other types of non-polar singularities. As a consequence, the structure of the minimal set $K_{0}(f,\infty)$ is in many respects special and also much simpler to describe than this is the case in general. All functions $f$ in the Examples \[e61\] - \[e65\] of Section \[s6\] are algebraic, and these examples are illustrations of what we can expect on special results. Since there exist only finitely many branch points, we have a direct connection between Problem $(f,\infty)$ and a certain type of Problem \[p81c\], which has already been discussed in Subsection \[s81\]. Details of the connection will be a major topic in the present subsection; another one will be the role of rational quadratic differentials, which are in some sense typical for Problem $(f,\infty)$ with $f$ being an algebraic function.
### \[s1011\]Sets of Minimal Hyperbolic Capacity
In the present subsection we investigate a special case of Problem \[p81c\] from Subsection \[s81\].
\[d101a\]*Let* $A=\{a_{1},\ldots,\allowbreak a_{n}\}\subset
\mathbb{D}$* be a set* of $n\geq2$ distinct points*. The task to find a continuum* $K\subset\mathbb{D}$* with* the property that $$a_{j}\in K\text{ \ \ \ \ \textit{for} \ \ }j=1,\ldots,n,\label{f101a1}$$ *and that the condenser capacity* $\operatorname*{cap}\left(
K,\partial\mathbb{D}\right) $* is minimal among all continua* $K\subset\mathbb{D}$* that satisfy (\[f101a1\]) is called Problem* $(A,\mathbb{D})$*. Its solution is denoted by* $K_{0}=K_{0}(A,\mathbb{D})$*.*
For a definition of the condenser capacity $\operatorname*{cap}\left(
K,V\right) $ with arbitrary compact sets $K,V\subset\overline{\mathbb{C}}$ we refer to Chapter II.5 in [@SaTo] or to [@Bagby]. In the special with $K\subset\mathbb{D}$ and $V:=\partial\mathbb{D}$, $\operatorname*{cap}\left(
K,\partial\mathbb{D}\right) $ is also known as the hyperbolic capacity of $K
$ in $\mathbb{D}$. For more details see [@Tsuji59]. Because of this terminology, the solution $K_{0}(A,\mathbb{D})$ of Problem $(A,\mathbb{D})$ is also called set of minimal hyperbolic capacity, and Problem $(A,\mathbb{D})$ can be seen as the hyperbolic analogue of Chebotarev’s Problem, which has been discussed in Section \[s8\] as Problem \[p81a\].
Problem $(A,\mathbb{D})$ can also be seen as a special case of Problem \[p81c\] from Section \[s81\]. This connection is established in the next proposition.
\[p101a\]A continuum $K_{0}\subset\mathbb{D}$ is a solution of Problem $(A,\mathbb{D})$ if, and only if, the pair of sets $(K_{0}$, $K_{0}^{-1})$ is a solution of Problem \[p81c\] formulated with the two sets of points $A=\{a_{1},\ldots,\allowbreak a_{n}\}\subset\mathbb{D}$ and $B=\{b_{1},\ldots,\allowbreak b_{n}\}:=A^{-1}\subset\overline{\mathbb{C}}\setminus
\overline{\mathbb{D}}$, i.e., $b_{j}:=1/\overline{a}_{j}$ for $j=1,\ldots,n$. By $S^{-1}$ we denote the reflection of a set $S$ on the unit circle $\partial\mathbb{D}$.
Proposition \[p101a\] follows from Theorem 3.1 in [@Kuzmina82], and the relevant elements in its deduction are also assembled in Theorem \[t101a\], below.
The capacity $\operatorname*{cap}\left( K,\partial\mathbb{D}\right) $ depends only on the outer boundary of $K\subset\mathbb{D}$, and therefore we have $$\operatorname*{cap}\left( K,\partial\mathbb{D}\right) =\operatorname*{cap}(\widehat{K},\partial\mathbb{D)},\label{f101b1}$$ where $\widehat{K}$ denotes the polynomial-convex hull of $K$. If $K\subset\mathbb{D}$ is a continuum with $K=\widehat{K}$, then $\mathbb{D}\setminus K $ is a ring domain, and in this special case $\operatorname*{cap}\left( K,\partial\mathbb{D}\right) $ is the reciprocal of the modulus of this ring domain (cf. [@Bagby]). If $K$ is not reduced to a single point, then there exists $1<r<\infty$ and a bijective conformal map $$\varphi:\mathbb{D}\setminus K\longrightarrow\{1<|z|<r\}\label{f101b2}$$ with $\varphi(1)=1$. The modulus of $\mathbb{D}\setminus K$ is then defined as $\log(r),$ and consequently, we have $\operatorname*{cap}(K,\partial
\mathbb{D})=1/\log(r)$.
The function $$p(z):=\left\{
\begin{array}
[c]{lcc}0\smallskip & \text{ \ for \ } & z\in K\\
\log|\varphi(z)|\smallskip & \text{ \ for \ } & z\in\mathbb{D}\setminus K\\
\log(r)=1/\operatorname*{cap}(K,\partial\mathbb{D}) & \text{ \ for \ } &
z\in\overline{\mathbb{C}}\setminus\mathbb{D}.
\end{array}
\right. \label{f101b3}$$ is known as the equilibrium potential of the condenser $(K,\partial
\mathbb{D})$. It is harmonic in $\mathbb{D}\setminus K$, and continuous throughout $\overline{\mathbb{C}}$.
Problem $(A,\mathbb{D})$ has a unique solution $K_{0}=K_{0}(A,\mathbb{D})\subset\mathbb{D}$. The continuum $K_{0}$ can be described very nicely by critical trajectories of a quadratic differential. In the next theorem we assemble these results together with other properties of the solution $K_{0}(A,\mathbb{D})$, which will be important for our further investigations. All results of the theorem have been proved in Chapter 3 of [@Kuzmina82].
\[t101a\]([@Kuzmina82], Theorem 3.1) Let $A=\{a_{1},\ldots
,a_{n}\}\subset\mathbb{D}$ be a set of $n\geq2$ distinct points.
\(i) There exists a continuum $K_{0}=K_{0}(A,\mathbb{D})\subset\mathbb{D}$, which is the unique solution of Problem $(A,\mathbb{D})$ as introduced in Definition \[d101a\].
\(ii) There exist $n-2$ points $b_{1},\ldots,b_{n-2}\in\mathbb{D}$ such that the continuum $K_{0}$ is the union of the closed critical trajectories of the quadratic differential $$q(z)\,dz^{2}\text{ \ \ with \ \ }q(z):=\frac{(z-b_{1})\ldots(z-b_{m-2})(1-\overline{b}_{1}z)\ldots(1-\overline{b}_{m-2}z)}{(z-a_{1})\ldots
(z-a_{m})(1-\overline{a}_{1}z)\ldots(1-\overline{a}_{m}z)}\label{f101c1}$$ in $\mathbb{D}$. There exist only finitely many critical trajectories.
\(iii) The equilibrium potential $p_{0}$ from (\[f101b3\]) which is associated with the extremal condenser $(K_{0},\partial\mathbb{D})$ satisfies relation $$\left( \frac{\partial}{\partial z}p_{0}(z)\right) ^{2}=\frac{1}{4}\,q(z)\text{ \ \ \ \ for \ \ \ }z\in\overline{\mathbb{D}}\label{f101c2}$$ with $\partial/\partial z$ denoting complex differentiation. The potential $p_{0}$ can be extended to a harmonic function in $\overline{\mathbb{C}}\setminus(K_{0}\cup K_{0}^{-1})$ by a reflection on the unit circle $\partial\mathbb{D}$, and relation (\[f101c2\]) holds true throughout $\overline{\mathbb{C}}$ for the harmonic extension of $p_{0}$.
\(iv) We have $$p_{0}(z):=\left\{
\begin{array}
[c]{lcc}1/\operatorname*{cap}(K_{0},\partial\mathbb{D})\smallskip & \text{ \ for \ } &
z\in\overline{\mathbb{C}}\setminus\mathbb{D}\\
0 & \text{ \ for \ } & z\in K_{0}\end{array}
\right. \label{f101d1}$$ and $$\frac{1}{2\,\pi}\oint_{\partial\mathbb{D}}\frac{\partial}{\partial n}p_{0}(\zeta)ds_{\zeta}=-1\label{f101d2}$$ with $\partial/\partial n$ the inward showing normal derivative on $\partial\mathbb{D}$ and $ds$ the line element on $\partial\mathbb{D}$.
### \[s1012\]Problem $(f,\infty)$ for Algebraic Functions
In the present subsection we study Problem $(f,\infty)$ for an algebraic function $f$. We will shed light on the connection between this problem and certain aspects of Problem $(A,\mathbb{D})$ from Definition \[d101a\].
Let $f$ be an algebraic function and assume that this function is meromorphic at infinity. Algebraic functions have only finitely many singularities, and the only non-polar singularities are branch points. Hence, in Problem $(f,\infty)$, only a finite number of points is of critical relevance. By $E_{0}\subset K_{0}(f,\infty)$ we denote the (finite) set of branch points of the function $f$ that can be reached on the minimal set $K_{0}(f,\infty)$ by meromorphic continuation of $f$ from within the extremal domain $D_{0}(f,\infty)$. In the discussion of the examples in Section \[s6\], this type of branch points have been called the active branch points for the determination of the minimal set $K_{0}(f,\infty)$. The sets $D_{0}(f,\infty)$ and $K_{0}(f,\infty)$ have been introduced in Definition \[d21b\].
\[l101a\]Let $f$ be an algebraic function that is meromorphic at infinity. Then the minimal set $K_{0}(f,\infty)$ for Problem $(f,\infty)$ has only finitely many components, which we denoted by $K_{1},\ldots,K_{m}$, i.e., we have $$K_{0}(f,\infty)=K_{1}\cup\ldots\cup K_{m}.\label{f1012a1}$$ Each component $K_{j}$, $j=1,\ldots,m$, contains at least two branch points of $f$. The Green function $g_{D_{0}}(\cdot,\infty)$ in the extremal domain $D_{0}=D_{0}(f,\infty)$ has only finitely many critical points, and we have $$g_{D_{0}}(z,\infty)=0\text{ \ \ \ for all \ \ \ }z\in K_{0}(f,\infty
).\label{f1012a3}$$
It follows from the two conditions (ii) and (iii) in Definition \[d21b\] that each component of $K_{0}(f,\infty)$ has to contain at least one non-polar singularity of $f$, and the single-valuedness of $f$ in $D_{0}(f,\infty)$ then further implies that each component of $K_{0}(f,\infty)$ has to contain at least two branch points.
After this conclusion, all other assertions of the lemma follow directly from of the finiteness of the set $E_{0}$ and the fact that a continuum has no irregular points with respect to the Dirichlet problem (cf. Subsection \[s1103\], further below).
Based on Lemma \[l101a\], we divide the set $E_{0}$ into $m$ subsets $E_{j}:=E_{0}\cap K_{j}$, $j=1,\ldots,m$. I.e., we have $$E_{0}=E_{1}\cup\ldots\cup E_{m}\text{ with\ \ }E_{j}\subset K_{j}\text{ \ for
\ }j=1,\ldots,m\text{.}\label{f1012a2}$$ For $c>0$ we define the open set $$U_{c}:=\left\{ \,z\in\mathbb{C}\,|\,g_{D_{0}}(z,\infty)(z)<c\,\right\}
\label{f1012b1}$$ with the Green function $g_{D_{0}}(\cdot,\infty)$ in $D_{0}=D_{0}(f,\infty)$. Since $g_{D_{0}}(\cdot,\infty)$ has only finitely many critical points in $D_{0}$, the open set $U_{c}$ consists of exactly $m$ components for $c>0$ sufficiently small. The number $m$ is the same as that in (\[f1012a1\]).
\[l101b\]Let the same assumptions hold true as in Lemma \[l101a\]. Then a constant $c_{0}>0$ can be chosen in such a way that the open set $U_{0}:=U_{c_{0}}$ from (\[f1012b1\]) has the following properties:
- $\overline{U}_{0}$ contains no critical point of the Green function $g_{D_{0}}(\cdot,\infty).$
- $U_{0}$ consists of exactly $m$ components $U_{j}$, $j=1,\ldots
,m$, i.e., $$U_{0}=U_{1}\cup\ldots\cup U_{m},\label{f1012b2}$$ and we have $$K_{j}\subset U_{j},\text{ \ \ for \ \ }j=1,\ldots,m\label{f1012b3}$$ with $K_{j}$ introduced in (\[f1012a1\]).
- Each component $U_{j}$, $j=1,\ldots,m$, in (\[f1012b2\]) is simply connected, and $\partial U_{j}$ is an analytic Jordan curve.
All three assertions of the lemma are rather immediate. Because of (\[f1012a3\]), $U_{0}$ is an open neighborhood of $K_{0}(f,\infty)$, and we can shrink $U_{0}$ as close to $K_{0}(f,\infty)$ as we wish. The first assertions (i) follows from the fact that the Green function $g_{D_{0}}(\cdot,\infty)$ has only finitely many critical points in $D_{0}(f,\infty)$.
The two other assertions (ii) and (iii) follow then immediately from (\[f1012a3\]) for $c_{0}>0$ sufficiently small.
In the next proposition we establish the connections between Problem $(f,\infty)$ and problems of the type of Problem $(A,\mathbb{D})$. These connections are the main topic in the present subsection.
\[p101b\]Let $f$ be an algebraic function that is meromorphic at infinity, and let $K_{0}$ be the minimal set $K_{0}(f,\infty)$ of Definition \[d21b\] for Problem $(f,\infty)$. Let further the sets $K_{j}$, $E_{j}$, and $U_{j}$, $j=1,\ldots,m$, be defined as in (\[f1012a1\]), (\[f1012a2\]), and (\[f1012b2\]), respectively, and let $\varphi_{j}:U_{j}\longrightarrow
\mathbb{D}$ be Riemann mapping functions, $j=1,\ldots,m$. We set $$A_{j}:=\varphi_{j}(E_{j})\text{, \ }K_{0,j}:=\varphi_{j}(K_{j})\text{,
\ }\alpha_{j}:=\omega_{K_{0}}(K_{j})\text{, \ }j=1,\ldots,m\text{,}\label{f1012c1}$$ with $\omega_{K_{0}}$ denoting the equilibrium distribution on $K_{0}$ as introduced in Subsection \[s1102\], further below. The following two assertions hold true:
- For each $j=1,\ldots,m$, the set $K_{0,j}$ is the minimal set $K_{0}(A_{j},\mathbb{D})$ that solves Problem $(A_{j},\mathbb{D})$ from Definition \[d101a\], which has been analyzed in Theorem \[t101a\].
- For each $j=1,\ldots,m$, we have $$g_{D_{0}}(z,\infty)(z)=\alpha_{j}(p_{0,j}\circ\varphi_{j})(z)\text{ \ \ for
\ \ \ }z\in U_{j}\text{,}\label{f1012c2}$$ where $p_{0,j}$ is the equilibrium potential (\[f101b3\]) for the extremal condenser $(K_{0,j},\allowbreak\mathbb{D})$ of Problem $(A_{j},\mathbb{D}),$ and $g_{D_{0}}(\cdot,\infty)$ is the Green function in the extremal domain $D_{0}(f,\infty)$.
The practical significance of Proposition \[p101a\] is that it shows a possibility to transplant specific properties of the solutions $K_{0}(A_{j},\mathbb{D})$ of the Problems $(A_{j},\mathbb{D})$, $j=1,\ldots,m$, to the solution of Problem $(f,\infty)$.
We start with assertion (i), which will be proved indirectly. For this purpose, we assume that at least one of the sets $K_{0,j}$, $j=1,\ldots,m$, is not a minimal solution $K_{0}(A_{j},\mathbb{D})$ of Problem $(A_{j},\mathbb{D})$. Without loss of generality we can assume that $$K_{0,1}\neq\widetilde{K}_{0,1}:=K_{0}(A_{1},\mathbb{D}).\label{f1012d3}$$ We define $$\widetilde{K}_{1}:=\varphi_{1}^{-1}(\widetilde{K}_{0,1})\text{, \ \ }\widetilde{K}_{0}:=(K_{0}\setminus K_{1})\cup\widetilde{K}_{1}\text{,
\ \ }\widetilde{D}_{0}:=\overline{\mathbb{C}}\setminus\widetilde{K}_{0},\label{f1012d4}$$ and show that the domain $\widetilde{D}_{0}$ is admissible for Problem $(f,\infty)$, i.e., $$\widetilde{D}_{0}\in\mathcal{D}(f,\infty)\text{.}\label{f1012d5}$$ From (\[f1012d3\]) we then deduce hat $$\operatorname*{cap}(\widetilde{K}_{0})<\operatorname*{cap}\left(
K_{0}(f,\infty)\right) .\label{f1012d1}$$
If (\[f1012d3\]) and (\[f1012d5\]) are proved, then with (\[f1012d1\]) we have a contradiction since, because of (\[f1012d5\]), inequality (\[f1012d1\]) clearly contradicts the minimality (\[f21a\]) in Definition \[d21b\] of the set $K_{0}=K_{0}(f,\infty)$. The contradiction shows that assumption (\[f1012d3\]) is false, and therefore assertion (i) is proved.
We start with the proof of (\[f1012d5\]). Using Cauchy’s formula, one can rewrite the function $f$ as $$f=f_{1}+\ldots+f_{m}\label{f1012d2}$$ with each $f_{j}$, $j=1,\ldots,m$, being meromorphic and single-valued in the simply connected domain $\overline{\mathbb{C}}\setminus K_{j}$. Since the sets $U_{1},\ldots,U_{m}$ are disjoint, we have only to consider the function $f_{1}$ if we want to understand the changes in the global behavior of $f$ that are caused by the exchange of the sets $K_{1}$ and $\widetilde{K}_{1}$ that is defined by (\[f1012d4\]).
From (\[f1012a2\]), (\[f1012c1\]), (\[f1012d3\]), and (\[f1012d4\]), we know that both sets $K_{1}$ and $\widetilde{K}_{1}$ contain the same set $E_{1}$ of branch points of the function $f$ on $K_{1}$. Consequently, the function $f_{1}$ can be continued meromorphically throughout the whole domain $\overline{\mathbb{C}}\setminus\widetilde{K}_{1}$. Since the domain $\overline{\mathbb{C}}\setminus\widetilde{K}_{1}$ is simply connected, it follows from the Monodromy Theorem that the continuation of $f_{1}$ is single-valued in $\overline{\mathbb{C}}\setminus\widetilde{K}_{1}$, and consequently, the function $f$ has also a single-valued meromorphic continuation to the domain $\widetilde{D}_{0}$, which proves (\[f1012d5\]).
In order to prove (\[f1012d1\]), we observe first that from the uniqueness of the solution $\widetilde{K}_{0,1}:=K_{0}(A_{1},\mathbb{D})$ of Problem $(A_{1},\mathbb{D})$, which has been established in Theorem \[t101a\], it follows that $$\operatorname*{cap}(\widetilde{K}_{0,1},\partial\mathbb{D)}<\operatorname*{cap}\left( K_{0,1},\partial\mathbb{D}\right) \label{f1012e2}$$ with $K_{0,1}$ defined in (\[f1012c1\]). Notice that $A_{1}\subset K_{0,1}$. We shall now show that inequality (\[f1012e2\]) implies (\[f1012d1\]).
Indeed, let $p_{01}$ and $\widetilde{p}_{01}$ be the equilibrium potentials (\[f101b3\]) of the two condensers $(K_{0,1},\mathbb{D})$ and $(\widetilde
{K}_{0,1},\mathbb{D})$, respectively. From (\[f101b3\]) it follows that $$p_{01}(z)=\frac{1}{\operatorname*{cap}\left( K_{0,1},\partial\mathbb{D}\right) }\text{, \ \ }\widetilde{p}_{01}(z)=\frac{1}{\operatorname*{cap}(\widetilde{K}_{0,1},\partial\mathbb{D)}}\text{\ \ for \ }z\in\partial
\mathbb{D}.\label{f1012e1}$$ From the definition of the open set $U_{0}$ in Lemma \[l101b\] together with the properties of the Green function $g_{D_{0}}(\cdot,\infty)$, the definition of the mapping $\varphi_{1}:U_{1}\longrightarrow\mathbb{D}$, the set $K_{0,1}$, and the number $\alpha_{1}$, which has been introduced in (\[f1012c1\]), we then deduce that the function $$\overset{\vee}{p}_{01}:=\frac{1}{\alpha_{1}}g_{D_{0}}(\cdot,\infty
)\circ\varphi_{1}^{-1}\label{f1012e3}$$ has the following four properties: (i) The function $\overset{\vee}{p}_{01} $ is harmonic in $\mathbb{D}\setminus K_{0,1}$. (ii) We have $\overset{\vee}{p}_{01}(z)=0$ for all $z\in K_{0,1}$. (iii) We have $$\overset{\vee}{p}_{01}(z)=\frac{c_{0}}{\alpha_{1}}\text{ \ \ for all \ \ }z\in\partial\mathbb{D}\label{f1012e4}$$ with the constant $c_{0}$ introduced in Lemma \[l101b\]. (iv) We have $$\frac{1}{2\pi}\int_{\partial\mathbb{D}}\frac{\partial}{\partial n}\overset{\vee}{p}_{01}ds=-1\label{f1012e4a}$$ because of the definition of $\alpha_{1}$ in (\[f1012c1\]). The normal derivative $\partial/\partial n$ on $\partial\mathbb{D}$ in (\[f1012e4a\]) is assumed to be inwardly oriented.
From these four properties together with (\[f101b3\]), it follows that in $\mathbb{D}$ the function $\overset{\vee}{p}_{01}$ is identical with the equilibrium potential $p_{01}$ of the condenser $(K_{0,1},\mathbb{D})$. From the first equality in (\[f1012e1\]) together with (\[f1012e4\]), it then follows that $$\frac{1}{\operatorname*{cap}\left( K_{0,1},\partial\mathbb{D}\right) }=\frac{c_{0}}{\alpha_{1}}.\label{f1012e5}$$ Motivated by (\[f1012e1\]) and (\[f1012e5\]), we define $$\widetilde{\alpha}_{1}:=\frac{\operatorname*{cap}(\widetilde{K}_{0,1},\partial\mathbb{D)}}{\operatorname*{cap}\left( K_{0,1},\partial
\mathbb{D}\right) }\alpha_{1}<\alpha_{1},\label{f1012e6}$$ where the inequality is a consequence of (\[f1012e2\]).
Next, we study the function $$\overset{\vee}{g}_{0}(z):=\left\{
\begin{array}
[c]{ccc}\widetilde{\alpha}_{1}(\widetilde{p}_{01}\circ\varphi_{1})(z)\smallskip &
\text{ \ for \ } & z\in\overline{U}_{1}\\
g_{D_{0}}(z,\infty) & \text{ \ for \ } & z\in\overline{\mathbb{C}}\setminus\overline{U}_{1},
\end{array}
\right. \label{f1012e7}$$ which is basically a modification of the Green function $g_{D_{0}}(z\cdot,\infty)$ in the neighborhood $U_{1}$ of $K_{1}$. The function is continuous in $\mathbb{C}$ since both partial functions in (\[f1012e7\]) are equal to $c_{0}$ on $\partial U_{1}$. Indeed, it follows from (\[f1012e6\]), (\[f1012e5\]), and the second equation in (\[f1012e1\]) that $$g_{D_{0}}(z,\infty)=\,\overset{\vee}{g}_{0}(z)=c_{0}\text{ \ \ \ for all
\ \ }z\in\partial U_{1}.\label{f1012e8}$$ The function $\overset{\vee}{g}_{0}$ has the following four properties: (i) It is harmonic in $\overline{\mathbb{C}}\setminus(\widetilde{K}_{0}\cup\partial
U_{1})$ because of the definitions made in (\[f1012d4\]). (ii) It has smooth normal derivatives from both sides of $\partial U_{1}$. (iii) We have $\overset{\vee}{g}_{0}(z)=0$ for all $z\in\widetilde{K}_{0}$ because of the first line in (\[f101b3\]). (iv) Near infinity we have $$\overset{\vee}{g}_{0}(z)=\log|z|+\log\frac{1}{\operatorname*{cap}\left(
K_{0}\right) }+\text{o}(1)\text{ \ \ as \ \ }z\rightarrow\infty
,\label{f1012e9}$$ which follows from (\[f1012e7\]) and Lemma \[l113b\] in Subsection \[s1103\], further below.
From the definition of $\overset{\vee}{g}_{0}$ in (\[f1012e7\]) together with the four properties of $\overset{\vee}{g}_{0}$ that have just been listed and with the use of Lemma \[l113f\] in Subsection \[s1103\], we deduce that $$g_{\widetilde{D}_{0}}(z,\infty)=\,\overset{\vee}{g}_{0}(z)+\int g_{\widetilde
{D}_{0}}(z,x)d\sigma(x),\text{ \ \ }z\in\overline{\mathbb{C}},\label{f1012e10}$$ where $\sigma$ is a signed measure with $\operatorname*{supp}(\sigma
)\subset\partial U_{1}$. From (\[f113d2\]) in Lemma \[l113f\], we know that the measure $\sigma$ is defined as the difference of the flux in $\overset{\vee}{g}_{0}$ that comes to the Jordan curve $\partial U_{1}$ from the both sites. Indeed, the total flux flowing into the set $\overline{U}_{1}$ from outside is equal to $\alpha_{1}$ because of the definition of $\alpha
_{1}$ in (\[f1012c1\]). On the other hand, the flux coming from within $U_{1}$ is equal to $\widetilde{\alpha}_{1}$ because of (\[f101d2\]) and (\[f1012e7\]). Hence, from (\[f1012e6\]) we conclude that $$\sigma(\partial U_{1})=\alpha_{1}-\widetilde{\alpha}_{1}>0.\label{f1012e11}$$ Putting all partial results of the last paragraphs together, we arrive at the following estimate: $$\begin{aligned}
\log\operatorname*{cap}(K_{0})-\log\operatorname*{cap}(\widetilde{K}_{0}) &
=(g_{\widetilde{D}_{0}}(z,\infty)-\overset{\vee}{g}_{0}(z))|_{z=\infty
}\nonumber\\
& =\int g_{\widetilde{D}_{0}}(x,\infty)d\sigma(x)\label{f1012e12}\\
& =\int\overset{\vee}{g}_{0}(x)d\sigma(x)+\int g_{\widetilde{D}_{0}}(x,y)d\sigma(x)d\sigma(x)\nonumber\\
& >c_{0}(\alpha_{1}-\widetilde{\alpha}_{1})>0.\nonumber\end{aligned}$$ Indeed, the first equality in (\[f1012e12\]) follows from (\[f1012e9\]) and representation (\[f113b1\]) in Lemma \[l113b\] in Subsection \[s1103\], further below. The second one is a consequence of (\[f1012e10\]) and the symmetry of the Green function with respect to both of its arguments. The third one follows again from (\[f1012e10\]). The first inequality in (\[f1012e12\]) is a consequence of (\[f1012e8\]) and (\[f1012e11\]) together with the positive definiteness of the Green kernel (cf. Lemma \[l113d\] in Subsection \[s1103\], further below). The last inequality follows again from (\[f1012e11\]).
With the inequalities in (\[f1012e12\]) we have proved (\[f1012d1\]). It has already been mentioned after (\[f1012d1\]) that the proof of assertion (i) is complete as soon as we have completed the deduction of (\[f1012d5\]) and (\[f1012d1\]).
The considerations made in (\[f1012e7\]) with respect to the function $\overset{\vee}{g}_{0}$ show that if each set $K_{0,j}$, $j=1,\ldots,m$, is the unique solution of Problem $(A_{j},\mathbb{D})$, therefore, identity (\[f1012c2\]) holds true for each $j=1,\ldots,m$. Hence, assertion (ii) is a consequence of assertion (i), and the proof of the whole Proposition \[p101a\] is complete.
### \[s1013\]The Minimal Set for Algebraic Functions
With Proposition \[p101a\] and Theorem \[t101a\] we are prepared to prove a detailed description of the minimal set $K_{0}(f,\infty)$ for Problem $(f,\infty)$ with an algebraic function $f$.
The next theorem covers most of the content in the main theorems in Section \[s4\] and \[s5\]. Since $f$ is assumed to be an algebraic function, we deal here only with a special version of Problem $(f,\infty)$, however, we remark that at the present point the results of Section \[s4\] and \[s5\] are still not proved, and more than that, the results in the next theorem will later be used as intermediate steps in the general proofs.
\[t101b\]Let $f$ be algebraic function that is not rational. We assume that the function is meromorphic at infinity. Let further $K_{0}(f,\infty)$ be the minimal set for Problem $(f,\infty)$.
\(a) The interior of $K_{0}(f,\infty)$ is empty, and there exist two finite sets $E_{0}$, $E_{1}$, and a finite family of open and analytic Jordan arcs $J_{j}$, $j\in I$, such that $$K_{0}(f,\infty)=E_{0}\cup E_{1}\cup\bigcup_{j\in I}J_{j}.\label{f1013a1}$$ The components in (\[f1013a1\]) correspond to those in Theorem \[t41a\] of Section \[s4\], but under the additional assumption that $f$ is algebraic we can give a more specific characterization:
- The set $E_{0}$ is finite, and it consists of all branch points of $f$ in $K_{0}(f,\infty)$ that can be reached by meromorphic continuation of $f$ out of the extremal domain $D_{0}(f,\infty)$.
- The set $E_{1}$ is finite, and it consists of all bifurcation points of $K_{0}(f,\infty)$ that do not belong to $E_{0}$.
- The family $\left\{ J_{j}\right\} _{j\in I}$ of analytic Jordan arcs is finite. All arcs $J_{j}$, $j\in I$, are pair-wise disjoint. The function $f$ has meromorphic continuations across each arc $J_{j}$, $j\in I$, from both sides. Each arc $J_{j}$, $j\in I$, is a trajectory of the quadratic differential (\[f1013a2\]) having end points that belong to $E_{0}\cup
E_{1}$, and all open trajectories of (\[f1013a2\]) starting and ending at a point of $E_{0}\cup E_{1}$ belong to the family $\left\{ J_{j}\right\}
_{j\in I}$.
\(b) The set $E_{0}$ contains at least $2$ points; we denote the points in $E_{0}$ by $a_{1},\ldots,a_{n}$. There exist $n-2$ points $b_{1},\ldots,b_{n-2}\in\mathbb{C}$ such that the Jordan arcs $J_{j}$, $j\in I$, are trajectories of the quadratic differential $$q(z)\,dz^{2}\text{ \ \ with \ \ }q(z):=\frac{(z-b_{1})\ldots(z-b_{n-2})}{(z-a_{1})\ldots(z-a_{n})}\label{f1013a2}$$ Not all points of the set $B=\{b_{1},\ldots,b_{n-2}\}$ are necessarily different, and not all of them are necessarily contained in $K_{0}(f,\infty)
$.
\(c) The minimal set $K_{0}(f,\infty)$ consists of finitely many components; we denote their number by $m$. Each of these components contains at least two elements of $E_{0}$. We have $E_{1}\subset B$. If $m>1$, then the Green function $g_{D_{0}}(\cdot,\infty)$, $D_{0}=D_{0}(f,\infty)$, possesses critical points of total order $m-1$, and each of these critical points appears in the set $B$ with a frequency of twice its order.
\[r101a\]The Examples \[e61\] - \[e64\] in Section \[s6\] belong to the class of problems covered by Theorem \[t101b\]. In the discussion of these examples one finds concrete and explicit examples for the sets $E_{0}$, $E_{1}$, for the families of Jordan arcs $\left\{ J_{j}\right\} _{j\in I}$, and also for the quadratic differentials (\[f1013a2\]).
There exists strong similarities between the two Theorems \[t101b\] and \[t53a\], but we note that the later one has a somewhat different orientation; it is focused only on the finiteness of the set $E_{0}$.
We define $$q(z):=\left( \frac{\partial}{\partial z}g_{D_{0}}(z,\infty)\right)
^{2}\text{ \ \ for \ \ }z\in D_{0}=D_{0}(f,\infty)\label{f1013b1}$$ with $D_{0}(f,\infty)$ the extremal domain for Problem $(f,\infty)$. It is immediate that $q$ is analytic in $D_{0}(f,\infty)$. From (\[f1013b1\]) and representation (\[f113b1\]) for the Green function in Lemma \[l113b\] of Subsection \[s1103\], further below, we deduce that at infinity the function $q$ has the development $$q(z)=z^{-2}+\text{O}(z^{-3})\text{ \ \ \ as \ \ \ }z\rightarrow\infty
.\label{f1013b2}$$
The function $q$ is different from zero everywhere in $D_{0}(f,\infty
)\cap\mathbb{C}$ except at the critical points of the Green function $g_{D_{0}}(\cdot,\infty)$, where it has zeros. It follows from (\[f1013b1\]) that the order of each of these zeros is twice the order of the critical point. Critical points and their order have been introduced in Definition \[d53b\] in Subsection \[s53\].
From Lemma \[l101a\] we know that $K_{0}(f,\infty)$ has only a finite number of components, which we denote again by $K_{j}$, $j=1,\ldots,m$. A combination of Proposition \[p101a\] and Theorem \[t101a\] shows that $q$ is meromorphic in a neighborhood of each component $K_{j}$, $j=1,\ldots,m$. Hence, the function is meromorphic throughout $\overline
{\mathbb{C}}$, and consequently it is a rational function with all its poles contained in $K_{0}(f,\infty)$.
For the deduction of more specific assertions we can without loss of generality restrict our attention to individual components $K_{j}$ and open neighborhoods $U_{j}$, $j=1,\ldots,m$, of these sets. Without loss of generality we will choose $j=1$ in the sequel.
It follows from (\[f1012c2\]) and (\[f1012c1\]) in Proposition \[p101a\] together with assertion (ii) and (iii) of Theorem \[t101a\] that all poles of $q$ on $K_{1}$ are simple, and they have to belong to the set $E_{1}$ from (\[f1012a2\]), i.e., they have to be branch points of $f$ on $K_{1}$.
Further it follows especially from assertion (ii) of Theorem \[t101a\] that on $K_{1}$ the function $q$ has exactly two zeros less than it has poles on $K_{1}$, where multiplicities of zeros have to be taken into account. The zeros in question constitute the set $E_{1}\cap K_{1}$.
In the application of assertion (ii) of Theorem \[t101a\] there may appear cancellations of numerator and denominator factors in the function (\[f101c1\]). If none of such cancellations occurs, which can be seen as the generic case, then every branch point of the function $f$ on $K_{1}$ corresponds to a simple pole of the function $q$ on $K_{1}$.
From what has been proved so far together with the definitions made in (\[f1012c1\]) and the identity (\[f1012c2\]) of Proposition \[p101a\], we deduce from assertion (ii) of Theorem \[t101a\] that all Jordan arcs $J_{j} $, $j\in I$, that belong to $K_{1}$ are transformed by the conformal map $\varphi_{1}$ of (\[f1012c1\]) into a critical trajectory $\varphi
_{1}(J_{j})$ of the quadratic differential (\[f101c1\]) of Theorem \[t101a\], and the reverse conclusion holds also true. With these last conclusions we have proved assertion (iii) of part (a) in the theorem for the component $K_{1}$.
All conclusions that have been proved so far for the component $K_{1}$ hold true in the same way on the other $m-1$ components $K_{j}$, $j=2,\ldots,m$, of the minimal set $K_{0}(f,\infty)$, which proves practically all assertions of the theorem.
We add that the number of zeros of the rational function $q$ on all $m$ component $K_{1},\ldots,K_{m}$ together with the $2(m-1)$ zeros at the critical points of the Green function $g_{D_{0}}(\cdot,\infty)$ add up to exactly two zeros less than the number of poles that $q$ has in $\mathbb{C}$. This account reaffirms exactly the behavior of the function $q$ at infinity, which is shown in development (\[f1013b2\]).
\[s102\]Some Technical Results
------------------------------
In the present subsection we prove some technical results which then are needed in the remainder of the section in proofs of results from the Sections \[s4\], \[s5\], and \[s7\]. Most important are here the proofs of the two Theorems \[t41a\] and \[t73a\]. In a first part of the subsection, the results will be formulated together with related definitions; proofs will then follow afterwards. Some of the results depend on rather subtle topological assumptions.
The first proposition is especially important for the proof of Theorem \[t73a\].
\[p102a\]Let $D_{1},D_{2}\in\mathcal{D}(f,\infty)$ be two admissible domains for Problem $(f,\infty)$. If we assume that $D_{1}$ possesses the $S-$property as introduced in Definition \[d71a\], and if we assume further that $D_{2}$ is elementarily maximal in the sense of Definition \[d71a0\], then we have either $D_{1}=D_{2}$ or $$\operatorname{cap}(\partial D_{1})<\operatorname{cap}(\partial D_{2}).\label{f102a1}$$
Besides of Proposition \[p102a\] we need a very similar result, which is not related to admissible domains $D\in\mathcal{D}(f,\infty)$; instead it is based on purely topological assumptions, which however comes to the same thing. We prepare the formulation of the result by some definitions.
\[d102a\]Let $K,E\subset\mathbb{C}$ be two polynomials-convex and compact sets with $\operatorname{cap}(K)>0$ and $E\subset K$. We say that the set $K$ possesses the $S-$property on the subset $K\setminus E$ if the following two assertions are satisfied:
- The set $K\setminus E$ is of the form $$K\setminus E=E_{1}\cup\bigcup_{j\in I}J_{j}\label{f102a2}$$ with $E_{1}$ a discrete set in $K\setminus E$ and $\{J_{j}\}_{j\in I}$ a family of smooth, open, and disjoint Jordan arcs $J_{j}$. Each point $z\in
E_{1}$ is an end point of at least three different arcs from $\{J_{j}\}_{j\in
I}$.
- The Green function $g_{D}(\cdot,\infty)$ with $D:=\overline
{\mathbb{C}}\setminus K$ satisfies the symmetry relation $$\frac{\partial}{\partial n_{+}}g_{D}(z,\infty)=\frac{\partial}{\partial n_{-}}g_{D}(z,\infty)\text{ \ for all \ }z\in J_{j},j\in I\label{f102a3}$$ with $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ denoting the normal derivatives to both sides of the arcs $J_{j}$, $j\in I$.
It is obvious that there exist many parallels between the Definitions \[d102a\] and Definitions \[d71a\] of Section \[s7\]; the special aspect of the new definition is the independence from Problem $(f,\infty)$ and the admissible domains $D\in\mathcal{D}(f,\infty)$. On the other hand, it is immediate that any admissible domain $D\in\mathcal{D}(f,\infty)$ with $K:=\overline{\mathbb{C}}\setminus D$ that possesses the $S-$property in the sense of Definition \[d71a\] possesses also the $S-$property in the sense of Definition \[d102a\] on the subset $K\setminus E_{0}$, where $E_{0}$ is the compact set from (\[f71a2\]) and assertion (i) in Definition \[d71a\].
Let $K_{1},K_{2},E\subset\mathbb{C}$ be three compact sets with $E\subset
K_{1}\cap K_{2}$. Two components $E_{1},E_{2}\subset E$ of $E$ are said to the connected in $K_{1}$ if both components are contained in the same component of $K_{1}$. In this sense, $K_{1}$ defines a connectivity relation on the components of $E$. We say (in the usual sense) that the connectivity of $E$ in $K_{1}$ is coarser than the connectivity in $K_{2}$ if the connectedness of two components $E_{1},E_{2}\subset E$ in $K_{2}$ implies their connectedness in $K_{1}$.
\[d102b\]Let $K_{1},K_{2},E\subset\mathbb{C}$ be three polynomial-convex and compact sets with $\operatorname{cap}(K_{1})>0$ and $E\subset K_{1}\cap
K_{2}$, and let us assume that $K_{1}$ possesses the $S-$property in the sense of Definition \[d102a\] on $K_{1}\setminus E$ with $\{J_{j}\}_{j\in I}$ denoting the family of Jordan arcs introduced in (\[f102a2\]). We say that the connectivity of $E$ in $K_{1}$ is minimally coarser than the connectivity of $E$ in $K_{2}$ if the following to assertions hold true:
- The connectivity of $E$ in $K_{1}$ is coarser than that in $K_{2} $.
- If $\widetilde{K}_{1}$ is the compact set that results from dropping one of the arcs $J_{j}$, $j\in I$, from $K_{1}$, then assertion (i) holds no longer true with $\widetilde{K}_{1}$ replacing $K_{1}$.
\[r102a\]Since the arcs $J_{j}$, $j\in I$, in (\[f102a2\]) are assumed to be open, one can drop any arc $J_{j_{0}}$, $j_{0}\in I$, from $K$, and the remaining set $\widetilde{K}=K\setminus J_{j_{0}}$ is still compact and polynomial-convex, but of course, the connectivity defined by $\widetilde{K}$ is finer than that defined by $K$.
\[p102b\]Let $K_{1},K_{2},E\subset\mathbb{C}$ be three polynomial-convex and compact sets with $\operatorname{cap}(K_{j})>0$, $j=1,2$, and $E\subset
K_{1}\cap K_{2}$. Let us assume further that $K_{1}$ possesses the $S-$property in the sense of Definition \[d102a\] on $K_{1}\setminus E$ and that the connectivity of $E$ in $K_{1}$ is minimally coarser than the connectivity in $K_{2}$. Then at least one of the two assertions $$K_{1}\subset K_{2}\text{ \ \ \ or \ \ }\operatorname{cap}(K_{1})<\operatorname{cap}(K_{2})\label{f102a4}$$ holds true.
The proof of Proposition \[p102b\] will follow in the footsteps of Proposition \[p102a\] only that at its beginning there are differences because of the different type of assumptions in Proposition \[p102a\].
In the next proposition a bridge is built between the set-up of Proposition \[p102b\] and the world of Problem $(f,\infty)$ with its admissible domains $D\in\mathcal{D}(f,\infty)$. The assumptions of the proposition are rather technical, but they are constructed in such a way that they fit well to the situation in the proof of Theorem \[t41a\], further below, where Proposition \[p102b\] is needed in a crucial way.
\[p102c\]Let the admissible domain $D\in\mathcal{D}(f,\infty)$ be elementarily maximal in the sense of Definition \[d71a0\] with $\operatorname{cap}(\partial D)>0$. Set $K:=\overline{\mathbb{C}}\setminus D$, and let $E_{0}\subset K$ denote the minimal compact and polynomial-convex set with the property that $\partial E_{0}$ contains all points for which assertion (i) of Definition \[d71a0\] holds true.
Let $U\subset\mathbb{C}$ be an open set with $E_{0}\subset U$, set $E:=\overline{U}\cap K$, and assumed that there exists a polynomial-convex and compact set $K_{1}\subset\mathbb{C}$ satisfying the following assertions:
- $\operatorname{cap}(K_{1})>0$ and $E\subset K_{1}$.
- $K_{1}$ possesses the $S-$property in the sense of Definition \[d102a\] on $K_{1}\setminus E$.
- The connectivity of $E$ in $K_{1}$ is minimally coarser than the connectivity of $E$ in $K$.
- We have $K_{1}\setminus K\neq\emptyset$.
If these assumptions are satisfied, then there exists an admissible domain $\widetilde{D}\in\mathcal{D}(f,\infty)$ with $$\operatorname{cap}(\partial\widetilde{D})<\operatorname{cap}(\partial
D).\label{f102a5}$$
We now come to a proposition, which will play a technical role in the proof of Theorem \[t41a\].
\[p102d\]Let $K,E\subset\mathbb{C}$ be two polynomial-convex and compact sets with $\operatorname{cap}(K)>0$ and $E\subset K$. We assumed that $K$ possesses the $S-$property in the sense of Definition \[d102a\] on $K\setminus E$, and by $E_{1}\subset K\setminus E$ and $\{J_{j}\}_{j\in I}$ we denote the compact discrete set and the family of open Jordan arcs introduced in (\[f102a2\]). We set $D:=\overline{\mathbb{C}}\setminus K$ and defined the function $q$ by $$q(z):=\left( 2\frac{\partial}{\partial z}g_{D}(z,\infty)\right)
^{2}\ \ \ \ \ \text{for \ \ }z\in\overline{\mathbb{C}}\setminus
E\label{f102a6}$$ with $\partial/\partial z=\frac{1}{2}\left( \partial/\partial x-i\,\partial
/\partial y\right) $ the usual complex differentiation and $g_{D}(\cdot,\infty)$ the Green function the domain $D$.
The function $q$ is analytic in $\overline{\mathbb{C}}\setminus E$ as a consequence of the assumed $S-$property of $K$ on $K\setminus E$, it has a zero at each point $z\in E_{1}$, the order of each of these zeros $z\in E_{1}
$ is equal to the number of different arcs from $\{J_{j}\}_{j\in I}$ that have $z$ as their end point minus $2$, i.e., it is of order $i(z)-2$ with $i(z)$ denoting the bifurcations index introduced in Definition \[d53a\], the function $q$ is different from zero in $\overline{\mathbb{C}}\setminus(E\cup
E_{1})$ except at the critical points of $g_{D}(\cdot,\infty)$ (cf. Definition \[d53b\]) and at infinity, further we have the estimate $$\left| q(z)\right| \leq\frac{3}{\operatorname{dist}(z,E)^{2}}\left(
\log(3\,r)+\log\frac{1}{\operatorname{cap}(K)}\right) \text{ \ for all
\ }z\in\{|z|\leq r\}\setminus E\label{f102a7}$$ and any $r>0$ sufficiently large so that $K\subset\{|z|\leq r\}$.
The most important part of Proposition \[p102d\] is the estimate (\[f102a7\]). We underlined that this estimate depends only on $\operatorname{cap}(K)$ and the set $E$, but not on the shape or extension of the set $K$ or the complementary domain $D$.
It has been stated in Proposition \[p102d\] that the $S-$property of a compact set $K$ on $K\setminus E$ implies the analyticity of the function $q$ defined in (\[f102a6\]), is a rather immediate conclusion of (\[f102a3\]) and (\[f102a6\]). It is interesting that also the reverse conclusion holds true, which is formulated in the next lemma.
\[l102a\]Let the function $q$ be defined by (\[f102a6\]) with the same notations as those introduced and use in Proposition \[p102d\], and assume further that $q$ is analytic in $\overline{\mathbb{C}}\setminus E$. Then the set $K$ possesses the $S-$property on $K\setminus E$ in the sense of Definition \[d102a\].
We next come to the proofs of the four Propositions \[p102a\] - \[p102d\] and of Lemma \[l102a\].
### Proof of Proposition \[p102a\]
The proof of Proposition \[p102a\] is rather involved and as an essential tool the Dirichlet integral of a Green function is used.
\[Proof of Proposition \[p102a\]\]We assume that $$D_{1}\neq D_{2},\label{f102b1}$$ and show then that this implies (\[f102a1\]).
Set $K_{j}:=\overline{\mathbb{C}}\setminus D_{j}$, $j=1,2$, and denote by $\widetilde{E}_{0,j}\subset K_{j}$, $j=1,2$, the two sets of points $z\in
K_{j}$ for which assertion (i) in Definition \[d71a\] and in Definition \[d71a0\] are satisfied for the domains $D_{1}$ and $D_{2}$, respectively. Let $E_{0,j}$ be the polynomial-complex hull of $\widetilde{E}_{0,j}$, i.e., $$E_{0,j}:=\widehat{\widetilde{E}_{0,j}},\,\ \ j=1,2.\label{f102b2}$$ Further, we defined $$K_{3}:=\widehat{K_{1}\cup K_{2}}\text{ \ and \ }D_{3}:=\overline{\mathbb{C}}\setminus K_{3}.\label{f102b3}$$ Since the domain $D_{1}$ has been assumed to possess the $S-$property, we know from (\[f71a2\]) in Definition \[d71a\] that $K_{1}$ can be represented in the form $$K_{1}=E_{1,0}\cup E_{1}\cup\bigcup_{j\in I}J_{j}\label{f102b4}$$ with the two sets $E_{1,0}$, $E_{1}$, and the family of Jordan arcs $J_{j}$, $j\in I$, with properties as described in Definition \[d71a\].
In the next step we study some properties of the two sets $K_{1}\setminus
K_{2}$ and $K_{2}\setminus K_{1}$ that follow immediately from assumption (\[f102b1\]). We have $$(\partial K_{3}\setminus K_{2})\cap E_{0,1}=\emptyset\text{ \ \ and
\ \ }(\partial K_{3}\setminus K_{1})\cap E_{0,2}=\emptyset.\label{f102b5}$$ Indeed, from (\[f102b3\]) it follows that $\partial K_{3}\subset\partial
K_{1}\cup\partial K_{2}$. Since $D_{3}\subset D_{1}\cap D_{2}$, and since both domains $D_{1}$ and $D_{2}$ are elementally maximal, $z\in\widetilde{E}_{0,1}\cap\partial K_{3}$ implies $z\in\widetilde{E}_{0,2}\cap\partial K_{3}$, and vice versa. Consequently, we have $$E_{0,j}\cap\partial K_{3}\subset K_{1}\cap K_{2}\cap\partial K_{3}\text{
\ \ for \ \ \ }j=1,2,\label{f102b6}$$ which proves (\[f102b5\]).
We have $$\operatorname{cap}(K_{2}\setminus K_{1})>0\text{ \ \ and \ \ }\operatorname{cap}(K_{1}\setminus K_{2})>0.\label{f102b7}$$ Here, we first prove that $\operatorname{cap}(K_{2}\setminus K_{1})>0$. This will be done in an indirect way, and we assume for this purpose that $$\operatorname{cap}(K_{2}\setminus K_{1})=0.\label{f102b8}$$
Let $f_{j}$ denote the meromorphic continuations of the function $f$ into the domains $D_{j}$, $j=1,2,3$. With the same arguments as used in the proof of Lemma \[l92a\] in Subsection \[s92\], we can show that assumption (\[f102b8\]) implies that all meromorphic continuations of the function $f_{2}$ out of $D_{1}\setminus K_{2}$ into $D_{1}$ lead to the same function in each component of the open set $D_{1}\setminus K_{2}$. These functions then are necessarily identical with the function $f_{1}$. Since the domain $D_{2}$ has been assumed to be elementarily maximal, it follows from assertion (ii) in Definition \[d71a0\] that $K_{2}\setminus K_{1}=\emptyset$, which implies that $K_{2}\subset K_{1}$. As a consequence of the assumed $S-$property of the domain $D_{1}$, we know that $D_{1}$ is also elementarily maximal (see the assertions (i) and (ii) in Definition \[d71a\]), and therefore $D_{2}\supset
D_{1}$ implies that $D_{1}=D_{2}$. This last conclusion contradicts (\[f102b1\]), and consequently we have proved $\operatorname{cap}(K_{2}\setminus K_{1})>0$ in (\[f102b7\]). A proof of $\operatorname{cap}(K_{1}\setminus K_{2})>0$ in (\[f102b7\]) can be done in exactly the same way.
From (\[f102b4\]) and the fact that the two sets $\partial K_{3}\setminus
K_{2}$ and $E_{0,1}$ are disjoint, which has been proved in (\[f102b5\]), we immediately conclude that $$\partial K_{3}\setminus K_{2}\subset E_{1}\cup\bigcup_{j\in I}J_{j}\label{f102b9}$$ i.e., $\partial K_{3}\setminus K_{2}$ is the union of open subarcs of arcs from the family $\{J_{j}\}_{j\in I}$ together with points from $E_{1}$. The dominant parts in this union are the open Jordan arcs since the set $E_{1}$ is countable, and therefore we have $\operatorname{cap}(E_{1})=0$.
We now continue our investigation with further definitions. We set $$V:=\overline{\operatorname{Int}(K_{3})\cap(K_{1}\setminus K_{2})},\text{
\ \ }\widetilde{D}_{2}:=D_{2}\setminus V\text{, \ \ }\widetilde{K}_{2}:=K_{2}\cup V=\mathbb{C}\setminus\widetilde{D}_{2}.\label{f102c1}$$
It is immediate that $\widetilde{D}_{2}$ is open, but it is not necessarily a domain. By $g_{j}(\cdot,\cdot)$ we denote the Green functions $g_{D_{j}}(\cdot,\cdot)$ in the domains $D_{j}$, $j=1,2$. Because of (\[f102b7\]), these two Green functions exist in a proper sense (see Subsection \[s1103\], further below).
Next, we show that $$K_{1}\cap\widetilde{D}_{2}\subset\bigcup_{j\in I}J_{j}\cap\partial
\operatorname{Int}(K_{3}),\label{f102c2}$$ or more precisely, we show that $K_{1}\cap\widetilde{D}_{2}$ consists only of open subarcs of the arcs $J_{j}$ from (\[f102b9\]) that are contained in $\partial\operatorname{Int}(K_{3})$. Indeed, since $K_{1}$ possesses the $S-$property of Definition \[d71a\], it follows from assertion (ii) in Definition \[d71a\] that $K_{1}\subset\overline{\operatorname{Int}(K_{3})}$. It further follows from the definitions in (\[f102c1\]) that $K_{1}\cap\widetilde{D}_{2}\subset\partial K_{3}\setminus K_{2}$. Because of (\[f102b9\]), it remains only to show that $K_{1}\cap\widetilde{D}_{2}\cap
E_{1}=\emptyset$. Let us assume that $z\in K_{1}\cap\widetilde{D}_{2}\cap
E_{1}$. From assertion (iv) in Definition \[d71a\] of the $S-$property we know that at least three different arcs of the family $\{J_{j}\}$ in (\[f102b4\]) have $z$ as endpoint. Since $K_{1}\cap\widetilde{D}_{2}$ lies in $\partial K_{3}\setminus K_{2}$, the meromorphic continuations of the two functions $f_{1}$ and $f_{2}$ out of the domain $D_{3}$ are identical, and therefore, at least one of the arcs ending at $z$ belongs to $V$; and consequently, we have $z\in V$, which contradicts $z\in K_{1}\cap\widetilde
{D}_{2}$. Thus, (\[f102c2\]) is proved.
A key role in the proof of the proposition is played by the function $\widetilde{g}_{1}$, which is defined as $$\widetilde{g}_{1}(z):=\left\{
\begin{array}
[c]{lll}g_{1}(z,\infty)\medskip & \text{ \ \ for \ \ } & z\in D_{3},\\
-g_{1}(z,\infty) & \text{ \ \ for \ \ } & z\in K_{3}\setminus K_{2}.
\end{array}
\right. \label{f102c3}$$ All discussions, so far, can be seen as preliminaries to an investigation of properties of the function $\widetilde{g}_{1}$. In this connection the $S-$property of the domain $D_{1}$ is very important since it implies that the two pieces in the definition of the function $\widetilde{g}_{1}$ are harmonic continuations of each other across the arcs in $K_{1}\cap\widetilde{D}_{2}$.
Indeed, from symmetry (\[f71a1\]) in Definition \[d71a\] together with (\[f102c2\]) and the remarks just after (\[f102c2\]), we conclude that $\widetilde{g}_{1}$ is harmonic in $\widetilde{D}_{2}$. Notice that the domain $D_{1}$ is assume to possess the $S-$property.
The function $\widetilde{g}_{1}$ is superharmonic in $D_{2}$. Indeed, from Lemma \[l113b\] in Subsection \[s1103\], further below, we know that the Green function $g_{1}(z,\infty)$ is subharmonic in $\mathbb{C}$. Since $V\subset K_{3}\setminus K_{2}$, the superharmonicity follows directly from (\[f102c3\]).
From the defining identity (\[f113a1\]) of the Green function in Subsection \[s1103\], further below, together with (\[f102c3\]) and the definition of $V$ in (\[f102c1\]), we conclude that $$\widetilde{g}_{1}(z)=0\ \ \ \text{for quasi every}\ \ \ z\in V.\label{f102c4}$$
From the properties of $\widetilde{g}_{1}$ that have just been discussed together with the Poison-Jensen Formula (cf. Theorem \[t113a\] in Subsection \[s1103\], further below) we get for $\widetilde{g}_{1}$ the representation $$\widetilde{g}_{1}(z)=\widetilde{h}_{1}(z)+g_{2}(z,\infty)+g_{0}(z)\text{
\ \ for \ \ }z\in D_{2},\label{f102c5}$$ where $g_{2}(\cdot,\infty)$ is the Green function in $D_{2}$, and $g_{0}$ the Green potential $$g_{0}(z)=\int_{V}g_{2}(z,v)d\omega_{1}(v)\label{f102c6}$$ with $\omega_{1}$ the equilibrium distribution on $K_{1}$, and $V$ the set from (\[f102c1\]). The function $\widetilde{h}_{1}$ in (\[f102c5\]) is the solution of the Dirichlet problem in $D_{2}$ with boundary values $$\widetilde{h}_{1}(z)=\widetilde{g}_{1}(z)\text{ \ \ for quasi every \ \ }z\in\partial K_{2}.\label{f102c7}$$ Identity (\[f102c5\]) can easily be verified by considering its values on $\partial D_{2}$, on $V$, and near infinity.
From (\[f102b7\]) and Lemma \[l113c\] in Subsection \[s1103\], further below, we deduce that the two Green functions $g_{1}(\cdot,\infty)$ and $g_{2}(\cdot,\infty)$ are essentially different, and consequently, we have $$g_{1}(z,\infty)>0\text{ \ \ for quasi every \ \ }z\in K_{2}\setminus
K_{1}.\label{f102c8}$$ From (\[f102c7\]) and (\[f102c3\]), we then conclude that $$\widetilde{h}_{1}\neq0.\label{f102c9}$$
By definition, we have $g_{0}(z)\geq0$ for all $z\in\overline{D}_{2}$, and $g_{0}(z)>0$ for $z\in D_{2}$ if, and only if, $\omega_{1}(V)>0$. However, this last condition may in general not be satisfied; even the case $V=\emptyset$ is possible.
In order to prove (\[f102a1\]), we prove that the identity $$\begin{aligned}
\log\frac{\operatorname{cap}(K_{2})}{\operatorname{cap}(K_{1})} &
=D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))+D_{D_{2}}(\widetilde{h}_{1})+2\,g_{0}(\infty)\label{f102d1}\\
& +\int_{V}\int_{V}g_{2}(v,w)d\omega_{1}(v)d\omega_{1}(w)\nonumber\end{aligned}$$ holds true, where $D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))$ and $D_{D_{2}}(\widetilde{h}_{1})$ are Dirichlet integrals that have been introduced in (\[f113f3\]) in Subsection \[s1103\], further below.
Since we know from (\[f102c9\]) that the harmonic function $\widetilde
{h}_{1}$ is not identical zero in $D_{2}$, it follows from the definition of the Dirichlet integral in (\[f113f3\]) that $$D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))\geq0\ \ \ \text{and
}\ \ D_{D_{2}}(\widetilde{h}_{1})>0.\label{f102d2}$$
We have already mentioned that the Green potential $g_{0}$ is always non-negative. It follows from (\[f102c6\]) and the positivity of the Green function as kernel function (see Lemma \[l113d\] in Subsection \[s1103\], further below) that $$g_{0}(\infty)\geq0,\text{ \ \ \ }\int_{V}\int_{V}g_{2}(v,w)d\omega
_{1}(v)d\omega_{1}(w)\geq0,\label{f102d3}$$ and we have proper inequalities in both cases of (\[f102d3\]) if, and only if, $\omega_{1}(V)>0$.
When identity (\[f102d1\]) is proved, then inequality (\[f102a1\]) follows immediately from identity (\[f102d1\]) together with (\[f102d2\]) and (\[f102d3\]). Hence, it remains only to prove that (\[f102d1\]) holds true, which will be done next.
From Lemma \[l113b\], Lemma \[l113g\], and Corollary \[c113g1\] in Subsection \[s1103\], further below, we deduce that $$\begin{aligned}
\log\frac{\operatorname{cap}(K_{2})}{\operatorname{cap}(K_{1})} & =\left(
g_{1}(\cdot,\infty)-g_{2}(\cdot,\infty)\right) (\infty)\label{f102e1}\\
& =D_{D_{1,r}}(g_{1}(\cdot,\infty))-D_{D_{2,r}}(g_{2}(\cdot,\infty
))+\text{O}(\frac{1}{r})\nonumber\end{aligned}$$ as $r\rightarrow\infty$, where $D_{j,r}$ denotes the bounded domain $$D_{j,r}:=D_{j}\cap\{|z|<r\}\text{ \ \ for \ \ }j=1,2,\label{f102e3}$$ and $r>0$ so large that $K_{j}\subset\{|z|<r\}$ for $j=1,2,3$.
From the definition of the Dirichlet integral in (\[f113f3\]), further below, and the definition of the function $\widetilde{g}_{1}$ in (\[f102c3\]), we deduce that $$\begin{aligned}
D_{D_{1,r}}(g_{1}(\cdot,\infty)) & =D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))+D_{\widetilde{D}_{2,r}}(g_{1}(\cdot,\infty))\label{f102e2}\\
& D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))+D_{\widetilde{D}_{2,r}}(\widetilde{g}_{1}).\nonumber\end{aligned}$$ In (\[f102e2\]), the open set $\widetilde{D}_{2,r}$ is defined as $\widetilde{D}_{2,r}:=$ $\widetilde{D}_{2}\cap\{|z|<r\}$ in analogy to (\[f102e3\]). It has already been mentioned in (\[f102d2\]) that $D_{K_{2}\setminus K_{1}}(g_{1}(\cdot,\infty))\geq0$.
We will now have a closer look on the Dirichlet integral $D_{\widetilde
{D}_{2,r}}(\widetilde{g}_{1})$ in (\[f102e2\]). From the representations (\[f102c5\]), (\[f102c6\]), and equality (\[f102c4\]), it follows with the help of Lemma \[l113j\] in Subsection \[s1103\], further below, that $$D_{\widetilde{D}_{2,r}}(\widetilde{g}_{1})=D_{D_{2,r}}(\widetilde{h}_{1}+g_{2}(\cdot,\infty)+g_{0,r})+\int_{V}\int_{V}g_{2}(v,w)d\omega_{1}(v)d\omega_{1}(w)\label{f102e4}$$ with $\omega_{1}$ and $V$ defined like in (\[f102c6\]). In (\[f102e4\]), the function $g_{0,r}$ denotes the solution of the Dirichlet problem in $D_{2,r}$ with boundary values $g_{0,r}(z)=g_{0}(z)$ for quasi every $z\in\partial D_{2,r}$. We know therefore from (\[f102c6\]) that $$g_{0,r}(z)=\left\{
\begin{array}
[c]{lll}0\medskip & \text{ \ \ for quasi every \ \ } & z\in\partial D_{2},\\
g_{0}(z) & \text{ \ \ for \ \ } & |z|=r.
\end{array}
\right. \label{f102e5}$$
Next, we investigate the Dirichlet integral $D_{D_{2,r}}(\widetilde{h}_{1}+g_{2}(\cdot,\infty)+g_{0,r})$ in (\[f102e4\]) in more detail. Using the notations introduced in (\[f102c5\]), (\[f102e4\]), and also in (\[f113f2\]), further below, we prove the identity $$\begin{aligned}
& D_{D_{2,r}}(\widetilde{h}_{1}+g_{2}(\cdot,\infty)+g_{0,r})\nonumber\\
& \ \ \ \ =D_{D_{2,r}}(\widetilde{h}_{1})+D_{D_{2,r}}(g_{2}(\cdot
,\infty))+D_{D_{2,r}}(g_{0,r})+\label{f102e6}\\
& \text{ \ \ \ \ \ \ \ \ }+2\,D_{D_{2,r}}(\widetilde{h}_{1},g_{2}(\cdot
,\infty))+2\,D_{D_{2,r}}(\widetilde{h}_{1},g_{0,r})+2\,D_{D_{2,r}}(g_{0,r},g_{2}(\cdot,\infty)).\nonumber\end{aligned}$$
Indeed, the positivity of the integrand in the Dirichlet integral implies that $$\lim_{r\rightarrow\infty}D_{D_{2,r}}(\widetilde{h}_{1})=D_{D_{2}}(\widetilde{h}_{1})>0,\label{f102e7}$$ and since $\widetilde{h}_{1}$ is harmonic and bounded in $D_{2}$, the Dirichlet integral on the right-hand side of (\[f102e7\]) is finite.
The Green potential $g_{0}$ is bounded near infinity, therefore it follows from the definition of $g_{0,r}$ as a solution of a Dirichlet problem in $D_{2,r}$ with boundary values (\[f102e5\]) that $$\lim_{r\rightarrow\infty}g_{0,r}(z)=0\text{ \ \ locally uniformly for
\ \ \ }z\in D_{2},\label{f102e8}$$ and the same conclusion holds also for the first derivatives in $\nabla
g_{0,r}$ and $r\rightarrow\infty$; these derivatives converge also to zero locally uniformly in $D_{2}$. As a consequence, we have $$\lim_{r\rightarrow\infty}D_{D_{2,r}}(g_{0,r})=0.\label{f102e9}$$ With the Cauchy-Schwartz inequality and the boundedness of $D_{D_{2,r}}(\widetilde{h}_{1})$ we deduce from (\[f102e9\]) that we also have $$\lim_{r\rightarrow\infty}D_{D_{2,r}}(\widetilde{h}_{1},g_{0,r})=0.\label{f102e10}$$
The function $\widetilde{h}_{1}$ is harmonic in $D_{2}$, and therefore it follows from Lemma \[l113h\] in Subsection \[s1103\], further below, that $$\lim_{r\rightarrow\infty}D_{D_{2,r}}(\widetilde{h}_{1},g_{2}(\cdot
,\infty))=0.\label{f102e11}$$
Since the Green potential $g_{0}$ is harmonic in $\{|z|>r\}$, we deduce from (\[f102e5\]) and Lemma \[l113i\] in Subsection \[s1103\], further below, that $$D_{D_{2,r}}(g_{0,r},g_{2}(\cdot,\infty))=\frac{1}{2\pi}\int_{\{|z|=r\}}g_{0}(z)\frac{\partial}{\partial n}g_{2}(z,\infty)ds_{z}=g_{0}(\infty
)\label{f102e12}$$ for $r>0$ sufficiently large.
From identity (\[f102e6\]) together with (\[f102e7\]), (\[f102e9\]), (\[f102e10\]), (\[f102e11\]), and (\[f102e12\]), we get $$\lim_{r\rightarrow\infty}\left( D_{D_{2,r}}(\widetilde{h}_{1}+g_{2}(\cdot,\infty)+g_{0,r})-D_{D_{2,r}}(g_{2}(\cdot,\infty))\right) =D_{D_{2}}(\widetilde{h}_{1})+2\,g_{0}(\infty).\label{f102e13}$$ Further, we then get from formula (\[f102e1\]) together with (\[f102e2\]), (\[f102e4\]), and (\[f102e13\]) that identity (\[f102d1\]) holds true, which completes the proof of the proposition.
### Proof of Proposition \[p102b\]
It has already been mentioned that the proof of Proposition \[p102b\] follows in the footsteps of that of Proposition \[p102a\] only that we have a modified opening since we have to start from a different type of assumptions. But after the introduction of the function $\widetilde{g}_{1} $ in (\[f102c3\]), we can use the argumentation of the last proof without any change.
\[Proof of Proposition \[p102b\]\]We assume $$K_{1}\nsubseteq K_{2},\label{f102f1}$$ and show then that this implies $\operatorname{cap}(K_{1})<\operatorname{cap}(K_{2})$.
Like in the proof of Proposition \[p102a\], we set $$K_{3}:=\widehat{K_{1}\cup K_{2}}\text{ \ \ and \ \ \ }D_{j}:=\overline
{\mathbb{C}}\setminus K_{j},j=1,2,3,\label{f102f2}$$ where $\widehat{\cdot}$ denotes the polynomial-convex hull (cf. Definition \[d111b\] in Subsection \[s1101\], further below). Since it has been assumed that the compact set $K_{1}$ possesses the $S-$property on $K_{1}\setminus E$ in the sense of Definition \[d102a\], in $K_{1}$ we have the compact set $E_{1}$ and the family of Jordan arcs $\{J_{j}\}_{j\in I}$ that have been introduced in (\[f102a2\]) of Definition \[d102a\], and from there we then know that $$K_{1}=E\cup E_{1}\cup\bigcup_{j\in I}J_{j}.\label{f102f3}$$ From assumption (\[f102f1\]), representation (\[f102f3\]), assumption $\operatorname{cap}(K_{1})>0$, $E\subset K_{2}$, and the fact that the capacity of an open Jordan arc is positive (cf. Lemma \[l111a\] in Subsection \[s1101\], further below), it follows that $$\operatorname{cap}(K_{1}\setminus K_{2})>0.\label{f102f4}$$
The boundary $\partial C$ of any component $C$ of the open set $$O:=\operatorname{Int}(K_{3})\setminus K_{2}\label{f102f5}$$ contains elements of $\partial K_{1}$ and $\partial K_{2}$ because of the assumed polynomial-convexity of $K_{1}$. We have $$(K_{1}\setminus K_{2})\cap\partial O\subset K_{1}\setminus E,\label{f102f6}$$ and if we define the set $V$ by $$V:=\overline{\operatorname{Int}(K_{3})\cap(K_{1}\setminus K_{2})},\label{f102f7}$$ then it follows from (\[f102f3\]), (\[f102f6\]), and (\[f102f7\]) that $$(K_{1}\setminus(K_{2}\cup V))\cap\partial O\subset\bigcup_{j\in I}J_{j}.\label{f102f8}$$ On the other hand, we have $$K_{1}\setminus(K_{2}\cup V)\subset\partial O,\label{f102f9}$$ since otherwise any arc $J_{j}$, $j\in I$, in $K_{1}\setminus(K_{2}\cup
V\cup\partial O)$ could be removed from $K_{1}$ without separating any pair of components of the set $E$ in the modified set $K_{1}$ if this pair is already connected in $K_{2}$. But such a situation would contradict the assumption that the connectivity of the set $E$ in $K_{1}$ is minimal coarser than the connectivity in $K_{2}$ (cf. Definition \[d102b\]).
Like in (\[f102c1\]) in the proof of Proposition \[p102a\], we define $$\widetilde{D}_{2}:=D_{2}\setminus V\text{ \ and\ \ }\widetilde{K}_{2}:=K_{2}\cup V=\mathbb{C}\setminus\widetilde{D}_{2}\label{f102f10}$$ with the compact set $V$ introduced in (\[f102f7\]). It follows from (\[f102f8\]) and (\[f102f9\]) that $$K_{1}\cap\widetilde{D}_{2}\subset\bigcup_{j\in I}J_{j}\cap\partial
\operatorname{Int}(K_{3}).\label{f102f11}$$ Indeed, because of (\[f102f9\]) we have $K_{1}\cap\widetilde{D}_{2}=K_{1}\setminus(K_{2}\cup V)\subset\partial O\setminus K_{2}\subset
\partial\operatorname{Int}(K_{3})$, and because of (\[f102f8\]) together with (\[f102f9\]) we have $K_{1}\cap\widetilde{D}_{2}=K_{1}\setminus
(K_{2}\cup V)\subset\bigcup_{j\in I}J_{j}$.
After these preparations we can now follow the argumentation in the proof of Proposition \[p102a\] word-for-word. Exactly, like in (\[f102c3\]) we define the function $\widetilde{g}_{1}$ in the domain $D_{2}$. With the same arguments as those used in the proof of Proposition \[p102a\] after (\[f102c3\]) we then arrive after many intermediate steps at the conclusion that $$\operatorname{cap}(K_{1})<\operatorname{cap}(K_{2}),\label{f102f12}$$ which proves Proposition \[p102b\].
### Proof of Proposition \[p102c\]
The proof of Proposition \[p102c\] relies strongly on Definition \[d95a\] and the subsequent Theorem \[t95a\] together with Proposition \[p102b\].
\[Proof of Proposition \[p102c\]\]We set $$K_{0}:=K,\text{ \ \ \ \ }K_{3}:=\widehat{K_{0}\cup K_{1}},\text{ \ \ and
\ \ \ }D_{j}:=\overline{\mathbb{C}}\setminus K_{j},j=0,1,3.\label{f102g1}$$ In (\[f102g1\]), $\widehat{\cdot}$ denotes the polynomial-convex hull (cf. Definition \[d111b\] in Subsection \[s1101\], further below). It is immediate that the $D_{j}$ are domains.
From Proposition \[p102b\] together with the assumptions (i) - (iv) of the proposition, it follows that $$\operatorname{cap}(K_{1})<\operatorname{cap}(K_{0}).\label{f102g2}$$
If we would know that $D_{1}=\overline{\mathbb{C}}\setminus K_{1}\in\mathcal{D}(f,\infty)$, then the proof of the proposition would be completed with (\[f102g2\]). However, in general we have $D_{1}\notin\mathcal{D}(f,\infty)$ since the meromorphic continuation of the function $f $ out of the domain $D_{3}$ into $D_{1}$ may hit non-polar similarities in $D_{1}\cap\operatorname{Int}(K_{3})$.
In order to overcome these difficulties we make use of a construction introduced in Definition \[d95a\]. We consider the whole family of compact sets $K_{h}$, $h\in\left[ 0,1\right] $, with complementary domains $D_{h}=\overline{\mathbb{C}}\setminus K_{h}$ that are defined by the relations (\[f95b1\]) through (\[f95b7\]) of Definition \[d95a\] starting from the two domains $D_{0}$ and $D_{1}$ defined in (\[f102g1\]), and with elements of the proof of Theorem \[t95a\] we then prove that there exist $h_{0}\in\left( 0,1\right) $ such that $$D_{h}\in\mathcal{D}(f,\infty)\text{ \ \ for all \ }0<h\leq h_{0},\label{f102g3}$$ and further that $$\log\operatorname*{cap}(K_{h})<(1-h)\,\log\operatorname*{cap}(K_{0})+h\,\log\operatorname*{cap}(K_{1})\text{ \ \ for \ \ }0<h<1.\label{f102g4}$$
From (\[f102g4\]) we deduce that $$\log\operatorname*{cap}(K_{h_{0}})<\log\operatorname*{cap}(K_{0})-h_{0}\log\frac{\operatorname*{cap}(K_{0})}{\operatorname*{cap}(K_{1})},\label{f102g5}$$ which together with (\[f102g2\]) proves that $$\operatorname{cap}(K_{h_{0}})<\operatorname{cap}(K_{0}).\label{f102g6}$$ If we set $\widetilde{D}:=D_{h_{0}}$, then the proposition follows from (\[f102g6\]) together with (\[f102g3\]). Hence, it remains only to prove that the two assertions (\[f102g3\]) and (\[f102g4\]) hold true.
The two assertions (\[f102g3\]) and (\[f102g4\]) have already been proved as assertion (i) and (ii) in Theorem \[t95a\], but under partly different assumptions. The difference is the following: In Theorem \[t95a\], it has been assumed that both domains $D_{0}$ and $D_{1}$ belong to $\mathcal{D}(f,\infty)$, while in the present situation, we do not know whether $D_{1}\in\mathcal{D}(f,\infty)$. Instead, we now have the assumptions (i) - (iv), of which (ii) and (iii) are the two most important ones.
In the a step we prove that (\[f102g3\]) holds true. As in the proof of Lemma \[l93c\], where an analogous result has been proved for Proposition \[p91a\], the main tool will again be Proposition \[p91a\].
It follows from assumption (iii) and the assumption that $D_{0}\in
\mathcal{D}(f,\infty)$ together with Proposition \[p91a\] that for every Jordan curve $\gamma\in\Gamma_{1}$, with $\Gamma_{1}$ introduced in Definition \[d91b\], and $\gamma\subset\overline{\mathbb{C}}\setminus E_{0}$, we have $$\gamma\cap K_{0}\neq\emptyset\text{ \ \ and \ \ }\gamma\cap K_{1}\neq
\emptyset.\label{f102g7}$$ From the defining relations (\[f95b1\]) - (\[f95b7\]) in Definition \[d95a\] for the sets $K_{h}$, we further conclude that besides of (\[f102g7\]) we also have $$\gamma\cap K_{h}\neq\emptyset\text{ \ \ for all \ \ \ }h\in\left( 0,1\right)
.\label{f102g8}$$ Details of the argumentation are the same as those in the proof of Lemma \[l93a\].
From Proposition \[p91a\] and (\[f102g8\]) it is clear that for a proof of (\[f102g3\]) it remains only to show that assertion (i) in Proposition \[p91a\] holds true for $0\leq h\leq h_{0}$, i.e., we have to show that there exists $h_{0}\in\left( 0,1\right) $ such that the function $f$ has a meromorphic continuation to each point of $D_{h}$ for $0\leq h\leq h_{0}$.
We defined the two sets $B_{jh}$, $j=0,1$, $h\in\left( 0,1\right) $, by $$\begin{aligned}
B_{0,h} & :=\{\,z\in D_{h}\cap K_{3}\,|\,(1-h)g_{0}(z)>hg_{1}(z)\,\},\label{f102g9}\\
B_{1,h} & :=\{\,z\in D_{h}\cap K_{3}\,|\,(1-h)g_{0}(z)<hg_{1}(z)\,\}.\label{f102g10}$$ where $g_{j}$ denotes the Green function $g_{D_{j}}(\cdot,\infty)$, $j=0,1$, in the same way as in Definition \[d95a\]. In the same way as in the proof of Lemma \[l93b\], it follows from (\[f95b4\]), (\[f95b5\]), (\[f95b6\]), and (\[f95b7\]) that the two sets $D_{3}\cap B_{jn}$, $j=0,1$, are domains, and we have $$B_{jh}\subset D_{j}\text{ \ \ \ for \ \ }j=0,1\text{, }h\in\left( 0,1\right)
.\label{f102g11}$$ Since it has been assumed that $D_{0}\in\mathcal{D}(f,\infty)$, the function $f$ processes a meromorphic continuation into the domain $D_{0}$, which we denote by $f_{0}$. From (\[f102g11\]) it follows that the function $f_{0}$ is defined throughout $D_{3}\cup B_{0,h}$.
An analogous argumentation is unfortunately not possible for the domain $D_{1}$. Here, we have to follow a different path of argumentation. From the assumed properties of the set $E_{0}\subset\overset{\circ}{E}$ it follows that there exists a domain $D_{2}$ with the property that $$D_{2}\supset D_{3},\text{ \ \ }K_{0}\setminus E_{0}\subset D_{2},\label{f102g12}$$ and the function $f$ can meromorphically be continued to the domain $D_{2}$. We denote this continuation by $f_{2}$.
In the next step we show that there exists $h_{0}\in\left( 0,1\right) $ such that $$K_{h}\setminus\operatorname{Int}(E)\subset D_{2}\text{ \ \ for all
\ \ \ }0\leq h\leq h_{0}.\label{f102g13}$$ Let $\widetilde{U}_{0}\subset D_{2}$ be an open set consisting only of simply connected components and assume that $K_{0}\setminus U\subset\widetilde{U}_{0}$, where $U$ is the open set introduced in the formulation of the proposition. There exists an open set $U_{0}\subset\mathbb{C}$ consisting only of simply connected components such that $$K_{0}\subset U_{0}\text{, \ \ \ }U_{0}\cap D_{0}\subset\widetilde{U}_{0}\text{, \ and \ }U_{0}\cap\partial U\subset\widetilde{U}_{0}.\label{f102g14}$$ With exactly the same arguments as those applied in the proof of assertion (iii) of Theorem \[t95a\] we then show that there exists $h_{0}\in\left(
0,1\right) $ such that $$K_{h}\subset U_{0}\text{ \ \ for all \ \ \ }0\leq h\leq h_{0}.\label{f102g15}$$ Notice that in the proof of assertion (iii) of Theorem \[t95a\] only topological assumptions about the two sets $K_{0}$, $K_{1}$, and their complementary domains $D_{0}$ and $D_{1}$ have been used. The inclusion (\[f102g13\]) follows directly from (\[f102g15\]).
Since $U_{0}$ consists only of simply connected components, it follows from (\[f102g13\]), (\[f102g15\]), and the definition of the set $B_{1,h}$ in (\[f102g10\]) that $$B_{1,h}\subset D_{2}\text{ \ \ for \ \ \ }0\leq h\leq h_{0},\label{f102g16}$$ and consequently the functions $f_{2}$ is defined throughout the domain $D_{3}\cup B_{1,h}$ for all $0\leq h\leq h_{0}$.
Since $D_{h}=D_{3}\cup B_{0,h}\cup B_{0,h}$, we have shown that the function $f$ possesses are meromorphic continuation to each point $z\in D_{h}$ for $0\leq h\leq h_{0}$. Hence, assumption (i) of Proposition \[p91a\] holds true for each $0\leq h\leq h_{0}$, and (\[f102g3\]) then follows from Proposition \[p91a\].
After the verification of (\[f102g3\]), it remains only to prove that the inequality (\[f102g4\]) holds true. Here, we copy the corresponding proof of (\[f95c2\]) from the proof of Theorem \[t95a\] word for word. The condition (\[f95c4\]) in Theorem \[t95a\] follows from the two assumptions (i) and (iv) in Proposition \[p102c\]. A detailed argumentation for this last conclusion has been given after (\[f102b7\]). With the proof of (\[f102g4\]), the whole proof of Proposition \[p102c\] is completed.
\[Proof of Proposition \[p102d\]\]It is rather immediate that the $S-$property of $K$ on $K\setminus E$ implies that the function $q$ is analytic in $\overline{\mathbb{C}}\setminus E$.
From the definition of $q$ in (\[f102a6\]) it follows that level-lines of the Green function $g_{D}(\cdot,\infty)$ are trajectories of the quadratic differential $q(z)dz^{2}$ (for a definition of trajectories see (\[f52a\]) in Section \[s52\]). The arcs $J_{j}$, $j\in I$, in $K\setminus E$ are critical trajectories of $q(z)dz^{2}$, and, of course, they are also level-lines of $g_{D}(\cdot,\infty)$ corresponding to the value $0$. From the local structure of trajectories, it follows that at each bifurcation point $z\in\overline{\mathbb{C}}\setminus E$ of trajectories, we have a zero of order $i(z)-2$, where $i(z)$ is the bifurcation index of Definition \[d53a\] (cf. [@Jensen75], Chapter 8.2, or [@Strebel84]).
That the only zeros of $q$ in $\overline{\mathbb{C}}\setminus(E\cup E_{1})$ are critical points of the Green function $g_{D}(\cdot,\infty)$ in the sense of Definition \[d53b\] is an immediate consequence of (\[f102a6\]), and the same is also true for the double zero of $q$ at infinity.
We now come to the proof of the inequality (\[f102a7\]). From (\[f102a6\]) and we deduce that $$\begin{aligned}
|q(z)| & =4\frac{\partial}{\partial z}g_{D}(z,\infty)\overline{\frac
{\partial}{\partial z}g_{D}(z,\infty)}\nonumber\\
& =(\frac{\partial}{\partial x}-i\,\frac{\partial}{\partial y})g_{D}(z,\infty)(\frac{\partial}{\partial x}+i\,\frac{\partial}{\partial y})g_{D}(z,\infty)\label{f102h1}\\
& =(\frac{\partial}{\partial x})g_{D}(z,\infty))^{2}+(\frac{\partial}{\partial
y})g_{D}(z,\infty))^{2}\nonumber\end{aligned}$$ for $z\in\overline{\mathbb{C}}\setminus E$, and consequently we have the estimate $$\begin{aligned}
|q(z)| & =\frac{1}{\pi d^{2}}\left\vert \iint_{\Delta(z,d)}q(\zeta)dm_{\zeta
}\right\vert \leq\frac{1}{\pi d^{2}}\iint_{\Delta(z,d)}|q(\zeta)|dm_{\zeta
}\nonumber\\
& =\frac{2}{d^{2}}D_{\Delta(z,d)}(g_{D}(\cdot,\infty))\label{f102h2}$$ for every $z\in\mathbb{C}\setminus E$ with $0<d<\operatorname{dist}(z,E)$, $\Delta(z,d):=\{\,\zeta\,|$ $|\zeta-z|\leq d\,\}$, $dm_{\zeta}$ the area element at the point $\zeta\in\mathbb{C}$, and $D_{\ldots}(\cdot)$ denotes the Dirichlet integral introduced in (\[f113f3\]) in Subsection \[s1103\], further below.
Let now $r>0$ be such that $K\subset\{|z|\leq r\}$, and let further $z\in\{|z|\leq r\}\setminus E$. Then we have $\operatorname{dist}(z,E)<2r$, and consequently $\Delta(z,d)\subset\{|z|\leq3r\}\setminus E$ for all $0<d<\operatorname{dist}(z,E)$. From (\[f102h2\]) and identity (\[f113g1\]) in Lemma \[l113g\] in Subsection \[s1103\], further below, we deduce that $$\begin{aligned}
|q(z)| & \leq\frac{2}{d^{2}}D_{\{|z|\leq3r\}\setminus E}(g_{D}(\cdot
,\infty))=\frac{2}{d^{2}}D_{\{|z|\leq3r\}\setminus K}(g_{D}(\cdot
,\infty))\nonumber\\
& =\frac{2}{d^{2}}\left( \log(3\,r)+\log\frac{1}{\operatorname{cap}(K)}+\text{O}(\frac{1}{r})\right) \text{ \ \ as \ \ r}\rightarrow
\infty.\label{f102h3}$$ The equality in the first line of (\[f102h3\]) is a consequence of the fact that the set $K\setminus E$ is of planar Lebesgue measure zero, which follows from the assumed $S-$property of $K$ on $K\setminus E$. The second equality in (\[f102h3\]) follows from (\[f113g1\]) in Lemma \[l113g\]. Since $d<\operatorname{dist}(z,E)$ can be chosen arbitrarily, the inequality (\[f102a7\]) follows directly from (\[f102h3\]).
\[Proof of Lemma \[l102a\]\]Let $z\in J_{j}$, $j\in I$, the an arbitrary point on the Jordan arc $J_{j}$. We have $g_{D}(z,\infty)=0$, and from the continuity in $\overline{\mathbb{C}}\setminus E$ of the function $q$ introduced in (\[f102a6\]), it follows that the two normal derivatives $\partial/\partial n_{+}$ and $\partial/\partial n_{+}$ of $g_{D}(\cdot
,\infty)$ to both sides of the Jordan arc $J_{j}$ at $z$ are equal in modulus.
Since $g_{D}(\cdot,\infty)\geq0$, and $g_{D}(z,\infty)=0$ for all $z\in J_{j}$, it follows also that the signs of the two normal derivatives are equal. Putting both conclusions together proves equality (\[f102a3\]) for all $z\in
J_{j}$, $j\in I$, and consequently the $S-$property on $K\setminus E$ in the sense of Definition \[d102a\] is proved.
\[s103\]Proofs of Results from Section \[s4\]
---------------------------------------------
The central result of Section \[s4\] is Theorem \[t41a\], the Structure Theorem. As a further result, we have Theorem \[t41b\], which addresses a special topological property of the minimal set $K_{0}(f,\infty)
$, and Theorem \[t42a\], which is the analog of Theorem \[t41b\] for Problem $(\mathcal{R},\infty^{(0)})$.
### \[s1031\]Proof of Theorem \[t41a\]
In the proof of Theorem \[t41a\] we start from Theorem \[t101b\], which covers the special case that the function $f$ in Problem $(f,\infty)$ is algebraic, and which therefore implies that the set $E_{0}$ of non-polar singularities is finite. The proof of Theorem \[t41a\] can then be seen as a lifting of the special result in Theorem \[t101b\] to the general situation. In this process the Propositions \[p102b\], \[p102c\], and \[p102d\] from the last subsection will play a decisive role.
\[Proof of Theorem \[t41a\]\]It is immediate that the extremal domain $D_{0}(f,\infty)=\overline{\mathbb{C}}\setminus K_{0}(f,\infty)$ is elementarily maximal in the sense of Definition \[d71a0\]. By $E_{00}\subset
K_{0}(f,\infty)$ we denote the subset of all $z\in\partial K_{0}(f,\infty)$ that satisfy condition (i) in Definition \[d71a0\], i.e., for each $z\in
E_{00},$ there exists at least one meromorphic continuation of the function $f$ out of the domain $D_{0}(f,\infty)$ that has a non-polar singularity at $z$. By $E_{0}$ we then denote the polynomial-convex hull of $E_{00}$, i.e., $$E_{0}:=\widehat{E_{00}}.\label{f103a1}$$ (for a definition of the polynomial-convex hull see Definition \[d111b\] in Subsection \[s1101\], further below.) Since $K_{0}(f,\infty)$ is polynomial-convex, we have $E_{0}\subset K_{0}(f,\infty)$.
In the sequel we assume that $E_{0}\cap K_{0}(f,\infty)\neq\emptyset$; for otherwise Theorem \[t41a\] is trivial.
Let $U_{n}\subset\mathbb{C}$, $n=1,2,\ldots$, be a sequence of open set with simply connected components and smooth boundaries $\partial U_{n}$ such that $$E_{0}\subset U_{n+1}\subset\overline{U}_{n+1}\subset U_{n}\,\ \text{and
\ \ }E_{0}=\bigcap_{n=1}^{\infty}U_{n}.\label{f103a2}$$ We define $E_{n}:=\overline{U}_{n}$ for $n\in\mathbb{N}$. It is immediate that all sets $E_{n}$ are polynomial-convex, and each of these sets consists only of finitely many components. The minimal set $K_{0}=K_{0}(f,\infty)$ defines a connectivity relation on the components of each set $E_{n}$; we say that two components of $E_{n}$ are connected in $K_{0}$, if they are connected in $K_{0}\cup E_{n}$. Let $E_{jn}$, $j=1,\ldots,j_{n}$, be the components of $E_{n}$ with respect to the connectivity in $K_{0}$, i.e., $$E_{n}=E_{1n}\cup\ldots\cup E_{j_{n},n},\label{f103a3}$$ and each $E_{jn}$ is connected in $K_{0}$.
In each component $E_{jn}$, $j=1,\ldots,j_{n}$, we select points $z_{jl}$ in the following manner: For a given $n\in\mathbb{N}$ and any $m\in\mathbb{N}$ we choose $j_{n}$ families of points $$Z_{jnm}=\{z_{jl}\}_{l=1}^{m_{j}},\text{ }j=1,\ldots,j_{n}\text{,
\ with\ \ }Z_{nm}:=\bigcup_{j=1}^{j_{n}}Z_{jnm}\text{, \ \ }m=\sum
_{j=1}^{j_{n}}m_{j}\text{,}\label{f103a4}$$ such that $$Z_{nm}\subset Z_{n,m+1}\text{ \ \ and \ \ }\bigcap_{M=1}^{\infty}\overline{\bigcup_{m\geq M}Z_{nm}}=\partial E_{n},\label{f103a5}$$ i.e., the sets $Z_{jnm}$, $j=1,\ldots,j_{n}$, are asymptotically ($m\rightarrow\infty$) dense in each $\partial E_{jn}$, $j=1,\ldots,j_{n}$. Associated with each point set $Z_{nm}$, we define the algebraic function $$f_{nm}(z):=\sum_{j=1}^{j_{n}}\left[ \prod_{l=1}^{m_{j}}(1-\frac{z_{jl}}{z})\right] ^{1/m_{j}}\text{ \ for \ }n,m=1,2,\ldots\label{f103a6}$$
Let now $K_{nm}:=K_{0}(f_{nm},\infty)\subset\mathbb{C}$ be the minimal set for Problem $(f_{nm},\infty)$, $D_{nm}:=\overline{\mathbb{C}}\setminus K_{nm}$ the corresponding extremal domain $D_{0}(f_{nm},\infty)$, $g_{nm}$ the Green function $g_{D_{nm}}(\cdot,\infty)$ in $D_{nm}$, and let $q_{nm}$ be the function $$q_{nm}(z)=\left( 2\frac{\partial}{\partial z}g_{nm}(z,\infty)\right)
^{2}\label{f103a7}$$ that is defined analogously to (\[f102a6\]), and this definition appears also already earlier in (\[f52b\]). Since $f_{nm}$ is an algebraic function, we know from Theorem \[t101b\] that $q_{nm}$ is a rational function. With Lemma \[l102a\], it follows from (\[f103a7\]) that $K_{nm} $ possesses the $S-$property on $K_{nm}\setminus Z_{nm}$ in the sense of Definition \[d102a\].
For $m\in\mathbb{N}$ sufficiently large, each component $E_{jn}$ in (\[f103a3\]) contains elements of the set $Z_{nm}$. The definition of the function $f_{nm}$ together with the definition of the components $E_{jn}$ in (\[f103a3\]) then imply that components of $E_{n}$ are connected in $K_{nm}
$ for $m$ sufficiently large if they are also connected in $K_{0}=K_{0}(f,\infty)$, i.e., the connectivity defined by $K_{nm}$ is coarser than that defined by $K_{0}$. Further, it follows from the minimality of $\operatorname{cap}(K_{nm})=\operatorname{cap}(K_{0}(f_{nm},\infty))$ that the connectivity defined by $K_{nm}$ is also only minimally coarser in the sense of Definition \[d102b\] than that defined by $K_{0}$. Notice that the connectivity relation defined by $K_{0}$ on $E_{n}$ stands in the background of the definition of the function $f_{nm}$.
We will now investigated in a first step what happens with the functions $f_{nm}$, $g_{nm}$, and $q_{nm}$ if $m\rightarrow\infty$. In a second step we then will also consider the limits for $n\rightarrow\infty$.
From Proposition \[p102d\] we deduce the upper estimates $$\left| q_{nm}(z)\right| \leq\frac{2}{\operatorname{dist}(z,E_{n})^{2}}\left( \log(3\,r)+\log\frac{1}{\operatorname{cap}(K_{nm})}\right)
\label{f103a8}$$ for the functions $q_{nm}$; they hold for all $z\in\{|z|\leq r\}\setminus
E_{n}$, all $m\geq m_{0}$, and $r>0$ sufficiently large. Notice that the assumption $E_{0}\cap K_{0}(f,\infty)\neq\emptyset$ at the beginning of this proof implies that $E_{0}$ contains at least two different components that are connected in $K_{0}(f,\infty)$, which then further implies that $\operatorname{cap}(K_{nm})\geq c_{0}>0$ for all $m\geq m_{0}$.
From (\[f103a8\]) together with Montel’s Theorem and the fact that all $q_{nm}$ are analytic outside of $E_{n}$ (cf. Theorem \[t101b\], part (b)), we deduce that there exists an infinite sequence $N_{n}\subset\mathbb{N}$ such that $$\lim_{m\rightarrow\infty,\text{ }m\in N_{n}}q_{nm}(z)=:q_{n}(z)\label{f103a9}$$ holds locally uniformly in $\overline{\mathbb{C}}\setminus E_{n}$.
For the Green functions $g_{nm}$ and the sets $K_{nm}$ we now prove the existence of limits that correspond to limit (\[f103a9\]). Using the same techniques as applied in the proof of Lemma \[l92b\] after (\[f92e1\]) and combining this with (\[f103a9\]) and (\[f103a7\]), we deduce that the limit $$\lim_{m\rightarrow\infty,\text{ }m\in N_{n}}g_{nm}(z)=:g_{n}(z)\label{f103a10}$$ exists locally uniformly in $\mathbb{C}\setminus E_{n}$. From the two relations in (\[f103a5\]) together with the fact that all points in each set $Z_{jnm}\subset E_{jn}$, $j=1,\ldots,j_{n}$, are connected by a subcontinuum in $K_{nm}$ and that these continua contain only regular boundary points, it follows that $$g_{n}(z)=0\text{ \ \ for all \ \ }z\in E_{n}.\label{f103a11}$$ With the two definitions $$K_{n}:=\bigcap_{M=1}^{\infty}\widehat{\bigcup_{m\geq M,\text{ }m\in N_{n}}K_{nm}}\text{ \ \ and \ \ }D_{n}:=\overline{\mathbb{C}}\setminus
K_{n},\label{f103a12}$$ it further follows from Lemma \[l114b\] from Subsection \[s1104\], further below, that the function $g_{n}$ introduced in (\[f103a10\]) is the Green function of the domain $D_{n}$, i.e., we have $$g_{n}=g_{D_{n}}(\cdot,\infty).\label{f103a13}$$
The two limits (\[f103a9\]) and (\[f103a10\]) together imply that analogously to (\[f103a7\]) we also have the relation $$q_{n}(z)=\left( 2\frac{\partial}{\partial z}g_{n}(z,\infty)\right)
^{2}\text{ \ \ for \ \ }z\in\overline{\mathbb{C}}\setminus E_{n},\label{f103a14}$$ from which we deduce again with Lemma \[l102a\] that the set $K_{n}$ possesses the $S-$property on $K_{n}\setminus E_{n}$ in the sense of Definition \[d102a\].
Like the sets $K_{nm}$, so also the set $K_{n}$ possesses the property that the connectivity relation defined on the components of $E_{n}$ by $K_{n}$ is minimally coarser in the sense of Definition \[d102b\] than that defined by $K_{0}=K_{0}(f,\infty)$. Indeed, both aspects of the property ”minimally coarser” carry over from $K_{nm}$ to $K_{n}$. With exactly the same techniques as those used in the proof of Lemma \[l92d\], we show that all connectivities defined by the sets $K_{nm}$ lead to identical connectivities defined by the set $K_{n}$. Consequently, also the new connectivity relation is coarser than that defined by the set $K_{0}$. On the other hand, as a consequence of convergence $\lim_{m}\operatorname{cap}(K_{nm})=\operatorname{cap}(K_{n})$, which follows from (\[f103a10\]), it then further follows that the connectivity defined on $E_{n}$ by $K_{n}$ is also only minimally coarser in the sense of Definition \[d102b\] than that defined by $K_{0}$.
In a second step we investigate limits for $n\rightarrow\infty$, where we can largely apply the same techniques as those just used after (\[f103a8\]) for the investigation of limits with $m\rightarrow\infty$, only that now the boundary behavior of the Green function can be complicated by irregular points on $\partial E_{0}$. Notice that the sets $E_{n}$, have been constructed with nice boundaries. We overcome these new difficulties by using Lemma \[l114a\] from Subsection \[s1104\], further below.
From the definition of the sets $E_{n}$ after (\[f103a2\]) we know that $$E_{0}=\bigcap_{n}E_{n}.\label{f103b1}$$ Analogously to (\[f103a8\]), we deduce from Proposition \[p102d\] that $$\left| q_{n}(z)\right| \leq\frac{2}{\operatorname{dist}(z,E_{n})^{2}}\left(
\log(3\,r)+\log\frac{1}{\operatorname{cap}(K_{n})}\right) \label{f103b2}$$ for all $z\in\{|z|\leq r\}\setminus E_{n}$, $n\in\mathbb{N}$, and $r>0$ sufficiently large, for the functions $q_{n}$ from (\[f103a9\]), which satisfy (\[f103a14\]). With the same argumentation as used after (\[f103a8\]), we conclude that $\operatorname{cap}(K_{n})\geq c_{0}>0$ for all $n\in\mathbb{N}$, which shows that the right-hand side of (\[f103b2\]) can be made independent of $n$. From (\[f103b2\]) it then follows as before in (\[f103a9\]) that there exists an infinite sequence $N\subset\mathbb{N}$ such that the limit $$\lim_{n\rightarrow\infty,\text{ }n\in N}q_{n}(z)=:\widetilde{q}(z)\label{f103b3}$$ exists locally uniformly for $z\in\overline{\mathbb{C}}\setminus E_{0}$, and the function $\widetilde{q}$ is analytic in $\overline{\mathbb{C}}\setminus
E_{0}$. With the same argumentation as used for the deduction of (\[f103a10\]), we now deduce that the limit $$\lim_{n\rightarrow\infty,\text{ }n\in N}g_{n}(z)=:\widetilde{g}(z)\label{f103b4}$$ exists locally uniformly for $z\in\mathbb{C}\setminus E_{0}$.
From (\[f103b4\]) together with Lemma \[l114a\] from Subsection \[s1104\] and (\[f103b1\]), we deduce that $$\widetilde{g}(z)=0\text{ \ \ for quasi every \ \ }z\in E_{0}.\label{f103b5}$$ Using now the definitions $$\widetilde{K}:=\bigcap_{m=1}^{\infty}\widehat{\bigcup_{n\geq m,\text{ }m\in
N}K_{n}}\text{ \ \ and \ \ }\widetilde{D}:=\overline{\mathbb{C}}\setminus\widetilde{K},\label{f103b6}$$ it follows in the same way as after (\[f103a12\]) with the help of Lemma \[l114b\] from Subsection \[s1104\] that the function $\widetilde{g}$ introduced in (\[f103b4\]) is the Green functions of the domain $\widetilde{D}$, i.e., we have $$\widetilde{g}=g_{\widetilde{D}}(\cdot,\infty).\label{f103b7}$$ Notice that the domain $\widetilde{D}$ may not be regular with respect to Dirichlet problems, there may exist irregular points (cf. Definition \[d113a\] in Subsection \[s1103\], further below) on $\partial
\widetilde{D}\cap E_{0}$, and consequently, the equality in (\[f103b5\]) holds only quasi everywhere on $E_{0}$.
The two limits (\[f103b3\]) and (\[f103b4\]) together with relation (\[f103a14\]) imply that $$\widetilde{q}(z)=\left( 2\frac{\partial}{\partial z}g_{\widetilde{D}}(z,\infty)\right) ^{2}\text{ \ \ for \ \ }z\in\overline{\mathbb{C}}\setminus
E_{0}.\label{f103b8}$$
With Lemma \[l102a\], we deduce from (\[f103b8\]) that the set $\widetilde{K}$ possesses the $S-$property on $\widetilde{K}\setminus E_{0}$ in the sense of Definition \[d102a\]. In the same way, as it has been shown before in the cases of the sets $K_{nm}$ and the set $K_{n}$, we also now show that the minimal coarseness of a connectivity relation in the sense of Definition \[d102b\] can be carried over from the connectivity relations defined by the sets $K_{n}$ to the relation defined by the set $\widetilde{K}$, i.e., we deduce that the connectivity defined by $K_{n}$ on $E_{0}$ is minimally coarser in the sense of Definition \[d102b\] than that defined by $K_{0}(f,\infty)$.
Since we have assumed at the beginning of the present proof that $E_{0}\cap
K_{0}(f,\infty)\allowbreak\neq\emptyset$, it follows that some components of $E_{0}$ are connected in $K_{0}(f,\infty)$, and consequently, those components are also connected in $\widetilde{K}$, which implies that $$\operatorname{cap}(\widetilde{K})>0.\label{f103b9}$$
Let us summarize what we have proved so far. Starting from the algebraic functions $f_{nm}$ introduced in (\[f103a6\]) and using the results proved in Theorem \[t101b\] for algebraic functions, we have shown that there exists a compact set $\widetilde{K}\subset\mathbb{C}$ of positive capacity with the following properties:
- $\widetilde{K}$ possesses the $S-$property in the sense of Definition \[d102a\] on $\widetilde{K}\setminus E_{0}$.
- The connectivity relation defined on $E_{0}$ by $\widetilde{K}$ is minimally coarser in the sense of Definition \[d102b\] than that defined by $K_{0}(f,\infty)$.
- The set $E_{0}\subset\widetilde{K}$ is compact and polynomial-convex, and its boundary $\partial E_{0}$ contains all non-polar singularities of the function $f$ that can be reached by meromorphic continuations of $f$ from within the extremal domain $D_{0}(f,\infty)$.
In the concluding part of the proof, Proposition \[p102c\] will play a crucial role. If in Proposition \[p102c\] we take $K_{0}(f,\infty)$ as $K$ and $\widetilde{K}$ from (\[f103b6\]) as $K_{1}$, then it is immediate from (\[f103b9\]) and the assertions (a), (b), and (c) that all assumptions of Proposition \[p102c\] are satisfied, except the assumption (iv).
Since the set $K_{0}(f,\infty)$ is of minimal capacity in the sense of (\[f21a\]) in Definition \[d21b\], it follows that conclusion (\[f102a5\]) of Proposition \[p102c\] cannot be true. Consequently, assumption (iv) of Proposition \[p102c\] has to be false, which proves with our choice of $K$ and $K_{1}$ that we have $$K_{0}(f,\infty)=\widetilde{K}.\label{f103b10}$$
Indeed, from the negation of assumption (iv) in Proposition \[p102c\] we conclude that $\widetilde{K}\subset K_{0}(f,\infty)$. Identity (\[f103b10\]) then follows from a combination of the fact that the connectivity relation defined by $\widetilde{K}$ on $E_{0}$ is coarser than that defined by $K_{0}(f,\infty)$ (cf. assertion (b)), and the fact that the extremal domain $D_{0}(f,\infty)$ is elementarily maximal in the sense of Definition \[d71a0\] (cf. Proposition \[p71a\]).
With identity (\[f103b10\]), all properties proved for the set $\widetilde{K}$ are now also valid for the minimal set $K_{0}(f,\infty)$. Hence, we have proved the following four assertions:
- The set $K_{0}(f,\infty)\setminus E_{0}$ consists of critical trajectories of the quadratic differential $\widetilde{q}(z)dz^{2}$ with $\widetilde{q}$ defined in (\[f103b8\]). In (\[f103b8\]), $\widetilde{D}$ is equal to the extremal domain $D_{0}(f,\infty)$ because of (\[f103b10\]).
Indeed, assertion ($\alpha$) follows immediately from (\[f103b8\]) and the definition of trajectories of quadratic differentials in (\[f52a\]) in Subsection \[s52\].
- At each $z\in\partial E_{0}$ at least one meromorphic continuation of the function $f$ out of the extremal domain $D_{0}(f,\infty)$ hits a non-polar singularity.
Indeed, assertion ($\beta$) is an immediate consequence of the definition of the set $E_{0}$ in and before (\[f103a1\]).
- Let $E_{1}$ be set of all zeros of the function $\widetilde{q}$ from (\[f103b8\]) on $K_{0}(f,\infty)\setminus E_{0}$. The set $E_{1}$ is discrete in $K_{0}(f,\infty)\setminus E_{0}$ since the function $\widetilde{q}$ is analytic in $\overline{\mathbb{C}}\setminus E_{0} $. The set $K_{0}(f,\infty)\setminus(E_{0}\cup E_{1})$ consists of open, analytic Jordan arcs, which are trajectories of the quadratic differential $\widetilde{q}(z)dz^{2}$.
Indeed, the first part of assertion ($\gamma$) follows directly from (\[f103b8\]). Since $\widetilde{q}(z_{0})\neq0$ for any $z_{0}\in
K_{0}(f,\infty)\setminus(E_{0}\cup E_{1})$, we also have $\frac{\partial
}{\partial z}g_{D_{0}(f,\infty)}(z_{0},\infty)\allowbreak\neq0$, and consequently, the equation $g_{D_{0}(f,\infty)}(z,\infty)=0$ defines an analytic Jordan arc in a neighborhood of any point $z_{0}\in K_{0}(f,\infty)\setminus(E_{0}\cup E_{1})$, which proves the second part of assertion ($\gamma$).
- Let $o(z)$, $z\in E_{1}$, be the order of the zero of $\widetilde{q}$ at $z$, then the point $z$ is endpoint of $o(z)+2$ different analytic Jordan arcs in $K_{0}(f,\infty)\setminus(E_{0}\cup E_{1})$.
Indeed, assertion ($\delta$) is an immediate consequence of the typical local structure of trajectories of quadratic differentials in a neighborhood of a zero of the differential (cf. [@Jensen75], Chapter 8.2, or [@Strebel84]).
- The function $f$ has meromorphic continuations to each point of $z\in K_{0}(f,\infty)\setminus E_{0}$ from all sides out of the extremal domain $D_{0}(f,\infty)$. These continuations lead to exactly two different function elements at each point $z\in K_{0}(f,\infty)\setminus
(E_{0}\cup E_{1})$, and it leads to more than two different function elements at each point $\in E_{1}$.
Indeed, the first part of assertion ($\varepsilon$) is a consequence of the definition of the set $E_{0}$ in (\[f103a1\]). The second part is a consequence of the minimality (\[f21a\]) in Definition \[d21b\] of the set $K_{0}(f,\infty)$.
With the four assertions ($\alpha$) - ($\varepsilon$) we have proved more than is needed for the proof of Theorem \[t41a\]. The additional results will be needed in subsequent proofs, most importantly in the proofs of the two Theorems \[t51a\] and \[t52a\].
Identity (\[f41a\]) in Theorem \[t41a\] follows directly from the three assertions ($\alpha$), ($\beta$), and ($\gamma$). The description of the set $E_{0}$ in assertions (i) of Theorem \[t41a\] is practically identical with assertion ($\beta$). The two remaining assertions (ii) and (iii) in Theorem \[t41a\] follow from the two assertions ($\gamma$) and ($\varepsilon$). With the last sentences the proof of Theorem \[t41a\] is completed.
### \[s1032\]Proof of Theorem \[t41b\]
The catchword in Theorem \[t41b\] is local connectedness. Among other things it will be shown in Theorem \[t41b\] that the most interesting part $K_{0}(f,\infty)\setminus E_{0}$ of the minimal set $K_{0}(f,\infty)$ is locally connected. Local connectedness is an important property in geometric function theory.
\[Proof of Theorem \[t41b\]\]Since we know that the function $\widetilde{q}$ of (\[f103b8\]) is analytic in $\overline{\mathbb{C}}\setminus E_{0}$, it follows from $\overline{\mathbb{C}}\setminus E_{0}$ that the Green function $g_{D_{0}(f,\infty)}(\cdot,\infty)$ is continuous throughout $\overline
{\mathbb{C}}\setminus E_{0}$. From Carathéodory’s theory about the boundary behavior of Riemann mapping functions (cf. Chapter 9 in [@Pommerenke75], and there especially Theorem 9.8) we know that the continuity of the Green function $g_{D_{0}(f,\infty)}(\cdot,\infty)$ is equivalent to the local connectedness of the set $\partial D_{0}(f,\infty)\setminus E_{0}$. From Theorem \[t41a\] it follows that the set $\partial D_{0}(f,\infty)\setminus E_{0}$ is equal to $K_{0}(f,\infty
)\setminus E_{0} $, which proves the first half of Theorem \[t41b\].
From the local behavior of trajectories of quadratic differentials, as it has been stated in Lemma \[l115a\] in Subsection \[s1105\], further below, together with assertion ($\gamma$) at the end of the proof of Theorem \[t41a\], we know that the bifurcation points $z\in E_{1}$ of $K_{0}(f,\infty)\setminus E_{0}$ are zeros of the analytic function $\widetilde{q}$ from (\[f103b8\]), and as such they are of finite order. From the connection between the zeros of quadratic differentials and the local structure of its trajectories (see again Lemma \[l115a\] in Subsection \[s1105\]), we then conclude that the finiteness of the order of the zeros of $\widetilde{q}$ implies that each $z\in E_{1}$ can be the endpoint of only finitely many Jordan arcs $J_{j}$, $j\in I$.
### \[s1033\]Proof of Theorem \[t42a\]
In Theorem \[t32a\] of Section \[s3\] it has been shown that the two Problems $(f,\infty)$ and $(\mathcal{R},\infty^{(0)})$ are equivalent if the Riemann surface $\mathcal{R}$ is the natural domain of definition for the function $f$. Because of this equivalence, Theorem \[t42a\] is practically a corollary to Theorem \[t41a\].
\[Proof of Theorem \[t42a\]\]In order to prove the characterization of the sets $E_{0}$, $E_{1}$, and the family of Jordan arcs $\left\{ J_{j}\right\}
_{j\in I}$ in (\[f42a0\]) of Theorem \[t42a\], we have only to keep in mind that a non-polar singularity of the function $f$ at a point $z\in\overline{\mathbb{C}}$ is either a branch point or a transcendental, essential singularity. In the first case, the point $z$ is an element of $Br(\mathcal{R})$, and in the second case, it is an element of the relative boundary $\partial\mathcal{R}$ of the Riemann surface $\mathcal{R}$ over $\overline{\mathbb{C}}$. With these observations we immediately verify identity (\[f42a\]) in Theorem \[t42a\].
The two assertions (ii) and (iii) in Theorem \[t42a\] follow then from the observation that meromorphic continuations of the function $f$ lead to different function elements at a point $z\in\overline{\mathbb{C}}$ if, and only if, the corresponding points $\zeta$ on the Riemann surface $\mathcal{R}$ lie on different sheets of this surface.
\[s104\]Proofs of Results from Section \[s5\]
---------------------------------------------
In Section \[s5\] the Jordan arcs $J_{j}$, $j\in I$, in the minimal set $K_{0}(f,\infty)$ have been characterized with the help of the $S-$property and with the help of quadratic differentials. Both concepts are very similar. These results are especially interesting if the function $f$ has only finitely many non-polar singularities; Proposition \[p53a\] and Theorem \[t53a\] deal with this situation.
### \[s1041\]Proof of Theorem \[t51a\]
Most of the work for a proof of Theorem \[t51a\] has already been done in the proof of Theorem \[t41a\].
\[Proof of Theorem \[t51a\]\]In assertion ($\gamma$) at the end of the proof of Theorem \[t41a\], it has been shown that each Jordan arc $J_{j}$, $j\in
I$, is a trajectory of the quadratic differential $\widetilde{q}(z)dz^{2}$ with the function $\widetilde{q}$ defined by (\[f103b8\]). Because of (\[f103b10\]), the domain $\widetilde{D}$ in (\[f103b8\]) is equal to the extremal domain $D_{0}(f,\infty)$. With Lemma \[l102a\], we then conclude from (\[f103b8\]) together with (\[f103b10\]) that $K_{0}(f,\infty)$ possesses the $S-$property in this sense of Definition \[d102a\] on $K_{0}(f,\infty)\setminus E_{0}$, which proves identity (\[f51a\]) because of (\[f102a3\]).
### \[s1042\]Proof of Theorem \[t52a\]
In Theorem \[t52a\] the Jordan arcs $J_{j}$, $j\in I$, of the minimal set $K_{0}(f,\infty)$ are characterized by quadratic differentials.
\[Proof of Theorem \[t52a\]\]From (\[f103b10\]) in the proof of Theorem \[t41a\] we know that the function $\widetilde{q}$ introduced in (\[f103b8\]) and the function $q$ introduced in (\[f52b\]) are identical. In assertion ($\alpha$) at the end of the proof of Theorem \[t41a\], it has been proved that the arcs $J_{j}$, $j\in I$, are trajectories of the quadratic differential $q(z)dz^{2}$.
Identity (\[f52c\]) in Theorem \[t52a\] follows from (\[f103b8\]) for $z\in\mathbb{C}\setminus E_{0}$, and it follows from the logarithmic pole of the Green functions $g_{D_{0}(f,\infty)}(\cdot,\infty)$ at infinity. From (\[f103b3\]), it further follows that the function $\widetilde{q}=q$ is analytic in $\overline{\mathbb{C}}\setminus E_{0}$. Thus, it only remains to prove that the function $\widetilde{q}$ from (\[f103b8\]) is meromorphic at isolated points of $E_{0}$. Actually, we shall prove slightly more and show that $\widetilde{q}$ has at most of simple pole at an isolated point of $E_{0}$.
If $z\in E_{0}$ is simultaneously an isolated point of $E_{0}$ and of the minimal set $K_{0}(f,\infty)$, then the Green function $g_{D_{0}(f,\infty
)}(\cdot,\infty)$ is harmonic in a neighborhood of $z$, and it follows from (\[f103b8\]) that $\widetilde{q}$ is analytic at $z$.
Let us now assume that $z_{0}\in E_{0}$ is an isolated point of $E_{0},$ but not of the minimal set $K_{0}(f,\infty)$. From assertion ($\alpha$) at the end of the proof of Theorem \[t41a\], we know that in a neighborhood of $z_{0}$ the set $K_{0}(f,\infty)\setminus\{z_{0}\}$ consists only of trajectories of the quadratic differential $\widetilde{q}(z)dz^{2}$. From the history of the definition of the set $\widetilde{K}$ before (\[f103b6\]), it then follows that only a finite number, let say $k_{0}\in\mathbb{N}$, of these Jordan arcs $J_{j}$, $j\in I$, in $K_{0}(f,\infty)\setminus E_{0}$ have $z_{0}$ as their endpoint. From this observation and the local structure of quadratic differentials, which has been reviewed in Lemma \[l115a\] in Subsection \[s1105\], further below, we conclude that near the point $z_{0}$ the function $\widetilde{q}$ of (\[f103b8\]) has the local development $$\widetilde{q}(z)=q_{0}(z-z_{0})^{k_{0}-2}+\ldots,\text{ \ \ }q_{0}\neq0,\label{f104a}$$ which shows that the function $\widetilde{q}=q$ is meromorphic at each isolated point of $E_{0}$, and poles are at most of order $1$.
### \[s1043\]Proof of Theorem \[t53a\]
In Theorem \[t53a\] the special case has been considered that the set $E_{0}$ of Theorem \[t41a\] is finite, which leads to a rational quadratic differential $q(z)dz^{2}$ for the determination of the Jordan arcs $J_{j}$, $j\in I$, in $K_{0}(f,\infty)\setminus E_{0}$ in the sense of Theorem \[t52a\]. Algebraic functions provide typical examples for this situation.
In Proposition \[p53a\] it has been stated that with the set $E_{0}$ also the two sets $E_{1}$ and $E_{2}$ in Theorem \[t41a\] and in Definition \[d53b\], respectively, are finite.
\[Proof of Proposition \[p53a\]\]If $E_{0}$ is finite, then all points of $E_{0}$ are isolated, and we know from Theorem \[t52a\] that the function $q$ from (\[f52b\]) is meromorphic throughout $\overline{\mathbb{C}}$, which implies that $q$ is a rational function. Together with (\[f52c\]) we then further know that its numerator degree is by 2 degrees smaller than its denominator degree. Let $m$ and $n$ denote the numerator and denominator degrees, respectively. Since it has been shown in Theorem \[t52a\] that $q$ has a most simple poles, which are all contained in $E_{0}$, it follows that $$m+2=n\leq\operatorname*{card}(E_{0}).\label{f104b1}$$ From the local structure of the trajectories of quadratic differentials, which has been reviewed in Lemma \[l115a\] in Subsection \[s1105\], further below, we know that at each bifurcation point $z$ of$\ K_{0}(f,\infty
)\setminus E_{0}$, the function $q$ has a zero of order $$\operatorname*{ord}(z)=\operatorname*{i}(z)-2\label{f104b2}$$ with $\operatorname*{ord}(z)$ denoting the order of the zero at $z$ and $\operatorname*{i}(z)$ denoting the bifurcation index of Definition \[d53a\]. Since $E_{1}$ is the set of all bifurcation points of $K_{0}(f,\infty)$, it follows from (\[f104b1\]) and (\[f104b2\]) that the set $E_{1}$ is finite.
From the definition of critical points of a Green function in Definition \[d53b\], it follows rather immediately that at such points several level lines of the Green function intersect. Consequently, the function $q$ has a zero at each critical point $z$ of the Green function $g_{D_{0}(f,\infty
)}(\cdot,\infty)$, but this also follows directly from (\[f52b\]), and more precisely, we have $$\operatorname*{ord}(z)=2\,\operatorname*{j}(z),\label{f104b3}$$ where $\operatorname*{ord}(z)$ denotes again the order of the zero at $z$, and $j(z)$ is the degree of the critical point introduced in Definition \[d53b\]. From (\[f104b1\]) and (\[f104b3\]), it then follows that the set $E_{2} $ is finite.
\[Proof of Theorem \[t53a\]\]Most work for the proof of Theorem \[t53a\] has already been done in the proof of Proposition \[p53a\]. From there we know that $q$ is a rational function. All zeros and poles of the function $q$ on the minimal set $K_{0}(f,\infty)$ are contained in $E_{0}\cup E_{1}$. The poles are at most of order $1$, and they appear at a point $z\in E_{0}$ if $z$ is endpoint of only one Jordan arc in $K_{0}(f,\infty)\setminus(E_{0}\cup
E_{1})$. For such points $z$, we have the bifurcation index $\operatorname*{i}(z)=1$. After putting the information from (\[f104b1\]) and (\[f104b2\]) together, we get the first product in (\[f53b\]).
The second product in (\[f53b\]) follows from (\[f104b3\]) and the observation that in the domain $D_{0}(f,\infty)$ the function $q$ is analytic and has zeros only at the critical points of the Green function $g_{D_{0}(f,\infty)}(\cdot,\infty)$ and at $\infty$. The order of these zeros is given by (\[f104b3\]) and (\[f52c\]).
\[s105\]Proofs of Results from Section \[s7\]
---------------------------------------------
The main results of Section \[s7\] are contained in Theorem \[t73a\], where a characterization of the extremal domain $D_{0}(f,\infty)$ has been given with the help of the $S-$property, and in Theorem \[t74a\], where several geometric estimates have been formulated with the help of convexity.
### \[s1051\]Proof of Theorem \[t73a\]
Most of the work for the proof of Theorem \[t73a\] has already been done by Proposition \[p102a\].
\[Proof of Theorem \[t73a\]\]Let $D\in\mathcal{D}(f,\infty)$ be an admissible domain that possesses the $S-$property in the sense of Definition \[d71a\]. We assume that $D$ is different from the extremal domain $D_{0}(f,\infty)$, i.e., $$D\neq D_{0}(f,\infty).\label{f105a1}$$
It is an immediate consequence of Theorem \[t41a\] that the extremal domain $D_{0}(f,\allowbreak\infty)$ is elementarily maximal in the sense of Definition \[d71a0\]. From Proposition \[p102a\] together with assumption (\[f105a1\]), we then conclude that $$\operatorname{cap}(\partial D)<\operatorname{cap}(\partial D_{0}(f,\infty)).\label{f105a2}$$ But the last inequality contradicts the minimality (\[f21a\]) in Definition \[d21b\], which proves Theorem \[t73a\].
### \[s1052\]Proof of Theorem \[t74a\]
All results in Theorem \[t74a\] are basically a consequence of the fact that the capacity is a monotonically decreasing functions under orthogonal projections, a result which has been reviewed in Lemma \[l111c\] in Subsection \[s1101\], further below.
\[Proof of Theorem \[t74a\]\]Let $$L=L_{z_{0},v}:=\{\text{\thinspace}z_{0}+v\,t\text{\ }|\text{\ }t\in
\mathbb{R}\text{\thinspace}\}\text{, \ \ }z_{0},v\in\mathbb{C}\text{,
\ }|v|=1\text{,}\label{f105b1}$$ be an arbitrary line in $\mathbb{C}$, and denote by $H_{\pm}=H_{\pm}^{L}$ the two complementary half-planes of $L$, i.e., $\mathbb{C}\setminus L=H_{+}\cup
H_{-}=H_{+}^{L}\cup H_{-}^{L}$, and $$H_{\pm}^{L}:=\{\text{ }z\in\mathbb{C}\text{\ }|\text{\ }\pm\operatorname{Im}(\overline{v}(z-z_{0}))>0\text{\ }\}\label{f105b2}$$ with $z_{0}$ and $v$ the same parameters as those used in (\[f105b1\]).
By $\varphi_{L}:\mathbb{C}\longrightarrow\mathbb{C}$ we denote the orthogonal projection (\[f111d3\]) on $L$ out of $H_{+}$. On $L\cup H_{-}$, $\varphi_{L}$ is the identity.
Before we come to the individual proofs of the five assertions of Theorem \[t74a\], we assemble and prove several preparatory assertions, in which $E_{0}$ is the compact set introduced in Theorem \[t41a\].
- Let $D$ be an admissible domain for Problem $(f,\infty)$, i.e., $D\in\mathcal{D}(f,\infty)$, and set $K:=\overline{\mathbb{C}}\setminus D$. If the line $L$ is such that $K\subset L\cup H_{-}^{L}$, $K\cap L\neq\emptyset$, and that the function $f$ has a meromorphic continuation out of $H_{+}^{L}$ into a neighborhood of each $z\in K\cap L$, then for every line $\widetilde
{L}$, which is parallel to $L$, and for which $f$ has a meromorphic continuation throughout $H_{+}^{\widetilde{L}}$, the compact set $\widetilde{K}:=\varphi_{\widetilde{L}}(K)$ and the domain $\widetilde
{D}:=\overline{\mathbb{C}}\setminus\widetilde{K}$ are admissible for Problem $(f,\infty)$, i.e., we have $$\widetilde{K}=\varphi_{\widetilde{L}}(K)\in\mathcal{K}(f,\infty)\text{ \ \ and
\ \ }\widetilde{D}=\overline{\mathbb{C}}\setminus\widetilde{K}\in
\mathcal{D}(f,\infty).\label{f105b3}$$ By $\varphi_{\widetilde{L}}$ we have denoted the orthogonal projection associated with $\widetilde{L}$ in accordance to (\[f111d3\]).
Indeed, assertion (a) and especially (\[f105b3\]) follows directly from Proposition \[p91a\] together with the specific properties of the orthogonal projection (\[f111d3\]) and the assumptions made in assertion (a).
- If the situation of assertion (a) is given, and if we have $H_{+}^{L}\subsetneq H_{+}^{\widetilde{L}}$, then it follows that $$\operatorname{cap}(\widetilde{K})\leq\operatorname{cap}(K).\label{f105b4}$$ and a strict inequality holds in (\[f105b4\]) if, and only if, $$\operatorname{cap}(K\cap H_{+}^{\widetilde{L}})>0.\label{f105b5}$$
Indeed, assertion (b) follows directly from Lemma \[l111c\]. For the necessity of condition (\[f105b5\]) one also needs Lemma \[l111b\].
- If the line $L$ is such that $$E_{0}\subset L\cup H_{-}^{L},\label{f105c1}$$ then we have $$K_{0}(f,\infty)\subset L\cup H_{-}^{L}.\label{f105c2}$$
Indeed, if (\[f105c1\]) holds true, but (\[f105c2\]) is false, then we concluded from Theorem \[t41a\] that $K_{0}(f,\infty)\cap H_{+}^{L}$ contains at least an open piece of on of the Jordan arcs $J_{j}$, $j\in I$, from (\[f41a\]) in Theorem \[t41a\]. With Lemma \[l111a\] in Subsection \[s1101\], further below, we then deduce that $$\operatorname{cap}(K_{0}(f,\infty)\cap H_{+}^{L})>0,\label{f105c3}$$ which together with the two assertions (a), (b), and assumption (\[f105c1\]) further implies that there exists a line $\widetilde{L}$, like that in assertion (a), such that $$\varphi_{\widetilde{L}}(K_{0}(f,\infty))\in\mathcal{K}(f,\infty
),\label{f105c5}$$ and, as in assertion (b), we further have $$\operatorname{cap}(\varphi_{\widetilde{L}}(K_{0}(f,\infty
)))<\operatorname{cap}(K_{0}(f,\infty)).\label{f105c6}$$ Inequality (\[f105c6\]) together with (\[f105c5\]) contradicts the minimality (\[f21a\]) of the set $K_{0}(f,\infty)$ in Definition \[d21b\], which completes the proof of assertion (c).
- Let $\operatorname{Ex}(K)\subset K$ denote the set of extreme points of a compact set $K\subset\mathbb{C}$. We have $$\operatorname{Ex}(K_{0}(f,\infty))\subset E_{0}.\label{f105d1}$$
Indeed, for each $z\in\operatorname{Ex}(K_{0}(f,\infty))$ there exists a straight line $L$ such that $L\cap K_{0}(f,\infty)=\{z\}$ and $K_{0}(f,\infty)\subset L\cup H_{-}^{L}$. From assertion (a) and condition (iii) in Definition \[d21b\], we then conclude that any meromorphic continuation of the function $f$ out of $H_{+}^{L}$ has a non-polar singularity at $z$. From the last conclusion we deduce (\[f105d1\]), but also the slightly stronger assertion (e), which is formulated next.
- Let $L$ be a straight line with the property that $K_{0}(f,\infty)\subset L\cup H_{-}^{L}$. If $K_{0}(f,\infty)\cap L\neq\emptyset$, then we also have $\operatorname{Ex}(K_{0}(f,\infty))\cap L\neq\emptyset$, and at each point $z\in\operatorname{Ex}(K_{0}(f,\infty))\cap L$ the restriction of the function $f$ to $H_{+}^{L}$ has a non-polar singularity.
For the proof of the assertions (iv) and (v) in Theorem \[t74a\] we need a refinement of assertion (e).
- Let $L$ be a straight line with the property that $\operatorname{cap}(K_{0}(f,\infty)\cap H_{+}^{L})=0$. Then the function $f$ is single-valued in $H_{+}^{L}\setminus K_{0}(f,\infty)$. If further $K_{0}(f,\infty)\cap L\neq\emptyset$, then we also have $\operatorname{Ex}(K_{0}(f,\infty)\setminus H_{+}^{L})\cap L\neq\emptyset$, and the restriction of the function $f$ to $H_{+}^{L}\setminus K_{0}(f,\infty)$ has a non-polar singularity at each point $z\in\operatorname{Ex}(K_{0}(f,\infty)\setminus
H_{+}^{L})\cap L$.
Indeed, the first part of assertion (f) follows directly from the fact that $H_{+}^{L}\setminus K_{0}(f,\infty)\subset D_{0}(f,\infty)$.
From the assumption that $\operatorname{cap}(K_{0}(f,\infty)\cap H_{+}^{L})=0$ together with Lemma \[l111a\] from Subsection \[s1101\], further below, we conclude that the set $K_{0}(f,\infty)\cap H_{+}^{L}$ contains no continuum, it is totally disconnected, and consequently, we have $K_{0}(f,\infty)\cap
H_{+}^{L}\subset E_{0}$ as a consequence of condition (iii) in Definition \[d21b\].
If $K_{0}(f,\infty)\cap L\neq\emptyset$, then it is immediate that we also have $\operatorname{Ex}(K_{0}(f,\infty)\setminus H_{+}^{L})\cap L\neq
\emptyset$. With the first part of assertion (f), we then conclude in the same way as in the proof of assertion (e) that the meromorphic continuation of the function $f$ out of the domain $H_{+}^{L}\setminus K_{0}(f,\infty)$ has a non-polar singularity at every $z\in\operatorname{Ex}(K_{0}(f,\infty)\setminus
H_{+}^{L})\cap L$.
We now come to the individual proofs of the five assertions in Theorem \[t74a\], where we will the assertions (a) - (f).
\(ii) Assertion (ii) is an immediate consequence of assertion (c).
\(i) We prove assertion (i) indirectly. Let us assume that there exists a convex and compact set $K\subset\mathbb{C}$ such that the function $f$ has a single-valued meromorphic continuation throughout $\overline{\mathbb{C}}\setminus K$ and that further $$K_{0}(f,\infty)\setminus K\neq\emptyset.\label{f105d2}$$ From (\[f105d2\]) and the convexity of $K$ it follows that also $$\operatorname{Ex}(K_{0}(f,\infty))\setminus K\neq\emptyset.\label{f105d3}$$ With assertion (e) we then conclude that the meromorphic continuation of the function $f$ out of $\overline{\mathbb{C}}\setminus(K_{0}(f,\infty)\cup
K)\subset D_{0}(f,\infty)$ has a non-polar singularity at each $z\in
\operatorname{Ex}(K_{0}(f,\infty))\setminus K$, which contradicts the assumption that the function $f$ is meromorphic throughout $\overline
{\mathbb{C}}\setminus K$. Hence, assertion (i) is proved.
\(iii) Assertion (iii) can be proved like assertion (i), only that now we have to use assertion (f) instead of assertion (e). We give more details since some of the conclusions will also be used in the proof of assertion (v), further below.
Assertion (iii) will be proved indirectly. We assume that $K\subset\mathbb{C}$ is a convex and compact set, $E\subset\mathbb{C}\setminus K$ a set that is relatively compact in $\mathbb{C}\setminus K$, $\operatorname{cap}(E)=0$, the function $f$ has a meromorphic and single-valued continuation throughout $\overline{\mathbb{C}}\setminus(K\cup E)$, and $$(K_{0}(f,\infty)\setminus E)\setminus K=K_{0}(f,\infty)\setminus(K\cup
E)\neq\emptyset.\label{f105d4}$$ From (\[f105d4\]) it follows that also $$\operatorname{Ex}(K_{0}(f,\infty)\setminus(K\cup E))\setminus(K\cup
E)\neq\emptyset.\label{f105d5}$$ Since $f$ is single-valued and meromorphic throughout $D_{0}(f,\infty
)=\overline{\mathbb{C}}\setminus K_{0}(f,\infty)$, it follows from assertion (f) that $f$ has to have a non-polar singularity at each $z\in
\operatorname{Ex}(K_{0}(f,\infty)\setminus(K\cup E))\setminus(K\cup E)$, which contradicts the assumption that $f$ is meromorphic throughout $\overline
{\mathbb{C}}\setminus(K\cup E)$.
\(iv) Let $K_{min}$ be the intersection of all compact sets $K\subset
\mathbb{C}$ that satisfy the assumptions made in assertion (iii). With the usual tools of planar topology one can show that $K_{min}$ can also be represented as a denumerable intersection of such sets $K$. Like these sets $K$, so the set $K_{min}$ is also convex and compact, and further it follows from the assumptions made in assertion (iii) together with Lemma \[l111b\] in Subsection \[s1101\] that $$\operatorname{cap}(K_{0}(f,\infty)\setminus K_{min})=0.\label{f105d6}$$ We define $E_{min}:=K_{0}(f,\infty)\setminus K_{min}$, and with this definition, assertion (iv) is proved.
\(v) Notice that in assertion (iii) we can choose $K=K_{min}$. With the same argumentation as used after (\[f105d5\]), we show that the meromorphic continuation of the function $f$ out of $D_{0}(f,\infty)\setminus K_{min}$ has a non-polar singularity at each $z\in\operatorname{Ex}(K_{0}(f,\infty)\cap
K_{min})$. Form the definition of $K_{min}$ as the intersection of all compact sets $K\subset\mathbb{C}$ that satisfy the assumptions made in assertion (iii), it follows that $K_{0}(f,\infty)\subset K_{min}$. Hence, we have proved that $\operatorname{Ex}(K_{min})\subset E_{0}$.
From $\operatorname{cap}(E_{min})=0$ together with Lemma \[l111a\] from Subsection \[s1101\], it follows that the set $E_{min}$ is totally disconnected, and consequently, it follows from the Structure Theorem \[t41a\] that $E_{min}\subset E_{0}$, which completes the proof of assertion (v).
\[s110\]Some Lemmas from Potential Theory and Geometric Function Theory
=======================================================================
In the present section we assemble definitions and lemmas concerning basic properties and tools from potential theory and from geometric function theory. These tools have been used at several places in the sections above. It is hoped that by its concentration in a separate section the flow of argumentation at earlier places has not been interrupted by argumentations of a rather different flavor, or by references to the literature together with the often necessary reformulations and adaptations of results. A separate compilation is also more convenient and economic with respect to a unified terminology, which unfortunately is not typical for the whole spectrum of the literature in this area. As general references to potential-theoretic results we have used [@SaTo], [@Ransford], and sometimes also [@Landkof]. Towards the end of the present section results become more specific, and some of them require rather technical proofs, which could not be found in the literature with the required specific orientation.
We start with topics related to the (logarithmic) capacity, continue then with logarithmic potentials, Green functions, some special results related to sequences of compact sets, and at last with some remarks on trajectories of quadratic differentials. In the penultimate subsection, Carathéodory’s Theorem about kernel convergence will be an important piece.
\[s1101\]Notations and Basic Properties of Capacity
---------------------------------------------------
The (logarithmic) capacity $\operatorname*{cap}\left( \cdot\right) $ is a set function defined on capacitable subsets of $\mathbb{C}$, which include Borel sets (cf. [@Ransford], Chapter 5 or [@SaTo], Chapter I.1). For a compact set $K\subset\mathbb{C}$ a definition with a strong intuitive flavor can be based on the principle of minimal energy: Let $\mu$ be a probability measure in $\mathbb{C}$; its energy is defined as $$I(\mu):=\int\int\log\frac{1}{|z-v|}d\mu(z)d\mu(v).\label{f111a1}$$ The capacity of the compact set $K\subset\mathbb{C}$ can then be defined as $$\operatorname*{cap}\left( K\right) :=\exp\left( -\inf_{\mu}I(\mu)\right)
,\label{f111a2}$$ where the infimum extends over all probability measures $\mu$ with $\operatorname*{supp}\left( \mu\right) \subset K$.
A special role is played by sets $E\subset\mathbb{C}$ of capacity zero, which are also known as polar sets. The property of being a set of capacity zero is invariant under Möbius transforms, and thanks to this property, the notion of ’capacity zero’ can be extended to the whole Riemann sphere $\overline
{\mathbb{C}}$.
\[d111a\]A property is said to hold quasi everywhere (written in short as ’qu.e.’) on a set $S\subset\overline{\mathbb{C}}$ if it holds for every $z\in
S\setminus E$ with $E$ a set of (outer) capacity zero (cf. [@SaTo], Chapter I.1).
The capacity is monoton with respect to an ordering by inclusions (cf. [@Ransford], Theorem 5.1.2). In our investigations we have needed some upper and lower estimates, which are formulated in the next lemma (cf. [@Ransford], Theorems 5.3.2, 5.3.4, and 5.3.5).
\[l111a\](i) Let $m(\cdot)$ denote the planar Lebesgue measure in $\mathbb{C}$. For any compact set $K\subset\mathbb{C}$ we have $$\sqrt{m(K)/\pi}\leq\operatorname*{cap}\left( K\right) \leq
\operatorname*{diam}(K)/2.\label{f111b1}$$ (ii) For a continuum $V\subset\mathbb{C}$ we have $$\operatorname*{diam}(K)/4\leq\operatorname*{cap}\left( V\right)
\leq\operatorname*{diam}(K)/2.\label{f111b2}$$
As a set function, the capacity is not additive, and does also not possess one of the usual subadditivity properties as a weaker substitute for the failing additivity. However, sets of capacity zero are an exception in this respect (cf. [@Ransford], Theorem 5.1.4).
\[l111b\]Let $K,E\subset\mathbb{C}$ be capacitable sets, and assume that $\operatorname*{cap}\left( E\right) =0$. Then we have $$\operatorname*{cap}\left( K\setminus E\right) =\operatorname*{cap}\left(
K\cup E\right) =\operatorname*{cap}\left( K\right) .\label{f111c1}$$ The denumerable union of capacitable sets of capacity zero is again a set of capacity zero.
Another property, which has been relevant in our investigations, concerns radial projections $\varphi_{r}:\mathbb{C}\longrightarrow\mathbb{C}$ onto a given disk $D:=\left\{ \text{ }|z|\leq r\text{ }\right\} $, $r>0$, and orthogonal projections on a line $L:=\left\{ \,z_{0}+v\,t\,|\,z_{0},v\in\mathbb{C},\text{ }|v|=1,\text{ }t\in\mathbb{R}\,\right\} $ from one side. Let the radial projection $\varphi_{r}$ be defined by $$z\mapsto\varphi_{r}(z):=\min(r,|z|)\frac{z}{|z|},\label{f111d1}$$ and the orthogonal projection $\varphi_{L}$ be defined by $$z\mapsto\varphi_{L}(z):=\left\{
\begin{array}
[c]{lll}z_{0}+\operatorname{Re}((z-z_{0})\overline{v})v\smallskip & \text{ \ \ \ if
\ \ \ } & \operatorname{Im}((z-z_{0})\overline{v})>0,\\
z & \text{ \ \ else. \ } &
\end{array}
\right. \label{f111d3}$$
\[l111c\](cf. [@Pommerenke92], Chapter 9.3, formula (11)) For any capacitable set $K\subset\mathbb{C}$ and radial projection $\varphi_{r}$ defined in (\[f111d1\]), we have $$\operatorname*{cap}\left( \varphi_{r}(K)\right) \leq\operatorname*{cap}\left( K\right) ,\label{f111d2}$$ and for the orthogonal projection $\varphi_{L}$ defined in (\[f111d3\]), we also have $$\operatorname*{cap}\left( \varphi_{L}(K)\right) \leq\operatorname*{cap}\left( K\right) .\label{f111d4}$$ We have a strict inequality in (\[f111d2\]) or (\[f111d4\]) if $\operatorname*{cap}(K\setminus\allowbreak\varphi_{r}\allowbreak(K))>0$ or $\operatorname*{cap}(K\setminus\allowbreak\varphi_{L}(K))>0$, respectively.
The capacity of a set depends only on the outer boundary of this set, which will become clear from the next definition and the follow-on lemma.
\[d111b\]For a bounded set $S\subset\mathbb{C}$ the polynomial-convex hull $\widehat{S}$ (also denoted by $\operatorname*{Pc}(S)$) is defined as the union of $\overline{S}$ with all bounded components of $\overline{\mathbb{C}}\setminus\overline{S}$. The set $\partial\widehat{S}$ is call the outer boundary, and $\Omega_{S}=\overline{\mathbb{C}}\setminus\widehat{S}$ the outer domain of $S$. A compact set $K\subset\mathbb{C}$ is call polynomial-convex if $K=\widehat{K}$.
The notion ’polynomial-convex hull’ hints to the possibility to define this hull by polynomial inequalities. We have $$\widehat{S}=\{\text{ }z\in\mathbb{C}\text{\ }|\text{ }|p(z)|\leq
||p||_{S}\text{\ for all }p\in\mathcal{P}\text{\ }\}\label{f111e1}$$ where $\mathcal{P}$ denotes the set of all polynomials and $||\cdot||_{S}$ the uniform norm on $S$.
\[l111d\](cf. [@Ransford], Theorem 5.1.2) For all compact sets $K\subset\mathbb{C}$ we have $$\operatorname*{cap}(K)=\operatorname*{cap}(\widehat{K}).\label{f111e2}$$
A special property of polynomial-convex sets is the fact that their complement is always a domain. The next lemma addresses a similar topic, but under different circumstances.
\[l111e\]Let $S\subset\mathbb{C}$ be a set of capacity zero and $D\subset\overline{\mathbb{C}}$ a domain, then $D\setminus S$ is connected. If in addition $S$ is assumed to be closed in $D$, then $D\setminus S$ is a domain.
The lemma has a certain degree of immediate evidence since sets of capacity zero are totally disconnected. However, a formal proof as to take care of the topological difficulties in one or the other way. We will use the tools provided by Lemma \[l91c\] in Subsection \[s91\], further above.
The assertion that $D\setminus S$ is connected will be proved indirectly, and for this purpose we assume that the opposite holds true. Then there exist two disjoint open sets $O_{1},O_{2}\subset\overline{\mathbb{C}}$ with $D\setminus
S\subset O_{1}\cup O_{2}$ and $O_{j}\cap(D\setminus S)\neq\emptyset$ for $j=1,2$. The set $\widetilde{K}:=D\setminus(O_{1}\cup O_{2}) $ is closed in $D$ and we have $\widetilde{K}\subset S$.
Let $z_{j}\in O_{j}\cap D$, $j=1,2$, be two points, and $\gamma_{0}$ a Jordan arc connecting $z_{1}$ with $z_{2}$ in $D$, and let further $U\subset D$ be a small, open, simply-connected neighborhood of $\gamma_{0}$ with $\overline
{U}\subset D$. The arc $\gamma_{0}$ can be extended to a closed Jordan curve $\gamma_{1}$ in $\mathbb{C}$, and correspondingly $U$ can be extended to a ring domain $R\subset\mathbb{C}$ with $\gamma_{1}\subset R$ separating the two components $A_{1}$ and $A_{2}$ of $\overline{\mathbb{C}}\setminus R$. This extension can be done in such a way that $R\cap\widetilde{K}=U\cap
\widetilde{K}$.
It is immediate that each Jordan curve $\gamma\subset R$ that separates $A_{1}$ from $A_{2}$ has to intersect the compact set $K:=R\cap\widetilde{K}$, for otherwise the two sets $O_{1}\cap D$ and $O_{2}\cap D$ would be connected.
After these preparations, we apply the tools offered in Lemma \[l91c\], which then shows that there exists a continuum $V\subset K$ which is not reduced to a single point, and consequently we have $$\operatorname{cap}(S)\geq\operatorname{cap}(\widetilde{K})\geq
\operatorname{cap}(V)>0\text{,}\label{f111f1}$$ which contradicts the assumption that $\operatorname{cap}(S)$.
If $S$ is closed in $D$, then $D\setminus S$ is open, and consequently it is a domain.
\[s1102\]Logarithmic Potentials
-------------------------------
Let $\mu$ be a (Borel) measure with compact $\operatorname*{supp}\left( \mu\right) \subset\mathbb{C}$. The (logarithmic) potential of the measure $\mu$ is defined as $$p(\mu;z):=\int\log\frac{1}{|z-x|}d\mu(x).\label{f112a1}$$ It is a superharmonic function in $\mathbb{C}$, and it is continuous quasi everywhere in $\mathbb{C}$ for every measure $\mu$ (cf. [@Landkof], Chapter III, Theorem 3.6). In the fine topology it is even continuous throughout $\mathbb{C}$, but in our investigations, the concept of fine topology has not been used. We shall address subtle questions about continuity only in connection with the Green function further below in Subsection \[s1103\].
Let $\{\mu_{n}\}_{n\in\mathbb{N}}$ be a weakly convergent sequence of measures with limit measure $\mu_{0}$; this is written as $$\mu_{n}\overset{\ast}{\longrightarrow}\mu_{0}\text{ \ \ as \ \ }n\rightarrow\infty.\label{f112b1}$$ With the convergence (\[f112b1\]) corresponds a specific asymptotic behavior of the potentials $p(\mu_{n};\cdot)$, $n\in\mathbb{N}$, (cf. [@SaTo], Chapter I.6.9), which is known as the Lower Envelope Theorem.
\[Lower Envelope Theorem\]\[t112a\]If $\sup(\mu_{n})\subset K$ for all $n\in\mathbb{N}$ with $K\subset\mathbb{C}$ compact, then from (\[f112b1\]) it follows that $$\liminf_{n\rightarrow\infty}p(\mu_{n};z)\geq p(\mu_{0};z)\label{f112b2}$$ for all $z\in\mathbb{C}$, and equality holds in (\[f112b2\]) quasi everywhere in $\mathbb{C}$.
On a compact set $K\subset\mathbb{C}$ of positive capacity, there uniquely exists an equilibrium measure $\omega_{K}$ (cf. [@Ransford], Chapter 3.3), which is the probability measure on $K$ that minimizes the energy (\[f111a2\]). Its potential has a typical behavior on $K$ (cf. [@Ransford], Theorem 3.3.4), we have $$p(\omega_{K};z)\left\{
\begin{array}
[c]{lll}=-\log\operatorname*{cap}\left( K\right) \smallskip & \text{ \ for quasi
every} & z\in\widehat{K}\\
>-\log\operatorname*{cap}\left( K\right) & \text{ \ for all \ } & z\in
\Omega_{K},
\end{array}
\right. \label{f112c1}$$ where $\Omega_{K}=\overline{\mathbb{C}}\setminus\widehat{K}$ is the outer domain and $\widehat{K}$ its polynomial-convex hull of $K$. Both objects have been introduced in Definition \[d111b\].$\smallskip$
In potential theory a special role is played by measures of finite energy, i.e., measures $\mu$ with $I(\mu)<\infty$ and $I(\cdot)$ defined by (\[f111a1\]). For instance, we have the following result ([@Ransford], Theorem 3.2.3).
\[l112a\]For any measure $\mu$ of finite energy and any bounded measurable set $S\subset\mathbb{C}$ with $\operatorname*{cap}\left( S\right) =0$, we have $\mu\left( S\right) =0$.
The equilibrium measure $\omega_{K}$ of a compact set $K\subset\mathbb{C}$ with $\operatorname*{cap}\left( K\right) >0$ is of finite energy.
In potential theory, a number of basic properties are known as principles; a first one has already been stated in Theorem \[t112a\]. In our investigations we have also needed the next one.
\[Principle of Domination\]\[t112b\]Let $\mu_{1}$ and $\mu_{2}$ be two (positive) measures with compact support in $\mathbb{C}$, let $\mu_{1}$ be of finite energy, and let $c\in\mathbb{R}$ be a constant. If the inequality $$p(\mu_{1};z)\leq p(\mu_{2};z)+c\label{f112g1}$$ holds true for $\mu_{1}$-almost every $z\in\mathbb{C}$, or if it holds true for quasi every $z\in\operatorname*{supp}(\mu_{1})$, then inequality (\[f112g1\]) holds true for all $z\in\mathbb{C}$.
The theorem has been proved in [@SaTo], Theorem II.3.2, under the assumption that (\[f112g1\]) is satisfied $\mu_{1}$-almost everywhere.
If (\[f112g1\]) holds true for quasi every $z\in\operatorname*{supp}(\mu
_{1})$, then it follows from Lemma \[l112b\] and from the assumption that $\mu_{1}$ is of finite energy that inequality (\[f112g1\]) holds true also $\mu_{1}$-almost everywhere.
The minimum of two potentials can again be represented by a logarithmic potential.
\[l112c\]Let $\mu_{1}$ and $\mu_{2}$ be two (positive) measures, then there exists a (positive) measure $\mu_{0}$ and a constant $r_{0}\in\mathbb{R}
$ such that $$\min\left( p(\mu_{1};\cdot),p(\mu_{2};\cdot)\right) =r_{0}+p(\mu_{0};\cdot)\label{f112e1}$$ with $\left\| \mu_{0}\right\| =\max(\left\| \mu_{1}\right\| ,\left\|
\mu_{2}\right\| )$. If the two measures $\mu_{1}$ and $\mu_{2}$ are of finite energy, then the same is true for $\mu_{0}$.
It is rather immediate that the minimum of two superharmonic functions is again superharmonic. One has only to check the definition of superharmonicity. The lemma then follows from the Poisson-Jensen Formula ( [@Ransford], Theorem 4.5.1). The determination of $\left\Vert \mu_{0}\right\Vert $ follows from a consideration of $\min(p(\mu_{1};\cdot),\,p(\mu_{2};\allowbreak\cdot))$ near infinity.
A broad variety of manipulations is possible in the class of logarithmic potentials if one allows signed measures $\sigma$ in (\[f112a1\]).
A signed measure $\sigma$ is of finite energy, i.e., $I(\sigma)<\infty$, if and only if each of its two components $\sigma_{+}$ and $\sigma_{-}$ ($\sigma=\sigma_{+}-\sigma_{-}$, $\sigma_{+},\sigma_{-}\geq0$) is of finite energy.
In order to keep our notations simple, we speak of logarithmic potentials also if there is an additive constant, as for instance, is the case on the right-hand side of (\[f112e1\]).
\[l112d\]Let the two potentials $p_{j}$, $j=1,2$, be given by $$p_{j}=r_{j}+p(\sigma_{j};\cdot),\text{ \ \ \ }j=1,2,\label{f112f1}$$ with $r_{j}\in\mathbb{R}$ and $\sigma_{j}$, $j=1,2$, signed measures in $\mathbb{C}$. The functions $p_{3}:=|p_{1}|$, $p_{4}:=\max(p_{1},0)$, $p_{5}:=\min(p_{1},0)$, $p_{6}:=\max(p_{1},p_{2})$, and $p_{7}:=\min
(p_{1},p_{2})$ then have representations of the same form as in (\[f112f1\]) with modified constants $r_{j}\in\mathbb{R}$ and signed measures $\sigma_{j}$, $j=3,\ldots,7$. If the two measures $\sigma_{1}$ and $\sigma_{2}$ are of finite energy, then the same is true for the five measures $\sigma_{3},\ldots,\sigma_{7}$.
For the positive and negative components of the two measures $\sigma_{j}$, $j=1,2$, we write $\sigma_{j+}$ and $\sigma_{j-}$, respectively, i.e., we have $\sigma_{j}=\sigma_{j+}-\sigma_{j-}$.We consider the potentials $p_{j+}=r_{j}+p(\sigma_{j+};\cdot)$, $p_{j-}=p(\sigma_{j-};\cdot)$, $j=1,2 $. Since we have $$|p_{1}|=p_{1+}+p_{1-}-2\,\min(p_{1+},p_{1-}),\label{f112f2}$$ representation (\[f112f1\]) for $p_{3}$ follows directly from Lemma \[l112c\]. The representations for $p_{4},\ldots,p_{7}$ follow then as further consequences since we have $p_{4}=\frac{1}{2}(p_{1}+|p_{1}|)$, $p_{5}=\frac{1}{2}(p_{1}-|p_{1}|)$, $p_{6}=p_{1}+\max(p_{2}-p_{1},0)$, and $p_{7}=p_{1}+\min(p_{2}-p_{1},0)$. The conclusion about the finite energy of the measures $\sigma_{3},\ldots,\sigma_{7}$ follows from the corresponding conclusion in Lemma \[l112c\].
An important tool for the work with logarithmic potentials is the balayage technique (sweeping out of a measure) (cf. [@SaTo], Chapter II.4). In case of logarithmic potentials, the balayage out of an unbounded domain requires special attention.
\[d112a\]Let $\mu$ be a measure in $\mathbb{C}$.
- For a bounded domain $D\subset\mathbb{C}$ with $\operatorname*{cap}\left( \partial D\right) >0,$ by $\widehat{\mu}$ we denote the balayage measure resulting from sweeping the measure $\mu$ out of the domain $D$; it has its support on $\partial D\cup\operatorname*{supp}(\mu)\setminus D$, and it is defined by the relation $$p(\widehat{\mu};z)=p(\mu;z)\label{f112c2}$$ for every $z\in\overline{\mathbb{C}}\setminus\overline{D}$ and for quasi every $z\in\partial D$. The balayage measure $\widehat{\mu}$ is uniquely determined by (\[f112c2\]) if we assume in addition to (\[f112c2\]) that $\widehat{\mu}(Ir(\partial D))=0$, where $Ir(\partial D)$ is the set of critical points of $\partial D$ that will be introduced in Definition \[d113a\] in the next subsection.
- For an unbounded domain $D\subset\overline{\mathbb{C}}$ with $\infty\in D$ and $\operatorname*{cap}\left( \partial D\right) >0,$ the concept of balayage is the same as in (i) only that relation (\[f112c2\]) now has the modified form $$p(\widehat{\mu};z)=p(\mu;z)+c_{1}\label{f112c3}$$ with a constant $c_{1}>0$ given by $$c_{1}=\int g_{D}(x,\infty)d\mu(x)\label{f112c4}$$ where $g_{D}$ is the Green function in $D$, which will be introduced in the next subsection.
With the help of the balayage technique, we can introduce an additional method for manipulating logarithmic potentials which in some sense complements the methods considered in Lemma \[l112d\], and which has also been used further above.
\[l112e\]Let the two logarithmic potentials $p_{j}$, $j=1,2$, be given in the form (\[f112f1\]) with signed measures $\sigma_{1}$ and $\sigma_{2}$ that are of finite energy, and let further $D\subset\overline{\mathbb{C}} $ be a (possibly unbounded) open set with connected complement. We define a new function $p_{0}$ in a piecewise manner by $$p_{0}(z):=\left\{
\begin{array}
[c]{lcl}p_{1}(z) & \text{ \ for \ }\smallskip & z\in D,\\
p_{2}(z) & \text{ \ for \ } & z\in\overline{\mathbb{C}}\setminus D.
\end{array}
\right. \label{f112h1}$$ If we have $$p_{1}(z)=p_{2}(z)\text{ \ \ \ for quasi every \ \ }z\in\partial
D,\label{f112h2}$$ then there exists a signed measure $\sigma_{0}$ in $\mathbb{C}$ and a constant $r_{0}\in\mathbb{R}$ such that $$p_{0}(z)=r_{0}+p(\sigma_{0};z)\text{ \ \ for quasi every \ \ }z\in
\mathbb{C}.\label{f112h3}$$ The measure $\sigma_{0}$ is of finite energy, and in (\[f112h3\]) we have equality everywhere in $\overline{\mathbb{C}}\setminus\partial D$. Further we have $$\sigma_{0}|_{D}=\sigma_{1}|_{D}\text{ \ \ and \ \ }\sigma_{0}|_{\mathbb{C}\setminus\overline{D}}=\sigma_{2}|_{\mathbb{C}\setminus\overline{D}}.\label{f112h4}$$
The function $$d:=p_{1}-p_{2}=r_{1}-r_{2}+p(\sigma_{1}-\sigma_{2};\cdot)\label{f112h5}$$ has the form (\[f112f1\]) with defining measure $\sigma_{1}-\sigma_{2}$. If we apply the balayage technique to the measure $\sigma_{1}-\sigma_{2}$ and sweep this measure out of the domain $\overline{\mathbb{C}}\setminus
\overline{D}$, than this leads to a balayage measure in $\overline{D}$ which we denote by $\widehat{\sigma}_{12}$. With an appropriately chosen constant $r_{12}\in\mathbb{R}$, we have $$\begin{aligned}
& \operatorname*{supp}(\widehat{\sigma}_{12})\subset\overline{D}\text{,
\ \ \ }\widehat{\sigma}_{12}|_{D}=(\sigma_{1}-\sigma_{2})|_{D}\text{,}\smallskip\label{f112h6a}\\
& \text{ \ \ }d(z)=r_{12}+p(\widehat{\sigma}_{12};z)\text{ \ \ for quasi every
}\ z\in\partial D\text{,}\label{f112h6b}$$ and the inequality in (\[f112h6b\]) holds also for all $z\in D$. The two statements in (\[f112h6a\]) and (\[f112h6b\]) are a consequence of part (ii) in Definition \[d112a\]. We define $$\widehat{d}:=r_{12}+p(\widehat{\sigma}_{12};\cdot).\label{f112h7}$$
Since logarithmic potentials are continuous quasi everywhere in $\mathbb{C}$ (cf. [@Landkof], Chapter III, Theorem 3.6), it follows from (\[f112h2\]) and (\[f112h5\]) that $d(z)=0$ for quasi every $z\in\partial D$, and hence, we deduce from (\[f112h6b\]) that $$\widehat{d}(z)=0\text{ \ \ \ for quasi every }\ z\in\partial D,\label{f112h8}$$ and because of (\[f112h6a\]) further that $$\widehat{d}(z)=0\text{ \ \ \ for all }\ z\in\overline{\mathbb{C}}\setminus\overline{D}.\label{f112h9}$$ From (\[f112h1\]), (\[f112h6b\]), (\[f112h7\]), (\[f112h8\]), and (\[f112h9\]), it then follows that $$p_{0}(z)=p_{2}(z)+\widehat{d}(z)=r_{2}+r_{12}+p(\sigma_{2}+\widehat{\sigma
}_{12};z)\label{f112h10}$$ for all $z\in\overline{\mathbb{C}}\setminus\partial D$ and for quasi every $z\in\partial D$, which proves (\[f112h3\]) if we set $$\sigma_{0}:=\sigma_{2}+\widehat{\sigma}_{12}.\label{f112h11}$$ Since $\operatorname*{cap}(\overline{D})>0$ and since $\sigma_{1}-\sigma_{2}$ is of finite energy, it follows from (\[f112c3\]) and (\[f112c4\]) that the measure $\widehat{\sigma}_{12}$ is of finite energy, and consequently the same is true for $\sigma_{2}+\widehat{\sigma}_{12}$. The identities in (\[f112h4\]) follow from (\[f112h6a\]).
We close the present subsection with some estimates of the logarithmic energy (\[f111a1\]) associated with signed measures. It is important here that the logarithmic kernel in (\[f111a1\]) is positive definite for signed measures $\sigma$ with $\operatorname*{supp}(\sigma)\subset K\subset\mathbb{C}$ if $K$ is a compact set with $\operatorname*{cap}(K)\leq1$. In the next lemma estimates have been put together that are relevant in this connection.
\[l112b\](i) Let $K\subset\mathbb{C}$ be a compact set of positive capacity. For all signed measures $\sigma$ with $\operatorname*{supp}(\sigma)\subset K$ we have $$I(\sigma)\geq\sigma(K)^{2}\log\frac{1}{\operatorname*{cap}(K)}.\label{f112d1}$$
\(ii) Let $\sigma$ be a signed measure in $\mathbb{C}$ with $$\sigma(\mathbb{C})=0,\label{f112d2}$$ and let either $\operatorname*{supp}(\sigma)$ be a compact set or let $\sigma$ be a signed measure of finite energy, then we have $$I(\sigma)\geq0,\label{f112d3}$$ and equality holds in (\[f112d3\]) if, and only if, $\sigma=0$.
Part (ii) of the lemma has been proved in [@Landkof], Theorem 1.6.
In a first step of the proof of part (i), we assume the set $K$ is regular (cf. Definition \[d113a\], below). We set $a:=\sigma(K)$, and define $$\sigma_{0}:=\sigma-a\,\omega_{K}\label{f112d4}$$ with $\omega_{K}$ the equilibrium measure on $K$. Consequently, we have $\sigma_{0}(\mathbb{C})=0$. From (\[f111a1\]) it follows that $$I(\sigma)=I(\sigma_{0})+a^{2}I(\omega_{K})+2a\,I(\sigma_{0},\omega
_{K}),\label{f112d5}$$ where $$I(\sigma_{0},\omega_{K})=\int\int\log\frac{1}{|z-v|}d\sigma_{0}(z)d\omega
_{K}(v)\label{f112d6}$$ is the mutual energy of the two measures $\sigma_{0}$ and $\omega_{K}$, which in case of the equilibrium distribution $\omega_{K}$ can be expressed as $$I(\sigma_{0},\omega_{K})=\int\left[ -g_{\Omega}(z,\infty)-\log
\operatorname*{cap}\left( K\right) \right] d\sigma_{0}(z)\label{f112d7}$$ with the help of Lemma \[l113b\], below. In (\[f112d7\]), $\Omega$ is the outer domain of $K$.
From the assumption that $K$ is regular, it follows that $g_{\Omega}(z,\infty)=0$ for all $z\in K$ (cf. the properties stated in (\[f113a1\]), further below). From $\sigma_{0}(\mathbb{C})=0$ and (\[f112d7\]), it then follows that $I(\sigma_{0},\omega_{K})=0$. From part (ii) we know that $I(\sigma_{0})\geq0$, which together with (\[f112d5\]) and (\[f111a2\]) proves (\[f112d1\]).
If the compact set $K\subset\mathbb{C}$ is not regular, then it can be approximated from the outside by open sets (cf. [@Ransford], Theorem 5.1.3). This implies that for any $\varepsilon>0$ there exists a compact set $\widetilde{K}\subset\mathbb{C}$ with $\partial\widetilde{K}$ consisting of piece-wise analytic arcs, $K\subset\operatorname*{Int}(\widetilde{K})$, and $\operatorname*{cap}(\widetilde{K})\leq\operatorname*{cap}(K)+\varepsilon$. Since $\varepsilon>0$ is arbitrary, (\[f112d1\]) holds also true in the non-regular case.$\medskip$
\[s1103\]The Green Function
---------------------------
By $g_{D}(\cdot,w)$ we denote the Green function in a domain $D\subset\overline{\mathbb{C}}$ with logarithmic singularity at $w\in D$ (for a definition see [@Ransford], Chapter 4.4, or [@SaTo], Chapter I.4). Somewhat different from the usual definitions, we assume that the Green function $g_{D}(\cdot,w)$ is defined throughout $\overline{\mathbb{C}} $ and also for domains $D\subset\overline{\mathbb{C}}$ with $\operatorname*{cap}\left( \partial D\right) =0$. If the domain $D$ has a boundary $\partial D$ of positive capacity, then for $w\in D$ we have $$g_{D}(z,w)\left\{
\begin{array}
[c]{lll}=0\smallskip & \text{ \ for quasi every} & z\in\partial D\\
>0\smallskip & \text{ \ for all \ } & z\in D\\
=0 & \text{ \ for all \ } & z\in\overline{\mathbb{C}}\setminus\overline{D}.
\end{array}
\right. \label{f113a1}$$ If $\operatorname*{cap}\left( \partial D\right) =0$ and $w\in D$, then we define $g_{D}(\cdot,w)\equiv\infty$.$\medskip$
Irregular points of $\partial D$ with respect to solutions of Dirichlet problems in the domain $D\subset\overline{\mathbb{C}}$ have required special attention at several places in our investigations. Irregular points are indeed an interesting topic in potential theory. This type of points can be defined in many different ways; one of the possibilities is based on the behavior of the Green function $g_{D}(\cdot,w)$ on $\partial D$ (cf. [@Ransford], Chapter 4.2). We use this approach in the next definition.
\[d113a\]A point $z\in\partial D$ is irregular with respect to Dirichlet problems in the domain $D$ (or short: it is an irregular point of $\partial
D$) if $g_{D}(z,w)>0$ for some $w\in D$. The set of all irregular points of $\partial D$ is denoted by $Ir(\partial D)$.
It follows from the existence of the Riemann mapping function (see also [@Ransford] Theorem 4.2.1) that if $D\subset\overline{\mathbb{C}}$ is a simply connected domain and $\partial D$ is not reduced to a single point, then $Ir(\partial D)=\emptyset$.
Often we have had to deal with the outer domains $\Omega_{K}$ of a compact set $K\subset\mathbb{C}$; the irregular points of $\partial\Omega_{K}$ are elements of $K$. In the next definition we repeat certain aspects of Definition \[d113a\], but with a refined and a partially new orientation.
\[d113a2\]Let $K\subset\mathbb{C}$ be a polynomial-convex set of positive capacity with outer domain $\Omega_{K}$. By $Ir(K)\subset K$ we denote the set $Ir(\partial\Omega_{K})$ of critical points. This set is broken down into the two subsets $$Ir_{I}(K):=Ir(K)\cap\overline{K\setminus Ir(K)}\text{ \ \ and \ }Ir_{II}(K):=Ir(K)\setminus(\overline{K\setminus Ir(K)}).\label{f113a2}$$ We further define the set of regular points of $K$ as $Rg(K):=K\setminus
Ir(K)$.
If $\operatorname*{cap}(K)=0$, then we defined $Ir_{II}(K):=Ir(K):=K$ and $Ir_{I}(K):=Rg(K):=\emptyset$.
We note that the set $Rg(K)$ introduced in Definition \[d113a2\] is more comprehensive then the set $Rg(K)\cap\partial\Omega_{K}$ of regular points with respect to solutions of Dirichlet problems in $\Omega_{K}$. An important result in potential theory is Kellog’s Theorem, which we state here in a somewhat specialized and at the same time also extended version (cf. [@Ransford], Theorem 4.2.5 together with Theorem 4.4.9).
\[l113a1\]For a polynomial-convex set $K\subset\mathbb{C}$ we have $\operatorname*{cap}(Ir(K))=0$, and the Green function $g_{\Omega}(\cdot
,w)$ is continuous in $\mathbb{C}\setminus Ir_{I}(K)$ for every $w\in
\Omega=\Omega_{K}$.
As a consequence of the Lemmas \[l111a\], \[l112a\], and \[l113a1\], we have the following results about irregular points and Green functions.
\[l113a2\]Let $K\subset\mathbb{C}$ be a polynomial-convex set with outer domain $\Omega=\Omega_{K}$.
\(i) The set $Ir_{II}(K)$ is totally disconnected.
\(ii) We have $\omega_{K}(Ir(K))=0$ for the equilibrium distribution $\omega_{K}$ on $K$.
\(iii) The Green function $g_{\Omega}(\cdot,\infty)$ is harmonic in $(\Omega\setminus\{\infty\})\cup Ir_{II}(K)=\mathbb{C}\setminus\allowbreak
\overline{\operatorname*{Rg}(K)}$.
\(iv) We have $\operatorname*{cap}(U_{z}\cap K)>0$ for every open neighborhood $U_{z}\subset\mathbb{C}$ of a point $z\in\overline
{\operatorname*{Rg}(K)}$, and $\operatorname*{cap}(U_{z}\cap K)=0$ for every open neighborhood $U_{z}\subset\mathbb{C}\setminus\overline{\operatorname*{Rg}(K)}$ of a point $z\in Ir_{II}(K)$.
The first three assertions are rather immediate. Assertion (iv) follows from [@Ransford], Theorem 4.2.3 and Theorem 4.2.4.
A connection between logarithmic potentials and Green functions is given by the representation formula in the next lemma (cf. [@Ransford], Theorem 4.4.7 together with Theorem 5.2.1).
\[l113b\]Let $K\subset\mathbb{C}$ be a compact set with $\operatorname*{cap}\left( K\right) >0$, $\Omega=\Omega_{K}$ its outer domain, and $\omega_{K} $ the equilibrium distribution on $K$. Then for the Green function $g_{\Omega}(\cdot,\infty)$, we have the representation $$g_{\Omega}(\cdot,\infty)=-p(\omega_{K};\cdot)+\log\frac{1}{\operatorname*{cap}\left( K\right) },\label{f113b1}$$ and near infinity we have $$g_{\Omega}(z,\infty)=\log|z|+\log\frac{1}{\operatorname*{cap}\left( K\right)
}+\text{O}(\frac{1}{|z|})\text{ \ \ as \ \ }z\rightarrow\infty.\label{f113b2}$$
The next result is related to Lemma \[l113b\].
\[l113c\]Let $K_{1},K_{2}\subset\mathbb{C}$ be polynomial-convex sets of positive capacity. Then we have $$g_{\overline{\mathbb{C}}\setminus K_{1}}(\cdot,\infty)\equiv g_{\overline
{\mathbb{C}}\setminus K_{2}}(\cdot,\infty)\label{f113c2}$$ if, and only if, $$\operatorname*{cap}\left( \left( K_{1}\setminus K_{2}\right) \cup\left(
K_{2}\setminus K_{1}\right) \right) =0,\label{f113c1}$$ i.e., if, and only if, the two sets $K_{1}$ and $K_{2}$ differ only in a set of capacity zero.
We assume that identity (\[f113c2\]) holds true. From (\[f113c2\]) together with the defining identity (\[f113a1\]) for the Green function and the definition of irregular points in Definition \[d113a\] it follows that $$\left( \partial K_{1}\setminus\partial K_{2}\right) \cup\left( \partial
K_{2}\setminus\partial K_{1}\right) \subset\operatorname*{Ir}(K_{1})\cup\operatorname*{Ir}(K_{2}),\label{f113c3}$$ which with Lemma \[l113a1\] and Lemma \[l111b\] implies (\[f113c1\]).
If, on the other hand, (\[f113c1\]) holds true, then identity (\[f113c2\]) is an immediate consequence of the defining identity (\[f113a1\]) for the Green function and the uniqueness of the Green function.$\medskip$
The balayage technique of Definition \[d112a\] can be made more concrete with the help of the Green function (cf. [@SaTo], Chapter II.4).
\[l113e\]Under the assumptions of Definition \[d112a\], we have $$p(\widehat{\mu};\cdot)=p(\mu;\cdot)-\int g_{D}(\cdot,x)d\mu(x)\label{f113e1}$$ if the domain $D$ is bounded, and otherwise we have $$p(\widehat{\mu};\cdot)=p(\mu;\cdot)-\int\left[ g_{D}(\cdot,x)-g_{D}(x,\infty)\right] d\mu(x).\label{f113e2}$$
Related to Lemma \[l113e\] is the Riesz Decomposition Theorem (cf., Theorem 3.1 of [@SaTo], Chapter II), and more definite the Poison-Jensen Formula, which has been used at several places in our investigations.
\[t113a\](Poison-Jensen Formula) Let $D\subset\overline{\mathbb{C}}$ be a domain with $\operatorname*{cap}\left( \partial D\right) \allowbreak>0$. We assume that the real-valued function $u$ is subharmonic on $\overline{D}$, not identical $-\infty$, and possesses an harmonic majorant in $\overline{D} $. Then there exists a nonnegative measure $\mu$ in $D$ of finite mass such that $$u(z)=-\iint_{D}g_{D}(z,v)d\mu(z)+\int_{\partial D}u(v)d\widehat{\delta_{z}}(v)\text{ \ \ for \ \ }z\in D\label{f113d3}$$ with $\widehat{\delta_{z}}$ denoting the balayage measure on $\partial D$ resulting from sweeping out the Dirac measure $\delta_{z}$ out of $D$. ($\widehat{\delta_{z}}$ is also known as the harmonic measure on $\partial D$ of the point $z\in D$.)
The theorem has been proved in [@Ransford] as Theorem 4.5.1 under stronger assumptions about the domain $D$ and the function $u$. The more general form of the theorem given here is the consequence of a combination of the two Theorems 4.5.1 and 4.5.4 in [@Ransford].
Analogously to the energy $I(\cdot)$ that has been defined in (\[f111a1\]) with a logarithmic kernel, one can also define an energy formula with a Green kernel, i.e., a formula like (\[f111a1\]) with a Green function as kernel. The new formula is called Green energy. A systematic investigation of Green energy and Green capacity together with the associated Green potentials can be found in [@SaTo], Chapter II.5.
In our investigations, we have needed the property of positive definiteness of the Green kernel. The result is contain in the next lemma, and it can be seen as a completion of the material in Lemma \[l112b\].
\[l113d\]Let $D\subset\overline{\mathbb{C}}$ be a domain with $\operatorname*{cap}\left( \partial D\right) >0$. For a signed measure $\sigma$ of finite energy we have $$\int\int g_{D}(z,v)d\sigma(z)d\sigma(v)\geq0\label{f113d1}$$ and equality holds in (\[f113d1\]) if, and only if, $\sigma|_{D\cup
Ir(\partial D)}=0$.
The lemma can be proved like the analogous result in [@Landkof], Theorem 1.6, together with the tools used in [@SaTo] for the proof of Lemma 5.4 in Chapter II.
We next come to some results that are connected with the Green formula. Let $D\subset\overline{\mathbb{C}}$ be a domain with a smooth and non-empty boundary $\partial D\subset\mathbb{C}$, and let further $u$ and $v$ be two real $C^{2}-$functions in $D$ with $L^{1}-$integrable second derivatives in $D$ and $C^{1}$ boundary functions with $L^{1}-$integrable first derivatives on $\partial D$. Under these assumptions, the Green identity $$\iint_{D}u\nabla\nabla v\,dm+\iint_{D}\nabla u\nabla v\,dm+\int_{\partial
D}u\frac{\partial}{\partial n}v\,ds=0\label{f113f1}$$ holds true (see Chapter VIII of [@Kellogg67] or [@HaymanKennedy76], Theorem 1.9). In (\[f113f1\]), $\nabla\nabla$ denotes the Laplace operator $\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$, $\nabla$ the napla or gradient operator $(\partial/\partial x,\partial/\partial y)$, $\partial/\partial n$ the inwardly showing normal derivation on $\partial D$, $dm$ the area element in $D$, and $ds$ the (positively oriented) line element on $\partial D$.
If the function $v$ is harmonic in $D$, then obviously the first term in (\[f113f1\]) vanishes. The second term is known as the Dirichlet integral of $u$ and $v$, and we use the abbreviation $$D_{D}(u,v):=\frac{1}{2\pi}\iint_{D}\nabla u\nabla v\,dm.\label{f113f2}$$ Using the same letter $D$ in one formula with two different meanings is certainly unlucky, but mix-ups should be avoidable. In comparison to the assumptions made in (\[f113f1\]) , we often relax assumptions for (\[f113f2\]); thus, for instance, with the help of an exhaustion technique we can admit arbitrary domains $D\subset\overline{\mathbb{C}}$. If not explicitly stated otherwise, then we assume that both functions $u$ and $v$ in (\[f113f2\]) have $L^{2}-$integrable first order derivatives almost everywhere in $D$.
In the special case that both functions $u$ and $v$ are identical, we write $$D_{D}(u):=D_{D}(u,u)=\frac{1}{2\pi}\iint_{D}\left( \nabla u\right)
^{2}dm.\label{f113f3}$$
Notice that in (\[f113f1\]) the Dirichlet integral $2\pi D_{D}(u,v)$ is the only term that is symmetric in both functions $u$ and $v$. The use of this fact leads to interesting special cases of the Green identity.Thus, for instance, one gets Formula (1.1) in Chapter II.1 of [@SaTo], which will also be used in the present subsection; it is the basis for the proof of Lemma \[l113f\], below, after the next paragraph.
Next, we come to several lemmas that are rather immediate consequences on the Green identity. The first one has been used at several places, where potentials have been defined in a piecewise manner. With respect to a proof of this result, we are in the lucky situation that most of the detailed work has already been done in [@SaTo], Chapter II, where similar results have been proved.
\[l113f\]Let $D\subset\overline{\mathbb{C}}$ be a domain with $\operatorname*{cap}\left( \partial D\right) >0$, $\gamma$ a $C^{1+\delta}-$smooth Jordan arc in $D$, $\delta>0$, and $u$ a bounded real-valued function that is continuous in $\overline{D}\setminus Ir(\partial D)$, harmonic in $D\setminus\gamma$, and which possesses $L^{1}-$integrable normal derivatives to both sides of $\gamma$.
If $u(z)=0$ for all $z\in\partial D\setminus Ir(\partial D)$, then we have $$u(z)=-\frac{1}{2\pi}\int_{\gamma}(\frac{\partial}{\partial n_{-}}+\frac{\partial}{\partial n_{+}})u(v)g_{D}(z,v)ds_{v}\text{ \ \ for \ }z\in
D.\label{f113d2}$$ In (\[f113d2\]), $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ denote the normal derivation to both sides of $\gamma$, $ds$ is the line element on $\gamma$, and $g_{D}(\cdot,\cdot)$ the Green function in $D$.
From Theorem 1.5 in Chapter II of [@SaTo] it follows that if we define the measure $\sigma$ on $\gamma$ by $$d\sigma(v):=-\frac{1}{2\pi}(\frac{\partial}{\partial n_{-}}+\frac{\partial
}{\partial n_{+}})u(v)ds_{v},\label{f113d22}$$ then the function $$d(z):=u(z)-\int g_{D}(z,v)d\sigma(v),\text{ \ \ \ \ \ }z\in D,\label{f113d21}$$ is harmonic in $D$. From the assumptions of the lemma and from (\[f113a1\]) together with Definition \[d113a\], we know that $d(z)=0$ for all $z\in\partial D\setminus Ir(\partial D)$. Since $d$ is bounded in $D$, it follows from the uniqueness of the Dirichlet problem under the given circumstances (cf. Theorem 3.1 in the Appendix A of [@SaTo]) that $d(z)=0$ for $z\in D$, which proves the lemma.
In the next lemmas, properties of Green functions are expressed with the help of Dirichlet integrals.
\[l113g\]Let $D\subset\overline{\mathbb{C}}$ be a domain with $\infty\in
D$, $r>0$, and $\operatorname*{cap}\left( \partial D\right) >0$, then we have $$D_{\{|z|<r\}\cap D}(g_{D}(\cdot,\infty))=\log(r)+\log\frac{1}{\operatorname*{cap}\left( \partial D\right) }+\text{O}(\frac{1}{r})\text{
\ \ as \ \ }r\rightarrow\infty.\label{f113g1}$$
The lemma is a consequence of the Green identity (\[f113f1\]). In a first step we assume that the domain $D$ has a sufficiently smooth boundary $\partial D$. If $\partial D$ is $C^{2}$ smooth, then the Green function $g_{D}(\cdot,\infty)$ has continuous first order derivatives on $\partial D$ (details can be found, for instance, in the proof of Theorem 4.11 in Chapter II of [@SaTo]). For $r>0$ sufficiently large, we define $$D_{r}:=D\setminus\{|z|\geq r\}.\label{f113g11}$$ Since $g_{D}(\cdot,\infty)$ is harmonic in $D_{r}$, we deduce from (\[f113f1\]) and (\[f113f3\]) that $$\begin{aligned}
& D_{\{|z|<r\}\cap D}(g_{D}(\cdot,\infty))=\frac{1}{2\pi}\iint_{D_{r}}(\nabla
g_{D}(\cdot,\infty))^{2}\,dm\nonumber\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=-\frac
{1}{2\pi}\int_{\partial D_{r}}g_{D}(\cdot,\infty)\frac{\partial}{\partial
n}g_{D}(\cdot,\infty)\,ds\label{f113g12}\\
& =-\frac{1}{2\pi}\int_{\partial D}g_{D}(\cdot,\infty)\frac{\partial}{\partial
n}g_{D}(\cdot,\infty)\,ds-\frac{1}{2\pi}\int_{\{|z|=r\}}g_{D}(\cdot
,\infty)\frac{\partial}{\partial n}g_{D}(\cdot,\infty)\,ds.\nonumber\end{aligned}$$
From the smoothness of $\partial D$ it follows that $\partial D$ is regular, and consequently that $g_{D}(z,\infty)=0$ for all $z\in\partial D$, which implies that the first integral in the last line of (\[f113g12\]) is identical zero.
From Lemma \[l113b\] we know that the function $$\widetilde{g}:=g_{D}(\cdot,\infty)-\log|\cdot|\label{f113g13}$$ is harmonic in $\{|z|>r\}$, and we have $\widetilde{g}(\infty)=-\log
\operatorname*{cap}\left( \partial D\right) $. From (\[f113g13\]) it follows that for the inward showing normal derivative on the circle $\{|z|=r\}$ we have $$\frac{\partial}{\partial n}g_{D}(z,\infty)=\frac{\partial}{\partial
n}\widetilde{g}(z)+\frac{1}{r}\text{ \ \ \ for \ \ }|z|=r.\label{f113g14}$$ It is rather immediate that $$\begin{aligned}
& \frac{1}{2\pi}\medskip\int_{\{|z|=r\}}\widetilde{g}(z)\frac{1}{r}\,ds_{z}=\widetilde{g}(\infty)=\log\frac{1}{\operatorname*{cap}\left(
\partial D\right) },\label{f113g14a}\\
& \frac{1}{2\pi}\int_{\{|z|=r\}}\log|z|\frac{1}{r}\,ds_{z}=\log
(r),\label{f113g14b}$$ and since $\widetilde{g}$ is harmonic in $\{|z|>r\}$, we further have $$\begin{aligned}
& \medskip\int_{\{|z|=r\}}\frac{\partial}{\partial n}\widetilde{g}(z)\,ds_{z}=0,\label{f113g14c}\\
& ||\frac{\partial}{\partial n}\widetilde{g}||_{\{|z|=r\}}=\text{O}(\frac
{1}{r^{2}})\text{ \ \ \ as \ \ \ }r\rightarrow\infty,\label{f113g14d}$$ where $||\cdot||_{\{|z|=r\}}$ denotes the sup-norm on $\{|z|=r\}$.
Using (\[f113g13\]) through (\[f113g14d\]), the only remaining integral in the last line of (\[f113g12\]) can be transformed in the following way: $$\begin{aligned}
& \text{ }-\frac{1}{2\pi}\medskip\int_{|z|=r}g_{D}(\cdot,\infty)\frac
{\partial}{\partial n}g_{D}(\cdot,\infty)\,ds=\nonumber\\
& \text{ \ \ \ \ \ }\medskip-\frac{1}{2\pi}\int_{|z|=r}(\log|\cdot
|\frac{\partial}{\partial n}\widetilde{g}+\log|\cdot|\frac{1}{r}+\widetilde
{g}\frac{\partial}{\partial n}\widetilde{g}+\widetilde{g}\frac{1}{r})ds=\text{
\ \ \ \ \ \ \ \ }\label{f113g15}\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ }\log(r)+\log\frac{1}{\operatorname*{cap}\left( \partial D\right) }+\text{O}(\frac{1}{r})\text{ \ \ as \ \ }r\rightarrow\infty,\nonumber\end{aligned}$$ which then proves (\[f113g1\]) under the assumption of a sufficiently smooth boundary $\partial D$.
In the general case, the domain $D$ is exhausted by a nested sequence of domains $D_{n}$, $n\in\mathbb{N}$, with sufficiently smooth boundaries $\partial D_{n}$. We assume that $$\overline{D_{n}}\subset D_{n+1}\subset D\text{ \ \ and \ \ \ }D=\bigcup
_{n}D_{n}.\label{f113g16}$$ By $g_{n}$ we denote the Green function $g_{D_{n}}(\cdot,\infty)$. Because of (\[f113g16\]) we have $$g_{n}(z)\geq g_{n+1}(z)\geq g_{D}(z,\infty)\text{ \ \ for \ \ }z\in
\mathbb{C}.\label{f113g17}$$ From the Harnack principle of monotonic convergence (cf. Theorem 4.10 in Chapter 0 of [@SaTo]) and the assumption that $\operatorname*{cap}\left(
\partial D\right) >0$, we then deduce that the sequence $\{g_{n}\}$ as well as their first order derivatives $\nabla g_{n}$ converge locally uniformly in $D$, i.e., we have $$\lim_{n\rightarrow\infty}\nabla g_{n}=\nabla g\label{f113g18}$$ locally uniformly in $D$. From the identity (\[f113g1\]) for the domains $D_{n}$ together with (\[f113g18\]), identity (\[f113g1\]) then follows also in the general case.
In (\[f113a1\]), the Green function $g_{D}(\cdot,\cdot)$ has been defined for the whole Riemann sphere $\overline{\mathbb{C}}$, and one could therefore consider the extension of the Dirichlet integral in (\[f113g1\]) from $\{|z|\leq r\}\cap D$ to the whole disc $\{|z|\leq r\}$. Such an extension would indeed be without problems if the planar Lebesgue measure of $\partial
D$ were zero. However, $\partial D$ may be of positive planar Lebesgue measure.
The combination of the assertion of Lemma \[l113g\] for two different domains yields the next corollary, which has been useful for the comparison of the capacities of the complements of two domains.
\[c113g1\]Let $D_{1},D_{2}\subset\overline{\mathbb{C}}$ be two domains with $\infty\in D_{j}$ and $\operatorname*{cap}\left( \partial D_{j}\right)
\allowbreak>0$ for $j=1,2$. Then for $r>0$ we have $$\begin{aligned}
& D_{\{|z|<r\}\cap D_{1}}(g_{D_{1}}(\cdot,\infty))-D_{\{|z|<r\}\cap D_{2}}(g_{D_{2}}(\cdot,\infty))=\medskip\label{f113g2}\\
& \text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\log\frac{\operatorname*{cap}\left( \partial D_{2}\right) }{\operatorname*{cap}\left( \partial D_{1}\right) }+\text{O}(\frac{1}{r})\text{ \ \ as \ \ }r\rightarrow\infty.\nonumber\end{aligned}$$
\[l113h\]Let the function $u$ be harmonic and bounded in the domain $D\subset\overline{\mathbb{C}}$ with $\infty\in D$ and $\operatorname*{cap}\left( \partial D\right) >0$. Then the Dirichlet integral $D_{D}(u,g_{D}(\cdot,\infty))$ exists, and we have $$\lim_{r\rightarrow\infty}D_{\{|z|<r\}\cap D}(u,g_{D}(\cdot,\infty
))=D_{D}(u,g_{D}(\cdot,\infty))=0.\label{f113g4}$$
Like in the proof of Lemma \[l113g\], in a first step, we assume that the domain $D$ has a $C^{2}$ smooth boundary $\partial D$. The subdomain $D_{r}$ is again defined by (\[f113g11\]) for $r>0$ sufficiently large. Analogously to (\[f113g12\]), we have the identities $$\begin{aligned}
& D_{D_{r}}(u,g_{D}(\cdot,\infty))\medskip=\frac{1}{2\pi}\iint_{D_{r}}\nabla
u\,\nabla g_{D}(\cdot,\infty)\,dm\nonumber\\
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\medskip=-\frac
{1}{2\pi}\int_{\partial D_{r}}u\frac{\partial}{\partial n}g_{D}(\cdot
,\infty)\,ds\label{f113g41}\\
& \text{ \ }=\medskip-\frac{1}{2\pi}\int_{\partial D}u\frac{\partial}{\partial
n}g_{D}(\cdot,\infty)\,ds-\frac{1}{2\pi}\int_{\{|z|=r\}}u\frac{\partial
}{\partial n}g_{D}(\cdot,\infty)\,ds\nonumber\\
& \text{ \ }=-u(\infty)+u(\infty)+\text{O}(\frac{1}{r})=\text{O}(\frac{1}{r})\text{ \ \ as \ \ }r\rightarrow\infty.\nonumber\end{aligned}$$ Indeed, the second equality is a consequence of the Green identity (\[f113f1\]). The penultimate equality in (\[f113g41\]) follows from two observations, which are concerned with the two integrals in the third line of (\[f113g41\]). In the first integral the normal derivative $(1/2\pi
)\allowbreak\partial g_{D}(\cdot,\infty)/\partial n$ is the density of the equilibrium distribution on $\partial D$ (cf. Theorem 4.11 of Chapter II in [@SaTo]), and it defines the balayage measure on $\partial D$ resulting from sleeping $\delta_{\infty}$ out of $D$, which implies that this first integral is equal to $-u(\infty)$.
The second integral in the third line of (\[f113g41\]) extends over the circle $\{|z|=r\}$. Using the definition of the function $\widetilde{g}$ in (\[f113g13\]) together with (\[f113g14\]) and (\[f113g14d\]) yields $$\begin{aligned}
-\frac{1}{2\pi}\int_{\{|z|=r\}}u\frac{\partial}{\partial n}g_{D}(\cdot
,\infty)\,ds & =-\frac{1}{2\pi}\int_{\{|z|=r\}}(u(z)\frac{\partial}{\partial
n}\widetilde{g}(z)+u(z)\frac{1}{r})\,ds_{z}\medskip\nonumber\\
& =\text{O}(\frac{1}{r})+u(\infty)\text{ \ \ \ as \ \ }r\rightarrow
\infty.\label{f113g42}$$ The last two observation together prove the penultimate equality in (\[f113g41\]).
We note that the two integrals in the third line of (\[f113g41\]) have opposite orientations with respect to the two domains $D$ and $\{|z|>r\}$. Like in the conclusions after (\[f113g13\]), and also in (\[f113g41\]), we have applied Theorem 3.1 of the Appendix A in [@SaTo].
With (\[f113g41\]) we have proved (\[f113g4\]) under the assumption of a sufficiently smooth boundary $\partial D$. We add that the existence of the integral $D_{D_{r}}(u,g_{D}(\cdot,\allowbreak\infty))$ follows from the Cauchy-Schwartz inequality $D_{D_{r}}(u,g_{D}(\cdot,\infty))^{2}\leq D_{D_{r}}(u)\allowbreak D_{D_{r}}(g_{D}(\cdot,\infty))$ together with $\operatorname*{cap}\left( \partial D\right) >0$ and the assumed boundedness of the function $u$.
Like in the proof of Lemma \[l113g\], for a general domain $D$ identity (\[f113g4\]) follows from exhausting the domain $D$ by a sequence of nested domains $D_{n}$, $n\in\mathbb{N}$, with sufficiently smooth boundaries $\partial D_{n}$.
In the next two lemmas we prove rather technical results, which have been used in Subsection \[s102\], further above.
\[l113i\]Let $D\subset\overline{\mathbb{C}}$ be a domain with $\infty\in
D$ and $\operatorname*{cap}\left( \partial D\right) >0$. Set $D_{r}:=\{|z|<r\}\cap D$ with $r>0$ sufficiently large so that $\partial
D\subset\{|z|<r\}$, and let $u$ be a real-valued function that is harmonic in $\{|z|>r\}$, and let further $\widehat{u}_{r}$ be the solution of the Dirichlet problem in $D_{r}$ with boundary function $$\widehat{u}_{r}(z)=\left\{
\begin{array}
[c]{lll}0\smallskip & \text{ \ \ \ for \ \ } & z\in\partial D,\\
u(z) & \text{ \ \ \ for \ } & |z|=r.
\end{array}
\right. \label{f113g51}$$ Under these assumptions, we have $$D_{\{|z|<r\}\cap D}(\widehat{u}_{r},g_{D}(\cdot,\infty))=u(\infty
)+\text{O}(\frac{1}{r})\text{ \ \ as \ \ }r\rightarrow\infty.\label{f113g5}$$
In a first step, we assume that $D$ has a $C^{2}$ smooth boundary $\partial D
$. Like in (\[f113g41\]), we deduce from (\[f113f1\]) that $$\begin{aligned}
& D_{D_{r}}(\widehat{u}_{r},g_{D}(\cdot,\infty))\medskip\label{f113g52}\\
& \text{ \ \ }=-\frac{1}{2\pi}\int_{\partial D}\widehat{u}_{r}\frac{\partial
}{\partial n}g_{D}(\cdot,\infty)\,ds-\frac{1}{2\pi}\int_{\{|z|=r\}}\widehat
{u}_{r}\frac{\partial}{\partial n}g_{D}(\cdot,\infty)\,ds.\nonumber\end{aligned}$$ It follows from the first line in (\[f113g51\]) that the first integral in the second line of (\[f113g52\]) is identical zero. For the second integral we deduce with the same arguments as applied in (\[f113g42\]) and with the use of the last line in (\[f113g51\]) that $$-\frac{1}{2\pi}\int_{\{|z|=r\}}\widehat{u}_{r}\frac{\partial}{\partial n}g_{D}(\cdot,\infty)\,ds=u(\infty)+\text{O}(\frac{1}{r})\text{ \ \ as
\ \ }r\rightarrow\infty.\label{f113g53}$$ Identity (\[f113g5\]) follows immediately from (\[f113g52\]) and (\[f113g53\]) for a domain $D$ with a sufficiently smooth boundary $\partial
D$.
For a general domain, identity (\[f113g5\]) can again be proved by exhausting the domain $D$ by a sequence of nested domains $D_{n}$, $n\in\mathbb{N}$, as it has been done in the proof of Lemma \[l113g\].
\[l113j\]Let $D\subset\overline{\mathbb{C}}$ be a domain with $\operatorname*{cap}\left( \partial D\right) >0$, $V\subset\overline{D}$ a compact set, $\mu$ a positive measure of finite energy with $\operatorname*{supp}(\mu)\subset V $, and $u$ a real-valued function defined by $$u(z):=h(z)+\int g_{D}(z,v)d\mu(v)\text{ \ \ for \ \ }z\in D\label{f113h1}$$ with $h$ a harmonic and bounded function in $D$. If we assume that $$u(z)=0\text{ \ \ \ for quasi every \ \ }z\in V,\label{f113h2}$$ then we have $$D_{D\setminus V}(u)=D_{D}(h)+\iint g_{D}(v,w)d\mu(v)d\mu(w).\label{f113h3}$$
We deduce from Lemma \[l113h\] that $$D_{D}(h,g_{D}(\cdot,v))=0\text{ \ \ for \ \ }v\in D,\label{f113h11}$$ since with the help of a Moebius transform any $v\in D$ can be transported to infinity. If we choose $g_{D}(\cdot,w)$, $w\in D$, instead of the function $h$ in (\[f113h11\]) and set $D_{r}:=D\setminus\{|z-w|\leq r\}$, for $r>0$ small, then we deduce from Lemma \[l113h\] that $$D_{D_{r}}(g_{D}(\cdot,v),g_{D}(\cdot,w))=0.\label{f113h12}$$ With the same argumentation as used after (\[f113g12\]) and later also in the proof of Lemma \[l113h\] after (\[f113g41\]), we show that $$\lim_{r\rightarrow\infty}D_{\{|z-w|<r\}}(g_{D}(\cdot,v),g_{D}(\cdot
,w))=g_{D}(w,v).\label{f113h13}$$ Putting (\[f113h12\]) and (\[f113h13\]) together proves that $$D_{D}(g_{D}(\cdot,v),g_{D}(\cdot,w))=g_{D}(v,w)\text{ \ \ for \ \ }v,w\in
D.\label{f113h14}$$
In a strict sense the Dirichlet integrals in (\[f113h11\]) and (\[f113h14\]) exist only as improper integrals, which is reflected in the removal of small disks around the points $\infty$ and $w$ in (\[f113g4\]) and (\[f113h12\]), respectively. There exist techniques to overcome this specific problem, as for instance, the use of local smoothing techniques at the singularity of the Green function, which is demonstrated in detail in [@Landkof], Chapter 1, §5.
In the next step of the proof we assume that $D\setminus V$ has a smooth boundary $\partial(D\setminus V)$. It follows then that the planar Lebesgue measure of $\partial V$ is zero, i.e., $m(\partial V)=0$, and further that in (\[f113h2\]) we have equality for all $z\in V$. By $g$ we denote the Green potential in (\[f113h1\]), i.e., $$g:=\int g_{D}(\cdot,v)d\mu(v).\label{f113h15}$$
Because of $m(\partial V)=0$, we have $$D_{D\setminus V}(u)=D_{D}(u),\label{f113h16}$$ and with (\[f113h1\]) and (\[f113h15\]), we rewrite the Dirichlet integral in (\[f113h16\]) as $$D_{D\setminus V}(u)=D_{D}(h)+2\,D_{D}(h,g)+D_{D}(g).\label{f113h17}$$ Then we deduce with Fubini’s Theorem from (\[f113h11\]) that $$D_{D}(h,g)=\int D_{D}(h,g_{D}(\cdot,v))d\mu(v)=0,\label{f113h18}$$ and analogously from (\[f113h14\]) that $$D_{D}(g)=\iint D_{D}(g_{D}(\cdot,v),g_{D}(\cdot,w))d\mu(v)d\mu(w)=\iint
g_{D}(v,w)d\mu(v)d\mu(w).\label{f113h19}$$ Putting (\[f113h17\]), (\[f113h18\]), and (\[f113h19\]) together, we have proved identity (\[f113h3\]) for the case that $\partial(D\setminus V)$ is sufficiently smooth.
In the general situation, identity (\[f113h3\]) follows, as in the proof of Lemma \[l113g\], by exhausting the open set $D\setminus V$ by a sequence of nested open sets with sufficiently smooth boundaries.
\[s1104\]Sequences of Compact Sets $K_{n}$
------------------------------------------
Let $K_{n}\subset\mathbb{C}$, $n\in\mathbb{N}$, be a sequence of compact sets of positive capacity. Because of the weak$^{\ast}$-compactness of the set of probability measures supported on a compact set, we know that any infinite subsequence $N\subset\mathbb{N}$ contains an infinite subsequence, which we continue to denote by $N$, such that the equilibrium measures $\omega_{n}=\omega_{K_{n}}$ of $K_{n}$ converge weakly in $\overline
{\mathbb{C}}$, i.e., there exists a probability measure $\omega_{0}=\omega_{0,N}$ in $\overline{\mathbb{C}}$ with $$\omega_{n}\overset{\ast}{\longrightarrow}\omega_{0}\text{ \ \ as
\ \ }n\rightarrow\infty\text{, }n\in N.\label{f114a1}$$ If in addition to (\[f114a1\]) also the limit $$\lim_{n\rightarrow\infty\text{, }n\in N}\operatorname*{cap}\left(
K_{n}\right) =:c_{0}>0\label{f114a2}$$ exists and the inequality in (\[f114a2\]) holds true, then it follows from the Lower Envelope Theorem \[t112a\] of potential theory that for the sequence of Green functions $g_{\Omega_{n}}(\cdot,\infty)$, $n\rightarrow
\infty$, $n\in N$, which is associated with the compact sets $K_{n}$ via the outer domains $\Omega_{n}=\Omega_{K_{n}}$, we have the asymptotic relation $$\limsup_{n\rightarrow\infty\text{, }n\in N}g_{\Omega_{n}}(\cdot,\infty
)\leq-p(\omega_{0};\cdot)-\log\operatorname*{cap}\left( c_{0}\right)
=:g_{0,N},\label{f114a3}$$ and equality holds in (\[f114a3\]) quasi everywhere in $\mathbb{C}$.$\smallskip$
Like the measure $\omega_{0}=\omega_{0,N}$ in (\[f114a1\]), so also the function $g_{0}=g_{0,N}$ in (\[f114a3\]) depends on the subsequence $N\subset\mathbb{N}$.
For all infinite subsequences $N\subset\mathbb{N}$, for which the two limits (\[f114a1\]) and (\[f114a2\]) exist, the potential $g_{0,N}$ in (\[f114a3\]) is well-defined, and it is an immediate consequence of (\[f113a1\]) and the inequality in (\[f114a3\]) that $$g_{0,N}(z)\geq0\text{ \ \ for all \ \ \ }z\in\mathbb{C}.\label{f114a4}$$
\[l114a\]Let $E\subset\mathbb{C}$ be a compact set, and let $N\subset
\mathbb{N}$ be an infinite subsequence for which the two limits (\[f114a1\]) and (\[f114a2\]) exist. The inclusion $E\subset K_{n}$ for all $n\in N$ implies that $$g_{0,N}(z)=0\text{ \ \ for quasi every \ }z\in E.\label{f114b1}$$
Without loss of generality we can assume that $\operatorname*{cap}\left(
E\right) >0$. From $E\subset K_{n}$, $n\in N$, it follows that $$g_{\Omega_{n}}(z,\infty)\leq g_{\Omega_{E}}(z,\infty)\text{ \ for \ }z\in\mathbb{C}\text{, \ }\Omega_{n}=\Omega_{K_{n}},\label{f114b2}$$ (cf. [@Ransford], Corollary 4.4.5), and consequently we have $$\limsup_{n\rightarrow\infty\text{, }n\in N}g_{\Omega_{n}}(z,\infty)\leq
g_{\Omega_{E}}(z,\infty)\text{ \ for \ }z\in\mathbb{C.}\label{f114b3}$$ From (\[f113a1\]), we know that $g_{\Omega_{E}}(z,\infty)=0$ for quasi every $z\in\widehat{E}$ ($\widehat{E}$ denotes the polynomial-convex hull of $E$), and from (\[f114a3\]) and (\[f114b3\]), we then conclude that $g_{0,N}(z)=0$ for quasi every $z\in\widehat{E}$, which proves (\[f114b1\]).
The next lemma has played a key role at several places in the proof of existence of an extremal domain in Subsection \[s92\]. The proof of the lemma relies heavily on Carathéodory’s Kernel Convergence Theorem, which will be stated just after the next lemma.
\[l114b\]Let $R\subset\overline{\mathbb{C}}$ be a ring domain with $\infty\in R$, $A_{1}$, $A_{2}\subset\mathbb{C}$ the two components of $\overline{\mathbb{C}}\setminus R$, let further $N\subset\mathbb{N}$ be an infinite subsequence for which the two limits (\[f114a1\]) and (\[f114a2\]) exist, and for which therefore the function $g_{0,N}$ in (\[f114a3\]) is well-defined, and let further $0<r<\infty$ be an appropriately chosen constant.
If for each $n\in N$ there exists a continuum $V_{n}\subset K_{n}\cap\{\,|z|\leq r\,\}$ that intersects the ring domain $R$, i.e., we have $$V_{n}\cap A_{j}\neq\emptyset\text{ \ \ \ for \ \ \ }j=1,2,\label{f114f10}$$ then $$V_{0}:=\overline{\mathbb{C}}\setminus\Omega\left( \bigcap_{m\in\mathbb{N}}\overline{\bigcup_{n\geq m,\text{ }n\in N}V_{n}}\right) \label{f114f11}$$ is a continuum with $V_{0}\subset\{\,|z|\leq r\,\}$ that also intersects $R$, i.e., we have $$V_{0}\cap A_{j}\neq\emptyset\text{ \ \ \ for \ \ \ }j=1,2,\label{f114f12}$$ and further we have $$g_{0,N}(z)=0\text{ \ \ for all \ \ }z\in V_{0}\label{f114c13}$$ where $g_{0,N}$ denotes the function defined in (\[f114a3\]). By $\Omega(\cdot)$ we denote the outer domain in (\[f114f11\]).
It has already been mentioned that the proof of Lemma \[l114b\] is based on Carathéodory’s Kernel Convergence Theorem from geometric function theory, which establishes an equivalence between an analytic and a geometric description of the convergence of a sequences of conformal mapping functions (cf. [@Pommerenke75], Chapter 1.4).
\[Carathéodory’s Kernel Convergence Theorem\]\[t114a\] Let $\allowbreak\left\{ \varphi_{n}\right\} _{n\in\mathbb{N}}$ be a sequence of univalent functions defined in $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ with $\varphi_{n}(\infty)=\infty$ and $$0<m_{0}\leq\varphi_{n}^{\prime}(\infty)\leq M_{0}<\infty\text{ \ \ for each
\ }n\in\mathbb{N}.\label{f114c2}$$ The sequence of functions $\varphi_{n}$, $n\in\mathbb{N}$, convergence locally uniformly in $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ to an univalent function $\varphi_{0}$ if, and only if, the domains $$D_{N}=\operatorname*{Ker}\left( \left\{ \varphi_{n}(\overline{\mathbb{C}}\setminus\overline{\mathbb{D}})\right\} _{n\in N}\right) :=\Omega\left(
\bigcap_{m\in\mathbb{N}}\overline{\bigcup_{n\geq m,\text{ }n\in N}\overline{\mathbb{C}}\setminus\varphi_{n}(\overline{\mathbb{C}}\setminus
\overline{\mathbb{D}})}\right) \label{f114c3}$$ are identical for all infinite subsequences $N\subset\mathbb{N}$. In (\[f114c3\]), the outer domain is denoted by $\Omega(\cdot)$. The domain $D_{N}$ associated with the sequence $N\subset\mathbb{N}$ is called kernel of the sequence of domains $\varphi_{n}(\overline{\mathbb{C}}\setminus
\overline{\mathbb{D}})$, $n\in N$.
If the sequence $\{\varphi_{n}\}_{n\in\mathbb{N}}$ converges locally uniformly in $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$, then the limit function $$\varphi_{0}=\lim_{n\rightarrow\infty}\varphi_{n}\label{f114c4}$$ is the Riemann mapping function of $\overline{\mathbb{C}}\setminus
\overline{\mathbb{D}}$ onto the domain $D_{\mathbb{N}}\subset\overline
{\mathbb{C}}$ with $\varphi_{0}(\infty)=\infty$ and $m_{0}\leq\varphi
_{0}^{\prime}(\infty)\leq M_{0}$.
The restrictions (\[f114c2\]) placed on $\varphi_{n}^{\prime}(\infty)$ make sure that degenerated cases are excluded.
\[Proof of Lemma \[l114b\]\]Let $\varphi_{n}$ be the Riemann mapping function from $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ onto $\overline
{\mathbb{C}}\setminus V_{n}$ for each $n\in\mathbb{N}$ with $$\varphi_{n}(\infty)=\infty\text{ \ \ and \ \ }\varphi_{n}^{\prime}(\infty)>0.\label{f114d1}$$ It is immediate that $$g_{\overline{\mathbb{C}}\setminus V_{n}}(z,\infty)=\log\left| \varphi
_{n}^{-1}(z)\right| \text{ \ \ for \ }z\in\overline{\mathbb{C}}\setminus
V_{n},\label{f114d2}$$ and we have $g_{\overline{\mathbb{C}}\setminus V_{n}}(z,\infty)=0$ for all $z\in V_{n}$ since continua are regular sets. From Lemma \[l113b\] it follows that $$\operatorname*{cap}(V_{n})=\varphi_{n}^{\prime}(\infty).\label{f114d3}$$ From Lemma \[l111a\] and (\[f114d3\]), we conclude that the assumptions $V_{n}\subset\{$$|z|\leq r\,\}$ and $V_{n}\cap A_{j}\neq\emptyset$ for $j=1,2$ and $n\in\mathbb{N}$ imply that $$\operatorname*{dist}(A_{1},A_{2})/4\leq\operatorname*{cap}(V_{n})\leq r\text{
\ for all \ }n\in\mathbb{N.}\label{f114d4}$$ From (\[f114d3\]) and (\[f114d4\]), it then follows that the restrictions (\[f114c2\]) in Theorem \[t114a\] are satisfied.
From (\[f114d4\]) and the assumption $V_{n}\subset\{\,|z|\leq r\,\}$ for all $n\in\mathbb{N}$, it further follows that there exists an infinite subsequence $N\subset\mathbb{N}$ such that the two limits (\[f114a1\]) and (\[f114a2\]) exist, and therefore also the limit function $g_{0,N}$ in (\[f114a3\]) exists with $K_{n}$ replaced by the sets $V_{n}$, and the outer domain $\Omega_{n}$ by $\overline{\mathbb{C}}\setminus V_{n}$ for each $n\in
N$.
Because of $V_{n}\subset\{\,|z|\leq r\,\}$ for $n\in\mathbb{N}$, we have a proper limit and locally uniform convergence in $\{\,|z|>r\,\}$ in (\[f114a3\]). With (\[f114d2\]), this implies that the sequence $\varphi_{n}$, $n\in N$, converges uniformly in a closed neighborhood of infinity. From the convergence together with the property that the $\varphi_{n}$ are mapping functions into $\overline{\mathbb{C}}\setminus
V_{n}$ and the property (\[f114d1\]), we deduce that $\{$$\varphi_{n}$, $n\in N\,\}$ forms a normal family in $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$, and by Montel’s Theorem together with the convergence in $\{\,|z|>r\,\}$, it then follows that $$\lim_{n\rightarrow\infty,\text{ }n\in N}\varphi_{n}(z)=:\varphi_{0,N}(z)\label{f114d5}$$ holds locally uniformly for $z\in\overline{\mathbb{C}}\setminus\overline
{\mathbb{D}}$.
From Carathéodory’s Kernel Convergence Theorem, we then concluded that the limit function $\varphi_{0}$ in (\[f114d5\]) is the Riemann mapping function of $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ onto the domain $D_{N}\subset\overline{\mathbb{C}}$ defined by (\[f114c3\]) with the subsequence $N$, which has been used in (\[f114d5\]), and from (\[f114c3\]) and (\[f114d5\]), we further know that $$V_{0}:=\overline{\mathbb{C}}\setminus D_{N}\label{f114d6}$$ is a continuum. From (\[f114c3\]) we learn that $$V_{0}=Pc(\bigcap_{m\in\mathbb{N}}\overline{\bigcup_{n\geq m,\text{ }n\in
N}V_{n}})=\overline{\mathbb{C}}\setminus\Omega\left( \bigcap_{m\in\mathbb{N}}\overline{\bigcup_{n\geq m,\text{ }n\in N}V_{n}}\right) \label{f114d7}$$ with $Pc(\cdot)$ denoting the polynomial-convex hull. For the two components $A_{1}$ and $A_{2}$ of $\overline{\mathbb{C}}\setminus R$ we have $$A_{j}\cap\bigcap_{m\in\mathbb{N}}\overline{\bigcup_{\substack{n\geq m, \\n\in
N }}V_{n}}=\bigcap_{m\in\mathbb{N}}\overline{(A_{j}\cap\bigcup
_{\substack{n\geq m, \\n\in N }}V_{n})}=\bigcap_{m\in\mathbb{N}}\overline{\bigcup_{\substack{n\geq m, \\n\in N }}\left( A_{j}\cap
V_{n}\right) },\label{f114d8}$$ $j=1,2$. Since we have assumed that $V_{n}\cap A_{j}\neq\emptyset$ for $j=1,2$ and all $n\in N$, we conclude from (\[f114d8\]) that $$A_{j}\cap\bigcap_{m\in\mathbb{N}}\overline{\bigcup_{n\geq m,\text{ }n\in
N}V_{n}}\text{ }\neq\text{ }\emptyset\text{ \ for \ }j=1,2\text{,}\label{f114d9}$$ since the intersections are nested. From (\[f114d9\]) and (\[f114d7\]), we immediately get $$A_{j}\cap V_{0}\neq\emptyset\text{ \ for \ }j=1,2\text{.}\label{f114d10}$$
Since we have assumed $V_{n}\subset K_{n}$, we have $$g_{\Omega_{n}}(z,\infty)\leq g_{\overline{\mathbb{C}}\setminus V_{n}}(z,\infty)\text{ \ for \ }z\in\mathbb{C}\label{f114d11}$$ and all $n\in N$ and $\Omega_{n}:=\Omega_{K_{n}}$ (cf. [@Ransford], Corollary 4.4.5). From the convergence (\[f114d5\]) together with the identities in (\[f114d2\]), we conclude that $$\lim_{n\rightarrow\infty,\text{ }n\in N}g_{\overline{\mathbb{C}}\setminus
V_{n}}(z,\infty)=g_{\overline{\mathbb{C}}\setminus V_{0}}(z,\infty
)\label{f114d12}$$ holds locally uniformly for $z\in\mathbb{C}$. From (\[f114d11\]) together with limit relation (\[f114c3\]) and (\[f114d12\]), it then follows that $$g_{0,N}(z)\leq g_{\overline{\mathbb{C}}\setminus V_{0}}(z,\infty)\text{ \ for
quasi every \ }z\in\mathbb{C,}\label{f114d13}$$ where $g_{0,N}$ is the limit function in (\[f114d12\]). Because of the first inequality in (\[f114d4\]), $V_{0}$ is of positive capacity, and therefore the equilibrium measure $\omega_{V_{n}}$ of $V_{n}$ is of finite energy. With the principle of domination in Theorem \[t112b\], it then follows that the inequality in (\[f114d13\]) holds for all $z\in\mathbb{C} $. Since $g_{\overline{\mathbb{C}}\setminus V_{0}}(z,\infty)=0$ for all $z\in V_{0}$, conclusion (\[f114c13\]) of Lemma \[l114b\] follows from (\[f114d13\]). The two other conclusions (\[f114f11\]) and (\[f114f12\]) are identical with (\[f114d7\]) and (\[f114d10\]), which completes the proof of the lemma.
\[s1105\]Critical Trajectories of Quadratic Differentials
---------------------------------------------------------
Let $q$ be a function meromorphic in a domain $D\subset\mathbb{C}$. In (\[f52a\]), trajectories of the quadratic differential $q(z)dz^{2}$ have been defined as smooth Jordan arcs $\gamma$ with parametrization $z:[0,1]\longrightarrow\overline{\mathbb{C}}$ that satisfy the relation $$q(z(t))\overset{\bullet}{z}(t)^{2}<0\text{ \ \ \ for \ \ }t\in
(0,1).\label{f115a}$$ Like in Section \[s52\], we use [@Strebel84] and [@Jensen75] as general reference for quadratic differentials.
Assertions about the global behavior of trajectories of a quadratic differential $q(z)dz^{2}$ are difficult to obtain, but the situation is dramatically different with respect to their local behavior; it depends only on the local form of the function $q$, and is basically a consequence of the degree of its poles and zeros. Further, it is not difficult to see that the qualitative behavior of trajectories is invariant under conformal maps.
All zeros and simple poles of the function $q$ are called *finite critical points* of the quadratic differential $q(z)dz^{2}$, and trajectories that end at zeros and poles are called *critical*. In the next lemma we assemble results about the local behavior of trajectories that have been used at several places of our analysis, further above. These results are not difficult to prove, and proofs can be found in [@Jensen75], Chapter 8.2.
\[l115a\]We consider a quadratic differential $q(z)dz^{2}$, and assume that $q$ is meromorphic in a domain $D\subset\mathbb{C}$.
\(i) If $q$ is analytic in a neighborhood $U$ of $z_{0}\in D$ and if $q(z)\neq0$ for all $z\in U$, then all trajectories of $q(z)dz^{2}$ are laminar in $U$.
(ii) Let $z_{0}\in D$ be a finite critical point of the quadratic differential $q(z)dz^{2}$, i.e., at $z_{0}$ the function $q$ has the local behavior $$q(z)=q_{0}(z-z_{0})^{l}+\text{O}((z-z_{0})^{l+1})\text{ \ \ as \ \ }z\rightarrow z_{0}\text{, \ }q_{0}\neq0,\label{f115b}$$ and let further $U$ be a neighborhood of $z_{0}$ with $q(z)\neq0,\infty$ for all $z\in U\setminus\{z_{0}\}$, then $l+2$ trajectories of $q(z)dz^{2}$ end at the point $z_{0}$, and they form a regular star at $z_{0}$, i.e., all angles between neighboring trajectories are equal to $2\pi/(l+2)$. All other (non-critical) trajectories of $q(z)dz^{2}$ are laminar in $U$.
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[^1]: The research has been supported by the grant STA 299/13-1 der DFG
|
---
abstract: 'Helicases are molecular motors that unwind double-stranded nucleic acids (dsNA), such as DNA and RNA). Typically a helicase translocates along one of the NA single strands while unwinding and uses adenosine triphosphate (ATP) hydrolysis as an energy source. Here we model of a helicase motor that can switch between two states, which could represent two different points in the ATP hydrolysis cycle. Our model is an extension of the earlier Betterton-Jülicher model of helicases to incorporate switching between two states. The main predictions of the model are the speed of unwinding of the dsNA and fluctuations around the average unwinding velocity. Motivated by a recent claim that the NS3 helicase of Hepatitis C virus follows a flashing ratchet mechanism, we have compared the experimental results for the NS3 helicase with a special limit of our model which corresponds to the flashing ratchet scenario. Our model accounts for one key feature of the experimental data on NS3 helicase. However, contradictory observations in experiments carried out under different conditions limit the ability to compare the model to experiments.'
author:
- 'Ashok Garai[[^1]]{}'
- 'Debashish Chowdhury[[^2]]{}'
- 'M. D. Betterton[[^3]]{}'
title: 'A two-state model for helicase translocation and unwinding of nucleic acids'
---
Helicases are enzymes that unwind double-stranded nucleic acids (dsNA) [@alberts02]. Helicase proteins typically translocate along one of the single strands and perform mechanical work while consuming chemical energy (usually supplied by the hydrolysis of ATP). Therefore, these NA translocases are molecular motors [@schliwa03; @mavroidis04] which share common features with cytoskeletal molecular motors [@lohman98; @howard01].
All helicases undergo a biochemical cycle which typically involves ATP binding, ATP hydrolysis, and release of the hydrolysis products adenosine diphosphate (ADP) and and inorganic phosphate (P$_i$). An important question in the study of helicase mechanisms is to understand how the ATP hydrolysis cycle is coupled to the binding state and the motion of the helicase [@lohman96; @delagoutte02]. Helicases may exhibit changes in helicase/NA binding affinity when the helicase is bound to ATP, ADP/P$_i$, or neither; coordination of hydrolysis between different helicase subunits, and conformational changes in the helicase triggered by different steps in the hydrolysis cycle. Some helicases form hexamers (which include six ATPase domains), while others are members of the non-hexameric (dimeric or monomeric) group; different types of mechanochemical cycle have been suggested for the different structural classes [@lohman96; @patel00]. In all cases, one seeks to explain how the helicase coordinates NA binding and hydrolysis to move along single-stranded NA and unwind double-stranded NA.
Here we develop a generic model of a helicase that switches between two biochemical states while translocating on ssNA. This is a simplified representation of the different states of the helicase during the ATP hydrolysis cycle. The model may be generally applicable to helicases for which the transition between two states is the key feature of the motion. In other words, this model should be a good approximation for helicases with more than two biochemical states if one transition is far slower than the others. We incorporate such a two-state picture by extending the original Betterton-Jülicher (BJ) model [@betterton03; @betterton05a; @betterton05b] of NA helicases [@note1].
Our work is also connected to two-state models that have been used extensively for a variety of molecular motors [@julicher97; @nishi05; @zeldo05; @fisher07]. Under a mean-field approximation, such models can be easily solved when periodic boundary conditions are imposed. However, the problem is usually more difficult with open boundary conditions. The model for helicase motion is even more complex because the position of one boundary (i.e., the ssNA-dsNA junction) varies randomly with time. Thus our work is also an extension of previous work on two-state models to the more difficult case of a fluctuating boundary.
The two-state model developed here is consistent with the observation that binding and hydrolysis of ATP can modulate the affinity of a helicase for the nucleic-acid track [@bjornson96; @kim98; @velankar99]. The flashing-ratchet mechanism suggested qualitatively for the hepatitis C virus non-structural protein 3 (HCV NS3) helicase [@levin03; @levin05] can be captured by a special case of the generic model proposed here. In the flashing-ratchet [@julicher97] picture, the motor protein switches between two states: one where the protein is tightly bound to the track, and another where the motor is weakly bound and can diffuse along the track. In this paper we make quantitative comparisons between our theoretical predictions for a passive helicase which follows the flashing-ratchet mechanism, and the experimental data for NS3 helicase.
In section \[sec:model\] we describe the ingredients of the model: the helicase, which can switch between two states and translocate on ssNA, and the fluctuating NA ss-ds junction. In section \[sec:sseqns\] we calculate the single-strand translocation rate of the helicase. Section \[sec:eqns\] contains the model equations for double-strand unwinding, the transformation of the equations using midpoint and difference variables, and the general solutions for the velocity and diffusion coefficient. We describe the results for a hard-wall interaction between the helicase and junction in section \[sec:soln\]. Using rate constants estimated from experiments on NS3 helicase, in section \[sec:results\] we specialize to the flashing-ratchet scenario and make predictions specific to NS3. In section \[sec:disc\] we summarize our results.
The model {#sec:model}
=========
Here we develop a physical model for a helicase that moves on ssNA while cycling between two chemical states (labeled 1 and 2). Levin et al. suggested such a two-state model for NS3 helicase motion [@levin03; @levin05]. In this paper, we first consider a general two-state model, and later focus on the specific flashing-ratchet picture.
![Schematic of the model. The protein can exist in either of two chemical states (labeled 1 and 2) at each lattice site (labeled $n$). Sliding transitions (where $n$ changes but the chemical state does not) occur at rate $s_{1f}$, etc., depending on the state and whether the transition is forward (toward increasing $n$) or backward (toward decreasing $n$). Chemical transitions (where the chemical state changes but $n$ does not) occur at rates $\omega_{12}$ (for the transition from 1 to 2) and $\omega_{21}$ (for the transition from 2 to 1). Coupled transitions, where both the chemical state and $n$ change, occur at rates $r_f$ (for the transition from 2 to 1 coupled to forward motion), $r_b$ (for the transition from 1 to 2 coupled to backward motion), $q_f$ (for the transition from 1 to 2 coupled to forward motion), and $q_b$ (for the transition from 2 to 1 coupled to backward motion). The nucleic acid single strand-double strand junction is at site $m$. The junction moves toward increasing $m$ when the NA opens by one base (rate $\alpha$) and toward decreasing $m$ when the NA closes (rate $\beta$). []{data-label="fig-model"}](model_sketch){width="3in"}
In the traditional continuous models of Brownian ratchets, one first writes a Fokker-Planck equation. We use a discrete model, so our approach is based on master equations. The discrete approach can be useful when comparing to experiments. In the Fokker-Planck approach, one needs the explicit functional form of the fluctuating potential, which has not been measured for any real motor. In the discrete model, we bypass this difficulty by capturing the motor mechanism through a choice of rate constants (or transition probabilities), many of which can be obtained from experiments (see section \[sec:results\]).
In the discrete model, we represent the ssNA by a one-dimensional lattice where each site corresponds to a single base. We label each site by the integer index $i$. As in the BJ model [@betterton03], we neglect the sequence inhomogeneity of the ssNA (in principle, the model can be extended to capture this feature, which may be important in some limits [@kafri04]). The position of the helicase is denoted by the integer $n$. Most helicases have a fixed direction of translocation, either $3'$ to $5'$ or $5'$ to $3'$ along the left-right asymmetric ssNA [@lohman96]. In our model the helicase translocates toward increasing $n$ (from left to right in fig. \[fig-model\]). At any spatial position $n$, the helicase can be either in biochemical state 1 or 2.
The model is fully described by the allowed transitions between states and the corresponding reaction rates. In general, we could have all transitions sketched in fig. \[fig-model\]. Helicase “sliding” corresponds to transitions along the ssNA without a change in biochemical state of the protein. In state 1, these sliding transitions occur at rate $s_{1f}$ (for increasing $n$) and $s_{1b}$ (for decreasing $n$). When the helicase is in state 2, the forward/backward sliding rates are $s_{2f}$ and $s_{2b}$. Physically, these transitions occur because of Brownian motion of the protein, decoupled from any biochemical state change.
The helicase can undergo “chemical” transitions which correspond to a change in biochemical state without physical translocation along the ssNA. At fixed $n$, the rate of transition from state 1 to 2 occurs at rate $\omega_{12}$, while the reverse transition occurs at rate $\omega_{21}$. Finally, “coupled” mechanochemical transitions are those where a change of biochemical state and physical translocation occur together. If the helicase is located at $n$ and is in state 2, then it can make a transition to state 1 while moving forward to site $n+1$ at rate $r_{f}$; the corresponding reverse rate is $r_b$. The transition of the helicase from state 1 to 2 while moving forward from $n$ to $n+1$ occurs at rate $q_f$; the corresponding reverse rate is $q_b$.
If any of these reactions is coupled to ATP hydrolysis, then the forward/reverse transitions may be out of equilibrium and break the detailed balance relation. The Levin et al. model of HCV NS3 helicase suggests that ATP binding is required to remove the helicase from the tightly bound state [@levin03; @levin05], implying that the $1 \to
2$ transition at rate $\omega_{12}$ is determined by the ATP concentration. In the Levin et al. flashing-ratchet model, ATP hydrolysis and product release is coupled to the translocation and chemical transition back to state $1$, which in our representation means that rates $\omega_{21}$ and $r_f$ would be coupled to ATP hydrolysis and would therefore be out of equilibrium (see section \[sec:results\]).
The junction between ssNA and dsNA is labeled by $m$ (see fig. \[fig-model\]). The dsNA opens and closes due to thermal fluctuations. When the helicase and junction are far apart, the opening rate is $\alpha$ and the closing rate $\beta$. We assume that these rates are independent of the NA base sequence and that the only fluctuations are those for which the NA opens or closes at the ss-ds fork. Following the BJ model [@betterton03], we neglect the possibility of any jump $>1$ bp in the position of the ssNA-dsNA junction. However, this approximation is justified because, at the temperatures of our interest (i.e., sufficiently below the melting temperature of the dsDNA) the spontaneous formation of bubbles is rare. Since the NA breathing results from thermal fluctuations, the rates $\alpha$ and $\beta$ satisfy detailed balance: $
\frac{\alpha}{\beta}=e^{-\Delta G}$, where $\Delta G$ is the free energy of one base-pair bond in units of $kT$.
The main quantity of interest is the speed of unwinding of dsNA by a helicase. We derive an analytical expression for the unwinding velocity. We compare the predicted velocity with the corresponding experimental data for a specific helicase, NS3 helicase of hepatitis C virus. Although we also derive an analytical expression for the diffusion constant of the helicase, we do not compare it with experimental data for any specific helicase.
In this work we analyze passive unwinding, which is equivalent to a hard-wall interaction potential in the BJ model [@betterton05a]. In passive unwinding, the helicase acts as a block to NA closing when adjacent to the junction. The protein moves forward only when thermal fluctuations open a basepair at the NA ss-ds junction. This means that when the helicase and junction are adjacent ($j=1$), the helicase cannot hop forward (all helicase forward rates, $s_{1f}(j=1)$, $s_{2f}(j=1)$, $r_f(j=1)$, and $q_f(j=1)$, are zero) and the NA cannot close ($\beta(j=1)=0$). Otherwise, the rates are unaffected by the helicase-junction interaction.
Single-strand translocation {#sec:sseqns}
===========================
In order to motivate our approach, we first formulate the equations for a helicase sufficiently far from the ssNA-dsNA junction so that it translocates on ssNA without any dsNA unwinding activity. Let ${\cal
P}_{\mu}(n,t)$ denote the probability that, at time $t$, the helicase is located at site $n$ and is in the chemical state $\mu$. We will drop the reference to the time dependence of ${\cal P}_{\mu}(n)$. The master equations governing the time evolution of ${\cal
P}_{\mu}(n)$ are $$\begin{aligned}
\frac{d{\cal P}_{1}(n)}{dt} &=&
-(\omega_{12}+s_{1f}+s_{1b}+q_f+r_b){\cal P}_{1}(n) +
s_{1f}{\cal P}_{1}(n-1) + r_f {\cal P}_{2}(n-1)
\nonumber \\
&+&s_{1b}{\cal P}_{1}(n+1) +q_b {\cal P}_{2}(n+1)
+ \omega_{21}{\cal P}_{2}(n),
\label{eq-master1ss}\end{aligned}$$ and $$\begin{aligned}
\frac{d{\cal P}_{2}(n)}{dt} &=&
-(\omega_{21}+s_{2f}+s_{2b}+r_f+q_b){\cal P}_{2}(n) +
s_{2f}{\cal P}_{2}(n-1) + q_f {\cal P}_{1}(n-1)
\nonumber \\
&+&s_{2b}{\cal P}_{2}(n+1) +r_b {\cal P}_{1}(n+1)
+ \omega_{12}{\cal P}_{1}(n).
\label{eq-master2ss}\end{aligned}$$ Summing these equations, we find the total probability ${\cal P}(n)={\cal
P}_1(n)+{\cal P}_2(n)$ satisfies $$\begin{aligned}
\frac{d{\cal P}(n)}{dt} &=&
-(s_{1f}+s_{1b}+q_f+r_b){\cal P}_{1}(n)
-(s_{2f}+s_{2b}+r_f+q_b){\cal P}_{2}(n) +
(s_{1f}+q_f){\cal P}_{1}(n-1) + (s_{2f}+r_f) {\cal P}_{2}(n-1)
\nonumber \\
&+&(s_{1b}+r_b){\cal P}_{1}(n+1) +(s_{2b}+q_b) {\cal P}_{2}(n+1).
\label{eq-masterss}\end{aligned}$$ These equations have a translationally invariant steady-state solution where ${\cal P}_{\mu}(n)$ is independent of $n$. In this case, we expect that the probability in state 2 is a multiple of the probability in state 1: $${\cal P}_{2}(n)=\sigma {\cal P}_{1}(n),
\label{eq-propss}$$ which means that ${\cal P}(n)=(1+\sigma){\cal P}_1(n)$.
In this case, the master equation for the total probability can be written as a hopping model with effective rates $k_f$ for forward transitions and $k_b$ for backward transitions. At steady state, $$\begin{aligned}
0 &=& k_f {\cal P}(n-1) - (k_f + k_b) {\cal
P}(n) + k_b {\cal P}(n+1),
\label{eq-hop}\end{aligned}$$ where $$\begin{aligned}
k_f &=& \frac{s_{1f}+q_f+\sigma(s_{2f}+r_f)}{1+\sigma}, \label{eq-kf}\\
k_b &=& \frac{s_{1b}+r_b+\sigma(s_{2b}+q_b)}{1+\sigma}.
\label{eq-kb}\end{aligned}$$ and the expression $$\sigma=
\frac{\omega_{12}+q_f+r_b}{r_f+q_b+\omega_{21}}.$$ has been obtained from eqn. at steady state, assuming translational invariance. The mean single-strand translocation velocity is $v_{ss}=k_f-k_b$.
Double-strand unwinding: Model equations {#sec:eqns}
========================================
In this section we extend the formulation of the preceding section by incorporating helicase-catalyzed dsNA unwinding. Let ${\cal
P}_{\mu}(n,m;t)$ denote the probability that, at time $t$, the helicase is at located at $n$ and is in the chemical state $\mu$, while the ss-ds junction is at $m$. We will drop the reference to the time dependence of ${\cal P}_{\mu}(n,m)$. The master equations governing the time evolution of ${\cal P}_{\mu}(n,m)$ are given by $$\begin{aligned}
\frac{d{\cal P}_{1}(n,m)}{dt} &=& -(\alpha + \beta+
\omega_{12}+s_{1f}+s_{1b}+q_f+r_b){\cal P}_{1}(n,m) +
s_{1f}{\cal P}_{1}(n-1,m) + r_f {\cal P}_{2}(n-1,m)
\nonumber \\
&+&s_{1b}{\cal P}_{1}(n+1,m) +q_b {\cal P}_{2}(n+1,m)
+ \omega_{21}{\cal P}_{2}(n,m)+\alpha {\cal P}_{1}(n,m-1)
+\beta P_{1}(n,m+1) ~~(m > n).
\label{eq-master1}\end{aligned}$$ and $$\begin{aligned}
\frac{d{\cal P}_{2}(n,m)}{dt} &=& -(\alpha + \beta+
\omega_{21}+s_{2f}+s_{2b}+r_f+q_b){\cal P}_{2}(n,m) +
s_{2f}{\cal P}_{2}(n-1,m) + q_f {\cal P}_{1}(n-1,m)
\nonumber \\
&+&s_{2b}{\cal P}_{2}(n+1,m) +r_b {\cal P}_{1}(n+1,m)
+ \omega_{12}{\cal P}_{1}(n,m)+\alpha {\cal P}_{2}(n,m-1)
+\beta {\cal P}_{2}(n,m+1) ~~(m > n).
\label{eq-master2}\end{aligned}$$ Note that the rates depend on the separation $m-n$; this notation is omitted for clarity. We assume the interaction potential is the same for both chemical states, so that the position-dependent NA opening and closing rates $\alpha$ and $\beta$ are independent of the chemical state.
Next we change variables to work with the difference $j = m-n$ and midpoint $l= 2 l^{'}=m+n$ positions of the helicase-junction complex. Rewriting eqns. and we have $$\begin{aligned}
\frac{d{\cal P}_{1}(j,l)}{dt} &=& -(\alpha + \beta+
\omega_{12}+s_{1f}+s_{1b}+q_f+r_b){\cal P}_{1}(j,l) +
s_{1f}{\cal P}_{1}(j+1,l-1) + r_f {\cal P}_{2}(j+1,l-1)
\nonumber \\
&+&s_{1b}{\cal P}_{1}(j-1,l+1) +q_b {\cal P}_{2}(j-1,l+1)
+ \omega_{21}{\cal P}_{2}(j,l)+\alpha {\cal P}_{1}(j-1,l-1)
+\beta {\cal P}_{1}(j+1,l+1) \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(j > 0).
\label{eq-master1j}\end{aligned}$$ and $$\begin{aligned}
\frac{d{\cal P}_{2}(j,l)}{dt} &=& -(\alpha + \beta+
\omega_{21}+s_{2f}+s_{2b}+r_f+q_b){\cal P}_{2}(j,l) +
s_{2f}{\cal P}_{2}(j+1,l-1) + q_f {\cal P}_{1}(j+1,l-1)
\nonumber \\
&+&s_{2b}{\cal P}_{2}(j-1,l+1) +r_b {\cal P}_{1}(j-1,l+1)
+ \omega_{12}{\cal P}_{1}(j,l)+\alpha {\cal P}_{2}(j-1,l-1)
+\beta {\cal P}_2(j+1,l+1) \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(j > 0).
\label{eq-master2j}\end{aligned}$$ Again, the rates vary with $j$. However, the rates are independent of $l$, so we can sum over the position of the complex center of mass: $$\begin{aligned}
P_{1}(j) = \sum_{l} {\cal P}_{1}(j,l) \nonumber \\
P_{2}(j) = \sum_{l}{\cal P}_{2}(j,l)
\label{eq-defp}\end{aligned}$$ Applying the sum over $l$ to eqns.(\[eq-master1j\]) and (\[eq-master2j\]) we find $$\begin{aligned}
\frac{d P_{1}(j)}{dt} &=& -(\alpha + \beta+
\omega_{12}+s_{1f}+s_{1b}+q_f+r_b) P_{1}(j) +
(s_{1f}+\beta) P_{1}(j+1) + r_f P_{2}(j+1)
\nonumber \\
&+&(s_{1b}+\alpha) P_{1}(j-1) +q_b P_{2}(j-1)
+ \omega_{21} P_{2}(j) .
\label{eq-dif1}\end{aligned}$$ and $$\begin{aligned}
\frac{d P_{2}(j)}{dt} &=& -(\alpha + \beta+
\omega_{21}+s_{2f}+s_{2b}+r_f+q_b) P_{2}(j) +
(s_{2f}+\beta) P_{2}(j+1) + q_f P_{1}(j+1)
\nonumber \\
&+&(s_{2b}+\alpha) P_{2}(j-1) +r_b P_{1}(j-1)
+ \omega_{12} P_{1}(j).
\label{eq-dif2}\end{aligned}$$ We consider the total probability by summing eqns. and . Defining the total probability current $$I(j)= \alpha P(j) -\beta P(j+1) + (s_{1b}+r_b) P_1(j)+(s_{2b}+q_b) P_2(j)
-(s_{1f}+q_f) P_1(j+1) - (s_{2f}+r_f) P_2 (j+1),
\label{eq-curr}$$ the total probability satisfies $$\frac{d P(j)}{dt} =-I(j)+I(j-1).
\label{eq-dif}$$
At steady state $P(j)$ is time independent, so $I(j) = I(j-1)$. Further, since $U(j) \rightarrow \infty$ as $j\rightarrow -\infty$, this constant probability flux must be zero, i.e., $I(j) = 0$ for all $j$.
Adding the two eqns. (\[eq-master1j\]) and (\[eq-master2j\]) and defining ${\cal P}(j,l) = {\cal P}_{1}(j,l) + {\cal P}_{2}(j,l)$, we get $$\begin{aligned}
\frac{d{\cal P}(j,l)}{dt} &=& -(\alpha + \beta) {\cal P}(j,l) + \alpha
{\cal P}(j-1,l-1) + \beta P(j+1,l+1)
+( s_{1f} + q_f){\cal P}_{1}(j+1,l-1) \nonumber \\
&+& (r_f+s_{2f}) {\cal P}_{2}(j+1,l-1) +(s_{1b}+r_b){\cal
P}_{1}(j-1,l+1) +(q_b+s_{2b}) {\cal P}_{2}(j-1,l+1) \nonumber \\
&+& \omega_{21}{\cal P}_{2}(j,l)+ \omega_{12}{\cal
P}_{1}(j,l) -(\omega_{12}+s_{1f}+s_{1b}+q_f+r_b) {\cal P}_{1}(j,l)
\nonumber \\
&-&(\omega_{21}+s_{2f}+s_{2b}+r_f+q_b){\cal P}_{2}(j,l).
\label{eq-masterjl}\end{aligned}$$ The probability distribution in $l$ at time $t$ is $$\Pi(l;t) = \sum_{j} {\cal P}(j,l;t)
\label{eq-defpi}$$ Note that, by definition, $\Pi(l;t)$ is independent of the chemical state of the helicase For times much longer than the relaxation time of the difference variable $j$, we can assume $${\cal P}_{\mu}(j,l) = P_{\mu}(j) ~\Pi(l) \quad (\mu = 1 ~{\rm or} ~2)$$ Starting from the eqn. (\[eq-masterjl\]), one can derive $$\frac{d\Pi(l)}{dt} = u \Pi(l-1) - (u+w) \Pi(l) + w \Pi(l+1)
\label{eq-pieqn}$$ where $$\begin{aligned}
u = \sum_{j} \alpha P(j) + (s_{1f}+q_f) { P}_{1}(j) +
(s_{2f}+r_f ) { P}_{2}(j),
\label{eq-u}\end{aligned}$$ and $$\begin{aligned}
w = \sum_{j} \beta { P}(j) + (s_{1b}+r_b) { P}_{1}(j) +
(s_{2b}+q_b) { P}_{2}(j).
\label{eq-w}\end{aligned}$$ Thus the motion of the helicase-junction complex is a combination of drift and diffusion. Note that in the special case $u = w$ the drift vanishes and the dynamics of $l$ becomes purely diffusive.
As in ref. [@betterton05a], the average speed of unwinding is $v =
\frac{1}{2}(u-w)$, or $$v=\frac{1}{2} \sum_{j} (\alpha-\beta) P(j) +
(s_{1f}+q_f-s_{1b}-r_b) P_{1}(j) +
(s_{2f}+r_f -s_{2b}-q_b) P_{2}(j).
\label{eq-v}$$ Similarly, the diffusion coefficient is $D =\frac{1}{4} (u+w)$, which is $$D=\frac{1}{4} \sum_{j} (\alpha+\beta) P(j) +
(s_{1f}+q_f+s_{1b}+r_b) P_{1}(j) +
(s_{2f}+r_f +s_{2b}+q_b) P_{2}(j).
\label{eq-d}$$ Note that if the sliding transitions represent unbiased diffusion, then the forward and backward rates $s_{\mu f}$ and $s_{\mu b}$ are equal. Then the terms involving the sliding rates drop out from the expression for $v$ but not from that for $D$.
Solution {#sec:soln}
========
In order to evaluate the expressions for the unwinding velocity and diffusion coefficient, we must determine $ P_{1}(j)$ and $ P_{2}(j)$ in terms of the rate constants. Consider the result of summing eqns. and over $j$ to determine equations for the total probability of being in state 1, $P_1$ and the total probability of being in state 2, $P_2$. We can write these equations as $$\begin{aligned}
\frac{d P_{1}}{dt} &=& -k_{12} P_{1} + k_{21}
P_{2}, \\
\frac{d P_{2}}{dt} &=& -k_{21} P_{2} + k_{12} P_1,
\label{eq-jsum}\end{aligned}$$ where the rate constant $k_{12}$ depends on $\omega_{12}$, $q_f$, and $r_b$, and $k_{21}$ depends on $\omega_{21}$, $r_f$, and $q_b$. The steady-state solution has $P_2 = k_{12}/k_{21} P_1$.
This observation suggests a translationally invariant solution for $P_1(j)$ and $P_2(j)$ when the rates are constant. We consider the case where the relative probability of being in state 1 or 2 is translationally invariant (independent of $j$). This must occur if the hopping rates are constant or spatially vary in the same way (for example, if states 1 and 2 have the same interaction potential with the dsNA). Since we are primarily interested in a passive helicase with constant rates, we will focus on this case. Because of the translational invariance, the probability in state 2 is a multiple of the probability in state 1, so that $$P_{2}(j)=\gamma P_{1}(j).
\label{eq-prop}$$
The zero-current relation requires that eqn. equal zero, which requires $$(\beta+s_{1f}+q_f) P_1(j+1) + (\beta +s_{2f}+r_f) P_2
(j+1)=(\alpha+s_{1b}+r_b) P_1(j)+(\alpha+s_{2b}+q_b) P_2(j) .
\label{eq-recur}$$ We can plug in to eqn. and solve for the unknown constant $\gamma$. We can rewrite eqn. as a recursion relation that relates $P_1(j+1)$ to $P_1(j)$: $$\frac{P_1(j+1)}{P_1(j)} = \frac{\alpha (1 +\gamma) +s_{1b}+r_b +\gamma
(s_{2b}+q_b) }{ \beta(1+\gamma)+s_{1f}+q_f + \gamma
(s_{2f}+r_f)} = c.
\label{eq-recur1}$$ Note that $c$ is a function of $\gamma$. While it is possible to solve coupled equations for $c$ and $\gamma$ in general, the resulting expressions are long and not useful for developing intuition. Instead, we use the approximation relevant for helicases that $\alpha$ and $\beta$, the opening and closing rates of the NA, are several orders of magnitude larger than the other rates in the problem (see reference [@betterton05a], where experimental data from reference [@bonnet98] was used to estimate the opening rate $\alpha \sim 10^7$ s$^{-1}$; other rates in the problem are of order $10^2$ s$^{-1}$). In this case, eqn. reduces to $$\label{eq-c}
c\approx \frac{\alpha}{\beta}.$$ Throughout the remainder of this paper, we will use this approximate value of $c$. Note that because $\alpha$ and $\beta$ are constant, $c$ is also constant and eqn. shows that $P_1(j)$ has power-law decay with increasing $j$ (as in the BJ model for a passive helicase [@betterton05a]).
Using eqns. and in eqn. at steady state, and imposing the requirement that $P_1(j)$ cannot vanish for arbitrary $j$, we find a unique expression for $\gamma$: $$\gamma= \frac{s_{1f}(1-c)-s_{1b}(c^{-1}-1)+r_b+q_f+\omega_{12}}
{cr_f+c^{-1}q_b+\omega_{21}}.
\label{eq-gamma}$$
With this result, we can evaluate eqns. and and express $v$ and $D$ in a fashion analogous to the expressions in the simpler BJ model: $$v = \frac{1}{2} \sum_j {\cal P}_1(j) (a + k^+ - b - k^-),
\label{eq-vfin}$$ $$D = \frac{1}{4} \sum_j {\cal P}_1(j) (a + k^+ + b + k^-),
\label{eq-dfin}$$ if we define the effective rates $$\begin{aligned}
a &=& \alpha (1+\gamma), \\
b &=& \beta (1+\gamma), \\
k^+ &=& \gamma (s_{2f}+r_f) + s_{1f}+q_f ,\\
k^- &=&\gamma (s_{2b}+q_b)+s_{1b}+r_b.\end{aligned}$$ Next we evaluate the sums in eqns. (\[eq-vfin\]) and (\[eq-dfin\]), noting that $P_1(j)=P_1 c^j$ and taking into account that for $j=1$ the rates $k^+$ and $b$ are zero. The result is $$v = \frac{c k^+ - k^-}{2(1+\gamma)},
\label{vel}$$ $$D = \frac{\alpha}{2} + \frac{c k^+ +k^-}{4(1+\gamma)}
\label{diff}$$ Equations (\[vel\]) and (\[diff\]) are the main results.
Note that under most conditions the NA opening and closing rate $\alpha$ is orders of magnitude larger than the other rates, and therefore $D \approx \alpha/2$.
Comparison with NS3 helicase {#sec:results}
============================
The NS3 helicase of the hepatitis C virus (HCV) is important for HCV replication, and is therefore a potential drug target [@frick07]. NS3 is also an interesting model helicase because it is the only currently known helicase capable of unwinding both dsRNA and dsDNA [@kim95; @tai96]. The flashing-ratchet mechanism proposed for NS3 helicase in ref. [@levin05] is a special case of the two-state model which we have developed in the preceding sections. In this section, we first briefly summarize the experimental data on NS3 helicase and their mutually contradictory interpretations which highlight the current debates in the literature. Then, we present analytical results for the special case of our model which captures the flashing ratchet mechanism. We compare these theoretical predictions with the corresponding experimental data for NS3 helicase. The comparisons are, however, limited by the contradictions between the observations in different experiments, many of which have been performed under different conditions.
Summary of experimental results on NS3 helicase
-----------------------------------------------
To compare our model to experiments on NS3 helicase, we would ideally like to know the enzyme step size, the single-strand translocation rate, and the double-strand unwinding rate—including information on how it varies with NA sequence or applied force. Interpretation of experimental data on NS3 is complicated by differences in experiments done by different research groups. Some groups study the full-length NS3 protein, including the helicase and protease domains [@serebrov04; @frick04c; @beran06; @dumont06; @myong07], while others study the helicase domain only [@preugschat96; @levin99; @levin02; @levin03; @levin04; @frick04c; @levin05]. Moreover, genetically different versions of NS3 can have different properties [@lam03b]. The NS3 protein can also function in different oligomeric states. In bulk solution experiments, full-length NS3 seems to function best as a dimer or higher-order oligomer [@tackett05], but single-molecule experiments can observe unwinding by NS3 monomers [@dumont06; @myong07]. The helicase domain NS3h appears not to form dimers in solution [@levin99; @kim98; @porter98b], but multiple copies of the protein can bind to ssNA and unwind dsNA [@levin99]. In at least one experiment, the kinetic parameters did not vary with the length of the ss tail used to load NS3h, suggesting that the helicase mechanism may not depend on whether the protein is a monomer or dimer [@levin04].
Contradictory claims have been made in the literature on the qualitative description of NS3 helicase as well as on its quantitative characteristics. First, we consider the empirical evidence for the stepping pattern and the step size of NS3 helicase. Recently a detailed computational model of NS3, based on known crystal structures, supported the idea of single-base “inchworm” motion taken by NS3 monomers. This model of Zheng et al. proposes a major protein conformational change which is triggered by ATP binding and is coupled to forward motion of the helicase [@zheng07]. Models based on structural studies of NS3 have suggested single-base steps [@yao97; @kim98]. Similarly, structures of the distantly related Hel308 helicase, which shows some structural similarities to NS3, supports the idea of a ratchet-like mechanism during the ATP cycle [@buttner07]. However, most experimental efforts to determine the step size don’t support single-base steps. Bulk kinetic experiments have given a kinetic step size of 9-17 basepairs, depending on protein form and unwinding substrate [@levin04; @serebrov04; @beran06]. Single-molecule experiments on monomers of full-length NS3 have suggested a step size of 11 basepairs with 3 basepair substeps [@dumont06] or 3 basepairs with 1 basepair substeps [@myong07]. The most recent single-molecule work has proposed that the fundamental step size is one basepair, with pauses occurring less frequently as part of the ssNA bound to the helicase occasionally “rips” off [@myong07].
Next we summarize the current estimates of ss translocation rate and the speed of double-strand unwinding by NS3 helicase. The maximum ss translocation rate can be estimated from experiments that measure the ATP hydrolysis rate. In one experiment, the NS3h rate of ATP hydrolysis had a maximum $k_{cat}$ of 80 s$^{-1}$ in the presence of the single-stranded oligo dU$_{18}$ [@preugschat96]. Assuming that during ss translocation the helicase hydrolyzes 1 ATP per step, this measurement sets an upper bound on the ss translocation velocity of 80 bases s$^{-1}$. The double-strand unwinding velocity of NS3 has been estimated from bulk and single-molecule experiments. In one single-turnover bulk kinetic study, the maximum unwinding rate of NS3h was 2.7 bp s$^{-1}$ [@levin04]; similar results were found by another group [@frick04c]. Full-length NS3 may unwind at higher velocities, up to 16.5 bp s$^{-1}$ [@serebrov04; @frick04c]. In single-molecule experiments with applied force, full-length NS3 monomers unwind at force-independent rates of 50 bp s$^{-1}$ [@dumont06]. This relatively high velocity may be possible because of the applied force that reduces the energetic cost of opening the NA. In single-molecule FRET experiments on full-length NS3 monomers where no force is applied, an unwinding rate of $k\approx 0.9$ s$^{-1}$ was measured for one base pair substeps [@myong07]—a value closer to the bulk value measured for NS3h.
Finally, we examine the experimental data to investigate whether the unwinding by NS3 helicase is active or passive. The dependence of the unwinding rate on the base-pair binding free energy was measured both in single-molecule and bulk experiments. In the work of Dumont et al., the RNA unwinding rate of full-length NS3 monomers was approximately independent of applied force in the range 9-17 pN [@dumont06]. In this experiment, the applied force was relatively high: the double strand melted at a force of 20 pN. In single-molecule experiments using a similar experimental setup, Cheng et al. [@cheng07] observed a significant effect of varying the RNA sequence on the NS3 unwinding rate. This observation of Cheng et al. indicates that a passive unwinding mechanism may not be adequate to explain the behavior of full-length NS3 helicase. Further, the apparent contradiction between the observations of Cheng et al. [@cheng07] and Dumont et al. [@dumont06] may be reconciled if we abandon the simple physical picture in which the base-pair binding free energy can be altered in a similar way by applied force or by changing the sequence. Recent bulk measurements examined the effects of sequence variation on the unwinding rate of NS3h [@donmez07]; this work is discussed below where we compare our theoretical predictions to experimental results.
In order to motivate our minimal model for the NS3 helicase, we now discuss the affinity of NS3h to NA and its modulation during the ATP hydrolysis cycle. Binding experiments on NS3h found that when the helicase is bound to an ATP analogue, it binds to NA more weakly than when not bound to ATP or ADP [@levin99; @levin02]. The change in binding free energy is approximately 6 $kT$ at room temperature (15 kJ mol$^{-1}$) [@levin05]. In addition, the affinity of NS3h for ADP is low, so release of hydrolysis products is expected to be rapid [@levin02]. These observations are the basis of the proposed flashing-ratchet mechanism of NS3h. (However, we note that another work has found no dependence of NA binding on the ATP hydrolysis state [@frick04c]; the source of this difference is unclear.)
![Schematic of the simplified model that represents a flashing ratchet. The protein can exist in either of two chemical states (labeled 1 and 2) at each lattice site (labeled $n$). Sliding transitions (where $n$ changes but the state does not) occur only in state 2 at rate $s_{2}$. Chemical transitions (where the state changes but $n$ does not) occur at rates $\omega_{12}$ (for the transition from 1 to 2) and $\omega_{21}$ (for the transition from 2 to 1). A coupled transition (where both the state and $n$ change) occurs at rate $r_f$ (for the transition from 2 to 1 coupled to forward motion). The nucleic acid single strand-double strand junction is at site $m$. The junction moves toward increasing $m$ when the NA opens by one base (rate $\alpha$) and toward decreasing $m$ when the NA closes (rate $\beta$). []{data-label="fig-ns3model"}](ns3_model){width="3.25in"}
Flashing-ratchet model of NS3 helicase
--------------------------------------
Here we consider a special case of our model which corresponds to a flashing ratchet mechanism. Levin et al. proposed that NS3 helicase switches between two states: one tightly bound to the ssNA, the other weakly bound [@levin03; @levin05]. This scenario is referred to in the physics literature as a flashing ratchet [@julicher97]. When applying the flashing ratchet scenario to NS3, the tightly bound state is represented by a periodic sawtooth potential (with periodicity of one ssNA base pair) and the weakly bound state is represented by a uniform (weakly position-independent) potential [@levin05]. When comparing to the flashing-ratchet scenario, we will consider state 1 to represent the strongly bound (S) state and state 2 the weakly bound (W) state. By comparing the theoretical predictions for this special case of our model with the experimental data for NS3 helicase, we test whether or not NS3 follows the flashing ratchet mechanism.
We assume that no sliding is possible in the tightly-bound state 1, so $s_{1f}=s_{1b}=0$, and that the sliding is unbiased in state 2, so $s_{2f}=s_{2b}=s_2$. To connect with the flashing-ratchet scenario and for simplicity, we assume that the rates $q_f=q_b=r_b=0$ (see fig. \[fig-ns3model\]). With these assumptions, we find that the rate of ss translocation is (from eqns. and ) $$v_{ss} = \omega_{12} \frac{r_f}{r_f+\omega_{12}+\omega_{21}},
\label{vssns3}$$ and the rate of ds unwinding is $$v_u= \frac{\omega_{12}}{2} \frac{ (cr_f-(1-c)s_2)}
{cr_f+\omega_{12} +\omega_{21} }. \label{vns3}$$
The excitation rate $\omega_{12}$ is associated with ATP binding, and so is assumed proportional to ATP concentration. Therefore we write $\omega_{12}=\omega_o \mbox{[ATP]}$. The rates $\omega_{21}$ and $r_f$ represent the relaxation from the weakly bound to the tightly bound state that occurs after ATP hydrolysis, product release, and diffusion in the weakly bound state. For a flashing ratchet, a high rate of forward motion will occur when the positions of the energy barriers and the time constants are such that forward movement (rate $r_f$) and return to the same place after one cycle (rate $\omega_{21}$) occur with equal probability. To match this optimal case, we therefore assume that $\omega_{21}=r_f$. Further, we assume that the sliding rate $s_2$ is small compared to the other rates; for concreteness we will suppose $s_2 = \epsilon r_f$ with $\epsilon=0.1$ unless otherwise stated. The velocities then become $$\begin{aligned}
v_{ss} &=& \frac{r_f \omega_o \mbox{[ATP]} }{\omega_o
\mbox{[ATP]}+2r_f}, \label{vssl}\\
v_u&=& \frac{(c-\epsilon(1-c))}{2} \frac{ r_f \omega_o \mbox{[ATP]}}
{\omega_o \mbox{[ATP]} + (1+c) r_f }. \label{vns3s} \end{aligned}$$
Both $v_{ss}$ and $v_u$ are consistent with the Michaelis-Menten equation for enzyme kinetics, but with slightly different forms. Their ratio is $$\frac{v_u}{v_{ss}} = \frac{(c-\epsilon(1-c))}{2} \frac{\omega_o
\mbox{[ATP]}+2r_f}{\omega_o \mbox{[ATP]} + (1+c) r_f }.$$ In other words, we predict that the ratio of the unwinding velocity to the single-strand translocation velocity depends on ATP concentration. If we average over sequence variation in DNA [@betterton03], we get the estimate $c=\alpha/\beta \approx 1/7$. For the purpose of quantitative illustration of the variation of $\frac{v_u}{v_{ss}}$ with ATP concentration, let us assume $\epsilon=1/10$. Then, $\frac{v_u}{v_{ss}} \approx 0.029$ at high ATP concentration and $\frac{v_u}{v_{ss}} \approx 0.05$ at low ATP concentration. This suggests that the ratio of the unwinding velocity to the single-strand translocation velocity could vary significantly with ATP concentration—the change is almost a factor of 2 for this example.
Next, we estimate $v_u$ and $v_{ss}$ for NS3 helicase. The single-strand translocation and unwinding velocities are fully determined by the parameters $c$, $r_f$, $\omega_o$, $\epsilon$, and ATP concentration; we now extract estimates of $r_f$ and $\omega_o$ from experimental data. In experiments at high ATP concentration and in the presence of ssNA, NS3h shows a maximum ATP hydrolysis rate of 80 s$^{-1}$ [@preugschat96]. If we take this value as the limiting ss-translocation rate and assume single base-pair steps, then $v_{ss}
= $ 80 nt s$^{-1}$ in the limit of high ATP concentration. Using this estimate of $v_{ss}$ in eqn. (\[vssl\]), we get the estimate $r_f=80$ s$^{-1}$. This, in turn, implies that at high ATP concentration the unwinding velocity $v_u \approx 0.029 v_{ss} \approx
2.3 $ bp s$^{-1}$. This value is comparable to the values of 2.7 bp s$^{-1}$ [@levin04] found for NS3h and $0.9$ bp s$^{-1}$ found for the one-bp substeps of full-length NS3 [@myong07]. We note that the unwinding velocity $v_u \ll v_{ss}$, as should be expected for this model which assumes a passive helicase mechanism. Experiments studying how NS3 ATPase activity [@preugschat96] and unwinding [@dumont06] vary with ATP concentration found a similar Michaelis constant $K_m \approx$ 90 $\mu$M. Using this value of $K_m$ in eqn. , we estimate $\omega_o=2 r_f/K_m \approx 1.8 \
\mu$M$^{-1}$ s$^{-1}$.
![Dependence of the unwinding velocity on the base-pair binding free energy. The reference state is a value $c=1/7$, which represents a sequence-averaged value for DNA. The additional destabilization energy $\Delta G$ (in units of $kT$) represents a free energy change that favors NA opening. When $\epsilon$ increases, the dependence of the velocity on $\Delta G$ becomes more pronounced. However, decreasing $\epsilon$ can not flatten the curve indefinitely.[]{data-label="fig-forcedep"}](forcedep){width="3.25in"}
The only remaining unknown parameter is $\epsilon=s_2/r_f$, the ratio of the sliding rate to the forward transition rate. A smaller value of $\epsilon$ means that the sliding transitions in the weakly bound state are less probable (see fig. \[fig-ns3model\]). A higher value of $\epsilon$ means that sliding transitions in the weakly bound state are more probable. This parameter has an important effect on the dependence of the helicase velocity on the base-pair binding free energy.
To study the effects of varying the base-pair binding free energy, we focus on the limit of high ATP concentration. In this case, if $\Delta
G$ is the free energy of destabilization of base-pair binding, the parameter $c=\alpha/\beta$ varies according to $c = c_o e^{\Delta G}$. Therefore, at high ATP concentration, the unwinding velocity varies as $$\lim_{[ATP] \to \infty}v_u= \frac{(c_o(1+\epsilon)e^{\Delta G}-\epsilon)}{2} r_f.$$ The unwinding velocity increases exponentially if the NA is destabilized, as one would expect for a passive helicase. However, the precise shape of the curve of unwinding velocity versus $\Delta G$ depends on $\epsilon$. In the limit $\epsilon \to 0$, which physically means no helicase sliding transitions occur in the weakly bound state, the unwinding velocity varies with $\Delta G$ as a simple exponential: $$\lim_{[ATP] \to \infty, \epsilon \to 0 }v_u= \frac{c_o r_f }{2}
e^{\Delta G}.$$
As $\epsilon$ increases, the helicase can slide in the weakly bound state. This allows more rapid unwinding by the helicase: when the dsNA is destabilized, the ds base just ahead of the helicase has an increased probability to be open. Rather than wait for the helicase chemical transitions to move forward, the helicase can take advantage of this increased junction open probability and slide forward. This allows the steeper rate of increase of $v_u$ with $\Delta G$ seen in fig. \[fig-forcedep\]. This prediction is qualitatively consistent with the result of Tackett et al. [@tackett01], who found that full-length NS3 unwound double strands with higher melting temperatures less efficiently. However, in the single-molecule experiments of Dumont et al. the unwinding rate of full-length NS3 monomers was practically independent of applied force in the range 9-17 pN [@dumont06]. This disagrees with the prediction of this model, if the only effect of the applied force is to change the binding free energy per base pair. However, this physical interpretation is clearly not valid, because recent experiments from the same lab find a significant variation in the RNA unwinding rate of full-length NS3 with the variation of the base composition of the RNA [@cheng07]. Reconciliation of the apparent contradictions in these experimental observations is possible by assuming an active helicase mechanism which, however, is not incorporated in the current version of our model. Analyzing data from bulk experiments, Donmez et al. [@donmez07] claimed that the the variation of NS3h unwinding velocity with base-pair binding free energy is inconsistent with a passive helicase mechanism. However, this conclusion is drawn from an analysis based on a reported single-strand translocation velocity of 6.4 bases s$^{-1}$, which is much lower than the value of 80 bases s$^{-1}$ mentioned above. A ss translocation rate of 80 bases s$^{-1}$ is an upper limit, assuming the helicase hydrolyzes 1 ATP per single-base step. If the helicase on average hydrolyzes $>1$ ATP per step, the ss translocation rate would be lower. A lower ss translocation rate would lead to an even larger disagreement between the passive helicase model we presented and the experimental data. We believe that a conclusive comparison between our model of a flashing-ratchet mechanism for NS3 helicase and the experimental data is not possible because of the contradictory reports of experimental studies.
Conclusion {#sec:disc}
==========
In this paper we have developed a general model of unwinding of a double-stranded nucleic acid molecule by a helicase motor. To capture some of the key features of the helicase mechanochemical cycle, we have modeled helicase switching between two chemical states. In this model, the sites of a discrete lattice represent the positions of the individual bases on the ssNA. At any spatial position, the helicase can exist in either of the two allowed chemical states. This model should be generally applicable to helicases where one of the transitions in the mechanochemical cycle is much slower than the other transitions. In this work, we have considered only a passive helicase mechanism—the helicase at the junction must wait for thermal fluctuations to open the dsNA before it can advance. In future work, it would be valuable to extend the model to include active destabilization of the dsNA by the helicase.
To compare the model in detail to experimental data, we focused on a special case which captures the flashing-ratchet mechanism proposed for the NS3 helicase [@levin05]. Solving the master equations for this model at steady state, we have calculated the speed of unwinding and the speed of single-strand translocation. The ratio of the unwinding velocity to the ss translocation velocity varies with ATP concentration as well as with the base-pair binding free energy.
Our comparison to experimental data on NS3 helicase suggests that the model captures some features of the experiments. However, the experimental literature on NS3 contains contradictory results. This may be a result of the different genetic variants, protein truncations, oligomeric states, substrates, and buffer conditions used by different laboratories. A set of detailed experiments by different labs under consistent conditions may be important to fully understand the unwinding mechanism of NS3 helicase.
[**Acknowledgments**]{}: The authors thank Frank Jülicher for several useful suggestions. DC also acknowledges support from the Council of Scientific and Industrial Research (India) and the Visitors Program of the Max-Planck Institute for Physics of Complex Systems, Dresden (Germany). MDB acknowledges support from the Alfred P. Sloan Foundation and the Butcher Foundation.
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[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
|
---
author:
- Hitoshi Yamamoto
- Isamu Okada
- Satoshi Uchida
- Tatsuya Sasaki
bibliography:
- 'refs\_knockout.bib'
title: 'A norm knockout method on indirect reciprocity to reveal indispensable norms[^1]'
---
Introduction {#introduction .unnumbered}
============
Reciprocity is a fundamental mechanism that underlies all cooperative societies. Theoretically it is well known that direct reciprocity, typified by the “I’ll help you if you help me” attitude, promotes cooperative regimes [@Trivers1971; @Axelrod1981]. However, in recent societies that have high relational mobility, indirect reciprocity such as “I’ll help you and somebody else will help me” plays a more important role in promoting cooperation. Indirect reciprocity has therefore been the focus of much research in the interdisciplinary fields in recent decades [@Alexander1987; @Sugden1986; @kandori1992; @wedekind2000; @Panchanathan2004].
Many theoretical studies on indirect reciprocity have explored norms that become evolutionarily stable against defection and the invasion of free riders, and several typical norms have been proposed [@Ohtsuki2004a; @Nowak2005; @Ohtsuki2006; @Takahashi2006]. These approaches have clarified the robust norms that can maintain the cooperative regime. The norms in the studies on the indirect reciprocity are regarded as assessment rules that label the other’s action as either Good or Bad. They include tolerant norms that assess cooperative behaviors toward defectors as good[@Sugden1986] and strict norms that assess such behaviors as bad[@pacheco2006]. Other theoretical studies analysing the global dynamics of norms assume that at most a few robust norms are shared in the population [@Ohtsuki2007; @uchida2010competition; @uchida2010effect].
Their approaches have clarified the robustness of the norms against invasion of other norms including free riders when the norms are acceptable in the population. However, little is known about a process by which gradual changes toward cooperation occur as new norms emerge and compete, which is to say, the co-evolutionary process of norm-diversity and cooperation. A study on the indirect reciprocity has dealt with co-existing different norms and has analysed their frequencies in the population as a consequence of a dynamical process[@Brandt2004]. In the study, each individual keeps a private image of everyone else and errors of perception and implementation are included in a limited strategy space. Although they have considered some action rules and assessment rules, all possible norms in indirect reciprocity have not been studied all-together. How cooperation evolves cannot be fully understood unless the evolution of norms is also considered.
It is thus a challenging task to theoretically understand how cooperation can be formed even under a collection of norms in a social system. How is the co-existence of cooperation and diversity possible at all? Are there any indispensable norms needed to facilitate the evolution of cooperation in the melting pot of norms, even though some norms never become dominant? Do norms that could be accepted as a result of the co-evolutionary process have common aspects? These questions can be addressed only if all possible norms are considered, and the combination of norms governing a group can evolve.
Here we explore the dynamics of co-evolution of cooperation by using different social norms. The process of the evolution of norms has a transition from stricter to tolerant norms. Additionally, we find a set of norms that seem not to have an impact on promoting cooperation, but are fundamental to allow a transition to a cooperative regime from a defective regime.
Results {#results .unnumbered}
=======
Agent-based simulations [@Nigel1999; @Roberts2008] are an optimal tool to tackle the challenge outlined above. See Methods for the details of our agent-based model described by the ODD protocol[@Grimm2010]. Using an evolutionary game theoretical framework and constructing an interaction model based on players’ private rules and local information, we model a giving game to elucidate the dynamics of the evolution of cooperation amid the coexistence of diverse norms (Fig. \[fig:framework\]). We conducted numerical simulations of all 16 possible norm combinations that could react to the four combinations of assessment criteria to clarify the dynamics of the evolution of cooperation from the melting pot of diverse norms. Figure \[fig:alternation1\]a shows time-series graphs of each norm’s population and cooperation ratio. As shown, the majority undergo an alternation from strict to tolerant norms, mostly in the order of $\rm{SH \rightarrow SJ \rightarrow ST}$. Figure \[fig:alternation2\]a shows the transition in the norm with the greatest population ratio. In many cases, the majority transitioned from the state where strict SH [@Takahashi2006] was the majority to SJ [@kandori1992; @pacheco2006]. Afterwards, the majority norm changed to tolerant ST [@Sugden1986; @leimar2001; @Panchanathan2003] and ALLG. In contrast, as shown in Figs. \[fig:alternation1\]b and \[fig:alternation2\]b, in an environment with errors, alternation from strict norms to tolerant norms was observed. However, the likelihood of going through SJ decreased. Alternation paths through IS [@nowak1998; @Nowak1998b], which could not be seen in an environment without errors [@Lotem99], increased. It is important to note here that similar paths toward cooperation are observed when only ALLB-individuals are initially assumed. New norms are created during the evolutionary process at the same time cooperation evolves. This indicates that cooperation and diversity of norms jointly evolve in the model.
Why does the alternation of norms emerge? For one thing, in states in which defection is dominant, ALLB (BBBB) and SH (GBBB) coexist and jointly form the majority. However, BGBB and IS (GGBB) continue to exist as the minority. The characteristic of these groups is having the evaluation rule of \*\*BB. Evaluation rule \*\*BB assesses donors that took D as B, regardless of the evaluation of the recipient. In states in which defection is dominant, those who adopt \*\*BB strategies consider many partners as B. As a result, cooperation does not occur for the most part. The ALLB and SH norms thus survive because they do not lower their own cost. On the other hand, after cooperation is achieved, ALLG (GGGG), ST (GGBG), IS (GGBB), and GGGB coexist. The common characteristic of these norms is having the evaluation rule of GG\*\*. Thus, reciprocally cooperating norms survive. Because SJ (GBBG), which becomes the majority temporarily when the cooperation ratio rises in an environment without errors, does not belong to either group, it cannot stably exist. Also, it is rare that SJ makes up the majority temporarily in an environment with errors. Meanwhile, because IS belongs to both norm groups with \*\*BB and GG\*\*, IS can constantly exist.
We discover several norms that are indispensable to the evolution of cooperation. Reputation-based cooperation cannot emerge without indispensable norms. To elucidate indispensable norms for the evolution of cooperation, we propose a novel analysis using the norm knockout method. This method enables us to determine which norms are indispensable for the evolution of cooperation. The norm knockout method is inspired by the targeted gene knockout technique used in genetic engineering [@Strepp1998]. Gene knockout, a genetic technique in which one of an organism’s genes is made inoperative, is used to research genes whose sequences are known but whose functions are not well-understood. Researchers infer the gene’s function from differences between the knockout animal and a normal animal. For simulating evolution, we utilized a method that removed only one particular norm from the population to understand whether that norm is an indispensable one that plays a critical role in the evolution of cooperation.
Figure \[fig:knockout\] shows the cooperation ratio when a particular norm is knocked out. Regardless of whether there is an error, if SH or IS is knocked out, cooperat
ion does not evolve at all. We define indispensable norms in the evolution of cooperation as the norms that, when knocked out, have an average cooperation ratio of less than 0.1 after 1,000 generations. In an environment with no errors, SH and IS are indispensable norms. In an environment with errors, SH, IS, and ST are indispensable norms.
When an indispensable norm is knocked out, cooperation does not evolve. When cooperation evolves, alternation from strict norms to tolerant norms was observed, as shown in Figs. \[fig:alternation1\] and \[fig:alternation2\]. To analyse whether alternation also occurs when a norm is knocked out, the population ratio of norms when typical norms are knocked out is displayed as time-series graphs (see Fig. \[fig:knockout\_time\]). Figure \[fig:knockout\_time\] shows the results in the cases where SH or IS were knocked out. We discovered that the first condition for the necessary process when cooperation evolves is whether SH can antagonize ALLB. No norm that resists the invasion of ALLB appears in a society in which SH does not exist. Also, in a society in which IS does not exist, SH cannot antagonize ALLB. We found that IS is a norm indispensable for SH to resist ALLB.
Discussion {#discussion .unnumbered}
==========
Our model offered two major findings on the evolution of cooperation on indirect reciprocity. On the one hand, the most essential contribution is the discovery of indispensable norms by the norm knockout method. By using the norm knockout method, we were able to elucidate the existence of norms indispensable for the evolution of cooperation from a melting pot of norms. Regardless of the existence of errors, SH and IS were indispensable norms. In addition, in an environment with errors, ST is an indispensable norm. Interestingly, SH and IS are reconciled to the minorities after the cooperative regime emerges while they temporarily become major norms in the process of dynamics. We call such minority norms required for the evolution of cooperation “unsung hero norms”. The results clearly illustrate the two roles of norms: one to catalyse a cooperative regime and the other to maintain the regime. Norms having the GG\*\* for the evaluation rule play the latter role.
On the other hand, we discovered alternation of norms. Recent analysis of evolutionary stability against the invasion of free riders could identify neither superiority among norms nor the process on the path to cooperation. Among studies on indirect reciprocity, ours is the first exhaustive theoretical analysis on all possible norms, although several studies have addressed the comparison of two types of reciprocal norms [@uchida2010competition; @uchida2010effect; @matsuo2014]. Others analyse the alternation of norms in direct reciprocity [@lindgren1992; @Zagorsky2013; @Berg2015]. We find the alternation of the norms and also discover the indispensable norms that are required to foster indirect reciprocity.
An empirical study [@Swakman2016] supports the co-existence of various norms in the cooperative regime and indicates that the ST norm plays a more important role in human cooperation than SJ, which is consistent with our simulation. This is because we show that the SJ norm cannot survive in the cooperative regime, while the ST one can. Our approach may provide deep insight on the evolution of cooperation because several norms absolutely play an essential role in order to evolve cooperation even though, on the surface, it seems as though they are not directly leading to the evolution of the cooperation.
The present work considers a single action rule (cooperate with Good, defect with Bad) to stress the role of multiple assessment rules. However, the other papers stress the role of multiple co-existing action rules [@Brandt2005; @Ohtsuki2007; @Santos2016; @Sasaki2016]. Integrating the multiple assessment and action rules may be a useful extension of this paper. We analyse what happens when one norm is absent from the population; however, we have not analysed all the indispensable combinations of norms yet. Extending the norm knockout method to combinations of norms may also be a useful extension of this paper.
Methods {#methods .unnumbered}
=======
In this section, we describe the details of our agent-based model that uses the norm knockout method. The following model description follows the ODD protocol[@Grimm2010].
Purpose {#purpose .unnumbered}
-------
The aim of the model is to understand the dynamics of norms during the evolution of cooperation, and to find indispensable norms without which cooperative societies could never emerge. In particular, we reveal the effect of these indispensable norms on indirect reciprocity using a new methodology we call the “norm knockout method”. We utilize the giving game framework[@Sigmund2010] for simulation.
Entities, state variables, and scales {#entities-state-variables-and-scales .unnumbered}
-------------------------------------
The entities in the model are agents who play as donor and recipient in the giving game with no spatial structures. The donor chooses cooperation or defection with a recipient using an image that the donor has to the recipient. An image is either Good or Bad. If a donor’s image to a recipient is Good, the donor cooperates with the recipient. If the image is Bad, the donor defects. The group size of the model is $N$. Each agent has it’s own norm and a list of images to other agents. The agent also has a probability of errors and a payoff of the game.
The norm of an agent is denoted as one of four possible “assessment combinations”, and there are two possible “alleles (G/B)” at the “locus” for each of the four assessment combinations. The first locus of the gene represents an assessment rule to an agent who cooperates with a Good recipient. The second locus represents an assessment rule to an agent who cooperates with a Bad recipient. The third locus represents an assessment rule to an agent who defects with a Good recipient. The fourth locus represents an assessment rule to an agent who defects with a Bad recipient. Incidentally, all agents evaluate themselves as Good. For instance, ALLG always assesses others as Good, and thus, its “genotype” is GGGG using the above mentioned definition of the four loci. Similarly, ALLB is described as BBBB, IS as GGBB, ST as GGBG, and SJ as GBBG.
Each agent has two different types of errors: one, the probability that the agent’s updating of its evaluation of others, Good/Bad, is inverted (errors in perception), described as $p$, and two, the probability to perform an action differently from the one prescribed by its action rule (errors in implementation), described as $q$. The evolution process of norms involves adopting a genetic algorithm [@Holland1975]. State variables and initialization in the simulation are shown in Table \[tab:variable\].
Process overview and scheduling {#process-overview-and-scheduling .unnumbered}
-------------------------------
Our simulation runs throughout $G$ generations. A generation consists of $R$ rounds. The agents play the giving game $R$ times as donor in each generation. At the end of a generation, they evolve their own norms using accumulated payoffs that are obtained in the generation. One round has two phases: (A) a phase to play giving games and (B) a phase to update images. After all agents play giving games in phase (A), all agents update their images in phase (B). In phase (A), each agent becomes a donor and each donor randomly chooses a recipient from $N-1$ players excluding itself. The donor chooses whether to give benefits to the recipient or not. At that time, the action of the donor is inverted with the probability $q$. The donor who cooperates pays cost $c$ and the recipient receives benefit $b$ $(b>c>0)$. In phase (B), each agent (set to $i$) evaluates and updates an image to the other agent (set to $j$). The new image to $j$ from $i$’s viewpoint depends on $j$’s action (C/D) as a donor in the last round and depends on the image to $k$ from the viewpoint of $i$ (G/B), where $k$ was a recipient of $j$ in the last round. At that time, the image to $j$ is inverted with the probability $p$. In the first round in a generation, $j$’s action (C/D) is regarded as random.
After $R$ rounds of the giving game are played in every generation, agents evolve their norm. The evolution process of norms involves adopting the genetic algorithm [@Holland1975]. Because each locus of norm has an independent meaning for assessment to others, the adaptive process should contain a combination of these elements rather than a string of norms. We have modeled a process of updating norms not as a string of norms but rather as four different assessment rules, which enables the norms to be interpreted as different situations depending on the norm genotype. The first, second, third, and fourth loci represent the assessment for pro-social behavior, tolerant behavior, anti-social behavior, and justified defection (punishment), respectively. Each agent randomly selects two agents from $N$ agents (including itself) to become its parents. For choosing parents, we adopt a roulette selection method. This roulette selection sets a probability distribution of all agents as $\Pi_i=(U_i - U_{min})^2/\sum_{j}^{} (U_j - U_{min})^2$ , where $U_i$ denotes the agent $i$’s accumulated payoff in a generation given by $U_i = bW - cV$, with $W$ being the number of donations $i$ received in the generation and $V$ the number of donations $i$ gave. $U_{min}$ means a minimum value of the accumulated payoffs among all. Finally, each agent updates its norm using a uniform crossover technique. With mutation rate $m$, each locus is inverted for maintaining the diversity of the norm space.
Design concepts {#design-concepts .unnumbered}
---------------
[**Basic principles**]{} An agent-based simulation is utilized to study indirect reciprocity. We explore how different combinations of norms interact to produce an evolutionary progression towards cooperation.
[**Emergence**]{} A cooperative regime in the situation of social dilemma emerges from interactions among agents who have various social norms.
[**Adaptation**]{} The agents of the model play the giving game using their images to others. The agents update their images to others using their norms every round. They evolve their own norms using accumulated payoffs that are obtained in the generation. A norm that can obtain a higher payoff can increase the population through the generation.
[**Objectives**]{} The objective of all agents is to maximize their own payoff. To maximize payoff, they change their own norm at the end of each generation.
[**Learning**]{} The agents change their norms in each generation using a genetic algorithm. The fitness of each agent is calculated from the accumulated payoff in the generation. To select the parents of the agent, the model utilizes a roulette selection method.
[**Interaction**]{} The interaction between the agents is one to one interaction. The giving game consists of the donor and the recipient. There are no spatial structures in the society.
[**Stochasticity**]{} The interaction between agents is a stochastic process because interaction partners are chosen randomly from the society. At the start of the simulation, each agent is randomly assigned a norm of all 16 norms.
[**Observation**]{} Three indexes are used for observation: average cooperation ratio in the society, the transition of norms with the greatest populations, and population ratio of each norm.
Initialization {#initialization .unnumbered}
--------------
At the start of simulation, the norm of each agent is chosen randomly from all 16 possible norm combinations. In the first round of each generation, the evaluation of all agents is initialized as Good [@Nowak1998b; @Takahashi2006; @Ohtsuki2015] and payoff of the agents is initialized as $0$.
Input data {#input-data .unnumbered}
----------
After initialization, the model does not include any external inputs, i.e., the number of agents ($N$), error ratio ($p, q$), benefit ($b$), and cost ($c$) are constant.
Submodels {#submodels .unnumbered}
---------
### The norm knockout method {#the-norm-knockout-method .unnumbered}
The norm knockout method is implemented as follows. When we knock out a particular norm, that norm is removed in the first round of each generation. Concretely, if the norm of an agent evolves into a norm that is knocked out as a result of the adopting process, the norm of the agent is changed to one of the other 15 norms randomly. In other words, the norm that is knocked out will never exist at all in the society.
Acknowledgements {#acknowledgements .unnumbered}
================
HY acknowledges Grant-in-Aid for Scientific Research (C) 15KT0133 and 26330387. HY also acknowledges Prof. Kurihara (UEC, Japan) for providing the computational resources. IO acknowledges Grant-in-Aid for Scientific Research (B) 16H03120. TS acknowledges the Austrian Science Fund (FWF): P27018-G11.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
HY initiated and performed the project. HY, IO, SU and TS designed the project, wrote the paper, and approved the submission. All authors reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
The authors declare no competing financial interests.
Tables {#tables .unnumbered}
======
Variable Description Type of variable Initial value
--------------------- --------------------------------------------- ------------------ -----------------
[**Agent**]{}
Norm Norm of agent 16 types chosen randomly
Image Images to other agents Binary (G/B) G
Payoff Accumulated payoff of the giving game Real Number 0
$p$ Errors in perception Constant {0, 0.001}
$q$ Errors in implementation Constant {0, 0.001}
[**Environment**]{}
$N$ Number of agents Constant 500
$G$ Generations of simulation Constant 1000
$R$ Times of playing giving game per generation Constant 500
$b$ The benefit of giving game Constant \[3.0, 6.0\]
$c$ The cost of giving game Constant 1
$m$ The mutation ratio Constant 0.01
: State variables and initialization in the simulation[]{data-label="tab:variable"}
Figures {#figures .unnumbered}
=======
![[**The norms of cooperation and simulation framework**]{}. [**a**]{}, (1) If the donor’s image of a recipient is Good, the donor gives the recipient something with personal cost $c$ and the recipient receives benefit $b$. Nothing happens otherwise. (2) In the Updating image phase, the observer updates the evaluation to the donor on the basis of the donor’s action (Cooperation \[C\] / Defection \[D\]) and the observer’s evaluation (Good \[G\] / Bad \[B\]) of the recipient. [**b**]{}, Each agent adopts an evaluation rule of the donor that depends on the donor’s action and the recipient’s image. This combination of Good/Bad is the norm held by the agent. There are a total of $2^4 = 16$ possible norms. In this phase, each agent evaluates and updates its image to all donors. [**c**]{}, Typical norms can be expressed in the manner shown in this table. Typical norms include Shunning \[SH\] = GBBB, Stern Judging \[SJ\] = GBBG, Image Scoring \[IS\] = GGBB, and Simple Standing \[ST\] = GGBG. SH is a strict norm where any action for a Bad recipient is assessed as Bad. ST is a tolerant norm where any action for a Bad recipient is assessed as Good. SJ is an intermediately strict norm where cooperation for a Bad recipient is assessed as Bad while defection is Good. In contrast, IS does not use an image to recipient but uses only donor’s action. If the donor’s previous action is C, then IS evaluates the donor as Good, otherwise IS evaluates the donor as Bad. []{data-label="fig:framework"}](fig_framework.eps){width="\linewidth"}
![[**Time series of typical simulation runs with all norms. With no error (left panel) and with errors (right panel).**]{} [**a**]{}, The average frequencies of 16 norms and the cooperation of the overall society. The black dotted line is the cooperation ratio. Parameters: $b = 5, c = 1, N = 500, R = 500, G = 1000, p = 0, q = 0$. When SH and ALLB coexist, cooperation does not emerge. When ALLB is completely driven out by SH, SJ invades and the cooperation ratio abruptly rises. At the same time, SH is driven out by SJ. After cooperation is completely achieved, SJ permits the invasion of ST, and also coexists with other tolerant norms (IS, ST, GGGB, and ALLG). Finally, strategies whose norm is expressed as GG\*\* (in other words, norms that constantly cooperate if cooperation has been selected in the past by the recipient) coexist. In [**b**]{}, both errors in perception and implementation were introduced, and simulation similar to [**a**]{} was run ($b = 5, c = 1, N = 500, R = 500, G = 1000, p = 0.001, q = 0.001$). As in [**a**]{}, when SH and ALLB coexist, cooperation does not emerge. However, cooperation is achieved without going through SJ.[]{data-label="fig:alternation1"}](fig_alternation1.eps){width="\linewidth"}
![[**The alternation patterns of the majority of norms with 50 replications. With no error (left panel) and with errors (right panel).**]{} [**a**]{}, The panel shows the transition of norms with the greatest populations in a round of 20 generations, before the cooperation ratio exceeds 0.8, and 100 generations, after the cooperation ratio exceeds 0.8 (for a total of 120 generations). For the sake of visibility, in a replication, we stop calculation when ALLG becomes the majority norm. This is because in a state in which tolerant norms coexist, the norms with the greatest population frequently change place. The thickness of the arrows corresponds to the number of times alternation of norms occurred. (See the Supplementary Information for details.) The alternation of norms SH $\to$ SJ $\to$ ST $\to$ ALLG was observed to be stable. In [**b**]{} , both errors in perception and implementation were introduced, and simulations similar to [**a**]{} were run ($b = 5, c = 1, N = 500, R = 500, G = 1000, p = 0.001, q = 0.001$). As shown in [**b**]{}, the transition of majority norms is not distinct compared to the times when there were no errors.[]{data-label="fig:alternation2"}](fig_alternation2.eps){width="\linewidth"}
![[**The cooperation ratio in the norm knockout method**]{}. Each graph shows the average cooperation ratio of 50 replications when a typical norm is knocked out. The basic parameter set is $c = 1, N = 500, R = 500, G = 1000$. To confirm the effects of errors in perception and errors in implementation, two simulations with and without error were executed. See the Supplementary Information for knockout analysis of all norms. [**a**]{}, The case when errors in perception ($p$) and errors in implementation ($q$) are 0. When SH or IS is knocked out, cooperation does not evolve at all. Also, when SJ, which becomes the majority for only a brief round during the process of alternation, is knocked out, cooperation evolves to the extent of only 30 percent, even when $b$ is large. Furthermore, when ST is knocked out, the range in which cooperation is achieved becomes narrow. Only when $b$ is sufficiently large can cooperation evolve. [**b**]{}, The case where $p = q = 0.001$. The indispensable norm is ST in addition to SH and IS. Conversely, when SJ is knocked out, cooperation evolves when $b$ is sufficiently large in the same manner as ST in [**a**]{}.[]{data-label="fig:knockout"}](fig_knockout.eps){width="\linewidth"}
![[**Time series of typical simulation runs in norm knockout method**]{}. The parameters are $b = 5,c = 1,N = 500,R = 500,p = q = 0$. [**a**]{}, When SH is knocked out, the strategy to eliminate ALLB does not exist. [**b**]{}, When IS is knocked out, SH exists only in a small population and cannot gain superiority over ALLB.[]{data-label="fig:knockout_time"}](fig_knockout_time.eps){width="\linewidth"}
Supplementary Information {#supplementary-information .unnumbered}
=========================
This section includes:
- Supplementary Text S1
- Supplementary Tables S1 – S6
- Supplementary Figure S1
Text S1 {#text-s1 .unnumbered}
-------
### The details of alternation of norms {#the-details-of-alternation-of-norms .unnumbered}
We present the details of alternating the majority of norms, as shown in Fig. 3 (in the Main Text). Table S1 shows the alternation with no error and Table S2 shows with errors.
### The population of each norm {#the-population-of-each-norm .unnumbered}
We show the average population ratio at the 1,000th generation (50 replications). As shown in Table S3, persistent norms are able to be described as GG\*\*.
### The details of results of norm knockout method {#the-details-of-results-of-norm-knockout-method .unnumbered}
We comprehensively explore the indispensable norms by knocking out each of the 16 norms. Table S4 shows the cooperation ratio in which each norm is knocked out. Without errors, SH and IS are the indispensable norms. If SJ or ST is knocked out, their standard deviations (S.D.) are larger than other cases. The reason for this is that to knock out SJ or ST produces two contrasting results: cooperation dominant or defection dominant. When indispensable norms are knocked out, no cooperative regime appears at all, and the standard deviation has a very small value.
### The analysis of the alternation of norms after cooperative regime achieved {#the-analysis-of-the-alternation-of-norms-after-cooperative-regime-achieved .unnumbered}
In this section, we have analysed the transition of norms with the greatest populations after a cooperation ratio exceeds 0.9 until the end of generations. For the sake of understanding the mechanism of co-evolution of norms and cooperation, we have focused on the duration of regime changes from defection to cooperation in the main text. Therefore, we stopped the calculation of the transition of the majority norm when ALLG becomes the majority norm in Fig. 3. Intuitively, it seems impossible to transit from a majority of ALLG to a majority of any other norm; however, it is well-known that a state where everyone cooperates indiscriminately is easily replaced by a state where everyone refuses to cooperate.
To clarify whether ALLG is a stable state and what would happen if more generations were considered, we show the transition of norms with the greatest population after a cooperative regime is achieved. The results (Tables S5 and S6, Fig. S1) show that the cooperation regime is maintained robustly and tolerant norms such as ALLG, GGGB, and ST coexist. Although ALLG forms the majority of the population, the other norms including GGGB, ST (GGBG), and IS (GGBB) protect against invasion from defective norms.
Tables {#tables-1 .unnumbered}
------
**The alternation patterns of dominant norms with no error correspond to the left panel of Fig. 3 (in the Main Text).** Each row shows the transition of norms with the greatest populations in a period of 20 generations, before the cooperation ratio exceed s 0.8, and 100 generations, after the cooperation ratio exceeds 0.8 (for a total of 120 generations). For the sake of visibility, we stop calculation when ALLG becomes the majority norm. Fifty replications are conducted. During this time, alternation in majority norms for a total of 156 times could be observed. A cooperative regime with a cooperation ratio exceeding 0.8 was achieved in 46 replications. For example, over 50 replications, the number of times the transition of greatest population followed (SH $\to$ SJ $\to$ ST $\to$ ALLG) was 31. Moreover, four (indicated by the dash) never had a cooperation ratio exceeding 0.8.
Transition pattern of dominant strategies No.
------------------------------------------------------------------------------------ -----
SH $\to$ SJ $\to$ ST $\to$ ALLC 31
SH $\to$ SJ $\to$ ST $\to$ SJ $\to$ ST $\to$ ALLC 5
$ - $ 4
SH $\to$ GBGB $\to$ GGGB $\to$ ALLC 3
SH $\to$ GBGB $\to$ SH $\to$ SJ $\to$ ST $\to$ ALLC 2
SH $\to$ IS $\to$ GGGB $\to$ ALLC 2
SH $\to$ IS $\to$ GGGB $\to$ SH $\to$ ALLD $\to$ SH $\to$ ALLD $\to$ SH $\to$ ALLD 1
SH $\to$ GGGB $\to$ ALLC 1
SH $\to$ GBGB $\to$ ST $\to$ ALLC 1
**The alternation patterns of dominant norms with errors correspond to the right panel of Fig. 3 (in the Main Text).** The setting of the table is the same as Table S1.
Transition pattern of dominant strategies No.
------------------------------------------------------------------------------------------------- -----
SH $\to$ IS $\to$ GGGB $\to$ ALLC 13
SH $\to$ GBGB $\to$ GGGB $\to$ ALLC 8
SH $\to$ SJ $\to$ ST $\to$ ALLC 7
SH $\to$ GGGB $\to$ ALLC 6
SH $\to$ IS $\to$ ST $\to$ ALLC 5
SH $\to$ ST $\to$ ALLC 2
SH $\to$ GBGB $\to$ GGGB $\to$ SH $\to$ GBGB $\to$ GGGB $\to$ SH $\to$ IS $\to$ GGGB $\to$ ALLC 1
SH $\to$ IS $\to$ ST $\to$ IS $\to$ GGGB $\to$ ALLC 1
SH $\to$ GBGB $\to$ SH $\to$ IS $\to$ GGGB $\to$ ALLC 1
SH $\to$ IS $\to$ GGGB 1
SH $\to$ GBGB $\to$ IS $\to$ GGGB $\to$ GBGB $\to$ GGGB $\to$ ALLC 1
SH $\to$ SJ $\to$ IS $\to$ ST $\to$ ALLC 1
SH $\to$ GBGB $\to$ IS $\to$ GGGB $\to$ ALLC 1
SH $\to$ GBGB $\to$ SH $\to$ IS $\to$ ST $\to$ ALLC 1
SH $\to$ GBGB $\to$ GGGB $\to$ IS $\to$ GGGB $\to$ ALLC 1
**The average population ratio at the 1,000th generation ($b$ = 5).** Each column shows without/with errors. All cells are obtained by averaging the results of 50 replications. These results show the population of each norm in which all norms exist (i.e., the norm knockout method is not used). The second row shows the average cooperation ratio ($C_{ratio}$) and standard deviation at the 1,000th generation. Below the fourth row is shown the population of each norm and its standard deviation. The norms described as GG\*\* can coexist stably while any norm that is not GG\*\* can barely exist. SH, which is an indispensable norm, also cannot survive. IS, which is included in four persistent norms GG\*\*, is the most in minority of the four.
$p = q = 0$ $p = q = 0.001$
-------------------- ------------------ ------------------
$c_{ratio}$ (S.D.) 0.939 (0.187) 0.980 (0.006)
Norms population (S.D) population (S.D)
BBBB \[ALLB\] 0.015 (0.085) 0.000 (0.001)
BBBG 0.001 (0.002) 0.000 (0.000)
BBGB 0.001 (0.003) 0.000 (0.001)
BBGG 0.000 (0.001) 0.000 (0.001)
BGBB 0.003 (0.007) 0.002 (0.002)
BGBG 0.003 (0.003) 0.002 (0.002)
BGGB 0.004 (0.004) 0.004 (0.003)
BGGG 0.005 (0.004) 0.005 (0.003)
GBBB \[SH\] 0.026 (0.109) 0.002 (0.003)
GBBG \[SJ\] 0.009 (0.006) 0.005 (0.004)
GBGB 0.020 (0.013) 0.007 (0.005)
GBGG 0.024 (0.012) 0.012 (0.006)
GGBB \[IS\] 0.132 (0.043) 0.148 (0.040)
GGBG \[ST\] 0.165 (0.073) 0.201 (0.071)
GGGB 0.271 (0.093) 0.271 (0.079)
GGGG \[ALLG\] 0.322 (0.090) 0.341 (0.064)
**Analysis of norm knockout method for each of the 16 norms.** The table shows the cooperation ratio at the 1,000th generation in which each of the 16 norms is knocked out ($b$ = 5). Each value shows the average cooperation ratio from 50 replications and standard deviation. The cells in which the average cooperation ratio is less than 0.1 are shown in red. In this paper, we call these norms “indispensable norms”. With no error, SH and IS are indispensable norms. With errors, these two plus ST are indispensable norms.
------------------ ------------------- -------------------
$p = q = 0$ $p = q = 0.001$
Knockouted norm Mean (S.D.) Mean (S.D.)
BBBB \[ALLB\] 0.816 (0.354) 0.745 (0.399)
BBBG 0.980 (0.008) 0.979 (0.007)
BBGB 0.978 (0.012) 0.979 (0.007)
BBGG 0.923 (0.226) 0.961 (0.134)
BGBB 0.920 (0.225) 0.922 (0.225)
BGBG 0.982 (0.008) 0.977 (0.006)
BGGB 0.959 (0.134) 0.978 (0.007)
BGGG 0.979 (0.011) 0.959 (0.134)
GBBB \[SH\] [0.025 (0.004)]{} [0.026 (0.005)]{}
GBBG \[SJ\] 0.120 (0.287) 0.616 (0.457)
GBGB 0.982 (0.007) 0.977 (0.006)
GBGG 0.941 (0.188) 0.978 (0.006)
GGBB \[IS\] [0.023 (0.006)]{} [0.022 (0.004)]{}
GGBG \[ST\] 0.412 (0.432) [0.055 (0.060)]{}
GGGB 0.915 (0.225) 0.961 (0.010)
GGGG \[ALLG\] 0.897 (0.179) 0.371 (0.431)
Without Knockout 0.939 (0.187) 0.980 (0.006)
------------------ ------------------- -------------------
**The number of transitions of norms with the greatest populations with no error after a cooperation ratio exceeds 0.9 until the end of generations.** The simulation runs 50 replications. The transition is counted when the most majority norm is superseded by other norms. Parameters: $b = 5, c = 1, N = 500, R = 500, G = 1000, p = 0, q = 0$. For example, the transition from ALLG to GGGB occurs 929 times. During this time, alternation of norms with the greatest populations for a total of 2,324 times could be observed.
From To No.
------ ------ -----
ALLG GGGB 929
GGGB ALLG 920
ST ALLG 225
ALLG ST 185
SJ ST 35
ST SJ 6
IS GGGB 3
SH ALLB 3
ALLB SH 3
ALLG IS 3
IS ST 3
ST IS 2
GGGB ST 2
GGGB SH 1
SH SJ 1
GGGB IS 1
ST GGGB 1
IS ALLG 1
**The number of transitions of norms with the greatest populations with errors after cooperation ratio exceeds 0.9 until the end of generations.** The setting of the table is the same as Table S5. Parameters: $b = 5, c = 1, N = 500, R = 500, G = 1000, p = 0.001, q = 0.001$.
From To No.
------ ------ -----
GGGB ALLG 968
ALLG GGGB 950
ST ALLG 434
ALLG ST 426
GGGB IS 41
IS GGGB 40
ST GGGB 15
GGGB ST 14
IS ST 12
ALLG IS 10
ST IS 9
IS ALLG 8
Figure {#figure .unnumbered}
------
{width="\linewidth"}
**The transition diagram of norms with the greatest populations with no error (A) and with errors (B) after cooperation ratio exceeds 0.9 until the end of generations.** Panel A is drawn using the data of Table S5 and panel B is drawn using the data of Table S6. Both panels show that the tolerant norms (such as ALLG, GGGB, and ST) coexist as the majority.
[^1]: The final version was published in [*Scientific Rreports*]{}. How to cite this article: Yamamoto, H., Okada, I., Uchida, S., & Sasaki, T. A norm knockout method on indirect reciprocity to reveal indispensable norms. [*Sci. Rep.*]{} 7, 44146; doi: 10.1038/srep44146 (2017).
|
---
abstract: 'We study the interference of C$_{70}$ fullerenes in a Talbot-Lau interferometer with a large separation between the diffraction gratings. This permits the observation of recurrences of the interference contrast both as a function of the de Broglie wavelength and in dependence of the interaction with background gases. We observe an exponential decrease of the fringe visibility with increasing background pressure and find good quantitative agreement with the predictions of decoherence theory. From this we extrapolate the limits of matter wave interferometry and conclude that the influence of collisional decoherence may be well under control in future experiments with proteins and even larger objects.'
author:
- Lucia Hackermüller
- Klaus Hornberger
- 'Björn Brezger[^1]'
- Anton Zeilinger
- and Markus Arndt
date:
title: 'Decoherence in a Talbot Lau interferometer: the influence of molecular scattering'
---
Introduction {#sec:intro}
============
Matter wave interferometry of small quantum objects has become an active field of research during the last decades [@Berman1997a]. The new field of coherent optics with large molecules is now exploring the technical and possibly fundamental limits of interferometry with quantum objects of high mass, high internal complexity and high internal excitation – i.e. with novel properties which allow to study in detail the quantum-classical transition [@Arndt1999a; @Arndt2002a].
Several challenges arise when one moves from small to large systems: At a given velocity the de Broglie wavelength $\lambda=h/(m v)$ shrinks with increasing mass, and in addition it becomes increasingly difficult to slow down massive systems which have a large kinetic energy. The requirements on interferometer technology for de Broglie wavelengths in the sub-picometer range are already rather demanding. But even more important are the potential mechanisms which may lead to decoherence, that is to a loss of visibility in the interference pattern due to the coupling of the quantum system to its environment (see e.g. [@Joos1985a; @Zurek1991a]).
The investigation of this apparent loss of quantum properties has become an important corner stone of modern quantum physics – not only due to its fundamental role in mesoscopic physics and its importance for the understanding of the quantum-classical transition, but also because of its potential impact on emerging quantum technologies, such as quantum computers.
Any increase in size and complexity generally opens new decoherence channels, and for large molecules one can think of many interactions with the environment, either by scattered radiation [@Chapman1995b], by collisions with particles [@Hornberger2003a], by an interaction with fluctuating quasi-static electro-magnetic fields [@Myatt2000a] or even by the interaction with gravitational waves [@Reynaud2002a].
While it is impossible to manipulate and track the details of the perturbations for really macroscopic systems, the environment of isolated mesoscopic quantum systems can still be efficiently controlled. In the present paper we focus on one particular interaction between large molecules and an environment, namely collisions between the coherently propagating molecules and various background gases. First results on this subject have already been discussed in a previous letter [@Hornberger2003a]. In the present work we give more detailed background information on our quantitative investigations of collisional decoherence of the fullerene C$_{70}$, and we study both experimentally and theoretically the influence of increased interaction times, which will be unavoidable in interferometry with proteins.
The Talbot Lau interferometer {#sec:1Setup}
=============================
![Artist’s view of the ’pressurized’ Talbot Lau interferometer. A beam of C$_{70}$ molecules is generated from fullerene powder at 900 K. The beam passes a series of three gold gratings each with a grating constant of $d=990\,$nm, an open width of 480 nm and a grating thickness of 500 nm. The grating separation $L$ was set to 38cm. The whole vacuum chamber is evacuated to $2\times 10^{-8}$mbar and can then be pressurized with different gases, typically up to $10^{-6}$mbar. The fullerenes are detected using a laser induced thermal ionization process [@Nairz2000a]. The interferogram is scanned by shifting the third grating along the grating vector.[]{data-label="fig:1"}](fig1.eps){width="1.1\columnwidth"}
One can conceive various experimental arrangements to demonstrate the wave-nature of material particles and many interferometers have already been built for atoms (see refs. in [@Berman1997a]). Also for small molecules a number of arrangements such as grating diffraction [@Schollkopf1994a; @Arndt1999a], Ramsey-Bordé interferometry [@Borde1994a; @Lisdat2000a], or Mach-Zehnder interferometry [@Chapman1995b; @Bruehl2002a] have been shown to work. However, all these arrangements need well collimated beams or experimentally distinguishable internal states in order to separate the various diffraction orders. This requirement makes them less suitable for large clusters and large molecules for which brilliant sources and highly efficient detection schemes still have to be developed.
A near-field interferometer of the Talbot Lau type, in contrast, does away with the collimation requirement. Unlike the far-field interferometers, it is more compact, rugged and allows a much higher transmission [@Clauser1992a].
The basic idea of such a device, the lens-less periodic imaging of molecular density distributions, can already be seen from a short discussion of the Talbot effect as used in light optics [@Patorski1989a]. Suppose that a plane wave $\psi_0=\exp(\rmi
k z)$ illuminates a grating located in the $(x,y)$-plane with grating function $t(x)$. The wave function at a distance $L$ behind the grating is then given in paraxial approximation by the Kirchhoff-Fresnel integral $$\psi_L = \rme^{\rmi k L} \left(\frac{k}{2\pi\rmi L}\right)^{\oh}
\int \rmd x' t(x')
%\rme^{\rmi k (x-x')^2/(2z)}
\exp\left(\rmi k\frac{(x-x')^2}{2L}\right) \PO$$ It is easily evaluated if the grating function is periodic, $$\label{eq:gratingfunction} t(x) = \sum_{\ell\in\Z} a_\ell \exp
\left( 2\pi\rmi \ell\frac{x}{d} \right) \CO$$ and one finds by Gaussian integration, $$\psi_L = \rme^{\rmi k L} \sum_\ell a_\ell \exp \left( 2\pi\rmi
\ell\frac{x}{d} \right) \exp \left( -\rmi\pi \ell^2
\frac{L\lambda}{d^2} \right) \PO$$ From this expression one observes immediately that at even multiples of the distance $$\LT = \frac{{d^2 }}{\lambda }
%L = 2 \frac{{d^2 }}{\lambda } \equiv 2 L_\lambda \CO m \in \Natural$$ the transverse part of the wave function is simply given by the grating function . The grating pattern is also repeated at [odd]{} multiples of the *Talbot length* $L_\lambda$, but there it is shifted along $x$ by half a grating period $d/2$.
This lens-less Talbot imaging is a pure interference effect and was already successfully applied to material objects [@Chapman1995c; @Deng1999a]. However, in the version described so far it still requires a plane wave, i.e., a parallel input beam. The full intensity gain of the Talbot effect is only deployed when it is applied to uncollimated and therefore much more intense molecular beams [@Clauser1994a; @Brezger2002a]. This is realized if the single diffraction grating is replaced by three gratings, which act – from front to end – as a multiplexing collimator, a diffraction grating and a detection mask (for details see, e.g., [@Brezger2003a]). Each point in the grating then acts as the source of an interference pattern and, even though there is no coherence between different source points, the independent interference patterns originating from each of them overlap in a position-synchronized manner to form a pattern of high fringe visibility. One may also regard the first grating as a tool to impose some coherence on the uncollimated molecular beam. The finite width of each opening in the first grating induces lateral coherence at the second grating which is of the order of 2-3 grating periods.
Our experimental setup is based on this idea, and a sketch of it is shown in Fig. \[fig:1\]. C$_{70}$ molecules are sublimated at 900 K to form an effusive beam with molecular de Broglie wavelengths in the range from 2pm to 5pm. The beam is essentially uncollimated in the horizontal direction, but it is selected by three spatially separated height delimiters, namely the oven aperture (200$\mu$m), a central height delimiter (alternatively 50 or 150$\mu$m), and the detector laser beam with a waist of 10$\mu$m. By shifting the oven vertically one can then select a well-determined free-flight parabola and thus a certain velocity class. This way, mean velocities could be chosen from 90m/s to 220m/s with the full width at half maximum of their distributions ranging from 7% to 17% of the mean velocity. The gravitational method is superior to a set of rotating slotted disks because it avoids additional vibrations and allows up to 100% throughput at the central selected velocity.
The transverse coherence of the beam is unprepared until the molecules pass the first grating. The three gold gratings have a period of 990nm, a nominal open fraction of $0.48\pm0.02$ (as specified by the manufacturer Heidenhain, Traunreut) and a flat open field of roughly 16mm diameter. We limit the lateral width of the molecular beam to 1mm which is also comparable to the width of the ionization range of the detecting laser beam [@Nairz2000a]. The distance between consecutive gratings was set to $L=0.38\,$m. This is almost twice as large as in our previous experiment [@Hornberger2003a] and enables us to observe two Talbot recurrences (see Fig. \[fig:2\]). All gratings can be rotated around the molecular beam to align them with an accuracy of about 1mrad both with respect to each other and to the direction of earth’s acceleration. The third grating G$_{3}$ masks the molecular density pattern behind the second grating. G$_{3}$ is mounted on a piezo translation stage (Piezosystem Jena) and is scanned perpendicular to the molecular beam in steps of 100 nm. Those molecules which are in phase with the openings of G$_{3}$ pass the grating and are heated by the crossing Ar$^+$ laser beam (488 nm, 15Watt). The positive ions which are generated by the laser induced thermionic emission are counted as a function of the lateral position $x_{\rm s}$ of G$_{3}$. As shown below, theory predicts an almost sine-wave shaped interferogram $S(x_{\rm s})$ for the transmitted molecules. This is indeed observed in our experiment, and the fringe contrast $V_\lambda=(S_{max}-S_{min})/(S_{max}+S_{min})$ serves to characterize the interference pattern.
![Interferometer visibility as a function of the mean molecular de Broglie wavelength. We observe a clear recurrence of the interference maximum both in the experiment (circles) and in the numerical model (lines). This is expected for the Talbot effect with varying wavelength or varying Talbot distance respectively. The pressure in the chamber was below $3\times
10^{-8}$mbar so that collisions are still negligible, and the central height limiter was set to 50 $\mu$m. The theoretical lines correspond to the quantum calculation at open fractions of $f=0.45$ (solid line) and $f=0.48$ (dashed line), while the corresponding classical expectation is shown by the dotted ($f=0.45$) and dash-dotted line ($f=0.48$). The arrows indicate the wavelengths where the Talbot criterion $L=m \LT,=m
g^2/\lambda\;, m\in\Natural$, is met with $m=1$ for $\lambda =
2.58\,$pm and $m=2$ for $\lambda = 5.14\,$pm. With respect to ideal gratings the true maxima are slightly shifted to smaller wavelengths and the two maxima have different heights. This is due to the interaction between the molecule and the wall.[]{data-label="fig:2"}](fig2.eps){width="\columnwidth"}
The quantum origin of the observed signal is confirmed by the characteristic dependence of its contrast $V_\lambda$ on the molecular wavelength. If there is a coherent evolution in the interferometer the expected fringe signal is easily calculated, using wave optics in paraxial approximation. Starting with an incoherent beam one finds, after a coherent passage from the first to the third grating [@Brezger2002a],
$$\label{eq:signal}
S(x_{\rm s}) \propto
\!\!\sum_{m\in\Z}
\left({B^{(0)}_m}^*\right)^2 B^{(\lambda)}_{2m}
\exp\left(2\pi\rmi m \frac{x_{\rm s}}{d}\right)$$
The coefficients $B_m$ are defined in terms of the Fourier components $a_\ell$ of the transmission function of each of the three equal gratings, $$\begin{aligned}
\label{eq:defBlambda} B_m^{(\lambda)}&=\sum_{\ell\in\Z} a_\ell
a^*_{\ell-m}
\exp\left(\rmi\pi\frac{m^2-2\ell
m}{2}\frac{L}{\LT}\right) \CO \intertext{and} \label{eq:defBnull}
B_m^{(0)}&=\sum_{\ell\in\Z} a_\ell a^*_{\ell-m} \PO\end{aligned}$$ The $B_m^{(\lambda)}$ describe the diffraction at the second grating, while the $B_m^{(0)}$ belong to the first and third grating serving to mask the molecular density. The Fourier coefficients $a_\ell$ also include the effect of the attractive interaction between molecule and grating in eikonal approximation [@Brezger2002a; @Brezger2003a]. The masking by G$_1$ and G$_3$ is essentially wavelength independent. But due to the finite flight time of the molecules through the grating slits the van der Waals force introduces a certain wavelength dependence also in the $B_m^{(0)}$.
Before taking the experimental signal as evidence for quantum interference one must note that a certain fringe contrast could also be explained by classical mechanics due to a shadow effect. The classical expectation can be calculated by propagating the classical phase space density of an uncollimated particle stream through the interferometer subjected to the same forces and approximations as in the quantum case. For ideal gratings the resulting expression is given by Eq. after simply replacing the $B_{2m}^{(\lambda)}$ by $B_{2m}^{(0)}$ [@Hornbergerinprep]. In the presence of van der Waals interactions the deflection tends to be underestimated in the classical analog of the eikonal approximation, so that our classical calculation gives an upper limit for the classical visibility. Hence, whenever the experimental contrast is significantly greater than the classical value one has evidence that quantum interference took place.
An even stronger proof is the characteristic wavelength dependence of the fringe visibility. Varying the mean molecular velocity corresponds to changing the mean molecular de Broglie wavelength. This dependence is used to scan the Talbot length $L_\lambda$ and to demonstrate the periodic wave nature of the molecular Talbot Lau effect in Fig. \[fig:2\].
The experimental data are shown as full circles and are generally well represented by the quantum theoretical calculation (solid line). Both clearly show the expected recurrence of the visibility with $\lambda$. The classical expectation (dash-dotted line) completely fails to reproduce the observed effects. Note that the visibility peaks are neither equally high nor symmetric around their maxima. Also, the peaks do not occur exactly at the Talbot length (indicated by arrows in Fig. \[fig:2\]). Instead the maxima are shifted to shorter wavelengths and the peaks have a broad shoulder towards the longer wavelengths. These deviations from the simple optical Talbot Lau effect appearing in the experiment are reproduced in the quantum calculation once we take into account the retarded van der Waals interaction [@Casimir1948a] between the polarizable molecules and the gold bars of the gratings. This interaction reduces the effective slit width. At fixed grating distance this corresponds to a shift of the maxima towards smaller wavelengths.
Fig. \[fig:2\] shows two theoretical predictions which represent the experimental data. The dashed line assumes a grating with an open fraction, i.e. a ratio of opening to grating constant, of 0.48 as originally specified when the gratings were purchased and mounted about two years ago. The solid line assumes and open fraction of 0.45. A possible explanation for the apparent shrinking of the openings might be a deposited layer of fullerenes which are also visible to the unaided eye, at least on the first grating. But also tiny mechanical grating deformations might be a reason.
We observe a notable difference between theory and observation in the peak height at a wavelength around 5pm, corresponding to molecules with a mean velocity of around 100m/s. We have evidence for the hypothesis that the experimental reduction of the interference contrast is mainly due to remaining vibrations of the setup with oscillation amplitudes of a few ten nanometers. Further investigations of this effect are currently under way.
Collisional decoherence: A quantum system interacting with the environment {#sec:2DecoherenceTheory}
==========================================================================
We now introduce a controlled source of decoherence by filling the vacuum chamber with various gases at low pressure ($p=0.05\ldots 2.5\times 10^{-6}\,$mbar) at room temperature. Each collision between a fullerene molecule and a gas particle entangles their motional states. Hence, the effect of a single collision on the molecular center-of-mass state is obtained by tracing over the state of the scattered molecule. One can safely assume that the mass of the fullerene molecule is much greater than the mass $\mg$ of the gas particle. We then find that the density operator, $\rho_0(\mathbf{r},\mathbf{r}')$, describing the quantum state of the fullerene molecules, changes simply by a multiplicative factor, $$\label{eq:rho} \rho(\mathbf{r},\mathbf{r}')=\rho_0(\mathbf{r},
\mathbf{r}')\, \eta(|\mathbf{r}-\mathbf{r}'|)\PO$$ This factor $\eta$ may be called the decoherence function since it describes the effective loss of coherence in the fullerene state. For elastic scattering with an isotropic potential and the gas initially in a thermal state it reads [@Hornberger2003b]
$$\begin{aligned}
\label{eq:eta} \eta(\Deltar)&=& \int_0^{\infty} \!\rmd \vg
\;\frac{g(\vg )}{\sigma(\vg )} \int\rmd\Omega
\left|f\big(\cos(\theta)\big)\right|^2 \nnn && \times
\sinc\!\Big(\sin\!\Big(\frac{\theta}{2}\Big)\frac{2 \mg \vg
\Deltar}{\hbar} \Big) \PO\end{aligned}$$
This expression involves an integration over the thermal distribution $g(\vg)$ of the gas velocities and an integral over the scattering angle $\Omega=(\theta,\phi)$. In the argument of the sinc function one finds the distance of the considered points times the momentum change in units of $\hbar$. Hence, the sinc function suppresses the integrand whenever the change in the state of the gas particle during a collision is able to resolve the distance $\delta r$. This leads to a reduction of the corresponding off-diagonal elements in when the gas particle transmits (partial) position information about the molecule to the environment. For small distances the sinc approaches unity so that the angular integral yields the total scattering cross section $\sigma(\vg)$. Hence, for $\delta r \to
0$ the decoherence function approaches unity as required from the conservation of the trace in .
By formulating the molecular evolution through the interferometer in the Wigner representation one finds that the effect of collisional decoherence can be treated analytically [@Hornbergerinprep]. It is completely described by a modification of the coefficients . To obtain the interference signal in the presence of a gas the $B_{2m}^{(\lambda)}$ in must be replaced by $$\begin{aligned}
\label{eq:Bdeco}
B_{2m}^{(\lambda)}
\exp\!\Big(-n\sigma_{\rm eff} \int_0^{2L}
\Big[1-\eta\Big({m}\frac{L-|z-L|}{\LT}d \Big)\Big]\rmd z\Big)
\PO\end{aligned}$$ Here $n\sigma_{\rm eff}$ is the number density of gas particles times the effective total cross section defined below in Eq. , describing the number of collisions per unit length. The integral in the exponent covers the various positions in the interferometer where a collision may occur. As discussed above we have $\eta(0)=1$, and the function decreases to zero for increasing arguments. It follows that the $m=0$ component, related to the mean flux through the interferometer, is not affected by decoherence. The other components of the interference signal are most sensitive to collisions occurring close to the second grating, at $z=L$, where the path separation is greatest. Indeed, if the Talbot criterion is met, $L=\ell\LT$, $\ell\in\Natural$, the distances entering the decoherence function are integer multiples of the grating period $d$. For the other $z$ positions the sensitivity decreases according to the path separation and, as one expects, a collision event will not contribute to decoherence directly at the first or at the third grating, at $z=0,\,2L$.
One can show that the general formula is equivalent[^2] to the solution in paraxial approximation of the master equation for the decoherence of a massive particle by interacting with a gas [@Gallis1990a; @Hornberger2003b]. The present form has the advantage that it separates the rate of the decoherence events, the factor $n\sigma_{\rm eff}$, from their effect described by the integrand in . This is particularly useful if the two processes should be treated with different degrees of accuracy, as is the case in the present experiment.
At the prevailing (room) temperature of our experiment each collision with a gas particle is so strong that it serves to localize the fullerene to the scale of a few nanometers, which is small compared to the typical path separation of 1$\mu$m. Therefore one can safely approximate the integral in by $2L$ if $m\neq 0$. On the other hand, the effective scattering cross section must be evaluated with care, since it must account for the longitudinal velocity $v_{\rm
m}\mathbf{e}_z$ of the fullerene and for the thermal distribution $\mu(\mathbf{v}_{\rm g})$ of the gas particle velocities. The general expression reads $$\begin{aligned}
\label{eq:sigmaeff1}
\sigma_{\rm eff}(\vm) =& \int \mu(\mathbf{v}_{\rm g})
\sigma(|v_{\rm m}\mathbf{e}_z-\mathbf{v}_{\rm g}|) \frac{|v_{\rm
m}\mathbf{e}_z-\mathbf{v}_{\rm g}|}{v_{\rm m}}
\,\rmd\mathbf{v}_{\rm g}
\PO\end{aligned}$$ In our experiment the interaction potential is well described by the isotropic London dispersion force (van der Waals force between polarizable molecules). The corresponding potential $U(r)=-C_6/r^6$ has a single parameter $C_6$ that can be found in [@Hornberger2003a] for a number of gases. The cross section $\sigma(v)$ for a fixed relative velocity follows from a semiclassical calculation [@Maitland1981a] and the remaining integration in can be performed asymptotically. One finds $$\begin{aligned}
\sigma_{\rm eff}(\vm) =
\frac{2(3\pi^6 C_6/8\hbar)^{2/5}}{\pi^\oh\Gamma(2/5)\sin(\pi/5)}
\,\frac{\vgt^{3/5}}{\vm}\, G\!\left(\frac{\vm}{\vgt }\right)
\label{eq:sigmaeff2}\end{aligned}$$ with $\vgt
= ({2\kB T/\mg})^\oh$ the most probable velocity in the gas and $$\begin{aligned}
G(u)=
\Gammafunction(9/5) (1-u^2)
+\frac{2}{3}\Gammafunction({14}/{5})u^2+\Or(u^4)
\PO\end{aligned}$$ This effective cross section exceeds the geometric one by two orders of magnitude at the velocities of our experiment ($\vm=80\ldots 240\, {\rm
m/sec}$).
A further simplification comes from the fact that for our experimental setup the visibility of the interference signal is essentially determined by the $m=0$ and $m=\pm1$ Fourier components only. The expected reduction of the contrast is therefore easily evaluated in terms of the coefficients and . Using $p=n
k_{\rm B}T$ one finds $$\begin{aligned}
\label{eq:Vdeco} V_\lambda(p)&=& \frac{2
|B^{(\lambda)}_{2}|(B^{(0)}_1)^2}{(B^{(0)}_0)^3}
\exp\left(-\frac{2L\sigma_{\rm eff}}{\kB T}p\right) \nnn &=&
V_\lambda(0)\exp(-p/\pv)
\PO\end{aligned}$$ Hence, we expect an exponential decrease of the visibility as a function of the gas pressure $p$. This is the expected experimental signature of collisional decoherence. It should not be confused with Beer’s law for absorption which predicts an exponential decrease of the mean signal at *constant* visibility.
Experimental decoherence: the pressurized interferometer
========================================================
A first experimental indication of collisional localization is presented in Fig. \[fig:3\]. It shows the change in the interference pattern of C$_{70}$ if a small amount of argon gas is added to the vacuum chamber. We observe a significant reduction in visibility from 42% to 34% if the pressure in the chamber is increased from $3 \times 10^{-8}$mbar to $5 \times
10^{-7}$mbar. The horizontal shift between the two curves is not significant since it can be explained by thermal drifts of the setup between the two recordings. In contrast to that, the values of the visibilities *are* significant. We tested that they were reproducible within $\pm$2 percent on different days over several weeks.
![Left: C$_{70}$ interference fringes at a pressure of $3\times 10^{-8}$mbar (residual background gases) shown as full circles. Right: the same signal in the presence of argon gas, at a pressure of $5\times 10^{-7}$ mbar. The lines are fits of a sine function. The mean velocity of the fullerene molecules was 189m/sec.[]{data-label="fig:3"}](fig3.eps){width="\columnwidth"}
We then record the visibility for a series of interferograms at different gas pressures. A typical result is given in Fig. \[fig:4\], which shows the pressure dependence of the interference visibility in the presence of thermal argon gas. As expected from Eq. (\[eq:Vdeco\]) we observe an exponential decay. Given the high initial contrast this is clear evidence for the occurrence of collisional decoherence.
![Visibility of the C$_{70}$ fringes as a function of the argon pressure at room temperature. The experimental data are given by full circles and the theoretical prediction gives the slope of the solid line. The data follow very nicely the expected exponential decay proving the occurrence of collisional decoherence. This experiment was done with L=22cm[]{data-label="fig:4"}](fig4.eps){width="\columnwidth"}
The good quantitative agreement with decoherence theory is obtained after taking into account a modification that is related to the particular method of velocity selection used in the experiment. As discussed in Sect. \[sec:1Setup\] we employ a gravitational velocity selection scheme by restricting the molecular beam to a free-flight parabola. If the apparatus is filled with a gas this velocity selection gets disturbed by collisions *outside* of the interferometer. After a collision each molecule gets slightly deflected so that now fullerenes with a ‘wrong’ velocity may fit through the setup. At the same time the molecule detector has only a finite size so that some of the molecules will pass it undetected after a collision. One has to take into account the small modification of the expected decay of visibility due to these effects. We do this by solving the *classical* phase space dynamics, i.e., the Boltzmann equation, effectively by a Monte Carlo method. The scattering angles are determined from the (diffraction limited) differential cross sections. Semiclassical expressions for the latter can be found in [@Helbing1964a]. Our predictions for the visibility are obtained by weighting with the classical velocity distribution in the detector – which corresponds to an averaging over a distribution of de Broglie wavelengths. Also the reduction of the mean count rate found in Fig. \[fig:2\] is well reproduced by our calculation.
[rcccccccccccc]{} Atom & H$_2$ & D$_2$& He & CH$_4$& Ne & N$_2$ & Air & Ar & CO$_2$ & Kr & Xe & SF$_6$\
mass/amu& 2 & 4 & 4 & 16 & 20.2& 28& 28.8& 39.9& 44&83.8&131.3&146\
$p_{0}$ (theo.)& 7.3 & 9.2 & 13.8 & 7.9& 16.0 &11.3&11.3&11.8&N.A.&12.4&11.5&N.A.\
$p_{0}$ (exp.)& 4.6 & 8.0 & 10.7 & 8.1& 13.2 &11.5&10.5&10.8&8.9&12.9&10.6&11.3\
$\Delta p_{0}$ (exp.)& 0.7& 1.2 & 1.6 & 1.2& 2.0& 1.7 & 1.6 & 1.6 & 1.3& 1.9&1.6& 1.7\
The loss of coherence with increasing pressure is conveniently described by the ’decoherence pressure’ $p_0$ defined in . Table \[tab:1\] compares the measured values of $p_0$ to the theoretical predictions for a number of gases. One observes satisfactory agreement over the whole mass range which covers two orders of magnitude. The experimental error is mainly due to the uncertainty in the pressure measurement, which is about 15%.
The most remarkable feature of the results reported in Tab. \[tab:1\] is the very weak dependence of the decoherence pressure on the type of gas used. This can be explained by assuming that the polarizability of the gas particle is proportional to its mass $\mg$. Then also $C_6$ is proportional to $\mg$, and one observes from that the mass dependencies of the interaction constant and of the most probable gas velocity $\vgt$ almost cancel out leaving $\sigma_{\rm
eff}\sim\mg^{1/10}$. The observed variations in Tab. \[tab:1\] are due to deviations from the assumed proportionality and reflect the specific electronic structures of the gas particle.
It should be noted that the best contrast for all experiments contributing to Tab. \[tab:1\] was systematically smaller than that of Fig. \[fig:2\]. Although the whole experiment was mounted on top of an optical table with active pneumatic vibration isolation we were able to identify tiny vibrations of the interferometer, induced by the water flow in the laser cooling system, as being the cause of a reduced contrast. The visibility was increased by about 10% when the laser was set on rubber feet. This simple remedy was applied for the experiments in the extended interferometer (L=38cm) of Figs.\[fig:1\],\[fig:2\],\[fig:3\],\[fig:5\] but not yet for the experiments of Fig. \[fig:4\] and Table \[tab:1\] (L=22cm).
However, it is important to note that the influence of external vibrations and the effect of collisional decoherence are independent of each other, and their total effect can therefore be obtained by multiplying their respective contributions to the reduction in visibility. This assumption was experimentally verified for several gases: a varying initial (low-pressure) contrast always led to the same slope in the visibility-vs-pressure curve, i.e., to the same decoherence pressure. The validity of Table \[tab:1\] is therefore not compromised by potential mechanical perturbations.
However, the total time of flight plays an important role for the absolute value of the fringe contrast. Clearly, for slow molecules the interaction time with the gratings is longer, and vibrations will be more detrimental. Also, the effective cross section increases for decreasing molecular velocities. In Fig. \[fig:5\] we observe the increasing influence of collisions in an interferometer filled with argon, for various fullerene velocities, i.e. various interaction times. In contrast to the experiment of Fig. \[fig:2\], the central height delimiter was set to 150$\mu$m (instead of 50 $\mu$m). This increased the flux but it also reduced the overall contrast by a few percent, both due to the increased sensitivity to imperfections in the grating alignment and due to an increased width of the velocity distribution.
Fig. \[fig:5\] shows the experimental visibility curves for $p_{\rm low}=3\times 10^{-8}$mbar (full circles) and $p_{\rm
high}=5\times 10^{-7}$mbar (hollow circles) and compares them to the quantum calculation (solid and dashed line, respectively) with the same model parameters as already used for Fig. \[fig:2\]. The remarks of the discussion of Fig. \[fig:2\] apply also here. This holds for the reduction of the visibility at long wavelengths due to vibrations, the shift of the maxima with respect to the Talbot length and the asymmetric line shapes — caused by the molecule-wall interaction. The new feature in this graph is the contrast reduction due to the scattering events in the Argon atmosphere and their dependence on the interaction time. For wavelengths in the 2.5pm regime ($v\sim 200\,$m/s) the pressure increase leads to $V_\lambda(p_{\rm high})/V_\lambda(p_{\rm low})=0.8$ whereas for wavelengths in the 5pm regime ($v\sim 100\,$m/s) the increased pressure results in $V_\lambda(p_{\rm
high})/V_\lambda(p_{\rm low})=0.5\,$. These ratios are identical for theory and experiment, both for the slow and for the fast molecules. However, the [*absolute*]{} values still differ at long wavelengths. Again this shows that the different decoherence mechanisms (collisions and vibrations) are independent of each other and their effects can be considered as separate multiplicative factors to the visibility.
\[htb\] ![Pressure dependence of the visibility in the stretched Talbot Lau interferometer (height delimiter at $150\,\mu$m). The main feature in this plot is the strong dependence of the interference visibility on the molecular wavelength, i.e. the molecular velocity. As expected, the slow molecules are much stronger affected by collisions than the fast ones. The experimental and theoretical curve for a background pressure of $p=3\times 10^{-8}$mbar are given by the full circles and a solid line. The hollow circles and the dashed line represent the data for argon at $p=5\times 10^{-7}$mbar. The overall good agreement between theory and experiment deteriorates markedly in the regime of long wavelengths, i.e. small velocities, where residual vibrations of the interferometer are expected to be relevant.[]{data-label="fig:5"}](fig5.eps "fig:"){width="\columnwidth"}
Conclusion
==========
Our experiments have demonstrated the periodic nature of the Talbot Lau effect in a molecule interferometer by observing a visibility recurrence in the elongated setup (L=38cm). The increased grating separation permitted a detailed quantitative study of decoherence due to collisions with the background gas. Our experiments show that decoherence by scattering of small particles, which is ubiquitous in our macroscopic world, can be understood and well controlled under high vacuum conditions. Based on the good agreement which we found in comparing our experiments with our numerical simulations we will now estimate the residual gas pressures required to observe the quantum nature of much larger objects.
To be specific, we consider a set of proteins of increasing size up to the mass of a rhinovirus, interacting with molecular nitrogen (300K, polarizability $\alpha/\Angstrom^3=4\pi\epsilon_0\times
1.75$). Since the static polarizability of large hydrocarbons is closely proportional to their mass $M$, i.e., $\alpha/\Angstrom^3=4\pi\epsilon_0\times 0.123\,M/{\rm amu}$, we can use the Slater-Kirkwood approximation [@Bernstein1979a].
For the observation of interference with supermassive molecules one needs low velocities in order to get de Broglie wavelengths larger than 100fm. For particles in the mass range of 10$^5$amu this requires velocities of the order of $\vm=10\,$m/s. Although this is a rather demanding requirement it seems not impossible to develop appropriate sources in the future. Moreover, a realistic earth-bound interferometer would be limited to a Talbot length of $L\sim 1\,$m.
Based on these assumptions we extrapolate in Table \[tab:2\] the decoherence pressures for insulin, green fluorescent protein, hemoglobin, ferritin and a human rhinovirus. It turns out that the vacuum conditions for quantum interference of these objects can be provided using commercially available technology.
[cccccccc]{} object & C$_{70}$ & insulin & GFP$^a$& hemoglobin& ferritin &virus$^b$\
mass (amu)& 840 & 5730 & $2.7\times 10^4$ & $6.4\times 10^4$ & $4.8\times 10^5$& $8\times 10^6$\
min/max extension (nm) & 1 & 3 & 3/4 & 5/7 & 10& 30\
estim. $\sigma_{\rm eff}$(nm$^2$) &730 & 1900 & 3700& 5200 & $1.1\times 10^4$& $3.6\times 10^4$\
estim. $p_{0}$ (mbar) &$3\times 10^{-8}$& $1\times 10^{-8}$ & $6\times 10^{-9}$& $4\times 10^{-9}$ & $2\times 10^{-9}$ & $6\times 10^{-10}$\
\
[a) green fluorescent protein\
b) rhinovirus HRV2 S150]{}
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful for useful discussions on decoherence with John Sipe, Toronto, and acknowledge experimental help in the setup of the experiment by Stefan Uttenthaler. This work was supported by the Austrian FWF in the programs START Y177 and SFB 1505. BB has been supported by a EU Marie Curie fellowship (No.HPMF-CT-2000-00797), and KH by the DFG Emmy Noether program. We acknowledge contributions by the EU under contract HPRN-CT-2002-00309.
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[^1]: present address: Fachbereich Physik, Universit[ä]{}t Konstanz, D-78457 Konstanz
[^2]: As shown in [@Hornberger2003b] the original master equation by Gallis and Fleming [@Gallis1990a] predicts a localization rate that is too large by a factor of $2\pi$. It would yield an additional $2\pi$ in the exponent of Eq. . The experimental results discussed in Section 4 are sensitive to this factor (and rule it out).
|
---
abstract: 'We consider a piecewise-deterministic Markov process (PDMP) with general conditional distribution of inter-occurrence time, which is called a general PDMP here. Our purpose is to establish the theory of measure-valued generator for general PDMPs. The additive functional of a semi-dynamic system (SDS) is introduced firstly, which presents us an analytic tool for the whole paper. The additive functionals of a general PDMP are represented in terms of additive functionals of the SDS. The necessary and sufficient conditions of being a local martingale or a special semimartingale for them are given. The measure-valued generator for a general PDMP is introduced, which takes value in the space of additive functionals of the SDS. And its domain is completely described by analytic conditions. The domain is extended to the locally (path-)finite variation functions. As an application of measure-valued generator, we study the expected cumulative discounted value of an additive functional of the general PDMP, and get a measure integro-differential equation satisfied by the expected cumulative discounted value function.'
address:
- 'School of Mathematics and Statistics, Central South University, Changsha, Hunan, China, 410083'
- 'Department of Applied Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, Hebei, China, 050043'
author:
- Zhaoyang Liu
- Yong Jiao
- Guoxin Liu
bibliography:
- 'Measure-valued\_generator\_of\_general\_PDMPs.bib'
title: 'Measure-Valued Generators of General Piecewise Deterministic Markov Processes'
---
piecewise-deterministic Markov process ,semi-dynamic system ,additive functional ,measure-valued generator ,measure integro-differential equation
Intruduction
============
Piecewise deterministic Markov processes (PDMPs in short) are a general class of Markov processes, for which the randomness is only on the jumping times and post-jump locations. Between two adjacent random jumps, a PDMP evolves as a semi-dynamic system (SDS in short). The pioneering work of PDMPs is Davis [@davis1984piecewise], which introduces the definition and present the theory of PDMPs. Since then PDMPs have attracted wide attention from many fields, in particular as far as application in finance, insurance, statistical physics and queuing are concerned. For applications in insurance see in particular Dassios and Embretchts [@Dassios1989Martingales] and Rolski et al [@rolski1998stochastic]. They point out that PDMPs theory provides a systematic toolet for the study of the ruin theory. Kirch and Runggaldier [@Kirch2004Efficient] use the reduction technique to solve a hedging problem in a continuous-time market (with uncontrolled draft). Dai [@dai1995on] develops a unified approach via fluid limit model by virtue of PDMPs in queueing networks. Further applications in queuing can be found in Bäuerle [@Bauerle2001Discounted], Graham and Robert [@Graham2009Interacting] and Rieder and Winter [@Rieder2009Optimal]. Applications in statistical physics see Faggionato et al. [@Faggionato2009Non] for example.
PDMPs theory and their optimization have been developed deeply in the last two decades. Davis [@davis1993markov] not only gives the framework of the model, but also, based on the local martingale theory of jumping processes, transplants the extended generator proposed by Stroock and Varadhan [@stroock1979multidimensional] for multidimensional diffusion processes into PDMPs theory and get the corresponding Itô formula. Meanwhile, he develops the optimal control theory for PDMPs also. Jacod and Skorokhod [@jacod1996jumping] generalize the concept of PDMPs to general cases, in which they call them jumping Markov processes (JMPs in short). They study the characteristics of the general PDMPs and the normal subjects of Markov processes, such as additive functionals, semimartingales and semimartingale functions. Palmowski and Rolski [@palmowski2002technique] obtain the general expression of exponential martingale with the extended generator for a wide range of Markov processes, including Davis’ PDMPs as special cases, and set up the corresponding theory of change of measure. The stability and ergodicity of PDMPs are developed by Costa [@costa1990stationary], Costa and Dufour [@costa1999stability; @costa2008stability], Jin and Amin [@Jin2016Stability]. Based on the theory of marked point processes, Jacobsen [@jacobsen2006point] represents not only the time-homogeneous PDMPs, but also the nonhomogeneous ones. Hou and Liu [@hou2005markov] discuss the analytic properties of the characteristic triple of the general PDMPs and develop the concept of extended generator with a discrete part. The comprehensive literature on the stochastic control of PDMPs in insurance can be found in Schmidli [@schmidli2008stochastic], Azcue and Muller [@azcue2014stochastic] and the references therein. Continuous average control of PDMPs is developed by Costa and Dufour [@costa2013continuous] and the references therein. de Saporta et al. [@deSaporta2016Optimal] study the impulse control of PDMPs. The numerical methods for stochastic control of PDMPs are studied by Brandejsky et al. [@Brandejsky2012Numerical], de Saporta and Dufour [@deSaporta2010Numerical-impulse], de Saporta et al. [@deSaporta2010Numerical-stopping; @deSaporta2015Numerical] and the references therein.
The theory of infinitesimal characteristics of Markov processes is always one of the core research problems of Markov processes. Kolmogorov obtained the Kolmogorov forward and backward equations when he was investigating a class of Markov processes with finite-dimensional Euclidean state space whose transition probabilities possess density functions. Meanwhile, he found that the processes are characterized by the coefficients of the corresponding equations which have simple probabilistic meanings and which are infinitesimal characteristics of the processes. The infinitesimal characteristics of the processes introduced by Kolmogorov allow us to determine not only the transition probabilities but also to evaluate the distributions of various functionals of the processes. Utilizing the theory of semigroups of operators, Feller pointed out that this semigroup completely determines the transition probabilities, and that the semigroup is in many cases uniquely determined by its infinitesimal generator. Feller suggested that the infinitesimal generator be considered as an infinitesimal characteristic of the process. Kolmogorov’s ideas constituted the basis of the mathematical theory of Markov processes and provided the direction for further investigations.
A PDMP is called to be *quasi-Itô*, in the terminology of [@jacod1996jumping], if the predictable brackets of all the (locally) square-integrable martingales are absolutely continuous with respect to the Lebesgue measure. Jacod and Skorokhod [@jacod1996jumping] pointed out that it is equivalent to that all the conditional jumping time distributions have densities. Jacod and Skorokhod [@jacod1996jumping] discuss the (weak) infinitesimal generators and extended generators. Jacobsen [@jacobsen2006point] extend the extended generators to the nonhomogeneous cases. The situations they deal with for extended generators are both quasi-Itô, which leads to that the domains of the extended generators are limited in the class of absolutely (path-)continuous functions (see [@jacobsen2006point]). Hou and Liu [@hou2005markov] introduce a new generator for the general PDMP, the extended generator coupled with a discrete part, which leads to the situation they deal with beyond the quasi-Itô case, for which there is no singularly continuous part in the conditional jumping time distributions. This makes the domain extended to the class of piecewise absolutely (path-)continuous functions instead of the one of absolutely (path-)continuous functions. Just as Jacobsen [@jacobsen2006point] pointed out, it is of course advantageous to have the domain of the generator as large as possible.
The purpose of this paper is to establish the theory of the so-called measure-valued generator for general PDMPs. To do this, we firstly introduce the definition of the additive functionals of an SDS. Some basic properties of the additive functionals are discussed. Especially, we give the characteristics of the absolutely continuous part and purely discontinuous part in the Lebesgue decomposition of an additive functional. The concept of additive functionals of SDSs is the analytic basis for the general PDMPs theory. Just like Jacod and Skorokhod said, the case of general PDMPs is much more technical. Based on the properties of the additive functionals of the SDS, the technical obstacle is overcome. We provide the representation theorem in terms of the additive functionals of the SDS and the necessary and sufficient conditions of being local martingales and special semimartingales for the additive functionals of the general PDMP. Therefore, we extend the results about additive functionals of *quasi-Hunt* JMPs in [@jacod1996jumping] to general PDMPs. Then, by virtue of the analytic theory of the additive functionals of SDSs and the ideas about extended generator, we introduce the concept of measure-valued generator which is valued in the space of additive functionals of the SDS with locally finite variation. We also give the description of its domain. Roughly speaking, the domain of generator is extended from absolutely (path-)continuous functions to functions with locally (path-)finite variation. And the corresponding Itô formula is followed by the way. As the direct application of measure-valued generator, the domain of the extended generator is extended further. For a PDMP in the sense of Davis [@davis1993markov], the results in this paper are almost the same except for a slight modification on his boundary condition. By the way, we also discuss the L-extended generator proposed by Kunita [@Kunita1969Absolute] for a general PDMP, which generalizes the corresponding results of Jacod and Skorokhod [@jacod1996jumping]. Finally, as an application of the theory of measure-valued generator, we get a new measure integro-differential equation which is satisfied by the expected cumulative discounted value of an additive functional of a general PDMP, and give the uniqueness condition of the solution to the measure integro-differential equation. For illustration, we point out that, in some certain cases, the measure integro-differential equation becomes an integro-differential equation or an impulse integro-differential equation.
This paper is organized as follow. We start with the definition of general piecewise deterministic Markov processes, and get their characteristic triple $(\phi,F,q)$ in Section 2. Roughly speaking, a general PDMP is a jumping Markov process introduced by Jacod and Skorokhod [@jacod1996jumping], i.e., a strong Markov process with natural filtration of discrete type. Different from Jacod and Skorokhod [@jacod1996jumping], we introduce the general PDMPs in the way of Hou and Liu [@hou2005markov]. By virtue of the concept of piecewise deterministic Markov skeleton processes, we get the necessary and sufficient conditions for such a process to be a strong Markov process in Theorem \[thm.PDP=>str.Markov\], and present the characterization of characteristic triple $(\phi,F,q)$ of a general PDMP, which generalize the corresponding results for quasi-Hunt cases in Jacod and Skorokhod [@jacod1996jumping]. Based on these, some basic properties of the characteristic triple $(\phi,F,q)$ are studied separately one by one.
Motivated by the representation theorem (Theorem 15 in [@jacod1996jumping]) for additive functionals of quasi-Hunt JMPs, we introduce the so-called additive functionals of a semi-dynamic system in Section 3. Some basic examples and properties are presented. Specially, the representations are given for the absolutely continuous part and the purely discontinuous part of an additive functional with locally finite variation, which depend only on the current state along the path of the SDS $\phi$. The concept of additive functionals of SDSs plays a key role as an analytic tool in this paper. In Section 4, we devote to generalize the results of [@jacod1996jumping] for the additive functionals of quasi-Hunt JMPs to the general PDMPs. Theorem \[thm.A=a+b\] shows that an additive functional of a general PDMP can be represented in terms of an additive functional $a$ of the SDS $\phi$ and a measurable function $b:E\times{\mathbb{R}}_+\times E\mapsto{\mathbb{R}}$ with property (\[eq.b.invariant\]). It is different from Theorem 15 of [@jacod1996jumping] that here $b$ is satisfied (\[eq.b.invariant\]) instead of (\[eq.b=bar.b\]). Furthermore, we present the necessary and sufficient conditions for an additive functional of a general PDMP to be a local martingale or a special semimartingale in terms of the corresponding $a$ and $b$.
In Section 5, we come to define the so-called measure-valued generators of general PDMPs. The analytic formulation is presented for the measure-valued generator, and its domain is completely described in an analytic way. By the way, we generalize the extended generators and L-extended generators for general PDMPs. Form which we can see that the measure-valued generator is a natural generalization of both the extended generator and L-extended generator for general PDMPs. As corollaries, we get the corresponding results for quasi-Hunt, quasi-Itô and quasi-step cases respectively. It shows that the concept of the extended generators is more suitable to the quasi-Itô PDMPs than the general cases. As an application of the measure-valued generator, we study the expected cumulated discounted value function of an additive functional of a general PDMP in Section 6. Due to the extended domain of the measure-valued generator, the expected value function with locally (path-)finite variation can be investigated. It yields a measure integro-differential equation in terms of the measure-valued generator in general (see Theorem \[thm.V.expectation.predictable\] and Corollary \[cor.V.expectation.optional\]). And, under some certain conditions, the solution of the measure integro-differential equation is exactly the expected value function (see Theorem \[thm.UniqueSolution\]). In the quasi-Itô case, the expected value function of the usual integral type functional (\[eq.V.expectation.Ito\]) is surely absolutely (path-)continuous and satisfies the usual integro-differential equation in terms of the extended generator but in the sense of (path-)almost everywhere (see Corollary \[cor.V.expectation.Ito\] and \[cor.UniqueSolution.Ito\]), which extend the results of Theorem 32.3 and 32.10 of Davis [@davis1993markov]. Especially, in some other special PDMPs, the expected value function may satisfy some integro-differential equations or impulse integro-differential equations (see Corollary \[cor.V.expectation.step\]-\[cor.UniqueSolution.nonsingular\]).
Definitions and properties
==========================
Definition of general PDMPs
---------------------------
Let $(\Omega, {\mathcal{F}}, {\mathbb{P}})$ be a complete probability space, $E$ be a Polish space, and ${\mathcal{E}}$ the Borel $\sigma$-algebra on $E$. And let $X=\{X_t\}_{0\leqslant t<\tau}$ be a càdlàg stochastic process with lifetime $\tau$ defined on $(\Omega, {\mathcal{F}}, {\mathbb{P}})$ and taking values in $(E, {\mathcal{E}})$, and $\{{\mathcal{F}}_t\}_{0\leqslant t<\tau}$ the natural filtration of $X$.
In order to introduce the general PDMPs, we need the concept of piecewise deterministic Markov skeleton processes (PDPs in short) introduced by Hou and Liu [@hou2005markov].
A càdlàg stochastic process $X=\{X_t\}_{0\leqslant t<\tau}$ is called a *piecewise deterministic Markov skeleton process* (PDP) if there exists a strictly increasing sequence of nonnegative random variables $\{\tau_n\}_{n\geqslant 0}$ (i.e., for any $n\geqslant 1$, $\tau_n<\tau_{n+1}$ if $\tau_n<\tau$) with $\tau_0=0$, $\tau_n\uparrow\tau$, and a sequence of measurable mappings $\phi_n:{\mathbb{R}}_+\times E\mapsto E$ such that $$\label{eq.PiecewiseDeterministic}
X_t=\sum_{n=0}^\infty\phi_n(t-\tau_n,X_{\tau_n})\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}},
\quad 0\leqslant t<\tau \quad \hbox{a.s.};$$ and for any $n\in{\mathbb{N}}$ and a bounded measurable function $f$ defined on $(E^{[0,\infty)},$ ${\mathcal{E}}^{[0,\infty)})$, $$\label{eq.MarkovSkeletonProperty}
{\mathbb{E}}\big[f(X_{\tau_n+\cdot})|{\mathcal{F}}_{\tau_n}\big]=
{\mathbb{E}}\big[f(X_{\tau_n+\cdot})|X_{\tau_n}\big].$$
The formula (\[eq.PiecewiseDeterministic\]) describes that the process is piecewise deterministic. The property (\[eq.MarkovSkeletonProperty\]) states that the process has Markov property at each time $\tau_n$, which is called *Markov skeleton property* as in Hou and Liu [@hou2005markov]. It is easily to see that the natural filtration ${\mathcal{F}}$ of the process $X$ is a *discrete type filtration* (or in other word, a *jumping filtration*), i.e., $${\mathcal{F}}_t=\bigcup_{n=0}^\infty\big({\mathcal{F}}_{\tau_n}\cap\{\tau_n\leqslant t<\tau_{n+1}\}\big),\quad 0\leqslant t<\tau,$$ where ${\mathcal{F}}_{\tau_n}=\sigma(X_0,\tau_1,X_{\tau_1},\dots,\tau_n,X_{\tau_n})$ (See Hou and Liu [@hou2005markov], or Jacod and Skorohod [@Jacod1994Jumping] for examples). Let ${\mathcal{F}}_\infty=\bigvee_{n=1}^{\infty}{\mathcal{F}}_{\tau_n}$. Suppose that there exists a family of probability measures ${\mathbb{P}}_x$ on $(\Omega, {\mathcal{F}})$ for $x\in E$ satisfying that for any fixed $B\in {\mathcal{F}}_{\infty}$, $x\mapsto {\mathbb{P}}_x\{B\}$ is ${\mathcal{E}}$-measurable, and for any fixed $x\in E$, $${\mathbb{P}}_x\{B\}={\mathbb{P}}\{B|X_0=x\}, \quad B\in{\mathcal{F}}_{\infty}.$$
For a given PDP $X=\{X_t\}_{0\leqslant t<\tau}$, set $\sigma_0=0$, $\sigma_n:=\tau_n-\tau_{n-1}\,(n\geqslant 1)$. The Markov skeleton property (\[eq.MarkovSkeletonProperty\]) implies that the sequence $\{(\sigma_n,X_{\tau_n})\}$ is a Markov sequence taking values in Polish space $({\mathbb{R}}_+\times E,{\mathcal{B}}({\mathbb{R}}_+)\times{\mathcal{E}})$, and its transition kernel is independent of the first component. We call it a *Markov skeleton sequence*. The transition kernel of the sequence $\{(\sigma_n,X_{\tau_n})\}$ is denoted by $\{G_n(x,{\mathrm{d}}t,{\mathrm{d}}y)\}_{n\geqslant 0}$. Let $F_n(x,{\mathrm{d}}t):=G_n(x,{\mathrm{d}}t,E)$. Then for any $B\in{\mathcal{E}}$, $G_n(x,{\mathrm{d}}t,B)\ll F_n(x,{\mathrm{d}}t)$. Thus, by Radon-Nikodym Theorem, there exists a sequence of $q_n:E\times{\mathbb{R}}_+\times{\mathcal{E}}\mapsto[0,1]$ such that $q_n(x,t,\cdot)$ is a probability measure on $(E,{\mathcal{E}})$ for any fixed $(x,t)$, and $q_n(\cdot,\cdot,B)$ is ${\mathcal{E}}\times{\mathcal{B}}({\mathbb{R}}_+)$-measurable for any fixed $B\in{\mathcal{E}}$. $\{(\phi_n,F_n,q_n)\}$ is called the *characteristic triple sequence* of the PDP $X$. A PDP $X$ is called to be *homogeneous* if the characteristic triple sequence is independent of $n$, i.e., $(\phi_n,F_n,q_n)\equiv(\phi,F,q)$. The triple $(\phi,F,q)$ is called the *characteristic triple* of the homogeneous PDP $X$.
For a homogeneous PDP $X$ with the characteristic triple $(\phi,F,q)$, let $F(x,t):=F(x,(t,\infty])$ for $x\in E$, which is called the *conditional survival function*, and $c(x):=\inf\{t>0:F(x,t)=0\}$ ($\inf\emptyset=+\infty$ by convention). Denote $$\mathcal{I}_x:=
\left\{
\begin{array}{ll}
{\mathbb{R}}_+, & c(x)=\infty; \\
\displaystyle[0,c(x)), & c(x)<\infty,\,F(x,c(x)-)=0; \\
\displaystyle[0,c(x)], & c(x)<\infty,\,F(x,c(x)-)>0.
\end{array}
\right.$$
Now we present the definition of general PDMPs.
A homogeneous PDP $X=\{X_t\}_{0\leqslant t<\tau}$ is called a *general PDMP* if it is a strong Markov process.
With the definitions above, we have the following theorem.
\[thm.PDP=>str.Markov\] A homogeneous PDP $X=\{X_t\}_{0\leqslant t<\tau}$ with characteristic triple $(\phi,F,q)$ is a homogeneous strong Markov process if and only if for any $x\in E$, $s,t\in{\mathbb{R}}_+$ with $s+t\in\mathcal{I}_x$ and $B\in{\mathcal{E}}$, the characteristic triple $(\phi,F,q)$ satisfies
Now we will come to prove the sufficiency. First we will prove the simple Markov property, that is to prove for any $s,t\in{\mathbb{R}}_+$, $${\mathbb{E}}_x[f(X_{t+s})|{\mathcal{F}}_t]={\mathbb{E}}[f(X_{t+s})|X_t], \quad x\in E$$ for each bounded measurable function $f$. By the definition of a homogeneous PDP and the property of the discrete type filtration, we have $${\mathbb{P}}\{\sigma_{n+1}>s\,|{\mathcal{F}}_{\tau_n}\}={\mathbb{P}}\{\sigma_{n+1}>s\,|X_{\tau_n}\}=F(X_{\tau_n},s).$$ Then by (ii), we have $$\begin{aligned}
& {\mathbb{P}}_x\{\tau_{n+1}>t+s\,|{\mathcal{F}}_t\}\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & \frac{F(X_{\tau_n},t+s-\tau_n)}{F(X_{\tau_n},t-\tau_n)}
\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}}\\
= & \frac{F(X_{\tau_n},t-\tau_n)\,F(\phi(t-\tau_n,X_{\tau_n}),s)}{F(X_{\tau_n},t-\tau_n)} \bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}}\\
= & F(\phi(t-\tau_n,X_{\tau_n}),s)\,\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}}\\
= & F(X_t,s)\,\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}}. \end{aligned}$$ Denote $\tilde{\tau}:=\tau_{n+1}$ if $\tau_n\leqslant t<\tau_{n+1}$, i.e., $\tilde{\tau}$ is the next random jumping time. Summing up the both sides, we have $$\label{eq.MarkovProperty.1}
{\mathbb{P}}_x\{\tilde{\tau}>t+s\,|{\mathcal{F}}_t\}=F(X_t,s), \quad t<\tau.$$ Notice that $$\begin{gathered}
F(x,t+{\mathrm{d}}s)=F(x,t)\,F(\phi(t,x),{\mathrm{d}}s), \\
q(x,t+s,B)=q(\phi(t,x),s,B).
\end{gathered}$$ And by the definition of a homogeneous PDP and the property of the discrete type filtration, we have $$\begin{aligned}
& {\mathbb{P}}_x\{X_{\tilde{\tau}}\in B\,|{\mathcal{F}}_t\}\,\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & {\mathbb{P}}_x\{X_{\tau_{n+1}}\in B\,|{\mathcal{F}}_t\}\,\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & \frac{\int_{(t-\tau_n,\infty)}q(X_{\tau_n},u, B)\,F(X_{\tau_n},{\mathrm{d}}u)}{F(X_{\tau_n},t-\tau_n)} \bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & \int_{(0,\infty)}q(\phi(t-\tau_n,X_{\tau_n}),s,B)F(\phi(t-\tau_n,X_{\tau_n}),{\mathrm{d}}s)
\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & \int_{(0,\infty)}q(X_t,s,B)F(X_t,{\mathrm{d}}s)\,
\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}} \\
= & {\mathbb{P}}_x\{X_{\tilde{\tau}}\in B\,|X_t\}\,\bbbone_{\{\tau_n\leqslant t<\tau_{n+1}\}}. \end{aligned}$$ Again, sum up the both sides, $$\label{eq.MarkovProperty.2}
{\mathbb{P}}_x\{X_{\tilde{\tau}}\in B\,|{\mathcal{F}}_t\}={\mathbb{P}}_x\{X_{\tilde{\tau}}\in B\,|X_t\}, \quad t<\tau.$$ for all $x\in E$.
From (\[eq.MarkovProperty.1\]) and (\[eq.MarkovProperty.2\]), the distributions of $\tilde{\tau}$ and $X_{\tilde{\tau}}$ conditioning on ${\mathcal{F}}_t$ both only depend on $X_t$. And any $\sigma_n$, the time length between two adjacent random jumps after $\tilde{\tau}$, and each post-jump state are only relate to $X_{\tilde{\tau}}$. While, $X_{\tilde{\tau}}$ is only dependent on $X_t$ and independent on the history before $t$. So we prove that if $f:(E^{[0,\infty)},{\mathcal{E}}^{[0,\infty)})\mapsto({\mathbb{R}},{\mathcal{B}}({\mathbb{R}}))$ is bounded and measurable, then $${\mathbb{E}}_x[f(X_{t+\cdot})|{\mathcal{F}}_t]={\mathbb{E}}_x[f(X_{t+\cdot})|X_t]$$ for each $x\in E$.
In order to get the strong Markov property, we use the character of stopping time of discrete type filtration. If we let $Y_n:=(X_0,\sigma_1,X_{\tau_1},\dots,\sigma_n,X_{\tau_n})$, then for each stopping time $T$, there exists a sequence of functions $s_1,s_2,\dots$ such that $$T\bbbone_{\{\tau_{n-1}<T\leqslant\tau_n\}}=\big[\big(\tau_{n-1}+s_n(Y_{n-1})\big)\land\tau_n\big] \bbbone_{\{\tau_{n-1}<T\leqslant\tau_n\}}.$$ This means that there are three cases on $\{T<\infty\}$: $T=0$, $T=\tau_n$ for some $n$, or $T=\tau_{n-1}+s_n(Y_{n-1})$ for some $n$. Furthermore, $\{T=0\}$, $\{T=\tau_n\}$ and $\{T=\tau_{n-1}+s_n(Y_{n-1})\}$ are all ${\mathcal{F}}_T$-measurable. Likewise, we define that $\tilde{\tau}:=\tau_n$ if $\tau_{n-1}\leqslant T<\tau_n$. Making the same deduction as above for the three cases we can show that $$\begin{gathered}
{\mathbb{P}}_x\{(\tilde{\tau}>T+s)\cap(T<\infty)|{\mathcal{F}}_T\}=F(X_T,s)\bbbone_{\{T<\infty\}}, \\ {\mathbb{P}}_x\{X_{\tilde{\tau}}\in{\mathrm{d}}y\,|{\mathcal{F}}_T\}={\mathbb{P}}_x\{X_{\tilde{\tau}}\in{\mathrm{d}}y\,|X_T\}. \end{gathered}$$ The same reasoning as the proof of Markov property can prove that, for any bounded and measurable function $f:(E^{[0,\infty)},{\mathcal{E}}^{[0,\infty)})\mapsto({\mathbb{R}},{\mathcal{B}}({\mathbb{R}}))$, we have $${\mathbb{E}}_x[f(X_{T+\cdot})\bbbone_{\{T<\infty\}}|{\mathcal{F}}_T]=
{\mathbb{E}}_x[f(X_{T+\cdot})|X_T]\bbbone_{\{T<\infty\}}.$$ Hence we proved the strong Markov property of the process $X$.
Now we are in the position to give the proof of necessity.
We begin with the property (ii). For each $x\in E$, $t\in\mathcal{I}_x$ and $s\geqslant 0$, we have $$\begin{aligned}
F(x, t+s) = & {\mathbb{P}}_x\{\tau_1>t+s\}\\
= & {\mathbb{P}}_x\{\tau_1>t\}\,{\mathbb{P}}_x\{\tau_1>t+s|\tau_1>t\}\\
= & F(x,t)\,{\mathbb{E}}_x[{\mathbb{P}}_x\{\tau_1>t+s|{\mathcal{F}}_t\}|\tau_1>t]\\
= & F(x,t)\,{\mathbb{E}}_x[{\mathbb{P}}_x\{\tau_1>t+s|X_t\}|\tau_1>t]\\
= & F(x,t)\,{\mathbb{E}}_x[{\mathbb{P}}_{\phi(t,x)}\{\tau_1>t+s\}|\tau_1>t]\\
= & F(x,t)\,{\mathbb{P}}_{\phi(t,x)}\{\tau_1>t+s\}\\
= & F(x,t)F(\phi(t,x),s).
\end{aligned}$$ The property (ii) is proved.
We turn to the proof of the property (i). For $x\in E$, $s,t\in \mathcal{I}_x$ with $t+s\in\mathcal{I}_x$, and a single point set $\{\phi(t+s,x)\}\in {\mathcal{E}}$, we consider the conditional expectation ${\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{t+s})\,\bbbone_{\{\tau_1>t+s\}}|{\mathcal{F}}_t]$. On the one hand, It follows from Markov property that $$\begin{aligned}
&{\mathbb{E}}_x[{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{t+s})\,\bbbone_{\{\tau_1>t+s\}}|{\mathcal{F}}_t]]\\
=&{\mathbb{E}}_x[{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{s}\circ \theta_t)\,\bbbone_{\{\tau_1>t\}}\,\bbbone_{\{\tau_1\circ\theta_t>s\}}|{\mathcal{F}}_t]]\\
=&{\mathbb{E}}_x[\bbbone_{\{\tau_1>t\}}]\,{\mathbb{E}}_x[{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{s}\circ \theta_t)\,\bbbone_{\{\tau_1\circ\theta_t>s\}}|X_t]]\\
=&F(x,t)\, {\mathbb{E}}_{\phi(t,x)}[\bbbone_{\{\phi(t+s,x)\}}(\phi(s,\phi(t,x)))\,\bbbone_{\{\tau_1>s\}}]\\
=&\bbbone_{\{\phi(t+s,x)\}}(\phi(s,\phi(t,x))\,F(x,t)\,F(\phi(t,x),s), \end{aligned}$$ where $\theta_t$ is the shift operator. On the other hand, $$\begin{aligned}
&{\mathbb{E}}_x[{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{t+s})\,\bbbone_{\{\tau_1>t+s\}}|{\mathcal{F}}_t]]\\
=&{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(X_{t+s})\,\bbbone_{\{\tau_1>t+s\}}]\\
=&{\mathbb{E}}_x[\bbbone_{\{\phi(t+s,x)\}}(\phi(t+s,x))\,\bbbone_{\{\tau_1>t+s\}}]\\
=&{\mathbb{E}}_x[\bbbone_{\{\tau_1>t+s\}}]\\
=&F(x,t+s).
\end{aligned}$$ Therefore, $$\bbbone_{\{\phi(t+s,x)\}}(\phi(s,\phi(t,x))\,F(x,t)\,F(\phi(t,x),s)=F(x,t+s).$$ In these cases, $F(x,t+s)>0$. It follows from (ii) that $$\bbbone_{\{\phi(t+s,x)\}}(\phi(s,\phi(t,x)))=1.$$ That is $\phi(t+s,x)=\phi(s,\phi(t,x))$. It is obviously that $\phi(0,x)=x$.
At last, for $x\in E$, $s,t\in \mathcal{I} _x$ with $t+s\in\mathcal{I} _x$ and $B\in {\mathcal{E}}$, we have $$\begin{aligned}
G(x, t+{\mathrm{d}}s, B) = & {\mathbb{P}}_x\{\tau_1\in t+{\mathrm{d}}s, X_{\tau_1}\in B\}\\
= & F(x,t)\,{\mathbb{P}}_x\{\tau_1\in t+{\mathrm{d}}s, X_{\tau_1}\in B|\tau_1>t\}\\
= & F(x,t)\,{\mathbb{E}}_x[{\mathbb{P}}_x\{\tau_1\in t+{\mathrm{d}}s, X_{\tau_1}\in B|{\mathcal{F}}_t\}|\tau_1>t]\\
= & F(x,t)\,{\mathbb{E}}_x[{\mathbb{P}}_{\phi(t,x)}\{\tau_1\in t+{\mathrm{d}}s, X_{\tau_1}\in B\}|\tau_1>t]\\
= & F(x,t)\,{\mathbb{P}}_{\phi(t,x)}\{\tau_1\in t+{\mathrm{d}}s, X_{\tau_1}\in B\}\\
= & F(x,t)\,G(\phi(t,x), {\mathrm{d}}s, B).
\end{aligned}$$ Hence, $$\begin{aligned}
q(x, t+s, B) = & \frac{G(x,t+{\mathrm{d}}s, B)}{F(x, t+{\mathrm{d}}s)}\\
= & \frac{F(x,t)G(\phi(t,x),{\mathrm{d}}s, B)}{F(x,t)F(\phi(t,x), {\mathrm{d}}s)}\\
= & q(\phi(t,x), s,B).
\end{aligned}$$
This completes the proof of the theorem.
Properties of the characteristic triple
---------------------------------------
Let $X=\{X_t\}_{0\leqslant t<\tau}$ is a general PDMP with characteristic triple $(\phi,F,q)$. It follows from the subsection above that the property (i), (ii) and (iii) of $\phi$, $F$ and $q$ respectively can serve as the starting point for the development of the theory of general PDMPs. We firstly focus on the mapping $\phi$ which describes the deterministic evolution between two adjacent random jumps. Theorem \[thm.PDP=>str.Markov\] (i) indicates that $\phi$ is in fact a semi-dynamic system (SDS) defined as follow.
Let $(E,{\mathcal{E}})$ be a Polish space, $\phi:({\mathbb{R}}_+\times E,{\mathcal{B}}({\mathbb{R}}_+)\times{\mathcal{E}})\mapsto(E,{\mathcal{E}})$ a measurable mapping. $\phi$ is called a *semi-dynamic system* (or a *semi-flow*) if for any $x\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t<c(x)$, $$\label{eq.SDS}
\phi(0,x)=x,\quad \phi(t,\phi(s,x))=\phi(s+t,x),$$ where $$\label{eq.killing time}
c(x)=c(\phi(t,x))+t,\quad 0\leqslant t<c(x).$$
For convenience, we extend the state space by adding an isolated point $\Delta$ to $E$ and a part of the boundary of $E$. Let $$ \phi(c(x),x):=\left\{
\begin{array}{ll}
\displaystyle\lim_{t\uparrow c(x)}\phi(t,x), &
\hbox{if $\displaystyle\lim_{t\uparrow c(x)}\phi(t,x)$ exists;} \\
\Delta, & \hbox{otherwise.}
\end{array}\right.$$ Thus, (\[eq.SDS\]) still holds for $s+t=c(x)$ if $c(x)<\infty$. Denote $$\partial^+E:=\big\{\phi(c(x),x):x\in E\big\}.$$ We call $\partial^+E$ the *flowing-out boundary* of the state space $E$, and let $\bar{E}:=E\cup\partial^+E$.
The subset of $\bar{E}$, $\Phi_x:=\big\{\phi(t,x):t\in\mathcal{I}_x\big\}$ for any $x\in E$, is called a *path* (or a *trajectory*) of the SDS $\phi$ starting from $x$ and killed at time $c(x)$.
A state $x\in E$ is called an *equilibrium state* if $\phi(t,x)\equiv x$ holds for all $t\in {\mathbb{R}}_+$. An equilibrium state is also called a *rest state* or a *fixed state* somewhere. In this case, the path $\Phi_x$ contains only one state, that is, $\Phi_x=\{x\}$. The set of all the equilibrium states of the SDS $\phi$ is denoted by $E_e$.
A state $x\in E$ is called a *periodic state* if $\phi(\cdot,x)$ is periodic. Denote minimal period by $T_x\in{\mathbb{R}}_+$. In this case, the path $\Phi_x$ is called to be *periodic*. The set of all periodic states of the SDS $\phi$ is denoted by $E_p$. Note that an equilibrium state is a special case for $T_x=0$, thus $E_e\subset E_p$. From (\[eq.killing time\]), it is easily to check that $c(x)=\infty$ for $x\in E_p$.
A state $x\in E\setminus E_p$ is called an *aperiodic state*. For convenience, we let $T_x=\infty$ when $x$ is an aperiodic state.
The essential difference between an SDS and a dynamic system is that, starting from the initial state, an SDS can predict the future, but can not recall the history exactly. That is, an SDS allows the situation that different paths starting from different states join together at some states.
\[def.confluent state\] A state $x\in E$ is called a *confluent state* of an SDS $\phi$ if either one of the following conditions holds:
(i) there exist $x_1,x_2\in E\setminus E_p$ with $x_1\notin \Phi_{x_2}$, $x_2\notin\Phi_{x_1}$ and $\Phi_{x_1}\cap\Phi_{x_2}\neq \emptyset$ such that $x=\phi(t^*(x_1,x_2),x_1)$;
(ii) there exists $x_1\in E\setminus E_p$ and $x_2\in E_p$ with $\Phi_{x_1}\supset\Phi_{x_2}$ such that $x=\phi(t^*(x_1,x_2),x_1)$,
where $t^*(x_1,x_2):=\inf\{t>0:\phi(t,x_1)\in\Phi_{x_2}\}$.
To consider the conditional survival function $F:E\times{\mathbb{R}}_+\mapsto[0,1]$ satisfying Theorem \[thm.PDP=>str.Markov\] (ii) together with $F(x,0)=1$, let us recall the Stieltjes versions of exponentials and logarithms (see Meyer (1966) or Sharpe (1988), A.4). In fact, $\{F(x,\cdot)\}_{x\in E}$ is a family of *M-functions*, i.e., for any $x\in E$, $F(x,\cdot):[0,\infty)\mapsto[0,1]$ is a right continuous and decreasing function with $F(x,0)=1$. The *Stieltjes logarithm* of the M-function $F(x,\cdot)$ is defined as $$\label{eq.slogF}
\Lambda(x,t)={\mathrm{slog}\,}F(x,t):=\int_{(0,t]}\!\frac{F(x,{\mathrm{d}}s)}{F(x,s-)},\quad t\in\mathcal{I}_x.$$ Then $\{\Lambda(x,\cdot)\}_{x\in E}$ be a family of *A-functions*, i.e., for any $x\in E$, $\Lambda(x,\cdot):[0,\infty)\mapsto[0,\infty]$ is a right continuous and increasing function such that $\Lambda(x,0)=0$, $\Delta\Lambda(x,t)<1$ for all $t$ with $\Lambda(x,t)<\Lambda(x,\infty)$, with $\Delta\Lambda(x,t)=1$ possible if $\Lambda(x,t)=\Lambda(x,\infty)<\infty$ (here $\Lambda(x,\infty):=\lim_{t\uparrow c(x)}\Lambda(x,t)$). The *Stieltjes exponential* of the A-function $\Lambda(x,\cdot)$ is defined as $$\label{eq.sexpLambda}
{\mathrm{sexp}\,}\Lambda(x,t):=e^{-\Lambda^c(x,t)}\prod_{0<s\leqslant t}\big[1-\Delta\Lambda(x,s)\big],\quad t\in\mathcal{I}_x,$$ where $\Lambda^c(x,\cdot)$ is the continuous part of $\Lambda(x,\cdot)$ and $$\Delta\Lambda(x,t):=\Lambda(x,t)-\Lambda(x,t-),\quad t\in\mathcal{I}_x\setminus\{0\},\,x\in E.$$ It is known that the mapping $F\mapsto{\mathrm{slog}\,}F$ is a bijective mapping of the class of M-functions onto the class of A-functions, and that $\Lambda={\mathrm{slog}\,}F$ if and only if $F={\mathrm{sexp}\,}\Lambda$.
The function $\Lambda$ defined by the (\[eq.slogF\]) is called the *conditional hazard function*. Since the conditional survival function $F$ satisfies Theorem \[thm.PDP=>str.Markov\] (ii), we have the following theorem.
\[thm.A add<=>F mult\] The conditional hazard function $\Lambda$ satisfies that for any $x\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t\in\mathcal{I}_x$, $$ \Lambda(x,0)=0,\quad \Lambda(x,s+t)=\Lambda(x,s)+\Lambda(\phi(s,x),t)$$ if and only if the conditional survival function $F$ satisfies that for any $x\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t\in\mathcal{I}_x$, $$ F(x,0)=1,\quad F(x,s+t)=F(x,s)\,F(\phi(s,x),t).$$
It follows directly from the definition of Stieltjes logarithm (\[eq.slogF\]) and Stieltjes exponential (\[eq.sexpLambda\]).
Since the characteristic $F$ is uniquely determined by $\Lambda$, the triple $(\phi, \Lambda, q)$ plays the same role. We call $(\phi, \Lambda,q)$ the characteristic triple too.
Now we consider the transition kernel $q:E\times{\mathbb{R}}_+\times{\mathcal{E}}\mapsto[0,1]$ which satisfies Theorem \[thm.PDP=>str.Markov\] (iii).
\[thm.q=Q\] Let the transition kernel $q:E\times{\mathbb{R}}_+\times{\mathcal{E}}\mapsto[0,1]$ satisfy Theorem [\[thm.PDP=>str.Markov\] (iii)]{.nodecor}. There exists a measurable function $Q:\bar{E}\times {\mathcal{E}}\mapsto[0,1]$ such that, for any $x\in E$, $t\in\mathcal{I}_x\setminus\{0\}$, if $\phi(t,x)$ is not a confluent state, then $$\label{eq.q=Q}
q(x,t,B)=Q(\phi(t,x),B).$$ Conversely, for the transition kernel $q$, if there exists a measurable function $Q$ such that [(\[eq.q=Q\])]{.nodecor} holds for all $t\in\mathcal{I}_x\setminus\{0\}$ and $x\in E$, then $q$ satisfies Theorem [\[thm.PDP=>str.Markov\] (iii)]{.nodecor}.
Suppose $x^*\in E\setminus E_e$ is not a confluent state. Then, for any $x_1,x_2\in E$, $t_1\in\mathcal{I}_{x_1}\setminus\{0\}$, $t_2\in\mathcal{I}_{x_2}\setminus\{0\}$ satisfying $x^*=\phi(t_1,x_1)=\phi(t_2,x_2)$, there exists a $t_0$ with $0<t_0<t_1\land t_2$ such that $$x_0:=\phi(t_1-t_0,x_1)=\phi(t_2-t_0,x_2).$$ Thus $$\begin{aligned}
q(x_1,t_1,B) & =q(\phi(t_1-t_0,x_1),t_0,B)=q(x_0,t_0,B) \\
& =q(\phi(t_2-t_0,x_2),t_0,B)=q(x_2,t_2,B),\quad B\in{\mathcal{E}}.
\end{aligned}$$ By the arbitrariness of $(x_1,t_1)$ and $(x_2,t_2)$, $q(x,t,B)$ is independent of the choice of $(x,t)$ with $\phi(t,x)=x^*$. Then there exists a measurable function $Q_1:\bar{E}\setminus E_e\times{\mathcal{E}}\mapsto[0,1]$ such that if $\phi(t,x)=x^*$ then $$\label{eq.q=Q1}
q(x,t,B)=Q_1(x^*,B),\quad B\in{\mathcal{E}}.$$
Suppose that $x^*\in E_e$ is not a confluent state. That is, $\phi(t,x^*)=x^*$ holds for all $t\in{\mathbb{R}}_+$. Thus, for any $s,t\in{\mathbb{R}}_+\setminus\{0\}$, $$q(x^*,s+t,B)=q(\phi(s,x^*),t,B)=q(x^*,t,B),\quad B\in{\mathcal{E}},$$ which is independent of the choice of $s$ and $t$. Then, there exists a measurable function $Q_2:E_e\times{\mathcal{E}}\mapsto[0,1]$ such that $$\label{eq.q=Q2}
q(x^*,t,B)=Q_2(x^*,B),\quad t\in{\mathbb{R}}_+,\,B\in{\mathcal{E}}.$$
Let $$Q(x,B)=\left\{\begin{array}{ll}
Q_1(x,B), \quad & x\in \bar{E}\setminus E_e, B\in{\mathcal{E}};\\
Q_2(x,B), \quad & x\in E_e, B\in{\mathcal{E}}.
\end{array}\right.$$ Then, for all $x\in E$, $t\in\mathcal{I}_x\setminus\{0\}$ and $B\in{\mathcal{E}}$, we have $q(x,t,B)=Q(\phi(t,x),B)$.
Conversely, if (\[eq.q=Q\]) holds for all $t\in\mathcal{I}_x\setminus\{0\}$ and $x\in E$, we have $$q(x,s+t,B)=Q(\phi(s+t,x),B)$$ and $$q(\phi(s,x),t,B)=Q(\phi(t,\phi(s,x)),B)=Q(\phi(s+t,x),B).$$ Thus, $q(x,s+t,B)=q(\phi(s,x),t,B)$ for $s+t\in\mathcal{I}_x$, $x\in E$ and $B\in{\mathcal{E}}$. This completes the proof.
Sub-classes of general PDMPs and the characteristic $\Lambda$
--------------------------------------------------------------
The two well-studied sub-classes of general PDMPs, in the terminology of Jacod and Skorokhod [@jacod1996jumping], are quasi-Hunt PDMPs and qusi-Itô PDMPs defined as follows.
A general PDMP $X$ is called a *quasi-Hunt* if there is a jumping sequence $\{\tau_n\}$ with (\[eq.PiecewiseDeterministic\]) such that the jumping time $\tau_n$ is totally inaccessible for all $n\geqslant 1$.
Let $X$ be a general PDMP with the characteristic $\Lambda$. If for all $x\in E$ $\Lambda(x,\cdot)$ is continuous on $\mathcal{I}_x$, then the $X$ is quasi-Hunt. Conversely, if $\tau_1$ is totally inaccessible then $\Lambda(x,\cdot)$ is continuous on $\mathcal{I}_x$ for all $x\in E$.
It follows directly from Jacod and Skorokhod [@jacod1996jumping] Theorem 5 (b) and Theorem 6 (c).
The quasi-Hunt PDMP $X$ is called *quasi-Itô* if for each $x\in E$ the predictable brackets of the locally square-integrable ${\mathbb{P}}_x$-martingales prior to $\tau$ are absolutely continuous w.r.t. Lebesgue measure.
Jacod and Skorokhod [@jacod1996jumping] presents the necessary and sufficient condition for a quasi-Hunt PDMP to be quasi-Itô in term of $\Lambda$ as follow.
The quasi-Hunt PDMP $X$ is quasi-Itô if and only if $\Lambda(x,\cdot)$ is absolutely continuous on $\mathcal{I}_x$ for all $x\in E$.
See Jacod and Skorokhod [@jacod1996jumping] Theorem 17.
As supplementary, we introduce a new sub-class of PDMPs, the quasi-step PDMPs.
A general PDMP $X$ is called *quasi-step* if the jumping time $\tau_n$ is accessible for $n\geqslant 1$.
Let $X$ be a general PDMP with the characteristic $\Lambda$. The $X$ is quasi-step if and only if $\Lambda(x,\cdot)$ is a step function for all $x\in E$, i.e., $$ \Lambda(x,t)=\sum_{0<s\leqslant t}\Delta\Lambda(x,s),\quad t\in\mathcal{I}_x,\,x\in E,$$ or equivalently $$ F(x,t)=\prod_{0<s\leqslant t}[1-\Delta\Lambda(x,s)],\quad t\in\mathcal{I}_x,\,x\in E.$$
Let $X$ be a quasi-step PDMP. Then the jumping time $\tau_1$ is accessible. It follows from the definition of an accessible time that there exists a sequence of predictable times $\{T_n\}$ such that $\tau_1\in \{T_1,T_2,\cdots\}$ ${\mathbb{P}}_x$-a.s. for all $x\in E$. Since $\{T_n\}$ is a sequence of predictable times, there exists a sequence of ${\mathcal{F}}_0=\sigma(X_0)$-measurable r.v.’s, $\{R_n\}$ such that $$T_n\bbbone_{\{T_n\leqslant \tau_1\}}=R_n\bbbone_{\{R_n\leqslant \tau_1\}}.$$ The $\sigma(X_0)$-measurability of $\{R_n\}$ implies that there exists a sequence of real measurable functions, $r_n: (E,{\mathcal{E}})\mapsto({\mathbb{R}}_+,{\mathcal{B}}({\mathbb{R}}_+))$, such that $R_n=r_n(X_0)$, or equivalently, $R_n=r_n(x)$. We have $$T_n\bbbone_{\{T_n\leqslant \tau_1\}}=r_n(x)\bbbone_{\{r_n(x)\leqslant \tau_1\}}.$$ These imply $\tau_1\in \{r_1(x),r_2(x),\cdots\}$ ${\mathbb{P}}_x$-a.s. for all $x\in E$. Hence, $$\Lambda(x,t)=\sum_{0<r_n(x)\leqslant t}\Delta\Lambda(x,r_n(x))$$ for all $t\geqslant 0$ with $$\Delta\Lambda(x,r_n(x))=\frac{{\mathbb{P}}_x\{\tau_1=r_n(x)\}}{1-\sum_{r_m(x)<r_n(x)}{\mathbb{P}}_x\{\tau_1=r_m(x)\}}.$$
Conversely, let $\Lambda(x,t)$ is a step function w.r.t. $t$ for all $x\in E$. Denote by $\{r_1(x), r_2(x),\cdots\}$ the set of all jumping points of $\Lambda(x,\cdot)$. And let $T_n=r_n(X_0)$. Then each $T_n$ is predictable and $\tau_1\in \{T_1,T_2,\cdots\}$ ${\mathbb{P}}_x$-a.s. That is to say, $\tau_1$ is accessible. Let $T_{mn}=\tau_{m-1}+r_n(X_{\tau_{m-1}})$ for $m,n\geqslant 1$. Obviously, each $T_{mn}$ is predictable and $$\begin{aligned}
{\mathbb{P}}_x\{\tau_m=T_{mn}\}&={\mathbb{P}}_x\{\tau_m=\tau_{m-1}+r_n(X_{\tau_{m-1}})\}\\
&={\mathbb{P}}_x\{\tau_m-\tau_{m-1}=r_n(X_{\tau_{m-1}})\}\\
&={\mathbb{E}}_x\big[{\mathbb{P}}_x\{\tau_m-\tau_{m-1}=r_n(X_{\tau_{m-1}})|{\mathcal{F}}_{\tau_{m-1}}\}\big]\\
&={\mathbb{E}}_x\big[{\mathbb{P}}_{X_{\tau_{m-1}}}\{\tau_1=r_n(X_{\tau_{m-1}})\}\big].
\end{aligned}$$ Taking summation on both ends in the equation above, we have $$\sum_{n=1}^{\infty}{\mathbb{P}}_x\{\tau_m=T_{mn}\}
={\mathbb{E}}_x\left[\sum_{n=1}^{\infty}{\mathbb{P}}_{X_{\tau_{m-1}}}\{\tau_1=r_n(X_{\tau_{m-1}})\}\right]=1.$$ These imply that $\tau_m\in \{T_{m1}, T_{m2}, \cdots\}$ ${\mathbb{P}}_x$-a.s. for all $x\in E$. Equivalently, each $\tau_m$ is accessible. Hence, $X$ is quasi-step.
Additive functionals of an SDS
==============================
Now we introduce the definition of additive functionals of an SDS which plays an important role in the study of PDMPs.
\[def.a.SDS\] (i) For a given SDS $\phi$, a measurable function $a:E\times{\mathbb{R}}_+\mapsto{\mathbb{R}}$ is called an *additive functional* of the SDS $\phi$ if the function $a(x,\cdot)$ is right-continuous on $\mathcal{I}_x$ for all $x\in E$ and $$\label{eq.a.SDS}
a(x,0)=0,\quad a(x,s)+a(\phi(s,x),t)=a(x,s+t),$$ for any $x\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t\in\mathcal{I}_x$. The set of all the additive functionals of the SDS $\phi$ is denoted by $\mathfrak{A}_\phi$.\
(ii) And additive functional $a$ of the SDS $\phi$ is called to be of *locally finite variation* if $a(x,\cdot)$ is of finite variation on any compact subinterval of $\mathcal{I}_x$ for all $x\in E$. And denote by $\mathfrak{A}_\phi^{loc}$, the set of all the additive functionals of the SDS $\phi$ with locally finite variation.
\[ex.integrable&summable\] (i) For a measurable function $g:\bar{E}\mapsto{\mathbb{R}}$, $$a(x,t):=\int_0^tg(\phi(s,x)){\mathrm{d}}s,\quad t\in\mathcal{I}_x,\,x\in E$$ is an additive functional of the SDS $\phi$ if it is well defined. In this case, $a\in\mathfrak{A}_\phi^{loc}$ if and only if the function $g$ is *locally absolutely path-integrable*, i.e., $$\int_0^t \big|g(\phi(s,x))\big|{\mathrm{d}}s<\infty\quad \hbox{for all }t\in\mathcal{I}_x,\,x\in E.$$ (ii) For a measurable function $g:\bar{E}\mapsto{\mathbb{R}}$, $$a(x,t):=\sum_{0<s\leqslant t}g(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E$$ is an additive functional of the SDS $\phi$ if it is well defined. In this case, $a\in\mathfrak{A}_\phi^{loc}$ if and only if the function $g$ is *locally absolutely path-summable*, i.e., $$\sum_{0<s\leqslant t}\big|g(\phi(s,x))\big|<\infty\quad \hbox{for all }t\in\mathcal{I}_x,\,x\in E.$$ (iii) For a measurable function $g:\bar{E}\mapsto{\mathbb{R}}$ and $a\in\mathfrak{A}_\phi^{loc}$, $$a^*(x,t):=\int_{(0,t]}g(\phi(s,x))a(x,{\mathrm{d}}s),\quad t\in\mathcal{I}_x,\,x\in E$$ is also an additive functional of the SDS $\phi$ if the integral above is well defined. In this case, $a^*\in\mathfrak{A}_\phi^{loc}$ if and only if $$\int_{(0,t]}\big|g(\phi(s,x))\big|\,\big|a\big|(x,{\mathrm{d}}s)<\infty\quad \hbox{for all }t\in\mathcal{I}_x,\,x\in E.$$
$\mathfrak{A}_\phi^{loc}$ is a linear space.
It can be directly followed from the definition of an additive functional of the SDS $\phi$.
Let $a\in\mathfrak{A}_\phi^{loc}$. Then there exists a function $g:E_e\mapsto{\mathbb{R}}$ such that for any equilibrium state $x\in E_e$, $$\label{eq.a.equilbrium}
a(x,t)=g(x)t,\quad t\in{\mathbb{R}}_+.$$
Note that $c(x)=\infty$ in this case. The equation $$a(x,s)+a(\phi(s,x),t)=a(x,s+t)\quad \hbox{for }s,t\in{\mathbb{R}}_+$$ becomes $$a(x,s)+a(x,t)=a(x,s+t)\quad \hbox{for } s,t\in{\mathbb{R}}_+.$$ Thus, by the right-continuity, the conclusion is proved.
Let $a\in\mathfrak{A}_\phi$. If $x\in E_p\setminus E_e$, that is, $x$ is a periodic state with minimal period $T_x\in{\mathbb{R}}_+\setminus\{0\}$, then $$\label{eq.a.periodic}
a(x,t)=\left\lfloor\frac{t}{T_x}\right\rfloor a(x,T_x)+a(x,t-\left\lfloor\frac{t}{T_x}\right\rfloor T_x),$$ where $\left\lfloor t\right\rfloor$ is the largest integer not exceeding $t$.
Set $n:=\left\lfloor\frac{t}{T_x}\right\rfloor$, $s:=t-nT_x$. Thus $$\begin{aligned}
a(x,t)&=a(x,nT_x+s)\\
&=\sum_{k=1}^na(\phi((k-1)T_x,x),T_x)+a(\phi(nT_x,x),s)\\
&=n\,a(x,T_x)+a(x,s).
\end{aligned}$$
\[lem.a.r-derivative\] Let $a\in\mathfrak{A}_\phi$. For any $x\in E$, $s\in[0,c(x))$, if the right derivative of function $a(x,t)$ at $t=s$ exists, then the right derivative of function $a(\phi(s,x),t)$ at $t=0$ exists, and $$\label{eq.a.r-derivative}
\frac{\partial^+a(x,t)}{\partial t}\bigg|_{t=s} =
\frac{\partial^+a(\phi(s,x),t)}{\partial t}\bigg|_{t=0}.$$
By the definition of additive functionals of an SDS, $$\begin{aligned}
\frac{\partial^+ a(x,t)}{\partial t}\bigg|_{t=s}
&= \lim_{h\downarrow 0} \frac{a(x,s+h)-a(x,s)}{h}\\
&= \lim_{h\downarrow 0} \frac{\big[a(x,s)+a(\phi(s,x),h)\big]-\big[a(x,s)+a(\phi(s,x),0)\big]}{h}\\
&= \lim_{h\downarrow 0} \frac{a(\phi(s,x),h)-a(\phi(s,x),0)}{h}\\
&= \frac{\partial^+ a(\phi(s,x),t)}{\partial t}\bigg|_{t=0}.
\end{aligned}$$
For any $x\in E$, denote $$\label{eq.Xa}
\mathcal{X}a(x):=
\begin{cases}
\displaystyle{\frac{\partial^+a(x,t)}{\partial t}\bigg|_{t=0}}, &
\textrm{if}\; \displaystyle{\frac{\partial^+a(x,t)}{\partial t}\bigg|_{t=0}} \textrm{exists};\\
0, & \textrm{otherwise}.\\
\end{cases}$$
Let $a\in\mathfrak{A}_\phi$. $a(x,\cdot)$ is absolutely continuous on $\mathcal{I}_x$ for a fixed $x\in E$ if and only if $$\label{eq.add.ac}
a(x,t)=\int_0^t \mathcal{X}a(\phi(s,x)) {\mathrm{d}}s,\quad t\in\mathcal{I}_x.$$
The sufficiency is obvious. Now we will prove the necessity. For $x\in E$, if $a(x,\cdot)$ be a absolutely continuous function on $\mathcal{I}_x$, then it is differential on $\mathcal{I}_x$ almost everywhere. By Lemma \[lem.a.r-derivative\], we have $$\frac{\partial a(x,t)}{\partial t}=\frac{\partial^+a(x,t)}{\partial t}
=\mathcal{X}a(\phi(t,x)) \quad \textrm{a.e. on }\mathcal{I}_x.$$ Define a function $$\tilde{a}(x,t)=\int_0^t\mathcal{X}a(\phi(s,x)) {\mathrm{d}}s,\quad t\in\mathcal{I}_x.$$ Apparently, $\tilde{a}(x,\cdot)$ is absolutely continuous on $\mathcal{I}_x$, and $$\frac{\partial\tilde{a}(x,t)}{\partial t}=\mathcal{X}a(\phi(t,x))
\quad\textrm{a.e. on }\mathcal{I}_x.$$ Thus, $$\frac{\partial\tilde{a}(x,t)}{\partial t}=\frac{\partial a(x,t)}{\partial t}\quad\textrm{a.e. on }\mathcal{I}_x,$$ or $$\frac{\partial}{\partial t}(\tilde{a}(x,t)-a(x,t))=0
\quad\textrm{a.e. on }\mathcal{I}_x.$$ Notice that $\tilde{a}(x,0)=a(x,0)=0$, by the absolute continuity of $\tilde{a}(x,\cdot)$ and $a(x,\cdot)$, we know that $\tilde{a}(x,\cdot)-a(x,\cdot)$ is absolutely continuous on $\mathcal{I}_x$, and that $$\tilde{a}(x,t)-a(x,t)=\tilde{a}(x,0)-a(x,0)=0\quad\hbox{for all }t\in\mathcal{I}_x.$$ Hence $$a(x,t)=\tilde{a}(x,t),\quad t\in\mathcal{I}_x.$$ This completes the proof.
Let $$J_a:=\big\{\phi(t,x):a(x,t)-a(x,t-)\neq 0,\; x\in E,\, t\in \mathcal{I}_x\!\setminus\!\{0\}\big\}.$$ A state $y\in \bar{E}$ is called an $a$-*jumping state* if $y\in J_a$.
\[lem.a.discrete\] Let $a\in\mathfrak{A}_\phi$. There exists a function $\Delta a$ such that, for any $x\in E$, $t\in\mathcal{I}_x$, if $\phi(t,x)$ is not a confluent state, then $$\label{eq.a.Delta}
a(x,t)-a(x,t-)=\Delta a(\phi(t,x)).$$ Furthermore, if $a\in\mathfrak{A}_\phi^{loc}$, then $\Delta a$ is locally absolutely path-summable.
For any $x\in E$, $t\in\mathcal{I}_x$, denote $y=\phi(t,x)$. Apparently, if $y\notin J_a$, then we always have $\Delta a(y)=a(x,t)-a(x,t-)=0$.
On the other hand, if $y\in J_a$ is not a confluent state, for any two different states $x_1,x_2\in E$ with $\phi(t_1,x_1)=\phi(t_2,x_2)=y$, there exists $s\in(0,t_1\land t_2)$ and $z\in E$ satisfying $$\phi(t_1-s,x_1)=\phi(t_2-s,x_2)=z\quad \hbox{and}\quad \phi(s,z)=y.$$ Thus, $$\begin{aligned}
&a(x_1,t_1)-a(x_1,t_1-)\\
=&\big[a(x_1,t_1-s)+a(z,s)\big]-\big[a(x_1,t_1-s)+a(z,s-)\big]\\
=&a(z,s)-a(z,s-).
\end{aligned}$$ Similarly, we have $$a(x_2,t_2)-a(x_2,t_2-)=a(z,s)-a(z,s-).$$ Hence, $$a(x_1,t_1)-a(x_1,t_1-)=a(x_2,t_2)-a(x_2,t_2-).$$ By the arbitrariness of $(x_1,t_1),(x_2,t_2)$, we can denote that $\Delta a(y)=a(x,t)-a(x,t-)$, which is a function of $y$ and independent of the choice of $(x,t)$ with $\phi(t,x)=y$.
Moreover, if $a\in\mathfrak{A}_\phi^{loc}$, then $$\sum_{0<s\leqslant t}\big|\Delta a(\phi(s,x))\big|\leqslant\int_{(0,t]}\big|a\big|(x,{\mathrm{d}}s)<\infty$$ for all $x\in E$, $t\in\mathcal{I}_x$. This completes the proof.
Next we will show the Lebesgue decomposition of an additive functional of the SDS $\phi$.
\[thm.a.Lebesgue\] Let $a\in\mathfrak{A}_\phi^{loc}$. Assume that $J_a$ contains no confluent state. Then there exists a locally absolutely path-integrable function $\mathcal{X}a$ and a locally absolutely path-summable function $\Delta a$ such that $a(x,\cdot)\,(x\in E)$ has the Legesgue decomposition $$\label{eq.a.Lebesgue}
a(x,\cdot)=a^{ac}(x,\cdot)+a^{sc}(x,\cdot)+a^{pd}(x,\cdot),$$ where $$a^{ac}(x,t)=\int_0^t\mathcal{X}a(\phi(s,x)){\mathrm{d}}s,\quad
a^{pd}(x,t)=\sum_{0<s\leqslant t}\Delta a(\phi(s,x)),\qquad t\in\mathcal{I}_x,$$ and $a^{sc}(x,\cdot)$ is the singularly continuous part of $a(x,\cdot)$. Furthermore, $a^{ac}$, $a^{sc}$ and $a^{pd}$ $\in\mathfrak{A}_\phi^{loc}$.
For any $x\in E$, let $\mathcal{X}a$ be a function defined as (\[eq.Xa\]). Since the derivatives of $a^{sc}(x,\cdot)$ and $a^{pd}(x,\cdot)$ are equal to $0$ almost everywhere, $$\frac{\partial a^{ac}(x,t)}{\partial t}=\frac{\partial^+a(x,t)}{\partial t}
=\mathcal{X}a(\phi(t,x))\quad\textrm{a.e. on }\mathcal{I}_x,$$ that is, $$a^{ac}(x,t)=\int_0^t\mathcal{X}a(\phi(s,x)){\mathrm{d}}s,\quad t\in\mathcal{I}_x.$$ On the other hand, $a^{ac}(x,\cdot)$ and $a^{sc}(x,\cdot)$ are both continuous, so all of $a$-jumping states coincide with $a^{pd}$-jumping states. By Lemma \[lem.a.discrete\], we have $$a^{pd}(x,t)=\sum_{0<s\leqslant t}a(x,s)-a(x,s-)
=\sum_{0<s\leqslant t}\Delta a(\phi(s,x)),
\quad t\in\mathcal{I}_x.$$ Moreover, according to Example \[ex.integrable&summable\], $a^{ac}$ and $a^{pd}$ are both additive functionals of the SDS $\phi$, hence $a^{sc}$ is also an additive functional of the SDS $\phi$. The additive functional $a$ is of locally finite variation, so are $a^{ac}$, $a^{sc}$ and $a^{pd}$. Thus, $\mathcal{X}a$ and $\Delta a$ are locally absolutely path-integrable and locally absolutely path-summable, respectively.
Notice that, by Theorem \[thm.A add<=>F mult\], the conditional hazard function $\Lambda$ defined as (\[eq.slogF\]) is an additive functional of the SDS $\phi$. Then we have the following corollary.
\[cor.Lambda.Lebesgue\] Let $\Lambda$ be the conditional hazard function defined by [(\[eq.slogF\])]{.nodecor}. Then $\Lambda\in\mathfrak{A}_\phi^{loc}$. Moreover, if $J_\Lambda$ contains no confluent state, then for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.Lambda.Lebesgue}
\Lambda(x,t)=\Lambda^{ac}(x,t)+\Lambda^{sc}(x,t)+\Lambda^{pd}(x,t),$$ where $$\Lambda^{ac}(x,t)=\int_0^t\lambda(\phi(s,x)){\mathrm{d}}s,\quad
\Lambda^{pd}(x,t)=\sum_{0<s\leqslant t}\Delta\Lambda(\phi(s,x)),$$ and $\Lambda^{sc}(x,\cdot)$ is the singularly continuous part of $\Lambda(x,\cdot)$. Or equivalently, $$\label{eq.F.Lebesgue}
F(x,t)=F^{ac}(x,t)\,F^{sc}(x,t)\,F^{pd}(x,t),$$ where $$F^{ac}(x,t)=\exp\left[-\int_0^t\lambda(\phi(s,x)){\mathrm{d}}s\right],\quad
F^{sc}(x,t)=\exp\big[-\Lambda^{sc}(x,t)\big],$$ $$F^{pd}(x,t)=\prod_{0<s\leqslant t}\big[1-\Delta\Lambda(\phi(s,x))\big].$$ Furthermore, $$\begin{aligned}
&\lambda(x)=
\left\{\begin{array}{ll}
\displaystyle \frac{\partial^+\Lambda(x,t)}{\partial t}\bigg|_{t=0}, &
\hbox{if }\displaystyle\frac{\partial^+\Lambda(x,t)}{\partial t}\bigg|_{t=0}\hbox{ exists;} \\
0, & \hbox{otherwise,}
\end{array}\right. \\
&\Delta\Lambda(\phi(t,x))=\Lambda(x,t)-\Lambda(x,t-),
\end{aligned}$$ are locally path-integrable and locally path-summable, respectively.
Following from Theorem \[thm.A add<=>F mult\], we get $\Lambda\in\mathfrak{A}_\phi$. In addition, for any fixed $x\in E$, $F(x,\cdot)$ is a decreasing function, so $\Lambda(x,\cdot)$ is an increasing function, hence $\Lambda\in\mathfrak{A}_\phi^{loc}$. Thus, by Theorem \[thm.a.Lebesgue\], we get (\[eq.Lambda.Lebesgue\]). And $\lambda$ and $\Delta\Lambda$ are locally path-integrable and locally path-summable, respectively. Applying (\[eq.sexpLambda\]) to $F(x,\cdot)={\mathrm{sexp}\,}\Lambda(x,\cdot)$, and noticing that $\Lambda^{ac}(x,\cdot)$ and $\Lambda^{sc}(x,\cdot)$ are continuous, we have (\[eq.F.Lebesgue\]) directly.
Actually, from the corollary above, we have $$F^{ac}={\mathrm{sexp}\,}\Lambda^{ac},\quad F^{sc}={\mathrm{sexp}\,}\Lambda^{sc},\quad F^{pd}={\mathrm{sexp}\,}\Lambda^{pd}.$$ Since $\Lambda^{ac}$, $\Lambda^{sc}$ and $\Lambda^{pd}$ are all additive functionals of the SDS $\phi$, by Theorem [\[thm.A add<=>F mult\]]{.nodecor}, $F^{ac}$, $F^{sc}$ and $F^{pd}$ all satisfy the condition [(ii)]{.nodecor} in Theorem [\[thm.PDP=>str.Markov\]]{.nodecor}.
Additive functionals of a general PDMP
======================================
\[def.A.PDMP\] Let $X=\{X_t\}_{0\leqslant t<\tau}$ be a general PDMP with natural filtration $\{{\mathcal{F}}_t\}$. A real valued càdlàg $\{{\mathcal{F}}_t\}$-adapted process $A=\{A_t\}_{0\leqslant t<\tau}$ is called an *additive functional* of $X$ if, for any stopping time $T$,
(i) $A_0=0$;
(ii) $A_{T+s}=A_T+ A_s\circ\theta_T $ a.s. on $\{T<\infty\}$,
where $\theta_t$ is the shift operator.
\[thm.A=a+b\] Let $X$ be a general PDMP. An ${\mathbb{R}}$-valued càdlàg ${\mathcal{F}}$-adapted process $A=\{A_t\}_{0\leqslant t<\tau}$ with $A_0=0$ is an additive functional of $X$, if and only if there exists $a\in\mathfrak{A}_\phi$ and a measurable function $b:E\times{\mathbb{R}}_+\times E\mapsto{\mathbb{R}}$ satisfying that for any $x,y\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t\in\mathcal{I}_x\setminus\{0\}$, $$\label{eq.b.invariant}
b(x,s+t,y)=b(\phi(s,x),t,y),$$ such that $$\label{eq.A=a+b}
A_t=A_{\tau_n} + a(X_{\tau_n},t-\tau_n)
+b(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}})\bbbone_{\{t=\tau_{n+1}\}}$$ for $t\in(\tau_n,\tau_{n+1}]$, $n\in{\mathbb{N}}$. Especially, for any $x\in E$, $t\in\mathcal{I}_x\setminus\{0\}$, if $\phi(t,x)$ is not a confluent state, then there exist a measurable function $\bar{b}:\bar{E}\times E\mapsto {\mathbb{R}}$ such that $$\label{eq.b=bar.b}
b(x,t,y)=\bar{b}(\phi(t,x),y),\quad y\in E.$$
The sufficiency is obvious. We only need to prove the necessity. By the additivity of $A$, It suffices to prove (\[eq.A=a+b\]) for n=0. Since ${\mathcal{F}}_t=\sigma(X_0)$ for all $t<\tau_1$, and $A$ is ${\mathcal{F}}$-adapted, there exists a measurable function $a:\bar{E}\times{\mathbb{R}}_+\mapsto {\mathbb{R}}$ such that $A_t=a(X_0,t)$ for all $t<\tau_1$. It is followed from Definition \[def.A.PDMP\], that $a$ is an additive functional of the SDS $\phi$. Define a process $A^-=\{A_t^-\}_{0\leqslant t<\tau}$ by $$\begin{aligned}
A_0^- &:= 0, \\
A_t^- &:= A_{\tau_n}+a(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}],\;n\in{\mathbb{N}}.
\end{aligned}$$ Apparently, $A^-$ is an additive functional of $X$. And denote $$\hat{A}_t:=A_t-A_t^-,\quad 0\leqslant t<\tau.$$ Since ${\mathcal{F}}_{\tau_1}=\sigma(X_0,\tau_1,X_{\tau_1})$, there exists a measurable function $b:E\times{\mathbb{R}}_+\times E\mapsto {\mathbb{R}}$ such that $$\hat{A}_{\tau_1}=b(X_0,\tau_1,X_{\tau_1})\quad\hbox{ a.s. on }\{\tau_1<\infty\}.$$ By the additivity of $A$ and $A^-$, $$\hat{A}_{\tau_1}= \hat{A}_{\tau_1}\circ\theta_s \quad\hbox{ a.s. on }\{s<\tau_1\},$$ then we get (\[eq.b.invariant\]). Hence, (\[eq.A=a+b\]) holds for $n=0$.
Following from (\[eq.b.invariant\]), to prove (\[eq.b=bar.b\]), we need to consider whether $\phi(t,x)$ is an equilibrium state or not for $x\in E$ and $t\in\mathcal{I}_x\setminus\{0\}$. If $x'\in E_e$ is not a confluent state, that is, $\phi(t,x')\equiv x'$ for all $t\in{\mathbb{R}}_+$, then $$b(x',s+t,y)=b(\phi(s,x'),t,y)=b(x',t,y)$$ holds for all $y\in E$, $s,t\in{\mathbb{R}}_+$ and $s+t>0$. Thus there exists a measurable function $\bar{b}_1:E_e\times E\mapsto{\mathbb{R}}$ such that $$b(x',t,y)=\bar{b}_1(x',y)$$ which is independent of $t$. On the other hand, if $x''\in E\setminus E_e$ is not a confluent state. For any $x_1,x_2\in E$, $t_1\in\mathcal{I}_{x_1}\setminus\{0\}$ and $t_2\in\mathcal{I}_{x_2}\setminus\{0\}$ satisfying $\phi(t_1,x_1)=\phi(t_2,x_2)=x''$, there exists $t_0\in(0,t_1\land t_2)$ such that $$\phi(t_1-t_0,x_1)=\phi(t_2-t_0,x_2)=:x_0.$$ Thus, for $y\in E$, $$\begin{aligned}
b(x_1,t_1,y)=&b(\phi(t_1-t_0,x_1),t_0,y)=b(x_0,t_0,y)\\
=&b(\phi(t_2-t_0,x_2),t_0,y)=b(x_2,t_2,y).
\end{aligned}$$ By the arbitrariness of $(x_1,t_1)$ and $(x_2,t_2)$, there exists a measurable function $\bar{b}_2:E\setminus E_e\times E\mapsto{\mathbb{R}}$ such that $$b(x,t,y)=\bar{b}_2(x'',y)$$ holds for all $x\in E$, $t\in\mathcal{I}_x\setminus\{0\}$ with $\phi(t,x)=x''$. Therefore, for all $x\in E$, $t\in\mathcal{I}_x\setminus\{0\}$, if $\phi(t,x)$ is not a confluent state, then we have (\[eq.b=bar.b\]). Actually, we can take $$\bar{b}(x,\cdot)=
\left\{
\begin{array}{ll}
\bar{b}_1(x,\cdot), & x\in E_e; \\
\bar{b}_2(x,\cdot), & x\in E\setminus E_e.
\end{array}
\right.$$ The proof is complete.
\[cor.A.pred=a\] Let $A$ be an additive functional of a general PDMP $X$. $A$ is predictable if and only if there exists $a\in\mathfrak{A}_\phi$ such that $$\label{eq.A.pred=a}
A_t=A_{\tau_n} + a(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}],\, n\in{\mathbb{N}}.$$
The sufficiency is obvious. Now we prove the necessity. By the additivity of $A$, it suffices to prove (\[eq.A.pred=a\]) for n=0. Since ${\mathcal{F}}_t=\sigma(X_0)$ for all $t<\tau_1$, and $A$ is predictable, there exists a measurable function $a:\bar{E}\times{\mathbb{R}}_+\mapsto {\mathbb{R}}$ such that $A_t=a(X_0,t)$ for all $t\leqslant\tau_1$. The additivity of $A$ implies that $a$ is an additive functional of the SDS $\phi$.
Here we define an auxiliary process $X^-=\{X_t^-\}_{0\leqslant t<\tau}$, $$X_t^-:=X_0\bbbone_{\{t=0\}}+\sum_{n=0}^\infty\phi(t-\tau_n,X_{\tau_n})
\bbbone_{\{\tau_n<t\leqslant\tau_{n+1}\}}, \quad 0\leqslant t<\tau.$$ Obviously, $X_t^-=X_t$ if $t\neq\tau_n$, $n=1,2,\dots$ In other words, $X^-$ modifies the values of $X$ only at the random jumping times. It is known that $X^-$ is predictable.
\[thm.A.Lebesgue\] Let $A$ be an optional additive functional of the general PDMP $X$ with the associated $a$ and $b$ as in Theorem [\[thm.A=a+b\]]{.nodecor}. Assume that $J_a$ contains no confluent state. If $a\in\mathfrak{A}_\phi^{loc}$, then $$\begin{aligned}
A_t=&\int_0^t\mathcal{X}a(X_s){\mathrm{d}}s+\sum_{n=0}^{N_t}a^{sc}(X_{\tau_n},t\land\tau_{n+1}-\tau_n)
+\sum_{0<s\leqslant t}\Delta a(X_s^-) \nonumber\\
&+\sum_{n=1}^{N_t}b(X_{\tau_{n-1}},\tau_{n}-\tau_{n-1},X_{\tau_{n}}),\quad\quad 0\leqslant t<\tau, \label{eq.A.Lebesgue}
\end{aligned}$$ where $N_t:=\max\{n\geqslant 0:\tau_n\leqslant t\}$, $a^{sc}\in\mathfrak{A}_\phi^{loc}$, $\mathcal{X}a$ is locally absolutely path-integrable, and $\Delta a$ is locally absolutely path-summable.
For an optional additive functional $A$, we have (\[eq.A=a+b\]). If $a\in\mathfrak{A}_\phi^{loc}$, applying Theorem \[thm.a.Lebesgue\], we get (\[eq.A.Lebesgue\]).
The process $M=\{M_t\}_{0\leqslant t<\tau}$ is a *local martingale prior to $\tau$* if there exist localizing times $\{T_n\}$ with $T_n\uparrow\tau$ such that $M_{t\land T_n}$ is a martingale for each $n\in{\mathbb{N}}$.
Let $X$ be a general PDMP with characteristic triple $(\phi,F,q)$. The corresponding *jumping measure* is $$\label{eq.mu}
\mu({\mathrm{d}}t,{\mathrm{d}}y)=\sum_{n=1}^\infty \delta_{(\tau_n,X_{\tau_n})}({\mathrm{d}}t,{\mathrm{d}}y) \bbbone_{\{\tau_n<\infty\}}.$$ The *predictable dual projection* (or *compensator*) of the jumping measure $\mu$ for a general PDMP $X$ is $$\nu({\mathrm{d}}t,{\mathrm{d}}y) = \sum_{n=0}^\infty\frac{F(X_{\tau_n},{\mathrm{d}}t-\tau_n)\, q(X_{\tau_n},t-\tau_n,{\mathrm{d}}y)}{F(X_{\tau_n},(t-\tau_n)-)} \bbbone_{\{\tau_n<t\leqslant\tau_{n+1}\}}.$$ Due to (\[eq.slogF\]), the definition of Stieltjes logarithm, we have $$\frac{F(X_{\tau_n},{\mathrm{d}}t-\tau_n)}{F(X_{\tau_n},(t-\tau_n)-)}
= \Lambda(X_{\tau_n},{\mathrm{d}}t-\tau_n).$$ Therefore, $$\label{eq.nu}
\nu({\mathrm{d}}t,{\mathrm{d}}y)=\sum_{n=0}^\infty\Lambda(X_{\tau_n},{\mathrm{d}}t-\tau_n)\,
q(X_{\tau_n},t-\tau_n,{\mathrm{d}}y)\bbbone_{\{\tau_n<t\leqslant\tau_{n+1}\}}.$$
\[thm.semimartingale\] Let $A$ be an additive functional of a general PDMP $X$, and $a,b$ the functions associated with $A$ as Theorem [\[thm.A=a+b\]]{.nodecor}.
[(i)]{.nodecor} $A$ is a local martingale prior to $\tau$ if and only if for any $x\in E$ and $t\in\mathcal{I}_x$, we have $$\label{eq.local-m 1}
\int_{(0,t]}\int_E \big|b(x,s,y)\big| q(x,s,{\mathrm{d}}y) \Lambda(x,{\mathrm{d}}s)<\infty,$$ and $$\label{eq.local-m 2}
a(x,t)=-\int_{(0,t]}\int_E b(x,s,y)\, q(x,s,{\mathrm{d}}y)\, \Lambda(x,{\mathrm{d}}s).$$
[(ii)]{.nodecor} $A$ is a special semimartingale prior to $\tau$ if and only if $a\in\mathfrak{A}_\phi^{loc}$, and [(\[eq.local-m 1\])]{.nodecor} is satisfied. In this case, $A$ has a canonical decomposition $A=M+B$, where $M$ is a local martingale prior to $\tau$ which is also an additive functional of $X$, and $B$ is a predictable additive functional of $X$ with $B_0=0$ and $$\label{eq.predictable B}
B_t=B_{\tau_n}+a^*(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}],\,n\in{\mathbb{N}},$$ where $$ a^*(x,t)=a(x,t)+\int_{(0,t]}\int_E b(x,s,y)\,q(x,s,{\mathrm{d}}y)\,\Lambda(x,{\mathrm{d}}s).$$
\(ii) It follows from Theorem 14 a) of [@jacod1996jumping] that $A$ is a local martingale prior to $\tau$ if and only if there exists a predictable function $\bar{A}:\Omega\times{\mathbb{R}}_+\times E\mapsto{\mathbb{R}}$ such that $$\label{eq.local-m.ineq}
\int_{(0,t]\times E}\big|\bar{A}(s,y)\big|\,\big|\mu-\nu\big|({\mathrm{d}}s,{\mathrm{d}}y)<\infty \quad {\mathbb{P}}_x\hbox{-a.s. for }0\leqslant t<\tau$$ and $$\begin{aligned}
A_t&=\int_{(0,t]\times E}\bar{A}(s,y)(\mu-\nu)({\mathrm{d}}s,{\mathrm{d}}y)\nonumber\\
&=\sum_{n=1}^{N_t}\bar{A}(\tau_n,X_{\tau_n})-\int_{(0,t]\times E}\bar{A}(s,y)\nu({\mathrm{d}}s,{\mathrm{d}}y),\quad 0\leqslant t<\tau. \label{eq.local-m.eq}
\end{aligned}$$
If $A$ is a local martingale prior to $\tau$, comparing (\[eq.local-m.eq\]) with (\[eq.A=a+b\]), we have $$\bar{A}(\tau_{n+1},X_{\tau_{n+1}})=b(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}}),$$ and $$\begin{aligned}
&a(X_{\tau_n},t-\tau_n)\\
=&-\int_{(\tau_n,t]\times E}\bar{A}(s,y)\nu({\mathrm{d}}s,{\mathrm{d}}y)\\
=&-\int_{(0,t-\tau_n]}\int_E b(X_{\tau_n},u,y)\,q(X_{\tau_n},u,{\mathrm{d}}y)\,\Lambda(X_{\tau_n},{\mathrm{d}}u), \quad\tau_n<t\leqslant\tau_{n+1},
\end{aligned}$$ for all $n\in{\mathbb{N}}$. Thus, we get (\[eq.local-m 2\]). The condition (\[eq.local-m.ineq\]) is equivalent to that $A$ is of locally finite variation. And that $A$ is of locally finite variation is also equivalent to that $a$ is of locally finite variation, which is the same as (\[eq.local-m 1\]).
Conversely, if A is an additive functional with $a$ and $b$, and (\[eq.local-m 1\]) and (\[eq.local-m 2\]) are satisfied, by taking $$\bar{A}(s,y)=\sum_{n=0}^\infty\bar{A}_n(s,y)\bbbone_{\{\tau_n<s\leqslant\tau_{n+1}\}},$$ where $\bar{A}_n(s,y)=b(X_{\tau_n},s-\tau_n,y),$ we have (\[eq.local-m.ineq\]) and (\[eq.local-m.eq\]). Then $A$ is a local martingale prior to $\tau$.
\(iii) Assume first that $A$ is a special semimartingale prior to $\tau$. Since all local martingales prior to $\tau$ are of locally finite variation, equivalently, all the semi-martingales prior to $\tau$ are of locally finite variation, which implies that $a$ has locally finite variation. And its canonical decomposition $A=M+B$ gives a local martingale $M$ prior to $\tau$ and a predictable process $B$, while both of them are additive functionals of $X$. Call $(a_M,b_M)$ and $(a_B,b_B)$ the terms associated with $M$ and $B$ in Theorem \[thm.A=a+b\]. Clearly, $a=a_M+a_B$ and $b=b_M+b_B$. Since $B$ is predictable, we have $\Delta B_{\tau_n}=0$, hence $\Delta M_{\tau_n}=\Delta A_{\tau_n}$ for $n\geqslant 1$. Therefore, $b_B=0$ and $b_M=b$. For local martingale $M$ prior to $\tau$, applying (i), we obtain (\[eq.local-m 1\]) and also that $a_M$ is given by the right-hand side of (\[eq.local-m 2\]). Therefore (\[eq.predictable B\]) follows from $a_B=a-a_M$ and $b_B=0$.
For the converse, assume that we have (\[eq.local-m 1\]) and $a\in\mathfrak{A}_\phi^{loc}$. Set $b_M=b$ and define $a_M$ by the right-hand side of (\[eq.local-m 2\]), then $(a_M,b_M)$ is associated with an additive local martingale $M$ prior to $\tau$ by (i). If $B$ is given by (\[eq.predictable B\]), which is a predictable process associated with $a_B=a-a_M$ and $b_B=0$, and since $a_B$ is an additive functional of the SDS $\phi$, $B$ is a predictable additive functional of $X$. The assumptions ensure that $a_B$ is of locally finite variation. Equivalently, $B$ is also of locally finite variation. Then $A=M+B$ is a special semimartingale prior to $\tau$.
Let the PDMP $X$ be quasi-Hunt. We prefer the characteristic triple ($\phi, \Lambda, Q$) to ($\phi, \Lambda, q$) in the quasi-Hunt cases in the following. The following theorem, which is Theorem 18 in [@jacod1996jumping], is a direct corollary of Theorem \[thm.semimartingale\].
\[cor.semimartingale.Hunt\] Let $A$ be an additive functional of a quasi-Hunt PDMP $X$ with the characteristic triple $(\phi, \Lambda, Q)$, and $a,\bar{b}$ the functions associated with $A$ as Theorem [\[thm.A=a+b\]]{.nodecor}.
[(i)]{.nodecor} $A$ is a local martingale prior to $\tau$ if and only if for any $x\in E$ and $t\in[0,c(x))$, we have $$\label{eq.local-m 1.Hunt}
\int_{(0,t]}\int_E \big|\bar{b}(\phi(s,x),y)\big| Q(\phi(s,x),{\mathrm{d}}y) \Lambda(x,{\mathrm{d}}s)<\infty,$$ and $$\label{eq.local-m 2.Hunt}
a(x,t)=-\int_{(0,t]}\int_E \bar{b}(\phi(s,x),y)\, Q(\phi(s,x),{\mathrm{d}}y)\, \Lambda(x,{\mathrm{d}}s).$$
[(ii)]{.nodecor} $A$ is a special semimartingale prior to $\tau$ if and only if the associated $a$ has locally finite variation and [(\[eq.local-m 1.Hunt\])]{.nodecor} is satisfied. In this case, $A$ has a canonical decomposition $A=M+B$, where $M$ is a local martingale prior to $\tau$ which is also an additive functional of $X$, and $B$ is a predictable additive functional of $X$ with $B_0=0$ and $$\label{eq.predictable B.Hunt}
B_t=B_{\tau_n}+a^*(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}],\,n\in{\mathbb{N}},$$ where $$ a^*(x,t)=a(x,t)+\int_{(0,t]}\int_E \bar{b}(\phi(s,x),y)\,Q(\phi(s,x),{\mathrm{d}}y)\,\Lambda(x,{\mathrm{d}}s).$$
\[cor.semimartingale.step\] Let $A$ be an additive functional of a quasi-step PDMP $X$ with the characteristic triple $(\phi,\Lambda,Q)$, and $a,\bar{b}$ the functions associated with $A$ as Theorem [\[thm.A=a+b\]]{.nodecor}. Assume that $J_\Lambda$ contains no confluent state.
[(i)]{.nodecor} $A$ is a local martingale prior to $\tau$ if and only if for any $x\in E$ and $t\in\mathcal{I}_x$, we have $$\label{eq.local-m 1.step}
\sum_{0<s\leqslant t}\Delta\Lambda(\phi(s,x))\int_E \big|\bar{b}(\phi(s,x),y)\big| Q(\phi(s,x),{\mathrm{d}}y)<\infty,$$ and $$\label{eq.local-m 2.step}
a(x,t)=-\sum_{0<s\leqslant t}\Delta\Lambda(\phi(s,x))
\int_E \bar{b}(\phi(s,x),y)\,Q(\phi(s,x),{\mathrm{d}}y).$$
[(ii)]{.nodecor} $A$ is a special semimartingale prior to $\tau$ if and only if the corresponding $a$ has locally finite variation and [(\[eq.local-m 1.step\])]{.nodecor} is satisfied. In this case, $A$ has a canonical decomposition $A=M+B$, where $M$ is a local martingale prior to $\tau$ which is also an additive functional of $X$, and $B$ is a predictable additive functional of $X$ with $B_0=0$ and $$\label{eq.predictable B.step}
B_t=B_{\tau_n}+a^*(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}],\,n\in{\mathbb{N}},$$ where $$ a^*(x,t)=a(x,t)+\sum_{0<s\leqslant t}\Delta\Lambda(\phi(s,x))\int_E \bar{b}(\phi(s,x),y)\,Q(\phi(s,x),{\mathrm{d}}y).$$
Following from Theorem \[thm.semimartingale\], and noticing that $$\Lambda(x,t)=\sum_{0<s\leqslant t}\Delta\Lambda(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E$$ for the quasi-step PDMP $X$, we get the conclusions.
A local martingale $M=\{M_t\}_{0\leqslant t<\tau}$ prior to $\tau$ is called to be *additive* if for any stopping time $T$, $$M_0=0,\quad M_{T+s}=M_T+M_s\circ\theta_T\quad \hbox{a.s. on }\{T<\infty\}.$$
Let $X$ be a general PDMP. Assume that $J_\Lambda$ contains no confluent state. $X$ is quasi-step if and only if for each $x\in E$ all the additive ${\mathbb{P}}_x$-local martingales prior to $\tau$ are step processes.
If $X$ is a quasi-step PDMP, we have $F=F^{pd}$, or equivalently, $\Lambda=\Lambda^{pd}$. Then, following from Theorem \[cor.semimartingale.step\] (i), for each $x\in E$, a ${\mathbb{P}}_x$-local martingale $M$ can be represented by $$\begin{aligned}
M_t= & -\sum_{0<s\leqslant t}\Delta\Lambda(X_s^-)\int_E\bar{b}(X_s^-,y)\,Q(X_s^-,{\mathrm{d}}y)+\sum_{n=1}^{N_t}\bar{b}(X_{\tau_n}^-,X_{\tau_n}),\quad t\in[0,\tau),
\end{aligned}$$ which is a step process.
Conversely, let $X$ be a general PDMP. For any ${\mathbb{P}}_x$-local martingale $M$ prior to $\tau$, by Corollary \[thm.semimartingale\] (i), we have $$\begin{aligned}
M_t= & M_{\tau_n}-\int_{(\tau_n,t]}\int_Eb(X_{\tau_n},s-\tau_n,y)\,q(X_{\tau_n},s-\tau_n,{\mathrm{d}}y)\,\Lambda(X_{\tau_n},{\mathrm{d}}s-\tau_n) \\
& +b(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}})\,\bbbone_{\{t=\tau_{n+1}\}},
\qquad\qquad t\in(\tau_n,\tau_{n+1}],
\end{aligned}$$ for any $n\in{\mathbb{N}}$. If $M$ is a step process, then the second term must be a summation along $(\tau_n,t]$, that is, $\Lambda=\Lambda^{pd}$. This completes the proof.
Measure-valued generator for PDMPs
==================================
Measure-valued generater
------------------------
A measurable function $f:\bar{E}\mapsto{\mathbb{R}}$ is called to be of *locally path-finite variation* if $f(\phi(\cdot,x))$ is right-continuous and of finite variation on any compact subinterval of $\mathcal{I}_x$ for any fixed $x\in E$. And denote by $\mathfrak{V}_{\phi}^{loc}$, the set of all locally path-finite variation function $f$’s.
For a given $f\in\mathfrak{V}_{\phi}^{loc}$, define an operator $D$ by $$Df(x,t):=f(\phi(t,x))-f(x),\quad t\in\mathcal{I}_x,\,x\in E.$$ Apparently, $f\in\mathfrak{V}_{\phi}^{loc}$ if and only if $Df\in \mathfrak{A}_{\phi}^{loc}$.
\[lem.f-f is A\] Let $X=\{X_t\}_{0\leqslant t<\tau}$ be a general PDMP, and $f:\bar{E}\mapsto{\mathbb{R}}$ a measurable function. Then the process $\{f(X_t)-f(X_0)\}_{0\leqslant t<\tau}$ is an additive functional of $X$ with $$\label{eq.A^f=Df+bf}
f(X_t)=f(X_{\tau_n})+Df(X_{\tau_n},t-\tau_n)+[f(X_{\tau_{n+1}})-f(X_{\tau_{n+1}}^-)]\, \bbbone_{\{t=\tau_{n+1}\}}$$ for $t\in(\tau_n,\tau_{n+1}]$, $n\in{\mathbb{N}}$. Furthermore, $\{f(X_t)-f(X_0)\}_{0\leqslant t<\tau}$ is a.s. of locally finite variation on $[0,\tau)$ if and only if $f\in\mathfrak{V}_{\phi}^{loc}$.
As $\tau_n<t<\tau_{n+1}$, $$f(X_t)-f(X_{\tau_n})=f(\phi(t-\tau_n,X_{\tau_n}))-f(X_{\tau_n})=Df(X_{\tau_n},t-\tau_n);$$ and, in view of $X_{\tau_{n+1}}^-=\phi(\tau_{n+1}-\tau_n,X_{\tau_n})$, we have $$\begin{aligned}
f(X_{\tau_{n+1}})-f(X_{\tau_n})&=f(X_{\tau_{n+1}})-f(X_{\tau_{n+1}}^-)+f(X_{\tau_{n+1}}^-)-f(X_{\tau_n})\\
&=f(X_{\tau_{n+1}})-f(X_{\tau_{n+1}}^-)+f(\phi(\tau_{n+1}-\tau_n,X_{\tau_n})-f(X_{\tau_n})\\
&=f(X_{\tau_{n+1}})-f(X_{\tau_{n+1}}^-)+Df(X_{\tau_n},\tau_{n+1}-\tau_n).
\end{aligned}$$ Hence, (\[eq.A\^f=Df+bf\]) holds.
Let $A_t^f:=f(X_t)-f(X_0)$ for $t\in[0,\tau)$. It is obvious that $$A_t^f=A_{\tau_n}^f+a_f(X_{\tau_n},t-\tau_n)+b_f(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}}) \bbbone_{\{t=\tau_{n+1}\}}$$ holds for $t\in(\tau_n,\tau_{n+1}]$, $n\in{\mathbb{N}}$. It follows from Theorem \[thm.A=a+b\] with $a_f=Df\in \mathfrak{A}_{\phi}$ and $b_f(x,t,y)=f(y)-f(\phi(t,x))$ that the process $\{f(X_t)-f(X_0)\}_{0\leqslant t<\tau}$ is an additive functional of $X$.
Moreover, the sufficient and necessary condition of $A^f$ being a.s. of locally finite variation on $[0,\tau)$ is that $Df\in\mathfrak{A}_\phi^{loc}$, which is equivalent to that $f\in\mathfrak{V}_\phi^{loc}$.
\[def.measure-valued generator\] Let $X=\{X_t\}_{0\leqslant t<\tau}$ be a general PDMP with characteristic triple $(\phi,F,q)$. ${\mathcal{D}}({\mathcal{A}})$ denotes the set of $f\in\mathfrak{V}_{\phi}^{loc}$ with the following property: there exists $a\in \mathfrak{A}_{\phi}^{loc}$ such that the process $$\label{eq.generator.def}
M_t^f:=f(X_t)-f(X_0)-\sum_{n=0}^{N_t} a(X_{\tau_n},t\land\tau_{n+1}-\tau_n), \quad t<\tau$$ is a local martingale prior to $\tau$. Then we denote that $a={\mathcal{A}}f$, and call $({\mathcal{A}},{\mathcal{D}}({\mathcal{A}}))$ the *measure-valued generator* of the general PDMP $X$.
For $a\in \mathfrak{A}_{\phi}^{loc}$, because $a(x,t)$ is of locally finite variation with respect to $t$, so $a(x,\cdot)$ can also be treated as a $\sigma$-finite signed measure on $\mathcal{I}_x$ for all $x\in E$. This is the reason why we call the operator ${\mathcal{A}}$ is measure-valued.
\[thm.measure-valued generator\] Let $X$ be a general PDMP. Then the domain ${\mathcal{D}}({\mathcal{A}})$ of the measure-valued generator ${\mathcal{A}}$ of $X$ consists of all $f\in\mathfrak{V}_{\phi}^{loc}$ such that for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.A-general}
\int_{(0,t]}\int_E\big|f(y)-f(\phi(s,x))\big|q(x,s,{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s)<\infty.$$ Moreover, if $f\in{\mathcal{D}}({\mathcal{A}})$, then for any $x\in E$ and $t\in\mathcal{I}_x$, $$\label{eq.measure-valued generator}
{\mathcal{A}}f(x,t)=Df(x,t)+\int_{(0,t]}\int_E\big[f(y)-f(\phi(t,x))\big]q(x,s,{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s).$$
Suppose that $f\in{\mathcal{D}}({\mathcal{A}})$. Then $$M^f_t=f(X_t)-f(X_0)-\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n),\quad t<\tau$$ is a local martingale prior to $\tau$. Moreover, by Lemma \[lem.f-f is A\], $$\begin{aligned}
M^f_t=&\sum_{n=0}^{N_t}Df(X_{\tau_n},t\land\tau_{n+1}-\tau_n) +\sum_{n=1}^{N_t}[f(X_{\tau_n})-f(X_{\tau_n}^-)] \\
&-\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n).
\end{aligned}$$ Meantime, both $Df$ and ${\mathcal{A}}f$ are additive functional of $\phi$ in $\mathfrak{V}_{\phi}^{loc}$. According to Theorem \[thm.A=a+b\], $M^f$ is an additive functional of $X$ with $$\begin{aligned}
& a_M(x,t)=Df(x,t)-{\mathcal{A}}f(x,t),\\ & b_M(x,t,y)=f(y)-f(\phi(t,x)). \end{aligned}$$ By Theorem \[thm.semimartingale\] (ii), since the additive functional $M^f$ is a local martingale prior to $\tau$, then we have (\[eq.domain.A-general\]) and $$a_M(x,t)=-\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]q(x,s,{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s),$$ which is equivalent to (\[eq.measure-valued generator\]).
Conversely, if (\[eq.domain.A-general\]) is satisfied with $f\in\mathfrak{V}_{\phi}^{loc}$, then ${\mathcal{A}}f\in\mathfrak{A}_{\phi}^{loc}$. Furthermore, by Lemma \[lem.f-f is A\], we have $$\begin{aligned}
M^f_t=&f(X_t)-f(X_0)-\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n)\\
=&-\sum_{n=0}^{N_t}\int_{(\tau_n,t\land\tau_{n+1}]}\int_E[f(y)-f(X_s^-)] q(X_{\tau_n},s-\tau_n,{\mathrm{d}}y)\Lambda(X_{\tau_n},{\mathrm{d}}s-\tau_n)\\
&+\sum_{n=1}^{N_t}[f(X_{\tau_n})-f(X_{\tau_n}^-)],\qquad\qquad t<\tau.
\end{aligned}$$ By Theorem \[thm.semimartingale\] (ii), it follows from (\[eq.domain.A-general\]) that $M^f$ is a local martingale prior to $\tau$. Hence $f\in{\mathcal{D}}({\mathcal{A}})$ and then $({\mathcal{A}},{\mathcal{D}}({\mathcal{A}}))$ is the measure-valued generator of $X$.
Based on Theorem \[thm.measure-valued generator\], we give the Itô formula for a general PDMP.
Let $X$ be a general PDMP. If $f\in{\mathcal{D}}({\mathcal{A}})$, then for $t\in[0,\tau)$, $$\begin{aligned}
\label{eq.Ito's formula}
f(X_t)-f(X_0)=&\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n)\nonumber\\
&+\int_{(0,t]\times E}\!\!\!\big[f(y)-f(X_s^-)\big](\mu-\nu)({\mathrm{d}}s,{\mathrm{d}}y),
\end{aligned}$$ where the last item on the right side is a local martingale prior to $\tau$. Especially, if $$\label{eq.martingale condition}
{\mathbb{E}}_x\Big[\sum_{n=1}^\infty\big|f(X_{\tau_n})-f(X_{\tau_n}^-)\big|\Big]<\infty,\quad x\in E,$$ then the last item on the right side of [(\[eq.Ito’s formula\])]{.nodecor} is a martingale.
For $f\in{\mathcal{D}}({\mathcal{A}})$, by (\[eq.generator.def\]), $$f(X_t)-f(X_0)=\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n)+M^f_t,\quad 0\leqslant t<\tau,$$ where $M^f$ is a local martingale prior to $\tau$. And, by Theorem \[thm.semimartingale\] (ii), $$M_t^f=\int_{(0,t]\times E}\!\!\!\big[f(y)-f(X_s^-)\big](\mu-\nu)({\mathrm{d}}s,{\mathrm{d}}y).$$ Especially, if (\[eq.martingale condition\]) holds, then, by [@davis1993markov Theorem 26.12], $M^f$ is a martingale. And the proof is complete.
Extended generator and L-extended generator
-------------------------------------------
In the followings we intend by virtue of the measure-valued generator to extend the domain of two familiar extensions of generator: the extended generator introduced to PDMPs by Davis [@davis1984piecewise] and the L-extended generator introduced by Kunita [@Kunita1969Absolute]. And we will present their connections with the measure-valued generator. For convenience, we firstly give the classification of measurable functions defined on $\bar{E}$.
A measurable function $f:\bar{E}\mapsto{\mathbb{R}}$ is called to be
(i) *path-continuous* if $f(\phi(\cdot,x))$ is continuous on $\mathcal{I}_x$ for all $x\in E$;
(ii) *absolutely path-continuous* if $f(\phi(\cdot,x))$ is absolutely continuous on $\mathcal{I}_x$ for all $x\in E$;
(iii) *piecewise absolutely path-continuous* if the continuous part of the function $f(\phi(\cdot,x))$ is absolutely continuous on $\mathcal{I}_x$ for all $x\in E$;
(iv) *path-step* if $f(\phi(\cdot,x))$ is a step function on $\mathcal{I}_x$ for all $x\in E$.
### Extended generator
\[def.extended generator\] Let $X=\{X_t\}_{0\leqslant t<\tau}$ be a general PDMP with characteristic triple $(\phi,\Lambda,Q)$. ${\mathcal{D}}({\mathcal{A}}')$ denotes the set of $f\in\mathfrak{V}_{\phi}^{loc}$ with the following property: there exist a measurable function $h:E\mapsto{\mathbb{R}}$ such that $$\int_0^t\big|h(X_s)\big|{\mathrm{d}}s<\infty \quad\hbox{for }t\in[0,\tau)\quad{\mathbb{P}}_x\hbox{-a.s. for each }x\in E,$$ and the process $$ M'_t:=f(X_t)-f(X_0)-\int_0^th(X_s){\mathrm{d}}s,\quad 0\leqslant t<\tau$$ is a local martingale prior to $\tau$. Then we write $h={\mathcal{A}}'f$. It is well known that $({\mathcal{A}}',{\mathcal{D}}({\mathcal{A}}'))$ is called the *extended generator* of the process $X$ (see [@davis1993markov] Definition 14.15).
\[thm.A’.general\] Let $X$ be a general PDMP with characteristic triple $(\phi,\Lambda,Q)$. Then the domain ${\mathcal{D}}({\mathcal{A}}')$ of the extended generator ${\mathcal{A}}'$ of $X$ consists of all $f\in\mathfrak{V}_{\phi}^{loc}$ such that for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.A'-general.Q}
\int_{(0,t]}\int_E\big|f(y)-f(\phi(s,x))\big|Q(\phi(s,x),{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s)<\infty,$$ with constraint conditions $$\begin{aligned}
&D^{sc}f(x,t)+\!\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]Q(\phi(s,x),{\mathrm{d}}y)\Lambda^{sc}(x,{\mathrm{d}}s)=0,\label{eq.A^sc=0.general}\\
&D^{pd}f(x,t)+\!\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]Q(\phi(s,x),{\mathrm{d}}y)\Lambda^{pd}(x,{\mathrm{d}}s)=0. \label{eq.A^pd=0.general}
\end{aligned}$$ For $f\in{\mathcal{D}}({\mathcal{A}}')$, ${\mathcal{A}}'f$ is given by $$\label{eq.A'f.general}
{\mathcal{A}}'f(x)=\mathcal{X}f(x)+\lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y),\quad x\in E,$$ where $$\mathcal{X}f(x)=\left\{\begin{array}{ll}
\displaystyle\frac{\partial^+f(\phi(t,x))}{\partial t}\bigg|_{t=0}, &
\hbox{if }\displaystyle\frac{\partial^+f(\phi(t,x))}{\partial t}\bigg|_{t=0}
\hbox{ exists;} \\
0, & \hbox{otherwise,}
\end{array}\right.$$ and $$\lambda(x)=\left\{\begin{array}{ll}
\displaystyle \frac{\partial^+\Lambda(x,t)}{\partial t}\bigg|_{t=0}, &
\hbox{if }\displaystyle\frac{\partial^+\Lambda(x,t)}{\partial t}\bigg|_{t=0}\hbox{ exists;} \\
0, & \hbox{otherwise.}
\end{array}\right.$$
Suppose that $f\in{\mathcal{D}}({\mathcal{A}}')$. Then $$\begin{aligned}
M'_t&=f(X_t)-f(X_0)-\int_0^t{\mathcal{A}}'f(X_s){\mathrm{d}}s\\
&=f(X_t)-f(X_0)-\sum_{n=0}^{N_t}\int_{\tau_n}^{t\land\tau_{n+1}}{\mathcal{A}}' f(\phi(s-\tau_n,X_{\tau_n})){\mathrm{d}}s,\quad t<\tau
\end{aligned}$$ is a local martingale prior to $\tau$. Moreover, by Lemma \[lem.f-f is A\], $$\begin{aligned}
M'_t=&\sum_{n=0}^{N_t}Df(X_{\tau_n},t\land\tau_{n+1}-\tau_n)
+\sum_{n=1}^{N_t}[f(X_{\tau_n})-f(X_{\tau_n}^-)]\\
&-\sum_{n=0}^{N_t}\int_{\tau_n}^{t\land\tau_{n+1}}{\mathcal{A}}' f(\phi(s-\tau_n,X_{\tau_n})){\mathrm{d}}s, \quad t<\tau.
\end{aligned}$$ Meantime, both $Df(x,t)$ and $\int_0^t{\mathcal{A}}' f(\phi(s,x)){\mathrm{d}}s$ belong to $\mathfrak{A}_{\phi}^{loc}$. According to Theorem \[thm.A=a+b\], $M'$ is an additive functional of $X$ with $$\begin{aligned}
&a_{M'}(x,t)=Df(x,t)-\int_0^t{\mathcal{A}}' f(\phi(s,x)){\mathrm{d}}s, \\ &b_{M'}(x,t,y)=f(y)-f(\phi(t,x)). \end{aligned}$$ Since the additive functional $M'$ is a local martingale prior to $\tau$, then we have (\[eq.domain.A’-general.Q\]) by Theorem \[thm.semimartingale\] (ii) and $$a_{M'}(x,t)=-\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]Q(\phi(s,x),{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s),$$ that is, $$\label{eq.Af=intA'f}
{\mathcal{A}}f(x,t)=\int_0^t{\mathcal{A}}' f(\phi(s,x)){\mathrm{d}}s\quad\hbox{ for all }x\in E,\, t\in\mathcal{I}_x.$$ Consider the Lebesgue decomposition of ${\mathcal{A}}f={\mathcal{A}}^{ac}f+{\mathcal{A}}^{sc}f+{\mathcal{A}}^{pd}f$. The the equation (\[eq.Af=intA’f\]) is equivalent to (\[eq.A’f.general\]) with (\[eq.A\^sc=0.general\]) and (\[eq.A\^pd=0.general\]).
Conversely, if $f\in\mathfrak{V}_{\phi}^{loc}$, and (\[eq.domain.A’-general.Q\]) with the constraint conditions (\[eq.A\^sc=0.general\]) and (\[eq.A\^pd=0.general\]) are satisfied, then $f\in {\mathcal{D}}({\mathcal{A}})$. Hence, $$M^f_t=f(X_t)-f(X_0)-\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n),\qquad t<\tau$$ is a local martingale prior to $\tau$. But, in this case, we have (\[eq.Af=intA’f\]) which implies that $$M^f_t=f(X_t)-f(X_0)-\int_0^t{\mathcal{A}}'f(X_s){\mathrm{d}}s,\quad t<\tau$$ is a local martingale prior to $\tau$. Hence $f\in{\mathcal{D}}({\mathcal{A}}')$ and then $({\mathcal{A}}',{\mathcal{D}}({\mathcal{A}}'))$ is the extended generator of $X$.
Let $X$ be a quasi-Hunt PDMP with characteristic triple $(\phi,\Lambda,Q)$. Then the domain ${\mathcal{D}}({\mathcal{A}}')$ of the extended generator ${\mathcal{A}}'$ of $X$ consists of each path-continuous function $f\in\mathfrak{V}_{\phi}^{loc}$ such that for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.A'-Hunt}
\int_{(0,t]}\int_E\big|f(y)-f(\phi(s,x))\big|Q(\phi(s,x),{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s)<\infty,$$ with constraint condition $$\label{eq.A^sc=0.Hunt}
D^{sc}f(x,t)+\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]Q(\phi(s,x),{\mathrm{d}}y)\Lambda^{sc}(x,{\mathrm{d}}s)=0.$$ For $f\in{\mathcal{D}}({\mathcal{A}}')$, ${\mathcal{A}}'f$ is given by $$\label{eq.A'.Hunt}
{\mathcal{A}}'f(x)=\mathcal{X}f(x)+\lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y),\quad x\in E.$$
In the case of quasi-Hunt, the discrete part $\Lambda^{pd}=0$ of the characteristic $\Lambda$. The constraint condition (\[eq.A\^pd=0.general\]) becomes $D^{pd}f(x,t)=0$, or equivalently that $f$ is path-continuous. This completes the proof.
Let $$\Gamma:=\{\phi(c(x),x):F(x,c(x)-)>0,\,c(x)<\infty,\,x\in E\}.$$ It is obviously that $\Gamma\subset\partial^+E$ and $\Delta\Lambda(x)=1$ for $x\in\Gamma$. Here $\partial^+E$ is the flowing-out boundary of $E$. $\Gamma$ is used to be called the ‘active boundary’ in the PDMPs theory. Davis [@davis1993markov] deals essentially with the quasi-Itô PDMPs apart from possibly certain force jumps at the active boundary. Strictly speaking, a general PDMP $X$ with the characteristic triple $(\phi,\Lambda,Q)$ is called a PDMP in the sense of Davis [@davis1993markov] if $$\label{eq.Lambda.Davis}
\Lambda(x,t)=\int_0^t\lambda(\phi(s,x)){\mathrm{d}}s+\bbbone_{\{\phi(t,x)\in\Gamma\}},\quad x\in E,\, t\in \mathcal{I}_x.$$ Now we represent [@davis1993markov Theorem 26.14] for the extended generator of PDMPs as a corollary of Theorem \[thm.A’.general\].
\[cor.A’.Davis\] Let $X$ be a general PDMP with characteristic triple $(\phi,\Lambda,Q)$, in which $\Lambda$ satisfies [(\[eq.Lambda.Davis\])]{.nodecor}. Then the domain ${\mathcal{D}}({\mathcal{A}}')$ of the extended generator ${\mathcal{A}}'$ of $X$ consists of all $f\in\mathfrak{V}_{\phi}^{loc}$ such that for $x\in E$, $f(\phi(\cdot,x))$ is absolutely continuous on $[0,c(x))$, and for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.A'-Davis}
\int_0^t\lambda(\phi(s,x))\int_E\big|f(y)-f(\phi(s,x))\big|Q(\phi(s,x),{\mathrm{d}}y){\mathrm{d}}s<\infty,$$ with boundary conditions $$\label{eq.boundary condition.Davis}
\Delta f(x)+\int_E\big[f(y)-f(x)\big]Q(x,{\mathrm{d}}y)=0,\quad x\in \Gamma.$$ For $f\in{\mathcal{D}}({\mathcal{A}}')$, ${\mathcal{A}}'f$ is given by $$\label{eq.A'.Davis}
{\mathcal{A}}'f(x)=\mathcal{X}f(x)+\lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y),\quad x\in E.$$
In view of (\[eq.Lambda.Davis\]), the constraint conditions (\[eq.A\^sc=0.general\]) and (\[eq.A\^pd=0.general\]) become that $f(\phi(\cdot,x))$ is absolutely continuous on $[0,c(x))$ for $x\in E$ with the boundary condition (\[eq.boundary condition.Davis\]). In this case, (\[eq.domain.A’-general.Q\]) is equivalent to (\[eq.domain.A’-Davis\]).
Compare Corollary \[cor.A’.Davis\] with Theorem 26.14 in Davis [@davis1993markov], there are two differences for ${\mathcal{D}}({\mathcal{A}}')$. Firstly, in Theorem 26.14 of [@davis1993markov], $f\in{\mathcal{D}}({\mathcal{A}}')$ satisfies $${\mathbb{E}}_x\left[\int_{{\mathbb{R}}_+\times E}\big[f(y)-f(X_{s-})\big]\bbbone_{\{t<T_n\}}\nu({\mathrm{d}}s,{\mathrm{d}}y)\right]<\infty$$ for some sequence of stopping times $\{T_n\}$ with $T_n\uparrow\infty$ a.s. instead of (\[eq.domain.A’-Davis\]). While, (\[eq.domain.A’-Davis\]) is an analytic condition, which is easier to be verified. Secondly, in Theorem 26.14 of [@davis1993markov], the boundary condition is $$f(x)=\int_Ef(y)Q(x,{\mathrm{d}}y),\quad x\in\Gamma.$$ But, in Corollary \[cor.A’.Davis\], we allow $\Delta f(x)\neq 0$ with (\[eq.boundary condition.Davis\]). Thus, we get a larger domain of extended generator for the PDMPs in the sense of Davis [@davis1993markov].
### L-extended generator
Consider a basic additive functional $L$ of the general PDMP $X$ defined as $$L_0=0,\quad L_t=L_{\tau_n}+\Lambda(X_{\tau_n},t-\tau_n) \quad \hbox{for }t\in(\tau_n,\tau_{n+1}],\,n\in{\mathbb{N}}.$$
\[def.L-extended generator\] Let $X=\{X_t\}_{0\leqslant t<\tau}$ be a general PDMP with characteristic triple $(\phi,\Lambda,Q)$. ${\mathcal{D}}({\mathcal{A}}'')$ denotes the set of $f\in\mathfrak{V}_{\phi}^{loc}$ with the following property: there exist a measurable function $h:E\mapsto{\mathbb{R}}$ such that $$\int_{(0,t]}|h(X^-_s)|{\mathrm{d}}L_s<\infty\quad\hbox{for }t\in[0,\tau)\quad{\mathbb{P}}_x\hbox{-a.s. for each }x\in E$$ and the process $$ M''_t:=f(X_t)-f(X_0)-\int_{(0,t]}h(X^-_s){\mathrm{d}}L_s,\quad 0\leqslant t<\tau,$$ is a local martingale prior to $\tau$. Then we write $h={\mathcal{A}}''f$. $({\mathcal{A}}'',{\mathcal{D}}({\mathcal{A}}''))$ is called the *L-extended generator* of the process $X$.
Let $X$ be a general PDMP with characteristic triple $(\phi,\Lambda,Q)$. Then the domain ${\mathcal{D}}({\mathcal{A}}'')$ of the L-extended generator ${\mathcal{A}}''$ of $X$ consists of all $f\in\mathfrak{V}_{\phi}^{loc}$ such that for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.LA"-general}
\int_{(0,t]}\int_E\big|f(y)-f(\phi(s,x))\big|Q(\phi(s,x),{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s)<\infty,$$ and there exists a measurable function $Kf(\cdot)$ such that $$\label{eq.K-condition.general}
Df(x,t)=\int_{(0,t]}Kf(\phi(s,x))\Lambda(x,{\mathrm{d}}s),\quad t\in\mathcal{I}_x,\,x\in E.$$ For $f\in{\mathcal{D}}({\mathcal{A}}'')$, ${\mathcal{A}}''f$ is given by $$\label{eq.LA".general}
{\mathcal{A}}''f(x)=Kf(x)+\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y),\quad x\in \bar{E}.$$
Suppose that $f\in{\mathcal{D}}({\mathcal{A}}'')$. Hence $$\begin{aligned}
M''_t&=f(X_t)-f(X_0)-\int_{(0,t]}{\mathcal{A}}''f(X^-_s){\mathrm{d}}L_s\\
&=f(X_t)-f(X_0)-\sum_{n=0}^{N_t}\int_{(\tau_n,t\land\tau_{n+1}]}\!\!\!{\mathcal{A}}'' f(\phi(s-\tau_n,X_{\tau_n}))\Lambda(X_{\tau_n},{\mathrm{d}}s),\quad t<\tau
\end{aligned}$$ is a local martingale prior to $\tau$. Moreover, by Lemma \[lem.f-f is A\], $$\begin{aligned}
M''_t=&\sum_{n=0}^{N_t}Df(X_{\tau_n},t\land\tau_{n+1}-\tau_n) +\sum_{n=1}^{N_t}[f(X_{\tau_n})-f(X_{\tau_n}^-)]\\
&-\sum_{n=0}^{N_t}\int_{(\tau_n,t\land\tau_{n+1}]}{\mathcal{A}}''f (\phi(s-\tau_n,X_{\tau_n}))\Lambda(X_{\tau_n},{\mathrm{d}}s),\quad t<\tau.
\end{aligned}$$ Meantime, both $Df(x,t)$ and $\int_{(0,t]}{\mathcal{A}}''f(\phi(s,x))\Lambda(x,{\mathrm{d}}s)$ belong to $\mathfrak{A}_{\phi}^{loc}$. According to Theorem \[thm.A=a+b\], $M''$ is an additive functional of $X$ with $$\begin{aligned}
& a_{M''}(x,t)=Df(x,t)-\int_{(0,t]}{\mathcal{A}}''f(\phi(s,x))\Lambda(X_{\tau_n},{\mathrm{d}}s),\\ & b_{M''}(x,t,y)=f(y)-f(\phi(t,x)). \end{aligned}$$ By Theorem \[thm.semimartingale\] (ii), since the additive functional $M''$ is a local martingale prior to $\tau$, then we have (\[eq.domain.LA"-general\]) and $$a_{M''}(x,t)=-\int_{(0,t]}\int_E\big[f(y)-f(\phi(s,x))\big]Q(\phi(s,x),{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s),$$ that is, $$\label{eq.Af=intA"f}
{\mathcal{A}}f(x,t)=\int_{(0,t]}{\mathcal{A}}'' f(\phi(s,x))\Lambda(x,{\mathrm{d}}s)\quad\hbox{ for all }x\in E,\, t\in\mathcal{I}_x.$$ Equivalently, $$Df(x,t)=\int_{(0,t]}\!\left[{\mathcal{A}}''f(\phi(s,x))-\int_E[f(y)-f(\phi(s,x)]Q(\phi(s,x),{\mathrm{d}}y)\right]\!\Lambda(x,{\mathrm{d}}s)$$ for all $x\in E$, $t\in\mathcal{I}_x$. It follows $Df(x,t)=\int_{(0,t]}Kf(\phi(s,x))\Lambda(x,{\mathrm{d}}s)$ with $$Kf(x)={\mathcal{A}}''f(x)-\int_E[f(y)-f(x)]Q(\phi(s,x),{\mathrm{d}}y),\quad x\in \bar{E},$$ which is (\[eq.LA".general\]).
Conversely, if $f\in\mathfrak{V}_{\phi}^{loc}$ satisfies (\[eq.domain.LA"-general\]), then $f\in {\mathcal{D}}({\mathcal{A}})$. Hence, $$M^f_t=f(X_t)-f(X_0)-\sum_{n=0}^{N_t}{\mathcal{A}}f(X_{\tau_n},t\land\tau_{n+1}-\tau_n),\quad t<\tau$$ is a local martingale prior to $\tau$. But, in this case, we have (\[eq.Af=intA"f\]) which implies $$M^f_t=f(X_t)-f(X_0)-\int_0^t{\mathcal{A}}''f(X_s){\mathrm{d}}L_s,\quad t<\tau$$ is a local martingale prior to $\tau$. Hence $f\in{\mathcal{D}}({\mathcal{A}}'')$ and then $({\mathcal{A}}'',{\mathcal{D}}({\mathcal{A}}''))$ is the L-extended generator of $X$.
Let $X$ be a quasi-step PDMP with characteristic triple $(\phi,\Lambda,Q)$. Then the domain ${\mathcal{D}}({\mathcal{A}}'')$ of the L-extended generator ${\mathcal{A}}''$ of $X$ consists of each path-step function $f\in\mathfrak{V}_{\phi}^{loc}$ with $J_{Df}\subseteq J_\Lambda$ such that for any $x\in E$, $t\in\mathcal{I}_x$, $$\label{eq.domain.LA"-step}
\sum_{0<s\leqslant t}\Delta\Lambda(x,{\mathrm{d}}s)\int_E\big|f(y)-f(\phi(s,x))\big|Q(\phi(s,x),{\mathrm{d}}y)<\infty,$$ and there exists a measurable function $Kf(\cdot)$ such that $$\label{eq.K-condition.step}
\Delta f(x)=Kf(x)\Delta\Lambda(x),\quad x\in J_\Lambda.$$ For $f\in{\mathcal{D}}({\mathcal{A}}'')$, ${\mathcal{A}}''f$ is given by $$\label{eq.LA".step}
{\mathcal{A}}''f(x)=Kf(x)+\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y),\quad x\in J_\Lambda.$$
In the case of quasi-step, the parts $\Lambda^{ac}=0$ and $\Lambda^{sc}=0$ of the characteristic $\Lambda$. The condition (\[eq.K-condition.general\]) becomes that $$Df(x,t)=\sum_{0<s\leqslant t}Kf(\phi(s,x))\Delta\Lambda(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E,$$ or equivalently, that $f$ is a path-step function with $J_{Df}\subseteq J_\Lambda$ and (\[eq.K-condition.step\]). This completes the proof.
Expectations of additive functionals and measure differential equations for a general PDMP
==========================================================================================
This section deals with calculating the expected cumulative discounted value of an additive functional of a general PDMP, $$\label{eq.V.general}
V(x):={\mathbb{E}}_x\left[\int_{(0,\tau]}e^{-\delta s}{\mathrm{d}}A_s\right],\quad x\in E,$$ where the discount rate $\delta>0$. This functional appears as a ‘value function’ (or, ‘reward function’, ‘cost function’, etc.) in stochastic service systems, stochastic control theory, and is sufficiently general to cover a wide variety of apparently different functionals as special cases.
\[thm.V.expectation.predictable\] Let $X$ be a general PDMP. Assume that $A=\{A_t\}_{0\leqslant t<\tau}$ is a predictable additive functional of $X$ with the associated $a\in\mathfrak{A}_{\phi}^{loc}$ . If $V\in{\mathcal{D}}({\mathcal{A}})$ defined as [(\[eq.V.general\])]{.nodecor}, then it satisfies the following measure integro-differential equation $$\label{eq.V.equation}
{\mathcal{A}}V(x,{\mathrm{d}}t)+a(x,{\mathrm{d}}t)=\delta V(\phi(t,x)){\mathrm{d}}t, \quad t\in\mathcal{I}_x,\,x\in E.$$
Note that if $A$ is predictable, for any fixed $x\in E$, by the strong Markov property, it follows from Corollary \[cor.A.pred=a\] that $$\begin{aligned}
V(x) =& {\mathbb{E}}_x\left[\int_{(0,t\land\tau_1]}e^{-\delta s}a(x,{\mathrm{d}}s) + e^{-\delta(t\land\tau_1)}V(X_{t\land\tau_1})\right] \\
=& F(x,t)\int_{(0,t]}e^{-\delta s}a(x,{\mathrm{d}}s) +\int_{(0,t]}\int_{(0,s]}e^{-\delta u}a(x,{\mathrm{d}}u)F(x,{\mathrm{d}}s) \\
& +e^{-\delta t}V(\phi(t,x))F(x,t) +\int_{(0,t]}e^{-\delta s}\int_EV(y)q(x,s,{\mathrm{d}}y)F(x,{\mathrm{d}}s).
\end{aligned}$$ By the formula for integration by parts, $$\begin{aligned}
& F(x,t)\int_{(0,t]}e^{-\delta s}a(x,{\mathrm{d}}s) \\
=& -\int_{(0,t]}\int_{(0,s]}e^{-\delta u}a(x,{\mathrm{d}}u)F(x,{\mathrm{d}}s)+\int_{(0,t]}e^{-\delta s}F(x,s-)a(x,{\mathrm{d}}s) .
\end{aligned}$$ Thus, $$V(x) = \int_{(0,t]}\!\!e^{-\delta s}F(x,s-)H(x,{\mathrm{d}}s) +e^{-\delta t} V(\phi(t,x))F(x,t),$$ which is equivalent to $$e^{-\delta t}V(\phi(t,x))=\frac{V(x)}{F(x,t)}-\frac{1}{F(x,t)}\int_{(0,t]}\!\!e^{-\delta s}F(x,s-)H(x,{\mathrm{d}}s),$$ where $H(x,{\mathrm{d}}s):=a(x,{\mathrm{d}}s)+\Lambda(x,{\mathrm{d}}s)\int_EV(y)q(x,s,{\mathrm{d}}y)$. Substitute the equation above into the following integration $$\begin{aligned}
& \int_{(0,t]}e^{-\delta s}V(\phi(s,x))\Lambda(x,{\mathrm{d}}s) \\
=& \int_{(0,t]}\left[\frac{V(x)}{F(x,s)}-\frac{1}{F(x,s)} \int_{(0,s]}\!\!e^{-\delta u} F(x,u-)H(x,{\mathrm{d}}u)\right] \Lambda(x,{\mathrm{d}}s) \\
=& V(x)\int_{(0,t]}\frac{\Lambda(x,{\mathrm{d}}s)}{F(x,s)}-\int_{(0,t]}e^{-\delta s}F(x,s-) \int_{[s,t]}\frac{\Lambda(x,{\mathrm{d}}u)}{F(x,u)}H(x,{\mathrm{d}}s) \\
=& \frac{V(x)}{F(x,t)}-V(x)-\frac{1}{F(x,t)}\int_{(0,t]}e^{-\delta s}F(x,s-)H(x,{\mathrm{d}}s)+\int_{(0,t]}e^{-\delta s}H(x,{\mathrm{d}}s) \\
=& e^{-\delta t}V(\phi(t,x))-V(x)+\int_{(0,t]}e^{-\delta s}H(x,{\mathrm{d}}s).
\end{aligned}$$ On the other hand, by the formula for integration by parts, $$e^{-\delta t}V(\phi(t,x))-V(x)=\int_{(0,t]}e^{-\delta s}DV(x,{\mathrm{d}}s)-\int_0^te^{-\delta s}\delta V(\phi(s,x)){\mathrm{d}}s.$$ Thus, we have $$\begin{aligned}
&\int_{(0,t]}e^{-\delta s}V(\phi(s,x))\Lambda(x,{\mathrm{d}}s)\\
=&\int_{(0,t]}e^{-\delta s}DV(x,{\mathrm{d}}s)-\int_0^te^{-\delta s}\delta V(\phi(s,x)){\mathrm{d}}s +\int_{(0,t]}e^{-\delta s}H(x,{\mathrm{d}}s)
\end{aligned}$$ for all $t\in\mathcal{I}_x$. Noticing that the both sides of the equation above are additive functionals of the SDS $\phi$, we have $$V(\phi(t,x))\Lambda(x,{\mathrm{d}}t)=DV(x,{\mathrm{d}}t)-\delta V(\phi(t,x)){\mathrm{d}}t+H(x,{\mathrm{d}}t)$$ for all $t\in\mathcal{I}_x$, which is equivalent to (\[eq.V.equation\]) by the arbitrary of $x$.
\[cor.V.expectation.optional\] Assume that $A$ is an optional additive functional of $X$, the associated function $a\in\mathfrak{A}_\phi^{loc}$ and $b$ satisfies $$\label{eq.b.martingale}
{\mathbb{E}}_x\left[\sum_{n=1}^{\infty}e^{-\delta\tau_n} \big|b(X_{\tau_{n-1}},\tau_n-\tau_{n-1},X_{\tau_n})\big|\right]<\infty,\quad x\in E.$$ If $V\in{\mathcal{D}}({\mathcal{A}})$, then $V$ satisfies [(\[eq.V.equation\])]{.nodecor} by replacing $a$ in Theorem [\[thm.V.expectation.predictable\]]{.nodecor} with $$\label{eq.V.equation.optional}
a^*(x,{\mathrm{d}}t)=a(x,{\mathrm{d}}t)+\Lambda(x,{\mathrm{d}}t) \int_Eb(x,t,y)q(x,t,{\mathrm{d}}y),\quad x\in E.$$
If $A$ is optional and associated with $a$ and $b$, then $$V(x)\! = \!{\mathbb{E}}_x\!\left[\sum_{n=0}^{\infty}\!\left(\int_{(\tau_n,\tau_{n+1}]}\!\!\!\!\!\!\!\!\!e^{-\delta s}a(X_{\tau_n},{\mathrm{d}}s-\tau_n) +e^{-\delta\tau_{n+1}}b(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}})\right)\right].$$ Notice that (\[eq.b.martingale\]) is satisfied, $$\begin{aligned}
&{\mathbb{E}}_x\left[\sum_{n=0}^{\infty}e^{-\delta\tau_{n+1}} b(X_{\tau_n},\tau_{n+1}-\tau_n,X_{\tau_{n+1}})\right]\\
=&{\mathbb{E}}_x\left[\sum_{n=0}^{\infty}\int_{(\tau_n,\tau_{n+1}]\times E} \!\! e^{-\delta s} b(X_{\tau_n},s-\tau_n,y)\mu({\mathrm{d}}s,{\mathrm{d}}y)\right]\\
=&{\mathbb{E}}_x\left[\sum_{n=0}^{\infty}\int_{(\tau_n,\tau_{n+1}]\times E} \!\! e^{-\delta s} b(X_{\tau_n},s-\tau_n,y)\nu({\mathrm{d}}s,{\mathrm{d}}y)\right]\\
=&{\mathbb{E}}_x\left[\sum_{n=0}^{\infty}\int_{(\tau_n,\tau_{n+1}]}\!\!\!\!\! e^{-\delta s} \int_Eb(X_{\tau_n},s-\tau_n,y)q(X_{\tau_n},s-\tau_n,{\mathrm{d}}y)
\Lambda(X_{\tau_n},{\mathrm{d}}s-\tau_n)\right],
\end{aligned}$$ and $\int_{(0,t]}\int_Eb(x,s,y)q(x,s,{\mathrm{d}}y)\Lambda(x,{\mathrm{d}}s)\in\mathfrak{A}_\phi^{loc}$. Hence, $a^*\in\mathfrak{A}_\phi^{loc}.$ Define a process $A^*$ by $A^*_0=0$ and $$A^*_t=A^*_{\tau_n}+a^*(X_{\tau_n},t-\tau_n),\quad t\in(\tau_n,\tau_{n+1}]\,n\in{\mathbb{N}}.$$ Thus $$V(x)={\mathbb{E}}_x\left[\int_{(0,\tau]}e^{-\delta s}{\mathrm{d}}A^*_s\right],\quad x\in E.$$ Then, applying Theorem \[thm.V.expectation.predictable\], we prove the corollary.
Theorem \[thm.V.expectation.predictable\] tells us that there exists one solution $V$ of the equation (\[eq.V.equation\]). And we would like to know that the equation has a unique solution. The following is a general result in this direction.
\[thm.UniqueSolution\] Suppose a measurable function $f\in{\mathcal{D}}({\mathcal{A}})$ satisfies the measure integro-differential equation $$\label{eq.f.eq.general}
{\mathcal{A}}f(x,{\mathrm{d}}t)+a(x,{\mathrm{d}}t)=\delta f(\phi(t,x)){\mathrm{d}}t, \quad t\in\mathcal{I}_x,\,x\in E,$$ where $a\in\mathfrak{A}^{loc}_{\phi}$ is associated with the predictable additive functional $A$ of $X$. Suppose further that $f$ satisfies that $$\label{eq.m-condition.general}
{\mathbb{E}}_x\left[\sum_{n=1}^{\infty}e^{-\delta\tau_n} \big|f(X_{\tau_n})-f(X_{\tau_n}^-)\big|\right]<\infty, \quad x\in E,$$ and that ${\mathbb{E}}_x\big[e^{-\delta t}f(X_{t\land\tau})\big]\to 0$ as $t\to\infty$ for all $x\in E$. Then $$\label{eq.unique solution}
f(x)={\mathbb{E}}_x\left[\int_{(0,\tau]}e^{-\delta s}{\mathrm{d}}A_s\right],\quad x\in E.$$
Since $f\in{\mathcal{D}}({\mathcal{A}})$, applying the formula for integration by parts and the Itô formula (\[eq.Ito’s formula\]), we get $$\begin{aligned}
& e^{-\delta t}f(X_t)-f(X_0)\\
=& \int_{(0,t]}e^{-\delta s}{\mathrm{d}}f(X_s)-\int_0^te^{-\delta s}\delta f(X_s){\mathrm{d}}s \\
=& \sum_{n=0}^{N_t}\int_{(\tau_n,t\land\tau_{n+1}]}\!\!\!e^{-\delta s}{\mathcal{A}}f(X_{\tau_n},{\mathrm{d}}s-\tau_n)-\int_0^te^{-\delta s}\delta f(X_s){\mathrm{d}}s\\
&+\int_{(0,t]\times E}\!\!e^{-\delta s}\big[f(y)-f(X_s^-)\big](\mu-\nu)({\mathrm{d}}s,{\mathrm{d}}y)
\end{aligned}$$ for any $t\in[0,\tau)$. Since $f$ satisfies (\[eq.m-condition.general\]), the last term above is a martingale with mean zero. In view of ${\mathcal{A}}f(x,{\mathrm{d}}t)=-a(x,{\mathrm{d}}t)+\delta f(\phi(t,x)){\mathrm{d}}t$, we get by taking expectations $$\begin{aligned}
f(x)=&{\mathbb{E}}_x\big[e^{-\delta t}f(X_{t\land\tau})\big]
+{\mathbb{E}}_x\left[\sum_{n=0}^{N_t}\int_{(\tau_n,t\land\tau_{n+1}]}\!\!\!e^{-\delta s}a(X_{\tau_n},{\mathrm{d}}s)\right]\\
=&{\mathbb{E}}_x\big[e^{-\delta t}f(X_{t\land\tau})\big]+{\mathbb{E}}_x\left[\int_{(0,t]}e^{-\delta s}{\mathrm{d}}A_s\right].
\end{aligned}$$ Thus (\[eq.unique solution\]) follows by the assumption that $\lim_{t\to\infty}{\mathbb{E}}_x\big[e^{-\delta t}f(X_{t\land\tau})\big]=0$.
\[thm.equation.Lebesgue\] Let $a\in\mathfrak{A}_\phi^{loc}$. Assume that $J_\Lambda\cup J_a$ contains no confluent state. Function $f\in{\mathcal{D}}({\mathcal{A}})$ satisfies the measure integro-differential equation $$\label{eq.measure.diff.equation}
{\mathcal{A}}f(x,{\mathrm{d}}t)+a(x,{\mathrm{d}}t)=\delta f(\phi(t,x)){\mathrm{d}}t,\quad t\in\mathcal{I}_x,\,x\in E$$ if and only if it satisfies the equations
\_0\^tf((s,x))s+\_0\^ta((s,x))s=\_0\^tf((s,x))s,\[eq.f.eq.ac\]\
\_[0<st]{}f((s,x))+\_[0<st]{}a((s,x))=0,\[eq.f.eq.pd\]\
\^[sc]{}f(x,t)+a\^[sc]{}(x,t)=0\[eq.f.eq.sc\]
for $t\in\mathcal{I}_x$, $x\in E$.
Notice that the measure integro-differential equation (\[eq.measure.diff.equation\]) is equivalent to $$\label{eq.f.eq.add}
{\mathcal{A}}f(x,t)+a(x,t)=\int_0^t\delta f(\phi(s,x)){\mathrm{d}}s,\quad t\in\mathcal{I}_x,\,x\in E.$$ The both sides of the equation (\[eq.f.eq.add\]) are additive functionals of the SDS $\phi$, and the right side is absolutely continuous on $\mathcal{I}_x$ for all $x\in E$. Since $J_\Lambda\cup J_a$ contains no confluent state, following from Theorem \[thm.a.Lebesgue\], we have the Lebesgue decompositions of ${\mathcal{A}}f$ and $a$. And, the absolutely continuous parts, the singularly continuous parts and the purely discontinuous parts of the both sides are equal, respectively.
Conversely, suppose that the equations (\[eq.f.eq.ac\])-(\[eq.f.eq.sc\]) hold. Summing up the both sides of the equations, we get (\[eq.f.eq.add\]) which is equivalent to the measure integro-differential equation (\[eq.measure.diff.equation\]).
Consider the absolutely continuous part of the equations above, i.e., equation (\[eq.f.eq.ac\]), which is equivalent to $$ \mathcal{X}{\mathcal{A}}f(\phi(t,x))+\mathcal{X}a(\phi(t,x))=\delta f(\phi(t,x))\quad\hbox{a.e. on }\mathcal{I}_x \hbox{ for all }x\in E.$$ It implies that, for a solution of the measure integro-differential equation (\[eq.measure.diff.equation\]), it is not required to satisfy the equation $$\label{eq.f.eq.ac.x}
\mathcal{X}{\mathcal{A}}f(x)+\mathcal{X}a(x)=\delta f(x)$$ for all $x\in E$. Here we introduce the concept of the so-called path-solution and almost everywhere path-solution.
We call $f$ a *path-solution* of the equation (\[eq.f.eq.ac.x\]) if it satisfies $$\mathcal{X}{\mathcal{A}}f(\phi(t,x))+\mathcal{X}a(\phi(t,x))=\delta f(\phi(t,x))\quad\hbox{for all } t\in\mathcal{I}_x,x\in E.$$ And $f$ is called an *almost everywhere path-solution* of (\[eq.f.eq.ac.x\]) if it satisfies $$\mathcal{X}{\mathcal{A}}f(\phi(t,x))+\mathcal{X}a(\phi(t,x))=\delta f(\phi(t,x))\quad\hbox{a.e. on }\mathcal{I}_x\hbox{ for all }x\in E.$$ In this sense, we say the equation (\[eq.f.eq.ac.x\]) holds *path-almost everywhere* (*path-a.e.* in short).
Considering the purely discontinuous part (\[eq.f.eq.pd\]), we get $$\Delta f(x)=\Delta\Lambda(x)\int_E[f(x)-f(y)]Q(x,{\mathrm{d}}y)-\Delta a(x),\quad x\in \bar{E}.$$ We observe that $J_{Df}\subseteq J_\Lambda\cup J_a$. Thus, (\[eq.f.eq.pd\]) is equivalent to the equation $$\Delta{\mathcal{A}}f(x)+\Delta a(x)=0,\quad x\in J_\Lambda\cup J_a,$$ with the condition $J_{Df}\subseteq J_\Lambda\cup J_a$.
Following from the singularly continuous part (\[eq.f.eq.sc\]), we have $$D^{sc}f(x,t)=\int_{(0,t]}\int_E[f(\phi(s,x))-f(y)]q(x,s,{\mathrm{d}}y)\Lambda^{sc}(x,{\mathrm{d}}s)-a^{sc}(x,t)$$ for $t\in\mathcal{I}_x$, $x\in E$. Hence, if $\Lambda^{sc}=a^{sc}=0$, then $D^{sc}f=0$, i.e., $f$ is piecewise absolutely path-continuous.
Now we will give three simplified versions of the two theorems above in case of quasi-Itô, quasi-step and ‘non-singular’, respectively.
\[cor.V.expectation.Ito\] Let $X$ be a quasi-Itô PDMP, and $l$ a locally absolutely path-integrable function. Define $$\label{eq.V.expectation.Ito}
V(x):={\mathbb{E}}_x\left[\int_0^\tau e^{-\delta s}l(X_s){\mathrm{d}}s\right],\quad x\in E.$$ If $V\in{\mathcal{D}}({\mathcal{A}})$, then it is absolutely path-continuous and satisfies $$\label{eq.V.eq.Ito}
\mathcal{X}V(x)+\lambda(x)\int_E[V(y)-V(x)]Q(x,{\mathrm{d}}y)+l(x)=\delta V(x)\quad \hbox{path-a.e.}$$
Let $A$ be a predictable additive functional of $X$ with the associated $a$ which is defined by $$a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s\quad t\in\mathcal{I}_x,\,x\in E.$$ By the locally absolutely path-integrability of $l$, we know that $a$ is of locally finite variation. Applying Theorem \[thm.V.expectation.predictable\], we get that $V$ satisfies the measure integro-differential equation (\[eq.V.equation\]). Thus, by Theorem \[thm.equation.Lebesgue\], we get that $V$ is absolutely path-continuous and satisfies (\[eq.V.eq.Ito\]). The proof is completed.
\[cor.UniqueSolution.Ito\] Let $X$ be a quasi-Itô PDMP. Suppose that $f\in{\mathcal{D}}({\mathcal{A}})$ is absolutely path-continuous and satisfies $$\label{eq.f.eq.Ito}
\mathcal{X}f(x)+\lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y)+l(x)=\delta f(x)\quad \hbox{path-a.e.},$$ where $l$ is a locally absolutely path-integrable function. Suppose further that $f$ satisfies that $${\mathbb{E}}_x\left[\sum_{n=1}^\infty e^{-\delta\tau_n} \big|f(X_{\tau_n})-f(X_{\tau_n}^-)\big|\right]<\infty,\quad x\in E$$ and that ${\mathbb{E}}_x[e^{-\delta t}f(X_{t\land\tau})]\to 0$ as $t\to\infty$ for all $x\in E$. Then $$\label{eq.unique.sol.Ito}
f(x)={\mathbb{E}}_x\left[\int_0^\tau e^{-\delta s}l(X_s){\mathrm{d}}s\right],\quad x\in E.$$
For a quasi-Itô PDMP $X$, $\Lambda$ is absolutely path-continuous. If $f$ is also absolutely path-continuous, then ${\mathcal{A}}f(x,\cdot)$ is absolutely continuous for all $x\in E$. Thus, following from (\[eq.f.eq.Ito\]), we get that $f$ satisfies the measure integro-differential equation (\[eq.f.eq.general\]) with $a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s$. Hence, applying Theorem \[thm.UniqueSolution\], we have $$f(x)={\mathbb{E}}_x\left[\int_{(0,t]} e^{-\delta s}{\mathrm{d}}A_s\right]={\mathbb{E}}_x\left[\int_0^\tau e^{-\delta s}l(X_s){\mathrm{d}}s\right],$$ where $A$ is the predictable additive functional of $X$ corresponding to $a$.
\[cor.V.expectation.step\] Let $X$ be a quasi-step PDMP. Define $$V(x):={\mathbb{E}}_x\left[\int_0^te^{-\delta s}l(X_s){\mathrm{d}}s+\sum_{0<s<\tau}e^{-\delta s}g(X_s^-)\right],\quad x\in E,$$ where $l$ is a locally absolutely path-integrable function, and $g$ is a locally absolutely path-summable function. Assume that $J_\Lambda$ contains no confluent state. If $V\in{\mathcal{D}}({\mathcal{A}})$, then it is piecewise absolutely path-continuous with $J_{DV}\subseteq J_\Lambda\cup \mathrm{Supp}_g$ and satisfies $$\label{eq.V.eq.step}
\left\{
\begin{array}{l}
\displaystyle\mathcal{X}V(x)+l(x)=\delta V(x)\quad\hbox{path-a.e.},\\
\displaystyle\Delta V(x)+\Delta\Lambda(x)\int_E[V(y)-V(x)]Q(x,{\mathrm{d}}y)+g(x)=0,\; x\in J_\Lambda\cup \mathrm{Supp}_g,
\end{array}
\right.$$ where $\mathrm{Supp}_g$ is the support of the function $g$.
Let $A$ be a predictable additive functional of $X$ with the associated $$a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s+\sum_{0<s\leqslant t}g(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E.$$ By the locally absolutely path-integrability of $l$ and the locally absolutely path-summability of $g$, we get $a\in\mathfrak{A}_\phi^{loc}$. Applying Theorem \[thm.V.expectation.predictable\], we have that $V$ satisfies the measure integro-differential equation (\[eq.V.equation\]). Noticing that $\Lambda^{ac}=\Lambda^{sc}=a^{sc}=0$, by Theorem \[thm.equation.Lebesgue\], we get the conclusion.
\[cor.UniqueSolution.step\] Let $X$ be a quasi-step PDMP, $l$ a locally absolutely path-integrable function, and $g$ a locally absolutely path-summable function. Assume that $J_\Lambda$ contains no confluent state. Suppose that $f\in{\mathcal{D}}({\mathcal{A}})$ is piecewise absolutely path-continuous with $J_{Df}\subseteq J_\Lambda\cup\mathrm{Supp}_g$ and satisfies $$\label{eq.f.eq.step}
\left\{
\begin{array}{l}
\displaystyle\mathcal{X}f(x)+l(x)=\delta f(x)\quad \hbox{path-a.e.},\\
\displaystyle\Delta f(x)+\Delta\Lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y)+g(x)=0,\quad x\in J_\Lambda\cup \mathrm{Supp}_g.
\end{array}
\right.$$ Suppose further that $f$ satisfies that $${\mathbb{E}}_x\left[\sum_{n=1}^\infty e^{-\delta\tau_n} \big|f(X_{\tau_n})-f(X_{\tau_n}^-)\big|\right]<\infty,\quad x\in E$$ and that ${\mathbb{E}}_x[e^{-\delta t}f(X_{t\land\tau})]\to 0$ as $t\to\infty$ for all $x\in E$. Then $$ f(x)={\mathbb{E}}_x\left[\int_0^te^{-\delta s}l(X_s){\mathrm{d}}s+\sum_{0<s<\tau}e^{-\delta s}g(X_s^-)\right],\quad x\in E.$$
For a quasi-step PDMP $X$, if $f$ is piecewise absolutely path-continuous with $J_{Df}\subseteq J_\Lambda\cup\mathrm{Supp}_g$, then ${\mathcal{A}}f(x,\cdot)$ is piecewise absolutely continuous for all $x\in E$ with $J_{{\mathcal{A}}f}\subseteq J_\Lambda\cup\mathrm{Supp}_g$. Thus, it is following from the equations (\[eq.f.eq.step\]) that $f$ satisfies the measure integro-differential equation (\[eq.f.eq.general\]) with $$a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s+\sum_{0<s\leqslant t}g(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E.$$ By Theorem \[thm.UniqueSolution\], the result is following from $$f(x)={\mathbb{E}}_x\left[\int_{(0,t]} e^{-\delta s}{\mathrm{d}}A_s\right],\quad x\in E.$$ This completes the proof.
\[cor.V.expectation.nonsingular\] Let $X$ be a general PDMP with $\Lambda^{sc}=0$. Assume that $J_\Lambda$ contains no confluent state. Define $$ V(x):={\mathbb{E}}_x\left[\int_0^\tau e^{-\delta s}l(X_s){\mathrm{d}}s +\sum_{0<s<\tau} e^{-\delta s} g(X_s^-)\right],\quad x\in E,$$ where $l$ is a locally absolutely path-integrable function, and $g$ is a locally absolutely path-summable function. If $V\in{\mathcal{D}}(A)$, then it is piecewise absolutely path-continuous with $J_{DV}\subseteq J_\Lambda\cup\mathrm{Supp}_g$ and satisfies $$\label{eq.V.eq.nonsinglar}
\left\{
\begin{array}{l}
\displaystyle\mathcal{X}V(x)+\lambda(x)\int_E[V(y)-V(x)]Q(x,{\mathrm{d}}y)+l(x)=\delta V(x)\quad \hbox{path-a.e.},\\
\displaystyle\Delta V(x)+\Delta\Lambda(x)\int_E[V(y)-V(x)]Q(x,{\mathrm{d}}y)+g(x)=0,\; x\in J_\Lambda\cup\mathrm{Supp}_g.
\end{array}
\right.$$
Let $A$ be a predictable additive functional of $X$ with the associated $$a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s+\sum_{0<s\leqslant t}g(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E.$$ By the locally absolutely path-integrability and path-summability of $l$ and $g$, we get $a\in\mathfrak{A}_\phi^{loc}$. Applying Theorem \[thm.V.expectation.predictable\], we have that $V$ satisfies the measure integro-differential equation (\[eq.V.equation\]). Note that $\Lambda^{sc}=a^{sc}=0$ in this case. Then, following from Theorem \[thm.equation.Lebesgue\], we get the conclusion.
\[cor.UniqueSolution.nonsingular\] Let $X$ be a general PDMP with $\Lambda^{sc}=0$. Assume that $J_\Lambda$ contains no confluent state. Suppose that $f\in{\mathcal{D}}({\mathcal{A}})$ is piecewise absolutely path-continuous with $J_{Df}\subseteq J_\Lambda\cup\mathrm{Supp}_g$ and satisfies $$\label{eq.f.eq.nonsingular}
\left\{\begin{array}{l}
\displaystyle\mathcal{X}f(x)+\lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y)+l(x)=\delta f(x)\quad \hbox{path-a.e.},\\
\displaystyle\Delta f(x)+\Delta\Lambda(x)\int_E[f(y)-f(x)]Q(x,{\mathrm{d}}y)+g(x)=0,\; x\in J_\Lambda\cup\mathrm{Supp}_g,
\end{array}\right.$$ where $l$ is a locally absolutely path-integrable function, and $g$ is a locally absolutely path-summable function. Suppose further that $f$ satisfies that $${\mathbb{E}}_x\left[\sum_{n=1}^\infty e^{-\delta\tau_n} \big|f(X_{\tau_n})-f(X_{\tau_n}^-)\big|\right]<\infty,\quad x\in E$$ and that ${\mathbb{E}}_x[e^{-\delta t}f(X_{t\land\tau})]\to 0$ as $t\to\infty$ for all $x\in E$. Then $$\label{eq.unique.sol.nonsingular}
f(x)={\mathbb{E}}_x\left[\int_0^\tau e^{-\delta s}l(X_s){\mathrm{d}}s +\sum_{0<s<\tau}e^{-\delta s} g(X_s^-)\right],\quad x\in E.$$
For a general PDMP $X$ with $\Lambda^{sc}=0$, if $f$ is piecewise absolutely path-continuous with $J_{Df}\subseteq J_\Lambda\cup\mathrm{Supp}_g$, then ${\mathcal{A}}f(x,\cdot)$ is piecewise absolutely continuous for all $x\in E$ with $J_{{\mathcal{A}}f}\subseteq J_\Lambda\cup\mathrm{Supp}_g$. Thus, it is following from the equations (\[eq.f.eq.nonsingular\]) that $f$ satisfies the measure integro-differential equation (\[eq.f.eq.general\]) with $$a(x,t)=\int_0^tl(\phi(s,x)){\mathrm{d}}s+\sum_{0<s\leqslant t}g(\phi(s,x)),\quad t\in\mathcal{I}_x,\,x\in E.$$ By Theorem \[thm.UniqueSolution\], the result is following from $$f(x)={\mathbb{E}}_x\left[\int_{(0,t]} e^{-\delta s}{\mathrm{d}}A_s\right],\quad x\in E.$$ This completes the proof.
|
---
abstract: |
We have computed radiative equilibrium models for the gas in the circumstellar envelope surrounding the hot, classical Be star $\gamma\,$Cassiopeia. This calculation is performed using a code that incorporates a number of improvements over previous treatments of the disk’s thermal structure by @mil98 and @jon04; most importantly, heating and cooling rates are computed with atomic models for H, He, CNO, Mg, Si, Ca, & Fe and their relevant ions. Thus, for the first time, the thermal structure of a Be disk is computed for a gas with a solar chemical composition as opposed to assuming a pure hydrogen envelope. We compare the predicted average disk temperature, the total energy loss in H$\alpha$, and the near-IR excess with observations and find that all can be accounted for by a disk that is in vertical hydrostatic equilibrium with a density in the equatorial plane of $\rho(R)\approx 3$ to $5\cdot\,10^{-11} (R/R_*)^{-2.5}\, \rm
g\,cm^{-3}$. We also discuss the changes in the disk’s thermal structure that result from the additional heating and cooling processes available to a gas with a solar chemical composition over those available to a pure hydrogen plasma.
author:
- |
T. A. A. Sigut & C. E. Jones\
Department of Physics and Astronomy, The University of Western Ontario, London, Ontario, N6A 3K7, Canada
title: 'The Thermal Structure of the Circumstellar Disk Surrounding the Classical Be Star $\gamma\,$Cassiopeia'
---
Introduction
============
Classical Be stars are non-supergiant B stars that possess circumstellar material in the form of an equatorial disk. While the circumstellar disk is almost certainly a decretion disk of material from the star’s atmosphere, the detailed mechanism that creates and maintains such a disk remains unclear [@por03; @owo04]. Rapid rotation of the central B star seems to play an important role, but there is still considerable debate as to the extent [@tow04; @fre05]. Historically, the observational evidence for such circumstellar material has been either spectroscopic or polarimetric in nature, and the accepted observational definition of a Be star has been the appearance (at the current or previous epoch) of emission in the hydrogen Balmer lines. It has been recognized since the days of @str31 that recombination in a flattened disk could reproduce the range of spectroscopically observed H$\alpha$ profiles. In addition, the net (continuum) linear polarization observed in Be stars is well explained by electron scattering from non-spherically distributed circumstellar gas [@coy69; @wat92].
Beginning with the resolution of $\phi\,$Persei (B2 Vpe) at radio wavelengths with the Very Large Array by @dou92, interferometry has increasingly been used to spatially resolve circumstellar material. Be star disks have been resolved at radio, near-IR, and optical wavelengths, with these observations conclusively revealing the disk [@qui93; @qui97; @tyc05; @tyc06]. The observations are consistent with the suggestion of @poe78 that Be star disks are geometrically quite thin with opening angles of only a few degrees.
Currently, optical interferometric observations require theoretical models of the emitting region in order to interpret the observed visibilities. Often observers fit simple models with free parameters to the data to describe the disk emissivity. However, this simple procedure can be considerably improved by using a detailed model for the thermal structure of the Be star disk. Such models naturally predict the emissivity and opacity of the gas required to produce theoretical spectroscopic images [@mil98; @jon04; @car05].
$\gamma\,$Cassiopeia (HD 5394; B0 IVe) is a interesting classical Be star which has a dense, cool, equatorial disk [@mil98]. This disk has been resolved with optical interferometry [@ste95; @tyc05]. $\gamma\,$Cas is likely the primary in a binary system [@har00; @mir02] and has unique X-ray characteristics [@smi04]. Although $\gamma\,$Cas is often quoted as “the prototypical" Be star, it has become clear that it possesses some unique characteristics. Nevertheless, it is a well-studied star making it an appropriate choice to test new codes and compare results with previously published work. In this current paper, we extend the radiative-equilibrium models of @mil98 and @jon04 for the early Be star $\gamma\,$Cas to a disk models with a solar chemical composition.
Calculations
============
Overview
--------
The calculations in this paper were performed with a new code, [bedisk]{}, which is loosely based upon the calculational approach of @mil98 and @jon04. Models for the circumstellar material were constructed assuming a density distribution falling as an $R^{-n}$ power-law in the equatorial plane, following the models of @wat86, @cot87, and @wat87. By comparing to observations of the infrared excesses of Be stars, these authors found power-law density exponents in the range $n\approx\,2.0$ to $3.5$. Perpendicular to the equatorial plane, it was assumed that the gas is in vertical, isothermal, hydrostatic equilibrium. Given the disk density distribution and the photoionizing radiation field from the central star, the equations of statistical equilibrium were solved for the ionization state and level populations of H, He, CNO, Mg, Si, Ca, and Fe subject to the constraints of charge and particle conservation. Radiative transfer was handled via the escape probability approximation, and it was assumed that the dominant escape route for photons was perpendicular to the exponentially stratified disk. From this solution, the rates of energy gain and energy loss at each computational grid point were obtained, and the local temperatures were iteratively adjusted to enforce radiative equilibrium.
The main features and assumptions of this code are now discussed in detail.
The [bedisk]{} Code {#sec:detail}
-------------------
The circumstellar disk was assumed to be axisymmetric about the star’s rotation axis and symmetric about the mid-plane of the disk. In all that follows, $R$ is used for the radial distance from the star’s rotation axis and $Z$, for the perpendicular height above the equatorial plane. Thus the cylindrical co-ordinates of any disk location are $(R,Z)$. If $R_*$ denotes the stellar radius, then the calculation domain is $R_* \le R_i \le R_{\max}$ with $i=1\ldots\,n_r$, and $0 \le Z_j \le
Z_{\max}(R_i)$ with $j=1\ldots\,n_z$. Typically, $n_r$ is set to 60 with $R_{\max}/R_*=50$, and $n_z$ is set to 40.
The code accepts a user-defined set of atomic models which list the energy levels and the bound and free radiative and collisional transitions for each atom and ion to be included in the calculation. The set of atoms and ions and the total number of atomic levels and radiative transitions used for the current calculations are listed in Table \[tab:atomic\_models\]. For this initial work, the number of energy levels included for each atom and ion is similar to the list of energy levels given by @mor68. Sources for the required atomic data are given in Appendix A. The abundances assumed for the various elements can be found in Table \[tab:abun\] and are taken from the accepted solar abundances of @and89 and @and93.
The number of atomic levels included for each atom/ion in Table \[tab:atomic\_models\] is fairly modest, although large enough to include most of the collisionally-excited lines seen in the optical and UV spectra of Be stars. While non-LTE solutions can be sensitive to the number of atomic levels included (see, for example, [@sig96] for a case-study of C[ii]{} in B stars), the computational time also increases sharply with the number of levels. Table \[tab:atomic\_models\] represents a compromise between realism and computational efficiency. Several techniques exist to group atomic levels into “super-levels" [see, for example, @hhl94] which will allow future work to utilize more complete atomic models.
[rlrr]{} 1 & H[i]{} & 15 & 77\
1 & H[ii]{} & 1 & 0\
2 & He[i]{} & 13 & 16\
2 & He[ii]{} & 1 & 0\
6 & C[i]{} & 15 & 23\
6 & C[ii]{} & 10 & 20\
6 & C[iii]{} & 21 & 45\
6 & C[iv]{} & 8 & 15\
6 & C[v]{} & 1 & 0\
7 & N[i]{} & 23 & 51\
7 & N[ii]{} & 15 & 24\
7 & N[iii]{} & 10 & 20\
7 & N[iv]{} & 1 & 0\
8 & O[i]{} & 15 & 19\
8 & O[ii]{} & 29 & 80\
8 & O[iii]{} & 15 & 24\
8 & O[iv]{} & 10 & 19\
8 & O[v]{} & 1 & 0\
12& Mg[i]{} & 17 & 28\
12& Mg[ii]{} & 12 & 25\
12& Mg[iii]{} & 1 & 0\
14& Si[i]{} & 37 &109\
14& Si[ii]{} & 8 & 13\
14& Si[iii]{} & 18 & 27\
14& Si[iv]{} & 11 & 23\
14& Si[v]{} & 1 & 0\
20& Ca[i]{} & 24 & 42\
20& Ca[ii]{} & 11 & 20\
20& Ca[iii]{} & 1 & 0\
26& Fe[i]{} & 40 & 90\
26& Fe[ii]{} & 39 &191\
26& Fe[iii]{} & 1 & 0\
& [total]{} & 425 & 1001\
[lr]{} H & 12.00\
He & 10.90\
C & 8.55\
N & 7.97\
O & 8.87\
Mg & 7.58\
Si & 7.55\
Ca & 6.36\
Fe & 7.51\
The density structure of the disk is chosen in an ad-hoc manner. All calculations assume that the density drops as an $R^{-n}$ power-law in the equatorial plane, and at each $R$, the gas is in vertical, isothermal hydrostatic equilibrium. To obtain the density above the equatorial plane at each radial distance, the user supplies a fixed set of density-drops from the equatorial plane, $$d_j\equiv\ln\{\rho(Z_j)/\rho(Z_j=0)\}$$ for $j=1\ldots n_Z$. Here $d_1=0$ by definition, and $d_{n_z}\equiv-4$. Then the $Z_j$ at which this density drop would occur, given the current value of $R_i$, is computed assuming vertical hydrostatic equilibrium with an isothermal temperature $T_o$, $$\frac{Z_j}{R_i}=\left\{\left(\frac{\alpha /R_i}{d_j+
\alpha /R_i}\right)^2-1\right\}^{\frac{1}{2}} \;.$$ Here $\alpha$ is given by $$\label{eq:scaleH}
\alpha=\frac{\mu\,m_{\rm H}}{k\,T_o}\,GM \,,$$ where $\mu$ is the mean-molecular weight, $\approx 0.5$ for an ionized, pure hydrogen disk, and M is the mass of the central B-star. Thus the density at $(R_i,Z_j)$ is given by $$\rho(R_i,Z_j)=\rho_0\left(\frac{R_i}{R_*}\right)^{-n}\,d_j \;.$$ In this expression, $\rho_o$, $n$, and $T_o$ (through equation \[eq:scaleH\]) are the parameters which define the density structure of the disk.
At each computational grid point, the photoionizing radiation field is required to evaluate the photoionization rates of all atoms/ions for the statistical equilibrium equations and to compute the photoionization heating rates for the radiative equilibrium solution. It is usual to divide this radiation field into a direct and diffuse contribution, $$\label{eq:jnu}
J_{\nu}=J^{\rm Dir}_{\nu} + J^{\rm Dif}_{\nu} \;.$$ The direct contribution represents the radiation from the central star while the diffuse contribution arises from the disk. Note that despite this division, the only energy input into the circumstellar disk is assumed to be from the star itself so that ultimately, the energy in the diffuse field has its origin in direct radiation from the star.
There is some evidence that $\gamma$ Cas has a binary companion with an orbital period of $\sim200$ days [@har00; @mir02]. However, the exact orbit and the nature of the companion, including its spectral type and luminosity, are unknown. Rather than introduce further uncertain parameters into the calculation (and invalidate the axisymmetric geometry), we shall assume that the energy input into the disk of $\gamma$ Cas from any potential companion is negligible.
The direct component from the central star to the photoionizing radiation field at grid location $(R_i,Z_j)$ is given by $$\label{eq:jdir}
J^{\rm Dir}_{\nu}(R_i,Z_j)=\int_{\Omega_*} I_{\mu\nu}(R_i,Z_j,\hat{n})\,d\Omega\,$$ where $d\Omega$ is an infinitesimal patch of solid angle centred around the direction $\hat{n}$. The integral is over the visible stellar surface. Typically the surface is divided into a few hundred patches and the transfer equation is solved along a ray from the centre of each patch to the grid location[^1]. The radiation field at the stellar surface was taken from an LTE stellar atmosphere of @kur93 which specified the mean intensity, $J_{\nu}(\tau_{\nu}=0)$, at the top of the photosphere over a grid of 1221 frequencies. This was turned into the required intensity at each surface element by using the limb-darkening law $$I_{\mu\nu}=I_o\,\left\{1-a_{\nu}(1-\mu)-b_{\nu}(1-\mu)^2\right\}$$ where the coefficients $a_{\nu}$ and $b_{\nu}$ were linearly interpolated from Table V of @wad85. Here $\mu$ is the usual cosine of the surface viewing angle. Computing the mean intensity from this expression and setting it to the LTE model atmosphere prediction, we find that $$I_o=\frac{J_{\nu}(\tau_{\nu}=0)}{1-a_{\nu}-(3/4)\,b_{\nu}} \,.$$ While this procedure is approximate, and the exact $I_{\mu\nu}$ for each $\mu$ predicted by the LTE model could have been used, this approximation seems commensurate with others made in the construction of these models. We also note the use of a more physically realistic non-LTE, line-blanketed atmosphere for $J_{\nu}(\tau_{\nu}=0)$ would be a useful future improvement.
In addition to limb darkening, gravity darkening induced by rapid stellar rotation can change the intensity distribution across the stellar disk and hence modify the direct component to the photoionizing radiation field. [@tyc05] interferometrically resolved $\gamma$ Cas’s disk and estimated its inclination angle to the sky. They conclude the $\gamma$ Cas rotates at $0.7\pm0.1$ of its critical velocity. As $\gamma\,$Cas does not seem to rotate particularly close to its critical velocity, we have not included gravity darkening (and the associated geometrical distortion of it’s surface) into the calculation of the direct photoionizing radiation field. This point is further discussed in Section \[sec:MM\] where the adopted stellar parameters for $\gamma\,$Cas are discussed.
The simplest treatment for the diffuse field is to employ the on-the-spot (OTS) approximation in which the recombination rate to level $n$ of hydrogen is written as $$\label{eq:OTS}
R_{\kappa,n}^{\rm H}(\tau_n)\,\equiv\,R_{\kappa,n}^{\rm H}\,e^{-\tau_n} \,.$$ Here $\tau_n$ is the optical depth at the continuum limit for photoionization from level $n$ along a vertical ray to the nearest edge of the disk (this is consistent with our assumption that the dominant photon escape route is perpendicular to the disk —see later discussion). Thus the principle assumption of the OTS approximation is that at high continuum optical depths, $\tau_n\gg1$, recombination to level $n$ produces a photon that is locally absorbed within the same volume element, essentially undoing the recombination. Including the continuum optical depth dependence in equation (\[eq:OTS\]) ensures that the OTS approximation is used only when $\tau_n\gg1$; the full recombination coefficient is employed when the gas becomes optically thin in the continuum. We have applied to OTS approximation only to recombination to level $n=1$ in hydrogen. The optical depths in the remaining continua are typically not large enough for a significant effect. We discuss a more complex, but still approximate, treatment for the diffuse field in section \[sec:diffuse\].
Given the photoionizing radiation field and current estimates of the electron temperature and electron density at each grid location, we solve the statistical equilibrium equations to obtain the level populations for all atoms and ions. For atom $k$, and all of its associated ions, these are $$\sum_{j\ne i}^{N^k_L} n^k_i R_{ij} - \sum_{j\ne i}^{N^k_L} n^k_j R_{ji} = 0 \,,$$ for $i=1,\ldots N^k_L$ where $N^k_L$ is the number of atomic levels included for the $k^{\rm th}$ atom and its ionization stages. Note that these equations must be supplemented by a particle conservation equation of the form $$\sum_{i}^{N^k_L} n^k_i = 10^{A_k-12} n_{\rm H}$$ for each atomic species. The elemental abundance, $A_k$, can be found from Table \[tab:abun\].
In the case of bound-bound transitions $i\rightarrow j$, $i<j$, the rates have the simple form $$R_{ij} = n_{\rm e}\,q_{ij}(T_e)$$ and $$R_{ji} = A_{ji}\,P_{\rm esc}(\tau_z) + n_{\rm e}\,q_{ji}(T_e) \,.$$ Here the factors $q_{ij}(T_e)$ and $q_{ji}(T_e)$ represent collisional excitation and de-excitation respectively and are proportional to the Maxwellian-averaged collision strength for $i\,\rightarrow\,j$ transition. $A_{ji}$ is the usual Einstein transition probability for spontaneous emission. This form of the statistical equilibrium equations handles radiative transfer in the line via the escape-probability approximation. The escape probability for each grid location was obtained by computing the line-centre optical depth to the nearest vertical edge of the disk (denoted $\tau_z$) and then using this optical depth to estimate the static, single-flight escape probability assuming complete redistribution in the spectral line.
We have assumed that the dominant loss route for photons is perpendicular to the disk because of the exponential density stratification implied by vertical hydrostatic equilibrium. As it is reasonable to assume that the main motion of the disk gas is Keplerian rotation about the central star, the Doppler shifts experienced by photons escaping roughly perpendicular to the disk will be small, and it is appropriate to approximate the escape probability, $P_{\rm esc}(\tau_z)$, by a [*static*]{}, single-flight escape probability, as opposed to employing the Sobolev approximation. This is consistent with the definition of $\tau_z$ as the optical along a ray perpendicular to the disk to the nearest edge; $\tau_z$ is not defined in terms of a local velocity gradient as in the Sobolev approximation. The form of the static, single-flight escape probability appropriate for complete-redistribution over a Doppler profile is $$P_{\rm esc}(\tau) = \frac{1} { 4\tau\left(\ln(\tau/\sqrt{\pi})\right)^{1/2} }\,,$$ where $\tau\gg1$ [@mil78; @can85]. As this result is an asymptotic expansion [essentially in the limit that the scale of variation of the line source function is large compared to the width of the scattering kernel – see @can85], an ad-hoc correction is needed to handle $\tau\ll1$. Here we have adopted the suggestion of @tie05 where the escape probability is taken to be $$P_{\rm esc}(\tau) = \frac{1-e^{-2.34\tau}}{4.68\tau}\,,$$ for $\tau<7$. This function is continuous with the asymptotic result at $\tau=7$ and tends to the limit of $1/2$ as $\tau\rightarrow 0$, a reasonable result if the collisional destruction probability of the line photon upon scattering is not too small.
In the case $i\rightarrow j\equiv\kappa$ is a bound-free transition, the photoionization and recombination (spontaneous plus stimulated) rates are given by $$R_{i\,\kappa}=4\pi\int_{\nu_o}^{\infty}\,\sigma_{i\kappa}(\nu)\,J_{\nu}\, \frac{d\nu}{h\nu}$$ and $$\label{eq:recom}
R_{\kappa\,i}=4\pi\left(\frac{n_i}{n_{\kappa}}\right)^{*}
\int_{\nu_o}^{\infty} \sigma_{i\kappa}(\nu) \left(\frac{2h\nu^3}{c^2}+J_{\nu}\right)\,\frac{d\nu}{h\nu} \,.$$ Here $(n_i/n_{\kappa})^*$ is the LTE population ratio found from the Saha-Boltzmann equation and it is proportional to the electron density, $n_e$ [@mil78]. Dielectronic recombination and autoionization are included by retaining the full resonance structure of the photoionization cross section $\sigma_{i\kappa}(\nu)$.
Given the solution for the atomic level populations, a new estimate for the electron density can be made by enforcing charge conservation, and the rates of heating and cooling for the various atomic processes can then be computed. Heating includes photoionization and collisional de-excitation while cooling includes radiative recombination and collisional excitation. Detailed expressions for all of these processes can be found in [@ost89]. However, in contrast to @ost89, transitions (and the implied cooling) due to radiative recombination were computed explicitly via equation (\[eq:recom\]) for each atomic level; total recombination co-efficients (summed over $n$) were not used.
Heating due to viscous dissipation in a Keplerian disk was also included [@lee91] but was always found to be negligible.
Net cooling due to free-free emission (in the fields of H[i]{} and He[ii]{}) was included via the expression of @ryb79, modified as suggested by @net90 to account for the reduction in the free-free cooling rate as the gas becomes optically thick to free-free radiation, $$L_{\rm ff}= 1.4\cdot10^{-27}\,\sqrt{T_e}\,n_e\,
\sum_{i=\rm H, He}\,n_i\,Z_i^2\,\overline{g_{ff}}\,e^{-h\nu_{\rm max}/kT_e}\,.$$ Here $Z_i=1$ and the frequency cut-off, $\nu_{\rm max}$, suggested by @net90, is the smallest frequency for which the optical depth to the nearest edge of the disk exceeds one. The Gaunt factor, $\overline{g_{ff}}$, which varies slowly with temperature, was set to a constant value of 1.2 which @ryb79 indicate will approximate the exact result to within 20%.
To find the equilibrium kinetic temperature, $T_e$, at each grid location, heating and cooling were balanced by searching for a zero in the net-cooling rate, $\eta_{\rm C}(T_e)$. The root was initially located via bisection and then refined with the secant method. In the rare case of multiple roots, the stable one satisfying $d\eta_{\rm C}/dT_e>0$ was chosen.
The overall flow of the calculation is to start at the inner boundary of the disk, closest to the star at $i=1$. Solutions proceed downward in $Z$, from the top of the disk ($j=n_z$) to the equatorial plane ($j=1)$. This allows the optical depths back to the star and to the nearest edge of the disk to be kept current with the solution level populations.
Computations
============
In this section, we first compare the predictions of our code with known results for $\gamma\,$Cas. Next we explore a wide range of disk parameters for $\gamma\,$Cas and investigate the effect on the temperature structure of the disk of using a solar chemical composition for the gas. We then examine the energy loss in the spectral lines included in the models and compute the near infrared spectral energy distribution. Finally, we examine the computation of the diffuse photoionizing radiation field generated by the disk itself in order to evaluate the use of the OTS approximation for most of the models computed in this work.
Comparison with Millar & Marlborough {#sec:MM}
------------------------------------
@mil98 constructed a pure-hydrogen radiative equilibrium model for $\gamma\,$ Cas. @mil99 (MM, hereafter) extended this work to include the OTS approximation for the diffuse radiation field, and it is this temperature distribution which we have chosen for comparison. We adopt the same stellar parameters for $\gamma\,$ Cas as MM, which are reproduced in Table \[tab:gamma\_cas\_star\]. As we use these fundamental parameters for all of the calculations in this work, some additional comment is in order, particularly concerning the adopted stellar effective temperature. @fre05 investigate of the effect of rapid rotation on fundamental parameter determinations of B stars. They find, by fitting the line spectrum between $4250$ and $4500\;$Å, “apparent stellar parameters" for $\gamma\,$Cas (those obtained by a best fit classical, plane-parallel model atmosphere) of $T_{\rm
eff} = 26,400\;$K and $\log(g)=3.8$ which are close to the parameters adopted in this work. Nevertheless, they do find significant effects of rotation in their best-fit rotating models which have a [*parent non-rotating counterpart*]{} $T_{\rm eff}$ (see their paper for details) of $\approx\,30,000\;$K. This result suggests that accounting for the rotation of $\gamma\,$Cas in a manner following @fre05 (but using non-LTE stellar atmospheres) would be a useful future improvement in the computation of the direct stellar contribution to the photoionizing radiation field.
The fixed density structure for the $\gamma\,$ Cas disk adopted by MM is described by @mar69 and is slightly different from the model described in section \[sec:detail\]. MM assume that the disk is in isothermal, hydrostatic equilibrium at only one radial distance from the central star and that the radial drop-off in the equatorial density follows from an assumed (radial) outflow velocity law and the equation of continuity. We have simply adopted the $(R_i,Z_j)$ grid of MM (which is 24 by 20) and their total density at each grid point as input to [bedisk]{}. We have used a 5-level hydrogen atom plus continuum for this comparison and have also used the OTS approximation (as described previously) for the diffuse field. Despite this, there are still some significant differences between the calculations: our approach uses the optical depths in the OTS approximation and in the line escape probabilities as opposed to the various cases of MM. We use a newer ATLAS stellar atmosphere to predict the photoionizing radiation field from the star and use many more rays from each grid point back to the star. No attempt was made to use identical atomic data: MM included collisional transitions for only transitions $n$ to $n\pm 1$ in hydrogen whereas we have included all collisional rates. Nevertheless, despite these differences, the comparison is a useful check.
[lrl]{} Spectral Class& B0 IVe &\
Radius & 10.0 & $R_{\sun}$\
Mass & 17.0 & $M_{\sun}$\
Luminosity & $3.4\,\,10^4$ & $L_{\sun}$\
$T_{\rm eff}$ & 25,000 & K\
$\log(g)$ & 3.50 & $\rm cm\,s^{-2}$ \
Distance$^{a}$& $188^{+22}_{-18}$ & pc\
[bedisk]{} predicts a density-weighted average temperature, defined as $$\overline{T_\rho} \equiv \frac{1}{M} \int_{\rm Disk}\, T\,\rho\,dV \,,$$ (where $M$ is the total mass of the disk), of $11\,300$ K. This is to be compared to the $14\,500$ K quoted by MM (in their Table 1). This significant difference is simply one of definition: the density-weighted average quoted by MM is actually defined as $(\sum \rho_{ij}\,T_{ij})/\sum \rho_{ij}$ where the sum is over all of the grid points in the calculation. Computing this quantity for the current [bedisk]{} model yields $13\,900$ K which agrees with the MM result to within 5%.
Figure \[fig:tratio\_PM\] compares the ratio of the [bedisk]{} temperature to the MM temperature throughout the entire circumstellar disk. Agreement is generally good; [bedisk]{} tends to be somewhat cooler near the equatorial plane in the inner portion of the disk, while somewhat hotter towards the upper edge. However, 80% of the grid points agree to within $\pm20$%. The largest differences tend to occur along with upper edge of the envelope where the optical depths (to the nearest vertical edge of the disk) are most rapidly changing. The treatment of the escape of line radiation, as noted above, and particularly of how the OTS approximation was implemented (MM applied OTS to the whole disk as opposed to including an optical depth dependence as in equation \[eq:OTS\]), are likely the origin of the more significant differences.
Effect of Adding Metals on the Thermal Structure
------------------------------------------------
Table \[tab:models\] gives the disk parameters for 16 disk models computed to compare with observations of $\gamma$ Cas. These models span a range of nearly two orders of magnitude in density (as obtained by varying $\rho_o$ from $2.5\cdot\,10^{-12}$ to $1.0\cdot\,10^{-10}$ $\rm gm\,cm^{-3}$) with two values assumed for the radial drop-off of the density in the equatorial plane, $R^{-2.5}$ and $R^{-3.5}$. All models assumed $T_o=13\,500\;$K for the isothermal temperature which sets the vertical density scale-height via Eq. \[eq:scaleH\]. Also given in the table are the predicted density-weighted temperatures, the disk emission measures (in $\rm cm^{-3}$), defined as $${\rm EM} \equiv \int_{\rm Disk} n_e^2 \,dV \,,$$ and the predicted total H$\alpha$ luminosities (in $\rm ergs\,s^{-1}$). Section \[sec:lum\] discusses how the H$\alpha$ luminosity was computed. The parameters of the central star were again those of Table \[tab:gamma\_cas\_star\].
[ccccc]{} 2.5 & $2.5\,10^{-12}$ & 14060 & 59.25 & 33.43\
2.5 & $5.0\,10^{-12}$ & 12870 & 59.84 & 33.79\
2.5 & $7.5\,10^{-12}$ & 12420 & 60.16 & 33.99\
2.5 & $1.0\,10^{-11}$ & 12170 & 60.40 & 34.12\
2.5 & $2.5\,10^{-11}$ & 11040 & 61.10 & 34.48\
2.5 & $5.0\,10^{-11}$ & 9420 & 61.50 & 34.69\
2.5 & $7.5\,10^{-11}$ & 8590 & 61.57 & 34.77\
2.5 & $1.0\,10^{-10}$ & 8140 & 61.62 & 34.80\
3.5 & $2.5\,10^{-12}$ & 13990 & 58.88 & 32.90\
3.5 & $5.0\,10^{-12}$ & 13740 & 59.48 & 33.17\
3.5 & $7.5\,10^{-12}$ & 13500 & 59.81 & 33.31\
3.5 & $1.0\,10^{-11}$ & 13290 & 60.05 & 33.40\
3.5 & $2.5\,10^{-11}$ & 12080 & 60.82 & 33.68\
3.5 & $5.0\,10^{-11}$ & 11050 & 61.35 & 33.85\
3.5 & $7.5\,10^{-11}$ & 10560 & 61.48 & 33.95\
3.5 & $1.0\,10^{-10}$ & 10240 & 61.55 & 34.01\
The density-weighted temperatures predicted by the models are plotted in Figure \[fig:denaverage\] as a function of $\rho_o$. These results are compared to the predicted density-weighted temperatures for a set of pure hydrogen disks with identical physical parameters. Also shown in the Figure is the observed disk temperature for $\gamma$ Cas of $9500\pm1000$ K as found by @hon00 by fitting the IR Humphrey’s bound-free jump at 3.4$\,\mu$m. As expected, denser disks predict lower density-weighted temperatures.
The $R^{-2.5}$ models are consistent with the @hon00 result for densities in the range of $3$ to $8\cdot\,10^{-11}\,\rm g\,cm^{-3}$. This agrees well with the density estimated by @hon00 using their observations and the disk models of @wat86. The $R^{-3.5}$ models are only just consistent with the @hon00 result for largest densities considered, $\rho_o\approx 10^{-11}\,\rm g\,cm^{-3}$
The predicted temperature trend as a function of $\rho_o$ has an interesting dependence on the metallicity of the gas. For low density disks, the solar composition disks are considerably cooler than the pure hydrogen models by 1-2000 K. However, the difference decreases for higher disk densities, $\rho_o>2\cdot\,10^{-11}\rm \,g\,cm^{3}$. Indeed, for the $R^{-2.5}$ models, the higher density solar and pure hydrogen disks predict nearly the same density-weighted temperatures. This behaviour can be understood in terms of the heating and cooling avenues introduced by metals. Metals can act to cool the gas due to the escape of collisionally-excited line radiation. However, metals can also help to heat the gas via photoionization. If the optical depths in the hydrogen continua (excluding the Lyman continuum) are low, then the additional heating provided by the photoionization of metals is negligible in comparison to hydrogen. In this case, it is the cooling due to collisionally-excited line radiation that dominates. In a low-density disk, line cooling is further enhanced by the small optical depths (and hence high escape-probabilities) in the lines.
At high densities, however, the optical depths in the hydrogen bound-free continua are much larger; photoionization heating due to metals can then become important, particularly as many abundant metals have bound-free thresholds in the short-wavelength region of the Balmer continuum which is near the photospheric flux maximum in B stars. In this case, metals add both heating and cooling and the net result is a very similar density-weighted temperature to the case of a pure hydrogen plasma. These trends help explain why [@mil98] where able to obtain a reasonable density-weighted temperature for $\gamma$ Cas despite using a pure hydrogen envelope.
Two-dimensional temperature distributions in the disk for several different density $R^{-2.5}$ models are shown in Figure \[fig:2Dtemp\]. The inner portion of the disk for $R/R_*<5$ is expanded in each case for clarity. These figures clearly show the development of a cool region near and in the equatorial plane for the higher density disks. These denser disks have fairly strong vertical temperature gradients perpendicular to the equatorial plane. Given this, it would seem prudent to re-integrate the equation of hydrostatic equilibrium at each radial distance, accounting for the vertical variation of the gas temperature, and then to iterate the pressure structure along with the thermal solution to produce a disk that is in both radiative and (vertical) hydrostatic equilibrium; we shall present such models in a future work [@mcg06]. However, as the focus of the present work is on the thermal structure of the disk, it is convenient to have a fixed density structure so that the thermal effect of the gas metallicity can be unambiguously seen.
Figure \[fig:add\_metals\] presents a detailed comparison of the temperature structure predicted by a gas with a realistic solar composition to that of a pure hydrogen model. The figure compares the temperature ratio $T^{\rm Solar}/T^{\rm Pure H}$ for two densities, $\rho_o=5\cdot\,10^{-11}$ and $\rho_o=5\cdot\,10^{-12}\,\rm
g\,cm^{-3}$. In both cases, the optically thin gas far above the equatorial plane is cooler with the inclusion of metals. However, in the lower density model, $\rho_o=5\cdot\,10^{-12}\,\rm g\,cm^{-3}$, the gas in the equatorial plane near the star is hotter for the solar composition gas out to $R/R_*\sim 6$. In the higher density model, the gas near the equatorial plane has nearly the same temperature in the solar and pure hydrogen models. These trends are consistent with the effects noted in the discussion of the density-weighted average disk temperatures.
Line Luminosities {#sec:lum}
-----------------
Figure \[fig:halpha\_flux\] plots the total energy lost in H$\alpha$ (in $\rm ergs\,s^{-1}$) by the 16 models of Table \[tab:models\]. The line luminosity was found by integrating the flux divergence, in the escape probability approximation[^2], over the volume of the disk, [*i.e.*]{} $$\label{eq:lum}
L=h\nu_{ij} A_{ji}\int_{\rm Disk} n_j\,P_{\rm esc}(\tau_{ij}) \, dV \,.$$ For H$\alpha$, $i$ is level $n=2$ of H[i]{} and $j$ is level $n=3$. Also shown in the Figure is the H$\alpha$ luminosity observationally determined by @kas89 and @ste95. The H$\alpha$ luminosity from $\gamma\,$Cas is known to be variable, but the cited values are typical of the current epoch. There is good agreement with the observed luminosity for $R^{-2.5}$ models with $\rho_o$ between $2.5\cdot\,10^{-11}$ and $10^{-11}\,\rm g\,cm^{-3}$. This result is consistent with the models that best fit the observed disk temperature of [@hon00]. However, the $R^{-3.5}$ models seem inconsistent with the total energy loss in H$\alpha$ for the range of disk densities considered.
Figure \[fig:line\_flux\] shows energy escaping in all of the included radiative transitions for the $R^{-2.5}$ model with $\rho_o=5\cdot\,10^{-11}\,\rm g\,cm^{-3}$. The fluxes are again found by integrating Eq. \[eq:lum\] over the disk. It is important to keep in mind that Figure \[fig:line\_flux\] is not a spectrum (which would be obtained by integrating the transfer equation along a series of rays through the computational domain); it is simply a plot of the energy loss per second in each line acting to cool the gas. Nevertheless, it gives a good indication of the expected strong emission lines in the disk spectrum. For this particular model, 94% of the energy loss is provided by the lines of H[i]{}. Contributing at the level of $\approx\,1$% percent are Fe[ii]{}, C[ii]{}, Mg[ii]{}, and He[i]{}. Next to the lines of H[i]{}, the largest energy losses are in the resonance lines $\lambda\,1333.6\,$Å line of C[ii]{} ($\rm 2s^2\,2p\,^2P^o\,-\,2s\,2p^2\,^2D$) and the h and k lines of Mg[ii]{} near $\lambda\,2800\,$Å. Although it does not possess a single strong line, cooling due to Fe[ii]{} dominates over all of the metals due to its rich spectrum with many collisionally-excited lines. It should be noted that Figure \[fig:line\_flux\] and the percentage contributons cited above represent a [*global*]{} picture of energy loss integrated over the entire disk. The impact of the heating and cooling contributions of metals can be much larger at individual grid locations as demonstrated by Figure \[fig:add\_metals\].
Predicted IR spectral energy distributions
------------------------------------------
In this section, we consider the predicted IR continuum energy distribution as emitted [*perpendicular*]{} to the disk[^3]. Such models have a zero inclination angle between the rotation axis and the observer’s line of sight. This spectrum is easily found by solving the radiative transfer equation vertically through the disk at each $R_i$. If $I_i\equiv I_{+1,\nu}(R_i)$ is the emergent intensity for the annulus of area $A_i=\pi(R^2_{i+1/2}-R^2_{i-1/2})$ then the SED seen by the observer will be $$L^{\rm Star+Disk}_{\nu}=I^{*}_{\nu}\,\pi R^{2}_{*} + \sum_{i=1}^{n_{R}}\,I_i A_i\,.$$ Here $R_*$ is the radius of the star, $I^*_{\nu}$ is the specific intensity corresponding to the stellar surface, and $I_i$ is the specific intensity of the $i^{th}$ disk annulus. It is well known that at IR wavelengths, Be stars possess an excess of radiation over that predicted by an appropriate stellar photosphere model [@cot87]. This excess comes mainly from free-free emission in the ionized disk[^4], with a small contribution from free-bound emission shortward (in wavelength) of the ionization edges.
The near-IR spectral energy distributions are shown in Figure \[fig:IR\] for the $R^{-2.5}$ models of Table \[tab:models\]. Plotted is the (monochromatic) IR excess expressed in magnitudes, $$Z_{\nu}\equiv 2.5 \log\left(\frac{L^{Star+Disk}_{\nu}}{L^{Star}_{\nu}}\right).$$ The discontinuous jumps at wavelengths less than $5\,\mu$m in this figure, particularly for higher densities, represent the hydrogen free-bound continua for recombination to $n=4$ ($1.4\,\mu$m) and $n=5$ ($2.2\,\mu$m). Unlike (plane-parallel) stellar photospheres which have bound-free edges in absorption (reflecting the inward increase in temperature), circumstellar material exhibits the bound-free edges in emission, reflecting the increase in the gas emissivity due to recombination. Thus the figure indicates a small contribution from free-bound emission to the infrared excess at these wavelengths. We also note that as the disk density is increased, the infrared excess steepens most strongly between the wavelengths of 1 and 5$\;\mu$m.
To compare with observations, we first show the $12$ and $25\,\mu$m IRAS IR excesses for $\gamma\,$Cas found by @cot87. The model that best matches the IRAS IR excess at these wavelengths has a slightly smaller density, $\rho_o\approx 10^{-11}\,\rm g\,cm^{-3}$, than the model that best matches the observed average disk temperature and energy loss in H$\alpha$, $\rho_o\approx 3\cdot\,10^{-11}\,\rm g\,cm^{-3}$ (Figure \[fig:denaverage\]). However, there is good evidence that $\gamma\,$Cas is seen at an inclination of $i>\approx\,55^o$ [@tyc06], as opposed to being viewed pole-on ($i=0^o$) as assumed by the models. Hence the models have an effective emitting area that is too large and the IR excess is likely overestimated. In addition, the spectra for non-zero inclination angles will reflect the contribution of a different set of rays passing through the disk; the different physical conditions along these rays will lead to differences in the gas opacity and emissivity and hence a different predicted intensity along each ray.
We also show in Figure \[fig:IR\] the 2.5 to 11.6$\,\mu$m Infrared Space Observatory (ISO) spectrophotometry[^5] for $\gamma\,$Cas. In order to compare the ISO fluxes at the Earth to our photospheric model (to set the reference flux to extract the excess), we require the angular diameter of the star. @tyc05 cite the major axis of the H$\alpha$ emitting region of $\gamma\,$Cas of $3.67\pm0.09\,$milli-arcseconds, but this has a large contribution from the circumstellar disk. To resolve this problem, we have chosen an angular diameter such that the ISO data reproduces the $12\,\mu$m IRAS excess. Combining this stellar angular diameter for $\gamma\,$Cas with the range of Hipparcos distances listed in Table \[tab:gamma\_cas\_star\], we find a required radius for $\gamma\,$Cas of between 6.8 and 8.4 solar radii. While this result in disagreement with the 10 solar radii listed in Table \[tab:gamma\_cas\_star\], the larger value is within the plausible error range for $\gamma\,$Cas’s radius. With this normalization, the ISO observations match reasonably well with the $\rho_o\approx
10^{-11}\,\rm g\,cm^{-3}$ model prediction over the considered wavelength range. The fit is actually worse closer to the normalization point; the shorter wavelength excess, between $2.5<\lambda<5\,\mu$m, is quite well reproduced. Similar caveats to those ending the previous paragraph apply here: a detailed comparison will be made with a model that correctly accounts for the viewing inclination of $\gamma\,$Cas in a future work.
An Approximate Treatment of the Diffuse Radiation Field {#sec:diffuse}
-------------------------------------------------------
While the OTS approximation is simplest treatment of the diffuse radiation field, Figure \[fig:2Dtemp\] shows that in the densest disks ($\rho_o\approx 5\cdot\,10^{-11}\rm g\,cm^{-3}$), grid locations near the equatorial plane can become quite cool due to large optical depths back to the central star. However, examining the thermal structure in the Z-direction at such a location typically shows that at heights above and below the equatorial plane, the gas can still be quite hot as it can be directly illuminated by at least a portion of the star. One might expect that the radiation emitted from these hot “sheaths" might be an important source of secondary photoionizing radiation for the equatorial gas [@car05].
In principle, the calculation of $J^{\rm Dif}_{\nu}$ in Eq. \[eq:jnu\] requires a solution of the transfer equation in the 2D cylindrical geometry[^6]. However, to estimate the potential effect of this diffuse field on the current work, we have tried a simpler and approximate approach. We solve, at each $R_i$, the radiative transfer equation perpendicular to the disk in the Z-direction using the method of short-characteristics [@ols87]. The mean intensity along this ray is then $$J_{\nu}(Z_j)=\frac{1}{2}\left\{I_{\mu=+1,\nu}(Z_j)+I_{\mu=-1,\nu}(Z_j)\right\} \,.$$ For this solution, zero incident radiation is assumed perpendicular to the disk, We then estimate the diffuse contribution to the photoionizing radiation field at each $Z_j$ by assigning $$\label{eq:add_diff}
J^{\rm Dif}_{\nu}(Z_j)=W_{\rm dif}\,J_{\nu}(Z_j) \,,$$ Here $W_{\rm dif}$ represents an ad-hoc dilution factor, between zero and one, for the perpendicular rays. The expectation is that $J^{\rm Dif}_{\nu}$ will potentially be most important in the cool equatorial regions that develop for higher density disks such as the one illustrated in the bottom panel of Figure \[fig:2Dtemp\]. In such a cool obscured region, the hot sheaths above and below cover approximately $1/2$ of the available $4\pi$ steradians so a dilution factor of $W_{\rm dif}=0.5$ is a reasonable approximation. In regions of the disk where the central star is not obscured, the choice of $W_{\rm dif}$ is irrelevant as $J^{\rm Dir}\gg J^{\rm Dif}$. For this reason, we have taken $W_{\rm dif}$ to be $1/2$ at all grid locations in the disk.
Figure \[fig:add\_diff\] shows the effect of adding such an approximate treatment of $J^{\rm Dif}$ to the $R^{-2.5}$ disk model with $\rho_o=5\cdot\,10^{-11}\rm\,g\,cm^{-3}$, the model which best matched the observed disk temperature and the total H$\alpha$ luminosity. With the diffuse field added, the amount of additional heating in the equatorial plane is fairly small, with only a $10-20$% increase in the temperature there. Hence, we feel that the previously computed models are reliable despite the approximate treatment of the diffuse field through the OTS approximation.
Note that this figure is not meant to imply that the effect of the diffuse field is negligible; in both runs, the on-the-stop (OTS) approximation, which approximates the local trapping of radiation with $\lambda<912\,$Å and hence the diffuse photoionizing radiation field for $\lambda<912\,$Å was employed following Eq. \[eq:OTS\] If the OTS approximation is turned off, there is considerable cooling of the equatorial regions[^7] [see also @mil99] and this would mask the effect (illumination of these same regions from above and below) that we were trying to investigate.
Conclusions
===========
For the first time, we have constructed radiative equilibrium models for the disk of $\gamma\,$Cas using a gas with a solar chemical composition. We find that a model with a power-law equatorial gas density of $\rho(R)\approx 3$ to $5\cdot 10^{-11} (R/R_*)^{-2.5}\, \rm
g\,cm^{-3}$, which is in vertical (isothermal) hydrostatic equilibrium, can reproduce several overall observed disk properties, including its (density-weighted) average temperature, the energy loss in H$\alpha$, and the near infrared flux excess.
Also in this work, we have investigated the differences between our solar composition radiative equilibrium models and the corresponding set of pure hydrogen models with the same disk parameters. The effect of the additional heating and cooling provided by elements heavier than hydrogen on global measures of the disk structure, such as the density-weighted average disk temperature or the total energy loss in H$\alpha$, depend strongly on the parameter $\rho_o$ (the assumed equatorial density near the stellar surface) which sets the overall density of the disk. For low disk densities, $\rho_o< ~ 10^{-11}\rm\,g\,cm^{-3}$, the density-weighted averaged disk temperature is generally 1–2000 K cooler with a solar composition, due mostly to enhanced collisionally-excited line cooling. At higher densities, absorption of ionizing radiation in the bound-free continua of elements heavier than hydrogen provides additional heating which offsets somewhat the additional line cooling.
In examining the detailed temperature at each position withing the disk, differences of up to $\approx\pm\,40$% are found in comparing the solar composition models to the pure hydrogen models. While the global energy loss in H$\alpha$ in the solar and pure hydrogen models is similar, these significant differences in the disk temperature distribution may manifest themselves in the detailed spectra. The next step in the comparison between these two sets of models is to present predicted spectra over a wide range of wavelengths for a wide range of viewing inclinations.
The [bedisk]{} code represents a compromise between computational efficiency and realism. The code’s most notable approximations are the treatment of the disk diffuse radiation field, the use of (first-order) static, escape probabilities for the line radiative transfer, the assumption of a spherical, non-rotating, central star, and the somewhat limited atomic models. Nevertheless, these approximations allow the code to efficiently explore a wide region of parameter space for the thermal structure of the circumstellar disks surrounding hot stars. Future work will proceed along two fronts: we will use the current version of [bedisk]{} to compute a series of Be disk thermal models that will serve as the basis for detailed spectral synthesis to produce line profiles, interferometric visibilities, and continuum polarization signatures which can be directly compared to observations. Along the second front, we shall improve some of the basic assumptions of the code. Most notably, we will enforce a vertical hydrostatic equilibrium density structure consistent with the thermal solution, allow for potential gravitational darkening (and geometric distortion) of the central star by rapid rotation, and account for the diffuse photoionizing radiation field from the disk by a direct solution of the 2-D radiative transfer problem.
We would like to thank Chris Tycner for many helpful discussions. We thank Rens Waters for clarifying the IR excess of $\gamma\,$Cas. We also thank Kyle Lawson, Anna Molak, and Laura Thomson for help as part of their NSERC Summer Student awards. Finally, we would like to thank the anonymous referee for a careful reading of the text and providing many helpful comments and suggestions. This work is supported by the Canadian Natural Sciences and Engineering Research Council through Discovery Grants to TAAS and CEJ.
The Atomic Data
===============
Energy levels for all atoms and ions were adopted from the NIST Atomic Spectra Database[^8]. Radiative transition probabilities and photoionization cross sections were adopted from the Opacity Project Database TOPbase [^9]. The photoionization cross sections, which include complex resonance features due to autoionization, were smoothed by convolution with a Gaussian down to the resolution of the ATLAS frequency grid. Thermally-averaged collision strengths for the electron impact excitation of hydrogen were adopted from @cal94, @agg91, and @cha91. Thermally-averaged collisional strengths for the remaining atoms and ions were adopted from the compilation of @pp95 where available. The Gaunt factor approximation was used for the remaining dipole-allowed transitions. The atomic data for iron and its ions were adopted from the extensive model atoms constructed by @sig03 and @sig04.
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[^1]: By choosing to solve the transfer equation along these rays, as opposed to simply applying exponential extinction of the stellar photospheric intensity, some contribution of the diffuse field is included.
[^2]: In the escape probability approximation, the net radiative bracket is replaced by the (single-flight) escape probability.
[^3]: We shall present detailed synthesis for the infrared hydrogen disk spectrum for arbitrary inclination angles in a future work.
[^4]: Note that the exact expression for the free-free Gaunt factor was used for this calculation.
[^5]: We have extracted this data from the ISO on-line archive. The spectrophotometry is from the PHT-40 instrument for observing series TDT 76803401.
[^6]: A direct implementation of this approach can be found in @car05 who use a Monte Carlo approach to radiative transfer to estimate the diffuse radiation field in the envelope of a somewhat later-type Be star.
[^7]: A subtle point is whether a model including $J^{\rm Dif}$ according to Eq. \[eq:add\_diff\] should also employ the OTS approximation. In principle, employing both over counts the local radiation field in each volume element.
[^8]: http://units.nist.gov/PhysRefData/ASD/index.html
[^9]: http://cdsweb.u-strasbg.fr/topbase.html
|
---
abstract: 'We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if $n\geq 3$, $g:\mathbb{S}^{n-1}\to\mathbb{R}$ is an even bounded measurable function, $U$ is an open subset of $\mathbb{S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)|_U=c''$, for some fixed constants $c,c''\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$. Here, ${\cal C}(g)$ denotes the cosine transform and ${\cal R}(g)$ denotes the Funk transform of $g$. However, we show that $g$ does not need to be equal to a constant almost everywhere in $U^\perp:=\bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp)$. For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.'
author:
- 'Ioannis Purnaras, Christos Saroglou'
title: '**Functions with isotropic sections**'
---
Introduction
============
Let us fix an orthonormal basis $\{e_1,\dots,e_n\}$ in $\mathbb{R}^n$. We write $\langle x, y\rangle$ for the standard inner product of $x$ and $y$ in ${\mathbb R}^n$. For $k=1,\dots,n-1$, the set of all $k$-dimensional subspaces of ${{\mathbb R}^n}$ is denoted by $G_{n,k}$. If $A\subseteq {{\mathbb R}^n}$, the orthogonal projection of $A$ onto a subspace $H\in G_{n,k}$, will be denoted by $A|H$. If $u\in{{\mathbb R}^n}$, we denote by $u^\perp$ the subspace of codimension 1 which is orthogonal to $u$. The notation $B_2^n$ stands for the standard unit ball in $\mathbb{R}^n$. Also, ${\mathbb{ S}^{n-1}}=\{x\in {{\mathbb R}^n}:|x|=1\}$ denotes the unit sphere in ${\mathbb R}^n$. The boundary of a set $A$ will be denoted by $\textnormal{bd}A$. A [*spherical cap*]{} $U\subseteq {\mathbb{ S}^{n-1}}$ is any set of the form $\{x\in {\mathbb{ S}^{n-1}}:\langle x,u\rangle>a\}$, $0<a<1$, $u\in {\mathbb{ S}^{n-1}}$. The point $u$ is called the [*center*]{} of the spherical cap $U$. Denote, also, by ${\cal H}^a$, the $a$-dimensional Hausdorff measure in ${{\mathbb R}^n}$, where $0<a\leq n$. We will say that a Borel measure on the sphere ${\mathbb{ S}^{n-1}}$ is absolutely continuous if it is absolutely continuous with respect to ${{{\cal H}^{n-1}}}$. For a Borel set $\omega$ in ${\mathbb{ S}^{n-1}}$, ${{\cal B}}(\omega)$ stands for the $\sigma$-algebra of Borel subsets of $\omega$. Any convergence of sets will be with respect to the Hausdorff metric. The orthogonal group in ${{\mathbb R}^n}$ is denoted by $O(n)$. For $u\in{\mathbb{ S}^{n-1}}$, we set $O(n,u):=\{T\in O_n:Tu=u\}$.
Let $\mu$ be a signed Borel measure on ${\mathbb{ S}^{n-1}}$ and $\zeta:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be an integrable function. The [*cosine transform*]{} ${\cal C}(\mu)$ of $\mu$ and the [*Funk transform*]{} (=Radon transform on the sphere) ${\cal R}(\zeta)$ of $\zeta$ are defined as follows. $${\cal R}(\zeta)(u)=\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\zeta (x)d{{{\cal H}^{n-2}}}(x),\qquad u\in {\mathbb{ S}^{n-1}},$$ $${\cal C}(\mu)(u)=\int_{{\mathbb{ S}^{n-1}}}|\langle x,u\rangle| d\mu(x),\qquad u\in {\mathbb{ S}^{n-1}}.$$ If $f$ is an integrable function on ${\mathbb{ S}^{n-1}}$, we define ${\cal C}(f):={\cal C}(fd{{{\cal H}^{n-1}}}(\cdot))$ and we simply say that ${\cal C}(f)$ is the cosine transform of $f$. A function $g:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ is called [*isotropic*]{} if the map $${\mathbb{ S}^{n-1}}\ni u\mapsto \int_{{\mathbb{ S}^{n-1}}}\langle x,u\rangle^2 g(x)d{{{\cal H}^{n-1}}}(x)$$is constant. The following problem was proposed in [@M-R-S].
\[prob1\] Assume that for a measurable subset $U$ of $S^{n-1}$ and for an even bounded measurable function $g:S^{n-1}\to\mathbb{R}$, the restriction $g|_{S^{n-1}\cap u^{\perp}}$ onto $S^{n-1}\cap u^{\perp}$ is isotropic, for almost all $u\in U$. Is it true that $g$ is almost everywhere equal to a constant on the set $U^\perp$?
Here, $U^\perp$ stands for the union of all great subspheres of ${\mathbb{ S}^{n-1}}$, which are orthogonal to a direction from $U$, i.e $U^\perp=\bigcup_{u\in U} (S^{n-1}\cap u^{\perp})$. The following was established in [@M-R-S].
\[thm-old-s\] Problem \[prob1\] has affirmative answer if $U={\mathbb{ S}^{n-1}}$.
Our goal is to prove that the answer to Problem \[prob1\] is in general negative but on the other hand, a local version of Theorem \[thm-old-s\] is still valid.
\[thm-counterexample\] Let $U$ be an open subset of ${\mathbb{ S}^{n-1}}$, that does not contain $U^\perp$. There exists a continuous function $g:{\mathbb{ S}^{n-1}}\to {\mathbb R}$, such that for any $u\in U$, $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$ is isotropic, but $g$ is not constant on $U^\perp$.
\[thm-meta-main-1\] Let $n\geq 3$, $U$ be an open subset of ${\mathbb{ S}^{n-1}}$ and $g:U\to{\mathbb R}$ be an even, bounded, measurable function. If for almost every $u\in U$, $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)=c'$, almost everywhere in $U$, for some fixed constants $c,c'\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$.
The fact that Theorem \[thm-meta-main-1\] is a local version of Theorem \[thm-old-s\] follows from the classical fact that if ${\cal C}(g)$ is constant on ${\mathbb{ S}^{n-1}}$, then $g$ has to be almost everywhere equal to a constant on ${\mathbb{ S}^{n-1}}$. In fact, the proof of Theorem \[thm-old-s\], is based on Theorem \[thm-meta-main-1\], proved in [@M-R-S] in the case $U={\mathbb{ S}^{n-1}}$. The proof of the latter relies on a quick “global" argument based on the Aleksandrov-Fenchel inequality (see next section). However, such arguments will not work in the local setting.
For a strictly convex body $K$ with $C^2$ smooth boundary and a direction $u\in {\mathbb{ S}^{n-1}}$, denote by $r_K^1(u),\dots,r_K^{n-1}(u)$ the principal radii of curvature of $K$ at $u$ (see next section). It is well known that $$\label{eq:rad-curv}
r_K^i(u)=\frac{1}{k_K^i(v_K(u))}, \qquad i=1,\dots,n-1,$$ where $k_K^1(x),\dots,k_K^{n-1}(x)$ are the principal curvatures of the hypersurface $\textnormal{bd}K$ at the point $x\in\textnormal{bd}K$. Here, $v_K:{\mathbb{ S}^{n-1}}\to\textnormal{bd}K$ denotes the inverse Gauss map , i.e. for $u\in{\mathbb{ S}^{n-1}}$, $v_K(u)$ is the (unique) point of intersection of $K$ with its supporting hyperplane whose outer unit normal vector is $u$.
The proof of the general case of Theorem \[thm-meta-main-1\] exploits the following observation that we believe is new: If $g$ is smooth enough and $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$ is isotropic for some $u\in{\mathbb{ S}^{n-1}}$, then the principal curvatures of the boundary of the zonoid $Z(g)$, whose generating measure is given by $S_{n-1}(Z(g),\cdot)=gd{{{\cal H}^{n-1}}}(\cdot)$ (see Section 4), at $v_K(u)$ are all equal. That is, the point $v_k(u)$ is an umbilic of the boundary of $Z(g)$. Therefore, if $g$ is smooth enough, one can use the following classical result (see e.g. [@Doc pp 183]) to prove Theorem \[thm-meta-main-1\].
\[thm-old\] Let $V$ be a hypersurface in ${{\mathbb R}^n}$, $n\geq 3$, of class $C^3$ (or according to [@S-V], of class $C^2$). If for all $x\in V$, it holds $0\neq k_1(x)=\dots=k_{n-1}(x)\in\mathbb{R}$, then $V$ is contained in a Euclidean sphere, where $k_1(x),\dots,k_{n-1}(x)$ are the principal curvatures of $V$ at $x$.
The reader might guess that, since we do not assume any regularity on $g$, Theorem \[thm-old\] cannot be used directly (to our knowledge, not even if we assume $g$ to be continuous) to prove Theorem \[thm-meta-main-1\]. Thus, we need somehow to relax the regularity assumptions in Theorem \[thm-old\], at least in the convex case. This is done in the following theorem, which we believe is of independent interest.
\[thm-main\] Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$, $U$ be an open connected subset of $\mathbb{S}^{n-1}$ and assume that the measure $S_1(K,\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous. If for almost every direction $u\in U$ it holds $$\label{eq:equal-princ-radii}
r_K^1(u)=\dots=r_K^{n-1}(u),$$ then $\tau(K,U)$ is contained in a Euclidean sphere.
Here, $S_1(K,\cdot)|_{{{\cal B}}(U)}$ denotes the order 1 area measure of $K$, restricted into the family of Borel subsets of $U$ and $\tau(K,U)$ is the inverse spherical image of $U$ with respect to $K$. We refer to the next section for definitions.
Theorem \[thm-main\] is in some sense optimal. This is demonstrated in the following examples.
\[ex-1\] One cannot replace (\[eq:equal-princ-radii\]) by the condition that for almost every point in an open subset of $\textnormal{bd}K$, the principal curvatures are equal. To see this, take $K$ to be the intersection of two Euclidean balls with different centers.
\[ex-2\] The assumption that $S_1(K,\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous cannot be removed. Indeed, take for instance $K$ to be the Minkowski sum of a Euclidean ball and a polytope and $U={\mathbb{ S}^{n-1}}$.
Nevertheless, we do not know whether the assumption of absolute continuity of the order 1 area measure (restricted in ${\cal B}(U)$) in Theorem \[thm-main\] can be replaced by the absolute continuity of the area measure of any other order.
The main tools for the proof of our results come from Convex and Integral Geometry. This paper is structured as follows. In Section 2, we provide the necessary background for the proof of our main results. Theorem \[thm-main\] is proved in Section 3. In Section 4, we prove Theorems \[thm-counterexample\] and \[thm-meta-main-1\] and, under some regularity assumptions on $g$, a local version of Theorem \[thm-meta-main-1\].
Preliminaries and notation
==========================
In this section we introduce notation and collect basic facts from classical theory of convex bodies that we use in the paper. As a general reference on the theory we use R. Schneider’s book “Convex bodies: the Brunn-Minkowski theory" [@Sc] (see also [@B-F] or [@G]).
Let $A,\ B$ be subsets of ${\mathbb R}^n$. The [*linear hull*]{} of $A$ is denoted by $\textnormal{span} A$. The [*Minkowski sum*]{} $A+B$ of $A$ and $B$ is the set $\{x+y:x\in A,y\in B\}$ .
A [*convex body*]{} $K$ in ${{\mathbb R}^n}$ is a convex compact set with non-empty interior. The function $h_K:{{\mathbb R}^n}\to{\mathbb R}$, with $h_K(u)=\max\{\langle x, u\rangle : x \in K\}$ is the [*support function*]{} of $K$. The support functional is known to be additive with respect to the Minkowski sum and 1-homogeneous. That is, $h_{\lambda K+\mu L}=\lambda h_K+\mu h_L$, for any compact convex sets $K,L$ and for any $\lambda,\mu\geq 0$. Moreover if $H$ is a subspace of ${{\mathbb R}^n}$ and $T:{{\mathbb R}^n}\to{{\mathbb R}^n}$ is any orthogonal map, then the following identities hold: $$h_{K|H}=(h_K)|_H\qquad\textnormal{and}\qquad h_{TK}=h_K\circ T^{\ast},$$where $T^\ast$ denotes the adjoint of $T$.
For a convex body $K$ and $u\in{\mathbb{ S}^{n-1}}$, the support set $F(K,u)$ of $K$ in the direction $u$ is defined by $F(K,u)=\{x\in K:\langle x,u\rangle=h_K(u)\}$. Similarly with the support functional, the support set functional is additive with respect to the Minkowski sum. That is, if $L$ is another convex body, then $$\label{eq:support-set}
F(K+L,u)=F(K,u)+F(L,u).$$
A classical theorem of Minkowski says that if $K_1, K_2, \dots, K_n$ are convex compact sets in ${\mathbb R}^n$ and $\lambda_1, \dots, \lambda_n \ge 0,$ then the volume of the set $\lambda_1 K_1 +\lambda_2 K_2+\dots + \lambda_nK_n$ is a homogeneous polynomial in $\lambda_1,\dots, \lambda_n$ of degree $n$, with non-negative coefficients. The coefficient of $\lambda_1\cdots \lambda_n$ is called the [*mixed volume* ]{}of $K_{1}, \dots, K_{n}$ and is denoted by $V(K_{1}, \dots, K_{n})$. We will also write $V(K_1[m_1],\dots, K_r[m_r])$ for the mixed volume of $K_1,\dots, K_r$ where each $K_i$ is repeated $m_i$ times and $m_1+\dots+m_r=n$.
The Aleksandrov–Fenchel inequality states the following $$\label{eq:af}
V(K_1,K_2, K_3, \dots, K_n)^2 \ge V(K_1,K_1, K_3, \dots, K_n)V(K_2,K_2, K_3, \dots, K_n).$$
It turns out that for given convex bodies $K_1,\dots,K_{n-1}$, there is a unique Borel measure $S(K_1,\dots,K_{n-1},\cdot)$ on the sphere ${\mathbb{ S}^{n-1}}$, such that for any convex body $L$, it holds $$\label{eq:mixarea}
V(L,K_1,\dots, K_{n-1})=\frac{1}{n}\int_{{\mathbb{ S}^{n-1}}}h_L(u)dS(K_1,\dots, K_{n-1},u).$$ Similarly, as with mixed volumes, the notation $S(K_1[m_1],\dots, K_r[m_r],\cdot)$ means that $K_i$ is repeated $m_i$ times, $i=1\dots,r$, where $m_1+\dots+m_r=n-1$. One of the fundamental properties of mixed area measures is additivity and homogeneity with respect to any of its arguments. That is, $$\begin{aligned}
\label{eq:mixedarea-additivity}
&&S(K_1,\dots,K_{m-1},\lambda K_m+\mu K'_m,K_{m+1},\dots,K_{n-1},\cdot)\nonumber\\
&=&\lambda S(K_1,\dots,K_{m-1},K_m,K_{m+1},\dots,K_{n-1},\cdot)+\mu S(K_1,\dots,K_{m-1},K'_m,K_{m+1},\dots,K_{n-1},\cdot),\end{aligned}$$ for any convex body $K'_m$ and any numbers $\lambda,\mu>0$.
A useful fact concerning mixed area measure is that if $\{L^{(m)}_{j}\}_{m=1}^\infty$ is a sequence of convex bodies, converging to $K_j$, in the Hausdorff metric, where $j=1,\dots,n-1$, then the corresponding sequence $ \{S(L^{(m)}_1,\dots,L^{(m)}_{n-1},\cdot)\}_{m=1}^\infty$ of mixed area measures converges weakly to $S(K_1,\dots,K_{n-1},\cdot)$. That is, for every continuous function $\varphi:{\mathbb{ S}^{n-1}}\to{\mathbb R}$, it holds $$\int_{{\mathbb{ S}^{n-1}}}\varphi dS(L^{(m)}_1,\dots,L^{(m)}_{n-1},\cdot)\xrightarrow{m\to\infty}\int_{{\mathbb{ S}^{n-1}}}\varphi dS(K_1,\dots,K_{n-1},\cdot).$$
Let $u\in S^{n-1}$ be a point at which $h_K$ is twice differentiable. If $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$ is an orthonormal basis of $u^\perp$, we denote by $Hess (h_{K})(u)$ the $(n-1)\times(n-1)$ Hessian matrix of the restriction of $h_K$ onto $T_u {\mathbb{ S}^{n-1}}$ (the tangent hyperplane of ${\mathbb{ S}^{n-1}}$ at $u$), where we differentiate with respect to the basis $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$. The eigenvalues $r_K^1(u),\dots,r_K^{n-1}(u)$ of this matrix are non-negative (since $h_K$ is convex), independent of the choice of the orthonormal basis $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$ of $u^\perp$ and are called “the principal radii of curvature" of $K$ at $u$.
We say that a convex body $K$ is of class ${\cal C}^2_+$ if $h_{K}$ is of class $C^2$ and if all the principal radii of curvature of $K$ at any $u\in {\mathbb{ S}^{n-1}}$ are strictly positive. If the convex bodies $K_1,\dots,K_{n-1}$ are of class ${\cal C}^2_+$, then the mixed area measure $S(K_1,\dots,K_{n-1},\cdot)$ is absolutely continuous and its density depends pointwise only on the Hessian matrices $Hess (h_{K_i})(u)$, $i=1,\dots,n-1$ but not on the (common) choice of the orthonormal basis $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$. In fact, $$\label{eq:mixed-discriminant}
\frac{dS(K_1,\dots,K_{n-1},\cdot)}{d{{{\cal H}^{n-1}}}(\cdot)}(u)=D(Hess(h_{K_1})(u),\dots,Hess(h_{K_{n-1}})(u)),$$ where the last expression is the mixed discriminant of the matrices $Hess(h_{K_1})(u),\dots,Hess(h_{K_{n-1}})(u)$ (see [@Sc Section 2.5] and the references therein).
If $\omega$ is a subset of ${\mathbb{ S}^{n-1}}$, define the [*inverse spherical image*]{} $\tau(K,\omega)$ of $\omega$ with respect to $K$ by $$\tau(K,\omega)=\big\{x\in\partial K:\exists u\in\omega,\textnormal{ such that }\langle x,u\rangle=h_K(u)\big\}.$$ Assume, furthermore that $K$ is of class ${\cal C}_+^2$. Since the inverse Gauss map $v_K:{\mathbb{ S}^{n-1}}\to \textnormal{bd}K$ is well defined and continuous, and since in this case it clearly holds $\tau(K,\omega)=v_K^{-1}(\omega)$, it follows that if $\omega$ is an open set in ${\mathbb{ S}^{n-1}}$ then $\tau(K,\omega)$ is also open in $\textnormal{bd}K$.
For $j=1,\dots,n-1$, the [*area measure of order $j$*]{} of a convex body $K$ is defined as $$S_j(K,\cdot):=S(K[j],B_2^n[n-1-j],\cdot).$$ In particular (as it follows from (\[eq:mixedarea-additivity\])), the order 1 area measure is additive and homogeneous, i.e. $S_1(\lambda K+\mu L,\cdot)=\lambda S_1(K,\cdot)+\mu S_1(L,\cdot)$, for any $\lambda,\mu>0$ and any convex bodies $K,\ L$.
The special case $j=n-1$ in the previous definition is better understood and of particular interest. The area measure $S_{n-1}(K,\cdot)$ is called the [*surface area measure*]{} of $K$. The following formula is valid $$\label{eq:surface-area-measure}
S_{n-1}(K,\omega)={\cal{H}}^{n-1}\big(\tau(K,\omega)\big),$$ for any Borel $\omega \subset {\mathbb{ S}^{n-1}}$. In addition, Minkowski’s Existence and Uniqueness theorem states that any Borel measure, whose center of mass is at the origin and is not concentrated in any great subsphere of ${\mathbb{ S}^{n-1}}$, is the surface area measure of a unique (up to translation) convex body.
The density of the absolutely continuous part (in its Lebesgue decomposition) of $S_j(K,\cdot)$ will be denoted by $f_K^{(j)}$. Densities of area measures behave well under the action of orthogonal maps. If $T\in O(n)$, then (see [@Lut]) $$\label{eq:density-orth-maps}
f^{(j)}_{TK}=f^{(j)}_K\circ T^\ast.$$
Recall the definition of the elementary symmetric functions $s_j$: If $a_1,\dots,a_{n-1}$ are positive reals, then $$s_j(a_1,\dots,a_{n-1}):={\displaystyle{\binom{n-1}{j}}}^{-1}\displaystyle{\sum_{1\leq i_1<\dots<i_j\leq n-1}a_{i_1}\dots a_{i_j}}.$$ The classical Newton inequality states that if $1\leq i<j\leq n-1$ $$\label{eq:Newton}
s_i(a_1,\dots,a_{n-1})^{1/i}\geq s_j(a_1,\dots,a_{n-1})^{1/j},$$ with equality if and only if $a_1=\dots=a_{n-1}$.
Recall that the support function $h_K$ of the convex body $K$ is twice differentiable for almost every $u\in {\mathbb{ S}^{n-1}}$. It is known (see [@Hug1], [@Hug2], [@Hug3] for additional information, references and related results concerning area measures and their densities) that $f^{(j)}_K$ is given by $$\label{eq:area-measure-density}
f^{(j)}_K(u)=s_j(r_K^1(u),\dots,r_K^{n-1}(u)),\qquad\textnormal{for almost every }u\in {\mathbb{ S}^{n-1}}.$$ In the case $j=1$, we can rewrite (\[eq:area-measure-density\]) as follows $$\label{eq:distributions}
f^{(1)}_K(u)=\frac{1}{n-1}\Delta_S h_K(u)+h_K(u),\qquad\textnormal{for almost every }u\in {\mathbb{ S}^{n-1}},$$ where $\Delta_S$ is the Laplacian (i.e. the Laplace-Beltrami operator) on the sphere. It is well known that the support function of a convex body, restricted on ${\mathbb{ S}^{n-1}}$ is contained in the Sobolev space $\mathbb{H}^1({\mathbb{ S}^{n-1}})$ (see [@Kid], where higher regularity is established). Moreover, as shown in [@Be], (\[eq:distributions\]) actually holds in the sense of distributions.
We have the following simple Lemmas.
\[l-1-Newton\] Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$, $\omega$ be a Borel subset of $\mathbb{S}^{n-1}$ and $1\leq i\leq j<n-1$. The following statements are equivalent.
i) $\left(f^{(i)}_{K}(u)\right)^{1/i}=\left(f^{(j)}_{K}(u)\right)^{1/j}$, for almost every $u\in \omega$.
ii) $\left(f^{(i)}_{K}(u)\right)^{1/i}\leq\left(f^{(j)}_{K}(u)\right)^{1/j}$, for almost every $u\in \omega$.
iii) $r^1_K(u)=\dots=r^{n-1}_K(u)$, for almost every $u\in \omega$.
Using Newton’s inequality (\[eq:Newton\]) together with the representation (\[eq:area-measure-density\]) of the densities $f^{(i)}_{K}$, $f^{(j)}_{K}$, we obtain $$\left(f^{(i)}_{K}(u)\right)^{1/i}= s_i\left(r_K^1(u),\dots,r_K^{n-1}(u)\right)^{1/i}\geq s_j\left(r_K^1(u),\dots,r_K^{n-1}(u)\right)^{1/j}=\left(f^{(j)}_{K}(u)\right)^{1/j},$$ for almost every $u\in \omega$. Therefore, if $(i)$ or $(ii)$ holds, then we have equality in Newton’s inequality (\[eq:Newton\]), which is only possible if $r^1_K(u)=\dots=r^{n-1}_K(u)$, for almost every $u\in \omega$. Conversely, if $(iii)$ holds, then by (\[eq:area-measure-density\]), $(i)$ and $(ii)$ trivially hold true.
\[l-2-Newton\] Let $K_1$, $K_2$ be convex bodies in ${{\mathbb R}^n}$, satisfying the assumptions of Theorem \[thm-main\] for some open set $U$ in ${\mathbb{ S}^{n-1}}$. Then, for $\lambda>0$, the convex body $\lambda(K_1+K_2)$ also satisfies the assumptions of Theorem \[thm-main\] for $U$.
Notice, first, that by the additivity and homogeneity of the order 1 area measure, we have $S_1(\lambda(K_1+K_2),\cdot)=\lambda S_1(K_1,\cdot)+\lambda S_2(K_2,\cdot)$. Hence, $S_1(\lambda(K_1+K_2),\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous. Moreover, it holds $r^1_{K_i}(u)=\dots=r^{n-1}_{K_i}(u)$, $i=1,2$, for almost every $u\in U$. Thus, $Hess(h_{K_i})(u)=r^1_{K_i}(u)I_{(n-1)\times (n-1)}$, for almost every $u\in U$, where $I_{(n-1)\times(n-1)}$ stands for the $(n-1)\times (n-1)$ identity matrix. This, together with the additivity and homogeneity of the support functional, gives $$\begin{aligned}
Hess(h_{\lambda(K_1+K_2)}(u))&=&Hess(\lambda h_{K_1}+h_{K_2})(u)=\lambda\left(Hess(h_{K_1})(u)+Hess(h_{K_2}(u)\right)\\
&=&\lambda (r^1_{K_1}(u)+r^1_{K_2}(u))I_{(n-1)\times(n-1)},\end{aligned}$$ for almost every $u\in U$, proving our claim.
We will also need two statements from basic measure theory (which of course hold in a much more general setting).
\[l-mt\] Let $\mu,\nu_1,\nu_2,\xi$ be Borel measures on an open set $U$ in ${\mathbb{ S}^{n-1}}$.
i) If $\int_U\varphi d\nu_1\leq \int_U\varphi d\nu_2$, for all continuous non-negative functions $\varphi$ supported on $U$, then $\nu_1\leq \nu_2$.
ii) If $\nu_i=f_id\mu$ (i.e. $\nu_i$ is absolutely continuous with density $f_i$ with respect to $\mu$), $i=1,2$ and $\mu$, $\xi$ are mutually singular measures and $\nu_1\leq \nu_2+\xi$, then $f_1\leq f_2$, $\mu$-almost everywhere.
We only prove $(ii)$, since $(i)$ is well known. Clearly, for $\varepsilon>0$, there exists a Borel set $A_\varepsilon\subseteq U$, such that $\mu(U\setminus A_\varepsilon)<1/\varepsilon$ and $\xi (A_\varepsilon)=0$. Then, for any Borel subset $B$ of $A_\varepsilon$, we have $\int_Bf_1d\mu=\nu_1(B)\leq \nu_2(B)=\int_Bf_2d\mu$. It follows that $f_1|_{A_\varepsilon}\leq f_2|_{A_\varepsilon}$, $\mu$-almost everywhere. Thus, $\mu(\{f_1>f_2\})<1/\varepsilon$ and, since $\varepsilon$ is arbitrary, our assertion follows.
Convex umbilical hypersurfaces
==============================
For the proof of Theorem \[thm-main\], we will show that if some pair $(K,U)$ satisfies the assumptions of the theorem, then $h_K$ is smooth enough. Theorem \[thm-main\] will then follow from Theorem \[thm-old\]. To this end, we will show that $f^{(1)}_K$ actually has to be harmonic on $U$, which by general theory of elliptic PDE’s, will give us the desired regularity of $h_K$.
Symmetrization
--------------
Let $f:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be a non-negative measurable function. The *radial symmetrization* $Sr(f)$ of $f$ with respect to the line ${\mathbb R}e_n$ is defined as follows. $$\label{eq-Sr-def}
Sr(f)(u):=\frac{\int_{\{x_n=u_n\}}f(x)d{{{\cal H}^{n-2}}}(x)}{{{{\cal H}^{n-2}}}(\{x_n=u_n\})}.$$ The operator $S_r(\cdot)$ corresponds to the so-called “Blaschke-Minkowski" symmetrization, when applied to the support function of a convex body. We refer to [@Bi-Ga-Gr] and [@B-F] for more information. In view of Lemma \[l-2-Newton\], one naturally expects that there is some sequence of averages of compositions of $f$ with maps from $O(n,e_n)$ that converges in some sense to $Sr(f)$. Since we are going to need convergence in $L^2$, we will do this process carefully.
It is clear that $Sr(f)$ is invariant under composition with maps from $O(n,e_n)$. Moreover, $Sr(g)=g$, for any function $g$ that is radially symmetric with respect to the line ${\mathbb R}e_n$; that is, $Sr$ is an idempotent operator. Furthermore, an immediate application of Hölder’s inequality yields $$\label{eq-Sr-CS}
Sr(f)(u)\leq (Sr(f^p)(u))^{1/p},\qquad p\geq 1,\qquad u\in {\mathbb{ S}^{n-1}}.$$ Later on, we will need the fact that the $L^1$-norm is preserved under the operator $Sr(\cdot)$ (this is mentioned in [@Bi-Ga-Gr]) and that if $f$ is in $L^2$, then $Sr(f)$ is also in $L^2$. This is done in the following lemma.
\[l-sr-l2\] Let $f:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be a non-negative measurable function. Then, for any $v\in {\mathbb{ S}^{n-1}}\cap e_n^\perp$, it holds $$\label{eq-sr-l2}
\|f\|_{L^1({\mathbb{ S}^{n-1}})}=(n+1)(n-1)\omega_{n-1}\int_{-1}^1\int_0^{\sqrt{1-t^2}}r^{n-2}\sqrt{r^2+t^2}Sr(f)\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right)drdt,$$ where $\omega_n$ is the volume of $B_2^n$. In particular, we have $\|f\|_{L^1({\mathbb{ S}^{n-1}})}=\|Sr(f)\|_{L^1({\mathbb{ S}^{n-1}})}$ and, for $p>1$, $\|f\|_{L^p({\mathbb{ S}^{n-1}})}\geq\|Sr(f)\|_{L^p({\mathbb{ S}^{n-1}})}$.
Fix $v\in {\mathbb{ S}^{n-1}}\cap e_n \equiv {{\mathbb S}^{n-2}}$ and let $r>0$, $t\in{\mathbb R}$, $\gamma\in {{\mathbb S}^{n-2}}$. Since $\langle(r\gamma,t)/|(r\gamma,t)|,e_n\rangle=t/\sqrt{r^2+t^2}$, an easy change of variables implies $$\begin{aligned}
\frac{1}{{{{\cal H}^{n-2}}}({{\mathbb S}^{n-2}})}\int_{{{\mathbb S}^{n-2}}}f\left(\frac{(r\gamma,t)}{|(r\gamma,t)|}\right)d{{{\cal H}^{n-2}}}(\gamma)&=&\frac{\int_{\left\{x_n=t/\sqrt{r^2+t^2}\right\}}f(x)d{{{\cal H}^{n-2}}}(x)}{{{{\cal H}^{n-2}}}(\{x_n=\sqrt{r^2+t^2}\})}\nonumber\\
&=&Sr(f)\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right).\label{eq-l-sr-l2-1}\end{aligned}$$ Extend $f$ to the whole ${{\mathbb R}^n}$, so that $f:{{\mathbb R}^n}\to{\mathbb R}$ is 1-homogeneous. Integrating in polar coordinates, we obtain $$\begin{aligned}
\int_{B_2^n}f(x)dx=\int_{{\mathbb{ S}^{n-1}}}\int_0^1f(r\gamma)r^{n-1}drd{{{\cal H}^{n-1}}}(\gamma)=\int_{{\mathbb{ S}^{n-1}}}f(\gamma)d{{{\cal H}^{n-1}}}(\gamma)\int_0^1r^ndr=\frac{1}{n+1}\int_{{\mathbb{ S}^{n-1}}}f(\gamma)d{{{\cal H}^{n-1}}}(\gamma).\end{aligned}$$ Therefore, using Fubini’s theorem, (\[eq-l-sr-l2-1\]) and again integration in polar coordinates, we get $$\begin{aligned}
\|f\|_{L^1({\mathbb{ S}^{n-1}})}&=&(n+1)\int_{B_2^n}f(x)dx=(n+1)\int_{-1}^1\int_{B_2^n\cap(e_n^\perp+te_n)}f(y,t)dydt\nonumber\\
&=&(n+1)\int_{-1}^1\int_0^{\sqrt{1-t^2}}\int_{{{\mathbb S}^{n-2}}}f(r\gamma,t)d{{{\cal H}^{n-2}}}(\gamma)r^{n-2}drdt\label{eq-l-sr-l2-2}\\
&=&(n+1)\int_{-1}^1\int_0^{\sqrt{1-t^2}}\sqrt{r^2+t^2}r^{n-2}\int_{{{\mathbb S}^{n-2}}}f\left(\frac{(r\gamma,t)}{|(r\gamma,t)|}\right)d{{{\cal H}^{n-2}}}(\gamma)drdt\nonumber\\
&=&(n+1)(n-1)\omega_{n-1}\int_{-1}^1\int_0^{\sqrt{1-t^2}}r^{n-2}\sqrt{r^2+t^2}Sr(f)\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right)drdt,\nonumber\end{aligned}$$ as required. The fact that $\|f\|_{L^1({\mathbb{ S}^{n-1}})}=\|Sr(f)\|_{L^1({\mathbb{ S}^{n-1}})}$ follows immediately from (\[eq-sr-l2\]) and the fact that $Sr$ is idempotent. Similarly, using (\[eq-Sr-CS\]), we get $$\begin{aligned}
\|Sr(f)\|^p_{L^p({\mathbb{ S}^{n-1}})}&=&(n+1)(n-1)\omega_{n-1}\int_{-1}^1\int_0^{\sqrt{1-t^2}}r^{n-2}\sqrt{r^2+t^2}Sr(Sr(f)^p)\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right)drdt\\
&=&(n+1)(n-1)\omega_{n-1}\int_{-1}^1\int_0^{\sqrt{1-t^2}}r^{n-2}\sqrt{r^2+t^2}Sr(f)^p\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right)drdt\\
&\leq &(n+1)(n-1)\omega_{n-1}\int_{-1}^1\int_0^{\sqrt{1-t^2}}r^{n-2}\sqrt{r^2+t^2}Sr(f^p)\left(v+\frac{t}{\sqrt{r^2+t^2}}e_n\right)drdt\\
&=&\|f\|^p_{L^p({\mathbb{ S}^{n-1}})}.\end{aligned}$$
Let $f:{\mathbb{ S}^{n-1}}\to{\mathbb R}$. For $T_1,\dots,T_m\in O(n,e_n)$, define the function $$M(f;T_1,\dots,T_m):=\frac{f\circ T_1+\dots+f\circ T_m}{m}.$$
\[prop-conv-sr\] Let $f_1,\dots,f_k:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be $L^2$-functions. Then, there exists a sequence $T_1^1,\dots ,T^1_{m_1},T_1^2,\dots, T_{m_2}^2,\dots \in O(n,e_n)$, such that $$M(f_i;T_1^j,\dots ,T_{m_j}^j)\xrightarrow{j\to\infty}Sr(f_i),\qquad i=1,\dots,k,$$ in $L^2({\mathbb{ S}^{n-1}})$.
Consider the linear space $X:=(L^2({\mathbb{ S}^{n-1}}))^k$ equipped with the natural norm given by $\|(w_1,\dots,w_k)\|^2=\sum_{i=1}^k\|w_i\|_{L^2({\mathbb{ S}^{n-1}})}$. Then, the pair $(X,\|\cdot\|)$ is a Hilbert space. Define the set $${\cal A}:=\{(M(f_1;T_1\dots,T_m),\dots,M(f_k;T_1\dots,T_m)):m\in \mathbb{N}, \ T_1,\dots,T_m\in O(n,e_n)\}$$ and observe that the closure ${\cal C}:=cl {\cal A}$ (with respect to the norm $\|\cdot\|$) of ${\cal A}$ is a convex set. To see this, notice that since ${\cal A}$ is clearly closed under rational convex combinations, its closure has to be closed under (any) convex combinations. Using a classical result from the theory of Hilbert spaces (see e.g. [@D Chapter 3]), we conclude that there exists a unique element $(g_1,\dots,g_k)\in{\cal C}$, such that $$\|(g_1,\dots,g_k)-(Sr(f_1),\dots,Sr(f_k))\|=\inf\left\{\|(w_1,\dots,w_k)-(Sr(f_1),\dots,Sr(f_k))\|:(w_1,\dots,w_k)\in{\cal C}\right\}=:d.$$ It suffices to prove that $g_i=Sr(f_i)$ almost everywhere in ${\mathbb{ S}^{n-1}}$. Indeed, then there will be a sequence from ${\cal C}$ that converges to $(Sr(f_1),\dots,Sr(f_k))$ in $L^2$. Observe that, by definition, for any $(w_1,\dots,w_k)\in {\cal A}$, it holds $$\int_{\{x_n=t\}}f_i(x)d{{{\cal H}^{n-2}}}(x)=\int_{\{x_n=t\}}w_i(x)d{{{\cal H}^{n-2}}}(x),\qquad i=1,\dots,k,$$for all $t\in[-1,1]$. This shows that $Sr(g_i)=Sr(w_i)=Sr(f_i)$, thus in fact, we only have to prove that $g_i$ is almost everywhere equal to a rotationally symmetric function with respect to the line ${\mathbb R}e_n$, $i=1,\dots,k$. For $u\in{\mathbb{ S}^{n-1}}\cap e_n^\perp$, let $T_u\in O(n,e_n)$ be the reflection with respect to the hyperplane $u^\perp$. Notice that if $(w_1,\dots,w_k)\in{\cal A}$, then the $k$-tuple $(M_u(w_1),\dots,M_u(w_k))$, also belongs to ${\cal A}$, where $M_u(w_i):=M(w_i;Id,T_u)$. Hence, if $\{(w_1^m,\dots,w_k^m)\}_{m=1}^\infty$ is a sequence from ${\cal A}$ that converges to $(g_1,\dots,g_k)$, then the sequence $\{(M_u(w_1^m),\dots,M_u(w_k^m))\}_{m=1}^\infty$ is also from ${\cal A}$ and converges to $(M_u(g_1),\dots,M_u(g_k))$. It follows that $(M_u(g_1),\dots,M_u(g_k))$ is also contained in ${\cal C}$. Using the trivial fact that for any $\varphi\in L^2({\mathbb{ S}^{n-1}})$, it holds $\|\varphi\circ T_u\|_{L^2}=\|\varphi\|_{L^2}$, the fact that $Sr(f_i)=Sr(f_i)\circ T_u$ and the triangle inequality, we obtain $$\begin{aligned}
&&\|(M_u(g_1),\dots,M_u(g_k))-(Sr(f_1),\dots,Sr(f_k))\|\\
&\leq& \frac{1}{2}\|(g_1,\dots,g_k)-(Sr(f_1),\dots,Sr(f_k))\|+\frac{1}{2}\|(g_1\circ T_u,\dots,g_k\circ T_u)-(Sr(f_1)\circ T_u,\dots,Sr(f_k)\circ T_u)\|\\
&=&\frac{1}{2}d+\frac{1}{2}d=d.\end{aligned}$$ It follows that $(M_u(g_1),\dots,M_u(g_k))=(g_1,\dots,g_k)$ (as elements of $X$), thus $g_i\circ T_u=g_i$ almost everywhere in $S^{n-1}$, for all $u\in {\mathbb{ S}^{n-1}}\cap e_n^\perp$. This is enough to prove our claim.
Reduction to surfaces of revolution
-----------------------------------
Let $K$ be a convex body in ${{\mathbb R}^n}$ and $U$ be an open subset of ${\mathbb{ S}^{n-1}}$. For technical reasons, we set $f^{(j)}_{K,U}:=f^{(j)}_K\mathbbm{1}_U$, where $\mathbbm{1}_U$ is the indicator function of $U$ and $j\in\{1,\dots,n-1\}$.
\[l-MK\] Let $K$ be a convex body in ${{\mathbb R}^n}$ and $U=\{x\in{\mathbb{ S}^{n-1}}:x_n>a\}$, for some $0<a<1$. Assume that $S_1(K,\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous and that for almost every direction $u$ in $U$, (\[eq:equal-princ-radii\]) holds. Then, $Sr(h_K)$ is the support function of a convex body of revolution $MK$, which has the properties that $S_1(MK,\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous and that for almost every direction $u$ in $U$, (\[eq:equal-princ-radii\]) holds for $MK$ at $u$.
Without loss of generality we may assume that $K$ contains the origin in its interior. Therefore, there exist Euclidean balls $B_1,B_2$, centered at the origin, such that $B_1\subseteq K\subseteq B_2$. Moreover, by assumption and by Lemma \[l-1-Newton\], we have $f^{(1)}_{K,U}=\left(f^{(2)}_{K,U}\right)^{1/2}$, almost everywhere in $U$. Since $f^{(2)}_{K,U}\in L^1$, it follows that $f^{(1)}_{K,U}\in L^2$. Moreover, by Proposition \[prop-conv-sr\], for $k=2$, there exists a sequence $T_1^1,\dots,T_{m_1}^1,T_1^2,\dots,T^1_{m_2},\dots\in O(n,e_n)$, such that $$h_j:=M(h_K;T_1^j,\dots,T_{m_j}^j)\xrightarrow{j\to\infty}Sr(h_K)$$ and $$M(f^{(1)}_{K,U};T_1^j,\dots,T_{m_j}^j)\xrightarrow{j\to\infty}Sr(f^{(1)}_{K,U})$$ in $L^2$ and (by taking subsequences) almost everywhere. Since $h_j=(1/m_j)(h_{(T_1^j)^\ast}+\dots+h_{(T_{m_j}^j)^\ast})$, $h_j$ is also a support function of some convex body $K_j$, where $B_1\subseteq K_j\subseteq B_2$, $j=1,2,\dots$. Thus, by the Blaschke Selection theorem, by taking a subsequence of $\{K_{_j}\}$ if necessary, we may assume that $\{K_{_j}\}$ converges to some convex body $\overline{MK}$ in the Hausdorff metric. Then, $h_{K_j}\to h_{\overline{MK}}$ (uniformly in ${\mathbb{ S}^{n-1}}$), which shows that $h_{\overline{MK}}=h_{Sr(h_K)}$ and $\overline{MK}=MK$. Next, notice that $$f^{(1)}_{K_j,U}=\frac{f^{(1)}_{(T^j_1)^\ast K,U}+\dots+f^{(1)}_{(T^j_{m_j})^\ast K,U}}{m_j}=M(f^{(1)}_{K,U};T_1^j,\dots,T_{m_j}^j),$$ which converges in $L^2$ and thus weakly to $Sr(f^{(1)}_{K,U})$. This, in particular, shows that $S_1(MK,\cdot)|_{{{\cal B}}(U)}$ is absolutely continuous and that $f^{(1)}_{MK,U}=Sr(f^{(1)}_{K,U})$. Moreover, using Lemma \[l-2-Newton\], we see that $f^{(1)}_{K_j,U}=\left(f^{(2)}_{K_j,U}\right)^{1/2}$, almost everywhere in $U$, thus $f^{(2)}_{K_j,U}$ converges to $Sr(f^{(1)}_{K,U})^2$, almost everywhere in $U$. Let $\varphi:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be any continuous non-negative function, supported inside $U$. Then, by Fatou’s lemma and by the fact that $S_2(K_j,\cdot)$ converges weakly to $S_2(MK,\cdot)$, we get $$\begin{aligned}
\int_{{\mathbb{ S}^{n-1}}}\left(f^{(1)}_{MK,U}\right)^2\varphi d{{{\cal H}^{n-1}}}&=&\int_{{\mathbb{ S}^{n-1}}}\left(Sr(f^{(1)}_{K,U})\right)^2\varphi d{{{\cal H}^{n-1}}}\leq \liminf_{j\to\infty}\int_{{\mathbb{ S}^{n-1}}}f^{(2)}_{K_j,U}\varphi d{{{\cal H}^{n-1}}}\\
&\leq & \liminf_{j\to\infty}\int_{{\mathbb{ S}^{n-1}}}\varphi dS_2(K_j,\cdot)=\int_{{\mathbb{ S}^{n-1}}}\varphi dS_2(MK,\cdot).\end{aligned}$$ Since $\varphi$ is arbitrary, we conclude by Lemma \[l-mt\] $(i)$ that $\left(f^{(1)}_{MK,U}\right)^2d{{{\cal H}^{n-1}}}|_{{{\cal B}}(U)}\leq S_2(MK,\cdot)|_{{{\cal B}}(U)}$, which by Lemma \[l-mt\] $(ii)$ gives $\left(f^{(1)}_{MK,U}\right)^2\leq f^{(2)}_{MK,U}$, almost everywhere in $U$. Thus, using Lemma \[l-1-Newton\], we see that for almost every direction $u$ in $U$, (\[eq:equal-princ-radii\]) holds for $MK$ at $u$, concluding our proof.
\[prop-bodies-of-rev\] Let $K_1,\dots,K_{n-1}$ be convex bodies of revolution with respect to the axis ${\mathbb R}e_n$ and let $U=\{x\in {\mathbb{ S}^{n-1}}:x_1>a\}$, for some $0<a<1$. For $i=1,\dots,n-1$, consider the Borel measure $\mu_i$ on the sphere, given by $$\label{eq-mu_i-def}
\mu_i(\omega)=S_{n-1}(K_i,\omega\cap U)+S_{n-1}(K_i,(-\omega)\cap U).$$ If none of the $K_1,\dots,K_{n-1}$ is a cylinder, then there are uniquely determined symmetric convex bodies $K_1^U,\dots,K_{n-1}^U$ of revolution with respect the the axis ${\mathbb R}e_n$, whose surface area measure equals $\mu_1,\dots,\mu_{n-1}$, respectively and $$\label{eq-mixed-area-bodies-of-rev}
S(K_1^U,\dots,K_{n-1}^U,\omega)=S(K_1,\dots,K_{n-1},\omega\cap U)+S(K_1,\dots,K_{n-1},(-\omega)\cap U),$$ for all $\omega\in{{\cal B}}({\mathbb{ S}^{n-1}})$.
Let $i\in\{1,\dots,n-1\}$. Since $K_i$ is not a cylinder, it is clear that $\mu_i$ is not concentrated on any great subsphere of ${\mathbb{ S}^{n-1}}$. Thus, by the Minkowski Existence and Uniqueness theorem, there exists a unique symmetric body of revolution (since $\mu_i$ is even and rotationally symmetric) $K_i^U$ with respect to the $x_n$-axis, whose surface area measure equals $\mu_i$. There is a simple geometric description of $K_i^U$: Since $U$ is contained in the hemisphere ${\mathbb{ S}^{n-1}}\cap\{x_n>0\}$, there is a continuous, concave, non-increasing function $\varphi_i:[0,d_ie_{n-1}]\to{\mathbb R}$, for some $d_i>0$, such that the surface of revolution $\tau(K_i,U)$ is obtained by revolving the graph of $\varphi_i|_{[0,d_ie_{n-1})}$ about the $x_n$-axis. It follows easily by (\[eq:surface-area-measure\]) that $(\textnormal{bd}K_i^U)\cap \{x_n\geq 0\}$ is obtained by rotating the graph of the function $\widetilde{\varphi_i}:=\varphi_i-\varphi_i(d_i)$ about the $x_n$-axis. In the case that $K_1,\dots,K_{n-1}\in {\cal C}_+^2$, $S(K_1,\dots,K_{n-1},\cdot)|_{{{\cal B}}(U)}$ has density given by (\[eq:mixed-discriminant\]) and since $h_{K_i}$ at any point in $U$ depends only on the function $\varphi_i$, $i=1,\dots,n-1$, it follows that $S(K_1^U,\dots,K_{n-1}^U,\cdot)|_{{{\cal B}}(U)}$ also has density; the same as the density of $S(K_1,\dots,K_{n-1},\cdot)|_{{{\cal B}}(U)}$. In the general case, one can approximate $K_1,\dots,K_{n-1}$ by sequences of ${\cal C}_+^2$ bodies of revolution. Since the corresponding sequence of mixed area measures converges weakly to $S(K_1,\dots,K_{n-1},\cdot)$, we conclude that for any continuous function $\phi:{\mathbb{ S}^{n-1}}\to{\mathbb R}$, supported inside $U$, we have $$\int_U\phi dS(K_1,\dots,K_{n-1},\cdot)=\int_U\phi dS(K_1^U,\dots,K_{n-1}^U,\cdot).$$ Hence, by Lemma \[l-mt\] $(i)$, it follows that $S(K_1^U,\dots,K_{n-1}^U,\omega)=S(K_1,\dots,K_{n-1},\omega)$, for any $\omega\in{{\cal B}}(U)$. The fact that (\[eq-mixed-area-bodies-of-rev\]) holds for all $\omega\in {{\cal B}}(U\cup -U)$ follows trivially by symmetry.
It remains to prove that $S(K_1^U,\dots,K_{n-1}^U,{\mathbb{ S}^{n-1}}\setminus (U\cup-U))=0$. Notice that for any $u\in ({\mathbb{ S}^{n-1}}\setminus U)\cap\textnormal{span}\{e_{n-1},e_n\}\cap\{x_n\geq 0\}$, the intersection of the supporting line to the graph of $\widetilde{\varphi_i}$, whose outer unit normal vector is $u$, with the graph of $\widetilde{\varphi_i}$, contains only the point $d_ie_{n-1}$, $i=1,\dots,n-1$. Hence, by the rotational symmetry and central symmetry of $K_i^U$, we conclude that for any $u\in {\mathbb{ S}^{n-1}}\setminus(U\cup-U)$, it holds $F(K_i^U,u)\subseteq d_iS^{n-1}\cap e_n^\perp$, $i=1,\dots,n-1$. The additivity of the support set functional (\[eq:support-set\]) gives $F(K_1^U+\dots+K_{n-1}^U,u)\subseteq (d_1+\dots+d_{n-1})S^{n-1}\cap e_n^\perp$. In other words, $\tau(K_1^U+\dots K_{n-1}^U,{\mathbb{ S}^{n-1}}\setminus (U\cup-U))=(d_1+\dots+d_{n-1})S^{n-1}\cap e_n^\perp$, which by (\[eq:surface-area-measure\]) gives $S_{n-1}(K_1^U+\dots+K_{n-1}^U,{\mathbb{ S}^{n-1}}\setminus(U\cup-U))=0$. It follows immediately by (\[eq:mixedarea-additivity\]) that $S(K_1^U,\dots,K_{n-1}^U,{\mathbb{ S}^{n-1}}\setminus (U\cup-U))=0$, as asserted.
Regularity
----------
\[l-umb-bodies-of-revolution\] Let $K$ be a convex body in ${{\mathbb R}^n}$ and $U$ be a spherical cap, centered in $e_n$. If $K$ and $U$ satisfy the assumptions of Theorem \[thm-main\], then $Sr(f^{(1)}_{K,U})$ equals to a constant, almost everywhere in $U$.
Recall that by Lemma \[l-MK\], it holds $Sr(f_{K,U}^{(1)})=f^{(1)}_{MK,U}=\left(f^{(2)}_{MK,U}\right)^{1/2}$, almost everywhere in $U$. Also, by Proposition \[prop-bodies-of-rev\], (\[eq:mixarea\]) and the Alesandrov-Fenchel inequality (\[eq:af\]), we have $$\begin{aligned}
\frac{1}{n}\int_Uf^{(1)}_{MK}d{{{\cal H}^{n-1}}}&=&\frac{1}{n}\int_UdS(MK,B_2^n[n-2],\cdot)=\frac{1}{n}\int_UdS((MK)^U,(B_2^n)^U[n-2],\cdot)\\
&=&\frac{1}{2}V(B_2^n,(MK)^U,(B_2^n)^U[n-2])\\
&\geq& \frac{1}{2}\left(V(B_2^n,(MK)^U[2],(B_2^n)^U[n-3])V(B_2^n,(B_2^n)^U[n-1])\right)^{1/2}\\
&=& \left(\frac{1}{n}\int_UdS((MK)^U[2],(B_2^n)^U[n-3],\cdot)\frac{1}{n}\int_UdS((B_2^n)^U[n-1],\cdot)\right)^{1/2}\\
&\geq& \frac{1}{n}\left({{{\cal H}^{n-1}}}(U)\right)^{1/2}\left(\int_Uf^{(2)}_{MK}d{{{\cal H}^{n-1}}}\right)^{1/2}\\
&=&\frac{1}{n}\left({{{\cal H}^{n-1}}}(U)\right)^{1/2}\left(\int_U\left(f^{(1)}_{MK}\right)^2d{{{\cal H}^{n-1}}}\right)^{1/2}\end{aligned}$$ On the other hand, the Cauchy-Schwartz inequality gives $$\label{eq-CS-last}
\int_Uf_{MK}^{(1)}d{{{\cal H}^{n-1}}}\leq \left({{{\cal H}^{n-1}}}(U)\right)^{1/2}\left(\int_U\left(f^{(1)}_{MK}\right)^2d{{{\cal H}^{n-1}}}\right)^{1/2}.$$ Therefore, there must be equality in the Cauchy-Schwartz inequality (\[eq-CS-last\]), which is only possible if $f^{(1)}_{MK}$ is equal to a constant almost everywhere in $U$, proving our claim.
**\
Proof of Theorem \[thm-main\].\
Let $K$, $U$ be as in the statement of Theorem \[thm-main\]. Without loss of generality, we may assume that $U$ is a spherical cap centered at $e_n$. By Lemma \[l-umb-bodies-of-revolution\], $Sr(f_{K,U}^{(1)})=f^{(1)}_{MK,U}$ can be taken to be equal to a constant $c\geq 0$ in $U$. For $u\in U$, define the following quantity (if it exists) $$F(u):=\lim_{\textnormal{diam}(U')\to 0}\frac{\int_{U'}f^{(1)}_{K}d{{{\cal H}^{n-1}}}}{{{{\cal H}^{n-1}}}(U')},$$ where $U'$ runs over all spherical caps $U'\subseteq U$, whose center is $u$. First assume that $u=e_n$ and let $U'\subseteq U$ be a spherical cap centered at $e_n$. Notice, also, that $Sr(f^{(1)}_{K,U'})|_{U'}=Sr(f^{(1)}_{K,U})|_{U'}$. Then, by Lemma \[l-sr-l2\], it follows that $\int_{U'}f^{(1)}_{K}d{{{\cal H}^{n-1}}}=\int_{U'}Sr(f^{(1)}_{K,U})d{{{\cal H}^{n-1}}}=c{{{\cal H}^{n-1}}}(U')$. In particular, $F(e_n)$ exists and equals to $c$. Moreover, notice that if $e_n$ is a Lebesgue point of $f^{(1)}_{K}$, then $F(e_n)=f^{(1)}_{K}(e_n)$. Next, take any spherical cap $V$ inside $U$, centered at some $v\in U$. Since the pair $(K,V)$ also satisfies the assumptions of Theorem \[thm-main\] and since $e_n$ can clearly be replaced by any other point on the sphere, our previous discussion shows that $F(v)$ exists and $$\label{eq-mvp}
\frac{\int_Vf^{(1)}_{K}d{{{\cal H}^{n-1}}}}{{{{\cal H}^{n-1}}}(V)}=F(v),$$ while $F(v)$ equals $f^{(1)}_{K}(v)$ if $v$ is a Lebesgue point of $f^{(1)}_{K}$. In particular, the function $F:U\to{\mathbb R}$ is well defined in $U$. Notice, however, that since almost every $v\in U$ is a Lebesgue point of $f^{(1)}_{K}$, $F$ equals $f^{(1)}_{K}$ almost everywhere in $U$. Thus, by (\[eq-mvp\]), it holds $$\frac{\int_VFd{{{\cal H}^{n-1}}}}{{{{\cal H}^{n-1}}}(V)}=F(v),$$ for all $v\in U$ and for all spherical caps $V\subseteq U$, centered at $v$. Thus, $F$ has the so-called mean value property, which on the sphere (just like in the Euclidean case) implies that $F$ is harmonic [@Wil]. It follows using e.g. [@Ta Proposition 1.6], that $F$ is $C^\infty$-smooth (actually real analytic). Consequently, $f^{(1)}_{K}$ is almost everywhere equal to a $C^\infty$ function in $U$. Since (\[eq:distributions\]) holds in the sense of distributions in $U$, it follows again by [@Ta Proposition 1.6] that $h_K$ is of class $C^\infty$ on $U$. Next, notice that, by Lemma \[l-2-Newton\], the pair $(K+B_2^n,U)$ also satisfies the assumptions of Theorem \[thm-main\]. Since $f^{(1)}_{K+B_2^n}\geq 1>0$ in $U$, it follows that all principal radii of curvature of $K+B_2^n$ are strictly positive, thus (since $h_{K+B_2^n}$ is smooth) as in [@Sc pp 120] we conclude that $\tau(K+B_2^n,U)$ is smooth as a manifold. This, together with (\[eq:rad-curv\]) and Theorem \[thm-old\], shows that $\tau(K+B_2^n)$ is contained in a Euclidean sphere. Therefore, and since $\tau(K+B_2^n,U)$ is open in $\textnormal{bd}K$, we conclude that $h_{K+B_2^n}+\langle a,\cdot\rangle $ is constant on $U$, for some fixed vector $a$, and hence $h_K+\langle a,\cdot\rangle $ is constant on $U$, ending the proof of Theorem \[thm-main\]. $\square$
Even functions with isotropic sections
======================================
A [*zonoid*]{} $Z$ is a convex body whose support function is the cosine transform of some (positive) Borel measure $\mu$ on ${\mathbb{ S}^{n-1}}$. The measure $\mu$ is called the [*generating measure*]{} of $Z$.
Let $Z_1,\dots,Z_{n-1}$ be zonoids in ${{\mathbb R}^n}$ with corresponding generating measures $\mu_1,\dots,\mu_{n-1}$. If $\mu_1,\dots, \mu_{n-1}$ are absolutely continuous with corresponding densities $g_1,\dots,g_{n-1}$, then there is an integral-geometric formula, essentially due to W. Weil [@W] (see also [@Sc Section 5.3]) that gives the density of the mixed area measure $S(Z_1,\dots,Z_{n-1},\cdot)$.
& (u)&
$$\label{eq-Weil-1}
=\frac{2^{n-1}}{(n-1)!}\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\dots\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\textnormal{det}(x_1,\dots,x_{n-1})^2g_1(x_1)\dots g_{n-1}(x_{n-1})d{{{\cal H}^{n-2}}}(x_1)\dots d{{{\cal H}^{n-2}}}(x_{n-1}).$$
In the particular case that $Z_1=\dots=Z_k=Z$, $g_1=\dots=g_k=g$, $Z_{k+1}=\dots=Z_{n-1}=B_2^n$, $k=1,\dots, n-1$, we have $$h_{Z_i}(u)=a_n\int_{{\mathbb{ S}^{n-1}}}|\langle x,u\rangle|d{{{\cal H}^{n-1}}}(x),\qquad\textnormal{where}\qquad a_n=\left(\int_{{\mathbb{ S}^{n-1}}}|x_1|d{{{\cal H}^{n-1}}}(x)\right)^{-1},$$ $i=j+1,\dots,n-1$. Hence, (\[eq-Weil-1\]) becomes $ f^{(j)}_Z(u)=$ $$\label{eq-Weil-2}
\frac{a_n^{n-j-1}2^{n-1}}{(n-1)!}\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\dots\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\textnormal{det}(x_1,\dots,x_{n-1})^2g(x_1)\dots g(x_{j})d{{{\cal H}^{n-2}}}(x_1)\dots d{{{\cal H}^{n-2}}}(x_{n-1}).$$ In particular, area measures of any order of the zonoid $Z$ are absolutely continuous, if the generating measure of $Z$ is absolutely continuous. Notice, also that (\[eq-Weil-2\]) implies that $$\label{eq-Weil-3}
f^{(1)}_Z(u)=b_n\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}g(x)d{{{\cal H}^{n-2}}}(x)=b_n{\cal R}(g),$$ where $b_n>0$ is a constant that depends only on the dimension.
\[l-last-1\] Let $n\geq 3$, $U$ be an open set in ${\mathbb{ S}^{n-1}}$ and $g:{\mathbb{ S}^{n-1}}\to{\mathbb R}_+$ be a bounded measurable function. Denote by $Z(g)$ the zonoid with generating measure $gd{{{\cal H}^{n-1}}}(\cdot)$. The following are equivalent.
i) The restriction $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$ is isotropic for almost every $u\in U$.
ii) For almost every $u\in U$, it holds $$\label{eq-last-s1-s2}
\left(f^{(1)}_{Z(g)}(u)\right)^2=f^{(2)}_{Z(g)}(u).$$
Assume that $(i)$ holds. For any $u\in{\mathbb{ S}^{n-1}}$, for which $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$, it holds (just expand the determinant and use the fact that $\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\langle x,e_i\rangle\langle x,e_j\rangle d{{{\cal H}^{n-2}}}(x)=0$, for $i\neq j$) $$\begin{aligned}
&&\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\dots\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}\textnormal{det}(x_1,\dots,x_{n-1})^2g(x_1)g(x_2)d{{{\cal H}^{n-2}}}(x_1)\dots d{{{\cal H}^{n-2}}}(x_{n-1})\\
&=&c_n\left(\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}g(x)d{{{\cal H}^{n-2}}}(x)\right)^2,\end{aligned}$$ where $c_n$ is a positive constant that depends only on the dimension $n$. Combining with (\[eq-Weil-2\]), (\[eq-Weil-3\]) and the assumption, we arrive at $$\left(f^{(1)}_{Z(g)}(u)\right)^2=d_nf^{(2)}_{Z(g)}(u),$$ for almost every $u\in U$, where $d_n>0$ again depends only on $n$. However, if $g\equiv a_n$ on ${\mathbb{ S}^{n-1}}$, that is $Z(g)=B_2^n$, we already know that (\[eq-last-s1-s2\]) holds, thus $d_n=1$. This proves $(ii)$. The proof that $(ii)$ implies $(i)$ is similar and we omit it.
**\
Proof of Theorem \[thm-counterexample\].\
We know (see [@Sc Theorem 3.5.4]) that if $G:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ is an even smooth enough function, then there exists an even continuous function $w:{\mathbb{ S}^{n-1}}\to{\mathbb R}$, such that $$\label{eq-w(x)}
G={\cal C}(w).$$ Let $V$ be a spherical cap which is disoint from $U$ and $G$ be an even function of class $C^\infty$, such that $G|_U\equiv 1$ and $G|_V\equiv 2$ and let $w$ be the corresponding function in the integral representation (\[eq-w(x)\]). Set $g:=w+1+\max_{u\in S_n}|w(u)|$. Then, $g>0$ on ${\mathbb{ S}^{n-1}}$ and ${\cal C}(g)$ is the support function of the zonoid $Z(g)$, which is constant on the open sets $U$ and $V$. Hence, $\tau(Z(g),U)$ and $\tau(Z(g),V)$ are contained in spheres of radii 1 and 2 respectively. Thus, $f^{(1)}_{Z(g)}(u)=1$, for all $u\in U$ and $f^{(1)}_{Z(g)}(v)=2$, for all $v\in V$. On the other hand, $\left(f^{(1)}_{Z(g)}(u)\right)^2=f^{(2)}_{Z(g)}(u)$, for all $u\in U$. This, together with Lemma \[l-last-1\], shows that $g|_{{\mathbb{ S}^{n-1}}\cap U^\perp}$ is isotropic for all $u\in U$. However, since $U^\perp\cap V^\perp\neq \emptyset$, (\[eq-Weil-3\]) shows that $g$ cannot be constant on $U^\perp$ (or in $V^\perp$). $\square$\
\
Proof of Theorem \[thm-meta-main-1\].\
Let us first extend $g$ to the whole ${\mathbb{ S}^{n-1}}$, so that $f|_{{\mathbb{ S}^{n-1}}\setminus U}\equiv 0$. Since for any two spherical caps $V_1,V_2\subseteq S^{n-1}$, it holds $V_1^\perp\cap V_2^\perp\neq \emptyset$, we may assume that $U$ is a spherical cap. Notice that if $g$ satisfies the assumptions of Theorem \[thm-meta-main-1\], then $g+c$ also satisfies the assumptions of Theorem \[thm-meta-main-1\], so since $g$ is bounded, we may assume $g$ to be non-negative. Denote, again, by $Z(g)$ the zonoid with generating measure $gd{{{\cal H}^{n-1}}}(\cdot)$. Lemma \[l-last-1\] and the assumption show that $$\left(f^{(1)}_{Z(g)}\right)^2=f^{(2)}_{Z(g)} \ ,$$ almost everywhere in $U$. Since $S_1(Z(g),\cdot)$ is absolutely continuous, it follows by Theorem \[thm-main\] that $\tau(Z(g),U)$ is contained in a sphere. In particular, ${\cal C}(g)|_U=h_{Z(g)}|_U=c+\langle a,\cdot\rangle$ and $b_n{\cal R}(g)=f^{(1)}_{Z(g)}=c'$, for some constants $c,c'>0$ and for some vector $a\in{{\mathbb R}^n}$. $\square$\
\
Before ending this note, we would like to state, under some regularity assumptions on $g$, a local version of Theorem \[thm-meta-main-1\].
\[thm-meta-main-2\] Let $n\geq 4$ and $g:{\mathbb{ S}^{n-1}}\to{\mathbb R}$ be a smooth enough function, so that the cosine transform of the measure $gd{{{\cal H}^{n-1}}}(\cdot)$ is of class $C^2$. Assume, furthermore, that there exist $k\geq 3$, $H\in G_{n,k}$ and an open set $U$ in $H$, such that $g|_{{\mathbb{ S}^{n-1}}\cap u^\perp}$ is isotropic, for all $u\in U$. Then, $({\cal R}g)|_U$ is constant.
Again, we may assume that $g>0$. Then, $Z(g)$ is of class ${\cal C}^2_+$ (the same holds of course for $Z(g)|H$) and therefore it is meaningful to consider (\[eq:equal-princ-radii\]) for $Z(g)$ pointwise. Let $u\in U$. As in Lemma \[l-last-1\], we see that (\[eq:equal-princ-radii\]) holds for $Z(g)$ at $u$. Let $\{\varepsilon_1,\dots,\varepsilon_{k-1}\}$ be an orthonormal basis of $H\cap u^\perp$ and extend it to an orthonormal basis $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$ of $u^\perp$. It holds $$Hess (h_Z(g))(u)_{(n-1)\times(n-1)}=r(u)I_{(n-1)\times(n-1)},$$ where the differentiation is with respect to the basis $\{\varepsilon_1,\dots,\varepsilon_{n-1}\}$ (or any orthonormal basis in $u^\perp$) and $r(u)>0$ is the common value of the principal radii of $\textnormal{bd}Z(g)$ at $u$. This shows that $Hess (h_{(Z|H)(g)})(u)_{(k-1)\times(k-1)}$ is also $r(u)$ times the $(k-1)\times(k-1)$ identity matrix, when the differentiation is with respect to the basis $\{\varepsilon_1,\dots,\varepsilon_{k-1}\}$. Consequently, for any $u\in U$, (\[eq:equal-princ-radii\]) holds for $Z(g)|H$ at $u$. Using Theorem \[thm-main\], we conclude that $\tau(Z(g)|H,U)$ is contained in a $k$-dimensional sphere, thus $r(u)$ is constant in $U$. Finally, as in the proof of Lemma \[l-last-1\], one can easily see that $$r(u)=\frac{1}{n-1}\int_{{\mathbb{ S}^{n-1}}\cap u^\perp}g(x)d{{{\cal H}^{n-2}}}(x),$$ which by Theorem \[thm-old\] completes our proof.
**\
[**Acknowledgement.**]{} We are grateful to Daniel Hug for his help and interest in this work and for discovering errors in the statement and proof of previous version of Theorem \[thm-main\]. In particular, Example \[ex-1\] is due to him. We would also like to thank Dmitry Ryabogin for some excellent discussions concerning problems related to Problem \[prob1\] and Andreas Savas-Halilaj for providing us references [@Doc] and [@S-V] and for related discussions.
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Ioannins Purnaras\
Department of Mathematics\
University of Ioannina\
Ioannina, Greece, 45110\
E-mail address: [email protected]
Christos Saroglou\
Department of Mathematics\
University of Ioannina\
Ioannina, Greece, 45110\
E-mail address: [email protected] & [email protected]
|
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abstract: |
=0.7cm
Atomic and molecular transitions for high Z objects in the early universe give bounds to the possible existence of extra spatial dimensions $D = 3+\epsilon$. We review the theory and present observational data, based on Lyman and Balmer hydrogen transitions of distant quasars and CO rotational transitions of far away giant molecular clouds.
---
USM-TH-146
[Review of Spectroscopic Determination of Extra Spatial Dimensions in the Early Universe]{}
Douglas J. Buettner\
Aerospace Corporation\
P.O. Box 92957 - M1/112\
Los Angeles, CA 90009-2957 [^1]
P.D. Morley\
General Dynamics Corporation\
National Systems Group\
14700 Lee Road Chantilly, VA 20151[^2]
Ivan Schmidt\
Department of Physics\
Universidad Técnica Federico Santa María\
Casilla 110-V, Valparaíso, Chile[^3]
The work of I. S. is supported in part by Fondecyt (Chile) grant 1030355.
Introduction
============
We live in a four dimensional space-time world. This has been checked experimentally with great precision \[1\]. Nevertheless, in principle there does not seem to be a reason for this, and in fact the universe might have any number of dimensions.
The physics of extra dimensions began with the work of Kaluza and Klein. They proposed uniting Maxwell’s theory of electromagnetism and Einstein’s theory of gravitation by embedding them into a generally covariant five-dimensional space-time, whose fifth dimension was curled up into a tiny ring which was not experimentally observable. More complicated non-abelian theories can be obtained in much the same way, by starting with more dimensions and compactifying them in various ways.
In recent years the idea of extra dimensions has been resurrected. The main reason is that the leading candidate for providing a framework in which to build a theory which unifies all interactions, superstrings, has been found to be mathematically consistent only if there are six or seven extra spatial dimensions. Otherwise the theory is anomalous. In conventional string theory the compactification of these extra dimensions occurs at very high scales, close to the Planck scale. Here the extra dimensions manifest themselves mainly through threshold effects of heavy states with masses close to the Planck mass. These models contain the Standard Model (SM) of particle interactions gauge group, and can also accommodate the Minimal Supersymmetric Standard Model. Their main interesting feature is that there is an automatic unification of gauge couplings at a scale $M_{un}$. Nevertheless, there is still a hierarchy problem because we have no way of explaining why the scales of particle physics are so different from those of gravity.
In the last years there have been several attempts to solve the hierarchy problem by using extra dimensions \[2\]. If space-time is fundamentally $(4+n)$-dimensional, the $4$-dimensional Planck mass $M_{Pl}^{(4)} \simeq 1.22 \times 10 ^{19} GeV$ depends on the $(4+n)$- dimensional Planck mass $M_{Pl}$ and the volume $V_c$ of the compact extra dimensions through
$$M_{Pl}^{(4)^2} = M_{Pl}^{n+2} V_c.$$
Since no extra dimensions have been detected experimentally, the compactification scale ($\sim 1/V_c^{1/n}$) would have to be much smaller than the weak scale $(100 GeV)^{-1}$, and the particles and forces of the SM (except for gravity) must be confined to the four dimensional world volume of a three-brane. By taking $V_c$ large enough it is possible to eliminate the hierarchy problem between the weak and Planck scales.
In all the above scenarios it is essential to know as precise as possible the number of space dimensions. Since at present this number is three \[1\], and since in the very early universe there exists the possibility of a larger number, then this indicates that it should be possible to define an effective number of space dimensions which changes continuously with time, from its very early universe value to the present one. Moreover, by looking at relics of the early universe, such as the cosmic microwave background or the light emitted by quasars with a very high redshift, it should be possible to measure deviations of the number of spatial dimensions, $D$, from its present (epoch) value of 3. This review looks at the spectroscopic method of determining $\epsilon = D -3$ in the early universe.
Quantum Mechanical Energy Levels in D-dimensions
================================================
The spacing of atomic and molecular energy levels varies with the dimension of space. An intuitive way to understand this is the following. Both the kinetic energy term and the Coulomb potential in the Schrödinger equation are related to the Laplacian operator. This operator has the property of measuring the difference between the average value of the field in the immediate neighborhood of a point and the precise value of the field at the point. How the field spreads out from the point depends on the dimension of space it resides in. In particular, $\epsilon$ differences between two fractal geometries give rise to corresponding $\epsilon$ differences in their Laplacians.
Under some circumstances, it is possible to solve the Schrödinger equation in $D =
3+\epsilon$ dimension of space ($4 + \epsilon$ dimension of space-time), using a Taylor expansion \[3\] about $D = 3$
$$<|E|>|_{D=3+ \epsilon} =<|E|>|_{D=3} + \frac{d <|E|> }{d
D}|_{D=3}\epsilon + \cdots \; .$$
The Hellmann-Feynman theorem is
$$\frac{d <|E|> }{d D}|_{D=3}=<| \frac{\partial H}{\partial
D}|_{D=3} |>$$
where $H$ is the D-dimensional Hamiltonian. In the succeeding sections, we will give the quantum mechanical hydrogen transitions and the linear molecular rotational energies in D-dimensions, but first we turn to the discussion of the technique needed to measure $\epsilon$.
How to Determine $\epsilon$
===========================
Using ancient light, we are faced with the problem of disentangling the cosmological redshift to obtain the light’s true rest-frame wavelength, $\lambda_{epoch}$, associated with the earlier time-epoch. In order to do this, we use the fact that all light from the same astronomical object has the same cosmological $Z_{c}$:
$$Z_{c} = (\lambda_{observed} - \lambda_{epoch}) / \lambda_{epoch} \; .$$
In this equation, we are trying to deduce $\lambda_{epoch}$ to see if $\lambda_{epoch}\neq \lambda_{present}$. Clearly, an observation of a single line, even if that line can be identified (e.g. Ly$_{\alpha}$), cannot determine $\lambda_{epoch}$ because the value of $Z_{c}$ is unknown. The problem is solvable if we have [*two*]{} lines from the same object, which then allows us to uniquely determine $\lambda_{epoch}$, and thus $\epsilon$:
$$\epsilon = \frac{\tau_{M} \lambda_{1} - \lambda_{0}}
{a - b \tau_{M}}$$
where
$$\tau_{M} = \frac{\lambda_{0M}}{\lambda_{1M}}$$
with $\lambda_{0M}$, $\lambda_{1M}$ the two measured redshifted lines, which have early universe epoch rest-frame wavelengths $\lambda_{0} + a\epsilon$, $\lambda_{1}+b\epsilon$, where $\lambda_{0}$, $\lambda_{1}$ are the present epoch (i.e. laboratory) transition wavelengths. In this equation, $a$ and $b$ are determined by solving the Schrödinger equation in $D = 3+\epsilon$ dimensions.
Thus the game is to find observational data of high quality that has at least two identifiable transitions from the same deep-space emitter.
Rotational Energy Levels in D-dimensions
========================================
We are interested in linear molecules, such as CO, because, being the simplest molecules, they are the ones most likely to be identified in distant, giant molecular clouds. Also, it turns out that the Schrödinger equation in $D = 3+\epsilon$ dimensions is solvable in this case.
The Hamiltonian is
$$H_{rot} = B(L)L^{2}$$
where $L$ is the body (molecule)-fixed rotational angular momentum and $B$ is the principal rotational constant (equal to 1/2I, where I is the principal moment of inertia). Because of centrifugal stretching, $B = B(L)$. We need to generalize this energy to $D =
3 + \epsilon$ fractal space. In quantum mechanics, $L^{2}$ is a second order Casimir invariant operator, $C_2$. Thus the generalized rotational operator is
$$H_{rot} = B(C_{2})C_{2}$$
The piece $B(C_{2})$ has a simple form when no vibrations are excited \[4\]
$$B(C_{2}) = B_{0} + B_{1} C_{2} \; .$$
In general, the second order Casimir invariant is \[5\]:
$$C_2 = f^i_{jk} f^j_{il} X^k X^l = H_i G_{ij} H_j + \sum_{\rm all\
roots} E^{\alpha} E_{-\alpha}$$
where $f^i_{jk}$ are the structure constants, $X_k$ are generators, and $C_2$ commutes with all generators. The Racah formula for the eigenvalue of $C_2$ for any irreducible representation is easily derived by letting $C_2$ act on the state with highest weight $\Lambda$. The result is:
$$C_2 = (\Lambda,\Lambda+2\delta)$$
where $\delta=(1,1,....,1)$ in the Dynkin basis. The scalar product of any two weights can be written as:
$$(\Lambda,\Lambda') = \sum_{ij} a'_j G_{ij} a _j$$
where $G_{ij}$ is a symmetric tensor whose elements can be computed for each simple group, and which are given in Table 7 of \[5\], and the $a_i$ are the Dynkin components of $\Lambda$.
In our case we want to find the Casimir invariant for the $(L,0,
...0)$ totally symmetric representation. The choice of representation depends, of course, on the way in which we want to generalize angular momentum, and the correct choice, we argue, would preserve the symmetry properties (in this case this would mean to keep the completely symmetric coupling) when generalizing to larger dimensions.
For odd space dimensions $D=2n+1, n=1,2,...$, the algebra is $B_n$, and for even space dimensions $D = 2n, n=1,2,...$, it is $D_n$. We calculate $C_2$, using the equations above, and the $a_j$ values given by $(L,0, ...0)$:
$$C_2 = (L,0,....,0) G(B_n\ {\rm or}\ D_n) (2+L,2,2,....,2)$$
which gives, by simple matrix multiplication, both for $B_n$ and $D_n$:
$$C_2 = L (L+D-2) \ .$$
The pure (no vibrational quanta) rotational energies in $D$-dimension space are then:
$${\cal H}_{rot} = [B_{0} + B_{1}(L(L+1))]L(L+1) + \{2B_{1}L^{2}(L+1) + B_{0}L \} \epsilon \; .$$
Hydrogen Lyman Transitions
==========================
The hydrogen atom is the only other system that is amendable to solution in $D = 3 + \epsilon$ dimensions. By means of the Taylor expansion, we obtain in Table 1 the Lyman transitions as a function of $\epsilon$, for $\epsilon \ll 1$. If $\epsilon \neq 0$, the result is a change in each transition by a unique amount. The effect is unmistakable: even a small shift $\epsilon \sim 0.03$ will be detectable.
Lyman data has been analyzed in reference \[3\]. Except for anomalous data associated with one data set @ $Z_{c} \sim 4$, $<\epsilon> \approx 0$. High $Z_{c}$ quasars are extremely rare, but the Sloan Digital Sky Survey (http://www.sdss.org) has discovered several $Z_{c} \geq
5$. Looking at each Sloan spectrogram, one has difficulty in identifying the original center-line of the Lyman series. The spectrum is a superposition of a priori unknown set of cosmological red shifts. Even the largest or primary shift introduces a complicated transposed spectrum. Next, uneven absorption of radiation as it moves through one intergalactic cloud to another can remove completely the center of a spectral line or one of its wings and may lead to complete masking of the location of the original radiated line. Lastly, line overlaps, which are such a rare occurrence in terrestrial plasmas is the norm for high $Z_{c}$ spectra. These circumstances have lead to the observation that no astronomy group has been able to locate the matching (same cloud or put in another way, identical $Z_{c}$ values) Lyman series Ly$_{\alpha}$ to Ly$_{\epsilon}$ in one $Z_{c} \geq 4$ spectrogram. Due to absorption, the Lyman series in hydrogen becomes difficult to use for $<\epsilon>$ determination for very high $Z_{c}\geq 4$.
[cc]{}\
Lyman line & wavelength\
&\
Ly$_{\alpha}$ & 1215.67 + $1418.27\epsilon$\
Ly$_{\beta}$ & 1025.72 + $1111.18\epsilon$\
Ly$_{\gamma}$ & 972.537 + 1021.155$\epsilon$\
Ly$_{\delta}$ & 949.743 + $981.391\epsilon$\
Ly$_{\epsilon}$ & 937.803 + $960.122\epsilon$
Hydrogen Balmer Transitions
===========================
The Balmer lines start off in the optical (for small $\epsilon \ll
1$) and get redshifted to the infrared for high $Z_{c}$. By using the atmospheric infrared windows where absorption is small, the infrared Balmer lines can evade the difficulties that potentially plague the Lyman series. In Table 2, we give the hydrogen Balmer epsilon-dependent transitions. The measured redshift of $(Z_{c}+1)\lambda_{0}$ means, for example, a $Z_{c}=3$ quasar has Balmer lines in the infrared.
Various online databases were searched for papers containing emission spectra from high Z$_{c}$ quasars that contained coincident hydrogen Balmer lines. Of the many references identified in these databases, only one had tabular data \[6\]. Another, \[7\], had a single Balmer line pair measured. In the future, infrared data will be more plentiful as various observatories bring on-line sophisticated infrared spectrometers. We use eq(4) above to determine $\epsilon$. The uncertainty in $\epsilon$, $\delta \epsilon$, is determined by the equation in reference \[3\].
[cc]{}\
Balmer line & wavelength\
&\
H$_{\alpha}$ & 6562.8 + $4155.24\epsilon$\
H$_{\beta}$ & 4861.36 + $2834.99\epsilon$\
H$_{\gamma}$ & 4340.49 + $2417.58\epsilon$
[cccccc]{}\
Quasar & transition & wavelength (Å) & FWHM (km/s) & FWHM (Å) & $\sigma$\
& & & & &\
Q0007-000 & $H_{\alpha}$ & 21572.1 & 4500 & 323.8 & 116.6\
& $H_{\beta}$ & 15896.5 & 3500 & 185.6 & 66.9\
Q0027+018 & $H_{\alpha}$ & 21952.7 & 4500 & 329.5 & 118.7\
& $H_{\beta}$ & 16091 & 4500 & 241.5 & 87.0\
Q0237-233 & $H_{\alpha}$ & 21152.1 & 7500 & 529.2 & 190.6\
& $H_{\beta}$ & 15702.1 & 3000 & 157.1 & 56.6\
Q1623-268 & $H_{\alpha}$ & 23258.7 & 2600 & 201.7 & 72.7\
& $H_{\beta}$ & 17063.3 & 2500 & 142.3 & 51.3\
Q1816+475 & $H_{\alpha}$ & 21132.4 & 2600 & 183.3 & 66.0\
& $H_{\beta}$ & 15653.5 & 2600 & 135.8 & 48.9
[cccc]{}\
Quasar & transition & wavelength (Ångstroms) & $\sigma$\
& & &\
1937-101 & $H_{\alpha}$ & 23228.2 & 57.2\
& $H_{\beta}$ & 20708.6 & 64.2
It is seen that $<\epsilon> \simeq 0$ is favored by this small sample of $Z_{c} \sim 3$ Balmer data. Balmer data for $Z_{c} \simeq 4$ would be especially interesting.
[ccc]{}\
Quasar & $\epsilon$ & $\Delta\epsilon$\
& &\
Q0007-000 & 0.11 & 0.15\
Q0027+018 & 0.24 & 0.2\
Q0237-233 & -0.04 & 0.18\
Q1623-268 & 0.22 & 0.12\
Q1816+475 & 0 & 0.09\
1937-101 & 0.059 & 0.16
Molecular Rotational Transitions
================================
The Balmer quasar spectra allow a better determination of $<\epsilon>$ than the Lyman data for high $Z_{c}$ objects, but the best spectroscopic data would be lines starting off already in the microwave, and redshifted towards the radio, namely rotational spectra.
The linear molecule $CO$ ($C^{12}O^{16}$) is the main non-hydrogen emitter in distant, giant molecular clouds. In Table 6, we give its (present epoch) laboratory transition frequencies \[8\]. It is a simple matter to determine the constants that appear in eq(8): $B_{0}=57.635968 \; {\rm GHz}$ and $B_{1}=-1.835 \times 10^{-4} \; {\rm GHz}$ for $CO$. By doing a literature search, we have identified 6 tabulations of $CO$ data that are usable, including data from the presently known farthest quasar J114816.64+525150.3 @ Z$_{c}$ = 6.42.
[cc]{}\
transition & MHz\
&\
$1 \rightarrow 0$ & 115271.202\
$2 \rightarrow 1$ & 230538\
$3 \rightarrow 2$ & 345795.991\
$4 \rightarrow 3$ & 461040.77\
$5 \rightarrow 4$ & 576267.922\
$6 \rightarrow 5$ & 691473.09\
$7 \rightarrow 6$ & 806651.719
[ccc]{}\
identification & observed transitions & source\
& &\
QSO J114816.64+525150.3 & $7 \rightarrow 6, 6 \rightarrow 5, 3 \rightarrow 2$ & \[9\]\
QSO H1413+117 & $5 \rightarrow 4, 4 \rightarrow 3, 3 \rightarrow 2$ & \[10\]\
QSO PSS 2322+1944 & $5 \rightarrow 4, 4 \rightarrow 3, 2 \rightarrow 1, 1 \rightarrow 0$ & \[11\]\
QSO BR1202-0725 & $7 \rightarrow 6, 5 \rightarrow 4$ & \[12\]\
QSO F10214+4724 & $6 \rightarrow 5, 3 \rightarrow 2$ & \[12\]\
QSO HR10 (J164502+4626.4) & $5 \rightarrow 4, 2 \rightarrow 1$ & \[13\]
[cccccc]{}\
QSO & approx Z$_{c}$ & transition & obs (Ghz) & FWHM & channel width\
& & & & &\
J114816.64+525150.3 & 6.42 & $7 \rightarrow 6$ & 108.729 & 279 km/s & 5 MHz\
& & $6 \rightarrow 5$ & 93.204 & 279 km/s & 5 MHz\
& & $3 \rightarrow 2$ & 46.61 & 320 km/s & 50 MHz\
H1413+117 & 2.56 & $5 \rightarrow 4$ & 161.964 & 398 km/s & 512 MHz\
& & $4 \rightarrow 3$ & 129.576 & 375 km/s & 512 MHz\
& & $3 \rightarrow 2$ & 97.199 & 362 km/s & 512 MHz\
PSS 2322+1944 & 4.12 & $5 \rightarrow 4$ & 112.55 & 273 km/s & 35 MHz\
& & $4 \rightarrow 3$ & 90.05 & 375 km/s & 35 MHz\
& & $2 \rightarrow 1$ & 45.035 & 200 km/s & 6.25 MHz\
& & $1 \rightarrow 0$ & 22.515 & 200 km/s & 50 MHz\
BR1202-0725 & 4.71 & $7 \rightarrow 6$ & 141.2 & 300 km/s & equivalent 60 km/s\
& & $5 \rightarrow 4$ & 101.3 & 350 km/s & equivalent 60 km/s\
F10214+4724 & 2.29 & $6 \rightarrow 5$ & 210.5 & 300 km/s & equivalent 80 km/s\
& & $3 \rightarrow 2$ & 105.2 & 220 km/s & equivalent 80 km/s\
HR10 & 1.44 & $5 \rightarrow 4$ & 235.982 & 380 km/s & equivalent 75 km/s\
& & $2 \rightarrow 1$ & 94.405 & 400 km/s & equivalent 50 km/s
[cccc]{}\
QSO & line pairs & $\epsilon$ & $\Delta \epsilon$\
& & &\
J114816.64+525150.3 & 7 $\rightarrow$ 6, 6 $\rightarrow$ 5 & -0.000012043 & 0.118937801\
& 7 $\rightarrow$ 6, 3 $\rightarrow$ 2 & -0.000004379 & 0.084521824\
& 6 $\rightarrow$ 5, 3 $\rightarrow$ 2 & -0.000003283 & 0.096631339\
H1413+117 & 5 $\rightarrow$ 4, 4 $\rightarrow$ 3 & -8.1128E-04 & 7.2375E-01\
& 5 $\rightarrow$ 4, 3 $\rightarrow$ 2 & 1.6508E-03 & 3.2882E-01\
& 4 $\rightarrow$ 3, 3 $\rightarrow$ 2 & 3.1291E-03 & 5.6347E-01\
PSS 2322+1944 & 5 $\rightarrow$ 4, 4 $\rightarrow$ 3 & 2.1489E-03 & 1.1919E-01\
& 5 $\rightarrow$ 4, 2 $\rightarrow$ 1 & 1.3294E-03 & 1.3841E-02\
& 5 $\rightarrow$ 4, 1 $\rightarrow$ 0 & 1.7313E-04 & 2.8838E-02\
& 4 $\rightarrow$ 3, 2 $\rightarrow$ 1 & 1.1654E-03 & 2.0347E-02\
& 4 $\rightarrow$ 3, 1 $\rightarrow$ 0 & 4.1410E-05 & 3.1010E-02\
& 2 $\rightarrow$ 1, 1 $\rightarrow$ 0 & -5.2042E-04 & 4.5728E-02\
BR1202-0725 & 7 $\rightarrow$ 6, 5 $\rightarrow$ 4 & 0.149826 & 0.0765425\
F10214+4724 & 6 $\rightarrow$ 5, 3 $\rightarrow$ 2 & -0.00775587 & 0.0210779\
HR10 & 5 $\rightarrow$ 4, 2 $\rightarrow$ 1 & -3.00361E-05 & 0.0132079
Conclusion
==========
We take the high resolution data @ Z = 6.42 and perform the statistical Z-test \[14\] by taking $< \Delta \epsilon> $ as the standard deviation. This predicts that the probability of $\epsilon \neq 0$ is 1 in 7794, only 850 million years (using the standard cosmology) after the Big Bang.
The experimental spectroscopic data from ancient light shows that the dimension of space was 3 (present value) very soon after the Big Bang. The extra dimensions that some theories predict must either occur at very early time or somehow be restricted such that ordinary baryonic matter cannot couple to it.
Zeilinger, Anton and Svozil, Karl, 1985, Phy. Rev. Lett., 54, 2553; Müller, Berndt and Schäfer, Andreas, 1986, Phys. Rev. Lett., 56, 1215; 1986, J. Phys. A, 19, 3891. For a review of extra dimensions physics see: Hewett, JoAnne and Spiropulu, Maria, 2002, Ann. Rev. Nucl. Part. Sci. 52, 397 and references therein. Buettner, Douglas J., Morley, P. D. and Schmidt, Ivan, 2000, Phys. Lett., B474, 41. Kroto, H. W., 1992, Molecular Rotation Spectra, Dover Publications, New York. Slansky, R., 1981, Phys. Rep., 79, 1. Baker, A.C., et al., 1996, Mon. Not. R. Astron. Soc. 282, 704 and Erratum. Taniguchi, Y., et al., arXiv: astro-ph/ 9705075. http://physics.nist.gov. Bertoldi, F. et al., 2003, Astronomy & Astrophysics (in press), arXiv: astro-ph/0307408; Walter, Fabian et al, 2003, Nature, 424, 406, arXiv: astro-ph/0307410. Barvainis, Richard, et al., 1997, arXiv: astro-ph/9702118. Cox, P. et al., 2002, arXiv: astro-ph/0203355; Carilli, C.L., et al., 2002, arXiv: astro-ph/0204253. Omont, Alain et al, 1996, arXiv: astro-ph/9608006. Andreani, Paola et al, 2000, arXiv: astro-ph/0001254. Kanji, Gopal K., 1999, 100 Statistical Tests, Sage publications, Thousand Oaks, California.
[^1]: e-mail address: [email protected]
[^2]: e-mail address: [email protected]
[^3]: e-mail address: [email protected]
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Introduction
============
Natural surfaces with texture provide organisms the ability to control liquid transport in fascinating ways. For example, lotus leaves have hierarchically structured surfaces that ease droplet removal[@Liu2017NRM; @GUO20071103]. The “pitcher plant” [*Nepenthes alata*]{} has wettable asymmetric microridges on their peristomes that make the surface slippery and aid in catching insects efficiently[@Chen2016; @Bohn2004]. Ryegrass leaves have inclined protrusions to help to shed off water unidirectionally[@Guo2012]. Cacti in deserts gather water through thorns with surface structures that collect fog effectively[@Ju2012].
Understanding how liquids act on non-smooth surfaces has also become increasingly important in technological processes, such as printing, coating, adhesion, mixing, and sorting[@Cohen1]. In particular, natural surfaces have inspired the fabrication of micro- and nanoscopic topographies where asymmetric structures are introduced to enable capillary-driven directional liquid transport[@Chu2010; @Holmes2015; @Malvadkar2010]. Chu *et. al.*[@Chu2010] first demonstrated that droplets can spread unidirectionally on arrays of asymmetric nanorods. Various applications of asymmetric-structured surfaces have been reported since, including mixing in microreactors[@Lin2018], oil-water separation[@Li2016], and transport in microfluidic channels[@Zhang2017].
The spreading of a liquid droplet over a textured surface has thus far been analyzed in the slow late spreading regime ($\approx$ 100 ms to few seconds)[@Bonn2009; @Tanner1979; @Chu2010; @Lin2018; @Yu2018]. The details and mechanisms of such spreading have been described in terms of contact-line pinning[@Yu2018; @Chen2016], Laplace pressure[@Yu2018; @Si2018; @Lin2018], and gradients in surface energy[@Shastry2006; @Liu2017]. Also, numerical simulations have provided valuable insights into the spreading mechanisms in the late regime[@Cavalli2013; @Chamakos2016]. However, the rapid wetting that precedes this late spreading is not well understood. Rapid wetting is crucial for droplet impact on solid surfaces[@Li2016_Janus; @Agapov2014b], or in situations where time scales are imposed externally, such as wetting phenomena in the presence of vibrations[@Habibi2016].
Here, we theoretically, experimentally and numerically investigate rapid wetting on microstructures. We consider the dynamics of the leading contact line on slanted ridges in a two dimensional plane. We elucidate how the contact line follows the microstructures and explain and predict the effect of the geometry on the spreading speed.
![ Spreading regimes during wetting of microstructured surfaces. Cross section of a hypothetical structured surface (grey) as a liquid front moves from left to right. Dashed blue lines mark the instantaneous position of the liquid-air interface and blue arrows indicate the movement direction of the interface. The surface that is wetted by the liquid is marked with red. The contact line spreads on flat and downward-sloping sections, whereas it may stick and/or leap at sections with corners. []{data-label="fig:map:a"}](Sketch_softmatter.pdf){width="47.00000%"}
We hypothesize that three wetting regimes determine the contact line velocity and that wetting can be understood from the succession of these regimes as the contact line travels over the surface (Fig. \[fig:map:a\]). The first regime, called [*spread*]{}, is the movement of the contact line on a homogeneous surface with a velocity determined by surface chemistry and liquid-gas properties[@Vo2018; @Do-Quang2015]. The second regime, called [*stick*]{}, refers to the pinning of the contact line on surface corners, resulting in a temporarily stationary contact line[@Mori1982]. Both spread and stick regimes are well understood[@Bird2008; @Tanner1979; @Mori1982] in contrast to the third phenomenon we observed and that we call [*leap*]{}. Contact line leaping involves the establishment of a new contact line downstream of the wetting front when the liquid leaps over specific sections of the solid surface, trapping gas between the solid and the liquid. As the surface wets, the wetting regimes can follow each other in any order, as illustrated in Fig. \[fig:map:a\].
Experiments and methods
=======================
Experimental set-up
-------------------
Spontaneous droplet spreading is recorded with a high-speed camera (DANTEK dynamics Speedsense M) at a frame rate of 52,044 per second. A water droplet develops from a needle with outer diameter of 0.31 mm (Hamilton, gauge 30, point style 3) and starts to spread spontaneously immediately after it touches a substrate. Water is pumped by a syringe pump (Cetoni, neMESYS 1000N) at a flow rate 0.04 $\mu$l/s. The flow rate is so small that a static state is assumed before the droplet touches the surface. The initial radius of the droplet $R_0$ is determined by the distance between the needle and the surface and fixed to 0.4 mm. The scope is composed of 128 $\times$128 pixels, which leads to the spatial resolution $\approx$8$\mu$m. As seen in Fig. \[fig:spread\](c), owning to the limited spatial resolution, the observed droplet has finite spreading radius even before it actually touches the substrate. In order to estimate the initial time $t_{0}$ in which the droplet starts to spread, a power-law curve $r= C(t-t_{0})^{\gamma} $ is fitted to the time history of observed spreading radius where $C$, $\gamma$, and $t_0$ are scalar parameters. The initial time is then determined by shifting the original spreading time by $t_0$.
Sample preparation
------------------
The microstructured surfaces are made from Ostemer 220 (Mercene Labs, Stockholm, Sweden), a UV-curing Off-Stoichiometry-Thiol-Ene (OSTE) resin, with excellent lithographic patterning, previously used to create complex slanted structures[@Hansson2016]. The samples are manufactured as follows. First, a flat Ostemer layer is manufactured on a smooth plastic film. Second, slanted Ostemer ridges are patterned on the base layer by exposing slanted UV light through a patterned photomask. After development in Aceton, a surface modification is performed to hydrophilize the surface (equilibrium contact angle $\theta_{e} \approx 50^\circ$ on a flat substrate) in order to achieve partial wetting. Surface structures are characterized with scanning electron microscopy as shown in Supplementary Fig. 1. The inclination of the ridges $\beta$ is 60$^\circ$ for all the structures. The samples used in this work are labeled as ($W$, $P$) where $W$ and $P$ are the width and pitch of the ridge in micrometers, respectively, and listed in Table 1.
Label $W$ $P$ $H$ $S_{against}$ $S_{with}$
-------------- ----- ----- ----- ----------------- -----------------
(10, 15) 10 15 14 1.39 $\pm$ 0.10 1.21 $\pm$ 0.06
(10, 20) 10 20 14 1.61 $\pm$ 0.15 1.30 $\pm$ 0.11
(10, 30) 10 30 14 3.07 $\pm$ 0.50 4.66 $\pm$ 0.71
(15, 23) 15 23 14 1.30 $\pm$ 0.12 1.13 $\pm$ 0.07
(15, 30) 15 30 14 1.43 $\pm$ 0.11 4.12 $\pm$ 0.58
(15, 45) 15 45 14 3.11 $\pm$ 0.67 6.3 $\pm$ 1.4
(20, 30) 20 30 15 1.99 $\pm$ 0.33 5.19 $\pm$ 0.84
(20, 40) 20 40 17 3.77 $\pm$ 0.44 5.5 $\pm$ 1.7
**(20, 60)** 20 60 17 1.80 $\pm$ 0.36 5.06 $\pm$ 0.62
: Geometry and passage times for different surface structures investigated experimentally. The sample with bold font is also investigated numerically. The numerical values given for width ($W$), pitch ($P$) and height ($H$) are in micrometers. The last two columns show the passage time ratio with and against inclination estimated from experiments.
\[tab1\]
Navier-Stokes-Cahn-Hillard equations
------------------------------------
The Navier-Stokes equations combined with the phase-field approach (Cahn-Hilliard equation) are solved to simulate the droplet wetting on a comparable geometry to the experiments. The incompressible Navier-Stokes equations are given by $$\rho(C) \frac{D\textit{\textbf{u}}}{Dt} = \frac{1}{Re} \nabla p+ \frac{1}{Re} \nabla \mu(C) (\nabla \textit{\textbf{u}}+\nabla^T \textit{\textbf{u}})-\frac{C \nabla \phi(C)}{Ca\cdot Cn \cdot Re},$$ $$\nabla \cdot \textit{\textbf{u}}=0.$$ The non-dimensional numbers characterizing the system are the capillary number $Ca$=$\mu U /\sigma$, the Reynolds number $Re$=$\mu UL/\rho$, and the Cahn number $Cn$=$\epsilon/L$. Here, $\rho$ and $\mu$ are the density and viscosity of the liquid phase, $\sigma$ is the surface tension of the liquid, and $\epsilon$ is the width of the liquid-gas interface. Moreover, $U$and $L$ are the characteristic velocity and length of the system, respectively. The scalar $C$ is the phase field variable, where $C =1$ represents the liquid phase, and $C =-1$ the vapor phase. The effect of gravity is negligible since the Bond number $Bo$=$\rho gL^{2}/\sigma$ is small ($\approx$0.03 for $L$=0.4 mm).
The Cahn-Hilliard equation is given by $$\frac{DC}{Dt} = \frac{1}{Pe} \nabla^{2} \phi(C),$$ where $\phi$ is the chemical potential of the system defined as $\phi=\Psi^{\prime}(C)-Cn \nabla^2 C$. The Peclet number is defined as $Pe=UL/D$ where $D$ is a mass diffusivity. Here, $\Psi(C)= (C+1)^{2}(C-1)^{2}/4$ is the double well function, where the minimum represents the stable phases for gas ($C =-1$) and liquid ($C =1$). The boundary condition for $C$ on a solid surface is given by[@CarlsonPoF2009; @CarlsonPRE2012] $$-\epsilon \mu_{f} \frac{\partial C}{\partial t} =\epsilon \sigma \nabla C \cdot \textit{\textbf{n}} - \sigma {\rm cos} (\theta _{e})g^{\prime} (C),$$ where $\theta_{e}$ is equilibrium contact angle and $g(C)= 0.5+0.75C-0.25C^3$ is a polynomial which rapidly shifts from $0$ (vapor phase $C =-1$) to $1$ (liquid phase $C =1$). The line friction parameter $\mu_{f}$ is associated with molecular-origin energy dissipation at the moving contact line.
{width="70.00000%"}
### Numerical simulations and parameters
The Navier-Stokes-Cahn-Hilliard equations are solved using in-house software called FemLego[@CarlsonPoF2009]. The physical properties are chosen to be comparable to the experiments. The characteristic density $\rho$ and viscosity $\mu$ are set to the values of pure water (0.992 kg/m$^{3}$ and 0.997 Pa$\cdot$s respectively), and the length scale $L$ is set to 0.4 mm to match with the initial radius of the droplet in the experiments, which leads to $Re$=29200. Surface tension $\sigma$ is fixed to 0.073 N/m, which gives the capillary speed $\sigma/\mu$=73 m/s employed as the characteristic velocity $U$, and leads to $Ca$=1. The mass diffusivity $D$ in the Peclet number is set to 5.7$\times$ 10$^{-6}$ m$^{2}$/s, which leads to $Pe$=5120. The choice of Peclet number does not influence the results. The line friction parameter $\mu_{f}$ is obtained by fitting[@CarlsonPRE2012; @Vo2018] the simulated spreading radius with the experiments on a smooth substrate and found to be $\mu_{f}$=0.10 Pa$\cdot$s (See Supplementary Fig. 2). The interface thickness $\epsilon$ is set to 2 $\mu$m, and results in $Cn$=5$\times$10$^{-3}$. The interface is thicker than the actual physical interface but it is sufficiently thin compared to the structures in order not to influence the simulated results.
Wetting mechanism
=================
Wetting regimes {#sec:regime}
---------------
For wetting of structured surfaces, the combination of spread, stick and leap that ultimately determines the contact-line speed depends on the relation between the globally observed apparent dynamic contact angle $\theta_g$, and the corner angle of the surface, $\alpha$ (Fig. \[fig:map:b\]). The former angle is the apparent angle between the solid and the liquid-gas interface measured from the wetted side in the vicinity of the contact line. The corner angle defines the local corner of the surface structure that is approached by a moving contact line (see insets in Fig. \[fig:map:b\]). For the rapid wetting considered here, contact line movement is driven by Young’s force per unit length of a contact line $F_Y \sim \sigma (\cos \theta_e -\cos \theta_g)$. During spontaneous spreading, we have $\theta_e<\theta_g<\pi$. Three different spreading mechanisms can be distinguished (Fig. \[fig:map:b\]); (i) a spread-and-leap behavior of the contact line when $\theta_g>\alpha$; (ii) a stick-and-leap behavior of the contact line when $\theta_g < \alpha-\pi+\theta_e$, and; (iii) continuous spreading driven by $F_Y$ for intermediate corner angles $\alpha-\pi+\theta_e<\theta_g<\alpha $. For spread-and-leap, the moving contact line leaps from the valley to the nearest ridge, leaving behind some dry surface. For stick-and-leap, the contact line is held pinned[@Quere2008] at the corner while the liquid-air interface above the pinning site bulges due to inertial forces. If there exists a rise of the surface sufficiently nearby, the interface eventually makes contact with the next rise in the texture, resulting in a leap that leaves dry the entire valley between the pinning position and the new contact line. For intermediate angles, where no leaping occurs and the liquid wets the dry solid, the contact line speed varies with the slope of the surface. In particular, a reduced speed is expected in downward-sloping sections of the surface, compared to flat sections, due to a reduced Young’s force $F_Y$.
{width="59.00000%"} {width="69.00000%"}
Experimental and numerical observations
---------------------------------------
We experimentally investigated wetting of asymmetric ridges by depositing a water drop of radius $0.4$ mm on the surfaces and recording the first $1.2$ ms of wetting using a high-speed video camera (Fig. \[fig:spread\]a). We also numerically simulated spreading of droplets on axisymmetric textured surfaces with the same height, pitch and angle as in our experiments using a phase-field approach[@CarlsonPoF2009; @Wang2015]. Figure \[fig:spread\](c) shows a time sequence of a water drop spreading on a surface with $W=20$ $\mu$m, $P= 60$ $\mu$m, $H=17$ $\mu$m. The experiments reveal the asymmetric evolution of the droplet spreading on the structure; the contact line travels faster against the inclination (passing five ridges in $0.5$ ms) than with the inclination (passing two ridges in $0.5$ ms). Our numerical simulations reproduce the experimental droplet shapes and the apparent contact line velocity with respect to the spreading radius (see Supplementary Fig. 3 and Supplementary Movie 1). This agreement between experiment and numerical model allows us to rely on the simulations for understanding the detailed contact line movement across the textured surface.
The difference between the wetting dynamics in the two directions results from the difference in wetting regimes. Figure \[fig:spread\](b) shows a time sequence of the liquid-gas interface as the droplet travels against the direction of inclination over one ridge. Figure \[fig:spread\](e) shows how the corresponding contact line velocity varies during the same time interval. We observe a fast spread on ridge tips (A1$\rightarrow$A2), slower spread while descending into the valley (A2$\rightarrow$A3), and finally an increase in speed with a spread-and-leap to the next ridge (A3$\rightarrow$A4$\rightarrow$A5). During the process of moving from A1 to A5, the simulation-predicted apparent dynamic contact angle is in the range of $\theta_g=140^\circ$ to $\theta_g=120^\circ$ (Supplementary Fig. 4), whereas the corner angles encountered by the moving contact line have the values $\alpha=180^\circ$(A1), $240^\circ$(A2), $120^\circ$(A3), and $60^\circ$(A4-A5). Thus, our numerical observations conform with the kinematic map in Fig. \[fig:map:b\], explaining the mechanisms behind the peaks and troughs of the contact line speed.
Figure \[fig:spread\](d) shows spreading in the direction with the inclination over one periodic structure. The corresponding contact line speed (Fig. \[fig:spread\]f) has a distinct slow-fast-slow spreading velocity going from point W1 to point W4. This velocity profile corresponds to stick-and-leap (W1$\rightarrow$W2) at the corner W1, followed by fast spread on the flat top of the ridge (W2$\rightarrow$W3) and then again followed by stick-and-leap (W3$\rightarrow$W4). As the liquid front moves from W1 to W3, the apparent dynamic contact angle varies from $\theta_g=160^\circ$ to $\theta_g=130^\circ$ (Supplementary Fig. 4). The stick-and-leap follows from $\theta_g<\alpha-\pi+\theta_e=170^\circ$ at the corners W1 and W3.
Further insight into the contact-line speed can be gained by characterizing the driving and resisting forces during initial wetting. The Young’s force driving the contact line can be balanced by an inertial force $F_I\sim\rho {U_{flat}}^2 R$ or a contact line friction force $F_f\sim\mu_f {U_{flat}}$, where ${U_{flat}}$ is a characteristic contact line speed on a smooth surface. The contact line friction is related to non-hydrodynamic energy dissipation at the contact line, connected to molecular scale processes represented by the line friction parameter, $\mu_f$[@Johansson2017; @CarlsonPoF2009]. The viscous resistance associated with bulk liquid viscosity ($\mu$) can be neglected in the early stages of wetting since $\mu_f\gg \mu$[@Do-Quang2015].
The Ohnesorge number based on the line friction parameter, $Oh_f = \mu_f/\sqrt{\sigma\rho R_0}$, expresses the relative importance of line friction and fluid inertia. For a water droplet of radius $R_0=0.4$ mm, $Oh_f \approx 0.6$, where we computed the contact line friction to $\mu_f=0.10$ Pa$\cdot$s by combining experiments and numerical simulations (see Section 2 and Supplementary Fig. 2 for details). Since $Oh_f$ is of order one, both inertia and contact line friction may be involved in resisting the spreading. However, they play very different roles during spread-and-leap compared to stick-and-leap. As we will show in the following, in the former regime line friction dominates, whereas, in the latter, inertial forces determine the contact-line speed.
{width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"}
Influence of the pitch on spreading speed
=========================================
Passage time ratio
------------------
Figure \[fig:pitch\](a) shows the experimental spreading radius evolution in the direction against the inclination for surfaces with different width ($W$) and pitch ($P$). We observe that the spread on structured surfaces is always slower compared to a smooth surface (solid line). To characterize the spread-and-leap regime quantitatively, we define the passage time ratio, $$S = \frac{{T_s}}{{T_{flat}}},
\label{eq:Sa}$$ where ${T_s}$ and ${T_{flat}}$ correspond to the liquid front travel time over the distance $P$ on structured surfaces and smooth surfaces, respectively. Large values of $S$ thus indicate slow spreading on structured surfaces compared to smooth surfaces. Figure \[fig:pitch\](b) shows that passage time $S$ increases nearly by a factor three when the pitch is increased from $15$ $\mu$m to $30$ $\mu$m for W=10 $\mu$m. A similar increase in $S$ is also observed for W=15 $\mu$m. However, for wider structures W=20 $\mu$m (Fig. \[fig:pitch\]b), the same rapid increase is observed when the pitch is increased from $30$ $\mu$m to $40$ $\mu$m; but the passage time reduces for very large pitch ($60$ $\mu$m). Figure \[fig:pitch\](c) shows how the experimental spreading radius evolves in the direction with the inclination for surfaces with a different pitch. Compared to the smooth surface, we observe a slow spreading, which rapidly decreases with the pitch. From the corresponding passage times $S$ (symbols in Fig. \[fig:pitch\](d)), $S$ increases monotonically for small $P$ and saturates to $S \sim 6$ for large $P$. We note that the travel time can be a factor five higher compared to a smooth surface, and thus also significantly larger compared to the spread-and-leap regime considered in the previous section (c.f. Fig. \[fig:pitch\](b, d)).
![ (a) Sketches illustrating wetting situation in the direction against the inclination for small (left, $r_1\approx r_2\approx 0$), intermediate (center, $r_1\approx1/2, r_2\approx0$) and large (right, $r_1\approx 1/2, r_2\approx 1$) pitch. Wetted sections are colored with red. (b) The passage time ratio $S$ in the direction against the inclination based on Eq. (\[Eq:S\_ag\]) for $W=10$ $\mu$m as a function of the surface pitch. The blue lines correspond to $S$ obtain from the theoretical model for different apparent dynamic contact angles ($\theta_g=125^\circ, 135^\circ, 145^\circ$). The black line corresponds to passage times $S$ that would exists without leaping, i.e. assuming that the structured surface is wetted everywhere. Without leaping, spreading becomes very slow for small pitch, which contradicts experimental observations where leaping will significantly increase the spreading speed. (c) Sketches illustrating wetting situation in the direction with the inclination for small(left) and large (right) pitch. (d) The passage time ratio $S$ in the direction with the inclination based on Eq. (\[eq:Sw\]) for surface ridges of different widths as a function of the surface pitch. []{data-label="fig:model"}](Fig4a.pdf "fig:"){width="35.00000%"} ![ (a) Sketches illustrating wetting situation in the direction against the inclination for small (left, $r_1\approx r_2\approx 0$), intermediate (center, $r_1\approx1/2, r_2\approx0$) and large (right, $r_1\approx 1/2, r_2\approx 1$) pitch. Wetted sections are colored with red. (b) The passage time ratio $S$ in the direction against the inclination based on Eq. (\[Eq:S\_ag\]) for $W=10$ $\mu$m as a function of the surface pitch. The blue lines correspond to $S$ obtain from the theoretical model for different apparent dynamic contact angles ($\theta_g=125^\circ, 135^\circ, 145^\circ$). The black line corresponds to passage times $S$ that would exists without leaping, i.e. assuming that the structured surface is wetted everywhere. Without leaping, spreading becomes very slow for small pitch, which contradicts experimental observations where leaping will significantly increase the spreading speed. (c) Sketches illustrating wetting situation in the direction with the inclination for small(left) and large (right) pitch. (d) The passage time ratio $S$ in the direction with the inclination based on Eq. (\[eq:Sw\]) for surface ridges of different widths as a function of the surface pitch. []{data-label="fig:model"}](Fig4b.pdf "fig:"){width="35.00000%"} ![ (a) Sketches illustrating wetting situation in the direction against the inclination for small (left, $r_1\approx r_2\approx 0$), intermediate (center, $r_1\approx1/2, r_2\approx0$) and large (right, $r_1\approx 1/2, r_2\approx 1$) pitch. Wetted sections are colored with red. (b) The passage time ratio $S$ in the direction against the inclination based on Eq. (\[Eq:S\_ag\]) for $W=10$ $\mu$m as a function of the surface pitch. The blue lines correspond to $S$ obtain from the theoretical model for different apparent dynamic contact angles ($\theta_g=125^\circ, 135^\circ, 145^\circ$). The black line corresponds to passage times $S$ that would exists without leaping, i.e. assuming that the structured surface is wetted everywhere. Without leaping, spreading becomes very slow for small pitch, which contradicts experimental observations where leaping will significantly increase the spreading speed. (c) Sketches illustrating wetting situation in the direction with the inclination for small(left) and large (right) pitch. (d) The passage time ratio $S$ in the direction with the inclination based on Eq. (\[eq:Sw\]) for surface ridges of different widths as a function of the surface pitch. []{data-label="fig:model"}](Fig4c.pdf "fig:"){width="33.00000%"} ![ (a) Sketches illustrating wetting situation in the direction against the inclination for small (left, $r_1\approx r_2\approx 0$), intermediate (center, $r_1\approx1/2, r_2\approx0$) and large (right, $r_1\approx 1/2, r_2\approx 1$) pitch. Wetted sections are colored with red. (b) The passage time ratio $S$ in the direction against the inclination based on Eq. (\[Eq:S\_ag\]) for $W=10$ $\mu$m as a function of the surface pitch. The blue lines correspond to $S$ obtain from the theoretical model for different apparent dynamic contact angles ($\theta_g=125^\circ, 135^\circ, 145^\circ$). The black line corresponds to passage times $S$ that would exists without leaping, i.e. assuming that the structured surface is wetted everywhere. Without leaping, spreading becomes very slow for small pitch, which contradicts experimental observations where leaping will significantly increase the spreading speed. (c) Sketches illustrating wetting situation in the direction with the inclination for small(left) and large (right) pitch. (d) The passage time ratio $S$ in the direction with the inclination based on Eq. (\[eq:Sw\]) for surface ridges of different widths as a function of the surface pitch. []{data-label="fig:model"}](Fig4d.pdf "fig:"){width="35.00000%"}
Model and scaling analysis
--------------------------
### A model for spread-and-leap regime
In order to understand and to predict the observed wetting behaviour, we build simple models based on estimates of the relevant forces.
The non-monotonic behavior in Fig. \[fig:pitch\](b) for W=20 $\mu$m can be explained by characterizing leaping over the microstructures in more detail. For a smooth flat surface, the passage time over a distance $P$ can be estimated as ${T_{flat}}\sim P/{U_{flat}}$, where ${U_{flat}}$ is the average contact-line velocity. For a surface with ridges, the wetted distance $W_s$ per period can be estimated as ${W_s}\approx 2r_1 H + r_2 P+ (1-r_2)W$. Here, $r_1$ and $r_2$ represent the wetted portions of the vertical walls of the ridge and the valley between the ridges, respectively (see Fig. \[fig:model\]a). The time it takes for the contact line to move a distance $P$ is ${T_s}\sim {W_s}/{U_{s}}$, where ${U_{s}}$ is the pitch-averaged contact-line velocity. Inserting the scaling estimates for ${T_s}$ and ${T_{flat}}$ in Eq. (\[eq:Sa\]), we obtain, $$S \approx \frac{{U_{flat}}}{{U_{s}}}\left (2r_1\frac{H}{P} + r_2+ (1-r_2)\frac{W}{P}\right).
\label{eq:S2}$$ For a very large pitch (rightmost frame in Fig. \[fig:model\]a), nearly all the structured surface is wetted ($r_1\approx1$ and $r_2\approx 1$); we have very small leaping and $S\sim 1+2H/P\approx 1$. For a very small pitch (leftmost frame in Fig. \[fig:model\]a) the entire valleys between ridges remain dry ($r_1\approx 0$ and $r_2\approx 0$); we have maximum leaping and $S\sim W/P\approx 1$. For intermediate values of the pitch $P$, where $r_1\approx 1/2$ and $r_2\approx 1/2$, a maximum value of the travel-time ratio $S$ exists.
To be more quantitative, we consider local contact line velocity based on phase field theory. As discussed in Section \[sec:regime\], the spreading in the direction against the inclination is in spread-and-leap regime. In this regime, we assume Young’s force is driving the contact line and the line friction is the resistive force. Assuming that wetting resistance for spread-and-leap is dominated by line friction, we can develop a theoretical model of the contact-line velocity based on the Navier-Stokes-Cahn-Hilliard equations[@Yue2011; @Lee2019], $$U_{cl,i} = \frac{\sigma}{\mu_f}\frac{3}{2\sqrt{2}} \frac{\cos\theta_e-\cos{\theta_{l,i}}}{\sin{\theta_{l,i}}}.
\label{eq:cl}$$ The wetted part of ridged surface is divided into $i=1,\dots, N$ smooth sections. Here, ${\theta_{l,i}}$ in Eq. (\[eq:cl\]) is the local dynamic contact angle formed between the liquid interface and the $i$th section of the structured surface. We assume that Eq. (\[eq:cl\]) is a valid model of liquid front at any surface point [@Yue2011; @Lee2019]. Considering the local dynamic contact angle and velocity, we sum up the time to pass sections $i=1,\dots,N$ $$T_{s} = \sum_{i=1}^N{L_i/U_{cl,i}} \\
=\sum_{i}{\frac{2\sqrt{2} \mu_{f}}{3 \sigma}\frac{L_{i}{\rm sin}\theta_{l,i}} {{\rm cos}\theta_{e}-{\rm cos}\theta_{l,i}}},
\label{eq:Sl}$$ and $$\begin{aligned}
S & = T_{s}/T_{flat} \\
& = \sum_{i}{(\frac{L_{i}{\rm sin}\theta_{l,i}} {{\rm cos}\theta_{eq}-{\rm cos}\theta_{l,i}})}\frac{{\rm cos}\theta_{eq}-{\rm cos}\theta_{g}}{L{\rm sin}\theta_{g}},
\label{Eq:S_ag}
\end{aligned}$$ where ${\theta_{l,i}}$, $U_{cl,i}$ and $L_i$ are the local dynamic contact angle, contact line velocity, and length of section $i$ respectively, and $\theta_g$ is the global apparent dynamic contact angle. The above effective spreading model is similar to that used in Lee et al.[@Lee2019]. Compared to the sawtooth geometries analyzed by Lee et. al.[@Lee2019]., the geometry studied here is more complicated as it includes backward facing sections and sharper angles. The present model accounts for the backward facing part of the structure. The droplet interface is assumed to be linear with a constant angle $\theta_g$, and the surface after the leap point is assumed to be not wetted. The wetted area is colored red in Fig. \[fig:model\](a).
Figure \[fig:model\](b) shows the dependence of $S$ on the pitch obtained from the model for three different local dynamic contact angles. We observe that the model captures the trend of the resistance due to surface texture relatively well, confirming that line-friction is the dominating physical resistance for the spreading. The spread-and-leap model explains the – perhaps counter-intuitive – fast spreading for small pitch. Without taking into account the leaping, a spreading model where all solid is wetted will overestimate $S$ significantly (Fig. \[fig:model\]b). The model also predicts the pitch that gives the maximum $S$, $P_{peak}$. For W= 10 $\mu$m, $P_{peak}$ is 32 $\mu$m, 39 $\mu$m for W=15 $\mu$m, and 43 $\mu$m for W=20 $\mu$m (obtained assuming $\theta_g=135^\circ$). The estimate of $P_{peak}$ explains why the non-monotonic behaviour is observed only for W=20 $\mu$m. For W=10 $\mu$m and 15 $\mu$m, each $P$ is below $P_{peak}$. However for W=20 $\mu$m, $P_{peak}$ is close to the intermediate pitch 40 $\mu$m, therefore the non-monotonic behavior is observed.
Scaling analysis for stick-and-leap regime
------------------------------------------
The spreading in the direction with the inclination is in stick-and-leap regime. The contact line travels on the tip of the ridges and is pinned on the corner intermittently, and the interface bulges by the liquid inertia until it makes a new contact to the next structure. To understand the role of pinning, we perform a simple scaling analysis. In the direction with the inclination, Young’s force is balanced both by inertia and line friction. Here, Young’s force accelerates the spreading motion when the contact line is pulled over the structure (W2 $\rightarrow$ W3 in Fig. \[fig:spread\]f), and then adds to the droplet inertia. When the contact line reaches the acute corner at W3, it is momentarily pinned there. The contact line remains pinned until the inertia in the droplet brings the local dynamic contact angle out of equilibrium such that the droplet interface can make contact with the solid in the forward direction.
Assuming the surface energy is distributed to the kinetic energy over a whole droplet during the spreading on the top of the ridge, we have $$2 \pi R_0 \sigma W \sim \frac{4}{3} \pi \rho {R_{0}}^3 U^2,$$ and $$U \sim \frac{\sqrt{3}}{R_{0}} \sqrt{\frac{\sigma W}{\rho}},
\label{eq:Uinertial}$$ where $R_0$ is the initial radius of the droplet, $W$ is the width of the ridge, $L$ is the space between the ridges, and $U$ is a characteristic velocity for inertia. The prefactor $\sqrt{3}$ in Eq. (\[eq:Uinertial\]) can be neglected for simplicity.
Based on this physical insight as depicted in Fig. \[fig:model\](c), we can model the passage time for stick-and-leap as $${T_s}\sim \frac{\mu_f W}{\sigma}+\frac{(P-W) R_0}{\sqrt{\sigma W/\rho}}.
\label{eq:TSW}$$ Here, the first term quantifies the balance between Young’s force and line friction, i.e., the travel time for the interface to move over the top of the ridge of width $W$. The second term quantifies the pinning time, i.e., the time needed for inertia to push the interface to the next ridge, leaping the distance $P-W$. Recalling that ${T_{flat}}\sim P/{U_{flat}}\approx P\mu_f/\sigma$, we can formulate the passage time $S$ as $$S = \frac{W}{P}+\frac{1}{Oh_f}\sqrt{\frac{R_0}{W}}\frac{(P-W)}
{P} .
\label{eq:Sw}$$ We observe that the second term dominates when inertia is much larger than the line friction ($Oh_f\ll 1$). For $P\gg W$, $S$ in Eq. (\[eq:Sw\]) saturates to the constant value $1/Oh_f \sqrt{R_0/W}$. In our scaling analysis, we have assumed that this saturation value is reached when the liquid-gas interface remains pinned at the acute corner infinitely long.
Figure \[fig:model\](d) presents $S$ obtained from Eq. (\[eq:Sw\]) for different ridge widths. We observe that the scaling estimates for stick-and-slip correctly capture the trend of increasing passage time with increasing pitch. The scaling analysis shows that inertia plays a central role in the stick-and-leap regime, in contrast to spread-and-leap in the direction against inclination. Surface energy is converted to the kinetic energy of the droplet ($\sim$ inertia) while the contact line spreads on the top of the ridge, and inertia drives the liquid front to reach the next rise of the surface during the stick-and-leap at corners (W2 in Fig. \[fig:spread\]f).
We note that the advancement of the contact line is primarily determined by the line friction and the geometrical details of the structure when moving against the inclination, but that it is inertial when moving with the inclination.
Conclusions
===========
We have presented a comprehensive study of rapid wetting on complex asymmetric microstructures. We identified three wetting regimes, denoted as spread, stick, and leap. By coordinated simulations and experiments we can follow the passage of the contact line over the microstructure in detail and formulate a model as well as scaling estimates that explain how the geometrical features determine the macroscopic wetting speed. We showed that when wetting proceeds against the inclination of the ridges, a spread-and-leap behavior underpins the wetting and the driving Young’s force is primarily balanced by contact line friction. In contrast, when the spreading direction is with the inclination, a stick-and-leap behavior is observed, and it is the liquid inertia that limits the wetting speed. Our experiments and theory show that leaping phenomenon plays a central role in increasing the spreading speed compared to a surface textured without leaping. We believe that this newly identified spreading mechanism forms the foundation to design surface structures for controlling wetting under realistic unsteady environments.
Conflicts of interest {#conflicts-of-interest .unnumbered}
=====================
There are no conflicts to declare.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Swedish Research Council (VR 2015-04019). S.B acknowledges the support of the Swedish Foundation of Strategic research (SSF-FFL6). We thank Calvin J. Brett at KTH for his kind help for SEM measurements.
|
---
abstract: '\[Abstract\] In many problems, agents cooperate locally so that a leader or fusion center can infer the state of every agent from probing the state of only a small number of agents. Versions of this problem arise when a fusion center reconstructs an extended physical field by accessing the state of just a few of the sensors measuring the field, or a leader monitors the formation of a team of robots. Given a link cost, the paper presents a polynomial time algorithm to design a minimum cost coordinated network dynamics followed by the agents, under an observability constraint. The problem is placed in the context of structural observability and solved even when up to $k$ agents in the coordinated network dynamics fail.'
author:
- 'Stephen Kruzick, Sérgio Pequito, Soummya Kar, José M. F. Moura, and A. Pedro Aguiar [^1]'
bibliography:
- 'StructurallyObservableDistributedNetworks.bib'
title: Structurally Observable Distributed Networks of Agents under Cost and Robustness Constraints
---
\[Keywords\] distributed networks, linear dynamics, structural systems, observability, minimum cost, Steiner subgraph
Introduction {#Introduction}
============
is important in many distributed settings to appreciate the impact of local cooperation. Much work has been done in distributed inference [@bracamaranomatta-08; @karmouraramanan-tit-2012], distributed diffusion [@lopessayed-08], and distributed optimization [@nedicozdaglar-09; @jakoveticxaviermoura-tac-14], to name a few applications and references. This paper is concerned with a design question under the following different but related scenario. A group of agents share a common goal. They cooperate at a cost to update their individual state, but only a few agents share their state with a leader or fusion center. This leads to network dynamics that we will describe more formally below. The mission of the leader is to reconstruct the network state, i.e., the state of all the agents. We consider how to design the minimal cost network of (local) cooperation among agents and with which agents the leader interacts, i.e., the design of minimal cost coordinated network dynamics, subject to the constraint that the leader can reconstruct the network state–an observability constraint. Before being more specific, we illustrate with two motivating examples. The first is static, while the second is dynamic like the problem we address. References [@schmidtmoura2; @schmidtmoura3] consider the problem of reconstructing a field from information (state) provided by only a few agents (sensors) in a network of sensors. For instance, the field may be the contamination level of a pollutant over a wide area. It turns out that, through local cooperation (sensors updating iteratively their state with the state of their neighbors), the fusion center can, under appropriate conditions, reconstruct (through a basis pursuit type algorithm) the entire field from a snapshot of the states of a very sparse subset of the agents. A question of interest is to design the cooperation dynamics and to determine a minimal set of sensors to be probed so that a fusion center can reconstruct the field.
The second considers formation with flocks of birds or schools of fish that can be described by models in which each individual constantly monitors or senses the distance to neighboring birds or fish and reacts to ensure that it maintains a suitable position. For example, each individual might accelerate to move closer if the distance to neighbors increases or slow down if the distance to neighbors becomes dangerously close. The collective networked dynamics of birds or fish and their qualitative properties are discussed in [@reynolds-87; @VicsekCzirokBenJacobShochet-95; @CuckerSmale-07], among others. These references provide several interesting and revealing examples. The self-organization of these adaptive networks has also been studied in [@tusayed-08]. Similarly, in several robotic problems like in formation control of multi-robot teams and in the coordination of groups of mobile autonomous agents, robots move by approaching or gaining distance from their neighbors in order to maintain the formation. This leads the multi-robot team to follow coordinated network dynamics. Furthermore, the robots do not explicitly communicate with each other in many of these applications. Each robot finds its own relevant information with respect to neighboring robots [@Moshtagh2009], for example, range and bearing from on-board sensors, such as cameras. However, for reasons derived from the application, the robots do not directly communicate this information to other individuals. The resulting networked dynamics have been used in the literature to obtain simplified linear distributed control or update laws achieving mission specific formation or flocking objectives [@cao2011formation; @sun2015rigid]. Finally, we assume that a leader is charged with monitoring that the robots remain in formation by probing only the state of a few agents and then computing the (global) network state. This task is trivial if the fusion center accesses every agent or node in the network. It becomes interesting when at the same time it is desirable to reduce the costs to a certain minimum, see below, including agents measuring their relative position to neighbors and the direct access to node states by the leader. It is then of interest to design the network structure of the coordinated dynamics and to determine which node states should be measured by the leader so that it can track the states of all the nodes in the network while minimizing the cost of network coordination and measurements.
The above are difficult questions. We cast problems of these types below in the context of structural observability, which entails designing the structure of a dynamical system with observable dynamics, a framework studied in [@JiEgerstedt; @SundaramHadjicostis1; @SundaramHadjicostis2]. Once the system is observable, the actual problem of continuously monitoring the state of the nodes by the remote fusion station can then be achieved by implementing a recursive observer such as, for example, of the Luenberger type [@ellis2002observers].
We emphasize that there may not be explicit communication among the nodes in the applications we envision, even though we use the consensus context to describe the coordination among agents. Accordingly, in this paper, we replace communication costs usually assumed in consensus problems by link costs. Link costs subsume costs associated with local neighbor sensing interactions (proxies for costs associated with sensing and extracting from measurements the state of neighbors), as well as costs incurred by a fusion center to learn (also, possibly by sensing) the current state (position) of a few selected agents. For simplicity, the interactions between the fusion center and nodes are described as through a *backbone* network.
We model the coordinated network dynamics followed by the aggregate of all the agents by a linear system $$\begin{aligned}
{4}
&\mathbf{x}({n+1})&&=A\mathbf{x}(n)\label{dyn1.1} \\
&\mathbf{y}({n}) &&=C\mathbf{x}(n)\label{dyn1.2}.
\end{aligned}$$ The system dynamics matrix $A$ captures the network graph of interactions among agents. Its nonzero entries represent interaction links between corresponding nodes or agents. Each agent or sensor node maintains a single scalar state variable $\mathbf{x}_i(n)$ initially set to $\mathbf{x}_i(0)$. The vector of agent or sensor states at a given iteration $n$ is denoted by $\mathbf{x}(n)$. The nodes update their states according to the coordinated dynamics given by , with initial state vector $\mathbf{x}(0)$. The nodes that are monitored by the fusion center, referred to as backbone nodes, are collected in the output vector $\mathbf{y}(n)$ given by . Because network structure must be respected, disallowed interactions among nodes or among the fusion center and nodes restrict corresponding entries of the $A$ and $C$ matrices to equal zero, whereas entries corresponding to network links represent design parameters.
We use structural systems theory, a survey of which can be found in [@StructSysSurvey], to find network dynamics $(A,C)$ requiring a minimum set of link costs while guaranteeing that the initial state can be recovered from the backbone outputs collected over time at the fusion center despite some number of sensor node failures, which are known to the central node and occur before the dynamics begin. Networks that operate according to this framework are discussed in [@JiEgerstedt]. This reference provides a necessary topological condition for the initial state to be reconstructed from the backbone outputs for a specific choice of dynamics related to the network graph structure. These networks are also studied under a structural systems context in [@SundaramHadjicostis1] and [@SundaramHadjicostis2], which describe types of networks where this can be achieved. Our paper focus on designing robust networks that operate according to the above model and that minimize a specified cost function. This work extends a preliminary version of the optimal network dynamics design problem that we previously introduced in [@PequitoKruzickEUSIPCO2013], by considering arbitrary costs, arbitrary backbone topologies, and sensing robustness requirements. From a technical standpoint, the methodology and combinatorial optimization tools are also more general and computationally efficient. Practical implementable dynamic systems of the optimal network structures that we obtain rely on results presented in [@SundaramHadjicostis2]. While the optimization problem involving cost minimization under robustness constraints that appears in this paper does not have a direct comparison in the existing structural systems literature, other related works regarding structural systems may be of interest for further reading. The structural systems framework for networks appears in [@JiEgerstedt], [@SundaramHadjicostis1], and [@SundaramHadjicostis2]. Other optimal network design problems involving this structural systems framework with significantly differing objectives and constraints can be found in [@SergioA], [@SergioB], and [@SergioD].
The paper is organized as follows. Section \[BackgroundConcepts\] introduces key background concepts, including relevant information from graph theory, combinatorial optimization, and systems theory. Section \[NetworkDescription\] formally describes our network operation model. Section \[DesignProblem\] examines our minimum cost design problem and provides a solution algorithm followed by a proof of correctness and practical discussion. Finally, Section \[Conclusions\] concludes the paper.
Background Concepts {#BackgroundConcepts}
===================
This section provides supporting background information. Of particular importance are graph theory definitions and concepts used to describe network structure as well as topics from combinatorial optimization relevant to the minimum cost design problem solutions. Network operation closely relates to system theory concepts, which are also introduced. Finally, results from structural system theory provide a bridge between network structure, described by graphs, and network operation, described by dynamical systems.
Graph Theory Concepts {#BackgroundConcepts:GraphTheory}
---------------------
This section introduces terminology and concepts regarding graphs. A [**directed graph**]{} $\mathcal{G}$ is an ordered pair $(\mathcal{V},\mathcal{E})$ in which $\mathcal{V}$ denotes a set of [**nodes (vertices)**]{} and $\mathcal{E}$ denotes a set of directed [**links (edges)**]{}. These directed links are ordered pairs $(v_i,v_j)$ of nodes $v_i,v_j\in \mathcal{V}$. Note that [**self-loops**]{}, links formed as $\left(v,v\right)$ for $v\in \mathcal{V}$, are not excluded from this definition. Furthermore, an [**directed, weighted graph**]{} $\mathcal{G}$ is the ordered triple $(\mathcal{V},\mathcal{E},w)$ in which $w:\mathcal{E}\rightarrow\mathbb{R}^+$ assigns a cost or [**weight**]{} $w(e)$ to each link $e\in \mathcal{E}$. Any graph $\mathcal{G}_S=(\mathcal{V}_S,\mathcal{E}_S)$ with $\mathcal{V}_S\subseteq \mathcal{V}$ and $\mathcal{E}_S\subseteq \mathcal{E}$ is called a [**subgraph**]{} of $\mathcal{G}$. Furthermore, if $\mathcal{V}_S=\mathcal{V}$, then $\mathcal{G}_S$ [**spans**]{} $\mathcal{G}$. Two graphs $\mathcal{G}_1=(\mathcal{V}_1,\mathcal{E}_1)$ and $\mathcal{G}_2=(\mathcal{V}_2,\mathcal{E}_2)$ are said to be [**isomorphic**]{}, written as $\mathcal{G}_1\simeq \mathcal{G}_2$, if there is bijective function $f:\mathcal{V}_1\rightarrow \mathcal{V}_2$ such that $(u,v)\in \mathcal{E}_1$ if and only if $(f(u),f(v))\in \mathcal{E}_2$.
A sequence $(v_1,v_2),(v_2,v_3),...,(v_{k-1},v_k)$ of directed links in which the [**head**]{} (destination) node of the previous link is the [**tail**]{} (origin) node of the subsequent link constitutes a [**directed path**]{}. Provided $v_i\neq v_j$ for all $i\neq j$, it comprises an [**directed elementary path**]{}. When $v_1=v_k$ but all other nodes are distinct, the path forms a [**directed cycle**]{}. Two directed paths are [**internally node-disjoint**]{} if they share no nodes apart from the start node and end node. Likewise, two directed paths are [**link-disjoint**]{} if they share no links. The [**directed local node-connectivity**]{} $\kappa_{\mathcal{G}}(u,v)$ from node $u$ to node $v$ gives the minimum number of nodes that must be removed from the graph $\mathcal{G}$ such that there is no directed path from $u$ to $v$, equal to the number of internally node-disjoint directed paths from $u$ to $v$. The [**directed local link-connectivity**]{} $\lambda_{\mathcal{G}}(u,v)$ from node $u$ to node $v$ gives the minimum number of links that must be removed from the graph such that there is no directed path from $u$ to $v$, equal to the number of link-disjoint directed paths from $u$ to $v$.
Several notions of connectedness exist for directed graphs. In this paper, a specially labeled [**root**]{} node $r\in \mathcal{V}$ is given, and a graph is [**$\mathbf{r}$-rooted connected**]{} if there is an elementary directed path from each node $v\in \mathcal{V}$ to $r$. An [**arborescence**]{}, also known as a directed rooted tree, is a directed graph in which there exists exactly one elementary directed path from each node to $r$. A collection of disjoint arborescences with root nodes collected into a set $R$ is called a [**branching**]{}. A directed graph is [**$\mathbf{r}$-rooted $\mathbf{k}$-node-connected**]{} if removing fewer than $k$ nodes leaves at least one elementary directed path from each node to $r$, that is $\kappa_{\mathcal{G}}(v,r)\geq k$ for all $v\in \mathcal{V}$. Similarly, a directed graph is [**$\mathbf{r}$-rooted $\mathbf{k}$-link-connected**]{} if removing fewer than $k$ links leaves at least one elementary directed path from each node to $r$, that is $\lambda_{\mathcal{G}}(v,r)\geq k$ for all $v\in \mathcal{V}$.
Optimal Connectivity Problems {#BackgroundConcepts:OptimizationProblems}
-----------------------------
Network design problems often involve finding optimal subgraph structures within a graph of possible network connections with the restriction that some notion of connectivity be ensured. The following discussion introduces problems involving optimality objectives and connectedness requirements that will prove useful in solving the problem presented in Section \[DesignProblem\].
Consider identification of the least costly set of directed links that connects all the nodes of a network to a root node. The [**minimum spanning arborescence**]{}, also known as a minimum directed rooted spanning tree, of a directed, weighted graph $\mathcal{G}=(\mathcal{V},\mathcal{E},w)$ rooted at a node $r\in \mathcal{V}$ formalizes this concept. The problem consists of finding a subgraph of minimum cost $\operatorname{MSpA}(\mathcal{G},r)$ that has exactly one directed path from each node $v\in \mathcal{V}$ to $r$. Note that the solution must be an $r$-rooted arborescence and that $\mathcal{G}$ must be $r$-rooted connected for a solution to exist. The minimum spanning arborescence can be computed by the Chu-Liu/Edmonds Algorithm, which can be implemented with complexity $O(|\mathcal{E}|+|\mathcal{V}|\log |\mathcal{V}|)$ [@Gabow]. Generalizations for higher connectivity, such as the minimum $r$-rooted $k$-node-connected spanning subgraph and the minimum $r$-rooted $k$-link-connected spanning subgraph are similarly defined. Unlike the corresponding undirected counterparts, these directed, rooted problems are solvable in polynomial time using a maximum weight matroid intersection formulation [@Edmonds], [@FrankRootedKConnections].
In a problem related to the minimum spanning arborescence, consider identification of the least costly set of directed links that connect a required subset of the network nodes to a root node while possibly involving other nodes. The general [**minimum Steiner arborescence**]{} problem, also known as the minimum directed rooted Steiner tree, formalizes this concept. Let a directed, weighted graph $\mathcal{G}=(\mathcal{V},\mathcal{E},w)$ with root node $r\in\mathcal{V}$ be given along with a set $\mathcal{S}\subseteq \mathcal{V}$ of [**terminal nodes**]{}. The problem consists of finding a minimum cost subgraph $\operatorname{MStA}(\mathcal{G},\mathcal{S},r)$ of $\mathcal{G}$ that has exactly one directed path from each node $v\in \mathcal{S}$ to $r$. Note that non-terminal nodes in $\mathcal{V}\backslash\mathcal{S}$ may or may not be used, and that a path must exist from each $v\in \mathcal{V}$ to $r$ for a solution rooted at $r$ to exist. The general minimum Steiner arborescence problem is NP-hard, so no polynomial time solution algorithm is known. However, polynomial time approximation algorithms with nontrivial performance guarantee ratios exist and may be used to obtain approximate solutions [@DirectedSteiner].
A generalization to greater connectivity requirements, the [**minimum $\mathbf{r}$-rooted $\mathbf{k}$-node-connected Steiner subgraph**]{} problem, provides the minimum cost connectivity problem of greatest utility to this paper. For a directed, weighted graph $\mathcal{G}=(\mathcal{V},\mathcal{E},w)$, terminal node set $\mathcal{S}$, and root node $r$, the problem consists of finding a minimum cost subgraph $\operatorname{MStNCS}(\mathcal{G},\mathcal{S},r,k)$ that has $k$ internally node-disjoint directed paths from each node $v\in \mathcal{S}$ to $r$. Note that non-terminal nodes in $\mathcal{V}\backslash\mathcal{S}$ do not necessarily have $k$ node-disjoint paths to $r$. It is clear that at least $k$ internally node-disjoint paths must exist from each $v\in \mathcal{S}$ to $r$ for a solution rooted at $r$ to exist. The problem is, in general, at least as computationally difficult as the minimum Steiner arborescence problem. However, in the particular restricted case in which all links with positive weight, called [**augmenting links**]{}, originate in $\mathcal{S}$, the problem is solvable in polynomial time using a submodular flow algorithm [@FrankRootedKConnections].
For networks that engage in data forwarding, the problem of finding the path between two nodes with the least costly total link sum often finds relevance. The concept of each node connecting to a root node over the least costly path possible is formalized by the [**shortest path spanning arborescence**]{}, which will also contribute to solving the design problem in Section \[DesignProblem\]. Let a directed, weighted, and $r$-rooted connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E},w)$ with nonnegative link weights $w$ and a root node $r\in \mathcal{V}$ be given. The problem consists of finding a path from each $v\in \mathcal{V}$ to the root node $r$ that has the minimum possible link sum. The solution set of links is not necessarily unique. However, at least one possible set of links forms a spanning tree called the shortest path spanning arborescence $\operatorname{SPSpA}(\mathcal{G},r)$ rooted at $r$. The shortest path lengths, using which the arborescence may be built starting at the root, can be computed using methods such as Dijkstra’s algorithm, which can be implemented with complexity $O(|\mathcal{E}|+|\mathcal{V}|\log|\mathcal{V}|)$ [@CombinatorialOpt].
System Theory Concepts {#BackgroundConcepts:SystemTheory}
----------------------
The weighted networks discussed in this paper follow the coordinated dynamics given by the discrete time dynamical system and . The concept of system observability plays a key role in understanding the communication scheme under consideration. A system is said to be [**observable**]{} if the initial state $\mathbf{x}(0)$ can be uniquely determined from the collected outputs over a finite time period with knowledge of the system matrices and input values. For discrete linear, time-invariant systems with zero-input such as in -, the output at time $n$ is $\mathbf{y}(n)=CA^n\mathbf{x}(0)$. The collected outputs $\mathbf{y}=[\mathbf{y}^\intercal(0),\cdots,\mathbf{y}^\intercal(N-1)]^\intercal$ over $N$ iterations are described by a linear transformation of the initial state $$\label{OMatrix1}
\mathbf{y}=O_{(A,C)}\mathbf{x}(0)$$ where $O_{(A,C)}$ is the [**observability matrix**]{} $$\label{OMatrix2}
O_{(A,C)}=\left[\begin{array}{ccc}(CA^0)^\intercal & \cdots & (CA^{N-1})^\intercal\end{array} \right]^\intercal$$ and $N$ is the number of state variables. Observability of a system can be inferred from the observability matrix as stated in Theorem \[BackgroundConcepts:SystemTheory:ObservabilityThm\].
\[BackgroundConcepts:SystemTheory:ObservabilityThm\]
The $N$ state linear, time-invariant system described by matrices $(A,C)$ is observable if and only if $\operatorname*{rank}(O_{(A,C)})=N$.
In contexts involving an interval $[0,T-1]$ other than the first $N$ iterations, the following notations apply. The collected outputs $\mathbf{y}_{[0,T-1]}=[\mathbf{y}^\intercal(0),...,\mathbf{y}^\intercal(T-1)]^\intercal$ with no input over $T$ iterations are described by a linear transformation of the initial state $$\label{OMatrix1-1}
\mathbf{y}_{[0,T-1]}=O_{(A,C),[0,T-1]}\mathbf{x}(0)$$ such that $$\label{OMatrix2-2}
O_{(A,C),[0,T-1]}=\left[\begin{array}{ccc}(CA^0)^\intercal & \cdots & (CA^{T-1})^\intercal\end{array} \right]^\intercal$$ with $[0,T-1]$ explicitly written for specificity.
A well known dual to the concept of observability is the concept of controllability for systems driven by inputs. Under appropriate conditions on the system matrices, these concepts are dual and results for observability hold with corresponding adaptation to controllability. This paper is primarily concerned with ensuring that the network systems designed are observable. Duality allows the techniques developed here to apply in situations requiring the design of controllable coordinated network systems. In the subsequent text, we consider only observability and will not address controllability.
Structural System Theory {#BackgroundConcepts:StructuralSystems}
------------------------
Because the dynamical system that describes network coordination must respect local network connections as represented in a directed graph, analysis of how network structure affects system properties provides design insights. Denote by $(\tilde{A},\tilde{C})$ a pair of structural matrices composed of entries that are zero or one, with $\tilde{A}\in \{0,1\}^{N\times N}$ and $\tilde{C} \in \{0,1\}^{M\times N}$. Structural system theory examines the general system properties of all dynamic matrix pairs $(A,C)$ that respect the structure $(\tilde{A},\tilde{C})$ in the following sense. An entry of $(A,C)$ is zero if the corresponding entry of $(\tilde{A},\tilde{C})$ is zero, while an entry of $(A,C)$ is an arbitrary parameter if the corresponding entry of $(\tilde{A},\tilde{C})$ is one. In a straightforward way, $(\tilde{A},\tilde{C})$ may be summarized by a directed graph $\mathcal{D}(\tilde{A},\tilde{C})$. Nodes of the graph for each of the $N$ system state variables have connections described by $\tilde{A}$, with a directed link from state $j$ to state $i$ if and only if $\tilde{A}_{ij}=1$. Nodes of the graph for each of the $M$ system output variables have connections to states as described by $\tilde{C}$, with a directed link from state $j$ to output $i$ if and only if $\tilde{C}_{ij}=1$.
Specifically, conditions on the network structure that ensure recoverability of the initial state are desired. Several results concerning structural systems are discussed in the survey paper [@StructSysSurvey], including conditions for structural observability. The pair $(\tilde{A},\tilde{C})$ is said to be [**structurally observable**]{} if there is an observable pair $(A,C)$ that respects the structure $(\tilde{A},\tilde{C})$ [@StructControl]. Additionally, if $(\tilde{A},\tilde{C})$ is structurally observable, nearly all realizations that respect the structure over suitable fields, such as $\mathbb{R}$ or $\mathbb{C}$, are observable in the sense that the set of feasible unobservable realizations must have measure zero [@StructControl]. A result critical to this work, Theorem \[BackgroundConcepts:StructuralSystems:StructuralObservabilityThm\] shows that structural observability of $(\tilde{A},\tilde{C})$ is equivalent to the existence of an output cactus patch, a specific type of graph defined below, that spans $\mathcal{D}(\tilde{A},\tilde{C})$ [@StructSysSurvey]. [**Output cacti**]{} are defined recursively. Consider a collection of nodes labeled either as state nodes or as output nodes. An [**output stem**]{} graph consists of an output node and a single directed elementary path potentially containing several state nodes rooted at the output node, with all links directed toward the output node. All output stems are defined to be output cacti. Furthermore, any existing output cactus to which a directed cycle of state nodes has been attached via a directed link from one node in the cycle to any node of the output cactus is also an output cactus. Finally, a union of node-disjoint output cacti is an [**output cactus patch**]{}.
\[BackgroundConcepts:StructuralSystems:StructuralObservabilityThm\]
The structural system matrix pair $(\tilde{A},\tilde{C})$ is structurally observable if and only if the structural system graph $\mathcal{D}(\tilde{A},\tilde{C})$ is spanned by an output cactus patch.
Formal Network Description {#NetworkDescription}
==========================
To formally describe the operation of the networks under consideration, we distinguish between two closely related network structures, namely, the physical network $\mathcal{G}_{P}$ describing link connections and the underlying dynamic system network $\mathcal{G}_D$ describing the relationships among sensor states and network outputs. The physical network is composed of three types of nodes: a set $\mathcal{X}$ of [**sensor nodes**]{}, a set $\mathcal{Q}$ of [**backbone nodes**]{}, and a [**fusion center node**]{} $\mathcal{Z}=\{z\}$. In this network, sensor nodes contain data and form a local state dynamics. Backbone nodes accept outputs from the sensor nodes and route them to the fusion center node, which aggregates data for state observation tasks. Network connectivity is described by a directed, weighted graph $\mathcal{G}_P=(\mathcal{V}_P,\mathcal{E}_P,w_P)$ where $\mathcal{V}_P=\mathcal{X}\cup\mathcal{Q}\cup\mathcal{Z}$ is the set of all nodes, $\mathcal{E}_P$ is the set of directed feasible links, and $w_P:\mathcal{E}_P\rightarrow \mathbb{R}^+$ is the weight function describing individual directed link cost. Self-links are assumed to always be possible at zero-cost $w_P(x,x)=0$ for all $x\in \mathcal{X}$. Figure \[NetworkExampleA\] provides an example of such a network, showing the sensor nodes (black), backbone nodes (green), and fusion center node (red). Conceptually, this network divides into two subnetworks, the sensing subnetwork and the backbone subnetwork.
\[\]\[\]\[t\]
![Dynamic system graph $\mathcal{G}_D$ showing feasible computation links among state nodes $\mathcal{X}$ (black), output nodes $\mathcal{Y}$ (blue), and central node $\mathcal{Z}$ (red).[]{data-label="NetworkExampleB"}](NetworkExampleGP_Script.png){width="\linewidth"}
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![Dynamic system graph $\mathcal{G}_D$ showing feasible computation links among state nodes $\mathcal{X}$ (black), output nodes $\mathcal{Y}$ (blue), and central node $\mathcal{Z}$ (red).[]{data-label="NetworkExampleB"}](NetworkExampleGD_Script.png){width="\linewidth"}
The [**sensor subnetwork**]{} $\mathcal{G}_S=(\mathcal{V}_S,\mathcal{E}_S,w)$ consists of all directed links between two sensor nodes or from a sensor node to a backbone node. Thus, $\mathcal{V}_S=\mathcal{X}\cup\mathcal{Q}$ and $\mathcal{E}_S=\mathcal{E}_{\mathcal{XX}}\cup\mathcal{E}_{\mathcal{XQ}}$ where $\mathcal{E}_{\mathcal{XX}}=\mathcal{E}_P\cap(\mathcal{X}\times\mathcal{X})$ and $\mathcal{E}_{\mathcal{XQ}}=\mathcal{E}_P\cap(\mathcal{X}\times\mathcal{Q})$. Each sensor $x_i\in\mathcal{X}$ makes a single initial scalar measurement $\boldsymbol\alpha_i$ from field $\mathbb{F}$ and maintains a single scalar variable of state $\mathbf{x}_i(n)$ over discrete time, with initial state $\mathbf{x}_i(0)={\boldsymbol\alpha}_i$. At every iteration, the network updates the state variable of each sensor node using local data according to a linear combination as described by a matrix $A\in \mathbb{F}^{N\times N}$ such that $\mathbf{x}(n+1)=A\mathbf{x}(n)$ where $N=|\mathcal{X}|$. Each row of $A$ indicates which neighboring states are used to compute the updated value of a given state and the coefficients of the linear combination. In this way, each sensor need only know the values in the corresponding row of $A$ rather than the entire network topology and dynamics. When $A_{ij}\neq 0$, sensor node $x_j$ is linked to $x_i$, incurring cost. Otherwise, no link exists between the two nodes.
The [**backbone subnetwork**]{} $\mathcal{G}_B=(\mathcal{V}_B,\mathcal{E}_B,w)$ consists of all directed links between two backbone nodes or from a backbone node to the fusion center. Thus, $\mathcal{V}_B=\mathcal{Q}\cup\mathcal{Z}$ and $\mathcal{E}_B=\mathcal{E}_{\mathcal{QQ}}\cup\mathcal{E}_{\mathcal{QZ}}$ where $\mathcal{E}_{\mathcal{QQ}}=\mathcal{E}_P\cap(\mathcal{Q}\times \mathcal{Q})$ and $\mathcal{E}_{\mathcal{QZ}}=\mathcal{E}_P\cap(\mathcal{Q}\times \mathcal{Z})$. Some of the states are output (sensed) through the network backbone nodes in $\mathcal{Q}$ and available at the central node $z$ as the vector $\mathbf{y}(n)$. It is assumed that the backbone subnetwork operates much faster than the sensor subnetwork iterations, such that $\mathbf{y}(n)$ is available to $z$ without delay. For link cost efficiency, the backbone network uses the least costly path to the central node. These outputs can be described by a matrix $C\in\mathbb{F}^{M\times N}$ such that $\mathbf{y}(n)=C\mathbf{x}(n)$ where $M=|\mathcal{E}_{\mathcal{XQ}}|$ is the number of feasible connections between the sensors and backbone nodes. Each row $i$ of $C$ either is composed of all zeros, indicating an output is not made over the corresponding connection and incurring no cost, or has a single nonzero entry in column $j$, indicating state $x_j$ is output over the connection and incurring cost.
Consider the system dynamics network $\mathcal{G}_D$ that describes the relationships among sensor states, backbone outputs, and fusion center in the physical network. The sensor nodes $\mathcal{X}$ of the physical network correspond to the [**state nodes**]{}, and maintain the same links $\mathcal{E}_{\mathcal{XX}}$ with $w_D(x_1,x_2)=w_P(x_1,x_2)$ for $x_1,x_2\in\mathcal{X}$. Additionally, the fusion center node $\mathcal{Z}=\{z\}$ appears in this network. Because the backbone nodes may output the states from multiple sensor nodes, backbone nodes do not correspond to only one output. Rather, each link in $\mathcal{E}_{\mathcal{XQ}}$ from a sensor node to a backbone node represents a potential output of the system. Thus, let a set $\mathcal{Y}$ of **output nodes** be indexed by $\mathcal{E}_{\mathcal{XQ}}$ such that $y_{(x,q)}\in\mathcal{Y}$ corresponds to $(x,q)\in \mathcal{E}_{\mathcal{XQ}}$. Each output node $y_{(x,q)}$ will be linked to the corresponding sensor node $x$ with weight $w_D(x,y_{(x,q)})=w_P(x,q)$ and to the central node with weight $w_D(y_{(x,q)},z)=w_{\operatorname*{sp}}(q,z;\mathcal{E}_{B})$ where $w_{\operatorname*{sp}}(q,z;\mathcal{E}_{B})$ is the weight of the shortest path from $q$ to $z$ over the backbone links $\mathcal{E}_B$. Hence, $\mathcal{E}_{\mathcal{XY}}=\{(x,y_{(x,q)})|(x,q)\in\mathcal{E}_{\mathcal{XQ}}\}$ and $\mathcal{E}_{\mathcal{YZ}}=\{(y_{(x,q)},z)|(x,q)\in\mathcal{E}_{\mathcal{XQ}}\}$. Note that state nodes may connect to multiple output nodes, but each output node corresponds to only one state node. Also, note that output nodes do not have links to each other, so the output nodes together with the fusion center form a star topology subnetwork. Thus, the graph that describes the state and output update dependencies is $\mathcal{G}_D=(\mathcal{X}\cup\mathcal{Y}\cup\mathcal{Z},\mathcal{E}_{\mathcal{XX}} \cup\mathcal{E}_{\mathcal{XY}} \cup\mathcal{E}_{\mathcal{YZ}},w_D)$. As with the physical network, a link only contributes cost to the network operation if it is required for updating states and outputs. The nodes, links, and weights of this network are entirely derived from the physical network $\mathcal{G}_P$, and Figure \[NetworkExampleB\] provides an example of such a modification. The figure shows the organization into state nodes (black), output nodes (blue), and the central node (red).
The network state $\mathbf{x}(n)$ and the network output $\mathbf{y}(n)$ behave according to the dynamical system - with $\mathbf{x}(0)={\boldsymbol\alpha}$ being the initial state. Observability of the system must be ensured when designing the matrix pair $(A,C)$ in -, which translates to the full rank condition on $O_{(A,C)}$. Furthermore, the fusion center must be aware of the entire network dynamics so that the observability matrix is known and state observation tasks may be performed.
Additionally, each node must only access the state from its neighbors in $\mathcal{G}_P$ to update its state or output values but does not necessarily use information from every neighbor. Let $\mathcal{G}_P(A,C)=(\mathcal{V}_P,\mathcal{E}_P(A,C),w_P)$ be the subgraph of $\mathcal{G}_P$ defined by the set $\mathcal{E}_P(A,C)$ of physical links required for the computation of - and outputs along the least costly backbone path. Note that $\mathcal{E}_P(A,C)=\mathcal{E}_{\mathcal{XX}}(A)\cup \mathcal{E}_{\mathcal{XQ}}(C) \cup \mathcal{E}_{B}(C)$ with links in $\mathcal{E}_{\mathcal{XX}}(A)$ used to compute , links in $\mathcal{E}_{\mathcal{XQ}}(C)$ used in , and links in $\mathcal{E}_{B}(C)$ used in the shortest path from every used backbone node $q$ directly connected to some sensor $x$ by $(x,q)\in\mathcal{E}_{\mathcal{XQ}}(C)$. For feasibility of network operations, the subset constraints $\mathcal{E}_{\mathcal{XX}}(A)\subseteq \mathcal{E}_{\mathcal{XX}}$, $\mathcal{E}_{\mathcal{XQ}}(C)\subseteq \mathcal{E}_{\mathcal{XQ}}$, and $\mathcal{E}_{B}(C)\subseteq \mathcal{E}_B$ must hold.
Hence, these link feasibility and system observability constraints must be applied when designing the network dynamics $(A,C)$. Furthermore, minimization of the sum total cost $$\label{objective}
F(A,C)=\sum_{\begin{array}{c}e\in \mathcal{E}_P(A,C)\end{array}}{ w_P(e)}$$ of all used network physical links is desired, including sensor state updates, sensor output production, and backbone activity. Section \[DesignProblem\] formulates a robust version of the minimum cost network design problem with this objective function and set of constraints.
Network Design Problem {#DesignProblem}
======================
The problem examined in this paper concerns optimal design of networks operating according to the description in Section \[NetworkDescription\]. As equation shows, the underlying system dynamics determines the physical link cost of operating such networks. A less complete formulation of this problem appeared in [@PequitoKruzickEUSIPCO2013], which provides an efficient solution algorithm based on minimum spanning tree methods for the simpler and much more fragile case in which only one backbone node exists and no node failures may occur. In contrast, this paper significantly generalizes this previous work by formulating the optimal design problem in the context of nontrivial backbone subnetworks and sensor node failures.
Therefore, this section examines minimization of the objective function in with respect to the dynamic matrices $(A,C)$ subject to the aforementioned system observability and link-connectivity constraints under added robustness requirements arising from sensor failures. Specifically, consider optimal design of a link feasible network where system observability must be guaranteed, while any subset of sensor nodes $\mathcal{U}\subset \mathcal{X}$ of size $|\mathcal{U}|\leq k$ less than the robustness design parameter $k$ experiences total failure before coordination begins. In this way, the original states of nodes that did not fail could still be recovered from the collected outputs. Sensor node failure results in the elimination of associated rows and columns in the network dynamics matrices. Note that the row of $A$, columns of $A$, and columns of $C$ are indexed by the sensor nodes $\mathcal{X}$, while the rows of $C$ are indexed by $\mathcal{E}_{\mathcal{XQ}}$. Denote by ${A}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}$ and ${C}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]}$ submatrices of $A$ and $C$, respectively, in which rows and columns corresponding to $\mathcal{U}$ have been removed. This yields the optimization problem appearing below. $$\begin{aligned}
{4}
&{\operatorname*{argmin}\limits}_{{A},{C}} \quad && F({A},{C}) \label{DesignProblem:V1:Eq1} \\
&\operatorname*{s.t.}\quad &&\mathcal{E}_{\mathcal{XX}}({A})\quad\subseteq \quad \mathcal{E}_{\mathcal{XX}} \label{DesignProblem:V1:Eq2} \\
& \quad &&\mathcal{E}_{\mathcal{XQ}}({C})\quad\subseteq \quad \mathcal{E}_{\mathcal{XQ}} \label{DesignProblem:V1:Eq3}\\
& \quad &&\mathcal{E}_{B\hphantom{B}}({C})\quad\subseteq \quad \mathcal{E}_{B} \label{DesignProblem:V1:Eq4}\\
& \quad &&({A}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]},{C}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]}) \label{DesignProblem:V1:Eq5}\\
& \quad &&\mathrlap{\textrm{observable}}\hphantom{\textrm{spanned by output cactus patch}} \nonumber\\
& \quad &&\textrm{for all }\mathcal{U}\subset \mathcal{X}\textrm{ with }|\mathcal{U}|\leq k\nonumber
\end{aligned}$$
The optimization described by - appears seemingly difficult due to the large number of observability constraints that must be satisfied. Recall that Theorem \[BackgroundConcepts:SystemTheory:ObservabilityThm\] states the equivalence between observability of $(A,C)$ and the condition $\operatorname*{rank}(O_{(A,C)})=N$. Thus, the constraint given in translates to $\operatorname*{rank}(O_{({A}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}, {C}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]}})=N-|\mathcal{U}|$ for all $\mathcal{U}\subset \mathcal{X}$ with $|\mathcal{U}|\leq k$. Because there are potentially very many ways to form subsets of $\mathcal{X}$ of size at most $k$, a large number of rank constraints must be satisfied. As such, it is not immediately clear that this problem is efficiently solvable. Furthermore, it is not obvious how to check that a solution exists for a given $k$ or to determine the largest $k$ for which a solution exists. Later in the paper, after additional discussion of this problem, Remark \[DesignProblem:ExistenceRmk\] addresses existence of the solution in terms of flow problems. However, the optimization problem must first be related more closely to the combinatorial structure of the graph.
A structural systems approach renders the problem tractable by decoupling the problem of finding the optimal structure for the dynamic matrices from the problem of finding an observable instantiation of the optimal structure. Note that the objective function in only depends on which links were used, as do the link feasibility constraints -. Thus, it depends on only the zero-nonzero structure $(\tilde{A},\tilde{C})$ of $(A,C)$ and can be restated in structural terms. By the definition of structural observability, if $(\tilde{A},\tilde{C})$ is structurally observable then an observable instantiation $(A,C)$ respecting that structure must exist and may be found subsequently. Therefore, the final constraint can be rewritten to guarantee structural observability under limited node failures. The problem appears below with this reformulation. $$\begin{aligned}
{4}
&{\operatorname*{argmin}\limits}_{\tilde{A},\tilde{C}} \quad && F(\tilde{A},\tilde{C}) \label{DesignProblem:V2:Eq1} \\
&\operatorname*{s.t.}\quad &&\mathcal{E}_{\mathcal{XX}}(\tilde{A})\quad\subseteq \quad \mathcal{E}_{\mathcal{XX}} \label{DesignProblem:V2:Eq2} \\
& \quad &&\mathcal{E}_{\mathcal{XQ}}(\tilde{C})\quad\subseteq \quad \mathcal{E}_{\mathcal{XQ}} \label{DesignProblem:V2:Eq3}\\
& \quad &&\mathcal{E}_{B\hphantom{B}}(\tilde{C})\quad\subseteq \quad \mathcal{E}_{B} \label{DesignProblem:V2:Eq4}\\
& \quad &&({\tilde{A}}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]},{\tilde{C}}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]}) \label{DesignProblem:V2:Eq5}\\
& \quad &&\mathrlap{\textrm{structurally observable}}\hphantom{\textrm{spanned by output cactus patch}} \nonumber\\
& \quad &&\textrm{for all }\mathcal{U}\subset \mathcal{X}\textrm{ with }|\mathcal{U}|\leq k\nonumber
\end{aligned}$$
Replacement of the observability constraint with a structural observability constraint may not appear, at first, to suggest a solution. However, by appealing to Theorem \[BackgroundConcepts:StructuralSystems:StructuralObservabilityThm\], the structural observability constraint can be replaced by the equivalent condition that the associated directed graph be spanned by an output cactus patch after any set of at most $k$ sensor node deletions are applied. Hence, the original analytic rank constraint has been transformed to a combinatorial structural constraint in the problem below. $$\begin{aligned}
{4}
&{\operatorname*{argmin}\limits}_{\tilde{A},\tilde{C}} \quad && F(\tilde{A},\tilde{C}) \label{DesignProblem:V3:Eq1} \\
&\operatorname*{s.t.}\quad &&\mathcal{E}_{\mathcal{XX}}(\tilde{A})\quad\subseteq \quad \mathcal{E}_{\mathcal{XX}} \label{DesignProblem:V3:Eq2} \\
& \quad &&\mathcal{E}_{\mathcal{XQ}}(\tilde{C})\quad\subseteq \quad \mathcal{E}_{\mathcal{XQ}} \label{DesignProblem:V3:Eq3}\\
& \quad &&\mathcal{E}_{B\hphantom{B}}(\tilde{C})\quad\subseteq \quad \mathcal{E}_{B} \label{DesignProblem:V3:Eq4}\\
& \quad &&D({\tilde{A}}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]},{\tilde{C}}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]}) \label{DesignProblem:V3:Eq5}\\
& \quad &&\textrm{spanned by output cactus patch} \nonumber\\
& \quad &&\textrm{for all }\mathcal{U}\subset \mathcal{X}\textrm{ with }|\mathcal{U}|\leq k \nonumber
\end{aligned}$$
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This optimization problem is solved by Algorithm 1, which takes as input the graph $\mathcal{G}_P$ of all possible physical network links. Through six steps, illustrated in Figure \[Alg1Fig\], it produces as output an optimal dynamics structure $(\tilde{A}^*,\tilde{C}^*)$ for the operation of the network. This also defines the subgraph $\mathcal{G}_P(\tilde{A}^*,\tilde{C}^*)$ of $\mathcal{G}_P$ that is necessary for the operation of the network, assuming backbone forwarding occurs along the least costly path. Practical discussion of how to find an observable instantiation of this structure appears later in this section.
Algorithm 1 begins by first finding the cost of the fusion center accessing a single output in the physical links cost graph $\mathcal{G}_P$ from each backbone node by computing the shortest path spanning arborescence of the backbone subnetwork in Step 1. Subsequently, Step 2 generates the dynamic system computation cost graph $\mathcal{G}_D$, which it accomplishes by removing the backbone nodes and creating an output node $y_{(x,q)}$ for every $(x,q)\in \mathcal{E}_{\mathcal{XQ}}$. This output node connects to only one sensor node $x$ with cost $w_D(x,y_{(x,q)})=w_P(x,q)$ and the fusion center $z$ with cost $w_D(y_{(x,q)},z)= w_{\operatorname*{sp}}(q,z;\mathcal{E}_B)$ obtained from the shortest path spanning arborescence found in Step 1. In order to frame the optimization problem as an efficiently solvable (by submodular flow) case of the Steiner subgraph problem with $\mathcal{X}$ as the terminal node set, all augmenting links must have tail in the terminal node set. Therefore, Step 3 produces the modified graph $\mathcal{G}_D'$ in which $w_D'(x,y_{(x,q)})=w_P(x,q)+w_{\operatorname*{sp}}(q,z;\mathcal{E}_B)$ and $w_D'(y_{(x,q)},z)=0$. Note that any minimal $z$-rooted Steiner subgraph of $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$ either include both $(x,y_{(x,q)})$ and $(y_{(x,q)},z)$ or includes neither $(x,y_{(x,q)})$ nor $(y_{(x,q)},z)$. Hence, this modification does not affect the total cost of any minimal solution. Step 4 performs the optimization step, which is a minimum $z$-rooted $(k+1)$-node-connected Steiner subgraph computation for $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$. Because all augmenting links have tail in $\mathcal{X}$ due to the modification in Step 3, the problem is solvable in polynomial time using submodular flows [@FrankRootedKConnections]. In fact, it can be solved through a maximum weighted matroid intersection algorithm [@FrankRootedKConnections] by the method in Remark \[DesignProblem:ComplexityRmk\]. Step 5 constructs a graph that is guaranteed to be spanned by an output cactus patch through the addition of zero-cost self loops to the sensor nodes. Finally, Step 6 interprets the graph from Step 5 as a structural system, allowing an optimal structurally observable network dynamics matrix pair to be output from the algorithm. Theorem \[DesignProblem:MainThm\] and the corresponding proof show that the supplied algorithm solves the minimum cost design problem for any input physical links cost network $\mathcal{G}_P$ and robustness requirement $k$ such that a solution exists.
\[DesignProblem:MainThm\]
For the physical links cost network $\mathcal{G}_P$ and robustness requirement $k$ the structural dynamics pair $(\tilde{A}^*,\tilde{C}^*)$ output by Algorithm 1 is a solution to the optimization problem in -, provided that for each $x\in \mathcal{X}$ there is a set of at least $k+1$ internally node-disjoint directed paths in $\mathcal{G}_P$ each beginning at $x$ and ending in $Q$ and that there is a directed path from each $q\in Q$ to $z$.
\[DesignProblem:MainPrf\]
We first demonstrate the feasibility and existence of the solution $(\tilde{A}^*,\tilde{C}^*)$. Subsequently, we show by contradiction that no other feasible solution incurring lesser cost exists. Hence, we conclude the solution found is optimal.
Note that the shortest path spanning arborescence of the backbone subnetwork computed in Step 1 must exist because there is a directed path from each $q\in \mathcal{Q}$ to $z$. There are $k+1$ internally node-disjoint directed paths in $\mathcal{G}_P$ beginning at $x$ and ending in $\mathcal{Q}$, each of which contains a distinct link $(x_i,q_i)\in \mathcal{E}_{\mathcal{XQ}}$. Thus, the graph $\mathcal{G}_D$ constructed in Step 2 has $k+1$ internally node-disjoint directed paths beginning at $x$ and ending in a distinct $y_{i}=y_{(x_i,q_i)}\in\mathcal{Y}$, which can be constructed from the paths in $\mathcal{G}_P$ by substituting $(x_i,y_{i})$ for the final link. Furthermore, because each $y_i\in\mathcal{Y}$ connects directly to $z$, this implies that $\kappa_{\mathcal{G}_D}(x,z)\geq k+1$ for all $x\in\mathcal{X}$. Since construction of $\mathcal{G}_D'$ in Step 3 does not alter connectivity, it also follows that $\kappa_{\mathcal{G}_D'}(x,z)\geq k+1$ for all $x\in\mathcal{X}$. Hence, the minimum $z$-rooted $(k+1)$-node-connected Steiner subgraph of $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$ exists and $\mathcal{T}^*$ can be found in Step 4. With deletion of any failing node set $\mathcal{U}\subset \mathcal{X}$ with $|\mathcal{U}|\leq k$, there is at least one path from each node in $\mathcal{X}\backslash\mathcal{U}$ to $z$ by $z$-rooted $(k+1)$-node-connectedness. These paths form a spanning tree $\mathcal{T}_{\mathcal{U}}^*$ for the subgraph of $\mathcal{T}^*$ in which $\mathcal{U}$ has been excluded. Because zero-cost self-loops are always permitted at every $x\in \mathcal{X}$, $\mathcal{P}^*$ may be formed in Step 5, and the subgraph of $\mathcal{P}^*$ that excludes $\mathcal{U}$ is spanned by $\mathcal{T}_{\mathcal{U}}^*$ with $z$ removed. Let $\mathcal{P}_{\mathcal{U}}^*$ be the graph formed from $\mathcal{T}_{\mathcal{U}}^*$ by addition of zero cost self-loops to all $x\in \mathcal{X}\backslash\mathcal{U}$ and with $z$ removed. Note that $\mathcal{P}_{\mathcal{U}}^*$ satisfies the recursive definition of an output cactus patch, with output nodes $\mathcal{Y}$, and that $\mathcal{P}_{\mathcal{U}}^*$ spans the subgraph of $\mathcal{P}^*$ with $\mathcal{U}$ excluded. By the construction in Step 6, $\mathcal{D}(\tilde{A}^*,\tilde{C}^*)$ is isomorphic to $\mathcal{P}^*$, and, consequently, $D({\tilde{A}^*}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}, {\tilde{C}^*}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]})$ is spanned by an output cactus patch isomorphic to $\mathcal{P}_{\mathcal{U}}^*$ (by the same isomorphism). Hence, by Theorem \[BackgroundConcepts:StructuralSystems:StructuralObservabilityThm\], $({\tilde{A}^*}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}, {\tilde{C}^*}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]})$ is structurally observable for all $\mathcal{U}\subset \mathcal{X}$ with $|\mathcal{U}|<k$. The other constraints, $\mathcal{E}_{\mathcal{XX}}(\tilde{A}^*)\subseteq \mathcal{E}_{\mathcal{XX}}$, $\mathcal{E}_{\mathcal{XQ}}(\tilde{C}^*)\subseteq \mathcal{E}_{\mathcal{XQ}}$, and $\mathcal{E}_{B}(\tilde{C}^*)\subseteq \mathcal{E}_{B}$, are also satisfied through subgraph constructions. Therefore, $(\tilde{A}^*,\tilde{C}^*)$ is a feasible solution.
Assume by way of contradiction that a feasible solution $(\tilde{A}^\dag,\tilde{C}^\dag)$ of lesser cost than $(\tilde{A}^*,\tilde{C}^*)$ with respect to the objective function $F$ exists. That is, $$\label{DesignProblem:MainPrf:Eq1}
F(\tilde{A}^\dag,\tilde{C}^\dag)<F(\tilde{A}^*,\tilde{C}^*).$$ Because $({\tilde{A}^\dag}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}, {\tilde{C}^\dag}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]})$ must be structurally observable for all $\mathcal{U}\subset\mathcal{X}$ with $|\mathcal{U}|\leq k$, $D({\tilde{A}^\dag}_{[\mathcal{X}\backslash\mathcal{U},\mathcal{X}\backslash\mathcal{U}]}, {\tilde{C}^\dag}_{[\mathcal{E}_{\mathcal{XQ}},\mathcal{X}\backslash\mathcal{U}]})$ must have a minimum cost subgraph $\mathcal{P}^\dag(\mathcal{X}\cup\mathcal{Y}, \mathcal{E}_{\mathcal{P}^\dag},w_D')$, taking into account sensor to output link costs, which is spanned by an output cactus patch when any such node set $\mathcal{U}$ is removed. Note that $$\label{DesignProblem:MainPrf:Eq2}
W(\mathcal{P}^\dag) \leq W(\mathcal{D}(\tilde{A}^\dag,\tilde{C}^\dag))=F(\tilde{A}^\dag,\tilde{C}^\dag)$$ where $W(\cdot)$ gives the total weight of a graph. Furthermore there are at least $k+1$ internally node-disjoint paths beginning at $x$ and ending in $\mathcal{Y}$ in $\mathcal{P}^\dag$ for all $x\in\mathcal{X}$ because any graph disconnected from all outputs by $k$ sensor node failures could not have that property. Reversing the process in Step 5, construct $\mathcal{T}^\dag(\mathcal{X}\cup\mathcal{Y}\cup\mathcal{Z}, \mathcal{E}_{\mathcal{T}^\dag},w_D')$ where $\mathcal{E}_{\mathcal{T}^\dag}=(\mathcal{E}_{\mathcal{P}^\dag} \backslash\{(x,x)|x\in\mathcal{X}\})\cup\{(y,z)|(x,y) \in\mathcal{E}_{\mathcal{P}^\dag}\textrm{ for some }x\in\mathcal{X}\}$. Because the self-loops have zero cost and output links connecting to $z$ have weight $0$ with respect to $w_D'$, it follows that $$\label{DesignProblem:MainPrf:Eq3}
W(\mathcal{T}^\dag)=W(\mathcal{P}^\dag).$$ Noting that $\kappa_{\mathcal{T}^\dag}(x,z)\geq k+1$ for all $x\in \mathcal{X}$, it is clear that $\mathcal{T}^\dag$ is a $z$-rooted $(k+1)$-node-connected Steiner subgraph of $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$. By - $$\label{DesignProblem:MainPrf:Eq4}
W(\mathcal{T}^\dag)\leq F(\tilde{A}^\dag,\tilde{C}^\dag).$$ Similarly, $$\label{DesignProblem:MainPrf:Eq5}
W(\mathcal{T}^*)=W(\mathcal{P}^*),$$ and, this time with equality, $$\label{DesignProblem:MainPrf:Eq6}
W(\mathcal{P}^*)=W(\mathcal{D}(\tilde{A}^*,\tilde{C}^*))= F(\tilde{A}^*,\tilde{C}^*).$$ Hence, by -, $$\label{DesignProblem:MainPrf:Eq7}
W(\mathcal{T}^*)=F(\tilde{A}^*,\tilde{C}^*),$$ so it follows from that $$\label{DesignProblem:MainPrf:Eq8}
W(\mathcal{T}^\dag)<W(\mathcal{T}^*).$$ This contradicts the fact that $\mathcal{T}^*$ is a minimum $z$-rooted $(k+1)$-node-connected Steiner subgraph for $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$. Thus, $(\tilde{A}^*,\tilde{C}^*)$ is a minimum cost solution to the optimization problem. $\small \blacksquare$
\[DesignProblem:ExistenceRmk\] The condition that for each $x\in \mathcal{X}$ there is a set of at least $k+1$ internally node-disjoint directed paths in $\mathcal{G}_P$ each beginning at $x$ and ending in $\mathcal{Q}$ may be efficiently verified through $N=|\mathcal{X}|$ maximum $(\{x_s\},\mathcal{Q})$-flow computations (with restricted vertex capacity) where the source node is $x_s=x$, non-source nodes in $\mathcal{X}$ have capacity 1, and links in $\mathcal{E}_{\mathcal{XX}}\cup\mathcal{E}_{\mathcal{XQ}}$ have capacity 1. This process can also be used to find the largest value of $k$ for which the solution exists. Among all the nodes $x\in \mathcal{X}$ find the one with smallest node-capacitated flow to $\mathcal{Q}$. The largest admissible value of $k$ is 1 less than that flow value.
Before applying Algorithm 1 to obtain an optimal dynamics structure $(\tilde{A}^*,\tilde{C}^*)$ that solves the problem in $\eqref{DesignProblem:V2:Eq1}-\eqref{DesignProblem:V2:Eq5}$, it would be prudent to ensure that a solution providing robustness factor $k$ exists for the input physical links cost graph $\mathcal{G}_P$. In order to accomplish this, it must be verified that for each $x\in \mathcal{X}$ there is a set of at least $k+1$ internally node-disjoint paths beginning at $x$ and ending in $\mathcal{Q}$, in which case Theorem \[DesignProblem:MainThm\] guarantees that a solution exists. This condition can be tested by computing the maximum flow from each $x_s$, treated as the only source, to $Q$, treated as a set of sinks, where all links $\mathcal{E}_{\mathcal{XX}}\cup\mathcal{E}_{\mathcal{XQ}}$ have capacity 1 and the nodes $x\in \mathcal{X}\backslash \{x_s\}$ also have capacity 1. Through node splitting techniques [@FrankRootedKConnections], the node capacities may be converted to link capacities. By connecting all sinks to a common sink by links of infinite capacity, the result can then be formulated as a standard maximum flow problem, which can be solved in polynomial time by a number of algorithms, such as the Ford-Fulkerson algorithm [@CombinatorialOpt]. Because each node can only be used once due to the node flow capacity, the desired condition holds if the maximum $(\{x_s\},\mathcal{Q})$-flow is at least $k+1$ for all $x_s\in \mathcal{X}$.
\[DesignProblem:ComplexityRmk\]
The optimization in Step 4 may be simplified to a minimum $z$-rooted $k$-node-connected spanning subgraph computation by inserting an additional $k$ zero-cost parallel (duplicate) links from each $y\in\mathcal{Y}$ to $z$ in $\mathcal{G}_D'$. This computation can be computed in polynomial time through maximum weighted matroid intersection methods detailed in [@FrankRootedKConnections].
While the optimization in Step 4 could be solved as a Steiner subgraph problem because all augmenting links have tail in the terminal nodes $\mathcal{X}$, it is simpler to convert the problem to a $z$-rooted $k$-node-connected spanning subgraph computation. All nodes in $\mathcal{X}$ have $k+1$ internally node-disjoint paths to $Q$ assuming a solution exists. Thus, each $x\in\mathcal{X}$ has $k+1$ internally node-disjoint paths to $\mathcal{Y}$, each ending at a different node. Nodes in $\mathcal{Y}$ have only one direct connection to $z$. Therefore, addition of $k$ zero-cost parallel links from each $y\in \mathcal{Y}$ to $z$ makes $\mathcal{G}_D'$ $z$-rooted $k$-node-connected as a whole. By computing the $z$-rooted $k$-node-connected spanning subgraph and discarding redundant zero-cost edges from $\mathcal{Y}$ to $\mathcal{Z}$, the solution is obtained.
Note that Step 4 is, by far, the most costly step and subsumes the complexity of the other steps. The $z$-rooted $k$-node-connected spanning subgraph can be computed in polynomial time as a maximum weighted common independent set of two matroids as described in [@FrankRootedKConnections]. Matroids provide rules that define which subsets of a set are independent. These rules must satisfy several axioms that will not be detailed in this paper. For two matroids over edge set $E$, the basic weighted matroid intersection algorithm involves $O(|S|^2||E|)$ calls to the two matroid independence oracles, functions that determine membership of a set in the matroid [@CombinatorialOpt], where $|S|$ is the maximum size of a common independent set. In this case $|S|$, the maximum size of a common independent set, is $|S|=(k+1)(|\mathcal{X}|+|\mathcal{Y}|)$. Links from $Y$ to $z$ are always included in the solution, so they can be trivially included in the final set, making $|S|$ is $O(k|\mathcal{X}|)$ for the purpose of counting calls to the matroid oracles. The set is $E=\mathcal{E}_{\mathcal{XX}}\cup\mathcal{E}_{\mathcal{XY}} \cup\mathcal{E}_{\mathcal{YZ}}$, but links from $Y$ to $z$ are always included in the final solution, so $|E|$ is $O(|\mathcal{E}_{\mathcal{XX}}|+|\mathcal{E}_{\mathcal{XY}}|)$ for purposes of counting. For this problem, one matroid oracle has relatively low complexity, while the other has complexity $O(|V||S|^2)$ [@FrankRootedKConnections] where $|V|$ is $O(|\mathcal{X}|+|\mathcal{Y}|)$. Thus the total complexity of the algorithm is $O(|V||E||S|^4)$ or $O(k^4|\mathcal{X}|^4(|\mathcal{X}|+|\mathcal{Y}|) (|\mathcal{E}_{\mathcal{XX}}|+|\mathcal{E}_{\mathcal{XY}}|))$. It is claimed in [@FrankRootedKConnections] that the matroid oracle can be computed in $O(|V|^3)$, which would reduce this to $O(|V|^3|E||S|^2)$ or $O(k^2|\mathcal{X}|^2(|\mathcal{X}|+|\mathcal{Y}|)^3 (|\mathcal{E}_{\mathcal{XX}}|+|\mathcal{E}_{\mathcal{XY}}|))$. Improved matroid intersection algorithms also exist.
\[DesignProblem:SpecialCaseRmk\] For the case in which $k=0$, the optimization in Step 4 reduces to a minimum $z$-rooted spanning arborescence of $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$.
The special case $k=0$ of the minimum cost network design problem, which requires no robustness to node failures, was originally proposed and solved in [@PequitoKruzickEUSIPCO2013] when symmetric cost structure is considered.
Under these conditions, the Steiner subgraph optimization step reduces to a minimum $z$-rooted Steiner arborescence problem on $\mathcal{G}_D'$ with terminal nodes $\mathcal{X}$. Because each $y\in \mathcal{Y}$ may be automatically connected directly to $z$ by the non-augmenting link $(y,z)$, this can be further reduced to a minimum spanning tree arborescence, leading to Remark \[DesignProblem:SpecialCaseRmk\]. Thus, this case may be computed with greater simplicity using Edmond’s algorithm [@PequitoKruzickEUSIPCO2013].
\[DesignProblem:InstantiationsRmk\] An observable instantiation $(A,C)$ of an optimal structurally observable solution $(\tilde{A}^*,\tilde{C}^*)$ output by Algorithm 1 over a finite field $\mathbb{F}_{p^n}$ can be found randomly with high probability for fields of large order $p^n$. As $p^n$ grows without bound, the probability of an observable random instantiation over $\mathbb{F}_{p^n}$ approaches 1 [@SundaramHadjicostis2].
Although the output of the algorithm gives an optimal structure $(\tilde{A}^*,\tilde{C}^*)$ of the system dynamics that satisfies the constraints, an observable instantiation $(A,C)$ of the dynamics must be obtained for any implementation. If the field in which the system operates is $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$, such instantiations of the structure are guaranteed to exist and the set of unobservable realizations over those fields has zero measure. Consequently, a random instantiation of the structure from a suitable distribution would be almost surely observable with probability one.
While interesting, this approach suffers from the possibility of producing systems with poorly conditioned observability matrices that are full rank yet present numerical problems. Additionally, physical devices cannot truly operate with general real or complex numbers. These problems may both be avoided through the use of finite fields $\mathbb{F}=\mathbb{F}_{p^n}$ where $p$ is prime and $n$ is a positive integer. It has been shown in [@SundaramHadjicostis2] that for an output tree with self-loops attached to the $N$ state nodes, an observable instantiation is guaranteed to exist over $\mathbb{F}_{p^n}$ if $p^n\geq N$. In fact, Corollary 1 of [@SundaramHadjicostis2] shows that this instantiation can be produced by assigning a distinct field element to each self-loop and the multiplicative identity to each link between distinct nodes. Instead, if elements are chosen randomly at each sensor, repetition can be avoided and observability achieved with high probability if the field is sufficiently large, as shown in Theorem 5 of [@SundaramHadjicostis2], approaching probability 1 as $p^n$ increases without bound. This provides a more advantageous approach to practical implementation and is summarized by Remark \[DesignProblem:InstantiationsRmk\].
![The plot shows simulated network failure probability plotted against the fraction of sensor node failures for designed robustness levels $k=0,\ldots,3$ in networks with $|\mathcal{X}|=50$. Results show a gentler slope for higher values of $k$, even beyond the robustness guarantee.[]{data-label="DesignProblem:RobustnessFig"}](RobustFig100.png){width="\linewidth"}
![This illustrative example shows a small network on the unit square designed with $|X|=30$ randomly placed sensor nodes (black), and $|Q|=4$ backbone nodes (green), central node $Z$ (red), and robustness level $k=2$, and distance squared costs with maximum radius $R_{lim}=.4$. The colors allow comparison to Figure \[NetworkExampleA\].[]{data-label="DesignProblem:ExampleFig"}](ExampleFig100.png){width=".75\linewidth"}
\[DesignProblem:RobustnessRmk\] While Algorithm 1 guarantees robustness to failure of any node subset $\mathcal{U}\subset \mathcal{X}$ with $|\mathcal{U}|\leq k$, the solution may be robust in practice to a larger number of randomly chosen node failures. Here, as in the rest of the paper, robustness refers to the fact that at least one spanning output cactus remains intact after $k$ node failures. That is, no surviving nodes are disconnected and, thus, unobservable. For more than $k$ node failures, the idea of robustness can be relaxed from this guarantee to examine the failure probability.
In practice, a network designed by Algorithm 1 may be robust to more sensor node failures than the guaranteed degree of robustness $k$. Although the algorithm guarantees that removal of any subset of failing sensors $\mathcal{U}\subseteq\mathcal{X}$ with $|\mathcal{U}|\leq k$ does not break the structural observability of the remaining network, it does not necessarily follow that the network fails if $|\mathcal{U}|=\ell>k$. For instance, the structure of the sensor subnetwork for $k=0$ is a $Q$ rooted branching with loops, and failure of any leaf node does not affect the remainder of the network. This effect becomes more pronounced for higher values of $k$. With $\ell$ node failures selected uniformly at random, the probability of network failure cannot be easily computed. However, it is possible to empirically simulate it for networks constructed from pseudo-randomly placed nodes with pseudo-random node failures.
Figure \[DesignProblem:RobustnessFig\] plots the empirical probability of network failure for pseudo-randomly generated node locations against the node failure ratio $\ell/|\mathcal{X}|$ for designed degree of robustness $k=0,...,3$. This simulation used $|\mathcal{X}|=50$ uniformly distributed sensor nodes, $|\mathcal{Y}|=3$ backbone nodes, and distance squared link cost. Figure \[DesignProblem:ExampleFig\] represents the graph of a smaller network designed in such a way, with the additional constraint on the communication radius. For each randomly generated network and failure ratio value, $\ell$ sensor nodes were randomly selected to fail. The network fails if any surviving node no longer has a directed path to a backbone node. The expected network failure probability was computed for each node failure ratio over 100 random graphs each with 1000 random sets of failing nodes. As seen in Figure \[DesignProblem:RobustnessFig\], the resulting networks can have low failure probability even beyond the robustness guarantee, where failure occurs with probability zero. The plot also demonstrates that when designing for higher guaranteed robustness level, the failure probability grows much more slowly. For instance, when designing for robustness level $k=3$, the failure probability is only approximately $0.2$ when $10$ nodes out of $50$ sensor nodes fail.
Conclusions {#Conclusions}
===========
The distributed networks of agents or sensors examined in this paper model situations in which nodes update their state variables using values obtained by sensing state information from nodes within their local neighborhoods. The state of some nodes is directly sensed by nodes in a network backbone, which are directly accessed by a central fusion center. The fusion center computes the (global) network state from the data it obtains. This type of scenario may arise in robotics formation control applications or in field inversion problems, for instance. By duality of observability and controllability, the same technique may be used to drive the sensor states to some desired target.
This paper posed the problem of finding minimum cost network update dynamics with respect to a link cost objective function under an observability constraint. The polynomial time algorithm proposed finds the optimal network that is structurally observable when any $k$ sensor nodes are removed by using combinatorial optimization algorithms, guaranteeing the existence of observable dynamics that can be found separately in a straightforward way by random instantiation over sufficiently large finite fields. In practice, networks may survive more node failures than the guaranteed amount, especially as $k$ increases. This paper improves the results presented in [@PequitoKruzickEUSIPCO2013], extending consideration to robustness requirements and designable network backbone topologies. Future efforts could address a number of interesting network design problems, such as finding optimal networks that are simultaneously both structurally observable and structurally controllable. Alternate problem formulations could decrease network iterations by attempting to reduce the observability index. Additional work could also focus on implementing the network design process in a fully distributed manner within the provided framework, using known distributed algorithms for computing minimum spanning arborescences in the simple $k=0$ case.
[^1]: Stephen Kruzick ([email protected]), Sérgio Pequito ([email protected]), Soummya Kar ([email protected]), and José M. F. Moura ([email protected]) affiliate with the Department of Electrical and Computer Engineering at Carnegie Mellon University in Pittsburgh, PA, USA. A. Pedro Aguiar ([email protected]) affiliates with the Department of Electrical and Computer Engineering at University of Porto in Porto, Portugal. The work of Kruzick was partially conducted with government support under and awarded by the DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a and by NSF grants CCF-1018509, CCF-1011903, and CCF-1513936. The work of Pequito was partially supported by grant SFRH/BD/33779/2009 from Fundação para a Ciência e a Tecnologia (FCT) and by the CMU-Portugal (ICTI) program.
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---
abstract: 'The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multiscale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.'
address: 'Fine Hall 208, Princeton University, Princeton, NJ 08544, USA'
author:
- Ramon van Handel
title: |
Chaining, Interpolation, and Convexity II:\
The contraction principle
---
Introduction
============
The development of sharp bounds on the suprema of random processes is of fundamental importance in diverse areas of pure and applied mathematics. Such problems arise routinely, for example, in probability theory, functional analysis, convex geometry, mathematical statistics, and theoretical computer science.
It has long been understood that the behavior of suprema of random processes is intimately connected with the geometry of their index sets. This idea has culminated in a remarkably general theory due to M. Talagrand that captures the precise connection between the underlying probabilistic and geometric structures for many interesting types of random processes. For example, the classic result in this theory, known (for historical reasons) as the majorizing measure theorem, provides a sharp geometric characterization of the suprema of Gaussian processes.
\[thm:mm\] Let $(X_x)_{x\in T}$ be a centered Gaussian process and denote by $d(x,y) = (\mathbf{E}|X_x-X_y|^2)^{1/2}$ the associated natural metric on $T$. Then $$\mathbf{E}\bigg[\sup_{x\in T}X_x\bigg]
\asymp
\gamma_2^*(T) :=
\inf\sup_{x\in T}\sum_{n\ge 0}2^{n/2}d(x,T_n),$$ where the infimum is taken over all sequences of sets $T_n$ with cardinality $|T_n|<2^{2^n}$.
The method behind the proof of Theorem \[thm:mm\] is called the generic chaining. It is by no means restricted to the setting of Gaussian processes, and full or partial analogues of Theorem \[thm:mm\] exist in various settings. We refer to the monograph [@Tal14] for a comprehensive treatment of this theory and its applications.
The majorizing measure theorem provides in principle a complete geometric understanding (up to universal constants) of the suprema of Gaussian processes. The chaining functional $\gamma_2^*(T)$ captures the relevant geometric structure: it quantifies how well the index set $T$ can be approximated, in a multiscale fashion, by increasingly fine discrete nets $T_n$. The apparently definitive nature of Theorem \[thm:mm\] belies the fact that this result is often very difficult to use in any concrete situation. The problem is that while Theorem \[thm:mm\] guarantees that there must exist some optimal sequence of nets $T_n$ that yields a sharp bound on the supremum of any given Gaussian process, the theorem does not explain how to find such nets. In many cases, straightforward discretization of the index set (Dudley’s inequality) gives rise to suboptimal bounds, and it is not clear how such bounds can be improved.
Even without going beyond the setting of Gaussian processes, there are plenty of challenging problems, for example, in random matrix theory [@RV08; @vH16b; @vH17], that remain unsolved due to the lack of understanding of how to control the supremum of some concrete Gaussian process; in fact, even in cases where the supremum of a Gaussian process can be trivially bounded by probabilistic means, the underlying geometry often remains a mystery, cf. [@Tal14 p. 50]. From this perspective, the generic chaining theory remains very far from being well understood. It is therefore of considerable interest to develop new mechanisms for the control of chaining functionals such as $\gamma_2^*(T)$. The aim of this paper is to take a further step in this direction.
The main (nontrivial) technique that has been used to date to control chaining functionals is contained in the proof of Theorem \[thm:mm\]. To show that $\gamma_2^*(T)$ is bounded above by the expected supremum of a Gaussian process, a sequence of nets $T_n$ is constructed by repeatedly partitioning the set $T$ in a greedy fashion, using the functional $G(A):=\mathbf{E}[\sup_{x\in A}X_x]$ to quantify the size of each partition element. It is necessary to carefully select the partition elements at each stage of the construction in order to control future iterations, which requires fairly delicate arguments (cf. [@Tal14 section 2.6]). It turns out, however, that the proof does not rely heavily on special properties of Gaussian processes: the only property of the functional $G(A)$ that is used is that a certain “growth condition” is satisfied. If one can design another functional $F(A)$ that mimics this property of Gaussian processes, then the same proof yields an upper bound on $\gamma_2^*(T)$ in terms of $F(T)$.
In principle, this partitioning scheme provides a canonical method for bounding chaining functionals such as $\gamma_2^*(T)$: it is always possible to choose a functional satisfying the requisite growth condition that gives a sharp bound on $\gamma_2^*(T)$. This observation has little practical relevance, as this conclusion follows from the fact that the chaining functional itself satisfies the growth condition (cf. [@Tal14 pp. 38–40]) which does not help to obtain explicit bounds on these functionals. Nonetheless, this observation shows that no loss is incurred in the partitioning scheme *per se*, so that its application is only limited by our ability to design good growth functionals that admit explicit bounds. Unfortunately, the latter requires considerable ingenuity, and has been carried out successfully in a limited number of cases.
In the first paper in this series [@vH16a], the author introduced a new method to bound chaining functionals that is inspired by real interpolation of Banach spaces. The technique developed in [@vH16a] is completely elementary and is readily amenable to explicit computations, unlike the growth functional method. This approach considerably simplifies and clarifies some of the most basic ideas in the generic chaining theory, such as the construction of chaining functionals on uniformly convex bodies. On the other hand, this basic method is not always guaranteed to give sharp bounds on $\gamma_2^*(T)$, as can be seen in simple examples (cf. [@vH16a section 3.3]). It is therefore natural to expect that the utility of the interpolation method may be restricted to certain special situations whose geometry is well captured by this construction.
The main insight of the present paper is that this is not the case: interpolation provides a canonical method for bounding chaining functionals. The problem with the basic method of [@vH16a] does not lie with the interpolation method itself, but is rather due to the fact that this method was inefficiently exploited in its simplest form. What is missing is a simple but apparently fundamental ingredient, a contraction principle, that will be developed and systematically exploited in this paper. Roughly speaking, the contraction principle states that we can control chaining functionals such as $\gamma_2^*(T)$ whenever we have suitable control on the entropy numbers of all subsets $A\subseteq T$. A precise statement of this principle will be given in section \[sec:contract\] below, and its utility will be illustrated throughout the rest of paper.
The combination of the interpolation method and the contraction principle provides a foundation for the generic chaining theory that yields significantly simpler proofs and is easier to use (at least in this author’s opinion) than the classical approach through growth functionals. This approach will be illustrated in a number of old and new applications. For example, we will fully recover the majorizing measure theorem with a remarkably short proof that does not involve any greedy partitioning scheme. The latter is somewhat surprising in its own right, as a greedy construction lies very much at the core of earlier proofs of Theorem \[thm:mm\].
This paper is organized as follows. In section \[sec:defn\], we set up the basic definitions and notation that will be used throughout. Section \[sec:contract\] develops the main idea of this paper, the contraction principle. This principle is first illustrated by means of some elementary examples in section \[sec:examples\]. In section \[sec:geom\], we develop a geometric principle that resolves a question posed in [@vH16a Remark 4.4]. We then use this principle to develop new results on the behavior of chaining functionals on Banach lattices, as well as to recover classical results on uniformly convex bodies. In section \[sec:mm\], we develop a very simple proof of Theorem \[thm:mm\] using the machinery of this paper. We also show that the growth functional machinery that lies at the heart of [@Tal14] can be fully recovered as a special case of our approach. Finally, in section \[sec:rmt\] we take advantage of the methods of this paper to develop new dimension-free bounds on the operator norms of structured random matrices.
Basic definitions and notation {#sec:defn}
==============================
The aim of this section is to set up the basic definitions and notation that will be used throughout the paper. We introduce a general setting that will be specialized to different problems as needed in the sequel.
Let $(X,d)$ be a metric space. We begin by defining entropy numbers.
For every $A\subseteq X$ and $n\ge 0$, define the *entropy number* $$e_n(A) := \inf_{|S|<2^{2^n}}\sup_{x\in A} d(x,S).$$ (In this definition, the net $S\subseteq X$ is not required to be a subset of $A$.)
Another way to interpret $e_n(A)$ is by noting that $A$ can be covered by less than $2^{2^n}$ balls of radius $e_n(A)$. It is useful to recall at this stage a classical observation (the duality between covering and packing) that will be needed below.
\[lem:packing\] Let $n\ge 0$ and $N=2^{2^n}$. Then for every $0<\delta<e_n(A)$, there exist points $x_1,\ldots,x_{N}\in A$ such that $d(x_i,x_j)>\delta$ for all $i\ne j$.
Select consecutive points $x_1,x_2,\ldots$ as follows: $x_1\in A$ is chosen arbitrarily, and $x_i\in A$ is chosen such that $d(x_i,x_j)>\delta$ for all $j<i$. Suppose this construction terminates in round $M$, that is, there does not exist $x\in A$ such that $d(x,x_j)>\delta$ for all $j\le M$. Then setting $S=\{x_1,\ldots,x_M\}$, we have $\sup_{x\in A}d(x,S)\le\delta$. Thus $M\ge N$, as otherwise $e_n(A)\le\delta$ which contradicts our assumption.
We now turn to the definition of chaining functionals. For the purposes of the present paper, it will be convenient to use a slightly different definition than is stated in Theorem \[thm:mm\] that uses partitions rather than nets. We also formulate a more general class of chaining functionals that are useful in the more general generic chaining theory (beyond the setting of Gaussian processes), cf. [@Tal14].
Let $T\subseteq X$. An *admissible sequence* of $T$ is an increasing sequence $(\mathcal{A}_n)$ of partitions of $T$ such that $|\mathcal{A}_n|<2^{2^n}$ for all $n\ge 0$. For every $x\in T$, we denote by $A_n(x)$ the unique element of $\mathcal{A}_n$ that contains $x$.
Let $T\subseteq X$. For $\alpha>0$ and $p\ge 1$, define the *chaining functional* $$\gamma_{\alpha,p}(T) :=
\Bigg[\inf \sup_{x\in T}\sum_{n\ge 0}(2^{n/\alpha}
{\mathop{\mathrm{diam}}}(A_n(x)))^p\Bigg]^{1/p},$$ where the infimum is taken over all admissible sequences of $T$. The most important case $p=1$ is denoted as $\gamma_\alpha(T) := \gamma_{\alpha,1}(T)$.
It is an easy fact that the chaining functional $\gamma_2^*(T)$ that appears in Theorem \[thm:mm\] satisfies $\gamma_2^*(T)\le\gamma_2(T)$: given an admissible sequence $(\mathcal{A}_n)$ of $T$, we may simply select a net $T_n$ by choosing one point arbitrarily in every element of the partition $\mathcal{A}_n$. As our interest in this paper is to obtain upper bounds on $\gamma_2(T)$, these trivially give upper bounds on $\gamma_2^*(T)$ as well. It is not difficult to show that these quantities are actually always of the same order, cf. [@Tal14 section 2.3]. This will also follow as a trivial application of the main result of this paper, see section \[sec:adnet\] below.
Let us emphasize that the definitions of $e_n(A)$ and $\gamma_{\alpha,p}(T)$ depend on the metric of the underlying metric space $(X,d)$. In some situations, we will be working with multiple metrics; in this case, the metric $d$ that is used to define the above quantities will be denoted explicitly by writing $e_n(A,d)$, $\gamma_{\alpha,p}(T,d)$, and ${\mathop{\mathrm{diam}}}(A,d)$.
Throughout this paper, we will write $a\lesssim b$ if $a\le Cb$ for a universal constant $C$, and we write $a\asymp b$ if $a\lesssim b$ and $b\lesssim a$. In cases where the universal constant depends on some parameter of the problem, this will be indicated explicitly.
The contraction principle {#sec:contract}
=========================
Statement of the contraction principle
--------------------------------------
At the heart of this paper lies a simple but apparently fundamental principle that will be developed in this section. The basic idea is that we can control the chaining functionals $\gamma_{\alpha,p}(T)$ whenever we have suitable control on the entropy numbers $e_n(A)$ of all subsets $A\subseteq T$.
\[thm:contr\] Let $s_n(x)\ge 0$ and $a\ge 0$ be chosen so that $$e_n(A) \le a{\mathop{\mathrm{diam}}}(A) + \sup_{x\in A}s_n(x)$$ for every $n\ge 0$ and $A\subseteq T$. Then $$\gamma_{\alpha,p}(T) \lesssim
a\,\gamma_{\alpha,p}(T) +
\Bigg[\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}s_n(x))^p
\Bigg]^{1/p},$$ where the universal constant depends only on $\alpha$.
Of course, this result is of interest only when $a$ can be chosen sufficiently small, in which case it immediately yields an upper bound on $\gamma_{\alpha,p}(T)$.
It should be emphasized that the only nontrivial aspect of Theorem \[thm:contr\] is to discover the correct formulation of this principle; no difficulties of any kind are encountered in the proof. What may be far from obvious at present is that this is in fact a powerful or even useful principle. This will become increasingly clear in the following sections, where we will see that the interpolation method of [@vH16a] provides a canonical mechanism for generating the requisite controls $s_n(x)$.
As Theorem \[thm:contr\] lies at the core of this paper, we give two slightly different proofs.
First proof
-----------
The idea of the proof is that the assumption of Theorem \[thm:contr\] allows us to construct from any admissible sequence a new admissible sequence that provides more control on the value of the chaining functional.
As $e_n(A)\le {\mathop{\mathrm{diam}}}(T)$ for every $A\subseteq T$, we can assume without loss of generality that $s_n(x)\le{\mathop{\mathrm{diam}}}(T)$ for all $n,x$.
Let $(\mathcal{A}_n)$ be an admissible sequence of $T$. For every $n\ge 1$ and partition element $A_n\in\mathcal{A}_n$, we construct sets $A_n^{ij}$ as follows. We first partition $A_n$ into $n$ segments $$\begin{aligned}
A_n^i &:= \{x\in A_n:2^{-2i/\alpha}{\mathop{\mathrm{diam}}}(T)<s_n(x)\le
2^{-2(i-1)/\alpha}{\mathop{\mathrm{diam}}}(T)\} \quad(1\le i<n),\\
A_n^n &:= \{x\in A_n:s_n(x) \le 2^{-2(n-1)/\alpha}{\mathop{\mathrm{diam}}}(T)\}.\end{aligned}$$ The point of this step is to ensure that $$\sup_{y\in A_n^i}s_n(y) \le
2^{2/\alpha}s_n(x)+2^{-2(n-1)/\alpha}{\mathop{\mathrm{diam}}}(T)
\quad\mbox{for all }x\in A_n^i, ~i\le n,$$ that is, that $s_n(x)$ is nearly constant on $A_n^i$. Using the assumption of the theorem, we can further partition each set $A_n^i$ into less than $2^{2^n}$ pieces $A_n^{ij}$ such that $${\mathop{\mathrm{diam}}}(A_n^{ij}) \le 2a{\mathop{\mathrm{diam}}}(A_n) +
2^{1+2/\alpha}s_n(x) +
2^{1-2(n-1)/\alpha}{\mathop{\mathrm{diam}}}(T)
\quad\mbox{for all }x\in A_n^{ij}.$$ Let $\mathcal{C}_{n+3}$ be the partition generated by all sets $A_k^{ij}$, $k\le n$, $i,j$ thus constructed. Then $|\mathcal{C}_{n+3}|<\prod_{k=1}^n k(2^{2^k})^2<2^{2^{n+3}}$. Defining $\mathcal{C}_k=\{T\}$ for $0\le k\le 3$, we obtain $$\begin{aligned}
&\gamma_{\alpha,p}(T) \le
\Bigg[
\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}{\mathop{\mathrm{diam}}}(C_n(x)))^p\Bigg]^{1/p} \\
&\lesssim
a
\Bigg[
\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}{\mathop{\mathrm{diam}}}(A_n(x)))^p\Bigg]^{1/p}
+
\Bigg[
\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}s_n(x))^p\Bigg]^{1/p}
+
{\mathop{\mathrm{diam}}}(T)\end{aligned}$$ where the universal constant depends only on $\alpha$. The last term on the right can be absorbed in the first two as ${\mathop{\mathrm{diam}}}(T)\le 2e_0(T) \le 2a{\mathop{\mathrm{diam}}}(A_0(x))+2\sup_{x\in T}s_0(x)$. As the admissible sequence $(\mathcal{A}_n)$ was arbitrary, the conclusion follows readily.
One way to interpret this proof is as follows. We used the assumption of Theorem \[thm:contr\] to define a mapping $\Gamma:\mathcal{A}\mapsto\mathcal{C}$ that assigns to every admissible sequence $\mathcal{A}=(\mathcal{A}_n)$ a new admissible sequence $\mathcal{C}=(\mathcal{C}_n)$. This mapping can be thought of as inducing a form of dynamics on the space of admissible sequences. If we define the value of an admissible sequence $\mathcal{A}$ and the target upper bound as $$\mathop{\mathrm{val}}(\mathcal{A}) =
\Bigg[
\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}{\mathop{\mathrm{diam}}}(A_n(x)))^p\Bigg]^{1/p},\quad
S =
\Bigg[
\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}s_n(x))^p\Bigg]^{1/p},$$ then we have shown in the proof that $$\mathop{\mathrm{val}}(\Gamma(\mathcal{A})) \le
Ca\mathop{\mathrm{val}}(\mathcal{A}) +
CS$$ for a universal constant $C$. If $a$ can be chosen sufficiently small that $Ca<1$, then the mapping $\Gamma$ defines a sort of contraction on the space of admissible sequences. This ensures that there exists an admissible sequence with value $\mathrm{val}(\mathcal{A})\lesssim S$, which is the conclusion we seek. This procedure is reminiscent of the contraction mapping principle, which is why we refer to Theorem \[thm:contr\] as the contraction principle.
Second proof
------------
The above proof of Theorem \[thm:contr\] ensures the existence of a good admissible partition without directly constructing this partition. This is in contrast to the partitioning scheme of [@Tal14], where a good admissible partition is explicitly constructed in the proof. We presently show that by organizing the proof in a slightly different way, we also obtain an explicit construction.
We construct an increasing sequence of partitions $(\mathcal{B}_n)$ of $T$ by induction. First, set $\mathcal{B}_0=\{T\}$. Now suppose partitions $\mathcal{B}_0,\ldots,\mathcal{B}_{n-1}$ have already been constructed. We first split every set $B_{n-1}\in\mathcal{B}_{n-1}$ into $n$ segments $$\begin{aligned}
B_n^i &:= \{x\in B_{n-1}:2^{-2i/\alpha}{\mathop{\mathrm{diam}}}(T)<s_n(x)\le
2^{-2(i-1)/\alpha}{\mathop{\mathrm{diam}}}(T)\} \quad(1\le i<n),\\
B_n^n &:= \{x\in B_{n-1}:s_n(x) \le 2^{-2(n-1)/\alpha}{\mathop{\mathrm{diam}}}(T)\},\end{aligned}$$ and then further subdivide each segment $B_n^i$ into less than $2^{2^n}$ pieces $B_n^{ij}$ such that $${\mathop{\mathrm{diam}}}(B_n^{ij}) \le 2a{\mathop{\mathrm{diam}}}(B_{n-1}) +
2^{1+2/\alpha}s_n(x) +
2^{1-2(n-1)/\alpha}{\mathop{\mathrm{diam}}}(T)
\quad\mbox{for all }x\in B_n^{ij}.$$ Now let $\mathcal{B}_n=\{B_n^{ij}:B_{n-1}\in\mathcal{B}_{n-1},i\le
n,j<2^{2^n}\}$. As $|\mathcal{B}_n|<\prod_{k=1}^n k2^{2^k}<2^{2^{n+2}}$, $(\mathcal{B}_n)$ is not itself an admissible sequence. We can however easily convert it to an admissible sequence $(\mathcal{A}_n)$ by defining $\mathcal{A}_0=\mathcal{A}_1=\{T\}$ and $\mathcal{A}_{n+2}=\mathcal{B}_n$.
Now note that by construction, we have $${\mathop{\mathrm{diam}}}(A_{n}(x)) \le
2a{\mathop{\mathrm{diam}}}(A_{n-1}(x)) +
2^{1+2/\alpha}s_{n-2}(x) + 2^{1-2(n-3)/\alpha}{\mathop{\mathrm{diam}}}(T)$$ whenever $n\ge 3$. Therefore, in the notation of the previous subsection, $$\mathop{\mathrm{val}}(\mathcal{A}) \le
Ca\mathop{\mathrm{val}}(\mathcal{A})+CS$$ for a universal constant $C$ depending only on $\alpha$, where we used the same argument as in the first proof of Theorem \[thm:contr\] to absorb the ${\mathop{\mathrm{diam}}}(T)$ term. We now consider two cases. If $Ca\le\frac{1}{2}$, say, then we obtain the desired bound $\gamma_{\alpha,p}(T)\le\mathop{\mathrm{val}}(\mathcal{A})\le 2CS$ (this is the interesting case). On the other hand, if $Ca>\frac{1}{2}$, we trivially have $\gamma_{\alpha,p}(T)\le 2Ca\gamma_{\alpha,p}(T)$. Thus the conclusion of Theorem \[thm:contr\] follows.
The second proof of Theorem \[thm:contr\] is reminiscent of the partitioning scheme of [@Tal14] to the extent that an admissible sequence is constructed by repeatedly partitioning the index set $T$. In contrast to the method of [@Tal14], however, the present approach is completely devoid of subtlety: the partitioning at each stage is performed in the most naive possible way by breaking up each set arbitrarily into pieces of the smallest possible diameter. We will nonetheless see in section \[sec:mm\] that the growth functional machinery of [@Tal14] can be fully recovered from Theorem \[thm:contr\] with a remarkably simple proof. In our approach, the growth functional plays no role in the partitioning process itself, but will only be used to produce controls $s_n(x)$ that yield good *a priori* bounds on the entropy numbers $e_n(A)$ for $A\subseteq T$.
Simple illustrations {#sec:examples}
====================
Before we can apply Theorem \[thm:contr\] in a nontrivial manner, we should develop some insight into the meaning of the numbers $s_n(x)$ and basic ways in which they can be constructed. To this end, we aim in this section to illustrate Theorem \[thm:contr\] in the simplest cases. All results developed here admit more direct proofs, but the present treatment is intended to help understand the meaning of Theorem \[thm:contr\].
Admissible sequences and nets {#sec:adnet}
-----------------------------
When the abstract statement of Theorem \[thm:contr\] is first encountered, it may be far from obvious why the assumption $$e_n(A) \le a{\mathop{\mathrm{diam}}}(A) + \sup_{x\in A}s_n(x)$$ is a natural one. The relevance of the numbers $s_n(x)$ can be immediately clarified by observing that a canonical choice is already built into the definition of $\gamma_{\alpha,p}(T)$.
\[lem:triv\] Let $(\mathcal{A}_n)$ be any admissible sequence of $T$. Then the choice $s_n(x)={\mathop{\mathrm{diam}}}(A_n(x))$ satisfies the assumption of Theorem \[thm:contr\] with $a=0$.
Any set $A\subseteq T$ can be covered by less than $2^{2^{n}}$ sets $\{A_n(x):x\in A\}$ of diameter at most $\sup_{x\in A}{\mathop{\mathrm{diam}}}(A_n(x))$. Thus $e_n(A) \le \sup_{x\in A}{\mathop{\mathrm{diam}}}(A_n(x))$.
Of course, with this choice, the conclusion of Theorem \[thm:contr\] is $\gamma_{\alpha,p}(T)\lesssim\gamma_{\alpha,p}(T)$ which is not very interesting. Nonetheless, Lemma \[lem:triv\] explains why bounding entropy numbers $e_n(A)$ in terms of controls $s_n(x)$ is entirely natural. Moreover, we see that Theorem \[thm:contr\] can in principle always give a sharp bound on $\gamma_{\alpha,p}(T)$.
As an only slightly less trivial example, let us show that the chaining functional $\gamma_2^*(T)$ defined in the introduction is always of the same order as $\gamma_2(T)$.
$\gamma_2^*(T)\asymp\gamma_2(T)$.
As was noted in section \[sec:defn\], the inequality $\gamma_2^*(T)\le\gamma_2(T)$ is trivial. To prove the converse inequality, let $T_n$ be arbitrary sets of cardinality $|T_n|<2^{2^n}$, and define $s_n(x)=d(x,T_n)$. The definition of entropy numbers instantly yields $e_n(A)\le \sup_{x\in A}s_n(x)$. We can therefore apply Theorem \[thm:contr\] with $a=0$ to obtain $$\gamma_2(T) \lesssim
\sup_{x\in T}\sum_{n\ge 0}2^{n/2}d(x,T_n).$$ Taking the infimum over all choices of $T_n$ yields the conclusion $\gamma_2(T)\le\gamma_2^*(T)$.
So far, we have only used Theorem \[thm:contr\] with $a=0$ and have not exploited the “contraction” part of the contraction principle. In the next section, we provide a first illustration of an improvement that can be achieved by exploiting contraction.
A local form of Dudley’s inequality
-----------------------------------
The most naive bound on $\gamma_2^*(T)$ is obtained by moving the supremum in its definition inside the sum. This yields the following result, which is known as Dudley’s inequality: $$\gamma_2^*(T) \le
\sum_{n\ge 0}2^{n/2}e_n(T).$$ Dudley’s inequality represents the simplest possible construction where each net $T_n$ in the definition of $\gamma_2^*(T)$ is distributed as uniformly as possible over the index set $T$. Unfortunately, such a simple construction proves to be suboptimal already in some of the simplest examples (cf. [@Tal14; @vH16a]). To attain the sharp bound that is guaranteed by Theorem \[thm:mm\], it is essential to allow for the nets $T_n$ to be constructed in a genuinely multiscale fashion. Nonetheless, Dudley’s inequality is widely used in practice due to the ease with which it lends itself to explicit computations.
It is no surprise that Dudley’s inequality is trivially recovered by Theorem \[thm:contr\].
\[lem:dudley\] There is a universal constant $C$ depending only on $\alpha$ such that $$\gamma_{\alpha,p}(T) \le
C\Bigg[
\sum_{n\ge 0}(2^{n/\alpha}e_n(T))^p
\Bigg]^{1/p}.$$
As $e_n(A)\le e_n(T)$, this follows using $s_n(x)=e_n(T)$ and $a=0$ in Theorem \[thm:contr\]. However, without much additional effort, we can do slightly better using a simple application of the “contraction” part of the contraction principle.
To exploit contraction, we note that if $e_n(A)\le a{\mathop{\mathrm{diam}}}(A)=r$, then the assumption of Theorem \[thm:contr\] is automatically satisfied; thus the numbers $s_n(x)$ only need to control the situation where this condition fails. As $A$ is contained in a ball of radius ${\mathop{\mathrm{diam}}}(A)$, this condition essentially means that a certain ball of radius $r/a$ can be covered by less than $2^{2^n}$ balls of proportional radius $r$, which is a sort of doubling condition on the metric space $(T,d)$. Let us consider the largest radius of a ball that is centered at a given point $x$ for which this doubling condition fails: $$e_n^{a,x}(T) :=
\sup\{r:e_n(T\cap B(x,r/a))>r\},$$ where $B(x,r):=\{y\in X:d(x,y)\le r\}$. Then clearly $e_n^{a,x}(T)\le
e_n(T)$, so that this quantity can be viewed as a local improvement on the notion of entropy numbers. We can now use Theorem \[thm:contr\] to show that Dudley’s inequality remains valid if we replace the (global) entropy numbers by their local counterparts.
\[lem:localdud\] There are universal constants $C,a$ depending only on $\alpha$ such that $$\gamma_{\alpha,p}(T) \le
C\Bigg[\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}e_n^{a,x}(T))^p
\Bigg]^{1/p}.$$
Let $n\ge 0$ and $x\in A\subseteq T$. If ${\mathop{\mathrm{diam}}}(A)>e_n^{a,x}(T)/a$, then by definition $$e_n(A) \le e_n(T\cap B(x,{\mathop{\mathrm{diam}}}(A))) \le a{\mathop{\mathrm{diam}}}(A).$$ On the other hand, if ${\mathop{\mathrm{diam}}}(A)\le e_n^{a,x}(T)/a$, then trivially $$e_n(A) \le {\mathop{\mathrm{diam}}}(A) \le \frac{e_n^{a,x}(T)}{a}.$$ Thus the assumption of Theorem \[thm:contr\] holds for any $a>0$ with $s_n(x) = e_n^{a,x}(T)/a$. The proof is readily concluded by choosing $a$ to be a small universal constant.
An almost identical proof yields a variant of Lemma \[lem:localdud\] given in [@Tal96 eq. (1.9)] that uses a regularized form of the local entropy numbers $e_n^{a,x}(T)$.[^1] While these bounds can improve on Dudley’s inequality in some esoteric (ultrametric) examples, they are not particularly useful in practice. The reason that Lemma \[lem:localdud\] is included here is to help provide some initial intuition for how one might use the “contraction” part of the contraction principle. The real power of the contraction principle will however arise when it is combined with the interpolation method of [@vH16a].
The simplest interpolation estimate {#sec:interp}
-----------------------------------
As the interpolation method will play a crucial role in the remainder of this paper, we must begin by recalling the main idea behind this method. The aim of this section is to provide a first illustration of how interpolation can be used to generate the controls $s_n(x)$ in Theorem \[thm:contr\] by recovering the main result of [@vH16a].
The interpolation method is based on the following construction. Let $f:X\to\mathbb{R}_+\cup\{+\infty\}$ be a given penalty function, and define the interpolation functional $$K(t,x) := \inf_{y\in X} \{f(y) + td(x,y)\}.$$ We will assume for simplicity that the infimum in this definition is attained for every $t\ge 0$ and $x\in T$, and denote by $\pi_t(x)$ an arbitrary choice of minimizer (if the infimum is not attained we can easily extend the results below to work instead with near-minimzers). We now define for every $t\ge 0$ the interpolation sets $$K_t := \{\pi_t(x):x\in T\}.$$ The key idea of the interpolation method is that the sets $K_t$ provide a multiscale approximation of $T$ that is precisely of the form suggested by Theorem \[thm:mm\].
\[lem:interp\] For every $a>0$, we have $$\sup_{x\in T}\sum_{n\ge 0}2^{n/\alpha}d(x,\pi_{a2^{n/\alpha}}(x))
\lesssim
\frac{1}{a}\sup_{x\in T}f(x),$$ where the universal constant depends only on $\alpha$.
As $0\le K(t,x)\le f(x)$ and $K(t,x)-K(s,x) \ge (t-s)d(x,\pi_t(x))$, we have $$\sum_{n\ge 0} a2^{n/\alpha} d(x,\pi_{a2^{n/\alpha}}(x)) \lesssim
\sum_{n\ge 0} \{K(a2^{n/\alpha},x)-K(a2^{(n-1)/\alpha},x)\}
\le f(x)$$ for every $x\in T$ and $a>0$.
Lemma \[lem:interp\] provides a natural mechanism to create multiscale approximations. However, the approximating sets $K_{a2^{n/\alpha}}$ are still continuous, and must therefore be discretized in order to bound the chaining functional that appears in Theorem \[thm:mm\]. The simplest possible way to do this is to distribute each net $T_n$ in Theorem \[thm:mm\] uniformly over the set $K_{a2^{n/\alpha}}$. This yields the basic interpolation bound of [@vH16a].
\[thm:interp\] For every $a>0$, we have $$\gamma_\alpha(T) \lesssim
\frac{1}{a}\sup_{x\in T}f(x) +
\sum_{n\ge 0} 2^{n/\alpha}e_n(K_{a2^{n/\alpha}}),$$ where the universal constant depends only on $\alpha$.
By the definition of entropy numbers, we can choose a set $T_n$ of cardinality less than $2^{2^n}$ such that $d(x,T_n)\le 2e_n(K_{a2^{n/\alpha}})$ for every $x\in K_{a2^{n/\alpha}}$. Then $$e_n(A) \le \sup_{x\in A}d(x,T_n) \le
\sup_{x\in A}d(x,K_{a2^{n/\alpha}}) +
2e_n(K_{a2^{n/\alpha}})$$ for every $A\subseteq T$. We can therefore invoke Theorem \[thm:contr\] with $a=0$ and $s_n(x)=d(x,K_{a2^{n/\alpha}})+2e_n(K_{a2^{n/\alpha}})$, and applying Lemma \[lem:interp\] completes the proof.
The utility of Theorem \[thm:interp\] stems from the fact that the sets $K_t$ are often much smaller than the index set $T$, so that this result provides a major improvement over Dudley’s bound. This phenomenon is illustrated in various examples in [@vH16a]. Nonetheless, there is no reason to expect that the particular multiscale construction used here should always attain the sharp bound that is guaranteed by Theorem \[thm:mm\]. Indeed, it is shown in [@vH16a section 3.3] that this is not necessarily the case.
There are two potential ways in which Theorem \[thm:interp\] can result in a suboptimal bound. First, the ability of this method to produce sufficiently “thin” sets $K_t$ relies on a good choice of the penalty function $f$. While certain natural choices are considered in [@vH16a], the best choice of penalty is not always obvious, and a poor choice of penalty will certainly give rise to suboptimal bounds. This is, however, not an intrinsic deficiency of the interpolation method.
The fundamental inefficiency of Theorem \[thm:interp\] lies in the discretization of the sets $K_t$. The interpolation method cannot itself produce discrete nets: it only reveals a multiscale structure inside the index set $T$. To obtain the above result, we naively discretized this structure by distributing nets $T_n$ as uniformly as possible over the sets $K_{a2^{n/\alpha}}$. While this provides an improvement over Dudley’s bound, such a uniform discretization can incur a significant loss. In general, we should allow once again for a multiscale discretization of the sets $K_{a2^{n/\alpha}}$. It is easy to modify the above argument to formalize this idea; for example, one can easily show that $$\gamma_\alpha(T) \lesssim
\frac{1}{a}\sup_{x\in T}f(x) + \inf \sup_{x\in T}
\sum_{n\ge 0}2^{n/\alpha}d(\pi_{a2^{n/\alpha}}(x),T_n),$$ where the infimum is taken over all nets $T_n$ with $|T_n|<2^{2^{n}}$. This bound appears to be rather useless, however, as the quantity on the right-hand side is just as intractable as the quantity $\gamma_\alpha(T)$ that we are trying to control in the first place.
The basic insight that gave rise to the results in this paper is that it is not actually necessary to construct explicit nets $T_n$ to bound the right-hand side of this inequality: it suffices to show that the quantity on the right-hand side is significantly smaller than $\gamma_\alpha(T)$. For example, if we could show that $$\inf \sup_{x\in T}
\sum_{n\ge 0}2^{n/\alpha}d(\pi_{a2^{n/\alpha}}(x),T_n)
\lesssim a\gamma_\alpha(T),$$ then the resulting bound $\gamma_\alpha(T) \lesssim a\gamma_\alpha(T)+
\frac{1}{a}\sup_{x\in T}f(x)$ would yield an explicit bound on $\gamma_\alpha(T)$ by choosing $a$ to be sufficiently small. Such a bound captures quantitatively the idea that the sets $K_t$ are much smaller than the index set $T$. The author initially implemented this idea in a special case (section \[sec:geomp\]) using the formulation described above. It turns out, however, that the same scheme of proof is applicable far beyond this specific setting and is in some sense canonical. The contraction principle of Theorem \[thm:contr\] is nothing other than an abstract formulation of this idea that will enable us to efficiently exploit the interpolation method.
For future reference, we conclude this section by recording a convenient observation: the mapping $x\mapsto\pi_t(x)$ can often be chosen to be a (nonlinear) projection. This was established in [@vH16a] in a more restrictive setting.
\[lem:proj\] Suppose that $T=\{x\in X:f(x)\le u\}$. Then $K_t\subseteq T$ for every $t\ge 0$, and the minimizers $\pi_t(x)$ may be chosen to satisfy $\pi_t(\pi_t(x))=\pi_t(x)$.
As $f(\pi_t(x))\le K(t,x)\le f(x)$, we clearly have $\pi_t(x)\in T$ whenever $x\in T$. This shows that $K_t\subseteq T$. Now consider the set $$K_t' := \{x\in T:K(t,x)=f(x)\}.$$ By construction, if $x\in K_t'$, then we may choose $\pi_t(x)=x$. If $x\not\in K_t'$, we choose $\pi_t(x)$ to be an arbitrary minimizer. We claim that $\pi_t(x)\in K_t'$ for every $x\in T$.
Indeed, suppose $\pi_t(x)\not\in K_t'$. Then there exists $z\in X$ such that $$f(z)+td(\pi_t(x),z) < f(\pi_t(x)).$$ But then we have $$\begin{aligned}
K(t,x) &= f(\pi_t(x)) + td(x,\pi_t(x)) \\
&> f(z) + td(\pi_t(x),z) + td(x,\pi_t(x)) \\
&\ge f(z) + td(x,z)\end{aligned}$$ by the triangle inequality. This contradicts the definition of $K(t,x)$.
As $\pi_t(x)\in K_t'$ for every $x\in T$, we have $K_t\subseteq K_t'$. On the other hand, as $x=\pi_t(x)$ for every $x\in K_t'$, it follows that $K_t=K_t'$ and $\pi_t(\pi_t(x))=\pi_t(x)$.
Banach lattices and uniform convexity {#sec:geom}
=====================================
In this section, we encounter our first nontrivial application of the contraction principle. We begin by developing in section \[sec:geomp\] a sharper version of a geometric principle that was obtained in [@vH16a], resolving a question posed in [@vH16a Remark 4.4]. We will use this principle in section \[sec:lattice\] to obtain a rather general geometric understanding of the behavior of the chaining functionals $\gamma_\alpha$ on Banach lattices. In section \[sec:unifc\], we discuss an analogous result for uniformly convex bodies.
A geometric principle {#sec:geomp}
---------------------
Throughout this section, we specialize our general setting to the case that $(X,\|\cdot\|)$ is a Banach space and $T\subset X$ is a symmetric compact convex set. We let $d(x,y):=\|x-y\|$, and denote the gauge of $T$ as $\|x\|_T :=\inf\{s\ge 0:x\in sT\}$. It is natural in the present setting to use a power of the gauge as a penalty function in the interpolation method: that is, we define $$K(t,x) := \inf_{y\in X}\{\|y\|_T^r+t\|x-y\|\}$$ for some $r>0$. The existence of minimizers $\pi_t(x)$ for $x\in T$ is easily established,[^2] and we define as in section \[sec:interp\] the interpolation sets $$K_t := \{\pi_t(x):x\in T\}.$$ We would like to impose geometric assumptions on the sets $K_t$ that will allow us to obtain tractable bounds on $\gamma_\alpha(T)$. To this end, we will prove a sharper form of a useful geometric principle identified in [@vH16a Theorem 4.1].
\[thm:geom\] Let $q\ge 1$ and $L>0$ be given constants, and suppose that $$\|y-z\|_T^q \le Lt\|y-z\|\quad\mbox{for all }y,z\in K_t,~t\ge 0.$$ Then $$\gamma_\alpha(T)\lesssim
\begin{dcases}
L^{1/q}\Bigg[\sum_{n\ge 0}(2^{n/\alpha}e_n(T))^{q/(q-1)}\Bigg]^{(q-1)/q}
&\quad (q>1),\\
L\sup_{n\ge 0}2^{n/\alpha}e_n(T)&\quad(q=1),
\end{dcases}$$ where the universal constant depends only on $\alpha$.
The message of this result is that one can improve substantially on Dudley’s inequality (which is the case $q=\infty$) if the geometric condition of Theorem \[thm:geom\] is satisfied. This condition is one manifestation of the idea that the sets $K_t$ are much smaller than $T$: under this condition, every small ball in $K_t$ is contained in a proportionally scaled-down copy of $T$. Of course, it is not at all obvious how to realize this condition, but we will see below that it arises very naturally from the interpolation method under suitable geometric assumptions on $T$.
For fixed $q>1$, it was shown in [@vH16a Theorem 4.1] that the conclusion of Theorem \[thm:geom\] can be deduced from Theorem \[thm:interp\]. However, this approach has a crucial drawback: the constant in the inequality obtained in this manner diverges as $q\downarrow 1$. The key improvement provided by Theorem \[thm:geom\] is that the constant does not depend on $q$, which allows us in particular to attain the limiting case $q=1$. The latter is particularly interesting, as the so-called Sudakov lower bound $$\gamma_\alpha(T) \ge \sup_{n\ge 0}2^{n/\alpha}e_n(T)$$ holds trivially for any $T$. Thus the case $q=1$ of Theorem \[thm:geom\] gives a geometric condition for the Sudakov lower bound to be sharp, as conjectured in [@vH16a Remark 4.4]. We will encounter in section \[sec:lattice\] an important example where this is the case.
Let $n\ge 0$ and $A\subseteq T$. We denote by $$A_t := \{\pi_t(x):x\in A\},\qquad\quad
s(t,A) := \sup_{x\in A}\|x-\pi_t(x)\|$$ the projection of $A$ on $K_t$ and the associated projection error.
We first note that the assumption of the theorem implies that $$A_t \subseteq (Lt{\mathop{\mathrm{diam}}}(A_t))^{1/q}(z+T)$$ for some $z\in X$. That is, the projection $A_t$ is contained in a “shrunk” copy of $T$. On the other hand, replacing $A_t$ by $A$ only costs the projection error: $$\begin{aligned}
e_n(A) &\le e_n(A_t) + s(t,A),\\
{\mathop{\mathrm{diam}}}(A_t) &\le {\mathop{\mathrm{diam}}}(A) + 2s(t,A).\end{aligned}$$ We can therefore estimate $$e_n(A) \le
(Lt)^{1/q}({\mathop{\mathrm{diam}}}(A) + 2s(t,A))^{1/q}e_n(T)
+ s(t,A).$$ We apply this bound with $t=a2^{n/\alpha}$. The idea is now that the interpolation lemma will take care of the projection error, while the “contraction” part of the contraction principle allows us to exploit the shrinkage created by the geometric assumption.
**Case $q=1$.** In this case, we can estimate $$e_n(A) \le
LSa{\mathop{\mathrm{diam}}}(A) + (2LSa+1)s(a2^{n/\alpha},A),\qquad
S:=\sup_{n\ge 0}2^{n/\alpha}e_n(T).$$ Applying the contraction principle of Theorem \[thm:contr\] gives $$\gamma_\alpha(T) \lesssim
LSa\,\gamma_\alpha(T) +
(2LSa+1)
\sup_{x\in T}\sum_{n\ge 0}2^{n/\alpha}\|x-\pi_{a2^{n/\alpha}}(x)\|.$$ But we can now use the interpolation Lemma \[lem:interp\] to bound the second term as $$\gamma_\alpha(T) \lesssim
LSa\,\gamma_\alpha(T) +
LS+\frac{1}{a}.$$ We conclude by setting $a= C/LS$ for a sufficiently small universal constant $C$.
**Case $q>1$.** The proof is very similar, but now we use Young’s inequality $uv \le u^p/C^{p/q} + Cv^q$ with $p=q/(q-1)$ to estimate $$e_n(A) \le
C{\mathop{\mathrm{diam}}}(A)
+ (2C+1)s(a2^{n/\alpha},A)
+ \bigg(\frac{La}{C}\bigg)^{p/q}
2^{np/\alpha q}e_n(T)^p.$$ If $C$ is chosen to be a sufficiently small universal constant, then the contraction principle and interpolation lemma give, respectively, $$\begin{aligned}
\gamma_\alpha(T) &\lesssim
\sup_{x\in T}\sum_{n\ge 0}2^{n/\alpha}\|x-\pi_{a2^{n/\alpha}}(x)\|
+
(La)^{p/q}
\sum_{n\ge 0}(2^{n/\alpha}e_n(T))^p \\
&\lesssim
\frac{1}{a} +
(La)^{p/q}
\sum_{n\ge 0}(2^{n/\alpha}e_n(T))^p.\end{aligned}$$ The proof is completed by optimizing over $a>0$.
Let us note that the choice of $r>0$ in the definition of $K(t,x)$ appears nowhere in the statement of proof of Theorem \[thm:geom\]. The ability to choose $r$ will be convenient, however, when we try to verify that the assumption of Theorem \[thm:geom\] is satisfied.
Banach lattices {#sec:lattice}
---------------
The aim of this section is to show that Theorem \[thm:geom\] provides a rather general understanding of the behavior of $\gamma_\alpha(T)$ on Banach lattices. All the relevant background on Banach lattices and their geometry can be found in [@LT79].
In the present section, we specialize the setting of the previous section to the case where $(X,\|\cdot\|)$ is a Banach lattice and where the compact convex set $T\subset X$ is solid, that is, $x\in T$ and $|y|\le
|x|$ implies $y\in T$. Solidity of $T$ is simply the requirement that the gauge $\|\cdot\|_T$ is also a lattice norm (on its domain).
We now introduce a fundamental property that plays an important role in the geometry of Banach lattices, cf. [@LT79 section 1.f].
Let $q\ge 1$. $T$ satisfies a *lower $q$-estimate* with constant $M$ if $$\Bigg[\sum_{i=1}^n\|x_i\|_T^q\Bigg]^{1/q} \le
M\Bigg\|\sum_{i=1}^n |x_i|\Bigg\|_T$$ for all $n\ge 1$ and vectors $x_1,\ldots,x_n\in X$.
We have the following result.
\[thm:lattice\] Let $q\ge 1$. If $T$ satisfies a lower $q$-estimate with constant $M$, then $$\gamma_\alpha(T)\lesssim
\begin{dcases}
M\Bigg[\sum_{n\ge 0}(2^{n/\alpha}e_n(T))^{q/(q-1)}\Bigg]^{(q-1)/q}
&\quad (q>1),\\
M\sup_{n\ge 0}2^{n/\alpha}e_n(T)&\quad(q=1),
\end{dcases}$$ where the universal constant depends only on $\alpha$.
We will prove this theorem by showing that the condition of Theorem \[thm:geom\] is satisfied if we choose $r=q$ in the previous section. There is a somewhat subtle point, however, that we must take care of first. The computations used in our proof rely crucially on the fact that a lower $q$-estimate is satisfied with constant $M=1$. However, we did not require this special situation to hold in Theorem \[thm:lattice\]. We will therefore make essential use of the observation that any Banach lattice that satisfies a lower $q$-estimate admits an equivalent renorming whose lower $q$-estimate constant is identically one [@LT79 Lemma 1.f.11]. Concretely, define the new norm $$\|x\|_{\tilde T} := \sup \Bigg[\sum_{i=1}^n\|x_i\|_T^q\Bigg]^{1/q},$$ where the supremum is taken over all possible decompositions of $x$ as a sum of $n\ge 1$ pairwise disjoint elements $x_1,\ldots,x_n$, and define $\tilde T:=\{x\in X:\|x\|_{\tilde T}\le 1\}$. It is readily verified using [@LT79 Proposition 1.f.6] that if $T$ satisfies a lower $q$-estimate with constant $M$, then $\tilde T$ satisfies a lower $q$-estimate with constant $1$ and $\tilde T\subseteq
T\subseteq M\tilde T$. This implies in particular that $\gamma_\alpha(T)\le M\gamma_\alpha(\tilde T)$ and $e_n(\tilde T)\le
e_n(T)$, so that we may assume without loss of generality in the proof of Theorem \[thm:lattice\] that $M=1$.
We assume without loss of generality that $M=1$, and apply the setting of the previous section with $r=q$. Fix $t\ge 0$ and $y,z\in K_t$, and define $$u := (y\wedge z)\vee 0 + (y\vee z)\wedge 0.$$ The point of this definition is that $$|y|-|u|=|y-u|\le|y-z|,$$ as well as the analogous property where the roles of $y$ and $z$ are exchanged.
Using that $T$ satisfies a lower $q$-estimate with constant one, we obtain $$\|y-u\|_T^q \le \|y\|_T^q-\|u\|_T^q.$$ On the other hand, Lemma \[lem:proj\] gives $$\|y\|_T^q = K(t,y) \le \|u\|_T^q + t\|y-u\|.$$ All the above properties hold if we exchange $y$ and $z$. We can therefore estimate $$\begin{aligned}
\|y-z\|_T^q &\le
2^{q-1}(\|y-u\|_T^q + \|z-u\|_T^q) \\
&\le
2^{q-1}t\,(\|y-u\| + \|z-u\|) \\
&\le
2^q t\|y-z\|,\end{aligned}$$ where we used the triangle inequality, $(a+b)^q\le 2^{q-1}(a^q+b^q)$, and that $\|\cdot\|$ is a lattice norm. The proof is concluded by applying Theorem \[thm:geom\].
An interesting example of Theorem \[thm:lattice\] is the the following. Let $X=\mathbb{R}^d$, let $\|\cdot\|$ be any $1$-unconditional norm (with respect to the standard basis), and let $T=B_1^d$ be the unit $\ell_1$-ball. It is immediate that the $\ell_1$-norm satisfies a $1$-lower estimate with constant one. Theorem \[thm:lattice\] therefore yields $$\gamma_\alpha(B_1^d) \asymp
\sup_{n\ge 0}2^{n/\alpha}e_n(B_1^d),$$ that is, Sudakov’s lower bound is sharp for the $\ell_1$-ball. In the special case where $\alpha=2$ and $\|\cdot\|$ is the Euclidean norm, this can be verified by an explicit computation using Theorem \[thm:mm\] (or using Theorem \[thm:interp\], cf. [@vH16a section 3.2]) and simple estimates on the entropy numbers; however, such a computation does not explain *why* Sudakov’s lower bound turns out to be sharp in this setting. Theorem \[thm:lattice\] provides a geometric explanation of this phenomenon, and extends it to the much more general situation where $\|\cdot\|$ is an arbitrary unconditional norm.
We conclude this section with a few remarks.
We have shown that Sudakov’s inequality is sharp for $B_1^d$ if $\|\cdot\|$ is a lattice norm (that is, unconditional with respect to the standard basis). It is worth noting that the lattice property is really essential for this phenomenon to occur: the analogous result for general norms is absolutely false. To see why this must be the case, note that if $T$ is the symmetric convex hull of $d$ points in $X$, then we always have $T=AB_1^d$ for some linear operator $A:\mathbb{R}^d\to X$. We can therefore write $\gamma_\alpha(T,\|\cdot\|)=
\gamma_\alpha(B_1^d,\|\cdot\|')$ with $\|x\|':=\|Ax\|$. Thus if Sudakov’s lower bound were sharp for $B_1^d$ when endowed with a general norm, then Sudakov’s lower bound would be sharp for any symmetric polytope, and therefore (by approximation) for every symmetric compact convex set. This conclusion is clearly false.
The case $q=1$ of Theorem \[thm:lattice\] proves to be somewhat restrictive. Suppose that $\|\cdot\|_T$ satisfies a $1$-lower estimate with constant one (as may always be assumed after equivalent renorming). Because of the triangle inequality, we must then have the rather strong condition $\|x\|_T+\|y\|_T = \|(|x|+|y|)\|_T$. A Banach lattice satisfying this condition is called an AL-space. It was shown by Kakutani that such a space is always order-isometric to $L^1(\mu)$ for some measure $\mu$ [@LT79 Theorem 1.b.2]. Thus $L^1$-balls are essentially the only examples for which Theorem \[thm:lattice\] applies with $q=1$. The case $q>1$ is much richer, however, and Theorem \[thm:lattice\] provides a very general tool to understand chaining functionals in this setting.
Theorem \[thm:lattice\] shows that Dudley’s inequality can be substantially improved for solid sets $T$ that satisfy a nontrivial lower $q$-estimate. On the other hand, a solid set $T$ that fails to satisfy any nontrivial lower $q$-estimate must contain $\ell_\infty^d$-balls of arbitrarily large dimension, cf. [@LT79 Theorem 1.f.12]. For cubes, the majorizing measure theorem and the results of [@Car81] can be used to show that Dudley’s inequality is sharp, and that no improvement as in Theorem \[thm:lattice\] can hold in general. Thus Theorem \[thm:lattice\] is essentially the best result of its kind.
Uniformly convex bodies {#sec:unifc}
-----------------------
The lower $q$-estimate property of a Banach lattice is closely related to the notion of uniform convexity in general Banach spaces, as is explained in [@LT79 section 1.f]. It is therefore not surprising that an analogue of Theorem \[thm:lattice\] holds in a general Banach space when $T$ is a uniformly convex body. Unlike the results of the previous section, however, this case is already well understood [@Tal14 section 4.1]. It will nonetheless be useful to revisit this setting in the light of the present paper, as the method that appears in the proof will play an essential role in the random matrix problems that will be discussed in section \[sec:rmt\].
To this end, we return to the setting where $(X,\|\cdot\|)$ is a general Banach space and $T\subset X$ is a symmetric compact convex set.
\[defn:ucvx\] Let $q\ge 2$. $T$ is said to be *$q$-convex* with constant $\eta$ if $$\bigg\|\frac{x+y}{2}\bigg\|_T \le 1-\eta\|x-y\|_T^q$$ for all vectors $x,y\in T$.
It was shown in [@vH16a Lemma 4.7] that the assumption of Theorem \[thm:geom\] holds in the present setting with $L=1/2\eta$; the proof of this fact is not unlike the one we used in the lattice case. Thus the conclusion in the case $q$-convex bodies matches verbatim the one obtained for lattices in the previous section. However, in this setting we are never near the boundary case of Theorem \[thm:geom\], as the $q$-convexity property can only hold for $q\ge 2$ (no body is more strongly convex than a Euclidean ball). This means that the machinery of this paper is not really needed to establish this result; it was shown in [@vH16a] that it already follows from Theorem \[thm:interp\].
However, the boundary case reappears if we consider the more general chaining functionals $\gamma_{\alpha,p}(T)$ rather than just $\gamma_\alpha(T)$. For example, the following sharp bound of [@Tal14 Theorem 4.1.4] cannot be recovered using the methods of [@vH16a].
\[thm:ucvx\] Let $q\ge 2$. If $T$ is $q$-convex with constant $\eta$, then $$\gamma_{\alpha,q}(T) \lesssim
\eta^{-1/q}\sup_{n\ge 0}2^{n/\alpha}e_n(T),$$ where the universal constant depends only on $\alpha$.
We will presently give a short proof of this result using the methods of this paper in order to highlight a couple of points that arise when bounding $\gamma_{\alpha,p}(T)$. Of course, one can obtain extensions of both Theorems \[thm:lattice\] and \[thm:ucvx\] that bound $\gamma_{\alpha,p}(T)$ with general $\alpha>0$ and $1\le p\le q$ (not just $p=q$ as in Theorem \[thm:ucvx\]); as no new ideas arise in this setting, we leave the details to the reader.
In order to bound $\gamma_{\alpha,q}$, we require in principle only a minor adaptation of the interpolation method: we simply modify the definition of $K(t,x)$ in section \[sec:geomp\] to $$K(t,x) := \inf_{y\in X}\{\|y\|_T^r + t^q\|x-y\|^q\}.$$ We denote once again by $\pi_t(x)$ the minimizer in this expression, and by $K_t$ the set of minimizers for $x\in T$. The appropriate analogue of the interpolation lemma in this setting is obtained by repeating verbatim the proof of Lemma \[lem:interp\].
\[lem:qinterp\] For every $a>0$, we have $$\sup_{x\in T}\sum_{n\ge 0}
(2^{n/\alpha}\|x-\pi_{a2^{n/\alpha}}(x)\|)^q
\lesssim
\frac{1}{a^q},$$ where the universal constant depends only on $\alpha$.
With these simple modifications, we can now essentially follow the same scheme of proof as for Theorem \[thm:lattice\], replacing the use of the lower $q$-estimate by the $q$-convexity property. There is, however, one minor issue that requires some care. In the proof of Theorem \[thm:lattice\] (as in the proof of [@vH16a Lemma 4.7] where the assumption of Theorem \[thm:geom\] is verified for $q$-convex sets), we used the fact that $\pi_t(x)$ possesses the projection property of Lemma \[lem:proj\]. This property is however quite special to interpolation functionals of the form $\inf\{f(y)+td(x,y)\}$, as it relies crucially on the triangle property of the distance. When the distance is raised to a power as in the present setting, the projection property no longer holds and we must take care to proceed without it. Fortunately, it turns out to that the projection property was not really used in an essential way in Theorem \[thm:lattice\] and can easily be avoided.
For $x,y\in T$, the $q$-convexity property can be formulated as $$\bigg\|\frac{x+y}{2}\bigg\|_T \le
\max(\|x\|_T,\|y\|_T) -\eta\|x-y\|_T^q$$ by applying Definition \[defn:ucvx\] to $x/\gamma$, $y/\gamma$ with $\gamma=\max(\|x\|_T,\|y\|_T)\le 1$. To exploit this formulation of $q$-convexity, we will choose $r=1$ in the definition of $K(t,x)$.
Let $n\ge 0$ and $A\subseteq T$. As in the proof of Theorem \[thm:geom\], we write $$A_t := \{\pi_t(x):x\in A\},\qquad\quad
s(t,A) := \sup_{x\in A}\|x-\pi_t(x)\|.$$ Note that $A_t\subseteq T$. If $y=\pi_t(x)$ for $x\in A$, we can estimate $$\begin{aligned}
\|y\|_T & \le \|y\|_T + t^q\|x-y\|^q = K(t,x) \\
&\le
\|u\|_T + t^q\|x-u\|^q \\
& \le
\|u\|_T + 2^{q-1}t^q\|y-u\|^q +
2^{q-1}t^q\|x-y\|^q \\
& \le
\|u\|_T + 2^{q-1}t^q\|y-u\|^q +
2^{q-1}t^qs(t,A)^q\end{aligned}$$ for any $u\in X$, where we used the triangle inequality and $(a+b)^q\le 2^{q-1}(a^q+b^q)$. Thus for any $y,z\in A_t$, choosing $u:=(y+z)/2$ in the above inequality shows that $$\max(\|y\|_T,\|z\|_T) \le
\bigg\|\frac{y+z}{2}\bigg\|_T
+ 2^{q-1}t^q\bigg\|\frac{y-z}{2}\bigg\|^q +
2^{q-1}t^qs(t,A)^q.$$ Applying the $q$-convexity property yields $$\eta\|y-z\|_T^q \le
2^{-1}t^q\|y-z\|^q +
2^{q-1}t^qs(t,A)^q.$$ for all $y,z\in A_t$. Note that this condition is very similar to the assumption of Theorem \[thm:geom\], except that an additional projection error term appears. The latter is the price we pay for avoiding the projection property, which does not hold in the present setting. However, this additional term introduces no further complications.
The above inequality shows that $$A_t \subseteq
\eta^{-1/q}t ({\mathop{\mathrm{diam}}}(A_t) + 2s(t,A))(z+T)$$ for some $z\in X$. Proceeding as in the proof of Theorem \[thm:geom\], we obtain $$e_n(A) \le Sa{\mathop{\mathrm{diam}}}(A) + (4Sa+1)s(a2^{n/\alpha},A),\qquad
S := \eta^{-1/q}\sup_{n\ge 0}2^{n/\alpha}e_n(T).$$ Applying Theorem \[thm:contr\] and Lemma \[lem:qinterp\] yields $$\begin{aligned}
\gamma_{\alpha,q}(T) &\lesssim
Sa\,\gamma_{\alpha,q}(T) +
(4Sa+1)
\Bigg[\sup_{x\in T}
\sum_{n\ge 0}(2^{n/\alpha}\|x-\pi_{a2^{n/\alpha}}(x)\|)^q
\Bigg]^{1/q} \\
&\lesssim
Sa\,\gamma_{\alpha,q}(T) +
S + \frac{1}{a}.\end{aligned}$$ We conclude by choosing $a=C/S$ for a sufficiently small universal constant $C$.
It is also possible to give a proof more in the spirit of Theorem \[thm:lattice\] where we choose $r=q$ in the definition of $K(t,x)$. In this case, one should replace Definition \[defn:ucvx\] by the following homogeneous form of the $q$-convexity property: $$\bigg\|\frac{x+y}{2}\bigg\|_T^q \le
\frac{\|x\|_T^q+\|y\|_T^q}{2}
-\tilde\eta\|x-y\|_T^q$$ for all $x,y\in X$. It can be shown that this alternative formulation is equivalent to that of Definition \[defn:ucvx\] [@BCL94 Proposition 7], and a more careful accounting of the constants (as in [@Nao12 Lemma 2.2]) shows that $\tilde\eta\ge c^q\eta$ for a universal constant $c$.
As was mentioned above, the analogue of Theorem \[thm:lattice\] for $q$-convex sets was already proved in [@vH16a] using only Theorem \[thm:interp\]: one can show in this case that the entropy numbers of the interpolation sets $e_n(K_t)$ can be controlled efficiently by the entropy numbers $e_n(T)$. It was even shown in [@vH16a] by explicit computation that Theorem \[thm:interp\] yields a sharp bound for the $\ell_1$-ball in the special case that $\|\cdot\|$ is the Euclidean norm, which is a boundary case of Theorem \[thm:lattice\]. This is simpler conceptually than the present approach, which relies on the contraction principle. One might therefore wonder whether the contraction principle is really needed in this setting, or whether it is possible that results such as Theorems \[thm:lattice\] and \[thm:ucvx\] could be recovered from Theorem \[thm:interp\] using a more efficient argument. We will presently argue that this is not the case: the entropy numbers $e_n(K_t)$ are generally too large, so the contraction principle is essential to attain sharp bounds.
To this end, consider the following illuminating example. We consider $X=\mathbb{R}^d$ with the Euclidean distance $\|\cdot\|$, and let $T\subset X$ be the ellipsoid defined by $$\|x\|_T^2 =\sum_{k=1}^d k x_k^2.$$ $T$ is $2$-convex by the parallelogram identity, and Theorem \[thm:ucvx\] gives $$\gamma_{2,2}(T) \asymp \sup_{n\ge 0} 2^{n/2}e_n(T)\asymp 1$$ as $e_n(T)\lesssim 2^{-n/2}$ by the estimates in [@Tal14 section 2.5].
It is trivial to adapt Theorem \[thm:interp\] the present setting, which yields $$\gamma_{2,2}(T) \lesssim
\frac{1}{a} +
\Bigg[
\sum_{n\ge 0} (2^{n/2}e_n(K_{a2^{n/2}}))^2
\Bigg]^{1/2} =: S(a).$$ We claim that this bound cannot recover the correct behavior of $\gamma_{2,2}(T)$. To see this, we must compute the interpolation sets $K_t$. It is particularly convenient in this setting to choose $r=2$ in the definition of $K(t,x)$, which is appropriate as explained after the proof of Theorem \[thm:ucvx\]. The advantage of this choice is that $K(t,x):=\inf_y\{\|y\|_T^2+t^2\|x-y\|^2\}$ involves minimizing a quadratic function, which is trivially accomplished. We readily compute that $K_t$ is another ellipsoid: $$(\pi_t(x))_k = \frac{t^2}{t^2+k} x_k,\qquad\quad
\|x\|_{K_t} = \sum_{k=1}^d \bigg(
\frac{t^2+k}{t^2}
\bigg)^2 kx_k^2.$$ Using the entropy estimate of [@Tal14 Lemma 2.5.4], we find that $$e_n(K_{a2^{n/2}})
\gtrsim
\frac{a^2}{a^2+1} 2^{-n/2}$$ for $2^n\lesssim d$. It follows that $$S(a) = \frac{1}{a} +
\Bigg[
\sum_{n\ge 0} (2^{n/2}e_n(K_{a2^{n/2}}))^2
\Bigg]^{1/2} \gtrsim
\frac{1}{a} +
\frac{a^2}{a^2+1}\sqrt{\log d}
\gtrsim (\log d)^{1/6}.$$ We have therefore shown that a sharp bound on $\gamma_{2,2}(T)$ cannot be attained by choosing nets that are distributed uniformly on the interpolation sets $K_t$, as is done in Theorem \[thm:interp\]. On the other hand, the same interpolation scheme yields a sharp bound when combined with the contraction principle in Theorem \[thm:ucvx\]. This example provides an explicit illustration of the assertion made in the introduction that the deficiency of Theorem \[thm:interp\] is not due to the interpolation method, but rather due to the fact that the interpolation method is being used inefficiently.
The majorizing measure theorem {#sec:mm}
==============================
In the previous sections, we introduced the contraction principle and illustrated its utility in combination with the interpolation method in several interesting situations. We will presently use the same machinery to give a surprisingly simple proof of the majorizing measure theorem (Theorem \[thm:mm\]). With some small modifications, this will also allow us to recover the main growth functional estimate of [@Tal14]. Beside providing simple new proofs of these results, the fact that they can be attained at all shows that the methods of this paper are not restricted to some special situations, but can in fact fully recover the core of the generic chaining theory.
Gaussian processes {#sec:thm1}
------------------
Let $(X_x)_{x\in T}$ be a centered Gaussian process, and denote by $d(x,y):=({\mathbf{E}}|X_x-X_y|^2)^{1/2}$ the associated natural metric on $T$. To avoid being distracted by minor measurability issues, let us assume for simplicity that the index set $T$ is finite. It is well understood in the theory of Gaussian processes that this entails no loss of generality in any reasonable situation.
Let us define for any subset $A\subseteq T$ the Gaussian width $$G(A) := \mathbf{E}\bigg[
\sup_{x\in A}X_x
\bigg].$$ The statement of the majorizing measure theorem is that $G(T)\asymp\gamma_2(T)$. The upper bound $G(T)\lesssim\gamma_2(T)$ is however completely elementary; see [@Tal14 section 2.2] for this classical and very simple chaining argument. It is the lower bound $\gamma_2(T)\lesssim G(T)$ in the majorizing measure theorem that is a deep result. In this section, we will give a simple proof of the latter bound using the machinery of this paper.
In its simplest form, the idea that allows us to bound $\gamma_2(T)$ by $G(T)$ is clear: we should use $G(T)$ to define the penalty function in the interpolation method, and then use Sudakov’s inequality for Gaussian processes (which gives an upper bound on $e_n(A)$ in terms of $G(A)$) to verify the assumption of the contraction principle. To implement this idea, it will be convenient to define the interpolation functional $K(t,x)$ in a somewhat different manner than we did previously: we set $$K(t,x):=\inf_{s\ge 0}\{ ts + G(T)-G(B(x,s))\},$$ where $$B(x,s) := \{y\in T:d(x,y)\le s\}$$ is the ball in $T$ with radius $s$ centered at $x$. As the function $s\mapsto G(B(x,s))$ is upper-semicontinuous, the infimum in the definition of $K(t,x)$ is attained. Denoting the minimizer as $s(t,x)\ge 0$, we obtain the following interpolation lemma.
\[lem:ginterp\] For every $a>0$ $$\sup_{x\in T}\sum_{n\ge 0}2^{n/2}s(a2^{n/2},x) \lesssim
\frac{G(T)}{a}.$$
The proof is identical to that of Lemma \[lem:interp\].
It may not be obvious that the present definition of $K(t,x)$ is an interpolation functional in the sense of section \[sec:interp\], except in some generalized sense. This is nonetheless the case. To see why, let $L^\infty(\Omega;T)$ be the space of $T$-valued random variables endowed with the metric $d_\infty(\sigma,\tau):=\|d(\sigma,\tau)\|_\infty$. Then $$G(B(x,s)) = {\mathbf{E}}\bigg[\sup_{y\in T:d(x,y)\le s}X_y\bigg] =
\sup_{\tau\in L^\infty(\Omega;T):d_\infty(x,\tau)\le s}
{\mathbf{E}}[X_\tau].$$ Substituting this expression in the definition of $K(t,x)$ and exchanging the order of the two infima shows that we can in fact write $$K(t,x) = \inf_{\tau\in L^\infty(\Omega;T)}
\{G(T) - {\mathbf{E}}[X_\tau] + td_\infty(x,\tau)\}.$$ Thus $K(t,x)$ is an interpolation functional, on the space $(L^\infty(\Omega;T),d_\infty)$ and with penalty function $f(\tau)=G(T) -
{\mathbf{E}}[X_\tau]$, of precisely the form given in section \[sec:interp\]. While this formulation guides our intuition, it is more convenient computationally to work with the definition in terms of $G(B(x,s))$ as this will allow us to directly apply inequalities for the suprema of Gaussian processes.
To prove the majorizing measure theorem, we will verify that the condition of Theorem \[thm:contr\] is satisfied with $s_n(x)\lesssim s(a2^{n/2},x)$. To this end, we must bound the entropy numbers $e_n(A)$ of all subsets $A\subseteq T$. As our interpolation functional involves the supremum of a Gaussian process, this should surely involve Sudakov’s inequality. The appropriate form for our purposes, which is a straightforward extension of Sudakov’s inequality, can be found in [@Tal14 Proposition 2.4.9].
\[lem:supersud\] For $\sigma,b>0$ and $x_1,\ldots,x_n\in T$ such that $d(x_i,x_j)\ge b$ for $i\ne j$ $$\min_{i\le n}G(B(x_i,\sigma)) +
C_1b\sqrt{\log n} \le
G(\cup_{i\le n}B(x_i,\sigma)) + C_2\sigma\sqrt{\log n},$$ where $C_1,C_2$ are universal constants.
This is in fact a form of the “growth condition” that forms the central ingredient in the generic chaining theory as developed in [@Tal14]. One of the advantages of the approach developed in this paper is that it makes it possible to bound chaining functionals without engineering such a condition, which does not always arise natually in a geometric setting. However, in the case of Gaussian processes, the growth condition arises in a completely natural manner and is essentially the reason why the majorizing measure theorem is true. It therefore seems likely that any proof of the majorizing measure theorem must exploit a form of Lemma \[lem:supersud\] at the crucial point in the argument. We will presently show that Lemma \[lem:supersud\] provides a very simple method for verifying the assumption of the contraction principle.
\[lem:gausscontr\] For every $n\ge 0$, $A\subseteq T$, and $a>0$, we have $$e_n(A) \lesssim a{\mathop{\mathrm{diam}}}(A) + (a+1)\sup_{x\in A}s(a2^{n/2},x).$$
Assume $e_n(A)>0$, otherwise the result is trivial. By Lemma \[lem:packing\], we can find $N=2^{2^n}$ points $x_1,\ldots,x_N\in A$ such that $d(x_i,x_j)>e_n(A)/2$ for all $i\ne j$. Let $$\sigma = \sup_{x\in A}s(a2^{n/2},x),\qquad\quad
r = {\mathop{\mathrm{diam}}}(A)+\sigma.$$ Then $\cup_{i\le N}B(x_i,\sigma)\subseteq B(x_k,r)$ for every $k\le N$. We can now estimate $$\begin{aligned}
G(T) - G(B(x_k,\sigma)) &\le
G(T) - G(B(x_k,s(a2^{n/2},x_k))) \\ &\le
K(a2^{n/2},x_k) \\
&\le a2^{n/2}r + G(T) - G(B(x_k,r)) \\
&\le a2^{n/2}r + G(T) - G(\cup_{i\le N}B(x_i,\sigma))\end{aligned}$$ for every $k\le N$. Applying Lemma \[lem:supersud\] gives $$2^{n/2}e_n(A) \lesssim
a2^{n/2}r + 2^{n/2}\sigma,$$ which readily yields the conclusion.
With this simple lemma in hand, the proof of the lower bound in the majorizing measure theorem follows immediately from the contraction principle.
$\gamma_2(T)\lesssim G(T)$.
The condition of Theorem \[thm:contr\] is verified by Lemma \[lem:gausscontr\]. It remains to apply Lemma \[lem:ginterp\] and choose $a>0$ to be a sufficiently small universal constant.
Growth functionals {#sec:growth}
------------------
Now that we have proved the majorizing measure theorem using the approach of this paper, it will come as no surprise that the general growth functional machinery that forms the foundation of the generic chaining theory as developed in [@Tal14] can also be recovered by the interpolation method. This shows that applicability of the interpolation method is not restricted to some special situations, but that it is in principle canonical: the generic chaining theory can be fully recovered in this manner. In our approach, growth functionals provide one possible method for creating the condition of the contraction principle.
In this section, we will modify the proof of the majorizing measure theorem to utilize one of the generalized growth functional conditions considered in [@Tal14]. While the basic idea of the proof is already contained in the previous section, this generalization is instructive in its own right: it will help clarify the relevance of the ingredients in the definition of a growth functional from the present perspective, and will also illustrate the use of different interpolation functionals for different scales. Of course, the same method of proof admits numerous generalizations, including several considered in [@Tal14] that can be analogously recovered by our methods.
We will work on a general metric space $(T,d)$. Let us begin by stating some basic definitions. The first is a notion of well-separated sets [@Tal14 Definition 2.3.8].
\[defn:sep\] $H_1,\ldots,H_m\subseteq T$ are *$(b,c)$-separated* if there are $x_1,\ldots,x_m,y\in T$ such that $d(x_i,x_j)\ge b$ for all $i\ne j$, and $d(x_i,y)\le cb$ and $H_i\subseteq B(x_i,b/c)$ for all $i$.
We also need the basic notion of a functional.
A *functional* on $T$ is a map $F$ that assigns to every set $H\subseteq T$ a number $F(H)\ge 0$ and is increasing, that is, $F(H)\le F(H')$ if $H\subseteq H'$. A sequence of functionals $(F_n)_{n\ge 0}$ is *decreasing* if $F_{n+1}(H)\le F_n(H)$ for every set $H$.
We now state the growth condition of [@Tal14 Definition 2.3.10].
\[defn:growth\] A decreasing sequence of functionals $(F_n)_{n\ge 0}$ satisfies the *growth condition* with parameters $c,L>0$ if for any $b>0$, $n\ge 1$ and every collection $H_1,\ldots,H_N\subseteq T$ of $N=2^{2^n}$ subsets of $T$ that are $(b,c)$-separated, we have $$F_{n-1}(\cup_{i\le N}H_i) \ge L2^{n/2}b + \min_{i\le N}
F_n(H_i).$$
A minor variation on Lemma \[lem:supersud\] shows that the choice $F_n(H)=G(H)$ satisfies the growth condition provided that the parameter $c$ is chosen sufficiently large: that is, the Gaussian width $G(H)$ is a growth functional. However, the growth condition as defined above allows more flexibility in the design of functionals.
The aim of this section is to prove the following result [@Tal14 Theorem 2.3.16].
\[thm:growth\] Suppose that the decreasing sequence of functionals $(F_n)_{n\ge 0}$ satisfies the growth condition with parameters $c,L>0$. Then $$\gamma_2(T) \lesssim \frac{c}{L}F_0(T) + {\mathop{\mathrm{diam}}}(T)$$ provided that $c\ge c_0$, where $c_0$ is a universal constant.
In the rest of this section, we fix parameters $c,L>0$ and a decreasing sequence of functionals $(F_n)_{n\ge 0}$ that satisfies the growth condition of Definition \[sec:growth\].
There are two additional ideas in the proof of Theorem \[thm:growth\] as compared to that of the majorizing measure theorem. First, we have not one growth functional $G$, but rather a separate functional $F_n$ for every scale. This flexibility introduces more room in the growth condition, making it easier to satisfy. The complication that arises is that we have to work with multiple interpolation functionals $$K_n(t,x) := \inf_{s\ge 0}\{
ts+F_0(T)-F_n(B(x,s))
\}.$$ However, as $F_n$ is a decreasing sequence of functionals, we readily recover a variant of the usual interpolation lemma. We dispose at the same time of the minor technical issue that it is unclear whether minimizers in the definition of $K_n(t,x)$ exist in the absence of regularity assumptions on $F_n$, so we must work with near-minimizers.
\[lem:multinterp\] For every $n\ge 1$, $a>0$ and $x\in T$, choose $s_n^a(x)\ge 0$ such that $$\begin{aligned}
K_n(La2^{n/2},x) &\le
La2^{n/2}s_n^a(x)+F_0(T)-F_n(B(x,s_n^a(x))) \\ &\le
K_n(La2^{n/2},x) + 2^{-n}F_0(T).\end{aligned}$$ Then for every $a>0$ $$\sup_{x\in T}
\sum_{n\ge 1}2^{n/2}s_n^a(x) \lesssim \frac{F_0(T)}{La}.$$
By definition of $K_{n-1}$ and as $F_n$ is a decreasing sequence, $$\begin{aligned}
&2^{-n}F_0(T) +
K_n(La2^{n/2},x) -
K_{n-1}(La2^{(n-1)/2},x)
\\ &\ge
(1-2^{-1/2})La2^{n/2}s_n^a(x)
+F_{n-1}(B(x,s_n^a(x)))
-F_n(B(x,s_n^a(x)))
\\ &\ge
(1-2^{-1/2})La2^{n/2}s_n^a(x).\end{aligned}$$ We conclude by summing over $n\ge 1$ and using $K_n(t,x)\le F_0(T)$ for all $n,t$.
The second new feature in the proof of Theorem \[thm:growth\] is that the separation condition of Definition \[defn:sep\] is rather restrictive: it requires the sets $H_i$ to have small diameter and all the points $x_i$ to be close together. This provides, once again, more room in the growth condition of Definition \[defn:growth\] (as the growth condition must only hold for separated sets satisfying these restrictive assumptions). However, we will see in the proof of Lemma \[lem:growthcontr\] below that these additional restrictions arise essentially for free: if either of these restrictions is violated, the condition of the contraction principle is automatically satisfied and there is nothing to prove.
\[lem:growthcontr\] Fix $a>0$. Let $s_0(x):={\mathop{\mathrm{diam}}}(T)$ and for $n\ge 1$ $$s_n(x) :=
(a+c)s_n^a(x) +
\frac{1}{L2^{n/2}}\{
K_n(La2^{n/2},x)-
K_{n-1}(La2^{(n-1)/2},x)+
2^{-n}F_0(T)\}.$$ Then we have for every $n\ge 0$ and $A\subseteq T$ $$e_n(A) \lesssim \bigg(a+\frac{1}{c}\bigg){\mathop{\mathrm{diam}}}(A)
+ \sup_{x\in A}s_n(x).$$
Assume $n\ge 1$ and $e_n(A)>0$, else the result is trivial. Let $b=e_n(A)/2$. Lemma \[lem:packing\] yields $N=2^{2^n}$ points $x_1,\ldots,x_N\in A$ with $d(x_i,x_j)>b$ for $i\ne j$. Let $$\sigma = \sup_{x\in A}s_n^a(x),\qquad\quad
r = {\mathop{\mathrm{diam}}}(A)+\sigma.$$ **Case 1.** If $\sigma> b/c$, then the conclusion is automatically satisfied as $$e_n(A) < 2c \sup_{x\in A}s_n^a(x) \lesssim
\sup_{x\in A} s_n(x).$$ **Case 2.** If ${\mathop{\mathrm{diam}}}(A)> cb$, then the conclusion is automatically satisfied as $$e_n(A) < \frac{2}{c}{\mathop{\mathrm{diam}}}(A).$$ **Case 3.** If $\sigma\le b/c$ and ${\mathop{\mathrm{diam}}}(A)\le cb$, then the sets $H_i=B(x_i,s_n^a(x_i))$, $i=1,\ldots,N$ are $(b,c)$-separated, so the growth condition can be applied. We now essentially repeat the proof of Lemma \[lem:gausscontr\], except that we must pay the price $$\Delta_n(x) :=
K_n(La2^{n/2},x)-K_{n-1}(La2^{(n-1)/2},x)$$ for switching between two interpolation functionals (notice that $\Delta_n(x)\ge 0$ as $F_n$ is a decreasing sequence of functionals). To be precise, we estimate $$\begin{aligned}
&F_0(T) - F_n(H_i) \\
&\le K_n(La2^{n/2},x_i) + 2^{-n}F_0(T) \\
&=
K_{n-1}(La2^{(n-1)/2},x_i) + \Delta_n(x_i)
+ 2^{-n}F_0(T) \\
&\le La2^{(n-1)/2}r + F_0(T) - F_{n-1}(B(x_i,r)) +
\Delta_n(x_i)
+ 2^{-n}F_0(T) \\
&\le La2^{(n-1)/2}r + F_0(T) - F_{n-1}(\cup_{k\le N}H_k) +
\Delta_n(x_i)
+ 2^{-n}F_0(T)\end{aligned}$$ for every $i\le N$. Rearranging and applying the growth condition gives $$L2^{n/2}b \le
F_{n-1}(\cup_{i\le N}H_i) - \min_{i\le N}F_n(H_i)
\le
La2^{(n-1)/2}r +
\sup_{x\in A}\Delta_n(x) + 2^{-n}F_0(T).$$ Dividing by $L2^{n/2}$ and using the definitions of $b,r,\Delta_n$ concludes the proof.
Note that the quantity $s_n(x)$ in Lemma \[lem:growthcontr\] has an extra term as compared to Lemma \[lem:gausscontr\]. This additional term is the price we pay for switching between different interpolation functionals. However, the additional term is completely innocuous: it gives rise to a telescoping sum when we apply the contraction principle.
Applying Lemma \[lem:growthcontr\] and Theorem \[thm:contr\] yields $$\gamma_2(T) \lesssim
\bigg(a+\frac{1}{c}\bigg)\gamma_2(T) +
{\mathop{\mathrm{diam}}}(T) +
(a+c)\sup_{x\in T}\sum_{n\ge 1}2^{n/2}s_n^a(x)
+ \frac{F_0(T)}{L},$$ where we used that $K_n(La2^{n/2},x)\le F_0(T)$ for every $n\ge 1$ and $x\in T$. Thus $$\gamma_2(T) \lesssim
\bigg(a+\frac{1}{c}\bigg)\gamma_2(T) +
\frac{1+c/a}{L}F_0(T)
+ {\mathop{\mathrm{diam}}}(T)$$ by Lemma \[lem:multinterp\]. We can evidently choose a universal constant $c_0$ sufficiently large such that the conclusion of the theorem holds if $c\ge c_0$ and $a=1/c_0$.
Dimension-free bounds on random matrices {#sec:rmt}
========================================
As was stated in the introduction, there are numerous challenging probabilistic problems that remain unsolved due to the lack of understanding of how to control the supremum of some concrete Gaussian process. Such problems arise routinely, for example, in the study of structured random matrices [@RV08; @vH16b; @vH17], whose fine properties fall outside the reach of classical methods of random matrix theory. Concrete problems of this kind constitute a particularly interesting case study for the control of inhomogeneous random processes, and provide concrete motivation for the development of new methods to control chaining functionals.
Of particular interest in the setting of structured random matrices are dimension-free bounds on matrix norms. Such bounds cannot be obtained by classical methods of random matrix theory such as the moment method, which are inherently dimension-dependent. This is explained in detail [@vH16b; @vH17] in the context of a tantalizing conjecture on Gaussian random matrices due to R. Lata[ł]{}a. In this section, we make further progress in this direction by developing a closely related result: a dimension-free analogue of a well-known result of M. Rudelson [@Rud96]. The proof provides another illustration of the utility of the contraction principle.
Statement of results
--------------------
Throughout this section, let $A_1,\ldots,A_m\in\mathbb{R}^{d\times d}$ be nonrandom symmetric matrices, and let $g_1,\ldots,g_m$ be independent standard Gaussian variables. We are interested in bounding matrix norms of the random matrix $$X = \sum_{k=1}^m g_kA_k$$ in terms of the coefficients $A_k$. A well-known result of M. Rudelson [@Rud96], which was proved using a generic chaining construction (see also [@Tal14 section 16.7]), states that $$\mathbf{E}\|X\|\lesssim
\Bigg\|\sum_{k=1}^m A_k^2\Bigg\|^{1/2}\sqrt{\log(m+1)}$$ in the important special case where each $A_k=x_kx_k^*$ has rank one (here and below $\|\cdot\|$ denotes the spectral norm of a matrix). Due to the rank-one assumption, the matrices $A_k$ act nontrivially only on the $m$-dimensional subspace of $\mathbb{R}^d$ spanned by the vectors $x_1,\ldots,x_m$, so that the above bound is overtly dimension-dependent. This dimension-dependence is not expected to be sharp when different vectors $x_k$ possess substantially different scales. Unfortunately, the dependence on dimension arises in an apparently essential manner in the approach of [@Rud96]. We will see in the sequel that the contraction principle makes it possible to avoid this inefficiency. For example, we can obtain the following dimension-free form of Rudelson’s bound.
\[thm:dimfreerud\] Suppose that each $A_k=x_kx_k^*$ has rank one. Then $$\mathbf{E}\|X\|\lesssim
\Bigg\|\sum_{k=1}^m A_k^2\log(k+1)\Bigg\|^{1/2}.$$
\[rem:khin\] The generic chaining approach to Rudelson’s dimension-dependent bound is essentially made obsolete by a much simpler and more general approach using the noncommutative Khintchine inequality of Lust-Piquard and Pisier [@Rud99]. The latter shows that an analogue of Rudelson’s bound actually holds without any assumption on the matrices $A_k$ (that is, the rank-one assumption is not needed); see [@vH17] for an elementary proof. However, it does not appear that such an approach could ever produce a dimension-free bound as in Theorem \[thm:dimfreerud\], as it relies crucially on the moment method of random matrix theory which is inherently dimension-dependent in nature [@vH16b]. In addition, the moment method is useless for bounding operator norms other than the spectral norm, which is important for applications in functional analysis [@GR07; @GMPT08; @RV08]. Chaining methods appear to be essential for addressing problems of this kind that are out of reach of classical random matrix theory.
Theorem \[thm:dimfreerud\] arises as a special case of a much more general result that is of broader interest, and that clarifies the geometric structure behind the results of this section. In the remainder of this section, we will fix a symmetric compact convex set $B\subset\mathbb{R}^d$ that is $2$-convex with constant $\eta$ in the sense of Definition \[defn:ucvx\]. We will be interested in controlling $\sup_{v\in T}\langle v,Xv\rangle$ for $T\subseteq B$. When $T=B=B_2^d$ is the Euclidean ball, this is simply the largest eigenvalue of $X$ which is readily related to the spectral norm. However, we allow in general to consider any subset $T\subseteq B$. In addition, following [@GR07; @GMPT08] we can consider any $2$-convex ball $B$ instead of the Euclidean ball, which will present no additional complications in the proofs.
As $X$ is a Gaussian random matrix, clearly $v\mapsto\langle v,Xv\rangle$ is a centered Gaussian process. It therefore suffices by Theorem \[thm:mm\] to bound the right-hand side of $$\mathbf{E}\bigg[\sup_{v\in T}\langle v,Xv\rangle\bigg]
\asymp \gamma_2(T,d),$$ where the natural distance $d(v,w)$ is given by $$d(v,w) := [\mathbf{E}|\langle v,Xv\rangle-
\langle w,Xw\rangle|^2]^{1/2} =
\Bigg[\sum_{k=1}^m \langle v+w,A_k(v-w)\rangle^2\Bigg]^{1/2}.$$ We will also define for $v,z\in\mathbb{R}^d$ $$\|v\|_z :=
\Bigg[\sum_{k=1}^m \langle z,A_kv\rangle^2\Bigg]^{1/2},
\qquad\quad
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}} := \Bigg[
\sum_{k=1}^m \langle v,A_kv\rangle^2\Bigg]^{1/4}.$$ The main result of this section is the following, which could be viewed as a sort of Gordon embedding theorem [@Tal14 Theorem 16.9.1] for structured random matrices.
\[thm:gordon\] Suppose $A_1,\ldots,A_m$ are positive semidefinite. Then for any $T\subseteq B$ $$\mathbf{E}\bigg[\sup_{v\in T}\langle v,Xv\rangle\bigg] \lesssim
\frac{1}{\sqrt{\eta}}
\Bigg[
\sup_{v\in T}
\sum_{n\ge 0}(2^{n/2}e_n(B,\|\cdot\|_v))^2
\Bigg]^{1/2} +
\gamma_{4,2}(T,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.$$
When Theorem \[thm:gordon\] is specialized to the case $T=B=B_2^d$, we obtain the following bound on the spectral norm of $X$ from which Theorem \[thm:dimfreerud\] follows easily.
\[cor:supernck\] Suppose that $A_1,\ldots,A_m$ are positive semidefinite. Then $$\mathbf{E}\|X\| \lesssim
\Bigg\|\sum_{k=1}^m A_k^2 \Bigg\|^{1/2}
+ \sup_{n\ge 0}2^{n/2}e_n(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.$$
The assumption that the matrices $A_k$ are positive semidefinite is a natural relaxation of the rank-one assumption in Rudelson’s approach [@Rud96]. This assumption ensures that ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$ is a norm. Whether the positive semidefinite assumption can be weakened in Theorem \[thm:gordon\] and Corollary \[cor:supernck\] is a tantalizing question. Indeed, the abovementioned conjecture of Lata[ł]{}a [@vH16b] would follow if Corollary \[cor:supernck\] were to hold for matrices $A_k$ that are not positive semidefinite. While one can partially adapt the proof of Theorem \[thm:gordon\] to general $A_k$, significant loss is incurred in the resulting bounds. These issues will be further discussed in section \[sec:disc\] below.
Proof of Theorem \[thm:gordon\]
-------------------------------
We will assume throughout this section that the matrices $A_1,\ldots,A_m$ are positive semidefinite. This implies, in particular, that ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$ is a norm and that $\|v\|_z\le {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$ by Cauchy-Schwarz.
Let us begin by explaining the basic geometric idea behind the proof through a back-of-the-envelope computation. Note that $$d(y,z) = \|y-z\|_{y+z} \le 2\|y-z\|_x +
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}y-z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}({{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}y-x {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}+{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z-x {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})$$ by the triangle inequality and Cauchy-Schwarz. Thus $${\mathop{\mathrm{diam}}}(A,d) \le 2{\mathop{\mathrm{diam}}}(A,\|\cdot\|_x) + 2{\mathop{\mathrm{diam}}}(A,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2$$ for any $A\subseteq T$ and $x\in A$. This suggests we might try to bound $\gamma_2(T,d)$ by the sum of two terms, one of the form $\sup_{x\in T}\gamma_2(T,\|\cdot\|_x)$ and another of the form $\gamma_2(T,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2)=\gamma_{4,2}(T,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2$. If that were possible, we would obtain a result far better than Theorem \[thm:gordon\]. The problem, however, lies with the first term: a direct application of the contraction principle yields not $\sup_{x\in T}\gamma_2(T,\|\cdot\|_x)$, but rather $$\inf_{(\mathcal{A}_n)}
\sup_{x\in T}\sum_{n\ge 0}2^{n/2}{\mathop{\mathrm{diam}}}(A_n(x),\|\cdot\|_x).$$ The latter could be much larger than $\sup_{x\in T}\gamma_2(T,\|\cdot\|_x)$: here a single admissible sequence $(\mathcal{A}_n)$ must control simultaneously every norm $\|\cdot\|_x$, while in the definition of $\sup_{x\in T}\gamma_2(T,\|\cdot\|_x)$ each norm is controlled by its own admissible sequence. The remarkable aspect of Theorem \[thm:gordon\] is that by exploiting the contraction theorem and 2-convexity of $B\supseteq T$, we will nonetheless achieve the same upper bound as would be obtained if we were to control $\sup_{x\in T}\gamma_2(B,\|\cdot\|_x)$ using Theorem \[thm:geom\].
We now proceed with the details of the proof. To exploit $2$-convexity, it will be useful to replace the natural metric $d$ by a regularized form $$\tilde d(v,w) := d(v,w) + {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v-w {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2.$$ While $\tilde d$ is not a metric, it is a quasi-metric (the triangle inequality holds up to a multiplicative constant). This will suffice for all our purposes; in particular, it is readily verified that the proof of the contraction Theorem \[thm:contr\] holds verbatim in a quasi-metric space up to the value of the universal constant. We will use this observation in the sequel without further comment. The advantage of $\tilde d$, as opposed to the natural metric, is that it behaves in some sense like a norm.
\[lem:quasi\] For every $v,w,z\in\mathbb{R}^d$, we have:
a. $\tilde d(v,w) \le 2(\tilde d(v,z)+\tilde d(z,w))$.
b. $\tilde d(v,\frac{1}{2}(v+w)) \le \frac{1}{2}\tilde d(v,w)$.
The first claim follows from the triangle inequality and $(a+b)^2\le 2(a^2+b^2)$. To prove the second claim, note that we can write $$v - \tfrac{1}{2}(v+w) = \tfrac{1}{2}(v-w),\qquad
v + \tfrac{1}{2}(v+w) = \tfrac{1}{2}(v-w) + (v+w).$$ Therefore $$\begin{aligned}
\tilde d(v,\tfrac{1}{2}(v+w)) &=
\tfrac{1}{2}
\|\tfrac{1}{2}(v-w) + (v+w)\|_{v-w}
+
\tfrac{1}{4}{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v-w {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2 \\
&\le
\tfrac{1}{2}(\|v+w\|_{v-w} + {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v-w {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2) =
\tfrac{1}{2}\tilde d(v,w),\end{aligned}$$ where we used the triangle inequality.
We now define the interpolation functional $$K(t,x) := \inf_{y\in\mathbb{R}^d}\{
\|y\|_B + t\tilde d(x,y)
\},$$ and as usual we let $\pi_t(x)$ be a minimizer in this expression. Due to the second property of Lemma \[lem:quasi\] (which was engineered precisely for this purpose), we can control the shrinkage of interpolation sets as in the proof of Theorem \[thm:ucvx\].
\[lem:gordonshrink\] Let $t\ge 0$ and $A\subseteq T$. Then $A_t :=\{\pi_t(x):x\in A\}$ satisfies $$A_t \subseteq \frac{L\sqrt{t}}{\sqrt{\eta}}
\bigg\{{\mathop{\mathrm{diam}}}(A,\tilde d)+\sup_{x\in A}\tilde d(x,\pi_t(x))
\bigg\}^{1/2}
(z+B)$$ for some point in $z\in\mathbb{R}^d$, where $L$ is a universal constant.
Let $x\in A$ and $y=\pi_t(x)$. Then $$\begin{aligned}
\|y\|_B \le K(t,x) \le
\|u\|_B + 2t(\tilde d(x,y) + \tilde d(y,u))\end{aligned}$$ for any $u\in\mathbb{R}^d$ by the definition of the interpolation functional and the first property of Lemma \[lem:quasi\]. Therefore, we have for every $y,z\in A_t$ and $u\in\mathbb{R}^d$ $$\max(\|y\|_B,\|z\|_B) \le
\|u\|_B
+ 2t \max(\tilde d(y,u),\tilde d(z,u))
+ 2t \sup_{x\in A}\tilde d(x,\pi_t(x)).$$ If we choose $u=\tfrac{1}{2}(y+z)$, then we obtain $$\max(\|y\|_B,\|z\|_B) \le
\bigg\|\frac{y+z}{2}\bigg\|_B
+ t\tilde d(y,z)
+ 2t \sup_{x\in A}\tilde d(x,\pi_t(x))$$ using the second property of Lemma \[lem:quasi\]. In particular, $$\eta\|y-z\|_B^2 \le
t\tilde d(y,z)
+ 2t \sup_{x\in A}\tilde d(x,\pi_t(x))$$ for all $y,z\in A_t$ by $2$-convexity of $B$. It follows that $${\mathop{\mathrm{diam}}}(A_t,\|\cdot\|_B) \le
\frac{\sqrt{t}}{\sqrt{\eta}}
\bigg\{
{\mathop{\mathrm{diam}}}(A_t,\tilde d)
+ 2\sup_{x\in A}\tilde d(x,\pi_t(x))
\bigg\}^{1/2}.$$ It remains to note that ${\mathop{\mathrm{diam}}}(A_t,\tilde d)\le 4 {\mathop{\mathrm{diam}}}(A,\tilde d) + 8\sup_{x\in A}\tilde
d(x,\pi_t(x))$.
We now arrive at the main step in the proof of Theorem \[thm:gordon\]: we must verify the assumption of the contraction principle.
\[lem:gordoncontr\] Let $(\mathcal{C}_n)$ be an admissible sequence of $T$ and $a,b>0$. Then $$e_n(A,\tilde d) \lesssim
b{\mathop{\mathrm{diam}}}(A,\tilde d) + \sup_{x\in A} s_n(x)$$ for every $n\ge 1$ and $A\subseteq T$, where $$s_n(x) :=
(b+1)\tilde d(x,\pi_{a2^{n/2}}(x)) +
\frac{a2^{n/2}}{b\eta} e_{n-1}(B,\|\cdot\|_x)^2 +
{\mathop{\mathrm{diam}}}(C_{n-1}(x),{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.$$
Fix $n\ge 1$ and $A\subseteq T$. For every set $C\in\mathcal{C}_{n-1}$, define $$A_{a2^{n/2}}^C := \{\pi_{a2^{n/2}}(x):x\in A\cap C\}$$ and choose an arbitrary point $x_C\in A\cap C$. Now choose, for every $C\in\mathcal{C}_{n-1}$, a net $T_{n-1}^C\subseteq A_{a2^{n/2}}^C$ of cardinality less than $2^{2^{n-1}}$ such that $$\inf_{z\in T_{n-1}^C}\|y-z\|_{x_C} \le
4e_{n-1}(A_{a2^{n/2}}^C,\|\cdot\|_{x_C})
\quad\mbox{for all }y\in A_{a2^{n/2}}^C.$$ Then $T_n := \bigcup_{C\in\mathcal{C}_{n-1}}T_{n-1}^C$ has cardinality less than $2^{2^n}$. It remains to show that $$\sup_{x\in A}
\tilde d(x,T_n) \lesssim b{\mathop{\mathrm{diam}}}(A,\tilde d)+\sup_{x\in A}s_n(x),$$ which concludes the proof.
To this end, fix $C\in\mathcal{C}_{n-1}$ and $x\in A\cap C$, and choose $z\in T_{n-1}^C$ such that $$\|\pi_{a2^{n/2}}(x)-z\|_{x_C} \le
4e_{n-1}(A_{a2^{n/2}}^C,\|\cdot\|_{x_C}).$$ We can estimate $$\begin{aligned}
\tilde d(x,T_n) &\le 2\tilde d(x,\pi_{a2^{n/2}}(x)) +
2\tilde d(\pi_{a2^{n/2}}(x),T_n)\\
&\le 2\tilde d(x,\pi_{a2^{n/2}}(x)) +
2\tilde d(\pi_{a2^{n/2}}(x),z) \\
&\le 2\tilde d(x,\pi_{a2^{n/2}}(x)) +
4\|\pi_{a2^{n/2}}(x)-z\|_{x_C} \\
&\quad +
2{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\pi_{a2^{n/2}}(x)-z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}(
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\pi_{a2^{n/2}}(x)-x_C {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}+{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z-x_C {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}
) \\
&\quad + {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\pi_{a2^{n/2}}(x)-z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2.\end{aligned}$$ As $z\in A_{a2^{n/2}}^C$ by construction, there is a point $x'\in A\cap C$ such that $z=\pi_{a2^{n/2}}(x')$. We therefore obtain, using that ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v-w {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2\le\tilde d(v,w)$, $$\begin{aligned}
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\pi_{a2^{n/2}}(x)-z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}} &\le
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}x-\pi_{a2^{n/2}}(x) {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}+{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}x-x' {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}+
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}x'-\pi_{a2^{n/2}}(x') {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}
\\
&\le
2\sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v))^{1/2} +
{\mathop{\mathrm{diam}}}(C,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}).\end{aligned}$$ Similarly, we can estimate $${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\pi_{a2^{n/2}}(x)-x_C {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}} +
{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z-x_C {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}
\le
2\sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v))^{1/2} +
2{\mathop{\mathrm{diam}}}(C,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}).$$ Putting together the above estimates, we obtain $$\tilde d(x,T_n) \lesssim
\sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v)) +
e_{n-1}(A_{a2^{n/2}}^C,\|\cdot\|_{x_C}) +
{\mathop{\mathrm{diam}}}(C,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2$$ for every $x\in A\cap C$. We now note that by Lemma \[lem:gordonshrink\], $$\begin{aligned}
&e_{n-1}(A_{a2^{n/2}}^C,\|\cdot\|_{x_C}) \\
&\lesssim
\frac{\sqrt{a2^{n/2}}}{\sqrt{\eta}}
\bigg\{
{\mathop{\mathrm{diam}}}(A,\tilde d) + \sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v))
\Bigg\}^{1/2}
e_{n-1}(B,\|\cdot\|_{x_C})
\\ &\lesssim
\frac{a2^{n/2}}{b\eta}
\sup_{v\in A}
e_{n-1}(B,\|\cdot\|_{v})^2 +
b{\mathop{\mathrm{diam}}}(A,\tilde d) + b\sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v)).\end{aligned}$$ As $x\in A\cap C_{n-1}(x)$ for every $x\in A$, we have shown that $$\begin{gathered}
\sup_{x\in A}\tilde d(x,T_n) \lesssim
b{\mathop{\mathrm{diam}}}(A,\tilde d) +
(b+1)\sup_{v\in A}\tilde d(v,\pi_{a2^{n/2}}(v)) \\ +
\frac{a2^{n/2}}{b\eta}
\sup_{v\in A}
e_{n-1}(B,\|\cdot\|_{v})^2 +
\sup_{v\in A}
{\mathop{\mathrm{diam}}}(C_{n-1}(v),{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.\end{gathered}$$ The proof is concluded using $\sup_v a_1(v) + \sup_v a_2(v) + \sup_v a_3(v) \le
3\sup_v(a_1(v)+a_2(v)+a_3(v))$ for any nonnegative functions $a_1(v),a_2(v),a_3(v)\ge 0$.
\[rem:propercov\] We used above the standard fact that for any metric space $(X,d)$ and $T\subseteq X$, there is a net $T_n\subseteq T$ with $|T_n|<2^{2^n}$ so that $\sup_{x\in T}d(x,T_n)\le 4e_n(T,d)$. We recall the proof for completeness. The definition of entropy numbers guarantees the existence of a net $S_n\subseteq X$ with $|S_n|<2^{2^n}$ so that $\sup_{x\in T}d(x,S_n)\le 2e_n(T,d)$, but $S_n$ need not be a subset of $T$. For every point $z\in S_n$, choose $z'\in T$ such that $d(z,z')\le 2e_n(T,d)$, and let $T_n\subseteq T$ be the collection of points thus constructed. Then $d(x,T_n) \le d(x,S_n) + d(S_n,T_n) \le
4e_n(T,d)$ for every $x\in T$ as desired. The fact that one can choose the net $T_n$ to be a subset of $T$ rather than of $X$ was essential in the above proof in order to ensure that $T_{n-1}^C\subseteq A_{a2^{n/2}}^C$.
We can now complete the proof of Theorem \[thm:gordon\].
By Theorem \[thm:mm\], we have $$\mathbf{E}\bigg[\sup_{v\in T}\langle v,Xv\rangle\bigg]
\lesssim
\gamma_2(T,d) \le \gamma_2(T,\tilde d).$$ Fix $a,b>0$ and an admissible sequence $(\mathcal{C}_n)$ of $T$. Then $$\begin{gathered}
\gamma_2(T,\tilde d) \lesssim
b\gamma_2(T,\tilde d) +
{\mathop{\mathrm{diam}}}(T,\tilde d) +
(b+1)\sup_{x\in T}\sum_{n\ge 1}2^{n/2}\tilde d(x,\pi_{a2^{n/2}}(x))
\\
+ \frac{a}{b\eta}\sup_{x\in T}\sum_{n\ge 1}
(2^{n/2}e_{n-1}(B,\|\cdot\|_x))^2
+ \sup_{x\in T}\sum_{n\ge 1}2^{n/2}
{\mathop{\mathrm{diam}}}(C_{n-1}(x),{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2\end{gathered}$$ by Theorem \[thm:contr\], where we used Lemma \[lem:gordoncontr\] to define $s_n(x)$ for $n\ge 1$ and the trivial choice $s_0(x)={\mathop{\mathrm{diam}}}(T,\tilde d)$. Choosing $b$ to be a sufficiently small universal constant and applying the interpolation Lemma \[lem:interp\] gives $$\begin{aligned}
\gamma_2(T,\tilde d) \lesssim
{\mathop{\mathrm{diam}}}(T,\tilde d) &+
\frac{1}{a} +
\frac{a}{\eta}\sup_{x\in T}\sum_{n\ge 0}
(2^{n/2}e_n(B,\|\cdot\|_x))^2 \\
&+ \sup_{x\in T}\sum_{n\ge 0}2^{n/2}
{\mathop{\mathrm{diam}}}(C_n(x),{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.\end{aligned}$$ Optimizing over $a$ and over admissible sequences $(\mathcal{C}_n)$ of $T$ yields $$\gamma_2(T,\tilde d)\lesssim
{\mathop{\mathrm{diam}}}(T,\tilde d)+
\frac{1}{\sqrt{\eta}}
\Bigg[\sup_{x\in T}\sum_{n\ge 0}
(2^{n/2}e_n(B,\|\cdot\|_x))^2\Bigg]^{1/2}
+ \gamma_{4,2}(T,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.$$ It remains to note that as ${\mathop{\mathrm{diam}}}(T,\tilde d)\le
2{\mathop{\mathrm{diam}}}(B,\|\cdot\|_x) + 2{\mathop{\mathrm{diam}}}(T,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2$ for any $x\in T$, the first term can be absorbed in the remaining two.
Proof of Corollary \[cor:supernck\] and Theorem \[thm:dimfreerud\]
------------------------------------------------------------------
Using Theorem \[thm:gordon\], the proof of Corollary \[cor:supernck\] follows from classical entropy estimates for ellipsoids.
Note that for any $v\in\mathbb{R}^d$, the norm $\|\cdot\|_v$ is a Euclidean norm defined by the inner product $\langle x,y\rangle_v :=
\langle x,\Sigma_vy\rangle$ with $\Sigma_v := \sum_{k=1}^mA_kvv^*A_k$. Thus $e_n(B_2^d,\|\cdot\|_v)$ are entropy numbers of ellipsoids in Hilbert space, which are well understood. Using the entropy estimates in [@Tal14 section 2.5], we readily obtain $$\sum_{n\ge 0} (2^{n/2}e_n(B_2^d,\|\cdot\|_v))^2 \asymp
\mathrm{Tr}[\Sigma_v] =
\Bigg\langle v,\Bigg(\sum_{k=1}^m A_k^2\Bigg)v\Bigg\rangle.$$ In particular, we obtain $$\Bigg[\sup_{v\in B_2^d}
\sum_{n\ge 0} (2^{n/2}e_n(B_2^d,\|\cdot\|_v))^2\Bigg]^{1/2}
\asymp \Bigg\|\sum_{k=1}^m A_k^2\Bigg\|^{1/2}.$$ On the other hand, by Theorem \[thm:ucvx\], we have $$\gamma_{4,2}(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}) \asymp
\sup_{n\ge 0}2^{n/4}e_n(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}).$$ Thus Theorem \[thm:gordon\] implies $$\mathbf{E}\bigg[\sup_{v\in B_2^d}\langle v,Xv\rangle\bigg]
\lesssim
\Bigg\|\sum_{k=1}^m A_k^2\Bigg\|^{1/2} +
\sup_{n\ge 0}2^{n/2}e_n(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2.$$ It remains to note that $$\|X\| = \sup_{v\in B_2^d}|\langle v,Xv\rangle|
\le \sup_{v\in B_2^d}\langle v,Xv\rangle +
\sup_{v\in B_2^d}\langle v,(-X)v\rangle$$ and that $X$ and $-X$ have the same distribution.
To deduce Theorem \[thm:dimfreerud\] from Corollary \[cor:supernck\], we need to estimate the entropy numbers $e_n(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})$. We will accomplish this using a classical result, the dual Sudakov inequality of N. Tomczak-Jaegermann [@Tal14 Lemma 8.3.6].
We use the trivial estimate $${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}^2 \le \|v\|_\sim :=
\sup_{z\in B_2^d}\|v\|_z$$ for $v\in B_2^d$. This implies, using Remark \[rem:propercov\] and the dual Sudakov inequality, that $$e_n(B_2^d,{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}})^2 \lesssim
e_n(B_2^d,\|\cdot\|_\sim) \lesssim
2^{-n/2}\mathbf{E}\|g\|_\sim,$$ where $g$ is a standard Gaussian vector in $\mathbb{R}^d$. Corollary \[cor:supernck\] yields $$\mathbf{E}\|X\| \lesssim
\Bigg\|\sum_{k=1}^m A_k^2\Bigg\|^{1/2} +
\mathbf{E}\|g\|_\sim.$$ Now suppose $A_k=x_kx_k^*$ have rank one. Then $$\begin{aligned}
\mathbf{E}\|g\|_\sim &=
\mathbf{E}\Bigg[
\sup_{z\in B_2^d}\sum_{k=1}^m
\langle z,x_k\rangle^2 \langle x_k,g\rangle^2
\Bigg]^{1/2} \\
&\le
\Bigg[\sup_{z\in B_2^d}
\sum_{k=1}^m
\langle z,x_k\rangle^2\|x_k\|^2\log(k+1)\Bigg]^{1/2}
\mathbf{E}\bigg[
\max_{k\le m}
\frac{|\langle x_k,g\rangle|}{\|x_k\|\sqrt{\log(k+1)}}
\bigg]
\\
&\lesssim
\Bigg\|
\sum_{k=1}^m A_k^2\log(k+1)
\Bigg\|^{1/2}, \end{aligned}$$ using $A_k^2 = x_kx_k^*\|x_k\|^2$ and that $\mathbf{E}[\max_{k}|G_k|/\sqrt{\log(k+1)}]\lesssim 1$ when $G_k$ are (not necessarily independent) standard Gaussian variables [@Tal14 Proposition 2.4.16].
Discussion {#sec:disc}
----------
The aim of this section is to briefly discuss the connection between Corollary \[cor:supernck\] and a conjecture of Lata[ł]{}a. Let us briefly recall this conjecture, which is discussed in detail in [@vH16b]. Let $X$ be a symmetric $d\times d$ matrix whose entries $\{X_{ij}:i\ge j\}$ are independent centered Gaussians with arbitrary variances $X_{ij}\sim N(0,b_{ij}^2)$. Lata[ł]{}a’s conjecture states that the spectral norm of such a matrix is always of the same order as the maximum of the Euclidean norm of its rows, $$\mathbf{E}\|X\| \stackrel{?}{\asymp}
\mathbf{E}\Bigg[\max_i\sqrt{\sum_j X_{ij}^2}\Bigg].$$ The lower bound is trivial, as the spectral norm of any matrix is bounded below (deterministically) by the maximal Euclidean norm of its rows. It is far from obvious, however, why the upper bound should be true.
The independent entry model can be equivalently written as $$X = \sum_{i\ge j} g_{ij}A_{ij},\qquad\quad
A_{ij} = b_{ij}(e_ie_j^*+e_je_i^*),$$ where $\{e_i\}$ denotes the standard basis in $\mathbb{R}^d$ and $\{g_{ij}\}$ are independent standard Gaussian variables. This model is therefore a special case of the general model considered in this section. Unfortunately, the matrices $A_{ij}$ are not positive semidefinite. If the conclusion of Corollary \[cor:supernck\] were to hold nonetheless for these matrices, then Lata[ł]{}a’s conjecture would follow readily. Indeed, arguing precisely as in the proof of Theorem \[thm:dimfreerud\], we would obtain in this case $$\begin{aligned}
\mathbf{E}\|X\| &\stackrel{?}{\lesssim}
\Bigg\|\sum_{i\ge j} A_{ij}^2\Bigg\|^{1/2} +
\mathbf{E}\bigg[\sup_{z\in B_2^d}\|g\|_z\bigg]
\\
&\lesssim
\max_i\sqrt{\sum_j b_{ij}^2} +
\mathbf{E}\Bigg[
\max_i
\sqrt{\sum_{j} b_{ij}^2 g_j^2}
\Bigg]
\lesssim
\mathbf{E}\Bigg[\max_i\sqrt{\sum_j X_{ij}^2}\Bigg],\end{aligned}$$ where the last inequality was established in [@vH16b]. In view of these observations, it is of significant interest to understand to what extent the positive semidefinite assumption made in this section could be weakened.
An inspection of the proof of Theorem \[thm:gordon\] shows that the positive semidefinite assumption was used only to ensure that ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$ is a norm and that $\|v\|_z\le{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$. All results in this section therefore continue to hold verbatim if we were to replace ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}$ in the statement and proof of Theorem \[thm:gordon\] and Corollary \[cor:supernck\] by an arbitrary (quasi)norm ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'$ such that $\|v\|_z\lesssim {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'$. This makes it possible, in principle, to prove much more general versions of these results. For example, the norm $${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}' = \Bigg[\sum_{k=1}^m \langle v,|A_k|v\rangle^2
\Bigg]^{1/4}$$ satisfies the requisite condition for arbitrary $A_1,\ldots,A_m$, so that we obtain a general variant of Theorem \[thm:gordon\] and Corollary \[cor:supernck\] without any assumption on the coefficient matrices. However, significant loss may be incurred when we replace $A_k$ by $|A_k|$. For example, in the independent entry model this yields a bound of the form $$\mathbf{E}\|X\| \lesssim
\max_i\sqrt{\log i}\sqrt{\sum_j b_{ij}^2},$$ which is far larger than the bound suggested by Lata[ł]{}a’s conjecture.
Other choices of ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'$ are possible in specific situations. For example, in the independent entry model, consider the choice $${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}' = \Bigg[\sum_{i,j=1}^d v_i^2b_{ij}^2v_j^2
\Bigg]^{1/4}.$$ This defines a norm if we assume that the matrix of entry variances $(b_{ij}^2)$ is positive semidefinite, in which case it is readily verified that $\|v\|_z\lesssim {{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}v {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'{{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}z {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'$. This choice suffices to establish Lata[ł]{}a’s conjecture under the highly restrictive assumption that $(b_{ij}^2)\succeq 0$, recovering a result proved in [@vH16b] by different means.
In more general situations, it is not clear that it is possible to introduce a suitable (quasi)norm ${{{\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}\cdot {\rvert{\hspace{-0.12em}}\rvert{\hspace{-0.12em}}\rvert}}}'$ without incurring significant loss, and it is likely that the resolution of Lata[ł]{}a’s conjecture will require some additional geometric insight. Nonetheless, beside their independent interest, the results of this section provide a further step toward better understanding of the multiscale geometry of random matrices, and suggest that further development of the methods of this paper could yield new insights on various open problems in this area.
It is worth noting that even when $A_1,\ldots,A_m$ are positive definite, the geometric approach developed here is not necessarily efficient. Consider, for example, the trivial case where $m=1$ and $A_1=I$ is the identity matrix. Then obviously $\mathbf{E}\|X\|\asymp 1$, but Corollary \[cor:supernck\] gives the terrible bound $$\mathbf{E}\|X\| \lesssim 1 + \sup_{n\ge 0}
2^{n/2}e_n(B_2^d,\|\cdot\|_2)^2 \asymp \sqrt{d}.$$ Thus the geometric principle behind this section cannot fully explain the noncommutative Khintchine inequality discussed in Remark \[rem:khin\], even though it actually improves on this inequality when the coefficient matrices have low rank. Discovering the correct geometric explanation of the noncommutative Khintchine inequality is closely related to another fundamental problem in the generic chaining theory [@Tal14 pp. 50–51] whose resolution may also shed new light on other random matrix problems (such as, for example, the problem of obtaining sharp bounds in [@RV08]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author is grateful to Richard Nickl for hosting a very pleasant visit to Cambridge during which some key results of this paper were obtained, and to Roman Vershynin and Subhro Ghosh for motivating discussions. This work was supported in part by NSF grant CAREER-DMS-1148711 and by the ARO through PECASE award W911NF-14-1-0094.
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[^1]: Consider $\tilde e_n^{a,x}(T) :=
\inf\{a^k{\mathop{\mathrm{diam}}}(T):\prod_{i=0}^k N(T\cap
B(x,a^{i-2}{\mathop{\mathrm{diam}}}(T)),a^i{\mathop{\mathrm{diam}}}(T)) < 2^{2^n}\}$, where $N(A,\varepsilon)$ is the covering number of $A$ by balls of radius $\varepsilon$. The details are left to the reader.
[^2]: As $K(t,x)\le \|x\|_T^r\le 1$, we may restrict the infimum to be taken over the compact set $y\in T$. But $\|y\|_T=\sup_{z\in T^\circ}\langle z,y\rangle$ by duality, so the gauge is lower-semicontinuous and the inf is attained.
|
---
abstract: 'Traditional person re-identification (ReID) methods typically represent person images as real-valued features, which makes ReID inefficient when the gallery set is extremely large. Recently, some hashing methods have been proposed to make ReID more efficient. However, these hashing methods will deteriorate the accuracy in general, and the efficiency of them is still not high enough. In this paper, we propose a novel hashing method, called eep ulti-ndex ashing (), to improve both efficiency and accuracy for ReID. DMIH seamlessly integrates hashing and based networks into the same framework. Furthermore, a novel hashing table construction approach and a (SAMI) loss are proposed in to improve the search efficiency. Experiments on three widely used datasets show that DMIH can outperform other baselines, including both hashing methods and methods, in terms of both efficiency and accuracy.'
author:
- |
Ming-Wei Li, Qing-Yuan Jiang and Wu-Jun Li\
National Key Laboratory for Novel Software Technology\
Collaborative Innovation Center of Novel Software Technology and Industrialization\
Department of Computer Science and Technology, Nanjing University, China\
`{jiangqy,limw}@lamda.nju.edu.cn, [email protected]`\
bibliography:
- 'ref.bib'
title: 'Deep Multi-Index Hashing for Person Re-Identification'
---
Introduction
============
Person re-identification (ReID) [@DBLP:conf/eccv/SunZYTW18; @DBLP:conf/cvpr/ChangHX18; @DBLP:conf/nips/GeLZYYWL18] has attracted much attention in computer vision. For a given probe person, the goal of ReID is to retrieve (search) in the gallery set for pedestrian images containing the same individual in a mode. Recently, has been widely used in many real applications including video retrieval, video surveillance, and so on.
Existing ReID methods can be divided into two main categories [@DBLP:journals/corr/ZhengYH16]. One category focuses on utilizing features to represent person images, especially for most early ReID approaches [@DBLP:conf/cvpr/KostingerHWRB12; @DBLP:conf/cvpr/ZhaoOW14; @DBLP:conf/cvpr/LiaoHZL15]. The other category [@DBLP:conf/mm/WangYCLZ18; @DBLP:conf/eccv/SunZYTW18; @DBLP:conf/cvpr/ChangHX18] adopts deep learning architectures to extract features. Most of these existing methods, including both deep methods and non-deep methods, typically represent person images as real-valued features. This real-valued feature representation makes inefficient when the gallery set is extremely large, due to high computation and storage cost during the retrieval (search) procedure.
Recently, hashing [@DBLP:conf/nips/WeissTF08; @DBLP:conf/nips/LiSHT17; @DBLP:conf/nips/SuZHT18; @DBLP:conf/nips/LiuMKC14; @DBLP:conf/icml/LiLSHD13; @DBLP:conf/icml/YuKGC14; @DBLP:conf/icml/LiuWKC11; @DBLP:conf/icml/DaiGKHS17; @DBLP:conf/icml/NorouziF11; @DBLP:conf/icml/WangKC10] has been introduced into ReID community for efficiency improvement due to its low storage cost and fast query speed. The goal of hashing is to embed data points into a Hamming space of binary codes where the similarity in the original space is preserved. Several hashing methods have been proposed for ReID [@DBLP:journals/tip/ZhangLZZZ15; @DBLP:conf/ijcai/ZhengS16; @DBLP:conf/cvpr/ChenWQLS17; @DBLP:journals/tip/ZhuKZFT17]. In [@DBLP:conf/ijcai/ZhengS16], deep regularized similarity comparison hashing (DRSCH) was designed by combining formulation and binary codes generation. In [@DBLP:conf/ijcai/ZhengS16], binary identifies (CBI) was learned by constructing two sets of discriminative hash functions. In [@DBLP:conf/cvpr/ChenWQLS17], semantic binary transformation (CSBT) employed subspace projection to mitigate variations. In [@DBLP:journals/tip/ZhuKZFT17], deep hashing (PDH) was proposed to incorporate formulation and image partitions to learn binary codes. Among these methods, CBI and CSBT focus on designing models to learn binary codes by using features. DRSCH and PDH are deep hashing methods which try to integrate deep feature learning and hash code learning into an framework. Recent efforts [@DBLP:conf/nips/LiSHT17; @DBLP:conf/nips/SuZHT18; @DBLP:journals/tip/ZhuKZFT17] show that the deep hashing methods can achieve better performance than hand-crafted feature based hashing methods.
However, existing methods usually cannot achieve satisfactory performance for ReID. Exhaustive linear search based on Hamming ranking cannot handle dataset. More specifically, although one can adopt hash lookup to achieve query speed, they [@DBLP:journals/tip/ZhuKZFT17; @DBLP:conf/cvpr/ChenWQLS17] usually need long binary codes to achieve reasonable accuracy due to the high complexity in ReID. In this situation, the retrieval speed will become extremely slow because the number of hash bins that need to be retrieved increases exponentially as code length increases. Hence, although existing hashing methods can achieve faster speed than traditional real-valued ReID methods, these hashing methods will typically deteriorate the accuracy because the binary code cannot be too long. Furthermore, the efficiency of existing hashing methods is still not high enough.
In this paper, we propose a novel hashing method, called eep ulti-ndex ashing (DMIH), to improve both retrieval efficiency and accuracy for ReID. Our main contributions are summarized as follows.
DMIH seamlessly integrates multi-index hashing () [@DBLP:journals/pami/0002PF14] and multi-branch based networks into the same framework. In DMIH, feature learning procedure and hash code learning procedure can facilitate each other. To the best of our knowledge, DMIH is the first hashing based ReID method to integrate hashing and deep feature learning into the same framework.
In DMIH, a novel block-wise multi-index hashing table construction approach and a search-aware multi-index (SAMI) loss are proposed to improve the retrieval efficiency.
Experiments on three widely used datasets show that DMIH can outperform other state-of-the-art baselines, including both hashing methods and real-valued methods, in terms of both efficiency and accuracy.
Related Work {#sec:related-work}
============
#### Multi-Index Hashing
In real applications, when facing long binary codes, hash lookup will suffer from low retrieval speed due to large number of hash bins that need to be retrieved. hashing (MIH) [@DBLP:journals/pami/0002PF14], which can enable efficient $k$-nearest neighbors search[^1] for long codes is proposed to deal with this situation. MIH divides the long binary codes into several disjoint but consecutive codes and builds multiple hash tables on shorter code substrings, which can enormously reduce the number of hash bins to be retrieved and improve the search efficiency.
However, MIH is based on the assumption that the binary codes should be distributed balanced between codes, which is usually not satisfied in real applications [@DBLP:conf/mm/ZhangGZL11]. So the time performance of MIH will be adversely affected when dealing with unbalanced distributed codes. Our method learns to adjust the distribution of binary codes by minimizing SAMI loss to enhance the time performance of MIH.
#### Multi-Branch Architectures
Multi-branch based networks [@DBLP:conf/cvpr/SzegedyLJSRAEVR15; @DBLP:conf/cvpr/HeZRS16] have been widely exploited in computer vision tasks. Recently, “grouped convolution” [@DBLP:conf/cvpr/XieGDTH17; @DBLP:conf/cvpr/HuangLMW18] has been proposed to construct multi-branch architectures. These building blocks can achieve stronger modeling capacity. In ReID, due to the cross-camera variations, the partial information is significant to improve the discriminative performances. Multi-branch based networks have been used to learn discriminative information with various granularities in previous works [@DBLP:journals/tip/ZhuKZFT17; @DBLP:conf/mm/WangYCLZ18; @DBLP:conf/eccv/SunZYTW18; @DBLP:conf/cvpr/ChangHX18].
Notation and Problem Definition {#sec:notation}
===============================
Notation
--------
We use boldface lowercase letters like $\w$ to denote vectors and boldface uppercase letters like $\W$ to denote matrices. $\Vert\w\Vert_2$ denotes the $L_2$-norm for the vector $\w$. $[\cdot]_{+}$ is defined as $[x]_+=\max\{0,x\}$. For an integer $C$, we use $[C]$ to denote the set $\{1,2,\dots,C\}$. $\sgn(\cdot)$ is an element-wise sign function where $\sgn(x) = 1$ if $x\geq 0$ else $\sgn(x)=-1$. Furthermore, $\Vert\b_i-\b_j\Vert_H$ denotes the Hamming distance between two binary vectors $\b_i$ and $\b_j$, i.e., $\Vert\b_i-\b_j\Vert_H=(R-\b_i^T\b_j)/2$. Here, $R$ is the code length of $\b_i$ and $\b_j$.
Hashing based ReID
------------------
Assume that we have $n$ training samples which are denoted as $\X=\{\x_i\}_{i=1}^n$. Furthermore, person identities for images are also available and denoted as $\y=\{y_i\;\vert\;y_i\in[C]\}_{i=1}^n$, where $C$ denotes the number of persons in the training set. Our target is to learn a deep hash function $H(\x)\in \{-1, +1\}^R$, which can transform the person images to binary codes with $R$ bits.
Deep Multi-Index Hashing for ReID {#sec:model}
=================================
Model
-----
The DMIH model is illustrated in Figure \[fig:framework\], which is an deep learning framework containing two components, i.e., network part and binary codes learning part. Furthermore, a novel hashing table construction approach and a novel (SAMI) loss are developed to improve search efficiency.
\[fig:framework\]
#### Multi-Branch Network Part
As a variety of deep methods for ReID [@DBLP:journals/tip/ZhuKZFT17; @DBLP:conf/mm/WangYCLZ18; @DBLP:conf/eccv/SunZYTW18; @DBLP:conf/cvpr/ChangHX18] have demonstrated that multi-branch architectures based network can learn more discriminative features, we adopt the network (MGN) architecture [@DBLP:conf/mm/WangYCLZ18] as the feature learning part of DMIH. This architecture integrates global and local features to get more powerful pedestrian representations. The MGN architecture is shown in the left part of Figure \[fig:framework\], which contains a [@DBLP:conf/cvpr/HeZRS16] network and three branches. The upper branch without any partition information learns the global feature representations. The middle and lower branches uniformly split feature maps into several stripes in horizontal orientation to learn the local feature representations. For a given pedestrian image $\x_i$, the output of all the three branches are denoted as $\{\f_i^{(1)}, \f_i^{(2)}, \f_i^{(3)}\}$, which contains the global and local feature representations. Please note that DMIH is general enough to adopt other multi-branch architectures since our objective is to improve the retrieval efficiency and accuracy, rather than designing a new multi-branch building block. In other words, our method is an extensive learning algorithm, which is independent of specific network architectures.
#### Binary Codes Learning Part
The principle of binary codes learning is to preserve the similarity of samples. We use loss to achieve this goal, which has been proved to be effective in deep ReID tasks [@DBLP:journals/tip/ZhangLZZZ15; @DBLP:journals/tip/ZhuKZFT17; @DBLP:conf/mm/WangYCLZ18]. Specifically, for the $i$-th input $\x_i$, we add a layer after each branch of network as a hash layer to project the global and local features, i.e., $\{\f_i^{(1)}, \f_i^{(2)}, \f_i^{(3)}\}$, into $\RB^r$. Then we employ the function $\sgn(\cdot)$ to get its corresponding binary codes $\{\d_i, \g_i, \h_i\}$, where $\d_i,\g_i,\h_i \in\{-1,+1\}^r$ and $r$ denotes the code length of each code[^2].
Then a triplet loss function is imposed on $\{\d_i, \g_i, \h_i\}$. For example, the loss function for a mini-batch $\{\d_i\}^N_{i=1}$ with $N$ samples can be formulated as follows: $$\begin{aligned}
L_{t}(\{\d_i\}_{i=1}^N) = \frac{1}{N}\sum_{i=1}^N\Big[& \alpha+\max\Vert\d_{i}-\d^{+}_{i}\Vert_H-\min\Vert\d_{i}-\d_{i}^{-}\Vert_H\Big]_{+},\nonumber\end{aligned}$$ where $\d_i, \d_i^{+}, \d_i^{-}$ respectively represent the generated binary codes from anchor, positive and negative samples, $\alpha$ is the margin . Here the pedestrian who has the same/different identity with the anchor is the positive/negative sample. For each pedestrian image in a , we treat it as an anchor and build the corresponding triplet input by choosing the furthest positive and the closest negative samples in the same batch. This improved version of the triplet loss enhances the robustness in metric learning [@DBLP:journals/corr/HermansBL17], and improves the accuracy at the same time. Then we can get the following total triplet loss function: $$\begin{aligned}
\label{loss:totaltriplet}
\LM_{t}(\{\d_i,\g_i,\h_i\}_{i=1}^N)=&L_{t}(\{\d_i\}_{i=1}^N)+L_{t}(\{\g_i\}_{i=1}^N)+L_{t}(\{\h_i\}_{i=1}^N).\end{aligned}$$
In order to learn more discriminative binary codes, we explore cross entropy loss for classification on the outputs of each branch. Specifically, we utilize the formula: $L_{c}(\{\f_i^{(j)}\}_{i=1}^N) = -\frac{1}{N}\sum_{i=1}^N{\text{softmax}}(\f_i^{(j)})$.
Then we can get the total classification loss function: $$\begin{aligned}
\label{loss:totalsoftmax}
\LM_{c}(\{\f_i^{(1)},\f_i^{(2)},\f_i^{(3)}\}_{i=1}^N)=&\sum_{j=1}^3L_{c}(\{\f_i^{(j)}\}_{i=1}^N).\end{aligned}$$
#### Multi-Index Hashing Tables Construction
MIH supposes that each binary code $\b$ with $R$ bits is partitioned into $m$ disjoint isometric codes $\b^{(1)},\cdots,\b^{(m)}$. Given a query code $\q$, we aim to find all binary codes with the Hamming distance from $\q$ being $k$. We call them $k$-neighbors. Let $k'=\lfloor k/m\rfloor$ and $a=k-mk'$. According to the Proposition \[pro:mih\] proved in [@DBLP:journals/pami/0002PF14], we only need to search the first $a+1$ hash tables at the radius of $k'$ and the remaining $m-(a+1)$ hash tables at the radius of $k'-1$ to construct a candidate set when performing retrieval procedure for a given query. After that, we remove the points which are not $k$-neighbors from the candidate set by measuring full Hamming distance.
\[pro:mih\] if $\Vert\b-\q\Vert_H \leq k = mk' + a$, then $\exists~1 \leq z \leq a + 1~\st\;\Vert\b^{(z)}-\q^{(z)}\Vert_H \leq k'$ or $\exists~a + 1 < z \leq m~\st\;\Vert\b^{(z)}-\q^{(z)}\Vert_H \leq k' - 1.$
[r]{}[0.5]{} {width="49.00000%"}
\[fig:MIH\]
Once we get the learned binary codes, one way to construct MIH tables is to divide the total binary codes into $m$ hash tables, where the total binary codes is defined as $[\d_i;\g_i;\h_i]^T$. By doing so, each hash table might suffer from binary codes problem, and thus leads to large difference between different hash tables. To alleviate this situation, we design a novel MIH tables construction strategy, which is shown in Figure \[fig:MIH\]. Specifically, we divide the learned binary codes $\{\d_i,\g_i,\h_i\}$ into $m$ disjoint codes *separately*. Then we concatenate all the $j$-th codes to construct $\b_i^{(j)}$ for the $j$-th hash table, i.e., $\b_i^{(j)}=[\d_i^{(j)};\g_i^{(j)};\h_i^{(j)}]^T$. That is to say, this partition strategy can ensure that each hash table contains binary codes.
#### Search-Aware Multi-Index Loss
Based on the retrieval procedure of MIH tables, we propose a novel loss, called (SAMI) loss, to give feedback to the training procedure. Firstly, we define the binary codes for $\x_i$ and $\x_j$ as $\b_i$ and $\b_j$, respectively. And we use $\b_i^{(l)}$ and $\b_j^{(l)}$ to denote the corresponding codes in the $l$-th hash tables. Then we define the Hamming distance between $\b_i^{(l)}$ and $\b_j^{(l)}$ as $\Theta_{ij}^{(l)} = \Vert\b_i^{(l)}-\b_j^{(l)}\Vert_H$. As the first $a+1$ hash tables will be searched firstly, we hope $\Theta_{ij}^{(l)}$ is larger than $\Theta_{ij}^{(l+1)}$ as much as possible. Then we can get the following SAMI loss function:
$$\begin{aligned}
\label{loss:hl-abc}
\LM_{s}(\{\b_i^{(1)},\dots,\b_i^{(m)}\}_{i=1}^N) = \frac{1}{N^2(m-1)}\sum_{i,j=1}^{N}\sum_{l=1}^{m-1}\Big[\Theta_{ij}^{(l+1)} - \Theta_{ij}^{(l)} \Big]_{+}.\end{aligned}$$
By minimizing the above loss function, we can avoid the situation where the false data points are chosen into the candidate set too early when we utilize MIH tables to perform the retrieval procedure. Here the false data points are those data points whose Hamming distance to a given query point $\q$ is larger than the distance we need to retrieval, but the Hamming distance between its codes and the sub-binary codes of the query in the first-searched hash tables is small. Then we can reduce the number of points which are actually not $k$-neighbors but are added into the candidate set. As a result, the time for measuring full Hamming distance and removing the points which are not $k$-neighbors will be saved.
Then we can get the final objective function for DMIH by combining (\[loss:totaltriplet\]), (\[loss:totalsoftmax\]) and (\[loss:hl-abc\]), which is formulated as follows: $$\begin{aligned}
\min\;\LM=\LM_{t}(\{\d_i,\g_i&,\h_i\}_{i=1}^N)+\beta\LM_{c}(\{\f_i^{(1)},\f_i^{(2)},\f_i^{(3)}\}_{i=1}^N)+\gamma\LM_{s}(\{\b_i^{(1)},\dots,\b_i^{(m)}\}_{i=1}^N)\nonumber\\
&\hspace{2pt}\st\;\d_i,\g_i,\h_i\in\{-1,+1\}^r,\forall i\in\{1,\dots, N\},\label{obj:DMIH}\end{aligned}$$ where $\beta, \gamma$ are hyper-parameters.
Learning
--------
The objective function in (\[obj:DMIH\]) is NP-hard due to the binary constraint. One common approach to avoid this problem is to use relaxation strategy [@DBLP:conf/ijcai/LiWK16; @DBLP:conf/iccv/CaoLWY17]. In this paper, we also adopt this strategy to avoid this NP-hard problem. Specifically, we utilize $\tanh(\cdot)$ to approximate the $\sgn(\cdot)$ function.
Then we can reformulate the problem in (\[obj:DMIH\]) as follows: $$\begin{aligned}
\min\;\LM=\LM_{t}(\{\widetilde\d_i,\widetilde\g_i&,\widetilde\h_i\}_{i=1}^N)+\beta\LM_{c}(\{\f_i^{(1)},\f_i^{(2)},\f_i^{(3)}\}_{i=1}^N)+\gamma\LM_{s}(\{\widetilde\b_i^{(1)},\dots,\widetilde\b_i^{(m)}\}_{i=1}^N)\nonumber\\
&\hspace{2pt}\st\;\widetilde\d_i,\widetilde\g_i,\widetilde\h_i\in[-1,+1]^r,\forall i\in\{1,\dots, N\},\label{obj:relaxDMIH}\end{aligned}$$ where $\widetilde\d_i,\widetilde\g_i,\widetilde\h_i$ denote the continuous codes after relaxation.
Now we can use back propagation to learn the parameters in (\[obj:relaxDMIH\]). The learning algorithm for our DMIH is summarized in Algorithm \[alg:DMIH\].
$\X=\{\x_i\}_{i=1}^n$: images for training; $\y$: person identities for training images; $R$: code length. Parameters of deep neural networks. : Initialize parameters of DNN, maximum iteration number $T$, mini-batch size $N$. Randomly sample $N$ samples from $\X$ to construct a mini-batch $\{\x_i\}^N_{i=1}$. $\forall \x_i\in\{\x_i\}^N_{i=1}$, calculate $\{\f_{i}^{(1)},\f_{i}^{(2)},\f_{i}^{(3)}\}$ and $\{\widetilde\d_i,\widetilde\g_i,\widetilde\h_i\}$ by forward propagation. For mini-batch $\{\x_i\}^N_{i=1}$, calculate corresponding gradient according to loss function in (\[obj:relaxDMIH\]). Update the parameters of deep neural network based on the gradient.
Experiments {#sec:exp}
===========
In this section, we conduct extensive evaluation of the proposed method on three widely used ReID datasets: Market1501 [@DBLP:conf/iccv/ZhengSTWWT15], [@DBLP:conf/iccv/ZhengZY17] and CUHK03 [@DBLP:conf/cvpr/LiZXW14] in a mode. DMIH is implemented with PyTorch [@paszke2017automatic] on a NVIDIA M40 GPU server. We use the C++ implementation of MIH provided by the authors of [@DBLP:journals/pami/0002PF14][^3] and conduct the retrieval experiments on a server with Intel Core CPU (2.2GHz) and 96GB RAM.
Datasets
--------
#### Market1501
Market1501 dataset consists of 32,688 bounding boxes of 1,501 persons from 6 cameras. These bounding boxes are cropped by the (DPM) detector [@DBLP:journals/pami/FelzenszwalbGMR10]. 12,936 images of 751 persons are selected from the dataset as training set, and the remaining 750 persons are divided into test set with 3,368 query images and 19,732 gallery images.
#### DukeMTMC-ReID
DukeMTMC-ReID dataset is a subset of the dataset [@DBLP:conf/eccv/RistaniSZCT16] for . It consists of 36,411 images of 1,812 persons from 8 cameras. The whole dataset is divided into training set with 16,522 images of 702 persons and test set with 2,228 query images and 17,661 gallery images of the remaining 702 persons.
#### CUHK03
CUHK03 dataset contains 14,097 images of 1,467 persons from 6 surveillance cameras. This dataset provides both manually labeled pedestrian bounding boxes and bounding boxes detected by the DPM detector. For this dataset, we choose the labeled images for evaluation. To be more consistent with real application, we adopt the widely recognized [@DBLP:conf/mm/WangYCLZ18; @DBLP:conf/cvpr/ChangHX18; @DBLP:conf/eccv/SunZYTW18] protocol proposed in [@DBLP:conf/cvpr/ZhongZCL17].
Experimental Setup
------------------
#### Baselines and Evaluation Protocol
Both hashing methods and real-valued methods are adopted as baselines for comparison. The hashing methods for comparison include: [ 1) *hashing methods*: ]{} COSDISH [@DBLP:conf/aaai/KangLZ16], SDH [@DBLP:conf/cvpr/ShenSLS15], KSH [@DBLP:conf/cvpr/LiuWJJC12], ITQ [@DBLP:conf/cvpr/GongL11], LSH [@DBLP:conf/compgeom/DatarIIM04]; [2) *deep hashing methods*: ]{}PDH [@DBLP:journals/tip/ZhuKZFT17], [@DBLP:conf/iccv/CaoLWY17], DPSH [@DBLP:conf/ijcai/LiWK16]. Among these baselines, PDH is designed specifically for ReID. DRSCH [@DBLP:journals/tip/ZhangLZZZ15] is not adopted for comparison because it has been found to be outperformed by PDH. The ReID methods for comparison include: [ 1) *metric learning methods*: ]{}KISSME [@DBLP:conf/cvpr/KostingerHWRB12]; [ 2) *deep learning methods*: ]{} deep convolutional (PDC) [@DBLP:conf/iccv/SuLZX0T17], Spindle [@DBLP:conf/cvpr/ZhaoTSSYYWT17], MGN [@DBLP:conf/mm/WangYCLZ18].
Following the standard evaluation protocol on ReID tasks [@DBLP:conf/mm/WangYCLZ18], we report the mean average precision (mAP) and Cumulated Matching Characteristic (CMC) to verify the effectiveness of our proposed method. Furthermore, to verify the high accuracy and fast query speed DMIH can achieve, we adopt and [@DBLP:journals/corr/Cai16b; @DBLP:journals/corr/abs-1711-06016] to evaluate DMIH and baselines. Specifically, after constructing the MIH tables, we conduct neighbor search by performing hash lookup with different $k$ and then do on the selected nearest neighbors according to the corresponding deep features () before the hash layer. Based on the re-ranking results, we choose the nearest neighbors which have the minimum Euclidean distance to the query image and then calculate the precision and recall. At last, we summarize the time of hash lookup and to draw the and curves [@DBLP:journals/corr/Cai16b; @DBLP:journals/corr/abs-1711-06016].
#### Implementation Details
For DMIH, we set $\beta=2.0, \gamma=0.5$ and $T=160$ for all the experiments based on cross-validation strategy. We use Adam algorithm [@DBLP:journals/corr/KingmaB14] for learning and choose the learning rate from $[10^{-5},10^{-3}]$. The initial learning rate is set to $4\times 10^{-4}$ and the weight decay parameter is set to $5\times 10^{-4}$. The network based on is on ImageNet dataset [@DBLP:conf/cvpr/DengDSLL009]. The input for the image modality is raw pixels with the size of $384 \times 128$. We fix the size of each to be 64, which is made up of 16 pedestrians and 4 images for each pedestrian.
For hashing methods, we use two image features. The first one is the Local Maximal Occurrence (LOMO) feature [@DBLP:conf/cvpr/LiaoHZL15]. After getting the LOMO feature, we use PCA to reduce the dimensionality to . The second one is the deep features extracted by on ImageNet. Among hashing baselines, KSH and SDH are methods. For these methods, 1,000 data points are randomly selected from training set as anchors to construct kernels by following the suggestion of the original authors. For deep hashing methods, we adopt the same network for a fair comparison. For all hashing based ReID methods, other hyper-parameters are set by following the suggestion of the corresponding authors. The source code is available for all baselines except PDH and MGN. We carefully re-implement PDH and MGN using PyTorch.
Accuracy
--------
#### Comparison with Hashing Methods
We report the mAP on three datasets in Table \[tab:mAP\], where “COSDISH”/ denotes COSDISH with LOMO/deep features, respectively. Other notations are defined similarly. The CMC results are moved to supplementary materials due to space limitation. From Table \[tab:mAP\], we can see that DMIH outperforms all baselines including deep hashing based ReID methods, deep hashing methods and hashing methods in all cases.
Furthermore, we also present the curves on three datasets in Figure \[fig:precision20-32\]. Due to the mAP results in Table \[tab:mAP\], hashing methods utilize LOMO features. The curves and more curves with other bits are moved to the supplementary materials due to space limitation. From Figure \[fig:precision20-32\], we can find that DMIH can achieve the highest precision among all the hash methods while costs less time in all cases. Hence, can significantly outperform existing hashing methods and deep hashing methods in terms of both *efficiency* and *accuracy*. In addition, we find that the methods with higher mAP or CMC do not necessarily have better and . That is to say, only using mAP and CMC to evaluate hashing methods may not be comprehensive. So we adopt mAP, CMC, and to comprehensively verify the promising efficiency and accuracy of our method.
\[fig:precision20-32\]
#### Comparison with Real-Valued ReID Methods
We also compare our DMIH with some representative ReID methods. As the dimension of the features is usually high, we increase the binary code length of DMIH for fair comparison. We report CMC@20 and the corresponding retrieval time in Table \[tab:compare\_with\_ReID\_methods\], where the results of BoW+KISSME [@DBLP:conf/cvpr/KostingerHWRB12], Spindle [@DBLP:conf/cvpr/ZhaoTSSYYWT17] and PDC [@DBLP:conf/iccv/SuLZX0T17] are directly copied from the original papers and “–" denotes that the result of the corresponding setting is not reported in the original papers. denotes the DMIH method of 32 bits with after hash lookup. Other variants of DMIH are named similarly. The retrieval time includes the time for both hash lookup and . Furthermore, we also report the speedup of DMIH relative to the best baseline MGN.
From Table \[tab:compare\_with\_ReID\_methods\], we can see that DMIH can still achieve the best performance in all cases when compared with ReID methods. In particular, can outperform the best real-valued baseline MGN in terms of both accuracy and efficiency with suitable code length. We can also find that DMIH can achieve higher accuracy by increasing binary code length. However, longer binary code typically leads to worse retrieval efficiency. In real applications, one can choose proper binary code length to get a good between efficiency and accuracy.
Ablation Study
--------------
We conduct experiments to study whether all the loss terms in DMIH are necessary by removing these loss terms separately. The result on Market1501 dataset with 32 bits and 96 bits is presented in Figure \[fig:hyper-parameter\]. Here, “DMIH/SAMI” denotes the DMIH variant without the SAMI loss $\LM_s(\cdot)$, i.e., $\gamma=0$, and other notations are defined similarly. We can find that softmax loss and triplet loss can significantly improve accuracy. Furthermore, by comparing DMIH with DMIH/SAMI, we can find that SAMI loss can further accelerate the query speed without losing accuracy. As the time shown in Figure \[fig:hyper-parameter\] contains the hash table lookup time and re-ranking time, we compare the hash lookup time separately of DMIH with DMIH/SAMI in Figure \[fig:speedup\]. From Figure \[fig:speedup\], we can see that DMIH can achieve $2\sim3$ times acceleration of hash lookup with the help of SAMI loss.
+:---------------:+:---------------:+:---------------:+:---------------:+
| ![Time cost | ![Time cost | | |
| obtained by | obtained by | | |
| $k$NN.[]{data-l | $k$NN.[]{data-l | | |
| abel="fig:speed | abel="fig:speed | | |
| up"}](figure/hy | up"}](figure/hy | | |
| per_parameter/m | per_parameter/m | | |
| arket_32.pdf "f | arket_96.pdf "f | | |
| ig:"){width="10 | ig:"){width="10 | | |
| 0.00000%"}\ | 0.00000%"}\ | | |
| [(a) | [(b) | | |
| Market1501@32 | Market1501@96 | | |
| bits]{} | bits]{} | | |
+-----------------+-----------------+-----------------+-----------------+
+:---------------:+:---------------:+:---------------:+:---------------:+
| ![Time cost | ![Time cost | | |
| obtained by | obtained by | | |
| $k$NN.[]{data-l | $k$NN.[]{data-l | | |
| abel="fig:speed | abel="fig:speed | | |
| up"}](figure/fu | up"}](figure/fu | | |
| rther/market_32 | rther/market_96 | | |
| .pdf "fig:"){wi | .pdf "fig:"){wi | | |
| dth="100.00000% | dth="100.00000% | | |
| "}\ | "}\ | | |
| [(a) | [(b) | | |
| Market1501@32 | Market1501@96 | | |
| bits]{} | bits]{} | | |
+-----------------+-----------------+-----------------+-----------------+
Effect of Block-Wise MIH Table Construction
-------------------------------------------
[r]{}[0.5]{}
+:---------------:+:---------------:+:---------------:+:---------------:+
|  | | |
| df){width="100. | {width="100.000 | | |
| 00000%"}\ | 00%"}\ | | |
| [(a) | [(b) | | |
| Market1501@32, | Market1501@32, | | |
| 96 bits]{} | 96 bits]{} | | |
+-----------------+-----------------+-----------------+-----------------+
\[fig:blockwise\]
To verify the effectiveness of the proposed MIH tables construction strategy, we compare DMIH with a variant without the MIH table construction strategy. More specifically, for the variant without the MIH table construction strategy, we directly divide the learned binary code $[\d_i;\g_i;\h_i]^T$ into $m$ codes to construct MIH tables. For fair comparison, we utilize the same learned binary code for both DMIH and the variant. The and curves on Market1501 dataset with binary code length being 32 bits and 96 bits are shown in Figure \[fig:blockwise\], where “DMIH(32)/BW” denotes the variant without the MIH construction strategy and other notations are defined similarly. From Figure \[fig:blockwise\], we can see that using block-wise strategy can get a speedup in retrieval efficiency without losing accuracy.
Conclusion {#sec:conclusion}
==========
In this paper, we propose a novel deep hashing method, called DMIH, for person ReID. DMIH is an deep learning framework, which integrates hashing and based networks into the same framework. Furthermore, we propose a novel hashing tables construction strategy and a loss to further improve the search efficiency. Experiments on real datasets show that DMIH can outperform other baselines to achieve the retrieval performance in terms of both [efficiency]{} and [accuracy]{}.
Additional Experiments
======================
Comparison with Hashing Methods
-------------------------------
We report the precision-time curves in Figure \[fig:precision20-96\] and Figure \[fig:precision-64-128\], and non-deep hashing methods utilize LOMO features. Again, we can see that DMIH can outperform all the hashing baselines to achieve the highest precision but costs less time in all cases. Furthermore, we present the recall-time curves in Figure \[fig:recall-32-96\] and Figure \[fig:recall-64-128\]. Again, we can find that DMIH can achieve the highest recall but costs less time in all cases from Figure \[fig:recall-32-96\] and Figure \[fig:recall-64-128\]. Hence, we can conclude that DMIH can significantly outperform all baselines in terms of both efficiency and accuracy.
We also report the CMC@20 on three datasets in Table \[tab:cmc20\], where “COSDISH”/“COSDISH+CNN” denotes COSDISH with LOMO/deep features, respectively. We can also find that DMIH can outperform all baselines to achieve the state-of-the-art CMC@20 results in all cases.
[^1]: Please note that here the $k$-nearest neighbors are defined based on Hamming distance [@DBLP:journals/pami/0002PF14].
[^2]: In this paper, we assume that $\{\d_i, \g_i, \h_i\}$ have the same binary code length, i.e., $R=3r$. We can also set the code length for different sub-binary codes to be different.
[^3]: <https://github.com/norouzi/mih>
|
---
abstract: 'We consider the noncommutative Standard Model that contains Lorentz symmetry violation as a subset of the Standard Model extension. We introduce a constant electromagnetic field as a background to derive mutual relations between the free parameters of both theories. As the Lorentz violation parameters of the Standard Model extension are extensively explored in different experiments and many stringent bounds on these parameters are available, we can find new bounds on the scale of noncommutativity of the order of a few to tens of teraelectron volts.'
---
[**[Lorentz violation parameters and noncommutative scale ]{}**]{} 4em[ [**S. Aghababaei$^{\dag}$**]{} [^1] , [^2]]{} 1em $^\dag$ Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran $^{\dag\dag}$ Department of Physics, Shiraz University, Shiraz 71454, Iran
Introduction
============
The Standard Model of particle physics has achieved a remarkable phenomenological success through the past decades, but there are still unresolved various issues. Such issues are often discussed in the context of new physics or beyond the Standard Model theories. Meanwhile, the Standard Model of particle physics as well as many other theories for describing beyond the Standard Model respect the Lorentz symmetry that is supported by many experimental inspections. Although in the lower energy limit the Lorentz symmetry is an almost exact symmetry of nature, it is natural to study theories involving Lorentz symmetry breaking. In fact, in the Planck scale, the Lorentz symmetry violation arises through quantum gravity. However, irrespective of the underlying fundamental theory, there is an appropriate prescription for considering both Lorentz and Charge conjugation-Parity-Time reversal (CPT) violation in the minimal Standard Model [@SME]. In the so-called Standard Model extension (SME) the Lorentz violation is assumed to be induced by a spontaneous Lorentz symmetry breaking. Therefore, the Lorentz violated terms in the SME contain Lorentz violated (LV) parameters that are Lorentz quantities and act as constant backgrounds. Furthermore, the SME preserves the observer Lorentz symmetry, whereas the particle Lorentz symmetry is violated. Meanwhile, the phenomenological aspects of the SME have been extensively considered by many authors [@SME-ph] that have been led to very tight bounds on the LV parameters [@data]. Furthermore, noncommutative (NC) space-time intrinsically breaks the Lorentz symmetry that in many works has resulted from considering the NC effects on the deviation in Lorentz symmetry invariance [@NC-LV]. Moreover, the Lorentz symmetry violation in the noncommutative Standard Model (NCSM) may be systematically compared with the SME to find various relations between the LV parameters and the parameter of noncommutativity $\theta_{\mu\nu}$. Although the NC field theories and their phenomenological aspects have been studied for many years [@NC], the obtained tight bounds on the LV parameters can provide new bounds on the value and even the components of $\theta_{\mu\nu}$. Actually, such relations in the QED and Higgs parts of NCSM and the SME have resulted in more restricted bounds on the NC parameter [@NCLV; @NCHiggs]. Here, we will study the Lorentz violation in the electroweak part of NCSM to find the corresponding relations between the LV and NC parameters. In this article, we briefly introduce the Lagrangian of the SME and NCSM, respectively, in Secs. 2 and 3. The mutual relations among the parameters of both theories are explored in Sec. 4. We study the components of LV parameters to find new bounds on the value and also the components of NC parameter in Sec. 5. Moreover, we examine the time and location dependence of LV parameters to give the location dependence of the NC parameter in different experiments. In Sec. 6, we give a summary and some concluding remarks.
Standard Model extension
========================
The Standard Model extension provides a framework for considering the violation of Lorentz symmetry via a spontaneous symmetry breaking (SSB) at a fundamental level. Regardless of the fundamental theory, it can be constructed by taking all possible Lorentz violating terms into account that preserve the gauge symmetry of the Standard Model and to be power-counting renormalizable. These additional terms are combinations of the ordinary SM fields and parameters with Lorentz indices acting as constant backgrounds that lead to particle Lorentz symmetry violation [@SME]. To this end, the Lagrangian density for the electroweak part of the standard model in natural units $\hbar=c=\epsilon_{0}=1$ can be introduced as follows: $$\begin{aligned}
\mathcal{L}^{SM}_{Fermion}=\frac{1}{2}i\overline{L}_A\gamma^{\mu}\overleftrightarrow {D_{\mu}}L_A+\frac{1}{2}i\overline{R}_A\gamma^{\mu}\overleftrightarrow {D_{\mu}}R_A,
\label{LeptonSM}\end{aligned}$$ $$\begin{aligned}
\mathcal{L}^{SM}_{Gauge}=-\frac{1}{2}Tr(W_{\mu\nu}W^{\mu\nu})-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}.
\label{GaugeSM}\end{aligned}$$ $$\begin{aligned}
\mathcal{L}^{SM}_{Higgs}=(D_{\mu}\phi)^{\dag} D^{\mu}\phi+\mu^2 \phi^{\dag} \phi -\frac{\lambda}{3!} (\phi^{\dag} \phi)^2,
\label{HiggsSM}\end{aligned}$$ $$\begin{aligned}
\mathcal{L}^{SM}_{Yukawa}=-[(G_L)_{AB}\overline L_{A}\phi R_{B}]+H.c.,
\label{YukawaSM}\end{aligned}$$ where $D_\mu$ denotes the appropriate covariant derivative in each term, $ A\overleftrightarrow{\partial_{\mu}}B\equiv A(\overrightarrow{\partial_{\mu}}B)-(\overrightarrow{\partial_{\mu}}A)B$, and $W_{\mu\nu}$ and $B_{\mu\nu}$ are the field strengths for the gauge groups $SU(2)$ and $U(1)$ with the gauge fields $W_\mu$ and $B_\mu$, respectively. In the Higgs and Yukawa parts, $\phi$ shows the Higgs doublet representation with coupling $ \lambda $ and $G_L$’s are the Yukawa couplings. Meanwhile, the left- and right-handed fermions are defined as $${L_A} =
\left(\begin{array}{c} \nu_A \\ \l_A
\end{array} \right )_L, R_A=(l_A)_R,$$ for leptons and $${L^\prime_A} =
\left(\begin{array}{c} U_A \\ D_A
\end{array} \right )_L, R^\prime_A=(Q_A)_R,$$ for quarks where $A=1,2,3$ labels the flavors for each generation, and $Q$ is up- or down-type quarks. Now, we add all possible Lorentz violating terms to the SM-Lagrangian that preserve the gauge symmetries and are power-counting renormalizable. These additional terms can be categorized into CPT-even that preserves the CPT symmetry and CPT-odd with the CPT symmetry violation. Therefore, the SME Lagrangian can be introduced as follows: The fermion sector $$\begin{aligned}
\mathcal{L}^{CPT-even}_{Fermion}=i\frac{1}{2}(c_L)_{\mu\nu AB}\overline{L}_{A_{}}\gamma^{\mu}\overleftrightarrow {D^{\nu}}L_{B}
+i\frac{1}{2}(c_R)_{\mu\nu AB}\overline{R}_{A}\gamma^{\mu}\overleftrightarrow {D^{\nu}}R_{B},
\label{CPTevenLepton}\end{aligned}$$ and $$\begin{aligned}
\mathcal{L}^{CPT-odd}_{Fermion}=-(a_L)_{\mu AB}\overline{L}_A \gamma^{\mu} L_B-
(a_R)_{\mu AB}\overline{R}_A \gamma^{\mu} R_B,
\label{CPToddLepton}\end{aligned}$$ where the free parameters $ c_L $ and $ a_L $ show the LV parameters in the fermion sector for the leptons. Meanwhile, the quark terms can easily be found by replacing $L$ and $R$, respectively, with $L^\prime$ and $R^\prime$ in an appropriate manner [@SME].\
The gauge sector $$\begin{aligned}
\mathcal{L}^{CPT-even}_{Gauge}=-\frac{1}{2}(k_W)_{\mu\nu\rho\sigma}Tr(W^{\mu\nu}W^{\rho\sigma})
-\frac{1}{4}(k_F)_{\mu\nu\rho\sigma}B^{\mu\nu}B^{\rho\sigma},
\label{CPTevenGauge}\end{aligned}$$ and $$\begin{aligned}
\mathcal{L}^{CPT-odd}_{Gauge}&=&(k_2)_\kappa \epsilon^{\kappa\lambda\mu\nu} Tr(W_\lambda W_{\mu\nu}+\frac{2}{3}igW_\lambda W_\mu W_\nu)\nonumber\\
&+&(k_1)_\kappa \epsilon^{\kappa\lambda\mu\nu} B_\lambda B_{\mu\nu}+(k_0)_\kappa W^\kappa,
\label{CPToddGauge}\end{aligned}$$ where the LV parameters in this sector are $k_W$, $k_F$, and $ k_{0,1,2} $ and $Tr$ mean the trace with respect to the SU(2) group. Nevertheless, the real parameters $ k_{0,1,2} $ are associated with a negative contribution to the energy, which leads to some instability in the SME, and one may assume them to be zero.\
The Higgs sector $$\begin{aligned}
\mathcal{L}^{CPT-even}_{Higgs}&=&\frac{1}{2}(k_{\phi\phi})^{\mu\nu}( D_{\mu} \phi^{\dag}) D_{\nu}\phi+H.c.\nonumber \\
&-&\frac{1}{2}(k_{\phi B})^{\mu\nu} \phi^{\dag} \phi B_{\mu\nu}-\frac{1}{2}(k_{\phi W})^{\mu\nu} \phi^{\dag} W_{\mu\nu} \phi,
\label{CPTevenHiggs}\end{aligned}$$ for CPT-even and for CPT-odd one has $$\begin{aligned}
\mathcal{L}^{CPT-odd}_{Higgs}=i (k_{\phi})^{\mu}\phi^{\dag} D_{\mu}\phi +H.c.,
\label{CPToddHiggs}\end{aligned}$$ where H.c shows the Hermitian conjugate and $k_{\phi\phi}$, $k_{\phi B}$, $k_{\phi W}$ and $k_{\phi}$ denote the LV parameters in the Higgs sector. Finally, the Yukawa sector can be cast into [@SME; @Yukawaterms] $$\begin{aligned}
\mathcal L^{CPT-even}_{Yukawa}&=&\frac{1}{2}(K_{L})_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi
-(h)_{AB}\overline{L}_{A}\phi R_{B}-i\gamma_5(h')_{AB}\overline{L}_{A}\phi R_{B}\nonumber\\
&-&\frac{1}{2}(H_L)_{\mu\nu AB}\overline{L}_{A}\phi \sigma^{\mu\nu}R_{B}+ H.c.,
\label{CPTevenYukawa}\end{aligned}$$ for the CPT-even part, and the CPT-odd part can be written as $$\begin{aligned}
\mathcal L^{CPT-odd}_{Yukawa}&=&-(I_L)_{\mu AB}\overline{L}_{A}\gamma_{\mu}\phi R_{B}-(J_L)_{\mu AB}\overline{L}_{A}\gamma_5\gamma_\mu \phi R_{B}+H.c.,
\label{CPToddYukawa}\end{aligned}$$ where $K_L$, $h$, $h'$, $ H_L $, $ I_L $, and $ J_L $ are the LV parameters in the Yukawa sector. The LV parameters that are introduced in (\[CPTevenLepton\])-(\[CPToddYukawa\]) are sensitive to different experiments. The current bounds on the values of different LV parameters are available in Ref. [@data].
Noncommutative Standard Model
=============================
@sh\[\#1\]\#2[ [fmsl@sh]{} [fmsl@sh]{} [fmsl@sh]{}]{} fmsl@sh\#1\#2\#3[@th]{} In noncommutative space-time, the coordinates are operators that in the canonical version obey a noncommutative relation as follows: $$\begin{aligned}
[x^{\mu},^{*}x^{\nu}]\equiv x^\mu \star x^\nu - x^\nu \star x^\mu = i \theta^{\mu\nu}=\frac{i\epsilon_{\mu\nu}}{\Lambda_{NC}^2},
\label{Noncommutative}\end{aligned}$$ where $\Lambda_{NC}$ denotes the NC scale of energy, and $\theta^{\mu \nu}$ is a real constant antisymmetric matrix that can be realized as two distinct constant vectors in a four-dimensional space-time. These constant vectors obviously violate the particle Lorentz symmetry that in turn relates the NCSM to a subset of the SME. Meanwhile, based on the Weyl-Moyal $\star$ product that can be defined as $$\begin{aligned}
(f \star g)(x)= \left.\exp\!\left(\frac{i}{2}
\theta^{\mu \nu}\frac{\partial}{\partial x^\mu}
\frac{\partial}{\partial y^\nu}\right) f(x) g(y)\right|_{y \to x},\end{aligned}$$ which in the leading order leads to $$\begin{aligned}
f \star g = f \cdot g + \frac{i}{2}\theta^{\mu\nu}(x) \partial_{\mu} f \cdot \partial_{\nu} g + \mathcal{O}(\theta^2) ,
\label{starproduc}\end{aligned}$$ one can construct the NCSM by two different approaches. As in the NC space only for a unitary group dose one have a closed Lie algebra for generators of the group; therefore, in the first approach the SM gauge group has been achieved through a two steps spontaneous symmetry breaking from the $U(3)\times U(2)\times U(1)$ symmetry group [@u3u2u1]. In the second approach, by extending the algebra [@NCSM], one can consider the $SU(n)$ gauge group via Seiberg-Witten (SW) maps [@S.W]. However, to find the relation between NCSM and SME, we consider the second approach in which the symmetry group, the number of particles, the couplings, and the gauge fields are the same as the SM in the commutative space. To this end, one can define the whole gauge potential $V_{\mu}$ in the noncommutative Standard Model as $${V_\mu}=g' {B}_\mu(x)Y+g \sum_{a=1}^{3} W_{\mu a}(x) T^a_L
+g_S \sum_{b=1}^{8} G_{\mu b}(x) T^b_S,$$ where Y, $T^{a}_{L}$, and $T^{b}_{S}$ are the generators of $U(1)_Y$, $SU(2)_L$, and $SU(3)_C$ with the corresponding nonphysical gauge fields $ B_\mu $, $ W_\mu $, and $ G_\mu $, respectively. Therefore, the full NCSM action can be written as follows: $$\begin{aligned}
S_{NCSM}&&=\int d^4x \sum_{i=1}^3 \overline{\widehat \Psi}^{(i)}_L \star i
\widehat{\fmslash D} \widehat \Psi^{(i)}_L
+\int d^4x \sum_{i=1}^3 \overline{\widehat \Psi}^{(i)}_R \star i
\widehat{\fmslash D} \widehat \Psi^{(i)}_R \nonumber\\
&& \nonumber -\int d^4x \frac{1}{2 g'}
\mbox{{\bf tr}}_{\bf 1} \widehat
F_{\mu \nu} \star \widehat F^{\mu \nu}
-\int d^4x \frac{1}{2 g} \mbox{{\bf tr}}_{\bf 2} \widehat
F_{\mu \nu} \star \widehat F^{\mu \nu}\\
&&\nonumber
-\int d^4x \frac{1}{2 g_S} \mbox{{\bf tr}}_{\bf 3} \widehat
F_{\mu \nu} \star \widehat F^{\mu \nu}
+ \int d^4x \bigg( \rho_0(\widehat D_\mu \widehat \Phi)^\dagger
\star \rho_0(\widehat D^\mu \widehat \Phi)
\\ && \nonumber
- \mu^2 \rho_0(\widehat {\Phi})^\dagger \star \rho_0(\widehat \Phi) - \lambda
\rho_0(\widehat \Phi)^\dagger \star \rho_0(\widehat \Phi)
\star
\rho_0(\widehat \Phi)^\dagger \star \rho_0(\widehat \Phi) \bigg)
\\ && \nonumber
+ \int d^4x \bigg (
-\sum_{i,j=1}^3 W^{ij} \bigg
( ( \bar{ \widehat L}^{(i)}_L \star \rho_L(\widehat \Phi))
\star \widehat e^{(j)}_R
+ \bar {\widehat e}^{(i)}_R \star (\rho_L(\widehat \Phi)^\dagger \star \widehat
L^{(j)}_L) \bigg )
\\ && \nonumber
-\sum_{i,j=1}^3 G_u^{ij} \bigg
( ( \bar{\widehat Q}^{(i)}_L \star \rho_{\bar Q}(\widehat{\bar\Phi}))\star
\widehat u^{(j)}_R
+ \bar {\widehat u}^{(i)}_R \star
(\rho_{\bar Q}(\widehat{\bar\Phi})^\dagger
\star \widehat Q^{(j)}_L) \bigg )
\\ &&
-\sum_{i,j=1}^3 G_d^{ij} \bigg
( ( \bar{ \widehat Q}^{(i)}_L \star \rho_Q(\widehat \Phi))\star
\widehat d^{(j)}_R
+ \bar{ \widehat d}^{(i)}_R \star (\rho_Q(\widehat \Phi)^\dagger
\star \widehat Q^{(j)}_L) \bigg ) \bigg),
\label{NCSM}\end{aligned}$$ where the hat denotes the field in the NC space that can be obtained in terms of the corresponding field in the ordinary space via the SW map. For the fermion fields, Higgs field, gauge potentials, and the field strengths on the NC space-time the corresponding relations can be found in Refs.[@NCSM; @LCNCSM]. The matrices $W^{ij}$, $G^{ij}_u$, and $G^{ij}_d$ show the Yukawa couplings and $tr_i$’s, and $i=1,2,3$ show traces with respect to U(1)$_Y$, SU(2)$_L$, and SU(3)$_C$, respectively, as is defined in [@NCSM]. Now we would like to explore the Lorentz violation in the electroweak part (EW) of the NCSM (NCEW). For this purpose, the electroweak part of NCSM should be expanded up to the first order of $ \theta $. To this end, one can introduce the corresponding action as follows: $$\begin{aligned}
S_{NCEW}&=&S^{NC}_{Fermion} +S^{NC}_{Gauge}+S^{NC}_{Higgs}+S^{NC}_{Yukawa}.\end{aligned}$$ As (\[NCSM\]) shows, one can easily see that $$\begin{aligned}
S^{NC}_{Fermion}= \int d^4x \left (\sum_{A} \overline{\widehat
\Psi}^{(A)}_{L} \star
i \fmslash{\widehat D} \widehat \Psi^{(A)}_{L}
+ \sum_{A} \overline{\widehat \Psi}^{(A)}_{R} \star i \fmslash{\widehat D}
\widehat \Psi^{(A)}_{R}\right),
\label{Fermionic}\end{aligned}$$ where $\widehat \Psi^{(A)}_{L}$ and $\widehat \Psi^{(A)}_{R}$ are the left-handed $SU(2)$ doublets and the right-handed $SU(2)$ singlets for the flavor $A$, respectively. To find the fermion part of action up to the first order of $ \theta $ the star product should be expanded in terms of $ \theta $ and the NC fields should be replaced by the ordinary fields up to the first order of $ \theta $ via the SW map. For instance, for the first generation of the lepton fields $\Psi_A=L_L,e_R $ and up to the leading order $\widehat \Psi_A=\Psi_A+\Psi_A^{(1)} $ where for the $i$th generation $\Psi_A^{(1)} $ can be obtained as $$\begin{aligned}
L_L^{(i)1}[{B}, W]
&=&-\frac{1}{2} g'\theta^{\mu \nu} {B}_\mu \partial_\nu L_L^{(i)}
-\frac{1}{2} g \theta^{\mu \nu} W_\mu \partial_\nu L_L^{(i)}\nonumber
\\
&
+&\frac{i}{4} \theta^{\mu \nu}
\left( g'{ B}_\mu +g W_\mu\right)
\left( g'{ B}_\nu +g W_\nu\right)
L_L^{(i)},
\label{L_Li}\end{aligned}$$ for the left-handed leptons and $$\begin{aligned}
e^{(i)1}_R[{ B}]&=&-\frac{1}{2}g' \theta^{\mu \nu}
{ B}_\mu \partial_\nu e^{(i)}_R,
\label{e_Ri}\end{aligned}$$ for the right-handed ones. Therefore, the leptonic part of the action can be rewritten as $$\begin{aligned}
S^{NC}_{Lepton}&=&
\int d^4x \bigg ( \sum_{i} \left(\bar
L^{(i)}_{L}+ \bar L^{(i)1}_{L} \right)
\star
i
\left(\fmslash D^{SM} + \fmslash \Gamma \right)
\star
\left(L^{(i)}_{L}+ L^{(i)1}_{L} \right) \nonumber\\ &+&
\sum_{i} \left(\bar e^{(i)}_{R}+ \bar e^{(i)1}_{R} \right)
\star i
\left(\fmslash D^{SM} + \fmslash \Gamma \right)
\star
\left( e^{(i)}_{R}+ e^{(i)1}_{R} \right)
\bigg ) + {\cal O}(\theta^2),
\label{NCf}\end{aligned}$$ in which for the vector potential in the NC space we have defined $ \widehat V_\mu=V_\mu+ i\Gamma_\mu $ where up to the first order of $\theta$ through the SW map one has $$\begin{aligned}
\Gamma_\mu & = & i\frac{1}{4}\theta^{\alpha \beta}
\{ g' { B}_\alpha + g W_\alpha,
g' \partial_\beta { B}_\mu + g \partial_\beta W_\mu + g' B_{\beta \mu} +g W_{\beta \mu} \},
\label{gamma_mu}\end{aligned}$$ with the field strengths $ B_{\mu\nu} $ and $ W_{\mu\nu} $ corresponding to the gauge groups $ U(1) $ and $ SU(2) $, respectively. By replacing (\[L\_Li\]), (\[e\_Ri\]), and (\[gamma\_mu\]) in (\[NCf\]) and expanding the star product up to the first order of $\theta$ and after a little algebra one can find the lowest NC corrections on the leptonic action as follows: $$\begin{aligned}
S^{NC}_{Lepton}&=& \int d^4x \sum_{i}
\bar L^{(i)}_{L} i \fmslash{D}^{SM} L^{(i)}_{L}
\nonumber\\ &-&\frac{1}{4} \theta^{\mu \nu}\int d^4x \sum_{i}
\bar L^{(i)}_{L} (g'B_{\mu \nu}+ gW_{\mu \nu})
i \fmslash{D}^{SM} {L^{(i)}_{L}}
\nonumber\\
&-&\frac{1}{2}\theta^{\mu \nu}\int d^4x \sum_{i}
\bar L^{(i)}_{L} \gamma^\alpha
(g'B_{\alpha \mu}+gW_{\alpha \mu}) i D^{SM}_\nu
L^{(i)}_{L}
\nonumber\\
&+& \int d^4x \sum_{i}
\bar e^{(i)}_{R} i \fmslash{D}^{SM} e^{(i)}_{R} \nonumber\\
&-&\frac{1}{4} \theta^{\mu \nu}\int d^4x \sum_{i}
\bar e^{(i)}_{R} g'B_{\mu \nu}
i \fmslash{D}^{SM} e^{(i)}_{R} \nonumber\\
&-&\frac{1}{2}\theta^{\mu \nu}\int d^4x \sum_{i}
\bar e^{(i)}_{R} \gamma^\alpha
g'B_{\alpha \mu} i D^{SM}_\nu e^{(i)}_{R} + {\cal O}(\theta^2),
\label{NCf2}\end{aligned}$$ where $D^{SM}$ shows the covariant derivative in the ordinary Standard Model. The quark part of the fermionic action has a similar structure to the leptonic part which can easily be obtained by inserting $\Psi_A=L'_A, R'_A $ for the left- and right-handed quarks with the appropriate SW map in the action (\[Fermionic\]) [@NCSM].\
In (\[NCSM\]) the gauge part of the EW action is $$\begin{aligned}
S^{NC}_{Gauge}&=&-\int d^4x \frac{1}{2 g'}
\mbox{{\bf tr}}_{\bf 1} \widehat
F_{\mu \nu} \star \widehat F^{\mu \nu}
-\int d^4x \frac{1}{2 g} \mbox{{\bf tr}}_{\bf 2} \widehat
F_{\mu \nu} \star \widehat F^{\mu \nu},
\label{NCg1}\end{aligned}$$ with the following expansion for the field strength: $$\begin{aligned}
\widehat F_{\mu \nu}&=&F_{\mu \nu}+ F^1_{\mu \nu} +{\cal O}(\theta^2),\end{aligned}$$ where $$\begin{aligned}
F_{\mu \nu}&=&g'{B}_{\mu \nu}+g W_{\mu \nu},\end{aligned}$$ and $$\begin{aligned}
F^1_{\mu \nu}&=& \frac{1}{2} \theta^{\alpha \beta} \{ F_{\mu \alpha},
F_{\nu \beta} \} -\frac{1}{4} \theta^{\alpha \beta}
\{ V_\alpha,(\partial_\beta+D_\beta) F_{\mu \nu} \}.\end{aligned}$$ By inserting the field strengths up to the lowest order in (\[NCg1\]) and expanding the star products one finds the EW gauge action up to the first order of $\theta$ in the NC space as $$\begin{aligned}
S^{NC}_{Gauge}&=&-\frac{1}{4} \, \int d^4x \, B_{\mu \nu} B^{ \mu \nu}
-\frac{1}{2} \, {\rm Tr} \int d^4x \, W_{\mu \nu} W^{ \mu \nu}\nonumber\\
&-&g \, \theta^{\mu
\nu} \, {\rm Tr} \int d^4x \, W_{\mu \rho} W_{\nu \sigma} W^{ \rho
\sigma}.
\label{NCg2}\end{aligned}$$ The NC action for the Higgs field in (\[NCSM\]) is $$\begin{aligned}
\label{NChiggs0}
S^{NC}_{Higgs}&=&\int d^4x \bigg (
\rho_0\left( D_\mu \widehat \Phi \right)^\dagger \star\rho_0 \left( D^\mu \widehat
\Phi \right)
\nonumber\\ & - &
\mu^2 \rho_0(\widehat \Phi)^\dagger \star \rho_0( \widehat \Phi) -
\lambda ( \rho_0(\widehat \Phi)^\dagger \star \rho_0( \widehat \Phi)) \star
(\rho_0( \widehat \Phi)^\dagger \star \rho_0(\widehat \Phi)) \bigg ), \end{aligned}$$ where up to the lowest order of $\theta$ one has $$\rho_0(\hat \Phi)=\phi+\rho_0(\phi^1)+\mathcal{O}(\theta^2),$$ with $$\begin{aligned}
\rho_0(\phi^1)=-\frac{1}{2}\theta^{\alpha\beta}
(g'{ B}_\alpha+g W_\alpha) \partial_\beta \phi
+i\frac{1}{4} \theta^{\alpha \beta}
(g'{B}_\alpha+g W_\alpha) (g'{B}_\beta+g W_\beta) \phi.
\label{NCRhofield}\end{aligned}$$ By retaining all terms in the action (\[NChiggs0\]) at the leading order of the expansion in $ \theta $ one can easily find $$\begin{aligned}
S^{NC}_{Higgs}&=& \int d^4x\Bigg( (D^{SM}_\mu\phi)^\dagger D^{SM \mu}\phi
-\mu^2 \phi^\dagger \phi
-\lambda (\phi^\dagger \phi) (\phi^\dagger \phi) \Bigg)
\nonumber \\
&+&
\int d^4x \Bigg ( (D^{SM}_\mu\phi)^\dagger
\left( D^{SM \mu}\rho_0(\phi^1) + \frac{1}{2}
\theta^{\alpha \beta} \partial_\alpha V^{\mu} \partial_\beta \phi
+ \Gamma^\mu \phi \right)
\nonumber\\ & +&
\left(D^{SM}_\mu \rho_0 (\phi^1) + \frac{1}{2}
\theta^{\alpha \beta} \partial_\alpha V_\mu \partial_\beta \phi
+ {\Gamma_\mu} \phi \right)^\dagger D^{SM \mu}\phi
\nonumber\\ &
+&\frac{1}{4} \mu^2
\theta^{\mu \nu} \phi^\dagger (g' B_{\mu \nu} + g W_{\mu \nu}) \phi
- \lambda i \theta^{\alpha \beta}
\phi^\dagger \phi (D^{SM}_\alpha \phi)^\dagger (D^{SM}_\beta \phi)
\Bigg) + {\cal O}(\theta^2).
\label{NChiggs}\end{aligned}$$ For the NC Yukawa action, we only consider the leptonic part while the quark part can be obtained by replacing the corresponding fields with the leptonic ones. In this case, the action is $$\begin{aligned}
\label{NCyuk}
S^{NC}_{Yukawa}&=&\int d^4x \bigg (
-\sum_{i,j=1}^3 W^{ij} \bigg
( ( \bar{ \widehat L}^{(i)}_L \star \rho_L(\widehat \Phi))\star
\widehat e^{(j)}_R
+ \bar {\widehat e}^{(i)}_R \star (\rho_L(\widehat \Phi)^\dagger \star \widehat
L^{(j)}_L)\bigg )\bigg),\nonumber\\\end{aligned}$$ where by keeping only terms up to the first order of $\theta$ and using the appropriate representation for $ \rho_L $ [@NCSM] the action (\[NCyuk\]) leads to $$\begin{aligned}
S^{NC}_{Yukawa}&=&S^{SM}_{Yukawa}
- \int d^4x \bigg ( \sum_{i,j=1}^3 W^{ij} \bigg (
( \bar L^{i}_L \phi) e^{1 j}_R +
( \bar L^{i}_L \rho_L(\phi^1)) e^{j}_R \nonumber\\
&+& ( \bar L^{1 i}_L \phi) e^{j}_R +
i\frac{1}{2}\theta^{\alpha \beta} \partial_\alpha L^i_L \partial_\beta
\phi e^{j}_R
+ \bar e^{i}_R (\phi^{ \dagger} L^{1 j}_L)\nonumber\\
&+ &\bar e^{i}_R (\rho_L(\phi^1)^{ \dagger} L^{j}_L)+
\bar e^{1 i}_R (\phi^{\dagger} L^{j}_L) +
i\frac{1}{2}\theta^{\alpha \beta}
\partial_\alpha \bar e^{i}_R \partial_\beta \phi^\dagger L^j_L
\bigg)\bigg),
\label{NCyukawa}\end{aligned}$$ which $S^{SM}_{Yukawa}$ is defined in (\[YukawaSM\]) and $ L^{(1)}_{L}$ and $e^{(1)}_{R} $ are given in (\[L\_Li\]) and (\[e\_Ri\]).\
The total action through the NC parameter $\theta_{\mu\nu}$ violates the particle Lorentz symmetry, which can be considered as a subset of SME. Now we are ready to explore the mutual relations between the LV parameters and the NC parameter.
Lorentz violating parameters in terms of NC parameter
=====================================================
@sh\[\#1\]\#2[ [fmsl@sh]{} [fmsl@sh]{} [fmsl@sh]{}]{} fmsl@sh\#1\#2\#3[@th]{} In the previous section, the electroweak part of the NCSM has been introduced up to the first order of the NC parameter. Since the parameter of noncommutativity is a constant tensor, consequently each sector of the action violates the Lorentz symmetry. Therefore, it is sensible to have some relation between the NCSM and the SME. In the following subsections, we explore the mutual relations among the parameters of both theories in each sector.
Fermion sector
--------------
In the fermion sector, the Lagrangian density for the CPT-even part of the SME is $$\begin{aligned}
\mathcal{L}^{CPT-even}_{F}=i\frac{1}{2}(c_L)_{\mu\nu}\overline{L}\gamma^{\mu}\overleftrightarrow {D^{\nu}}L
+i\frac{1}{2}(c_R)_{\mu\nu}\overline{R}\gamma^{\mu}\overleftrightarrow {D^{\nu}}R,
\label{CPTevenLepton1}\end{aligned}$$ where with respect to the Left- and Right-handed fields $ L=(\dfrac{1-\gamma_5}{2})\psi $ and $ R=(\dfrac{1+\gamma_5}{2})\psi $, one has
$$\begin{aligned}
\mathcal{L}^{CPT-even}_{F}=i\frac{1}{2}c_{\mu\nu}\overline{\psi}\gamma^{\mu}\overleftrightarrow {D^{\nu}}\psi
-i\frac{1}{2} d_{\mu\nu} \overline{\psi}\gamma^{\mu}\gamma^{5}\overleftrightarrow {D^{\nu}}\psi,
\label{CPTevenfermion}\end{aligned}$$
where $$\begin{aligned}
c_{{\mu \nu }}=\frac{1}{2}\,(c_L)_{\mu\nu}+\frac{1}{2}\,(c_R)_{\mu\nu},
\label{c_munu}
\end{aligned}$$ $$\begin{aligned}
d_{{\mu \nu }}=\frac{1}{2}\,(c_L)_{\mu\nu}-\frac{1}{2}\,(c_R)_{\mu\nu}.
\label{d_munu}
\end{aligned}$$ Meanwhile, the fermion part of the NCSM Lagrangian density up to the first order of $\theta$ can be written as $$\begin{aligned}
{\cal L}^{NC}_{F}&=& i\frac{1}{2}(c_L)_{\mu\nu}[B,W]\overline{L}\gamma^{\mu}\overleftrightarrow {D^{\nu}}L
+i\frac{1}{2}(c_R)_{\mu\nu}[B]\overline{R}\gamma^{\mu}\overleftrightarrow {D^{\nu}}R + {\cal O}(\theta^2),
\label{NCf22}\end{aligned}$$ in which in terms of the flat space metric $\eta_{\mu\nu} $ $$\begin{aligned}
(c_L)_{\mu \nu}[B,W]=-\frac{1}{2}\theta^{\alpha\beta}(g'{B}_{\alpha\beta}+gW_{\alpha\beta})\eta_{\mu\nu}-{\theta^{\alpha}}_{\nu}(g'{B}_{{\mu \alpha }}+gW_{{\mu\alpha }})
,
\label{C_L}\end{aligned}$$ and $$\begin{aligned}
(c_R)_{\mu \nu }[B]=-\frac{1}{2}\theta^{\alpha\beta}g'{B}_{\alpha\beta}\eta_{\mu\nu}-{\theta^{\alpha}}_{\nu}g'{B}_{\mu \alpha },
\label{C_R}\end{aligned}$$ or in a similar way as is defined in Eq. (\[CPTevenfermion\]), $$\begin{aligned}
c_{\mu \nu }[B,W]=-\frac{1}{4}\theta^{\alpha\beta}(2g'{B}_{\alpha\beta}+g W_{\alpha\beta})\eta_{\mu\nu}-\frac{1}{2}{\theta^{\alpha}}_{\nu}(2g'{ B}_{\mu \alpha}+gW_{\mu\alpha}),
\label{c_munu2}\end{aligned}$$ $$\begin{aligned}
d_{\mu \nu}[W]=-\frac{1}{4}\theta^{\alpha\beta}g W_{\alpha\beta}\eta_{\mu\nu}-\frac{1}{2}{\theta^{\alpha}}_{\nu} g W_{\mu\alpha}.
\label{d_munu2}\end{aligned}$$ It should be noted that $c_{\mu \nu }[B,W]$ and $d_{\mu \nu}[W]$ as defined in (\[c\_munu2\]) and (\[d\_munu2\]) are not the usual LV parameters $c$ and $d$, respectively. In fact, they depend on the dynamical fields $B$ and $W$, which are the gauge fields before spontaneous symmetry breaking. Therefore, to find the appropriate LV parameters from (\[c\_munu2\]) and (\[d\_munu2\]) one should perform the following steps:\
1-Replace the gauge fields $B$ and $W$ with the physical fields $A$ and $Z$ as follows: $$W^\pm_\mu=\frac{W^1_\mu \mp i W^2_\mu}{\sqrt{2}}, \quad
Z_\mu=\frac{-g'{B}_\mu+gW^3_\mu}{\sqrt{g^2+g'^2}}
\quad \mbox{and} \quad
{A}_\mu=\frac{g{B}_\mu+g' W^3_\mu}{\sqrt{g^2+g'^2}},
\label{physicalfields}$$ and for the field strength tensors $$B_{\mu\nu}=\cos\theta_{\omega} A_{\mu\nu}-\sin\theta_{\omega}Z_{\mu\nu},$$
$$W^{3}_{\mu\nu}=\sin\theta_{\omega} A_{\mu\nu}+\cos\theta_{\omega}Z_{\mu\nu},$$
where $ \theta_{\omega} $ is the Weinberg angle.\
2-Introduce a background electromagnetic field $A^{b}_{\mu\nu}$ via $ A_{\mu\nu}\rightarrow A^{b}_{\mu\nu}+A_{\mu\nu} $ in (\[c\_munu2\]) and (\[d\_munu2\]), which leads to $$\begin{aligned}
c_{{\mu \nu }}[A^{b}]=g\sin\theta_{\omega}(-\frac{3}{4}\theta^{\alpha\beta}A^{b}_{\alpha\beta}\eta_{\mu\nu}-\frac{3}{2}{\theta^{\alpha}}_{\nu}A^{b}_{\mu\alpha}),
\label{c_munu3}\end{aligned}$$ and $$\begin{aligned}
d_{{\mu \nu }}[A^{b}]=g\sin\theta_{\omega}(-\frac{1}{4}\theta^{\alpha\beta}A^{b}_{\alpha\beta}\eta_{\mu\nu}-\frac{1}{2}{\theta^{\alpha}}_{\nu}A^{b}_{\mu\alpha}),
\label{d_munu3}\end{aligned}$$ where the LV parameters are obtained in term of the NC parameter through the electromagnetic background field ${A^{b}}_{\mu\nu}$. In fact, (\[c\_munu3\]) and (\[d\_munu3\]) show that in the presence of the NC background the LV parameters in the fermion sector arise only when there is a background electromagnetic field. Meanwhile, as the NCEW respects the CPT symmetry, the coefficients $a_{\mu}(L, R)$ in this sector are absent.
Gauge sector
------------
There are two versions for the gauge sector of the NCSM as is given in (\[NCg1\]). The origin of freedom in the gauge sector is due to the fact that the commutator of two gauge parameters does not form a closed Lie algebra in noncommutative space. The only exception is the fundamental representation of the U(N) group. Therefore, for the gauge group of the Standard Model one needs to extend the algebra, which leads to an infinite number of undefined parameters. They can be limited to the right number of fields and parameters via Seiberg-Witten maps, which themselves cannot be uniquely determined in the model. However, these many degrees of freedom lead to a freedom in the kinetic term of the gauge fields. In fact, gauge invariance alone is not enough to pick one of the possible choices [@nmNCSM]. In the minimal noncommutative Standard Model (mNCSM), which has minimal modification with respect to the ordinary Standard Model, there is no cubic self-interaction term for photons. In this case, where $ \rm tr_{1}Y^{3}=0 $, there is not any LV coefficient for the gauge sector from mNCSM in comparison with the SME. However, if one uses the freedom in the choice of trace for the gauge fields, as a nonminimal version of NCSM (nmNCSM) one has [@nmNCSM] $$\begin{aligned}
S_{\mbox{\tiny Gauge}}^{\mbox{\tiny nmNCEW}}
&=& S^{\mbox{\tiny mNCEW}}_{\mbox{\tiny Gauge}}
\nonumber \\
&+&{g'}^3\kappa_1{\theta^{\rho\sigma}}\hspace{-2mm}\int \hspace{-1mm}d^4x\,
\left(\frac{1}{4}{B}_{\rho\sigma}{B}_{\mu\nu}-{B}_{\mu\rho}
{B}_{\nu\sigma}\right){B}^{\mu\nu}
\nonumber \\
&+&g'g^2\kappa_2 \, \theta^{\rho\sigma}\hspace{-2mm}\int
\hspace{-1mm} d^4x
\left[(\frac{1}{4}{B}_{\rho\sigma} W^a_{\mu\nu}-
{B}_{\mu\rho} W^a_{\nu\sigma})W^{\mu\nu,a}\!+c.p.\right]
\nonumber \\
&+&{\cal O}(\theta^2) \, ,
\label{nmNCSM}\end{aligned}$$ where $\kappa_1$ and $\kappa_2$ are constant parameters, c.p. denotes the cyclic permutations of field strength tensors with respect to the Lorentz indices, $ S^{\mbox{\tiny mNCEW}}_{\mbox{\tiny Gauge}} $ is given in (\[NCg2\]), and the field strengths ${B}_{\mu\nu}(={B}_{\mu\nu}Y)$ and $W_{\mu\nu}(=W_{\mu\nu}^aT_L^a)$ are $$\begin{aligned}
{B}_{\mu\nu}&=&\partial_{\mu}{B}_{\nu}-\partial_{\nu}{ B}_{\mu}\:,\nonumber\\
W_{\mu\nu}^a &=& \partial_{\mu}W^a_{\nu}-\partial_{\nu}W^a_{\mu}
+g\;\epsilon^{abc}W^b_{\mu} W^c_{\nu}\:.\nonumber\\\end{aligned}$$ Now, the triple photon coupling can be extracted from (\[nmNCSM\]) by rewriting ${B}_{\mu\nu}$ and $W_{\mu\nu}$ in terms of the physical fields as $$\begin{aligned}
{\cal L}^{nmNCGauge}_{\gamma\gamma\gamma}&=&\frac{e}{4} \sin2{\theta_\omega}\;{\rm K}_{\gamma\gamma\gamma}{\theta^{\rho\sigma}}\left(A_{\mu\nu}A_{\rho\sigma}-4A_{\mu\rho}A_{\nu\sigma}\right)A^{\mu\nu}
\, ,
\label{nmNCGauge}\end{aligned}$$ with $$\begin{aligned}
{\rm K}_{\gamma\gamma\gamma}&=&\frac{1}{2}\; g g'(\kappa_1 + 3 \kappa_2)
\, .\end{aligned}$$ Meanwhile, $ ZZ\gamma $ and $ WW\gamma $ interactions can be obtained as $$\begin{aligned}
{\cal L}^{nmNCGauge}_{ZZ\gamma}&=&\frac{e}{4} \sin2{\theta_W}\,{\rm K}_{ZZ \gamma}\,
{\theta^{\rho\sigma}}
\left[2A^{\mu\nu}\left(2Z_{\mu\rho}Z_{\nu\sigma}-Z_{\mu\nu}Z_{\rho\sigma}\right)\right.\nonumber\\
&+&\left. 8 A_{\mu\rho}Z^{\mu\nu}Z_{\nu\sigma} - A_{\rho\sigma}Z_{\mu\nu}Z^{\mu\nu}\right]\,,
\label{z}\end{aligned}$$ with a similar expression for ${\cal L}^{nmNCGauge}_{WW\gamma}$ by replacing $Z$ by $W$ and ${\rm K}_{ZZ \gamma}$ by ${\rm K}_{WW \gamma}$ where $$\begin{aligned}
{\rm K}_{ZZ\gamma}&=&\frac{-1}{2gg'}\; \left[{g'}^4\kappa_1 + g^2\left(g^2-2{g'}^2\right)\kappa_2\right]\,,\end{aligned}$$ and $$\begin{aligned}
{\rm K}_{WW\gamma}&=&
-\frac{g}{2 g'}\left[{g'}^2+g^2\right]\kappa_2 \,.\end{aligned}$$ Therefore, in the presence of the electromagnetic background, $ A_{\mu\nu}\rightarrow A^{b}_{\mu\nu}+A_{\mu\nu} $ in each NC Lagrangian term, which in comparison with the CPT-even gauge part of the SME as $$\begin{aligned}
\mathcal{L}^{CPT-even}_{Gauge}=-\frac{1}{4}(k_W)_{\mu\nu\rho\sigma}W^{\mu\nu}W^{\rho\sigma}
-\frac{1}{4}(k_F)_{\mu\nu\rho\sigma}A^{\mu\nu}A^{\rho\sigma},
\label{CPTevenGaugephysical}\end{aligned}$$ where $ W=[Z, W^{\pm}] $, one finds all appropriate LV coefficients for this sector as follows: $$\begin{aligned}
(k_F)^{\mu\nu\rho\sigma}[A^{b}]&=&-8\varepsilon \theta^{\rho\sigma}(A^b)^{\mu\nu}+16\varepsilon \theta^{\nu\sigma}(A^b)^{\mu\rho}+32\varepsilon \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho},
\label{K_F}\end{aligned}$$ where by rewriting (\[K\_F\]) as $$\begin{aligned}
(k_F)^{\mu\nu\rho\sigma}[A^{b}]&=&-2\varepsilon \theta^{\rho\sigma}(A^b)^{\mu\nu}+4\varepsilon \theta^{\nu\sigma}(A^b)^{\mu\rho}
+8\varepsilon \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho}\nonumber\\
&+&2\varepsilon \theta^{\rho\sigma}(A^b)^{\nu\mu}-4\varepsilon \theta^{\mu\sigma}(A^b)^{\nu\rho}-8\varepsilon \theta^{\lambda\sigma}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\rho}\nonumber\\
&+&2\varepsilon \theta^{\sigma\rho}(A^b)^{\mu\nu}-4\varepsilon \theta^{\nu\rho}(A^b)^{\mu\sigma}-8\varepsilon \theta^{\lambda\rho}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\sigma}\nonumber\\
&-&2\varepsilon \theta^{\rho\sigma}(A^b)^{\nu\mu}+4\varepsilon \theta^{\mu\rho}(A^b)^{\nu\sigma}+8\varepsilon \theta^{\lambda\rho}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\sigma},
\label{K_F2}\end{aligned}$$ one can see that the tensor $ k_F $ has the properties of the Riemann curvature tensor and zero double trace. Meanwhile, in a similar way one has $$\begin{aligned}
(k_Z)^{\mu\nu\rho\sigma}[A^{b}]&=&8\varepsilon' \theta^{\rho\sigma}(A^b)^{\mu\nu}-16\varepsilon' \theta^{\nu\sigma}(A^b)^{\mu\rho}-32\varepsilon' \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho},\nonumber\\
&=&2\varepsilon' \theta^{\rho\sigma}(A^b)^{\mu\nu}-4\varepsilon' \theta^{\nu\sigma}(A^b)^{\mu\rho}
-8\varepsilon' \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho}\nonumber\\
&-&2\varepsilon' \theta^{\rho\sigma}(A^b)^{\nu\mu}+4\varepsilon' \theta^{\mu\sigma}(A^b)^{\nu\rho}+8\varepsilon' \theta^{\lambda\sigma}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\rho}\nonumber\\
&-&2\varepsilon' \theta^{\sigma\rho}(A^b)^{\mu\nu}+4\varepsilon' \theta^{\nu\rho}(A^b)^{\mu\sigma}+8\varepsilon' \theta^{\lambda\rho}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\sigma}\nonumber\\
&+&2\varepsilon' \theta^{\rho\sigma}(A^b)^{\nu\mu}-4\varepsilon' \theta^{\mu\rho}(A^b)^{\nu\sigma}-8\varepsilon' \theta^{\lambda\rho}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\sigma},
\label{K_Z}\end{aligned}$$ and $$\begin{aligned}
(k_{W^{\pm}})^{\mu\nu\rho\sigma}[A^{b}]&=&8\varepsilon'' \theta^{\rho\sigma}(A^b)^{\mu\nu}-16\varepsilon'' \theta^{\nu\sigma}(A^b)^{\mu\rho}-32\varepsilon'' \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho},\nonumber\\
&=&2\varepsilon'' \theta^{\rho\sigma}(A^b)^{\mu\nu}-4\varepsilon'' \theta^{\nu\sigma}(A^b)^{\mu\rho}
-8\varepsilon'' \theta^{\lambda\sigma}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\rho}\nonumber\\
&-&2\varepsilon'' \theta^{\rho\sigma}(A^b)^{\nu\mu}+4\varepsilon'' \theta^{\mu\sigma}(A^b)^{\nu\rho}+8\varepsilon'' \theta^{\lambda\sigma}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\rho}\nonumber\\
&-&2\varepsilon'' \theta^{\sigma\rho}(A^b)^{\mu\nu}+4\varepsilon'' \theta^{\nu\rho}(A^b)^{\mu\sigma}+8\varepsilon'' \theta^{\lambda\rho}{(A^b)^{\mu}}_{\lambda}\eta^{\nu\sigma}\nonumber\\
&+&2\varepsilon'' \theta^{\rho\sigma}(A^b)^{\nu\mu}-4\varepsilon'' \theta^{\mu\rho}(A^b)^{\nu\sigma}-8\varepsilon'' \theta^{\lambda\rho}{(A^b)^{\nu}}_{\lambda}\eta^{\mu\sigma}
,
\label{K_W}\end{aligned}$$ where $\varepsilon=\frac{e}{4}\sin2\theta_{\omega}K_{\gamma\gamma\gamma}$, $\varepsilon'=\frac{e}{4}\sin2\theta_{\omega}K_{ZZ\gamma}$, and $\varepsilon''=\frac{e}{4}\sin2\theta_{\omega}K_{WW\gamma}$. One should note that by replacing $ q\rightarrow 32\varepsilon $ in (\[K\_F\]) the result given in [@NCLV] for the QED part of SME can be rederived. As one expects, the $k_{AF}$ parameter in this sector is absent.
Higgs sector
------------
In this sector the NC Higgs action is $$\begin{aligned}
\label{NCHiggs}
S^{NC}_{Higgs}&=& \int d^4x\Bigg( (D_\mu\phi)^\dagger D^{ \mu}\phi
-\mu^2 \phi^\dagger \phi
-\lambda (\phi^\dagger \phi) (\phi^\dagger \phi) \Bigg)
\nonumber \\ &
+&
\int d^4x \Bigg ( (D_\mu\phi)^\dagger
\left( D^{ \mu}\rho_o(\phi^1) + \frac{1}{2}
\theta^{\alpha \beta} \partial_\alpha V^{\mu} \partial_\beta \phi
+ \Gamma^\mu \phi \right)
\nonumber\\ & +&
\left(D_\mu \rho_o (\phi^1) + \frac{1}{2}
\theta^{\alpha \beta} \partial_\alpha V_\mu \partial_\beta \phi
+ {\Gamma_\mu} \phi \right)^\dagger D^{ \mu}\phi
\nonumber\\ &
+&\frac{1}{4} \mu^2
\theta^{\mu \nu} \phi^\dagger (g' B_{\mu \nu} + g W^L_{\mu \nu}) \phi
- \lambda i \theta^{\alpha \beta}
\phi^\dagger \phi (D_\alpha \phi)^\dagger (D_\beta \phi)
\Bigg) + {\cal O}(\theta^2),
\end{aligned}$$ where after a little algebra by inserting $ \Gamma_\mu $ from (\[gamma\_mu\]) and $ \rho_o(\phi^1) $ from (\[NCRhofield\]) in (\[NCHiggs\]) one has
$$\begin{aligned}
\label{NCHiggs1}
{\cal L}^{NC}_{Higgs}&=& \bigg(-\frac{1}{2}\theta^{\alpha\nu}\partial_{\alpha}B^{\mu}-i\lambda\theta^{\mu\nu}\phi^{\dagger}\phi\bigg)(D_{\mu}\phi)^{\dagger}D_{\nu}\phi\nonumber\\
&+&\bigg(\frac{1}{4}\mu^2 \sqrt{{g'}^2+g^2} \theta^{\mu\nu}\bigg)\phi^{\dagger}\phi B_{\mu\nu}\nonumber\\
&+&\bigg(-\frac{\sqrt{2}}{2} g \mu^2 \theta_{\mu\nu}\bigg)\phi^{\dagger}W_{\mu\nu}\phi.\end{aligned}$$
By replacing $W$ and $B$ in (\[NCHiggs1\]) with $A$ and $Z$ and by comparing the obtained Lagrangian with (\[CPTevenHiggs\]), one can easily read the LV parameters in this sector as $$\begin{aligned}
(k^S_{\phi\phi})^{\mu\nu}[A^{b}]&=&- \theta^{\alpha\nu} \partial _{\alpha}A^{b\mu},\end{aligned}$$ $$\begin{aligned}
(k^A_{\phi\phi})^{\mu\nu}[\phi]&=&-2\lambda v^{2}\theta^{\mu\nu},\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{\mu\nu}=-\frac{1}{2}\mu^2 \sqrt{{g'}^2+g^2} \theta_{\mu\nu},\end{aligned}$$ and $$\begin{aligned}
(k_{\phi W})_{\mu\nu}=-\frac{\sqrt{2}}{2} g \mu^2 \theta_{\mu\nu},\end{aligned}$$ where we have written $ k_{\phi\phi} $ in terms of symmetric and antisymmetric parts as $ (k_{\phi\phi})[A,\phi]=(k^S_{\phi\phi})[A]+i(k^A_{\phi\phi}[\phi]) $. We also set $ <\phi^{\dagger}\phi>\equiv v^{2} $ where $v$ is the vacuum expectation value of the Higgs field after SSB. These parameters have already been introduced in [@NCHiggs].
Yukawa sector
-------------
For the Yukawa sector, we first substitute (\[L\_Li\]) and (\[e\_Ri\]) in the NC Yukawa action. Then, by using the SW map given in [@NCSM] and after some manipulations, the action up to the first order of $\theta$ leads to $$\begin{aligned}
\label{NCyu}
S_{Yukawa}^{NCSM}&=&S^{SM}_{Yukawa}- \int d^4x \sum_{i,j=1}^3 W^{ij} \bigg(\nonumber\\
&&[ \frac{i}{4} \theta^{\mu\nu}i{g'}B_{\mu\nu}] (\bar L^{i}_L \phi e^{j}_R)
+[\frac{i}{2}\theta^{\mu\nu}]( D_{\mu} \bar L^{i}_L D_{\nu} \phi e^{j}_R)\nonumber\\
&&+[\frac{1}{4}\theta^{\mu\nu}{g'}B_{\nu}]( D_{\mu} \bar L^{i}_L \phi e^{j}_R) +[ \theta^{\mu\nu}(-3{g'} B_{\nu}+2g W_{\nu})](\bar L^{i}_L D_{\mu}\phi e^{j}_R)\bigg)\nonumber\\
&&+H.c.,\end{aligned}$$ for leptons and a similar relation for quarks. As (\[NCyu\]) shows, only the first term in the NC corrections can be cast into a power-counting renormalizable form and can be compared with its counterpart in the SME. By comparing (\[NCyu\]) after SSB with (\[CPTevenYukawa\]), one can find the coupling constant $ h $ in the NC space as follows: $$\begin{aligned}
h[A^{b}]= \frac{1}{4} \theta^{\mu\nu}g'\cos\theta_{\omega}A^{b}_{\mu\nu},\end{aligned}$$ where $A^b_{\mu\nu}$ is not a dynamical field and it should be considered as a constant background the same as the other LV parameters. Therefore, for a constant magnetic field about $1 G$ and $\Lambda\sim 1 TeV$, the LV-parameter $h\sim 10^{-27}$ that is minuscule is the same as the other LV parameters.
The components of LV parameters
===============================
In the previous section, we have found the LV parameters in the electroweak part of the SME in terms of the NC parameter. These relations can be used to find new bounds on the value of free parameters of each theory from the existing bound on the other theory. However, the SME free parameters are extensively examined, and there are stringent bounds on each component or some combinations of LV parameters [@data]. Therefore, by studying these components new bounds on the NC parameter is expected. To this end, we define the electromagnetic background $A^{b}_{\mu\nu}$ where the Lorentz indices are $T$ and $I=X, Y, Z$ as follows: $$\begin{aligned}
A^{b}_{TI}=(A^{b}_{TX},A^{b}_{TY},A^{b}_{TZ})=(E_{X},E_{Y},E_{Z}),\nonumber\\
A^{b}_{IJ}=(A^{b}_{YZ},A^{b}_{ZX},A^{b}_{XY})=(B_{X},B_{Y},B_{Z}),
\label{componentA}\end{aligned}$$ and for the $\theta_{\mu\nu}$ $$\begin{aligned}
\theta_{t}=(\theta_{TX},\theta_{TY},\theta_{TZ}),\nonumber\\
\theta_{s}=(\theta_{YZ},\theta_{ZX},\theta_{XY}),
\label{componenttheta}\end{aligned}$$ where $\theta_{t}$ and $\theta_{s}$ are the time-space and space-space components of the NC parameter, respectively. Consequently, for instance, $c_{XX}$ , $c_{YY}$, and $c_{ZZ}$ from (\[c\_munu3\]) are $$\begin{aligned}
c_{XX}&=& \alpha[-\theta_{TY}E_{Y}-\theta_{TZ}E_{Z}+\theta_{YZ}B_{X}],\end{aligned}$$ $$\begin{aligned}
c_{YY}&=& \alpha[-\theta_{TX}E_{X}-\theta_{TZ}E_{Z}-\theta_{XZ}B_{Y}],\end{aligned}$$ $$\begin{aligned}
c_{ZZ}&=& \alpha[-\theta_{TX}E_{X}-\theta_{TY}E_{Y}+\theta_{XY}B_{Z}],\end{aligned}$$ and a suitable combination of the LV parameters for the LV experiment as $c_{Q}=c_{XX}+c_{YY}-2c_{ZZ}$, which lead to $$\begin{aligned}
c_{Q}&=& \alpha[\theta_{TX}E_X+\theta_{TY}E_Y-2\theta_{TZ}E_Z\nonumber\\
&+&\theta_{YZ}B_X-\theta_{XZ}B_Y-2\theta_{XY}B_Z],\end{aligned}$$ where $\alpha=-\frac{3}{2}g\sin\theta_{\omega}$. The other important components and their relevant combinations for the fermion part are given in Table 1 and for the other sectors are found in Appendix A. As Table 1 shows, the bounds on the components of $\theta_{\mu\nu}$ can easily be obtained as shown in the third column of Table 1.
The Lorentz violating parameters are defined in a nonrotating frame. Since the laboratory frame rotates with the Earth’s rotation, the LV components should be time and location dependent. Therefore, similar experiments in different places should lead to some discrepancy that is caused by the noncommutativity. To this end, one needs some relation between the nonrotating basis $(X, Y, Z)$ and the rotating one $(x,y,z)$ where $Z$ is along the north direction parallel to the Earth’s axis and $z$ is normal to surface of the Earth, as follows: $$\left(\begin{array}{c} $ x$ \\$ y $\\ $z$
\end{array}
\right)
=\left(
\begin{array}{ccc}
\cos{\chi}\cos{\Omega t} & \cos{\chi}\sin{\Omega t} & -\sin{\chi} \\
-\sin{\Omega t} & \cos{\Omega t} & 0 \\
\sin{\chi}\cos{\Omega t} & \sin{\chi}\sin{\Omega t} & \cos{\chi}\\
\end{array}
\right)
\left(
\begin{array}{c}
X \\ Y \\ Z
\end{array}
\right),
\label{coordmatrix}$$\
where $\Omega \simeq{2\pi}/(23h \, 56 min)$ is the Earth’s sidereal rotation frequency and $\chi$ is the angle between $Z$ and $z$ [@Comag; @location]. By transforming the timelike and spacelike vectors of the tensors $\theta_{\mu\nu}$ and $A^b_{\mu\nu}$ as are given in (\[componentA\]) and (\[componenttheta\]), one has $$\begin{aligned}
&A^{b}_{yz}=B_{x}&=\cos\chi \cos\Omega t B_{X}+\cos\chi \sin\Omega t B_{Y}-\sin\chi B_{Z},\nonumber\\
&A^{b}_{zx}=B_{y}&=-\sin\Omega t B_{X}+\cos\Omega t B_{Y},\nonumber\\
&A^{b}_{xy}=B_{z}&=\sin\chi \cos\Omega t B_{X}+\sin\chi \sin\Omega t B_{Y}+\cos\chi B_{Z},
\label{Bxyz}\end{aligned}$$ and $$\begin{aligned}
&A^{b}_{tx}=E_{x}&=\cos\chi \cos\Omega t E_{X}+\cos\chi \sin\Omega t E_{Y}-\sin\chi E_{Z},\nonumber\\
&A^{b}_{ty}=E_{y}&=-\sin\Omega t E_{X}+\cos\Omega t E_{Y},\nonumber\\
&A^{b}_{tz}=E_{z}&=\sin\chi \cos\Omega t E_{X}+\sin\chi \sin\Omega t E_{Y}+\cos\chi E_{Z},
\label{Exyz}\end{aligned}$$ with similar relations for the $\theta_{\mu\nu}$ components. Meanwhile, the components of LV parameters depend on the time and location via $\theta_{\mu\nu}$ and $A^b_{\mu\nu}$ dependency. For instance, the combination $c_{YZ}+c_{ZY}$ leads to $$\begin{aligned}
c_{YZ}+c_{ZY}&=&\alpha\{-\theta_{tx}E_{x}\sin2\chi \sin\Omega t -\theta_{tx}E_{y}\sin\chi\cos\Omega t - \theta_{tx}E_{z}(\cos^2\chi-\sin^2\chi)\sin\Omega t\nonumber\\
&+&\theta_{tz}E_{x}(\cos^2\chi-\sin^2\chi)\sin\Omega t+\theta_{tz}E_{y}\cos\chi\cos\Omega t+\theta_{tz}E_{z}\sin2\chi\sin\Omega t\nonumber\\
&-&\theta_{ty}E_{x}\sin\chi\cos\Omega t+\theta_{ty}E_{z}\cos\chi\cos\Omega t+\theta_{yz}B_{x}\sin2\chi\sin\Omega t\nonumber\\
&-&\theta_{yz}B_{z}(\cos^2\chi-\sin^2\chi)\sin\Omega t-\theta_{xy}B_{x}(\cos^2\chi-\sin^2\chi)\sin\Omega t\nonumber\\
&-&\theta_{xy}B_{z}\sin2\chi\sin\Omega t+\theta_{zx}B_{x}\sin\chi\cos\Omega t-\theta_{zx}B_{z}\cos\chi\cos\Omega t\nonumber\\
&+&\theta_{yz}B_{y}\sin\chi\cos\Omega t-\theta_{xy}B_{y}\cos\chi\cos\Omega t\},\nonumber\\\end{aligned}$$ where its time average is zero as $ \overline{\sin\Omega t}=\overline{\cos\Omega t}=0 $. All time and location dependence of the LV components and their relevant combinations are given in Appendixes B and C. However, the nonzero parameters after the time averaging for the fermion and Higgs sectors are presented in Tables \[locationFermion\] and \[locationHiggs\]. In Table \[locationFermion\], for simplicity, the NC location dependence is given in terms of the physical parameters in the rotating frame. For this purpose, the magnetic vector in the rotating frame is $\overrightarrow{B}\equiv(A^{b}_{yz}, A^{b}_{zx}, A^{b}_{xy})$ which is obtained from (\[Bxyz\]) and similarly for the vector $\overrightarrow{\theta}\equiv(\theta_{yz}, \theta_{zx}, \theta_{xy})$ in the same frame as $$\begin{aligned}
\theta_1=\theta_{yz}&=&\cos\chi \cos\Omega t \theta_{YZ}+\cos\chi \sin\Omega t \theta_{ZX}-\sin\chi \theta_{XY},\nonumber\\
\theta_2=\theta_{zx}&=&-\sin\Omega t \theta_{YZ}+\cos\Omega t \theta_{ZX},\nonumber\\
\theta_3=\theta_{xy}&=&\sin\chi \cos\Omega t \theta_{YZ}+\sin\chi \sin\Omega t \theta_{ZX}+\cos\chi \theta_{XY},\end{aligned}$$ which leads to $\overrightarrow{\theta}.\overrightarrow{B}=\theta_1A^{b}_{yz}+\theta_2A^{b}_{zx}+\theta_3A^{b}_{xy}$ in Table 2.
Parameter NC location dependence
--------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------
$\overline{c}_{TT}$ $\alpha\{ -(\theta.B)_x-(\theta.B)_y-(\theta.B)_z\}$
$\overline{c}_{XX}$ $\alpha\{\frac{1}{2}\cos^2\chi(\theta.B)_x+\frac{1}{2}(\theta.B)_y+\frac{1}{2}\sin^2\chi(\theta.B)_z+\frac{1}{4}\sin2\chi(\theta_{x}B_{z}+\theta_{z}B_{x})\}$
$\overline{c}_{YY}$ $\alpha\{\frac{1}{2}\cos^2\chi(\theta.B)_x+\frac{1}{2}(\theta.B)_y+\frac{1}{2}\sin^2\chi(\theta.B)_z+\frac{1}{4}\sin2\chi(\theta_{x}B_{z}+\theta_{z}B_{x})\}$
$\overline{c}_{ZZ}$ $\alpha\{\sin^2\chi(\theta.B)_x+\cos^2\chi(\theta.B)_z-\frac{1}{2}\sin2\chi(\theta_{x}B_{z}+\theta_{z}B_{x})\}$
$\overline{c}_{XY}$ $\alpha\{\frac{1}{2}\sin\chi(\theta\times B)_x-\frac{1}{2}\cos\chi(\theta\times B)_z\}$
$\overline{c}_{YX}$ $\alpha\{-\frac{1}{2}\sin\chi(\theta\times B)_x+\frac{1}{2}\cos\chi(\theta\times B)_z\} $
$\overline{c}_{Q}$ $ \alpha\{(\cos^2\chi-2\sin^2\chi)(\theta.B)_x+(\theta.B)_y+(\sin^2\chi-2\cos^2\chi)(\theta.B)_z$
$ +\frac{3}{2}\sin2\chi(\theta_{x}B_{z}+\theta_{z}B_{x})\}$
: Location dependence of the nonzero LV parameters in the fermion sector. Here, $\alpha=-\frac{3}{2}g\sin\theta_{\omega}$ and the electric field has been ignored.[]{data-label="locationFermion"}
Parameter NC location dependence
-------------------------- ---------------------------------------------------------
$(k_{\phi B})_{TZ}$ $\alpha\{-\sin\chi \theta_{tx}+\cos\chi \theta_{tz}\}$
$(k_{\phi B})_{XY}$ $\alpha\{-\sin\chi \theta_{yz}+\cos\chi \theta_{xy}\}$
$(k_{\phi W})_{TZ}$ $\beta\{-\sin\chi \theta_{tx}+\cos\chi \theta_{tz}\}$
$(k_{\phi W})_{XY}$ $\beta\{-\sin\chi \theta_{yz}+\cos\chi \theta_{xy}\}$
$(k^A_{\phi \phi})_{TZ}$ $\gamma\{-\sin\chi \theta_{tx}+\cos\chi \theta_{tz}\}$
$(k^A_{\phi \phi})_{XY}$ $\gamma\{-\sin\chi \theta_{yz}+\cos\chi \theta_{xy}\} $
: Location dependence of the nonzero LV parameters in the Higgs sector. Here, $\alpha=-\frac{1}{2}\mu^2\sqrt{{g'}^2+g^2}$, $\beta=-\frac{\sqrt{2}}{2}g\mu^2$, and $\gamma=-2\lambda v^2$.[]{data-label="locationHiggs"}
Nevertheless, to find the bound on the components of $\theta_{\mu\nu}$, one should precisely examine how the electromagnetic background field affects the system under consideration. One of the experiments that leads to valuable bounds on the NC parameter is the clock comparison test. In such a system:\
1. The background electric field is usually of order of $10 V/cm\sim 10^{-22} GeV^2$, which is much smaller than the background magnetic field of order of $0.1-1 T\sim 10^{-17}-10^{-16}GeV^2$ [@Comag]. Therefore, the electric field can safely be ignored.\
2. Although the parameter of noncommutativity $\theta$ is fixed, the magnetic field rotates with respect to the fixed frame, $$\begin{aligned}
B_{X}&=&\cos\chi \cos\Omega t B_{x}-\sin\Omega t B_{y}+ \sin \chi \cos\Omega t B_{z},\nonumber\\
B_{Y}&=&\cos\chi \sin\Omega t B_{x}+\cos\Omega t B_{y}+ \sin \chi \sin\Omega t B_{z},\nonumber\\
B_{Z}&=&-\sin\chi B_{x}+\cos \chi B_{z},
\end{aligned}$$ where $ \chi\simeq 118^{\circ} $ is the angle between the magnetic field and the Earth’s axis of rotation in the Cs/Hg clock comparison test [@ClockLV]. In this experiment the time average $ \bar{B}_X=\bar{B}_Y=0$ while $\bar{B}_Z=-0.88 B_{x}-0.46 B_{z}$. Therefore, for $(B_x,B_y,B_z)=(0,0,B)$, one has $\bar{B}_Z=-0.46 B$, which puts a bound on $\theta_{XY}$ as $c_{Q}=\alpha (0.92B)\theta_{XY}\sim10^{-25} $ or $\mid\theta_{XY}\mid<(10TeV)^{-2}$ for $B\sim 1 T$. For the other clock comparison tests available in Ref. [@data], one can put new bounds on different components of the NC parameter as is given in Table \[tab:NCbound\].\
Conclusion
==========
We considered NCSM as a subset of SME to find the mutual relations between the parameters of both theories. For this purpose, the electroweak part of the NCSM up to the first order of the NC parameter has been expanded by using the SW maps. Although $\theta$-dependent terms violate particle Lorentz symmetry, except in the Higgs sector, they have not any counterparts in the SME. Consequently, NCSM is considered in the presence of a constant electromagnetic field as a background. Subsequently, a lot of relations between the LV parameters and the NC parameter in each sector of the SME have been found in Sec. 4. For the Yukawa sector, we found a power-counting renormalizable term that violates Lorentz symmetry and is proportional to the NC parameter. This term in a background field about $1G$ and for $\Lambda\sim 1 TeV$ leads to the corresponding LV parameters of the order of $10^{-27}$, which is very small like the other LV parameters. In Ref. [@data] the latest bounds from many precise measurements on the components or some combinations of LV parameters is collected, which led to new bounds on the components of NC parameters or some combinations of $\theta_{\mu\nu}$ components as is given in Tables \[tab:Fermion\] and \[tab:NCbound\]. For instance, in the clock comparison test a bound of order $(10TeV)^{-2}$ can be found on the $|\theta_{XY}|$. We also explored the time and location dependencies of the LV parameters to obtain the location dependence of different experiments on the NC parameter as is found in Tables \[locationFermion\] and \[locationHiggs\].
Appendix
========
The components of LV guage parameters
=====================================
In this appendix, we derive all the LV parameters in the gauge sector that are related to the $k_F$ in terms of the NC parameter and the electromagnetic background fields. $$\begin{aligned}
\widetilde{\kappa}_{tr}&=&-\frac{2}{3} \left[(k_F)^{TXTX}+(k_F)^{TYTY}+(k_F)^{TZTZ}\right]\nonumber\\
&=&-\frac{176}{3}\left[\theta^{TX}(A^{b})^{TX}+\theta^{TY}(A^{b})^{TY}+\theta^{TZ}(A^{b})^{TZ}\right]\nonumber\\
&=&-\frac{176}{3}\left[\theta^{TX}E^X+\theta^{TY}E^Y+\theta^{TZ}E^Z\right],\end{aligned}$$ $$\begin{aligned}
k^1&=&(k_F)^{TYXZ}\nonumber\\
&=&-8\varepsilon\theta^{XZ}(A^{b})^{TY}+16\varepsilon\theta^{YZ}(A^{b})^{TX}\nonumber\\
&=&-8\varepsilon\theta^{XZ}E^Y+16\varepsilon\theta^{YZ}E^X,\end{aligned}$$ $$\begin{aligned}
k^2&=&(k_F)^{TXYZ}\nonumber\\
&=&-8\varepsilon\theta^{YZ}(A^{b})^{TX}+16\varepsilon\theta^{XZ}(A^{b})^{TY}\nonumber\\
&=&-8\varepsilon\theta^{YZ}E^X+16\varepsilon\theta^{XZ}E^Y,\end{aligned}$$ $$\begin{aligned}
k^3&=&(k_F)^{TYTY}-(k_F)^{XZXZ}\nonumber\\
&=&24\varepsilon\theta^{TY}(A^{b})^{TY}+24\varepsilon\theta^{ZX}(A^{b})^{XZ}-32\varepsilon\theta^{ZT}(A^{b})^{TZ}+32\varepsilon\theta^{YX}(A^{b})^{XY}\nonumber\\
&=&24\varepsilon\theta^{TY}E^Y-24\varepsilon\theta^{ZX}B^Y+32\varepsilon\theta^{TZ}E^Z-32\varepsilon\theta^{XY}B^Z,\end{aligned}$$ $$\begin{aligned}
k^4&=&(k_F)^{TZTZ}-(k_F)^{XYXY}\nonumber\\
&=&24\varepsilon\theta^{TZ}(A^{b})^{TZ}-24\varepsilon\theta^{XY}(A^{b})^{XY}-32\varepsilon\theta^{YT}(A^{b})^{TY}+32\varepsilon\theta^{ZX}(A^{b})^{XZ}\nonumber\\
&=&24\varepsilon\theta^{TZ}E^Z-24\varepsilon\theta^{XY}B^Z+32\varepsilon\theta^{TY}E^Y-32\varepsilon\theta^{ZX}B^Y,\end{aligned}$$ $$\begin{aligned}
k^5&=&(k_F)^{TXTY}+(k_F)^{XZYZ}\nonumber\\
&=&24\varepsilon\theta^{TY}(A^{b})^{TX}+24\varepsilon\theta^{YZ}(A^{b})^{XZ}-32\varepsilon\theta^{XY}(A^{b})^{XY}\nonumber\\
&=&24\varepsilon\theta^{TY}E^X-24\varepsilon\theta^{YZ}B^Y-32\varepsilon\theta^{XY}B^Z,\end{aligned}$$ $$\begin{aligned}
k^6&=&(k_F)^{TXTZ}-(k_F)^{XYYZ}\nonumber\\
&=&-8\varepsilon\theta^{TZ}(A^{b})^{TX}-8\varepsilon\theta^{YZ}(A^{b})^{XY}\nonumber\\
&=&-8\varepsilon\theta^{TZ}E^X-8\varepsilon\theta^{YZ}B^Z,\end{aligned}$$ $$\begin{aligned}
k^7&=&(k_F)^{TYTZ}+(k_F)^{XYXZ}\nonumber\\
&=&-8\varepsilon\theta^{TZ}(A^{b})^{TY}-8\varepsilon\theta^{XZ}(A^{b})^{XY}\nonumber\\
&=&-8\varepsilon\theta^{TZ}E^Y-8\varepsilon\theta^{XZ}B^Z,\end{aligned}$$ $$\begin{aligned}
k^8&=&(k_F)^{TXXY}+(k_F)^{TZYZ}\nonumber\\
&=&-24\varepsilon\theta^{XY}(A^{b})^{TX}+24\varepsilon\theta^{YZ}(A^{b})^{TZ}\nonumber\\
&=&-24\varepsilon\theta^{XY}E^X+24\varepsilon\theta^{YZ}E^Z,\end{aligned}$$ $$\begin{aligned}
k^9&=&(k_F)^{TXXZ}-(k_F)^{TYYZ}\nonumber\\
&=&8\varepsilon\theta^{XZ}(A^{b})^{TX}-8\varepsilon\theta^{YZ}(A^{b})^{TY}\nonumber\\
&=&8\varepsilon\theta^{XZ}E^X-8\varepsilon\theta^{YZ}E^Y,\end{aligned}$$ $$\begin{aligned}
k^{10}&=&(k_F)^{TYXY}-(k_F)^{TZXZ}\nonumber\\
&=&-8\varepsilon\theta^{XY}(A^{b})^{TY}+8\varepsilon\theta^{XZ}(A^{b})^{TZ}\nonumber\\
&=&-8\varepsilon\theta^{XY}E^Y+8\varepsilon\theta^{XZ}E^Z,\end{aligned}$$ where $\varepsilon $ has been introduced in the gauge subsection of Sec. 4.
Time dependence on LV parameters
================================
In Sec. 5, $\theta_{\mu\nu}$ and $A^{b}_{\mu\nu}$ were introduced in terms of their electric- and magnetic-like components and, subsequently, their time and location dependence. Here, by using these relations we give the time dependence of the LV parameters in the fermion and Higgs sectors of the SME.
- Fermion sector
The time dependence of all components and some of their important combinations in the fermion sector are as follows: $$\begin{aligned}
c_{TT}&=&\alpha\{-\theta_{yz}A^{b}_{yz}-\theta_{xy}A^{b}_{xy}-\theta_{zx}A^{b}_{zx}\},\end{aligned}$$ $$\begin{aligned}
c_{XX}&=&\alpha\{-\theta_{tx}A^{b}_{tx}(\sin^2\chi+\cos^2\chi\sin^2\Omega t)-\frac{1}{2}\theta_{tx}A^{b}_{ty}\cos\chi\sin2\Omega t +\frac{1}{2}\theta_{tx}A^{b}_{tz}\sin2\chi\cos^2\Omega t\nonumber\\
&-&\frac{1}{2}\theta_{ty}A^{b}_{tx}\cos\chi \sin2\Omega t-\theta_{ty}A^{b}_{ty}\cos^2\Omega t-\frac{1}{2}\theta_{ty}A^{b}_{tz}\sin\chi \sin2\Omega t\nonumber\\
&+&\frac{1}{2}\theta_{tz}A^{b}_{tx}\sin2\chi \cos^2\Omega t-\frac{1}{2}\theta_{tz}A^{b}_{ty}\sin\chi \sin2\Omega t-\theta_{tz}A^{b}_{tz}(\cos^2\chi+\sin^2\chi \sin^2\Omega t)\nonumber\\
&+&\theta_{yz}A^{b}_{yz}\cos^2\chi \cos^2\Omega t-\frac{1}{2}\theta_{yz}A^{b}_{zx}\cos\chi \sin2\Omega t+\frac{1}{2}\theta_{yz}A^{b}_{xy}\sin2\chi \cos^2\Omega t\nonumber\\
&-&\frac{1}{2}\theta_{zx}A^{b}_{yz}\cos\chi \sin2\Omega t+\theta_{zx}A^{b}_{zx}\sin^2\Omega t-\frac{1}{2}\theta_{zx}A^{b}_{xy}\sin\chi \sin2\Omega t\nonumber\\
&+&\frac{1}{2}\theta_{xy}A^{b}_{yz}\sin2\chi\cos^2\Omega t-\frac{1}{2}\theta_{xy}A^{b}_{zx}\sin\chi\sin2\Omega t+\theta_{xy}A^{b}_{xy}\sin^2\chi \cos^2\Omega t\},\end{aligned}$$ $$\begin{aligned}
c_{YY}&=&\alpha\{-\theta_{tx}A^{b}_{tx}(\sin^2\chi+\cos^2\chi\cos^2\Omega t)+\frac{1}{2}\theta_{tx}A^{b}_{ty}\cos\chi\sin2\Omega t+\frac{1}{2}\theta_{tx}A^{b}_{tz}\sin2\chi \sin^2\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{ty}A^{b}_{tx}\cos\chi \sin2\Omega t-\theta_{ty}A^{b}_{ty}\sin^2\Omega t+\frac{1}{2}\theta_{ty}A^{b}_{tz}\sin\chi \sin2\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{tz}A^{b}_{tx}\sin2\chi \sin^2\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{ty}\sin\chi \sin2\Omega t-\theta_{tz}A^{b}_{tz}(\cos^2\chi +\sin^2\chi \cos^2\Omega t)\nonumber\\
&&+\theta_{yz}A^{b}_{yz}\cos^2\chi \sin^2\Omega t+\frac{1}{2}\theta_{yz}A^{b}_{zx}\cos\chi \sin2\Omega t+\frac{1}{2}\theta_{yz}A^{b}_{xy}\sin2\chi \sin^2\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{zx}A^{b}_{yz}\cos\chi \sin2\Omega t+\theta_{zx}A^{b}_{zx}\cos^2\Omega t+\frac{1}{2}\theta_{zx}A^{b}_{xy}\sin\chi \sin2\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{xy}A^{b}_{yz}\sin2\chi\sin^2\Omega t+\frac{1}{2}\theta_{xy}A^{b}_{zx}\sin\chi\sin2\Omega t+\theta_{xy}A^{b}_{xy}\sin^2\chi \sin^2\Omega t \},\end{aligned}$$ $$\begin{aligned}
c_{ZZ}&=&\alpha\{-\theta_{tx}A^{b}_{tx}\cos^2 \chi-\frac{1}{2}\theta_{tx}A^{b}_{tz}\sin2\chi-\theta_{ty}A^{b}_{ty}\nonumber\\
&&-\frac{1}{2}\theta_{tz}A^{b}_{tx}\sin2\chi-\theta_{tz}A^{b}_{tz}\sin^2\chi+\theta_{yz}A^{b}_{yz}\sin^2\chi\nonumber\\
&&-\frac{1}{2}\theta_{yz}A^{b}_{xy}\sin2\chi-\frac{1}{2}\theta_{xy}A^{b}_{yz}\sin2\chi+\theta_{xy}A^{b}_{xy}\cos^2\chi \},\end{aligned}$$
$$\begin{aligned}
c_{TX}&=&\alpha\{\theta_{yz}A^{b}_{tx}\sin2\chi \sin\Omega t +\theta_{yz}A^{b}_{ty}\sin\chi\cos\Omega t +\theta_{yz}A^{b}_{tz}(\sin^2\chi-\cos^2\chi)\sin\Omega t \nonumber\\
&&+\theta_{xy}A^{b}_{tx}(\sin^2\chi-\cos^2\chi)\sin\Omega t-\theta_{xy}A^{b}_{ty}\cos\chi\cos\Omega t-\theta_{xy}A^{b}_{tz}\sin2\chi\sin\Omega t\nonumber\\
&&+\theta_{zx}A^{b}_{tx}\sin\chi\cos\Omega t-\theta_{zx}A^{b}_{tz}\cos\chi\cos\Omega t\},\end{aligned}$$
$$\begin{aligned}
c_{XT}&=&\alpha\{\theta_{tx}A^{b}_{yz}\sin2\chi \sin\Omega t +\theta_{ty}A^{b}_{yz}\sin\chi\cos\Omega t +\theta_{tz}A^{b}_{yz}(\sin^2\chi-\cos^2\chi)\sin\Omega t \nonumber\\
&&+\theta_{tx}A^{b}_{xy}(\sin^2\chi-\cos^2\chi)\sin\Omega t-\theta_{ty}A^{b}_{xy}\cos\chi\cos\Omega t-\theta_{tz}A^{b}_{xy}\sin2\chi\sin\Omega t\nonumber\\
&&+\theta_{tx}A^{b}_{zx}\sin\chi\cos\Omega t-\theta_{tz}A^{b}_{zx}\cos\chi\cos\Omega t\},\end{aligned}$$
$$\begin{aligned}
c_{TY}&=&\alpha\{\theta_{yz}A^{b}_{ty}\sin\chi \sin\Omega t -\theta_{yz}A^{b}_{tz}\cos\Omega t +\theta_{xy}A^{b}_{tx}\cos\Omega t \nonumber\\
&-&\theta_{xy}A^{b}_{ty}\cos\chi\sin\Omega t-\theta_{zx}A^{b}_{tx}\sin\chi\sin\Omega t+\theta_{zx}A^{b}_{tz}\cos\chi\sin\Omega t\},\end{aligned}$$
$$\begin{aligned}
c_{YT}&=&\alpha\{\theta_{ty}A^{b}_{yz}\sin\chi \sin\Omega t -\theta_{tz}A^{b}_{yz}\cos\Omega t +\theta_{tx}A^{b}_{xy}\cos\Omega t \nonumber\\
&-&\theta_{ty}A^{b}_{xy}\cos\chi\sin\Omega t-\theta_{tx}A^{b}_{zx}\sin\chi\sin\Omega t+\theta_{tz}A^{b}_{zx}\cos\chi\sin\Omega t\},\end{aligned}$$
$$\begin{aligned}
c_{TZ}&=&\alpha\{\theta_{yz}A^{b}_{tx}\cos^2\chi \sin2\Omega t +\theta_{yz}A^{b}_{ty}\cos\chi(\cos^2\Omega t-\sin^2\Omega t) +\frac{1}{2}\theta_{yz}A^{b}_{tz}\sin2\chi\sin2\Omega t \nonumber\\
&+&\theta_{zx}A^{b}_{tx}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)-\theta_{zx}A^{b}_{ty}\sin2\Omega t+\theta_{zx}A^{b}_{tz}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&+&\frac{1}{2}\theta_{xy}A^{b}_{tx}\sin2\chi\sin2\Omega t+\theta_{xy}A^{b}_{ty}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)+\theta_{xy}A^{b}_{tz}\sin^2\chi\sin2\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{ZT}&=&\alpha\{\theta_{tx}A^{b}_{yz}\cos^2\chi \sin2\Omega t + \theta_{ty}A^{b}_{yz}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)+\frac{1}{2}\theta_{tz}A^{b}_{yz}\sin2\chi\sin2\Omega t \nonumber\\
&&+\theta_{tx}A^{b}_{zx}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)-\theta_{ty}A^{b}_{zx}\sin2\Omega t+\theta_{tz}A^{b}_{zx}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&+\frac{1}{2}\theta_{tx}A^{b}_{xy}\sin2\chi\sin2\Omega t+\theta_{ty}A^{b}_{xy}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)+\theta_{tz}A^{b}_{xy}\sin^2\chi\sin2\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{XY}&=&\alpha\{\frac{1}{2} \theta_{tx}A^{b}_{tx}\cos^2\chi \sin2\Omega t- \theta_{tx}A^{b}_{ty}\cos\chi\sin^2\Omega t+\frac{1}{4}\theta_{tx}A^{b}_{tz}\sin2\chi\sin2\Omega t \nonumber\\
&&+\theta_{ty}A^{b}_{tx}\cos\chi\cos^2\Omega t-\frac{1}{2}\theta_{ty}A^{b}_{ty}\sin2\Omega t+\theta_{ty}A^{b}_{tz}\sin\chi\cos2\Omega t\nonumber\\
&&+\frac{1}{4}\theta_{tz}A^{b}_{tx}\sin2\chi\sin2\Omega t-\theta_{tz}A^{b}_{ty}\sin\chi\sin^2\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin^2\chi\sin2\Omega t\nonumber\\
&&-\frac{1}{2}\theta_{yz}A^{b}_{yz}\cos^2\chi\sin2\Omega t-\theta_{yz}A^{b}_{zx}\cos\chi\cos^2\Omega t-\frac{1}{4}\theta_{yz}A^{b}_{xy}\sin2\chi\sin2\Omega t \nonumber\\
&&+\theta_{zx}A^{b}_{yz}\cos\chi\sin^2\Omega t+\frac{1}{2}\theta_{zx}A^{b}_{zx}\sin2\Omega t+\theta_{zx}A^{b}_{xy}\sin\chi\sin^2\Omega t\nonumber\\
&&-\frac{1}{4}\theta_{xy}A^{b}_{yz}\sin2\chi\sin2\Omega t-\theta_{xy}A^{b}_{zx}\sin\chi\cos^2\Omega t-\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin^2\chi\sin2\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{YX}&=&\alpha\{\frac{1}{2} \theta_{tx}A^{b}_{tx}\cos^2\chi \sin2\Omega t- \theta_{ty}A^{b}_{tx}\cos\chi\sin^2\Omega t+\frac{1}{4}\theta_{tz}A^{b}_{tx}\sin2\chi\sin2\Omega t \nonumber\\
&&+\theta_{tx}A^{b}_{ty}\cos\chi\cos^2\Omega t-\frac{1}{2}\theta_{ty}A^{b}_{ty}\sin2\Omega t+\theta_{tz}A^{b}_{ty}\sin\chi\cos2\Omega t\nonumber\\
&&+\frac{1}{4}\theta_{tx}A^{b}_{tz}\sin2\chi\sin2\Omega t-\theta_{ty}A^{b}_{tz}\sin\chi\sin^2\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin^2\chi\sin2\Omega t\nonumber\\
&&-\frac{1}{2}\theta_{yz}A^{b}_{yz}\cos^2\chi\sin2\Omega t-\theta_{zx}A^{b}_{yz}\cos\chi\cos^2\Omega t-\frac{1}{4} \theta_{xy}A^{b}_{yz}\sin2\chi\sin2\Omega t\nonumber\\
&&+\theta_{yz}A^{b}_{zx}\cos\chi\sin^2\Omega t+\frac{1}{2}\theta_{zx}A^{b}_{zx}\sin2\Omega t+\theta_{xy}A^{b}_{zx}\sin\chi\sin^2\Omega t\nonumber\\
&&-\frac{1}{4}\theta_{yz}A^{b}_{xy}\sin2\chi\sin2\Omega t-\theta_{zx}A^{b}_{xy}\sin\chi\cos^2\Omega t-\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin^2\chi\sin2\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{XZ}&=&\alpha\{-\frac{1}{2} \theta_{tx}A^{b}_{tx}\sin2\chi \cos\Omega t+ \theta_{tx}A^{b}_{ty}\sin\chi\sin\Omega t-\theta_{tx}A^{b}_{tz}\sin^2\chi\cos\Omega t \nonumber\\
&&+\theta_{tz}A^{b}_{tx}\cos^2\chi\cos\Omega t-\theta_{tz}A^{b}_{ty}\cos\chi\sin\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin2\chi\cos\Omega t\nonumber\\
&&-\frac{1}{2}\theta_{yz}A^{b}_{yz}\sin2\chi\cos\Omega t+\theta_{yz}A^{b}_{xy}\cos^2\chi\cos\Omega t+\theta_{zx}A^{b}_{yz}\sin\chi\sin\Omega t\nonumber\\
&&-\theta_{zx}A^{b}_{xy}\cos\chi\sin\Omega t-\theta_{xy}A^{b}_{yz}\sin^2\chi\cos\Omega t+\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin2\chi\cos\Omega t \},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{ZX}&=&\alpha\{-\frac{1}{2}\theta_{tx}A^{b}_{tx}\sin2\chi \cos\Omega t + \theta_{ty}A^{b}_{tx}\sin\chi\sin\Omega t-\theta_{tz}A^{b}_{tx}\sin^2\chi\cos\Omega t \nonumber\\
&&+\theta_{tx}A^{b}_{tz}\cos^2\chi\cos\Omega t-\theta_{ty}A^{b}_{tz}\cos\chi\sin\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin2\chi\cos\Omega t\nonumber\\
&&-\frac{1}{2}\theta_{yz}A^{b}_{yz}\sin2\chi\cos\Omega t+\theta_{xy}A^{b}_{yz}\cos^2\chi\cos\Omega t+\theta_{yz}A^{b}_{zx}\sin\chi\sin\Omega t\nonumber\\
&&-\theta_{xy}A^{b}_{zx}\cos\chi\sin\Omega t-\theta_{yz}A^{b}_{xy}\sin^2\chi\cos\Omega t+\frac{1}{2} \theta_{xy}A^{b}_{xy}\sin2\chi\cos\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{YZ}&=&\alpha\{-\frac{1}{2}\theta_{tx}A^{b}_{tx}\sin2\chi \sin\Omega t -\theta_{tx}A^{b}_{ty}\sin\chi\cos\Omega t - \theta_{tx}A^{b}_{tz}\sin^2\chi\sin\Omega t\nonumber\\
&&+\theta_{tz}A^{b}_{tx}\cos^2\chi\sin\Omega t+\theta_{tz}A^{b}_{ty}\cos\chi\cos\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin2\chi\sin\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{yz}A^{b}_{yz}\sin2\chi\sin\Omega t-\theta_{yz}A^{b}_{xy}\cos^2\chi\sin\Omega t+\theta_{zx}A^{b}_{yz}\sin\chi\cos\Omega t\nonumber\\
&&-\theta_{zx}A^{b}_{xy}\cos\chi\cos\Omega t+\theta_{xy}A^{b}_{yz}\sin^2\chi\sin\Omega t-\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin2\chi\sin\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{ZY}&=&\alpha\{-\frac{1}{2}\theta_{tx}A^{b}_{tx}\sin2\chi \sin\Omega t -\theta_{ty}A^{b}_{tx}\sin\chi\cos\Omega t -\theta_{tz}A^{b}_{tx}\sin^2\chi\sin\Omega t \nonumber\\
&&+\theta_{tx}A^{b}_{tz}\cos^2\chi\sin\Omega t+\theta_{ty}A^{b}_{tz}\cos\chi\cos\Omega t+\frac{1}{2}\theta_{tz}A^{b}_{tz}\sin2\chi\sin\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{yz}A^{b}_{yz}\sin2\chi\sin\Omega t-\theta_{xy}A^{b}_{yz}\cos^2\chi\sin\Omega t+\theta_{yz}A^{b}_{zx}\sin\chi\cos\Omega t\nonumber\\
&&-\theta_{xy}A^{b}_{zx}\cos\chi\cos\Omega t+\theta_{yz}A^{b}_{xy}\sin^2\chi\sin\Omega t-\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin2\chi\sin\Omega t\}.\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{XY}+c_{YX}&=&\alpha\{\theta_{tx}A^{b}_{tx}\cos^2\chi \sin2\Omega t+\theta_{tx}A^{b}_{ty}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&+\frac{1}{2}\theta_{tx}A^{b}_{tz}\sin2\chi\sin2\Omega t+\theta_{ty}A^{b}_{tx}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&-\theta_{ty}A^{b}_{ty}\sin2\Omega t+\theta_{ty}A^{b}_{tz}\sin\chi(\cos2\Omega t-\sin^2\Omega t)\nonumber\\
&&+\frac{1}{2}\theta_{tz}A^{b}_{tx}\sin2\chi\sin2\Omega t+\theta_{tz}A^{b}_{ty}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&+\theta_{tz}A^{b}_{tz}\sin^2\chi\sin2\Omega t-\theta_{yz}A^{b}_{yz}\cos^2\chi\sin2\Omega t\nonumber\\
&&-\theta_{yz}A^{b}_{zx}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)-\frac{1}{2}\theta_{yz}A^{b}_{xy}\sin2\chi\sin2\Omega t \nonumber\\
&&-\theta_{zx}A^{b}_{yz}\cos\chi(\cos^2\Omega t-\sin^2\Omega t)+\theta_{zx}A^{b}_{zx}\sin2\Omega t\nonumber\\
&&-\theta_{zx}A^{b}_{xy}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)-\frac{1}{2}\theta_{xy}A^{b}_{yz}\sin2\chi\sin2\Omega t\nonumber\\
&&-\theta_{xy}A^{b}_{zx}\sin\chi(\cos^2\Omega t-\sin^2\Omega t)-\theta_{xy}A^{b}_{xy}\sin^2\chi\sin2\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{XZ}+c_{ZX}&=&\alpha\{-\theta_{tx}A^{b}_{tx}\sin2\chi \cos\Omega t+ \theta_{tx}A^{b}_{ty}\sin\chi\sin\Omega t+\theta_{tx}A^{b}_{tz}(\cos^2\chi-\sin^2\chi)\cos\Omega t \nonumber\\
&&+\theta_{tz}A^{b}_{tx}(\cos^2\chi-\sin^2\chi)\cos\Omega t-\theta_{tz}A^{b}_{ty}\cos\chi\sin\Omega t+\theta_{tz}A^{b}_{tz}\sin2\chi\cos\Omega t\nonumber\\
&&+ \theta_{ty}A^{b}_{tx}\sin\chi\sin\Omega t-\theta_{ty}A^{b}_{tz}\cos\chi\sin\Omega t-\theta_{yz}A^{b}_{yz}\sin2\chi\cos\Omega t\nonumber\\
&&+\theta_{yz}A^{b}_{xy}\cos^2\chi\cos\Omega t+\theta_{zx}A^{b}_{yz}\sin\chi\sin\Omega t-\theta_{zx}A^{b}_{xy}\cos\chi\sin\Omega t\nonumber\\
&&+\theta_{xy}A^{b}_{yz}(\cos^2\chi-\sin^2\chi)\cos\Omega t+\theta_{xy}A^{b}_{xy}\sin2\chi\cos\Omega t \},\end{aligned}$$
$$\begin{aligned}
c_{YZ}+c_{ZY}&=&\alpha\{-\theta_{tx}A^{b}_{tx}\sin2\chi \sin\Omega t -\theta_{tx}A^{b}_{ty}\sin\chi\cos\Omega t - \theta_{tx}A^{b}_{tz}(\cos^2\chi-\sin^2\chi)\sin\Omega t\nonumber\\
&&+\theta_{tz}A^{b}_{tx}(\cos^2\chi-\sin^2\chi)\sin\Omega t+\theta_{tz}A^{b}_{ty}\cos\chi\cos\Omega t+\theta_{tz}A^{b}_{tz}\sin2\chi\sin\Omega t\nonumber\\
&& -\theta_{ty}A^{b}_{tx}\sin\chi\cos\Omega t+\theta_{ty}A^{b}_{tz}\cos\chi\cos\Omega t+\theta_{yz}A^{b}_{yz}\sin2\chi\sin\Omega t\nonumber\\
&&-\theta_{yz}A^{b}_{xy}(\cos^2\chi-\sin^2\chi)\sin\Omega t-\theta_{xy}A^{b}_{yz}(\cos^2\chi-\sin^2\chi)\sin\Omega t\nonumber\\
&&-\theta_{xy}A^{b}_{xy}\sin2\chi\sin\Omega t+\theta_{zx}A^{b}_{yz}\sin\chi\cos\Omega t-\theta_{zx}A^{b}_{xy}\cos\chi\cos\Omega t\nonumber\\
&&+\theta_{yz}A^{b}_{zx}\sin\chi\cos\Omega t-\theta_{xy}A^{b}_{zx}\cos\chi\cos\Omega t\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{XX}-c_{YY}&=&\alpha\{-\theta_{tx}A^{b}_{tx}\cos^2\chi(\cos^2\Omega t-\sin^2\Omega t)-\theta_{tx}A^{b}_{ty}\cos\chi\sin2\Omega t \nonumber\\ &&+\frac{1}{2}\theta_{tx}A^{b}_{tz}\sin2\chi(\cos^2\Omega t-\sin^2\Omega t)-\theta_{ty}A^{b}_{tx}\cos\chi \sin2\Omega t\nonumber\\
&&-\theta_{ty}A^{b}_{ty}(\cos^2\Omega t-\sin^2\Omega t)-\theta_{ty}A^{b}_{tz}\sin\chi \sin2\Omega t\nonumber\\
&&+\frac{1}{2}\theta_{tz}A^{b}_{tx}\sin2\chi (\cos^2\Omega t-\sin^2\Omega t)-\theta_{tz}A^{b}_{ty}\sin\chi \sin2\Omega t\nonumber\\
&&-\theta_{tz}A^{b}_{tz}\sin^2\chi(\cos^2\Omega t- \sin^2\Omega t)+\theta_{yz}A^{b}_{yz}\cos^2\chi( \cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&-\theta_{yz}A^{b}_{zx}\cos\chi \sin2\Omega t+\frac{1}{2}\theta_{yz}A^{b}_{xy}\sin2\chi (\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&+\frac{1}{2}\theta_{zx}A^{b}_{yz}\cos\chi \sin2\Omega t-\theta_{zx}A^{b}_{zx}(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&-\theta_{zx}A^{b}_{xy}\sin\chi \sin2\Omega t+\frac{1}{2}\theta_{xy}A^{b}_{yz}\sin2\chi(\cos^2\Omega t-\sin^2\Omega t)\nonumber\\
&&-\theta_{xy}A^{b}_{zx}\sin\chi\sin2\Omega t+\theta_{xy}A^{b}_{xy}\sin^2\chi( \cos^2\Omega t-\sin^2\Omega t)\},\nonumber\\\end{aligned}$$
$$\begin{aligned}
c_{Q}&=&\alpha\{\theta_{tx}A^{b}_{tx}(\cos^2\chi-2\sin^2\chi)+\frac{3}{2}\theta_{tx}A^{b}_{tz}\sin2\chi+\theta_{ty}A^{b}_{ty}\nonumber\\
&&+\frac{3}{2}\theta_{tz}A^{b}_{tx}\sin2\chi+\theta_{tz}A^{b}_{tz}(\sin^2\chi-2\cos^2\chi)+\theta_{yz}A^{b}_{yz}(\cos^2\chi-2\sin^2\chi)\nonumber\\
&&+\frac{3}{2}\theta_{yz}A^{b}_{xy}\sin2\chi+\theta_{zx}A^{b}_{zx}+\frac{3}{2}\theta_{xy}A^{b}_{yz}\sin2\chi+\theta_{xy}A^{b}_{xy}(\sin^2\chi-2\cos^2\chi)\},\nonumber\\\end{aligned}$$
where $\alpha=-\frac{3}{2}g\sin\theta_{\omega}$. The components of $d_{\mu\nu}$ are also the same as $c_{\mu\nu}$ components except for replacing $\alpha$ by $\beta$, which is equal to $-\frac{1}{2}g\sin\theta_{\omega}$.
- Higgs sector
In this sector the time dependence of the LV parameters are $$\begin{aligned}
(k_{\phi B})_{TX}&=&\alpha\theta_{TX}=\alpha\{\cos\chi\cos\Omega t \theta_{tx}-\sin\Omega t \theta_{ty}+\sin\chi\cos\Omega t \theta_{tz}\},\nonumber\\\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{TY}&=&\alpha\theta_{TY}=\alpha\{\cos\chi\sin\Omega t \theta_{tx}-\cos\Omega t \theta_{ty}+\sin\chi\sin\Omega t \theta_{tz}\},\nonumber\\\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{TZ}&=&\alpha\theta_{TZ}=\alpha\{-\sin\chi \theta_{tx}+\cos\chi \theta_{tz}\},\nonumber\\\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{YZ}&=&\alpha\theta_{YZ}=\alpha\{\cos\chi\cos\Omega t \theta_{yz}-\sin\Omega t \theta_{zx}+\sin\chi\cos\Omega t \theta_{xy}\},\nonumber\\\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{ZX}&=&\alpha\theta_{ZX}=\alpha\{\cos\chi\sin\Omega t \theta_{yz}-\cos\Omega t \theta_{zx}+\sin\chi\sin\Omega t \theta_{xy}\},\nonumber\\\end{aligned}$$ $$\begin{aligned}
(k_{\phi B})_{XY}&=&\alpha\theta_{XY}=\alpha\{-\sin\chi \theta_{yz}+\cos\chi \theta_{xy}\},\nonumber\\\end{aligned}$$ where $\alpha=-\frac{1}{2}\mu^2\sqrt{{g'}^2+g^2}$. The components of $(k_{\phi W})_{\mu\nu}$ are the same as $(k_{\phi B})_{\mu\nu}$ but replacing $\alpha$ with $\beta=-\frac{\sqrt{2}}{2}g\mu^2$.
Location dependence of LV parameters
====================================
The time averaging of the obtained LV parameters in Appendix B leads to their location dependencies as follows:
- Fermion sector
$$\begin{aligned}
\overline{c}_{TT}&=&{c}_{TT},\nonumber\\\end{aligned}$$
$$\begin{aligned}
\overline{c}_{XX}&=&\alpha\{-\theta_{tx}A^{b}_{tx}(\sin^2\chi+\frac{1}{2}\cos^2\chi) +\frac{1}{4}\theta_{tx}A^{b}_{tz}\sin2\chi\nonumber\\
&-&\frac{1}{2}\theta_{ty}A^{b}_{ty}+\frac{1}{4}\theta_{tz}A^{b}_{tx}\sin2\chi \nonumber\\
&-&\theta_{tz}A^{b}_{tz}(\cos^2\chi+\frac{1}{2}\sin^2\chi )+\frac{1}{2}\theta_{yz}A^{b}_{yz}\cos^2\chi \nonumber\\
&+&\frac{1}{4}\theta_{yz}A^{b}_{xy}\sin2\chi +\frac{1}{2}\theta_{zx}A^{b}_{zx}\nonumber\\
&+&\frac{1}{4}\theta_{xy}A^{b}_{yz}\sin2\chi+\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin^2\chi\},\end{aligned}$$
$$\begin{aligned}
\overline{c}_{YY}&=&\alpha\{-\theta_{tx}A^{b}_{tx}(\sin^2\chi+\frac{1}{2}\cos^2\chi)+\frac{1}{4}\theta_{tx}A^{b}_{tz}\sin2\chi \nonumber\\
&&-\frac{1}{2}\theta_{ty}A^{b}_{ty}+\frac{1}{4}\theta_{tz}A^{b}_{tx}\sin2\chi \nonumber\\
&&-\theta_{tz}A^{b}_{tz}(\cos^2\chi +\frac{1}{2}\sin^2\chi )+\frac{1}{2}\theta_{yz}A^{b}_{yz}\cos^2\chi \nonumber\\
&&+\frac{1}{4}\theta_{yz}A^{b}_{xy}\sin2\chi +\frac{1}{2}\theta_{zx}A^{b}_{zx}\nonumber\\
&&+\frac{1}{4}\theta_{xy}A^{b}_{yz}\sin2\chi+\frac{1}{2}\theta_{xy}A^{b}_{xy}\sin^2\chi\},\end{aligned}$$
$$\begin{aligned}
\overline{c}_{ZZ}&=&c_{ZZ},\end{aligned}$$
$$\begin{aligned}
\overline{c}_{XY}&=&\alpha\{-\frac{1}{2}\theta_{tx}A^{b}_{ty}\cos\chi+\frac{1}{2}\theta_{ty}A^{b}_{tx}\cos\chi+\frac{1}{2}\theta_{ty}A^{b}_{tz}\sin\chi\nonumber\\
&&-\frac{1}{2}\theta_{tz}A^{b}_{ty}\sin\chi-\frac{1}{2}\theta_{yz}A^{b}_{zx}\cos\chi+\frac{1}{2}\theta_{zx}A^{b}_{yz}\cos\chi\nonumber\\
&&+\frac{1}{2}\theta_{zx}A^{b}_{xy}\sin\chi-\frac{1}{2}\theta_{xy}A^{b}_{zx}\sin\chi\},\end{aligned}$$
$$\begin{aligned}
\overline{c}_{YX}&=&\alpha\{-\frac{1}{2}\theta_{ty}A^{b}_{tx}\cos\chi+\frac{1}{2}\theta_{tx}A^{b}_{ty}\cos\chi+\frac{1}{2}\theta_{tz}A^{b}_{ty}\sin\chi\nonumber\\
&&-\frac{1}{2}\theta_{ty}A^{b}_{tz}\sin\chi-\frac{1}{2}\theta_{zx}A^{b}_{yz}\cos\chi+\frac{1}{2}\theta_{yz}A^{b}_{zx}\cos\chi\nonumber\\
&&+\frac{1}{2}\theta_{xy}A^{b}_{zx}\sin\chi-\frac{1}{2}\theta_{zx}A^{b}_{xy}\sin\chi\},\end{aligned}$$
$$\begin{aligned}
\overline{c}_{Q}&=c_{Q}.\end{aligned}$$
It should be noted that the other components and their combinations have vanished with time averaging.
- Higgs sector
In the Higgs sector, the nonvanishing location dependencies can be obtained as follows: $$\begin{aligned}
\overline{(k_{\phi B})}_{TZ}&=&{(k_{\phi B})}_{TZ},\nonumber\\\end{aligned}$$ $$\begin{aligned}
\overline{(k_{\phi B})}_{XY}&=&{(k_{\phi B})}_{XY},\nonumber\\\end{aligned}$$ in which $\alpha=-\frac{1}{2}\mu^2\sqrt{{g'}^2+g^2}$. The location dependence of the other Higgs coefficients $\overline{(k_{\phi W})}$ and $\overline{(k^A_{\phi \phi})}$ are the same as $\overline{(k_{\phi B})}$ except replacing $\alpha$ with $\beta=-\frac{\sqrt{2}}{2}g\mu^2$ and $\gamma=-2\lambda v^2$, respectively.
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[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'The paper is concerned with the development of a gravitational field theory having locally a covariant version of the Galilei group. We show that this Galilean gravity can be used to study the advance of perihelion of a planet, following in parallel with the result of the (relativistic) theory of general relativity in the post-Newtonian approximation.'
author:
- 'S. C. Ulhoa'
- 'Faqir C. Khanna'
- 'A. E. Santana'
title: Galilean Covariance and The Gravitational Field
---
Introduction
============
Since the birth of general relativity, several studies have been addressing the problem of an analogous (n-form) formulation for the non-relativistic theory of gravitation [@Havas; @kunzle1; @kunzle; @Duval; @Carter1; @Carter2; @Carter3], which has as a fundamental structure the Galilei group. Beyond that, the interest in such a covariant description of the Galilei physics lies in the fact that some phenomena are restricted to Galilean regime. A known example is the superfluidity phenomenon, existing at low velocity, only [@Landau1]. Particularly in cosmology, in order to understand the large scale structures of the universe, the Newtonian gravity is required. Besides that, the rotation curve of galaxies is obtained with a Newtonian formalism [@Brihaye], which was the first step to point to the existence of dark matter [@Persic; @Gallagher; @Faber]. In a general sense, the Newtonian gravity theory is a natural choice to verify some insights on gravitation; thus if new ideas emerge from the development of general relativity, or even from a better theory than that, it has to be tested in order to reproduce the known results. In this sense, a geometric formulation of gravity based on the Galilei group (a Galilei gravity theory) may be of interest; and this is one of our goal here.
For such a purpose, it is important to develop a covariant form of Galilean transformations, since Galilei group acts as the symmetry group of Newtonian theory [@kunzle1; @Duval]. This approach has been achieved by considering the space-time transformation in the light-cone of a 5-dimensional Minkowski space-time [@Duval; @taka1; @taka2; @taka3; @taka4]. The 5-momentum vector is interpreted physically by considering 3 components describing the Euclidian momentum; one component standing for energy and the fifth component describing mass. In terms of space canonical coordinates, one has three components for the space coordinates; one coordinate for time and the fifth component is associated to velocity. The consequence has been several developments for the non-relativistic classical and quantum field theory [@taka1; @taka2; @taka3; @taka4; @taka5; @taka6; @Montigny; @Santos; @Kobayashi].
In this context of non-relativistic covariant physics, Duval et al [@kunzle] have addressed the problem of the gravitational field, using Bargmann structures, rather than Galilei group. In another direction, Carter and Chamel [@Carter1] adapted the procedures used in general relativity for application in purely Newtonian framework in order to provide new insights other than that of 3+1 decomposition of space-time. The formalism have been applied in the construction of a Newtonian fluid model to treat effects of superfluidity in neutrons stars [@Carter2; @Carter3]. Here we follow a different perspective, avoiding the use of such a decomposition, since we work in the five dimensional space exploring the Galilei group. We develop a geometric description of a Galilean gravity parallelizing the usual general relativity. As an application we study the advance of perihelion of a planet and compare it to the results derived in the post-Newtonian version of the theory of general relativity.
Although our approach is quite close to the general relativity, they differ from each other by the fact that one is locally Lorentzian and the other is Galilean. The structure is the same and the equations will assume a similar tensor form, but the physical meaning is different. Since Galilei transformations are described in five dimensions as linear, similar to the Lorentz group, the tensor formalism developed in the context of general relativity, may be used. As a central result, our conclusions are genuine manifestations of the Galilean gravity and do not follow approximations of any kind of Einstein’s equations, as it is the case, for example, in the post-Newtonian approximation [@Weinberg], which is an expansion in terms of $1/c$, where $c$ is the speed of light.
The paper is organized in the following way. In section 2, we set forth the notation and discuss briefly some aspects of the covariant formulation of Galilean transformations. In section 3 we establish a geometrical formulation of Galilean gravity and in section 4, we apply it to the case of Schwarzschild-like line element to calculate the precession of the perihelion of a planet. In section 5, we present some concluding remarks.
Covariant Galilean Transformations
==================================
The Galilean transformations are given by $$\begin{aligned}
\textbf{x}'&=& R\textbf{x}-\textbf{V}t+\textbf{a} \nonumber\\
t'&=&t+b\,, \label{0}\end{aligned}$$ where $R$ is a 3-dimensional Euclidian rotation, $\textbf{V}$ is the relative velocity defining the Galilei boost, $\textbf{a}$ stands for a space translations and $b$ a time translation. In this realm of Galilean symmetries describing low velocities process, one can introduces a linear space-time tensor structure by noticing the following.
In non-relativistic physics, the dispersion relation for a free non-relativistic particle is given by $E=\mathbf{p}^{2}/2m,$ where $E$ is the energy, $\mathbf{p}$ is the 3-dimensional momentum and $m$ is the mass. This dispersion relation can also be written as $$\mathbf{p}^{2}-2mH=0. \label{disp5}$$ Now let us consider the physical observable describing momentum as a quantity consisting of five entries, that is $p^{\mu
}=(\mathbf{p},p^{4},p^{5}),$ where $\mu =1,...,5, $ $\mathbf{p}$ is standing for the 3-vector momentum, $p^{4}=E/c'$ is the energy, and $p^{5}=c'm$ is mass. Here $c'$ is a constant with units of velocity. We take $c'=1$. Using this notation, and in order to recover Eq. (\[disp5\]), we write a general 5-dimensional dispersion relation, i.e. $p_{\mu }p^{\mu }=p^{\mu }p^{\nu
}{\eta}_{\mu \nu }=\mathbf{p}^{2}-2p^{4}p^{5}=k^{2},$ where $$\eta_{\mu\nu}= \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & -1 & 0 \\
\end{array}
\right) \,. \label{1}$$ This is taken as a metric tensor, that has been introduced in different ways in the literature, in particular, it was obtained in a (1+1) theory of gravitation by Cangemi and Jackiw [@Jackiw].
Let us define the set of canonical coordinates associated to $p^{\mu }$, by writing a 5-vector in $\mathcal M$ as $q^{\mu }=(\mathbf{q}%
,q^{4},q^{5}).$ The entries in $q^{\mu }$ are physically interpreted as follows: $\mathbf{q}$ is the canonical coordinate attached to $\mathbf{p}$; $q^{4}$ is the canonical coordinate associated to $E,$ and so it can be considered as the time coordinate; $q^{5}$ is the canonical coordinate associated to $m$, and is explicitly given in terms of $\mathbf{q}$ and $%
q^{4}$ according to the corresponding dispersion relation, leading to $%
q_{\mu }q^{\mu }=q^{\mu }q^{\nu }{\eta}_{\mu \nu }=\mathbf{q}%
^{2}-2q^{4}q^{5}=s^{2}.$ Since $p_{\mu }p^{\mu }=0$, we have to take $s=0$, leading to $q^{5}={\mathbf q}^2/2t$; or infinitesimally, we obtain $\delta q^{5}={\mathbf v}\cdot \delta
{\mathbf q}/2$. Therefore the fifth component is basically defined by the velocity.
Let us study in more detail the content of canonical coordinates, by introducing a symplectic structure in the cotangent bundle $T^{\ast }{\mathcal M}$, through the 2-form $\omega $ , $$\omega ={\eta}^{\mu \nu }dq_{\mu }\wedge dp_{v},\quad \,\,\mu
=1,2,\dots 5. \label{c4}$$ Defining the vector field, $$X_{f}=\frac{\partial f}{\partial p_{\mu }}\frac{\partial }{\partial q^{\mu }}%
-\frac{\partial f}{\partial q^{\mu }}\frac{\partial }{\partial
p_{\mu }},$$ where $f$ is a $C^{\infty }$ function in the 10-dimensional (phase space) manifold $\Omega $ with coordinates $(q^{\mu },p^{\nu })$, we have $$\begin{aligned}
\omega (X_{f},X_{h}) &=&\{f,h\} \\
\ &=&dq(X_{f})dp(X_{h})-dp(X_{f})dq(X_{h}), \\
\ &=&{\eta}^{\mu \nu }(\frac{\partial f}{\partial q^{\mu }}\frac{\partial g}{%
\partial p^{\nu }}-\frac{\partial g}{\partial q^{\mu }}\frac{\partial f}{%
\partial p^{\nu }})\ ,\end{aligned}$$ where $\{f,g\}$ is the Poisson bracket. Observe that, since $w(X_{h})=dh$, we have $\{f,h\}=df(X_{h})=\left\langle
df,X_{h}\right\rangle $.
Defining a flow by $$\partial _{\mu }f=-X_{p_{\mu }}f, \label{c335}$$ where $f(q,p)$ is a real ($C^{\infty })$ density distribution function in $%
\Omega $, then, in terms of components, we have from Eq. (\[c335\]), $$\begin{aligned}
\partial _{i}f &=&-X_{p_{i}}f\mapsto \partial _{i}f=\{p_{i},f\}, \label{c51}
\\
\partial _{4}f &=&-X_{p_{4}}f\mapsto \partial _{t}f=\{H,f\}, \label{c52} \\
\partial _{5}f &=&-X_{p_{5}}f\mapsto \partial _{5}f=0. \label{c53}\end{aligned}$$ Consistency relations are given by Eq. (\[c51\]) and (\[c53\]), while Eq. (\[c52\]) describes the Liouville equation. In this way we recover the classical theory in the Liouville-Poisson representation.
Let us now turn our attention to the set of linear non-homogeneous transformations in $\mathcal M$ of type $\overline{q%
}^{\mu }=\Lambda_{\ \nu }^{\mu }q^{\nu }+a^{\mu }\ $, leaving $(dq^{\mu }dq_{\mu }^{\prime })$ invariant. In addition, we consider transformations connected to the identity, such that$|\Lambda|=1$. In this case, for infinitesimal transformations we have $\Lambda_{\ \nu }^{\mu }=\delta _{\ \nu }^{\mu }+\epsilon
_{\ \nu }^{\mu }$. Then we identify 15 generators of transformations. Using the definition $$\left. \widehat{K}_{\alpha }=i\frac{\partial \overline{q}^{\mu
}}{\partial \alpha }\right| _{\alpha =0}\frac{\partial }{\partial
q^{\mu }}, \label{e12}$$ where $K_{\alpha }$ is the generator associated with the group parameter $%
\alpha $ (which also labels the group generators), we have $$\begin{aligned}
\widehat{J}_{3} &=&-i(\,q^{1}\partial _{2}-\,x^{2}\partial _{1}),
\label{e13} \\
\widehat{J}_{1} &=&-i(\,q^{2}\partial _{3}-\,q^{3}\partial _{2}),
\label{e14} \\
\widehat{J}_{2} &=&-i(\,q^{3}\partial _{1}-\,q^{1}\partial _{3}),
\label{e15} \\
\widehat{G}_{i} &=&\,i(\,q^{4}\partial _{i}+\,q^{i}\partial _{5}),
\label{e16} \\
\widehat{C}_{i} &=&\,i(\,q^{5}\partial _{i}+\,q^{i}\partial _{4}),
\label{e17} \\
\widehat{D} &=&\,\,\,i(\,q^{4}\partial _{4}-\,q^{5}\partial _{5}),
\label{e18} \\
\widehat{P}_{\mu } &=&\,\,\,i\partial _{\mu }, \label{e19}\end{aligned}$$ where $i=1,2,3$ and $\mu =1,2,...,5$. These generators satisfy the following commutation relations: $$\begin{aligned}
\lbrack \widehat{M}_{\mu \nu },\widehat{M}_{\rho \sigma }]
&=&-i[\eta _{\nu \rho }\widehat{M}_{\mu \sigma }-\eta _{\mu \rho
}\widehat{M}_{\nu \sigma
}+\eta _{\mu \sigma }\widehat{M}_{\nu \rho }- \nonumber\\
&-&\eta _{\nu \sigma }\widehat{M}_{\mu \rho }], \label{e25} \\
\lbrack \widehat{P}_{\mu },\widehat{M}_{\rho \sigma }] &=&-i[\eta
_{\mu \rho
}\widehat{P}_{\sigma }-\eta _{\mu \sigma }\widehat{P}_{\rho }], \\
\lbrack \widehat{P}_{\mu },\widehat{P}_{\nu }] &=&0, \label{e27}\end{aligned}$$ where $\widehat{M}_{\alpha \beta }$ ($\alpha ,\beta =1,...,5$) are defined by
$$\begin{aligned}
\widehat{M}_{ij} &=&-\widehat{M}_{ji}=\varepsilon
_{ijk}\widehat{J}_{k},
\label{e21} \\
\widehat{M}_{5i} &=&-\widehat{M}_{i5}=\widehat{G}_{i}, \label{22} \\
\widehat{M}_{4i} &=&-\widehat{M}_{i4}=\widehat{C}_{i}, \label{e23} \\
\widehat{M}_{54} &=&-\widehat{M}_{45}=\widehat{D}. \label{e24}\end{aligned}$$
The commutation relations given in Eqs. (\[e25\])–(\[e27\]) is a Lie algebra, that we denote by $\mathbf{g}$. A subalgebra of $\mathbf{g}$ is $$\begin{aligned}
\lbrack \widehat{L}_{i},\widehat{L}_{j}] &=&i\varepsilon _{ijk}\widehat{L}%
_{k},\,\,\,[\widehat{L}_{i},\widehat{P}_{j}]=i\varepsilon _{ijk}\widehat{P}%
_{k},\,\,\,[\widehat{L}_{i},\widehat{B}_{j}]=i\varepsilon _{ijk}\widehat{B}%
_{k}, \nonumber \\
\lbrack \widehat{B}_{i},\widehat{P}_{4}] &=&i\widehat{P}_{i},\,\,\,[\widehat{%
B}_{i},\widehat{P}_{j}]=i\widehat{P}_{5}\delta _{ij}, \label{alg2}\end{aligned}$$ corresponding to the Galilei-Lie algebra with the usual central charge $\widehat{P}_{5}$ describing mass. Notice that here the central charge arises naturally from the isometry in 5-dimensions.
The dispersion relation $p_\mu p^\mu=0$ defines a Galilean vector in the light-cone. However, in a more general case we have $$\begin{aligned}
p_\mu p^\mu&=&p^2-2mE=k^2 \nonumber\\
E+k^2/2m&=&\frac{p^2}{2m}. \label{3.1}\end{aligned}$$ The constant $k^2$ is absorbed into the energy by means of the definition $E'=E+k^2/2m$. Therefore, we recover the dispersion relation $E'=p^2/2m$, which is physically consistent. Then we can work with $k\neq 0.$
Geometric Approach to Galilean Gravity
======================================
In order to generalize this formalism to a curved Galilean space-time, we introduce Galilean tensor. Considering $$\frac{\partial x^{\mu}\,'}{\partial x^{\nu}}=\Lambda^{\mu}\,_{\nu}
\,,\label{3.2}$$ where $\Lambda^{\mu}\,_\nu $ is given explicitly by
$$\left(
\begin{array}{c}
\textbf{x}' \\
x^4\,' \\
x^5\,' \\
\end{array}
\right)= \left(
\begin{array}{ccc}
R & 0 & -\textbf{V} \\
-\textbf{V}\cdot R & 1 & \frac{1}{2}V^2 \\
0 & 0 & 1 \\
\end{array}
\right)\,\left(
\begin{array}{c}
\textbf{x} \\
x^4 \\
x^5 \\
\end{array}
\right)
\,, \label{3.3}$$
we define covariant and contravariant components of tensors as usual. Taking a non-flat manifold where locally the metric is $\eta$, we define a covariant derivative as $$\nabla_\mu
X^{\nu}=\partial_{\mu}X^{\nu}+\Gamma^{\nu}_{\lambda\mu}X^{\lambda}
\,,\label{3.4}$$ where $\Gamma^{\nu}_{\lambda\mu}$ is a connection that stipulates the nature of Galilean space-time. The covariant derivative of a scalar reduces to the normal derivative.
Let us give a definition for the curvature tensor. The current idea to define curvature resides on an intuitive concept. If we consider a vector field $X^\mu$ on a closed circuit on a manifold and if there is any change in direction of $X^\mu$ after a round around the circuit then we say that this manifold is curved. Mathematically it is stated as $$\nabla_{[\mu}
\nabla_{\lambda]}X^{\nu}=\frac{1}{2}R^{\nu}\,_{\gamma\mu\lambda}X^{\gamma}
\,,\label{3.5}$$ where $R^{\nu}\,_{\gamma\mu\lambda}$ is the curvature tensor. We assume that when there is no gravitational field, the curvature tensor vanishes.
The metric tensor is a covariant tensor of rank 2. It is used to define distances and lengths of vectors. The infinitesimal distance between two points $x^a$ and $x^a+dx^a$ in curved manifold defined from $\mathcal M$ is defined by $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}\,,\label{3.6}$$ where $g_{\mu\nu}$ is the metric tensor. The relation (\[3.6\]) represents the line element as well. We have to notice that the metric $\eta_{\mu\nu}$ in (\[1\]) defines a flat line element. The imposition that the covariant derivative of metric is zero yields the following expression for the connection $$\Gamma^{\nu}_{\lambda\mu}=\frac{1}{2}g^{\nu\delta}(\partial_{\lambda}
g_{\delta\mu}+\partial_{\mu}g_{\delta\lambda}-\partial_{\delta}g_{\lambda\mu})\,.\label{3.7}$$ When the connection is written as in Eq. (\[3.7\]) the manifold is said to be an affine manifold.
The curvature tensor defined in affine manifold has the following properties: $$\begin{aligned}
R_{\mu\nu\lambda\gamma}=-R_{\mu\nu\gamma\lambda}=-R_{\nu\mu\lambda\gamma}&=&
R_{\lambda\gamma\mu\nu}\nonumber \\
R_{\mu\nu\lambda\gamma}+R_{\mu\gamma\nu\lambda}+R_{\mu\lambda\gamma\nu}&\equiv&0.\label{3.8}\end{aligned}$$ These properties are derived from Eq. (\[3.5\]). If we perform a contraction of the indices of the curvature tensor then it is possible to define the Galilei-invariant curvature scalar $$R=g^{\mu\nu}g^{\gamma\lambda}R_{\gamma\mu\lambda\nu}.\label{3.9}$$
To generate the field equations, we write a Lagrangian invariant under Galilean transformations. A natural candidate is the curvature scalar, then the action in a general form is $$I=\int_\Omega d\Omega (\sqrt{-g}R+kL_m)\,,\label{3.10}$$ where $g=det g_{\mu\nu}$, $k$ is the coupling constant, $L_m$ is a matter lagrangian density and $d\Omega$ is the 5-dimensional element of volume. Varying the action with respect to $g_{\mu\nu}$, we obtain $$\begin{aligned}
\frac{\delta(\sqrt{-g}R)}{\delta
g_{\mu\nu}}&=&-\sqrt{-g}(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R)\,,\nonumber
\\
\frac{\delta L_m}{\delta
g_{\mu\nu}}&=&\sqrt{-g}T^{\mu\nu},\label{3.11}\end{aligned}$$ where the latter equation defines the energy-momentum tensor of matter fields and $R_{\mu\nu}=R^{\lambda}\,_{\mu\lambda\nu}$. Thus the field equation becomes $$R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=kT^{\mu\nu}\,.\label{3.12}$$ These equations have the same form as for the General Relativity equations, since the Galilean transformations are written in a way similar to Lorentz transformations. Let us note that, the quantity $k$ is just a coupling constant between the Galilean gravity and matter fields (it has nothing to do with Einstein’s constant).
We have to note that the equations in (\[3.12\]) with $\mu=4$, i.e. the equations $$R^{4}\,_{\nu}-\frac{1}{2}\delta^{4}_{\nu}R=kT^{4}\,_{\nu}\,,\label{3.13}$$ contain only the first order derivative of the components of $g_{\mu\nu}$ with respect to time. Actually in (\[3.13\]) the components of the form $R_{4i4j}$ drop out, where the indices i and j run from 1 to 3 and assume the value 5 as well. In face of this we note that some components of time derivative of the metric tensor are associated with the freedom of the choice of the system of coordinates. So we have to specify in a particular coordinate system only the time derivatives of $g_{ij}$ as initial conditions. Therefore we see the ten equations $$R^{i}\,_{j}-\frac{1}{2}\delta^{i}_{j}R=kT^{i}\,_{j}\,,\label{3.14}$$ where the indices i and j run from 1 to 3 and assume the value 5 as well, as dynamical equations and the five equations (\[3.13\]) as constraint equations.
The Schwarzschild-like Line Element: The Advance of the Perihelion of a Planet
==============================================================================
Since the metric determines every feature of a system described by a geometrical approach, we intend to get a spherically symmetric metric [@Dinverno]. Then we introduce a Galilean Schwarzschild solution. The Galilean Schwarzschild line element is defined by $$ds^2=f^{-1}(r)dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2-f(r)dsdt\,,
\label{4}$$ where as usual $f=1-2M/r$. This metric describes a system with spherical symmetry. In the following we consider this line element to study the movement of a planet.
The space-time in the exterior region of a massive body can be described by the line element in Eq. (\[4\]). For example, for the system composed of Sun and Mercury, only force acting on this system is the gravitational one. Therefore, the movement will be geodesic. A two-body system can be described by means of the reduced mass as a consequence we deal with a one body system. We perform the calculations considering the mass in the line element as the reduced mass.
The geodesic movement can be described by means a variational principle where the action is the interval between two events in space-time. Then the equation of movement is given by Euler-Lagrange equation, i.e. $$\begin{aligned}
\delta s&=&\int\delta ds=\int\delta L d\tau\nonumber\\
&=&\int\delta
{(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})}^{1/2}d\tau=0\,,\label{4.01}\end{aligned}$$ where the dot represents the derivative with respect to the proper time $\tau$. The meaning of proper time remain the same of that defined in General Relativity, once our theory share the same feature with respect to transformation of coordinates. In this context the proper time is
$$\tau=\int{(-g_{00})}^{1/2}dt\,, \label{4.02}$$
where the coordinate $t$ could not assume necessarily the meaning of time. Thus the meaning of each coordinate depends on which system of coordinates one is using.
Instead of working with $L={(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})}^{1/2}$, we consider the quantity defined by $$K=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\,,\label{4.1}$$ which obey the Euler-Lagrange equation as well. Of course $K$ can be set equal to a constant which takes the possible values 1, -1 or 0. This relation is completely similar to that given in Eq. (\[3.1\]) defined on the flat Galilean space-time. If we perform the sum in Eq. (\[4.1\]), the expression assumes the following form $$f^{-1}\dot{r}^2+r^2\dot{\theta}^2+r^2\sin^2\theta\dot{\phi}^2-2f\dot{t}\dot{s}=\alpha
\,.\label{4.2}$$
If we put $K$ into the Euler-Lagrange equation and consider this movement restricted to a plane ($\theta=\pi/2$), since the angular momentum is a constant, then we get for $\mu=(3,4,5)$ the following equations, respectively, $$r^2\dot{\phi}=h, \,\, f\dot{s}=\beta, \,\,
f\dot{t}=\kappa.\label{4.3}$$ These equations represent conservation laws. In fact, the quantities $\kappa$, $\beta$ and $h$ are related to energy, mass and angular momentum respectively. The relations given in Eq. (\[4.3\]) can be substituted into Eq. (\[4.2\]) with $\theta=\pi/2$, leading to $$f^{-1}\dot{r}^2+\frac{h^2}{r^2}-2\frac{\beta\kappa}{f}=\alpha
\,.\label{4.4}$$
At this point we find an equation for the trajectory by changing the variable in Eq. (\[4.4\]). We define $U=U(\phi)=\frac{1}{r}$; such that the derivative of $U$ with respect to $\phi$, which will be designated by $U'$, is equal to $h$ times the derivative of $r$ with respect to the proper time. The final equation is, $$U^{'}\,^{2}+fU^2-2\frac{\kappa\beta}{h^2}=\frac{\alpha f}{h^2}
\,.\label{4.5}$$ Taking the derivative of the above equation with respect to $\phi$ and remembering that $f=1-2M/r$, we obtain the trajectory equation, $$U^{''}+ U=3MU^2-\frac{\alpha M}{h^2}.\label{4.6}$$ Choosing $\alpha=-1$, we have $$U^{''}+ U= 3MU^2+\frac{M}{h^2} \,,\label{4.7}$$ which is the same equation obtained in General Relativity. In this context, however, there does not exist a relation of causality due to the constancy of velocity of light, since there is no such imposition in a Galilean theory. As a consequence, the advance of perihelion of a planet is given by the usual expression [@Landau] $$\delta\phi=6\pi\frac{M^2}{h^2} \,. \label{4.8}$$
It is important to rewrite Eq. (\[4.8\]) with the constants $G$ and $c'$. In this case we have $$\delta\phi=6\pi\frac{G^2M^2}{c^{'2}h^2}=\frac{24\pi^3a^2}{c^{'2}T^2(1-e^2)}
\,, \label{4.9}$$ where $c'$ is a typical velocity of the system to be fixed experimentally for the Galilean symmetry, $M$ is the reduced mass, $T$ is the period of the movement, $e$ is the eccentricity of the orbit and $a$ is semi-major axis of the ellipse. We have to note that $\delta\phi$ is dimensionless.
In order to get the well known Newtonian equation for the planetary movement under the influence of a force proportional to the inverse of square radius, we have to rewrite Eq. (\[4.7\]) with the constants $G$ and $c'$, such that $$U^{''}+ U=G\frac{M}{h^2}+ 3\frac{G}{c^{'2}}MU^2 .\label{4.10}$$ This assumes the post-Newtonian equation for the planetary movement, up to the second term on right-hand-side of above equation. This term is a correction of the classical equation and the parameter $c'$ can be taken experimentally. Therefore the post-Newtonian equation arises from the 5-dimensional Galilean gravity if we consider the parameter to adjust dimensionality greater than the other parameters in Eq. (\[4.10\]); thus establishing a weak field regime. It is important to notice that Eq. (\[4.9\]) gives a precession of $43''/century$ for planet Mercury, since the parameter $c'$ is taken to be velocity of light c. The consistence of this choice can be established by noticing that, taking the Lorentz definition of energy and the limit of $v/c$ much less than 1, then energy and momentum are related by $E=p^2/(2m) +mc^2$, corresponding to a dispersion relation as given in Section 2, $p_{\mu }p^{\mu }=p^{\mu }p^{\nu
}{\eta}_{\mu \nu }=\mathbf{p}^{2}-2p^{4}p^{5}=k^{2},$ with $k^2=2(mc)^2$.
Conclusion
==========
We have established a geometric theory of gravity in the framework of Galilean symmetry. These transformations are taken in a covariant form and the flat space-time is defined with the Galilean metric. With curved space-time we have associated this to the gravitation, in parallel to the usual relativistic case. Then we construct a Schwarzschild-like line element, which is spherically symmetric, and apply it to the movement of a planet. As a result, we derive the post-Newtonian result of the general relativity in a covariant form. We have shown that our equation assumes the Newtonian form in the weak field regime. One basic physical new fact to be learned from all this is that, the advance of perihelion of a planet is a geometric effect that can be described fully in a Galilean covariant theory.
It would be of interest to analyze other related problems in this context of Galilean gravity, as for instance the bending of light, other gravitation models such as cosmological models [@Ehlers], the hamiltonian approach which could reveal more about the structure of field equations and a Galilean gauge theory. These aspects will be discussed elsewhere.
[**Acknowledgement**]{}
We thank R. Cuzinatto and P. Pompéia for helpful discussions. This work was supported by CNPq (of Brazil) and NSERC (of Canada).
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|
---
author:
- Alexandre GonÇalves and Karen Uhlenbeck
title: 'Moduli space theory for constant mean curvature surfaces immersed in space-forms'
---
Introduction
============
The study of constant mean curvature surfaces in a space-form has been an active field since the work of H. Hopf in the 1920’s and H.Liebmann in the years around 1900. The questions which are generally of interest are global questions of existence and uniqueness in complete 3-manifolds. We deal in this short paper on a question of existence and uniqueness with respect to the complex structure and the quadratic Hopf differential of a compact surface in a constant curvature 3-manifold which is not necessarily complete.Our final result applies only to the case of surfaces embedded in a local 3-dimensional space of constant curvature $-1$, where the mean curvature constant $c$ satisfies $|c|< 1$. However, the technique suggests some approaches to the more interesting cases, for example the work of Bryant [@B] and Kenmotsu [@K].
The Gauss-Codazzi equations for constant mean curvature immersions of a surface into a 3-dimensional space-form are a $3\times 3$ system of partial differential equations of mixed order. Once a complex structure is chosen, the equations break down into two equations. The Codazzi equation on the second fundamental form yields the Cauchy-Riemann equation for a holomorphic quadratic differential first noticed and used by Heintz Hopf [@H]. The second is a real non-linear single elliptic equation for the length function of the metric which comes from the Gauss curvature equation.
These equations can be approached via a number of techniques in partial differential equations. In this short note, we improve upon results obtained by assuming the Riemann surface structure and postulating a fixed quadratic differential representing the $(2,0)$ part of a second fundamental form as solving the Codazzi equations. This leaves the problem of solving the elliptic Gauss equation for the length function of the metric. By analyzing the Gauss and Codazzi equations together, we are able to reformulate the equations in a form which completely identifies all local solutions in the case of negative curvature. We prove that the moduli space of solutions to the Gauss-Codazzi equations for a Riemann surface of genus greater than one immersed with mean curvature constant and less than $1$, in a not necessarily complete 3-manifold of constant curvature $-1$ is parameterized by cohomology classes of $(0,2)$ differentials.
The result is similar and proved in the same fashion as the results in gauge theory in a paper of the first author [@G]. In fact, the details of how the computations change with the change in base-point $g$ is comlicated, but it is familiar to geometers from the variational formulation of the Yamabe problem and will not be repeated. An abstract proof could be constructed along the lines of the convexity theory used in the gauge theory literature to describe bundle extensions by authors Bradlow-Garcia-Prada [@B-G] and Daskalopolous-Uhlenbeck-Wentworth [@D-U-W]. Also, as presented here, the construction is not as natural as it would be if viewed from the point of view of these authors. However, the convexity theory fails in cases of positive curvature, whereas the reformulation of the Gauss-Codazzi equations as the Euler-Lagrange equations of a single variational problem has potential to contribute to the more interesting cases of zero or even positive curvature.
It is entertaining to note that the second author came across these equations more than twenty years ago in looking at the possibility that minimal surfaces could be used to parameterize quasi-Fuchsian hyperbolic 3-manifolds [@U]. Although there is a fairly good existence theory, it seems as if uniqueness is unlikely [@V].
The Variational formulation
===========================
We assume in this note that $X$ is a surface immersed with mean curvature $c$ in a 3-dimensional manifold $N$ of constant sectional curvature $k$. We do not assume this 3-manifold is complete. The induced metric and second fundamental form of $X \subset N$ satisfy a system of Gauss-Codazzi equations which can be analyzed in a series of steps as follows.
1\. The induced metric on $X$ is of the form $h = h_{z\bar z} dz\,d\bar z$ where $(z,\bar z)$ are local complex coordinates in a complex structure $X_\sigma$ on $X$.
2\. The second fundamental form $\gamma$ has the structure $$\gamma = \alpha_{zz}(dz)^2 + ch_{z\bar z} dz\,d\bar z + \alpha_{\bar z\bar z}
(d\bar z)^2\ .$$ Here $c$ is the (constant) mean curvature. $\gamma$ is symmetric with respect to the metric $h$, which implies the relationship $$\alpha_{zz} = \bar\alpha_{\bar z\bar z}\ .$$
3\. The (2,0) part of $\gamma$ is a holomorphic quadratic differential $\alpha$ on $X$.
4\. The induced (scalar) curvature $K$ of $X$ satisfies the Gauss equation. $$K = k+c^{2} - 2|\alpha|^{2} .$$ Here the norm of $\alpha$ is assumed to be taken with respect to the metric $h$. By rescaling, we may assume that $k = (-1,0,1)$. Our main results pertain to the case $\lambda = k+c^2 <0$.
The solution to steps 1–3 is completely understood, so an automatic procedure would be to fix the Riemann surface, the curvatures and the (2,0) part alpha of the second fundamental form and attempt to solve the Gauss equation for the metric. There is a variational formulation of this problem. This last step results in an equation which can be solved for small $\alpha$ by the techniques of Kazden and Warner [@K-W]. However, a slightly different variational problem arises when we solve for the second fundamental form and the metric in one step.
For convenience, we combine in a suggestive notation $\lambda = k+c^2$.
Fix the Riemann surface. We will choose as a base point the constant curvature metric on $X$ and a holomorphic quadratic differential. Let $\beta_O$ be the $(0,2)$ form on $X$ which is dual to the chosen holomorphic differential in the constant curvature metric. Note that the identification between the $(2,0)$ and $(0,2)$ form depends on the metric. Now we compute with respect to an arbitrary element $\beta = \beta_0 + \bar\partial f_0$ and fixed metric $g$ whose conformal class determines the Riemann surface. Let $K(g)$ denote the Gauss curvature of $g$. Calculations are always easier at the base metric, so we will want to have the freedom of changing it.
We call our functional $\D$ for Donaldson, as it is in reality a form of the Donaldson functional which appears in the construction of Hermitian-Einstein metrics in holomorphic vector bundles. Usually there is no explicit formula for this functional. Due to the fact that we are in line bundles and are looking at an abelian gauge theory, the functional is for us explicit. Note that it is well-defined up to an overall constant, which we have fixed by assuming the functional $\D$ is $0$ on the constant curvature metric and the dual to the chosen holomorphic quadratic differential. Note that the holomorphic quadratic differential constructed from the variational principle is not the one that we started with.
\[prop:gauss-codazzi\] The metric and holomorphic quadratic differential pair $$(h,\alpha) = (e^{2u}g ,e^{2u} (\beta^*+\partial f^*))$$ solve the Gauss-Codazzi system (3–4) if and only if $u:x\to c$, $f:x\to T^{1,0}x$ are critical maps for the functional: $$\D (u,f) = \iint \left[|\partial u|^2 + K(g)u + e^{2u}
(-\lambda/2 + |\beta + \bar\partial f|^2)\right]\,d\mu + C(g)\ .$$
This is a calculation. All the metrics, the covariant derivative $\partial = \bar\partial^*$ and the density are computed using the base metric $g$, although we suppress this in the statement of the theorem. We have included the constant $C(g,\beta)$, since we wish to make this computation $(g,\beta)$ independent. The value of $C(g)$ is determined by finding the value of the Donaldson functional at $g$ using our original choice of constant curvature metric and holomorphic quadratic differential.
Now the equation which arises from varying $f$ is a simple linear equation (in $f$, not $u$) $$\bar\partial \Big( e^{2u} (\beta^* +\partial f^*)\Big) =0\ .$$ This yields the holomorphic quadratic differential $$\alpha = e^{2u} (\beta^* + \partial f^*)\ .$$ The equation obtained by varying $u$ is the equation $$0 = - \partial^*\partial u + K(g) + e^{2u}( - \lambda + 2|\beta +
\bar\partial f|^2)\ .$$ Recall that in the new metric $h = e^{2u}g$ the curvature $K(h) =
e^{-2u} (K(g) - 2\partial\bar\partial u)$, so that the equation which arises when varying $u$ is indeed the desired equation for mean curvature. It can be written as $$K(h) = \lambda - 2e^{-4u}|\alpha|_g^{2}\ .$$ Note that the norm of $\alpha$ is correctly computed in the new metric $h$.
Fix $\Vol(X) = \iint e^{2u}\, d\mu =T$. Then the critical points of $$\hat \D(u,f) = \iint |\partial u|^2 + k(g) u + e^{2u}
|\beta + \bar\partial f|^2\,d\mu + C(g,\beta)$$ with respect to the constraint provide solutions of the Gauss-Codazzi equations (3–4) with an unknown Lagrange multiplier $\lambda$.
The actual utilization of this minimization principle seems quite delicate. One would need to employ the Moser inequality carefully. Moreover, in many cases one would be interested in saddle points rather than minima. However, this variational principle which fixes the cohomology class of $\beta$ has definite advantages over the one which fixes the holomorphic differential $\alpha$.
Solutions for $\lambda < 0$
===========================
The main result of this short note is the following theorem:
\[thm:main\] If $\lambda = k+c^2 <0$, there exists a unique solution to the Gauss Codazzi equations (3–4) for a fixed Riemann surface and a (0,2) cohomology class $[\beta] = \{\beta + \bar\partial f\}$.
We will show that for every solution to the Gauss Codazzi equations with $\lambda = k+c^2 < 0$, the Hessian of $\D$ is positive definite. For small $[\beta]$, there will be a solution near a constant negative curvature metric of curvature $\lambda$ on $X$ which can be found using an implicit function theorem. Openness follows from the invertibility of the Hessian. Closedness is a rather easy calculation which we leave to the reader, It is similar to proofs in the literature [@K-W],[@U].
Uniqueness follows since every solution can be connected to one with small cohomology class representative $[\beta]$.This leaves the important step of showing the positive definiteness of the Hessian to finish the proof.
\[th:hessian\] If $(u,f)$ is a critical point of $\D$, for a fixed Riemann surface and cohomology class $[\beta]$ with $\lambda < 0$, then the Hessian is positive definite.
We might as well make the calculation with the solution metric and holomorphic quadratic differential $\beta^*=\alpha$ as base-point. (Here is where our comments about basepoint pay off. These calculations in gauge theory are where the idea for the proof comes from).
Since $(0,0)$ is a critical point, we have $$K(g) = \lambda - 2|\beta|^2$$ and $$\partial \beta=0\ .$$ The Hessian is easy to compute. $$H(v,f) = 2\iint \left[ |\partial v|^2 - v^2 K(g)
+ 4v \Re \langle \beta, \partial f^*\rangle
+ \langle \bar \partial f,\partial f^*\rangle\right]\,d\mu\ .$$ We note that $\partial\beta =0$, so we may replace $$2\iint v\Re \langle \beta,\partial f^*\rangle\,d\mu
= -2 \iint \Re \langle \partial v\otimes f^*,\beta\rangle\,
d\mu\ .$$ Also $$\frac12 \iint \langle \bar\partial f,\partial f^*\rangle\,d\mu
= \frac12\iint \langle\bar\partial f,\partial f^*\rangle
- K(g) \langle f,f^*\rangle\,d\mu\ .$$
We have $K(g) = \lambda - 2|\beta|^2$ where $\lambda <0$. We rewrite the Hessian as $$\begin{split}
H(v,f) & = 2\iint \bigg[ |\partial v|^2 + v^2 (-\lambda + 2|\beta|^2)
+ 2v\Re \langle \beta,\partial f^*\rangle - 2\Re \langle \partial u\otimes
f^*,\beta\rangle\\
&\qquad + \frac12 \left( |\bar\partial f|^2 +\frac12 |\partial f|^2
+ |f|^2 (-\lambda + 2|\beta|^2)\right) \bigg]\,d\mu\cr
& > 2\iint \bigg[ |\partial v|^2 + 2|v|^2 |\beta|^2
+ \frac12 |\bar\partial f|^2 + |f|^2 |\beta|^2
- 2|v|\, |\beta|\, |\bar\partial f|
- 2|\partial v|\, |f|\, |\beta|\bigg] \,d\mu \cr
&\ge 0\ .
\end{split}$$ Note that we may allow $\lambda =0$ in all but extremely degenerate cases.
[B]{}
Bryant, Robert: Surfaces of mean curvature one in hyperbolic space, Asterisque [**154-155**]{} (l987), 321-353.
Bradlow, Steve and Garcia-Prado, Oscar: Higher Cohomology Triples and Holomorphic Extensions, Comm. Anal. Geom. [**3**]{} (1995), 421–465.
Daskalopolous, Georgios, Uhlenbeck, Karen and Wentworth, Richard: Moduli of extensions of holomorphic bundles over Kaehler manifolds, Comm. Anal. Geom. [**3**]{} (1995), 479–522.
Gonçalves, Alexandre: An Elliptic Non-Linear Equation on a Riemann Surface, to appear in Differential Geometry and its Applications (2007).
Hopf, Heintz: Differential Geometry in the Large, Lecture Notes in Mathematis [**1000**]{}, SpringerVerlag (l989).
: Hopf, Heintz: Ueber Flachen mit einer Relation zwischen den Haupkreumungen, Math Nach [**4**]{} (l951), 232-249.
: Kazdan, .J and Warner,F.: Curvature Functions for Compact 2-Manifolds, Ann. of Math. [**99**]{} (l964), 14-47.
Kenmotsu, Katsuei: Surfaces of constant mean curvature, Translation of Math Monographs [**221**]{}, Amer. Math Soc(2004).
Uhlenbeck, Karen: Closed minimal surfaces in hyperbolic 3 manifolds, Seminar on Minimal Submanifolds, ed. Bombieri, Ann. Math Studies [**103**]{}, 147–168.
Velling, John: Limits on prescribing the Hopf differential for minimal surfaces in $\mathbf H$, Commun Partial Diff Equations [**27**]{} (2002),2513–2525.
|
---
abstract: |
Apoptosis is one of the most basic biological processes. In apoptosis, tens of species are involved in many biochemical reactions with times scales of widely differing orders of magnitude. By the law of mass action, the process is mathematically described with a large and stiff system of ODEs (ordinary differential equations). The goal of this work is to simplify such systems of ODEs with the PEA (partial equilibrium approximation) method. In doing so, we propose a general framework of the PEA method together with some conditions, under which the PEA method can be justified rigorously. The main condition is the principle of detailed balance for fast reactions as a whole. With the justified method as a tool, we made many attempts via numerical tests to simplify the Fas-signaling pathway model due to Hua et al. (2005) and found that nine of reactions therein can be well regarded as relatively fast. This paper reports our simplification of Hua at el.’s model with the PEA method based on the fastness of the nine reactions, together with numerical results which confirm the reliability of our simplified model.\
[**Keywords**]{}: Partial equilibrium approximation, apoptosis, biochemical reactions, the principle of detailed balance, sensitivity analysis
author:
- 'Ya-Jing Huang[^1],Wen-An Yong[^2]'
title: 'Partial equilibrium approximations in Apoptosis\'
---
[GBK]{}[song]{}
[1.5]{}
Introduction
============
Apoptosis is one of the most basic biological phenomena. It is a cellular suicide route that allows for the selective removal of superfluous and potentially dangerous cells. This genetically controlled process ensures normal embryonic development, tissue homeostasis and normal immune-system function in multicellular organisms. On the other hand, defects in apoptosis may cause serious diseases such as cancer, autoimmunity, and neurodegeneration [@Hengartner; @Horvitz; @Thompson; @PK]. For these reasons, understanding the mechanism of apoptosis is of fundamental importance.
The apoptotic process involves tens of biological molecules (species), which react within tens of biochemical reactions with time scales of widely differing orders of magnitude. When the law of mass action [@Keener] is employed, it is described mathematically by a simultaneous system of tens of ordinary differential equations (ODEs). Such a large scale and stiff system of ODEs can hardly help us to understand the mechanism of the apoptosis. The goal of this work is to derive mathematically reliable simplifications of the large apoptosis system proposed by Hua et al. in [@Hua] for human Jurkat T cells. Two widely used methods for simplifying chemical kinetics are the Quasi Steady-State Approximation (QSSA) [@Benson; @Bowen], also called the Bodenstein method, and Partial Equilibrium Approximation (PEA) [@Ramshaw; @Rein; @Goussis]. The former assumes that the concentrations of transient intermediate species reach steady states and thereby the rate equations for the intermediate species are replaced with algebraic relations. On the other hand, the PEA simply takes the fast reactions in equilibrium. In this manner, the stiffness is removed and some algebraic constraints are obtained. For both methods, the algebraic relations can be used to reduce the number of the rate equations and consequently the chemical kinetics is simplified. An unexpected benefit of such simplifications is that less parameters are needed for the simplified models than for the original ones. This is good because the parameters are often not reliably known (see [@Ramshaw]).
The QSSA, PEA and their combinations have been extensively used to simplify chemical kinetics mechanisms for many years, with great success [@Schott; @Miller; @Pope; @PW; @Pe; @PR; @Smooke]. They were also used by Okazaki et al. in [@Okazaki] to simplify the large apoptosis system in [@Hua] (see comments below). However, these methods seem to lack a systematically mathematical justification. Recently, we pointed out that the PEA method can be rigorously justified for reversible reactions obeying the principle of detailed balance [@Walls; @Yong1], by using the singular perturbation theory of initial-value problems for ODEs [@Yong2]. Thus, our simplification will base on this justified PEA method and therefore is reliable.
As commented in [@HY], Okazaki et al.’s simplification seems baseless. In fact, when applying the QSSA method for the intermediate species Casp8$_2^*$:Casp3, Okazaki et al. assumed that the concentration sum of the intermediate species and Casp8$_2^*$ (an activated initiator caspase) was conserved and obtained the Michaelis-Menten equation for the product (see Appendix A of [@Okazaki]). The latter is only true if the activated initiator caspase does not participate in other reactions. However, this is not the case here because the activated caspase is simultaneously, instead of consecutively, involved in other reactions. Similar conservation assumptions were used for several steps. This is why we question the simplified model in [@Okazaki], although it is successful in some sense.
This paper is a continuation of our previous work [@HY]. In [@HY], we showed that two molecules (Smac and XIAP) involved in the apoptotic process, neglected in [@Okazaki], are not negligible in general. Then we applied the justified PEA method to obtain a very preliminary simplified model by assuming only six reversible reactions to be fast. In the present paper, we will use the PEA method to further simplify the large apoptosis system [@Hua]. To do this, we firstly verify the principle of detailed balance for more fast reactions as a whole. Having such a verification, the singular perturbation theory of initial-value problems for ODEs can be employed to derive our simplified models. Then we use numerical simulations to compare the new models with the original one and Okazaki et al.’s simplified model from various aspects, including accuracy, sensitivity and the M-D transition behavior [@Okazaki]. Moreover, we introduce a new quantity to evaluate our simplifications. All numerical results confirm the reliability of both our simplified models and the PEA method.
Let us remark that, thanks to its reliability, the justified PEA could be used as a tool to determine whether or not a reversible reaction is relatively fast. To this end, one could numerically compare the solution of partial equilibrium computation with that of the fully non-equilibrium computation. If the two solutions are close to each other, the reaction can be claimed to be relatively fast.
The paper is organized as follows. In Section 2 we present a general framework of the PEA method, together with some conditions under which the method can be justified rigorously. The apoptosis process is introduced in Section 3. The PEA method is used in Section 4 to simplify the apoptosis system by checking the principle of detailed balance for nine reversible reactions as a whole. Numerical simulations are reported in Section 5. Finally, the main results of this paper are summarized in Section 5.
The PEA Method
==============
In this section we first present a general framework of the PEA method, together with some conditions under which the method can be justified rigorously. Then we take the simplest system for enzyme inhibition as an example to show how to use our PEA method.
A general framework of the PEA method
-------------------------------------
Consider a system with $N$ chemical species $C_i (i = 1, 2, \cdots, N)$ participating in $M$ reactions $$\label{21}
a_1^p C_1 + a_2^p C_2 + \cdots + a_N^p C_N \autorightleftharpoons{$k_{+p}$}{$k_{-p}$} b_1^p C_1 + b_2^p C_2 + \cdots + b_N^p C_N$$ for $p = 1,2, \cdots ,M$. Here the non-negative integers $a_i^p$ and $b_i^p$ are the stoichiometric coefficients of the $i^{th}$-species in the $p^{th}$-reaction, and $k_{+p}$ and $k_{-p}$ are the respective forward and backward rate constants of the $p^{th}$-reaction. The reversibility means that both $k_{+p}$ and $k_{-p}$ are positive.
Denote by $u_i=[C_i](t)$ the concentration of the $i^{th}$-species $C_1$ at time $t$. According to the law of mass action [@Keener], the evolution equation for $u_i$ is $$\label{22}
\frac{du_i}{dt} = \sum\limits_{p = 1}^M (b_i^p - a_i^p )v_p$$ with $$v_p= {k_{+p} u_1^{a_1^p } u_2^{a_2^p } \cdots u_N^{a_N^p } - k_{-p} u_1^{b_1^p } u_2^{b_2^p } \cdots u_N^{b_N^p } }$$ being the reaction rate of the $p$-th reaction.
Suppose the first $M' (\leq M)$ reactions in (\[21\]) are much faster than others. Then the kinetic equations in (\[22\]) can be rewritten in the vectorial form: $$\label{23}
\begin{array}{l}
\displaystyle{ \frac{{dV}}{{dt}} = \frac{1}{\varepsilon} Q_1(V) + Q_2(V,Z)}, \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(V,Z)}.
\end{array}$$ Here $V$ is an $N'$-vector consisting of those $u_i$ so that the $i^{th}$-species participates in the fast reactions, $Z$ consists of the rest $u_i$, $\varepsilon$ is a small positive parameter characterizing the fastness, and $Q_1(V), Q_2(V, Z)$ and $Q_{3}(V,Z)$ stand for the corresponding reaction rates. The introduction of $\varepsilon$ is to make $Q_1(V)$ have the same order of magnitude as $Q_2(V,Z)$ and $Q_3(V,Z)$. A special case is that $Z$ is void and $V$ contains all $u_i$.
Assume that there is a steady state $u^* =(u_1^*, u_2^*, \cdots u_N^*)$ satisfying $u_i^*>0$ for all $i$ and a zero net flux condition for each fast reaction: $$k_{+p} (u_1^* )^{a_1^p } (u_2^* )^{a_2^p } \cdots (u_N^* )^{a_N^p } - k_{-p} (u_1^* )^{b_1^p } (u_2^* )^{b_1^p } \cdots (u_N^* )^{b_1^p } = 0, \qquad \forall p = 1, 2, \cdots, M'.$$ This is just the principle of detailed balance [@Walls] for the partial system consisting of the fast reactions merely. Clearly, it can only be true when both $k_{+p}$ and $k_{-p}$ are positive, that is, reversible reactions.
Under this assumption, we know from [@Yong1] that there is a strictly convex function $\eta=\eta(V)$ so that $Q_1(V)$ can be written as $$Q_1(V) = S(V)\eta_V(V)$$ for $V$ with strictly positive components. Here $S(V)$ is a symmetric matrix with null-space independent of $V$ and $\eta_V(V)$ is the gradient of $\eta(V)$. Moreover, the singular perturbation theory [@Yong2] for initial-value problems of ODEs can be applied to the stiff system in (\[23\]). In particular, the solutions to initial-value problems of (\[23\]) converge uniformly to those of a corresponding reduced system, as $\varepsilon$ goes to zero, in any bounded time interval away from zero.
In order to derive the reduced system, we notice the $V$-independence of the null-space and denote by $\Pi$ the constant matrix whose rows span the left null-space of $S(V)$. Without loss of generality, we assume that $\Pi$ is of the form $$\Pi = (\Theta, I)$$ with $I$ the unit matrix of proper order. Accordingly, we introduce the partition $$V = \left( {\begin{array}{l}
X \\
Y
\end{array}} \right), \qquad Q_1(V) = \left( {\begin{array}{l}
\hat Q_1(X, Y) \\
\hat Q_2(X, Y)
\end{array}} \right), \qquad Q_2(V, Z) = \left( {\begin{array}{l}
\tilde Q_1(X, Y, Z) \\
\tilde Q_2(X, Y, Z)
\end{array}} \right).$$ This partition ensures that $X$ can be uniquely and globally obtained by solving $\hat Q_1(X, \tilde Y - \Theta X)= 0$ (see [@Yong1] if necessary).
Define $$\tilde Y = Y + \Theta X .$$ The kinetic equations in (\[23\]) can be rewritten as $$\label{24}
\begin{array}{l}
\displaystyle{ \frac{{dX}}{{dt}} = \frac{1}{\varepsilon}\hat Q_1(X,\tilde Y - \Theta X) + \tilde Q_1(X,\tilde Y - \Theta X,Z),} \\[0.2cm]
\displaystyle{ \frac{{d\tilde Y}}{{dt}} = \tilde Q_2(X, \tilde Y - \Theta X, Z) + \Theta \tilde Q_1(X, \tilde Y - \Theta X, Z)} ,\\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_3(X, \tilde Y - \Theta X, Z)}.
\end{array}$$ As $\varepsilon$ goes to zero, the reduced system for (\[24\]) is $$\label{25}
\begin{array}{l}
\hat Q_1(X, \tilde Y - \Theta X) =0, \\[0.2cm]
\displaystyle{ \frac{{d\tilde Y}}{{dt}} = \tilde Q_2(X, \tilde Y - \Theta X, Z) + \Theta \tilde Q_1(X, \tilde Y - \Theta X, Z)} ,\\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_3(X, \tilde Y - \Theta X, Z)}.
\end{array}$$ From $\hat Q_1(X, \tilde Y - \Theta X)=0$ we solve $X$ in terms of $\tilde Y$, say $X= \Phi(\tilde Y)$. Substituting this expression into the second and third equations in (\[25\]), we obtain $$\left\{ \begin{array}{l}
\displaystyle{ \frac{{d\tilde Y}}{{dt}} = \tilde Q_2(\Phi(\tilde Y), \tilde Y - \Theta\Phi(\tilde Y), Z)+ \Theta\tilde Q_1(\Phi (\tilde Y), \tilde Y - \Theta\Phi(\tilde Y), Z)}, \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(\Phi (\tilde Y), \tilde Y - \Theta\Phi(\tilde Y), Z)} .\\
\end{array} \right.$$ This is our simplified model by using the PEA method. Note that the original variables $X$ and $Y$ are recovered from $$X= \Phi(\tilde Y), \qquad Y = \tilde Y - \Theta\Phi(\tilde Y).$$
In case $X$ can be solved from $\hat Q_1(X, Y)=0$ in terms of $Y$, say $X= \Psi(Y)$, we recall the fact that $X$ can be obtained by solving $\hat Q_1(X, \tilde Y - \Theta X)= 0$ and may well assume that both the Jacobian matrices $\hat Q_{1X}$ (of $\hat Q_1(X, Y)$ with respect to $X$) and $[\hat Q_{1X} - \hat Q_{1Y}\Theta]$ are invertible. Then $[I - \hat Q_{1X}^{-1}\hat Q_{1Y}\Theta]=\hat Q_{1X}^{-1}[\hat Q_{1X} - \hat Q_{1Y}\Theta]$ is invertible. It is an easy exercise to show that the invertibility of $[I - \hat Q_{1X}^{-1}\hat Q_{1Y}\Theta]$ is equivalent to that of $[I - \Theta\hat Q_{1X}^{-1}\hat Q_{1Y}]$. On the other hand, we deduce from $\hat Q_1(\Psi(Y), Y)=0$ that $\hat Q_{1X}\Psi(Y)_Y + \hat Q_{1Y} = 0$ and thereby $\Psi(Y)_Y=-\hat Q_{1X}^{-1}\hat Q_{1Y}$. Now we compute from $\tilde Y = Y + \Theta\Psi(Y)$ that $$\displaystyle{\frac{d\tilde Y}{dt}=\frac{{dY}}{dt}+\Theta\Psi(Y)_{Y}\frac{dY}{dt} } =
[I - \Theta\hat Q_{1X}^{-1}\hat Q_{1Y}]\frac{{dY}}{dt}.$$ Thus we gain equations for $Y$: $$\displaystyle{ \frac{{dY}}{{dt}}=(I + \Theta\Psi(Y)_{Y})^{-1}(\tilde Q_2(\Psi(Y), Y, Z)+ \Theta\tilde Q_1(\Psi(Y), Y, Z))}.$$ Consequently, the reduced system can be written as $$\label{26}
\begin{array}{l}
\displaystyle{ \frac{{dY}}{{dt}}=(I +\Theta\Psi(Y)_Y^{-1})( \tilde Q_2(\Psi(Y),Y,Z) + \Theta \tilde Q_1(\Psi(Y),Y,Z))}, \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(\Psi(Y),Y,Z)}
\end{array}$$ together with the algebraic relation $X = \Psi(Y)$.
A simple example
----------------
In order to elucidate how to use the PEA method, we consider the simplest system for enzyme inhibition [@Keener]. This system is demonstrated graphically in Fig. \[Fig:The enzyme system.\] and reveals the competitively inhibitory mechanism, where the enzyme reaction is stopped when the inhibitor is bound to the active site of the enzyme. In Fig. \[Fig:The enzyme system.\], the symbols E, S, I, P, $C_1$ and $C_2$ stand for the enzyme, the substrate, the inhibitor, the product and two complexes, respectively. $k_1$ and $k_{-1}$ are the respective kinetic rate constants of the forward and backward reaction for substrate binding, while $k_3$ and $k_{-3}$ are those of the inhibitor binding reaction. $k_2$ is the rate constant of the substrate conversion reaction.
![The simplest mechanism for enzyme inhibition[]{data-label="Fig:The enzyme system."}](./figure2/enzyme2.eps){width="30.00000%"}
According to law of of mass action, the corresponding kinetic equations read as $$\label{27}
\begin{array}{l}
\displaystyle{ \frac{{d[C_1]}}{{dt}} = v_1 - v_2 } \\[0.2cm]
\displaystyle{ \frac{{d[C_2]}}{{dt}} = v_3 } \\[0.2cm]
\displaystyle{ \frac{{d[E]}}{{dt}} = -v_1 - v_3 + v_2 } \\[0.2cm]
\displaystyle{ \frac{{d[S]}}{{dt}} = -v_1 } \\[0.2cm]
\displaystyle{ \frac{{d[I]}}{{dt}} = -v_3 } \\[0.2cm]
\displaystyle{ \frac{{d[P]}}{{dt}} = v_2 }.
\end{array}$$ with $$\begin{array}{l}
v_1 = k_1[E][S] - k_{-1} [C_1], \\[0.15cm]
v_2 = k_2[C_1], \\[0.15cm]
v_3 = k_3[E][I] - k_{-3} [C_2] . \\[0.15cm]
\end{array}$$
Classically, the reactions for enzyme to bind to substrates and inhibitors are regarded as fast and reversible. Thus we rewrite (\[27\]) as $$\label{28}
\begin{array}{l}
\displaystyle{ \frac{{d[C_1]}}{{dt}} = \frac{1}{\varepsilon} \hat v_1 - v_2 } \\[0.2cm]
\displaystyle{ \frac{{d[C_2]}}{{dt}} = \frac{1}{\varepsilon} \hat v_3 } \\[0.2cm]
\displaystyle{ \frac{{d[E]}}{{dt}} = -\frac{1}{\varepsilon} \hat v_1 - \frac{1}{\varepsilon}\hat v_3 + v_2 } \\[0.2cm]
\displaystyle{ \frac{{d[S]}}{{dt}} = -\frac{1}{\varepsilon}\hat v_1 } \\[0.2cm]
\displaystyle{ \frac{{d[I]}}{{dt}} = -\frac{1}{\varepsilon}\hat v_3 } \\[0.2cm]
\displaystyle{ \frac{{d[P]}}{{dt}} = v_2 }.
\end{array}$$ where $\hat v_1 =\varepsilon v_1$, $\hat v_3 =\varepsilon v_3$, and $\hat v_1$ and $\hat v_3$ have the same order of magnitude as $v_2$. Thanks to the reversibility, it is obvious that there are positive numbers $[E]^*, [S]^*, [C_1]^*, [I]^*$ and $[C_2]^*$ such that $$v_1 = k_1[E]^*[S]^* - k_{-1} [C_1]^*=0, \qquad v_3 =k_3[E]^*[I]^* - k_{-3} [C_2]^*=0.$$ Thus the PEA method above can be well applied to the stiff system (\[28\]).
The reduced system for (\[28\]) can be derived as follows. Set $$\begin{array}{ll}
X =& \big([C_1], [C_2] \big)^T ,\\
Y =& \big([E],[S],[I]\big)^T ,\\
Z =& [P] .
\end{array}$$ the stiff system (\[28\]) can be rewritten as $$\label{29}
\begin{array}{l}
\displaystyle{ \frac{{dX}}{{dt}} = \frac{1}{\varepsilon}\left( {\begin{array}{*{20}l}
\hat v_1 \\
\hat v_3 \\
\end{array}} \right) + v_2\left( {\begin{array}{*{20}l}
-1\\
0\\
\end{array}} \right), } \\[0.2cm]
\displaystyle{ \frac{{d(Y+ \Theta X)}}{{dt}} = v_2\left( {\begin{array}{*{20}l}
0\\
-1 \\
0 \\
\end{array}} \right), } \\[0.2cm]
\displaystyle{ \frac{{d[P]}}{{dt}} = v_2 }
\end{array}$$ with $$\Theta =\left( {\begin{array}{*{20}l}
{1} & {1} \\
{1} & {0} \\
{0} & 1 \\
\end{array}} \right).$$ From $\hat v_1=\hat v_3=0$, we get $$\label{210}
[C_1] = K_1[E][S] , \qquad [C_2] = K_3[E][I]$$ with $K_j=k_j/k_{-j}$ for $j= 1, 3$. Substituting these into the last two equations in (\[29\]), we obtain the following ODEs $$\begin{array}{l}
\displaystyle{ \frac{{d([E]+[C_1]+[C_2])}}{{dt}} = 0, } \\[0.2cm]
\displaystyle{ \frac{{d([S]+[C_1])}}{{dt}} = -k_2 K_1 [E][S], } \\[0.2cm]
\displaystyle{ \frac{{d([I]+[C_2]) }}{{dt}} = 0, } \\[0.2cm]
\displaystyle{ \frac{{d[P]}}{{dt}} = k_{2} K_1 [E][S] }.
\end{array}$$ This, together with (\[210\]), is the simplified model by using the justified PEA method. This model can be further simplified by using the conservation laws indicated in the first and third equations. We omit it here and leave it to the interested reader.
Apoptosis Systems
=================
Here we introduce the large apoptosis system proposed by Hua et al. in [@Hua] for human Jurkat T cells. To begin with, we recall that there are at least two pathways to trigger apoptosis—intrinsic (mitochondrial) and extrinsic (death receptor) signalling pathways. Both induce death-associated proteolytic and/or nucleolytic activities. The intrinsic pathway is initiated when the cell is severely damaged or stressed, while the extrinsic one is activated when extracellular death ligands are bound by their cognate membrane-associated death receptors such as TNF-R1(DR1,p55), Fas(DR2,CD95), DR3(APO-3,TRAMP), DR4(APO-2,TRAIL-R1) and DR5(TRICK2,TRAOL-R2) [@Hengartner; @Ashkenazi; @Barnhart; @Samraj; @Lavrik]. The Fas-induced signaling pathway is among the best understood and can be schematically shown in Fig. \[Fig:The cell apoptosis picture\]. It begins with the binding of Fas ligands (FasL), Fas and FADD (Fas-associated death domain) to form the complex DISC (death-inducing signaling complex). The latter can recruits initiator caspases such as caspase-8 (Casp8) molecules to cleave and activate them. The activated initiator caspase (Casp8$_2^*$) can cleaves and activates the executor caspase-3 (Casp3) to form Casp3$^*$ directly. The amount of Casp3$^*$ is the indicator of apoptosis. This way to activate Casp3 is called D-channel. In addition, Casp3 can also be activated in a so-called M-channel. In this channel, Casp8$_2^*$ cleaves Bid to generate truncated (t)Bid. The tBid then binds to two molecules of Bax to form a complex tBid:Bax$_2$, which will induce the release of Cyto.c and Smac from the mitochondria. The released Cyto.c$^*$ will combine an adaptor protein Apaf-1, ATP and caspase-9 to form apoptosome and thereby activate caspase-9. The activated caspase-9 (Casp9$^*$) cleaves and activates Casp3. On the other hand, the M-channel can be blocked by XIAP (X-linked inhibitor of apoptosis protein) and Bcl2 through their bindings to the released Smac\*, Casp9, Casp3\*, Bax and tBid.
![The Fas-induced apoptotic pathway, including two channels.[]{data-label="Fig:The cell apoptosis picture"}](./figure2/apoptosis.eps){width="100.00000%"}
The Fas-signaling pathway model proposed by Hua et at. [@Hua] consists of biochemical reactions (H1)–(H25) given in Table 1.
\[Tab:the ISS model\]
Reaction $k_{i}$ $ k_{-i}$
---------- ------------------------------------------------------------------------------------------------------------------ ---------------------------------- ----------------------
(H1) $ FasL + Fas \autorightleftharpoons{$k_{H1}$}{$k_{-H1}$} FasC $ $9.09\times10^{-5}nM^{-1}s^{-1}$ $ 1.00\times10^{-4}$
(H2)$^a$ $ FasC:FADD_p:Casp8_q:FLIP_r + FADD \autorightleftharpoons{$k_{H2}$}{$k_{-H2}$} Fas:FADD_{p+1}:Casp8_q:FILP_r $ $5.00\times10^{-4}nM^{-1}s^{-1}$ 0.2
(H3)$^b$ $ FasC:FADD_p:Casp8_q:FILP_r + Casp8 \autorightleftharpoons{$k_{H3}$}{$k_{-H3}$} Fas:FADD_p:Casp8_{q+1}:FILP_r $ $3.50\times10^{-3}nM^{-1}s^{-1}$ 0.018
(H4)$^b$ $ FasC:FADD_p:Casp8_q:FLIP_r + FILP \autorightleftharpoons{$k_{H4}$}{$k_{-H4}$} Fas:FADD_p:Casp8_q:FILP_{r+1} $ $3.50\times10^{-3}nM^{-1}s^{-1}$ 0.018
(H5)$^c$ $ FasC:FADD_p:Casp8_q:FLIP_r \autorightarrow{$k_{H5}$}{} Casp8*_2:p41 + FasC:FADD_p:Casp8_{q-1}:FILP_r $ $0.3s^{-1}$
(H6) $ Casp8_{_2 }^*:p41 \autorightarrow{$k_{H6}$}{} Casp8_{_2 }^* $ $0.1s^{-1}$
(H7) $Casp8_{_2 }^* + Casp3 \autorightleftharpoons{$k_{H7}$}{$k_{-H7}$} Casp8_{_2 }^* :Casp3 $ $1.00\times10^{-4}nM^{-1}s^{-1}$ 0.06
(H8) $ Casp8_{_2 }^* :Casp3 \autorightarrow{$k_{H8}$}{} Casp8_{_2 }^* + Casp3^* $ $0.1s^{-1}$
(H9) $ Casp8_{_2 }^* + Bid \autorightleftharpoons{$k_H9$}{$k_{-H9}$} Casp8_{_2 }^* :Bid $ $5.00\times10^{-4}nM^{-1}s^{-1}$ 0.005
(H10) $ Casp8_{_2 }^* :Bid \autorightarrow{$k_{H10}$}{} Casp8_{_2 }^* + tBid $ $0.1s^{-1}$
(H11) $ tBid + Bax \autorightleftharpoons{$k_{H11}$}{$k_{{-H11}}$} tBid:Bax $ $2.00\times10^{-4}nM^{-1}s^{-1}$ 0.02
(H12) $ tBid:Bax + Bax \autorightleftharpoons{$k_{H12}$}{$k_{-H12}$} tBid:Bax_2 $ $2.00\times10^{-4}nM^{-1}s^{-1}$ 0.02
(H13) $ Smac + tBid:Bax_2 \autorightarrow{$k_{H13}$}{} Smac^* + tBid:Bax_2 $ $1.00\times10^{-3}nM^{-1}s^{-1}$
(H14) $ Smac^* + XIAP \autorightleftharpoons{$k_{H14}$}{$k_{{-H14}}$} Smac^*:XIAP $ $7.00\times10^{-3}nM^{-1}s^{-1}$ $2.21\times10^{-3}$
(H15) $ Cyto.c + tBid:Bax_2 \autorightarrow{$k_{H15}$}{} Cyto.c^* + tBid:Bax_2 $ $1.00\times10^{-3}nM^{-1}s^{-1}$
(H16) $ Cyto.c^* + Apaf + ATP \autorightleftharpoons{$k_{H16}$}{$k_{-H16}$} Cyto.c^* :Apaf:ATP $ $2.78\times10^{-7}nM^{-1}s^{-1}$ $5.70\times10^{-3}$
(H17) $ Cyto.c^* :Apaf:ATP + Casp9 \autorightleftharpoons{$k_{H17}$}{$k_{-H17}$} Cyto.c^* :Apaf:ATP:Casp9 $ $2.84\times10^{-4}nM^{-1}s^{-1}$ 0.07493
(H18) $ Cyto.c^* :Apaf:ATP:Casp9 + Casp9 \autorightleftharpoons{$k_{H18}$}{$k_{-H18}$} Cyto.c^* :Apaf:ATP:Casp9_2 $ $4.41\times10^{-4}nM^{-1}s^{-1}$ 0.1
(H19) $ Cyto.c^* :Apaf:ATP:Casp9_2 \autorightarrow{$k_{H19}$}{} Cyto.c^* :Apaf:ATP:Casp9 + Casp9^* $ $0.7s^{-1}$
(H20) $ Casp9^* + Casp3 \autorightleftharpoons{$k_{H20}$}{$k_{-H20}$} Casp9^* :Casp3 $ $1.96\times10^{-5}nM^{-1}s^{-1}$ 0.05707
(H21) $ Casp9^* :Casp3 \autorightarrow{$k_{H21}$}{} Casp9^* + Casp3^* $ $4.8s^{-1}$
(H22) $ Casp9 + XIAP \autorightleftharpoons{$k_{H22}$}{$k_{{-H22}}$} Casp9:XIAP $ $1.06\times10^{-4}nM^{-1}s^{-1}$ $1.00\times10^{-3}$
(H23) $ Casp3^* + XIAP \autorightleftharpoons{$k_{H22}$}{$k_{{-H22}}$} Casp3^*:XIAP $ $2.47\times10^{-3}nM^{-1}s^{-1}$ $2.40\times10^{-3}$
(H24) $ Bcl_2 + Bax \autorightleftharpoons{$k_{H24}$}{$k_{-H24}$} Bcl_2 :Bax $ $2.00\times10^{-4}nM^{-1}s^{-1}$ 0.02
(H25) $ Bcl_2 + tBid \autorightleftharpoons{$k_{H25}$}{$k_{-H25}$} Bcl_2 :tBid $ $2.00\times10^{-4}nM^{-1}s^{-1}$ 0.02
: The Fas-signaling pathway model due to Hua et at. (2005)
\
[The index $(p, q, r)$ in reactions (a) takes values (0,0,0),(1,0,0),(1,0,1),(1,1,0),(2,0,0),(2,0,1),(2,0,2),(2,1,0),(2,1,1) and (2,2,0). In reactions (b) it takes values (1,0,0),(2,0,0),(2,0,1),(2,1,0),(3,0,0),(3,0,1),(3,0,2),(3,1,0),(3,1,1) and (3,2,0), while it takes values (2,2,0),(3,2,0),(3,2,1) and (3,3,0) in reactions (c). ]{}
From this table we see that the process activating the initiator caspase-8 (Casp8) consists of the reactions from (H1) to (H6), which is initiated by FasL. The activated Casp$8_2^*$ enzymatically cleaves caspase-3(Casp3) to produce activated executor Casp$3^*$ ((H7) and (H8)) and Bid to generate tBid ((H9) and (H10)) simultaneously. Then the tBid associates with two Bax to form tBid:Bax$_2$ through (H11) and (H12), which induces the release of Cyto.c and Smac from mitochondrial to cytosol ((H15) and (H13)). The released Cyto.c (Cyto.c$^*$) combines Apaf (Apaf-1) and ATP to form an apoptosome (Cyto.c$^*$:Apaf:ATP) in (H16), which recruits two caspase-9(Casp9) and generates the activated caspase-9 (Casp$9^*$) through (H17) to (H19). The activated Casp$9^*$ can also enzymatically cleaves and activates caspase-3 ((H20) and (H21)). On the other hand, the roles of Casp9, Casp$3^*$, tBid and Bax can be inhibited by binding to XIAP((H22) and (H23)) and Bcl$_2$ ((H24) and (H25)), while the released Smac (Smac\*) can suppress the function of XIAP (H14). Table 1 also contains all the forward/backward rate constants $k_{\pm i}(i =1, 2, \cdots, 25)$.
Observe that not every reaction in Table 1 is reversible and the reactions activating Casp8 are independent of the rest. As in [@Okazaki], we call the downstream process, consisting of reactions from (H7) to (H25), as the intracellular-signaling subsystem (ISS). Moreover, we follow [@Okazaki] and assume that the concentration of ATP is a fixed constant. Thus, there are 28 species and 19 biochemical reactions involved in the downstream process.
According to the law of mass action, the dynamics of the ISS is governed by 28 ordinary differential equations $$\label{31}
\displaystyle{ \frac{{dU}}{{dt}} = Q(U)}.$$ Here $U=U(t)$ is a column vector with 28 components representing the concentrations of all the 28 species in the ISS: $$\begin{array}{ll}
U =& \big([Casp8^*_2], [Casp8^*_2:Casp3], [Casp8^*_2:Bid], [Bid], [tBid], [tBid:Bax], [tBid:Bax_2], \\
& \ \ [Bcl_2:tBid], [Bax], [Bcl_2:Bax], [Bcl_2], [Cyto.c], [Cyto.c^*], [Cyto.c^* :Apaf:ATP], \\
& \ \ [Cyto.c^*:Apaf:ATP:Casp9], [Cyto.c^*:Apaf:ATP:Casp9_2], [Apaf], [Casp9^*],\\
& \ \ [Casp9], [Casp3], [Casp9*:Casp3], [Casp3^*], [Smac], [Smac^*], [XIAP], \\
& \ \ [Smac^*:XIAP], [Casp9:XIAP], [Casp3^*:XIAP] \big)^T ,
\end{array}$$ each element of the vector-valued function $Q(U)$ of $U$ is the change rate of concentration for the corresponding species $$\begin{array}{ll}
Q(U) = & \big( -v_{7} + v_{8} - v_{9} + v_{10} + v_0, v_{7}-v_{8}, v_{9}-v_{10}, -v_{9}, v_{10} - v_{11}- v_{25}, v_{11}-v_{12}, v_{12}, v_{25}, \\
& \ \ -v_{11}-v_{12}-v_{24},
v_{24}, -v_{24}-v_{25}, -v_{15}, v_{15}-v_{16}, v_{16}-v_{17}, v_{17}-v_{18}+v_{19}, \\ & \ \ v_{18}-v_{19}, -v_{16}, v_{19}-v_{20}+v_{21},
-v_{17}-v_{18}-v_{22}, -v_{7}-v_{20}, v_{20}-v_{21}, \\ & \ \ v_{8}+v_{21}-v_{23}, -v_{13}, v_{13}-v_{14}, -v_{14}-v_{22}-v_{23}, v_{14}, v_{22},v_{23} \big)^T
\end{array}$$ with $v_i (i=7\cdots25)$ the rate of the $i$-th reaction in Table 1: $$\begin{array}{l}
v_7 = k_7 [Casp8_{_2 }^* ][Casp3] - k_{-7} [Casp8_{_2 }^* :Casp3], \\[0.2cm]
v_8 = k_{8} [Casp8_{_2 }^* :Casp3], \\[0.2cm]
v_9 = k_{9} [Casp8_{_2 }^* ][Bid] - k_{ - 9} [Casp8_{_2 }^* :Bid], \\[0.2cm]
v_{10} = k_{10} [Casp8_{_2 }^* :Bid], \\[0.2cm]
v_{11} = k_{11} [tBid][Bax] - k_{ - 11} [tBid:Bax], \\[0.2cm]
v_{12} = k_{12} [tBid:Bax][Bax] - k_{ - 12} [tBid:Bax_2 ], \\[0.2cm]
v_{13} = k_{13} [Smac][tBid:Bax_2] \\[0.2cm]
v_{14} = k_{14} [Smac^*][XIAP] - k_{-14} [Smac^*:XIAP] , \\[0.2cm]
v_{15} = k_{15} [Cyto.c][tBid:Bax_2 ] ,\\[0.2cm]
v_{16} = k_{16} [Cyto.c^* ][Apaf][ATP] - k_{ - 16} [Cyto.c^* :Apaf:ATP] ,\\[0.2cm]
v_{17} = k_{17} [Cyto.c^* :Apaf:ATP][Casp9] - k_{ - 17} [Cyto.c^* :Apaf:ATP:Casp9] ,\\[0.2cm]
v_{18} = k_{18} [Cyto.c^* :Apaf:ATP:Casp9][Casp9] - k_{ - 18} [Cyto.c^* :Apaf:ATP:Casp9_2 ], \\[0.2cm]
v_{19} = k_{19} [Cyto.c^* :Apaf:ATP:Casp9_2 ], \\[0.2cm]
v_{20} = k_{20} [Casp9^* ][Casp3] - k_{ - 20} [Casp9^* :Casp3], \\[0.2cm]
v_{21} = k_{21} [Casp9^* :Casp3], \\[0.2cm]
v_{22} = k_{22} [Casp9][XIAP] - k_{-22} [Casp9:XIAP] , \\[0.2cm]
v_{23} = k_{23} [Casp3^*][XIAP] - k_{-23} [Casp3^*:XIAP] , \\[0.2cm]
v_{24} = k_{24} [Bcl_2 ][Bax] - k_{ - 24} [Bcl_2 :Bax], \\[0.2cm]
v_{25} = k_{25} [Bcl_2 ][tBid] - k_{ - 25} [Bcl_2 :tBid],
\end{array}$$ $v_0$ is the constant rate of generation for Casp8$^*_2$ from the upstream process and its value was suggested in [@Okazaki] as $v_0=0.001nMs^{-1}$. In addition, the non-zero initial concentrations for $U$ are taken as in [@Hua; @Okazaki] and are given in Table \[Tab:The initial concentrations of each species in ISS model\].
\[Tab:The initial concentrations of each species in ISS model\]
Species Initial concentration(nM)
--------- ---------------------------
Casp3 200.00
Bid 25.00
Bcl2 75.00
Bax 83.33
Cyto.c 100.00
Smac 100.00
XIAP 30.00
Casp9 20.00
ATP 10000.00
Apaf 100.00
: Non-zero initial concentrations of the species in the ISS model (Hua. et al. 2005)
In [@Okazaki], Okazaki et al. claimed that Smac and XIAP have little effect on the reaction process of the ISS by studying the M-D transition behavior of both the ISS model and ISS(wo/S,X) (without Smac and XIAP) model. So they did not consider reactions (H13), (H14), (H22) and (H23) in their simplification. However, in our previous paper [@HY] we found that Smac and XIAP should not be ignored because the numerical results of these two models are quite different if initial concentrations are changed. Therefore, our sequel discussion will base on the entire ISS system.
We conclude this section by explaining the M-D transition behavior. It means a D-channel and M-channel switching behavior and the quantity of \[Casp8$_2^*$\] is a control parameter. When a large amount of Casp8$_2^*$ is activated from the upstream process, it will directly induce cell death through the D-channel; otherwise, the M-channel plays more important role for cell death. In [@Hua], the authors claimed that the effects of D-channel and M-channel can be altered by varying the amount of Casp8$_2^*$ generated by DISC [@Barnhart; @Samraj; @Scaffidi], which is consistent with previous experiments.
Model reduction
===============
In this section we use the PEA method to investigate the large apoptosis system (\[31\]). This system contains 13 reversible reactions: (H7), (H9), (H11), (H12), (H14), (H16), (H17), (H18), (H20), (H22), (H23), (H24) and (H25). In our preliminary work [@HY], we showed that the six reactions (H11), (H12), (H16), (H17), (H24) and (H25) are fast. Because the PEA method has a solid mathematical basis, it can be used as a tool to determine whether or not a reversible reaction is fast. After many attempts by assuming some of the rest 7 reactions to be fast too, we find that the nine reversible reactions (H7), (H9), (H11), (H12), (H16), (H17), (H18), (H24) and (H25) can be well regarded as fast.
Here we derive the corresponding simplified model under the assumption that the nine reversible reactions are fast. According to the framework in Section 2, we decompose the concentration vector $U$ as $$U = \left( {\begin{array}{l}
X \\
Y \\
Z
\end{array}} \right).$$ Here $X$ stands for the products of the nine reactions, $Y$ for the reactants and $Z$ for the rest: $$\label{41}
\begin{array}{ll}
X =& \big( [Casp8^*_2:Casp3], [Casp8^*_2:Bid], [tBid:Bax], [tBid:Bax_2], \\
& ~ [Bcl_2:Bax], [Bcl_2:tBid], [Cyto.c^* :Apaf:ATP], \\
& ~ [Cyto.c^*:Apaf:ATP:Casp9],[Cyto.c^*:Apaf:ATP:Casp9_2] \big)^T, \\
Y = & \big( [Casp8^*_2], [Bid], [tBid], [Bax], [Bcl_2], [Cyto.c^*], [Apaf], [Casp9], [Casp3] \big)^T, \\
Z = & \big( [Cyto.c], [Casp9^*] , [Casp9*:Casp3],[Casp3^*] , [Smac], [Smac]^*, [XIAP],\\
& ~[Smac^*:XIAP], [Casp9:XIAP], [Casp3^*:XIAP] \big)^T.
\end{array}$$ With this decomposition, the kinetic equations in (\[31\]) can be rewritten as $$\label{42}
\begin{array}{l}
\displaystyle{ \frac{{dX}}{{dt}} = \frac{1}{\varepsilon}\hat Q_{1}(X,Y) + Q_{1}(X,Y,Z)} \\[0.2cm]
\displaystyle{ \frac{{dY}}{{dt}} = \frac{1}{\varepsilon}\hat Q_{2}(X,Y) + Q_{2}(X,Y,Z)} \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(X,Y,Z)}.
\end{array}.$$ Here the small parameter $\varepsilon$ characterizes the fastness of the reversible reactions as in Section 2, $$\begin{array}{ll}
\hat Q_1(X,Y) = & \varepsilon \big( v_7, v_9, v_{11}-v_{12}, v_{12},v_{24},v_{25},v_{16}-v_{17},v_{17}-v_{18},v_{18} \big)^T, \\
\hat Q_2(X,Y) = & \varepsilon \big( -v_7-v_9, -v_9, -v_{11}-v_{25}, -v_{11}-v_{12}-v_{24}, -v_{24}-v_{25}, -v_{16}, -v_{16}, -v_{17}-v_{18},-v_7 \big)^T,
\end{array}$$ stand for the change rates of concentration due to the rapid reactions, and $$\begin{array}{ll}
Q_{1}(X,Y,Z) = & \big( -v_{8}, -v_{10}, 0,0,0,0,0, v_{19}, -v_{19} \big)^T, \\
Q_{2}(X,Y,Z)=& \big( v_{8}+v_{10}+v_0, 0, v_{10}, 0, 0, v_{15}, 0,-v_{22}, -v_{20} \big)^T, \\
Q_{3}(X,Y,Z) = & \big( -v_{15}, v_{19}-v_{20}+v_{21}, v_{20}-v_{21}, v_8 + v_{21}-v_{23} , -v_{13}, v_{13}-v_{14},-v_{14}-v_{22}-v_{23},v_{14}, v_{22},v_{23}\big)^T,
\end{array}$$ are those for the other reactions. It is direct to check that $$\label{43}
\hat Q_{2}(X,Y) + \Theta\hat Q_{1}(X,Y)\equiv 0$$ with $\Theta$ the following constant 9x9-matrix $$\Theta =\left( {\begin{array}{*{20}l}
{1} & {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\
{0} & {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\
{0} & {0} & {1} & {1} & {0} & {1} & {0} & {0} & {0} \\
{0} & {0} & {1} & {2} & {1} & {0} & {0} & {0} & {0} \\
{0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} & {0} \\
{0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} & {1} \\
{0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} & {1} \\
{0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {2} \\
{1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\
\end{array}} \right).$$
Recall that $$\label{44}
\begin{array}{l}
v_7 = k_{7} [Casp8_{_2 }^* ][Casp3] - k_{- 7} [Casp8_{_2 }^* :Casp3], \\[0.1cm]
v_9 = k_{9} [Casp8_{_2 }^* ][Bid] - k_{-9} [Casp8_{_2 }^* :Bid], \\[0.1cm]
v_{11} = k_{11} [tBid][Bax] - k_{-11} [tBid:Bax] , \\[0.1cm]
v_{12} = k_{12} [tBid:Bax][Bax] - k_{ -12} [tBid:Bax_2] , \\[0.1cm]
v_{16} = k_{16} [Cyto.c^* ][Apaf][ATP] - k_{ -16} [Cyto.c^* :Apaf:ATP] ,\\[0.1cm]
v_{17} = k_{17} [Cyto.c^* :Apaf:ATP][Caps9] - k_{ -17} [Cyto.c^* :Apaf:ATP:Casp9] , \\[0.1cm]
v_{18} = k_{18} [Cyto.c^* :Apaf:ATP:Casp9][Casp9] - k_{ - 18} [Cyto.c^* :Apaf:ATP:Casp9_2 ], \\[0.1cm]
v_{24} = k_{24} [Bcl_2][Bax] - k_{ -24} [Bcl_2 :Bax] , \\[0.1cm]
v_{25} = k_{25} [Bcl_2][tBid] - k_{ -25} [Bcl_2 :tBid].
\end{array}$$ Then for any given $Y = \big( [Casp8^*_2], [Bid], [tBid], [Bax], [Bcl_2], [Cyto.c^*], [Apaf], [Casp9], [Casp3] \big)^T$ with positive components, we use (\[43\]) to get $$\begin{array}{ll}
X =& \big( [Casp8^*_2:Casp3], [Casp8^*_2:Bid], [tBid:Bax], [tBid:Bax_2], [Bcl_2:Bax], [Bcl_2:tBid], \\
&~[Cyto.c^* :Apaf:ATP], [Cyto.c^*:Apaf:ATP:Casp9],[Cyto.c^*:Apaf:ATP:Casp9_2] \big)^T
\end{array}$$ with positive components such that $v_7=v_9=v_{11}=v_{12}=v_{16}=v_{17}=v_{18}=v_{24}=v_{25}=0$. Thus, the principle of detailed balance is verified.
After verifying the conditions for the PEA method to be reliable, we turn to write down the simplified model. In view of (\[43\]), we define $\tilde Y = Y + \Theta X.$ Then the ODEs in (\[42\]) become $$\label{45}
\begin{array}{l}
\displaystyle{ \frac{{dX}}{{dt}} = \frac{1}{\varepsilon}\hat Q_1(X,Y) + Q_1(X,Y,Z),} \\[0.2cm]
\displaystyle{ \frac{{d\tilde Y}}{{dt}} = Q_2(X, Y, Z) + \Theta Q_1(X, Y, Z)} ,\\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_3(X, Y, Z)}.
\end{array}$$ Guided by the framework in Section 2, we solve $\hat Q_1(X, Y)=0$, namely, $$\left\{ \begin{array}{l}
v_7 = k_{7} [Casp8_{_2 }^* ][Casp3] - k_{ -7} [Casp8_{_2 }^* :Casp3]=0, \\[0.1cm]
v_9 = k_{9} [Casp8_{_2 }^* ][Bid] - k_{ - 9} [Casp8_{_2 }^* :Bid]=0, \\[0.1cm]
v_{11} = k_{11} [tBid][Bax] - k_{ -11} [tBid:Bax]=0 , \\[0.1cm]
v_{12} = k_{12} [tBid:Bax][Bax] - k_{ -12} [tBid:Bax_2]=0 , \\[0.1cm]
v_{16} = k_{16} [Cyto.c^* ][Apaf][ATP] - k_{ -16} [Cyto.c^* :Apaf:ATP]=0 ,\\[0.1cm]
v_{17} = k_{17} [Cyto.c^* :Apaf:ATP][Caps9] - k_{ -17} [Cyto.c^* :Apaf:ATP:Casp9] =0, \\[0.1cm]
v_{18} = k_{18} [Cyto.c^* :Apaf:ATP:Casp9][Casp9] - k_{ - 18} [Cyto.c^* :Apaf:ATP:Casp9_2 ]=0, \\[0.1cm]
v_{24} = k_{24} [Bcl_2][Bax] - k_{ -24} [Bcl_2 :Bax]=0 , \\[0.1cm]
v_{25} = k_{25} [Bcl_2][tBid] - k_{ -25} [Bcl_2 :tBid]=0.
\end{array} \right.$$ From these algebraic equations we can easily solve $X$ in terms of $Y$: $$\label{46}
\begin{array}{rl}
[Casp8_{_2 }^* :Casp3] = & \frac{{k_{7} [Casp8_{_2 }^* ][Casp3]}}{{k_{ -7} }} = {K_{7} [Casp8_{_2 }^* ][Casp3]}, \\[0.15cm]
[Casp8_{_2 }^* :Bid] = & \frac{{k_{9} [Casp8_{_2 }^* ][Bid]}}{{k_{ - 9} }} = {K_{9} [Casp8_{_2 }^* ][Bid]}, \\[0.15cm]
[tBid:Bax] = & \frac{{k_{11} [tBid][Bax]}}{{k_{ -11} }} = K_{11} [tBid][Bax], \\[0.15cm]
[tBid:Bax_2] = & \frac{{k_{12} [tBid:Bax][Bax]}}{{k_{ -12} }} = K_{12} K_{11} [tBid][Bax][Bax], \\[0.15cm]
[Cyto.c^* :Apaf:ATP] = & \frac{{k_{16} [Cyto.c^* ][Apaf][ATP]}}{{k_{ -16} }} = K_{16} [Cyto.c^* ][Apaf][ATP] , \\[0.15cm]
[Cyto.c^* :Apaf:ATP:Casp9] = & \frac{{k_{17} [Cyto.c^* :Apaf:ATP][Casp9]}}{{k_{ -17} }} \\[0.15cm]
= & K_{17} K_{16} [Cyto.c^* ][Apaf][ATP][Casp9], \\[0.15cm]
[Cyto.c^* :Apaf:ATP:Casp9_2 ]= & \frac{{k_{18} [Cyto.c^* :Apaf:ATP:Casp9][Casp9]}}{{k_{ -18} }} \\[0.15cm]
= & K_{18}K_{17} K_{16} [Cyto.c^* ][Apaf][ATP][Casp9]^2, \\[0.15cm]
[Bcl_2 :Bax] = & \frac{{k_{24} [Bcl_2 ][Bax]}}{{k_{ -24} }} = K_{24} [Bcl_2 ][Bax], \\[0.15cm]
[Bcl_2 :tBid] = & \frac{{k_{25} [Bcl_2 ][tBid]}}{{k_{ -25} }} = K_{25} [Bcl_2 ][tBid]
\end{array}$$ with $K_i=k_i/k_{-i} $ for $i = 7, 9, 11, 12, 16, 17, 18, 24, 25$. Denote these relations by $X = \Psi(Y)$.
It is remarkable that the relations rely only on the 9 constants $K_i$, instead of the 18 constants $k_{\pm i}$. The latter are often not reliably known.
Substituting $X= \Psi(Y)$ into the second and third equations in (\[45\]), we obtain $$\left\{ \begin{array}{l}
\displaystyle{ \frac{{d\tilde Y}}{{dt}} = Q_2(\Psi (Y), Y, Z)+ \Theta Q_1(\Psi (Y), Y, Z)}, \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(\Psi (Y),Y,Z)}
\end{array} \right.$$ and thereby gain equations for $Y$: $$\displaystyle{ \frac{{dY}}{{dt}}= (I + \Theta\Psi(Y)_Y)^{-1}\frac{{d\tilde Y}}{{dt}}=(I + \Theta\Psi(Y)_Y)^{-1} (Q_2(\Psi(Y), Y, Z)+ \Theta Q_1(\Psi(Y), Y, Z))}.$$ Consequently, the original system (\[31\]) of 28 ODEs can be approximated by the following 19 ODEs $$\label{47}
\begin{array}{l}
\displaystyle{ \frac{{dY}}{{dt}}=(I +\Theta\Psi(Y)_Y)^{-1}(Q_2(\Phi (Y),Y,Z) + \Theta Q_1(\Psi(Y),Y,Z))}, \\[0.2cm]
\displaystyle{ \frac{{dZ}}{{dt}} = Q_{3}(\Psi (Y),Y,Z)}
\end{array}$$ together with nine algebraic relations $$X = \Psi(Y)$$ being detailed in (\[46\]). Recall that $Y$ and $Z$ are defined in (\[41\]). We call this new simplified model as ISS-2.
Numerical simulations
=====================
The purpose of this section is to show the reliability of our ISS-2 model by resorting to numerical simulations. Precisely, we compare the ISS-2 model (\[46\])–(\[47\]) with the entire ISS model (\[31\]) and Okazaki et al.’s ISS skeleton model [@Okazaki] in several aspects, including the accuracy, M-D transition behavior and sensitivity. For the reader’s convenience, the skeleton model is given in Table \[Tab:the ISS skeleton model\].
\[Tab:the ISS skeleton model\]
Reaction Rate constant
----------------- -------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------ --
(S1) $ Casp8_{_2 }^* + Casp3 \autorightarrow {}{} Casp8_{_2 }^* + Casp3^* $ $6.25.00\times10^{-6}nM^{-1}s^{-1}$
\[0.1cm\] (S2a) $ Casp8_{_2 }^* + Bid \autorightarrow {}{} Casp8_{_2 }^* + 0.0328tBid:Bax_2 $ $ v_{S2a}=\frac{{k_a[Casp8^*_2][Bid]}}{{[Casp8^*_2]+K_a}}(k_a=0.1s^{-1},K_a=20nM)$
\[0.1cm\] (S2b) $ Cyto.c + tBid:Bax_2 \autorightarrow {}{} 0.867Cyto.c^*:Apaf:ATP + tBid:Bax_2 $ $1\times10^{-3}nM^{-1}s^{-1}$
\[0.1cm\] (S2c) $ Cyto.c^* :Apaf:ATP + 2Casp9 \autorightarrow {}{} Cyto.c^* :Apaf:ATP + Casp9 + Casp9^* $ $1.46\times10^{-6}nM^{-1}s^{-1}$
\[0.1cm\] (S2d) $ Casp9^* + Casp3 \autorightarrow {}{} Casp9^* + Casp3^* $ $1.96\times10^{-5}nM^{-1}s^{-1}$
\[0.1cm\]
: The ISS skeleton model due to Okazaki et al. (2008)
The simulations were carried out with Matlab.
Accuracy of the ISS-2 model
---------------------------
We compute the concentration of each species as functions of time $t$ for the entire ISS model, the ISS skeleton model and our ISS-2 model, with initial concentrations from Table \[Tab:The initial concentrations of each species in ISS model\]. Fig. \[Fig.Concentration of FSISS.1\] displays three curves of Casp3$^*$ as functions of time $t$ corresponding to the three models. In this figure, the equilibrium value of Casp3$^*$—the indicator of apoptosis— of the ISS-2 model is almost the same as that of the ISS model, whereas that of the skeleton model is slightly larger. The equilibrium values of all other species for the ISS model and our ISS-2 model are also very close to each other. The curves of Casp3, Casp9$^*$ and Bid as functions of time $t$ are given in Fig. \[Fig.Concentration of FSISS.2\], Fig. \[Fig.Concentration of FSISS.3\], and Fig. \[Fig.Concentration of FSISS.4\], respectively. These numerical results show that our ISS-2 model is a reliable simplification of the entire ISS model.
M-D transition behavior
-----------------------
The M-D transition behavior is explained at the end of Section 3. In [@Okazaki], it was reported that initial concentrations of Casp9 also have considerable impacts to the transition behavior. In order to study this behavior, Okazaki et al. introduced two quantities $\gamma_D$ and $v^C_0$ in [@Okazaki]. The former was defined as the ratio of the net production of Casp3$^*$ via the D-channel to its total production, while the latter represents the critical value of $v_0$ (the generation rate for Casp8$_2^*$) corresponding to $\gamma_D=0.5$. Note that, at $\gamma_D=0.5$, the effect of the M-channel is same as that of the D-channel.
As previously, we use the initial concentrations from Table \[Tab:The initial concentrations of each species in ISS model\] and numerically solve the three models to obtain three curves of $\gamma_D$ as a function of $v_0$. The results are shown in Fig. \[Fig.M-D transition of FSISS.1\]. From this figure, we see that the curve given by the ISS-2 model matches that by the ISS model quite well and is obviously better than that by the skeleton model.
The curves of $v^C_0$ as a function of the initial concentration of Casp9 are shown in Fig. \[Fig.M-D transition of FSISS.2\]. From this figure we see that when the initial concentrations of Casp9 are small, the values of $v^C_0$ for the three models are almost the same. However, when the initial concentrations of Casp9 are large, the ISS skeleton model behaves quite different from the ISS model. But our ISS-2 model still matches the ISS model very well. All these indicate that our ISS-2 model can well describe the actual M-D transition behavior and are much better than the ISS skeleton model.
Sensitivity Analysis
--------------------
Now we present some results on the sensitivity of our ISS-2 model. Because the half-time—the time for Casp3$^*$ to attain half of its equilibrium value—is an important quantity to characterize how fast a cell will die [@Hua], we compute this quantity for the three models with initial data changed by one or two orders of magnitude higher and lower than the baseline values given in Table \[Tab:The initial concentrations of each species in ISS model\]. The numerical results are shown in Fig. \[Fig.Half-time for $Casp3^*$ activation.\].
![The change of $\alpha _{C3^*}$ against initial concentrations. The overexpression or knockdown level of each species is changed one or two orders of magnitude of the baseline values while the others are unchanged.[]{data-label="Fig:The matching of FSISS to ISS for different initial concentration."}](./figure2/Ten20sen_ince.eps){width="60.00000%"}
From Fig. \[Fig.Half-time for $Casp3^*$ activation.\] we see that, like the full apoptosis model due to Hua et al. [@Hua], our ISS-2 model possesses the symmetrical or asymmetrical properties of varying each species to the outcome. The result by the ISS-2 model is very similar to that by the ISS model. A bit difference is that the half-time for the ISS-2 model is a little shorter than that for the ISS model, which is same as for the ISS skeleton model. This is expected because the assumption of fast reactions slightly speeds up the whole apoptotic process.
To evaluate our simplified model, we follow our previous work [@HY] and introduce the quantity $$\alpha_{C3^*}=\frac{{equilibrium\ value\ of\ Casp3^*\ for\ ISS\mbox{-2}}}{{equilibrium\ value\ of\ Casp3^*\ for\ ISS}}$$ to examine how different are the equilibrium values of Casp3$^*$ for the two models when changing initial concentrations of a certain species. When initial concentrations of some species are changed, $\alpha_{C3^*}$ will likely change too. For a good simplified model, such a quantity should be close to one.
We compute $\alpha_{C3^*}$ for changing initial concentrations of each species, including Casp3$^*$, by one or two orders of magnitude higher and lower than the baseline values as before. The result is given in Fig. \[Fig:The matching of FSISS to ISS for different initial concentration.\]. This result illustrates that $\alpha_{C3^*}$ is insensitive to most of initial concentration changes, except a little sensitivity for Bcl2 and Bax. In conclusion, our ISS-2 model well retains the main features of the ISS model and therefore can be viewed as a reliable simplification to the original ISS model.
Summary
=======
In this paper, we develop a general framework of the PEA method together with two conditions, under which the method can be justified rigorously. These conditions are the fastness assumption and the principle of detailed balance for fast reactions as a whole. Under these conditions, we simplify a general system of chemical reactions governed by the law of mass action. This simplification clearly has a solid mathematical basis.
Then we follow the general framework and study the ISS (intracellular–signaling subsystem) model as the downstream process of the Fas-signaling pathway model proposed by Hua et al. (2005) for human Jurkat T cells. Because the framework has a solid mathematical basis, it can be used as a tool to determine whether or not a reaction is relatively fast. After many attempts via numerical tests, we found that nine of reactions in the ISS model can be well regarded as fast.
Knowing that the nine reactions are faster than others, we use the justified PEA method and simplify the ISS model to derive a so-called ISS-2 model. It is remarkable that the reversible reactions in apoptosis obey the principle of detailed balance naturally. With numerical simulations, we compare the ISS-2 model with the ISS model as well as Okazaki’s ISS skeleton model in several aspects, including the accuracy, M-D transition behavior and sensitivity analysis. All the simulations show that the ISS-2 model is reliable. In particular, the new model can very well capture the M-D transition behavior of the ISS model at large initial concentrations of Casp9 and therefore improves Okazaki’s ISS skeleton model considerably (see Fig. \[Fig.M-D transition of FSISS.2\]).
At present, we are trying to simplify the upstream process with the justified PEA method. In the future, we will also try to simplify the whole process by correctly combining the PEA method and the QSSA method.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the National Natural Science Foundation of China (NSFC 10971113) and by Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100002110085).
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[^1]: Zhou Pei-Yuan Center for Appl. Math., Tsinghua university, Beijing 100084, China; Email: [email protected]
[^2]: Zhou Pei-Yuan Center for Appl. Math., Tsinghua university, Beijing 100084, China; Email: [email protected]
|
---
abstract: |
Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc ${\mathbb{D}}$ and satisfies $$\label{eq:dag}
\sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$}$$ By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space $\mathcal{B}$. In this paper it is shown that any coefficient condition implying fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient $A$ if all solutions of $f''+Af=0$ belong to $\mathcal{B}$? An overall revised look into slowly growing solutions is presented, emphasizing function spaces $\mathcal{B}$, ${\rm BMOA}$ and ${\rm VMOA}$.
address: |
Department of Physics and Mathematics, University of Eastern Finland\
P.O. Box 111, FI-80101 Joensuu, Finland
author:
- Janne Gröhn
title: Slowly growing solutions of ODEs revisited
---
[^1]
Introduction
============
Let $\mathcal{H}({\mathbb{D}})$ denote the collection of analytic functions in the (open) unit disc ${\mathbb{D}}$ of the complex plane ${\mathbb{C}}$. It is well-known that the growth of the coefficient $A\in\mathcal{H}({\mathbb{D}})$ controls the growth of solutions $f\in\mathcal{H}({\mathbb{D}})$ of the linear differential equation $$\label{eq:de2}
f''+Af=0,$$ and vice versa. The recent study [@GHR:preprint] concerns conditions, given in terms of the coefficient $A$, which imply that all solutions of belong to a given space of slowly growing analytic functions. Special attention is paid to ${\mathcal{B}}$ (Bloch space), ${\rm BMOA}$ (analytic functions of bounded mean oscillation) and ${\rm VMOA}$ (analytic functions of vanishing mean oscillation). These coefficient conditions have in common that they all imply $${\lVertA\rVert}_{\mathcal{L}^1} = \sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|^2)^2\log\frac{e}{1-|z|} < \infty,$$ which is the subject of this research. The operator theoretic approach in [@GHR:preprint] is based on duality relations, in contrast to this paper, where more classical tools are employed.
The search for coefficient conditions forcing all solutions of to be of slow growth has been active for many years. In the 1997 summer school *Function Spaces and Complex Analysis* (Mekrijärvi Research Station, Finland), N. Danikas posed the following problem:
1. Find a sharp condition for the coefficient $A$ which implies that all solutions of belong to $\mathcal{B}$.
It is known that, if ${\lVertA\rVert}_{\mathcal{L}^1}$ is sufficiently small, then all solutions of belong to $\mathcal{B}$. This result was recently discovered with the best possible upper bound for ${\lVertA\rVert}_{\mathcal{L}^1}$ in [@HKR:2016 Corollary 4(b) and Example 5(b)]. This means that in the language of $\mathcal{L}^1$-norms, the problem (Q) has been solved. The alternative approach in [@GHR:preprint] produces a family of coefficient conditions, which all fall into the category $A\in\mathcal{L}^1$, see [@GHR:preprint Theorems 10 and 11].
Our intention is to take a revised look into slowly growing solutions of , and in particular, to concentrate to the borderline case $A\in\mathcal{L}^1$. We show that any coefficient condition implying $A\in\mathcal{L}^1$ is not sufficiently delicate to detect certain special cases when Bloch solutions do appear. In this sense, the problem (Q) remains open as the most natural description is yet to be found. The converse problem is addressed in Section \[sec:converse\].
Results
=======
Growth of solutions
-------------------
Our first result solves the problem (Q) in terms of the maximum modulus $M_\infty(r,A) = \max_{|z|=r} |A(z)|$, $0\leq r<1$.
\[thm:imp\] Let $A\in\mathcal{H}({\mathbb{D}})$. If there exists $0\leq r_0<1$ such that $$\label{eq:impass}
\sup_{r_0<r<1} \, M_\infty(r,A) (1-r)^2 \exp \!\left( \, \int_{r_0}^r M_\infty(t,A) (1-t) \, dt \right) < \infty,$$ then all solutions of belong to $\mathcal{B}$.
Theorem \[thm:imp\] is based on a representation formula for solutions of and the following elementary observation. If $f$ is a solution of for $A\in\mathcal{H}({\mathbb{D}})$, then $f$ belongs to the Bloch space $$\mathcal{B} = \Big\{ f\in\mathcal{H}({\mathbb{D}}) : {\lVertf\rVert}_{\mathcal{B}} = \sup_{z\in{\mathbb{D}}} |f'(z)| (1-|z|^2) < \infty \Big\}$$ if and only if $$\label{eq:bb}
\sup_{z\in{\mathbb{D}}} |f(z)| |A(z)| (1-|z|^2)^2< \infty.$$ Theorem \[thm:imp\] sharpens [@HKR:2016 Corollary 4(b)], but fails to be an optimal solution to the problem (Q) as it shares the same defects with other known solutions; see Remarks \[remark:defects\] and \[remark:imp\] below.
The growth space $\mathcal{L}^\alpha$ for $0\leq \alpha<\infty$ consists of those $A\in\mathcal{H}({\mathbb{D}})$ for which $${\lVertA\rVert}_{\mathcal{L}^\alpha} = \sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|^2)^2 \left( \log\frac{e}{1-|z|} \right)^\alpha < \infty.$$ The space $\mathcal{L}^0$ appears several times in the literature, and is usually denoted by $H^\infty_2$ or $\mathcal{A}^{-2}$. In the sense of it seems to be the correct ballpark for the study of Bloch solutions of . However, even if ${\lVertA\rVert}_{\mathcal{L}^0}$ is arbitrarily small, it is possible that all non-trivial solutions ($f\not\equiv 0$) of lie outside $\mathcal{B}$; see Example \[ex:nonbloch\] below. If $A\in\mathcal{L}^\alpha$ for $1<\alpha<\infty$, then all solutions of are bounded in ${\mathbb{D}}$ by [@H:2000 Theorem 4.2]. As explained in the Introduction, if ${\lVertA\rVert}_{\mathcal{L}^1}$ is sufficiently small, then all solutions of belong to the Bloch space, while the weaker condition $A\in\mathcal{L}^1$ allows some solutions to lie outside $\mathcal{B}$. The following result is in line with the heuristic principle which claims that *small change in ${\lVertA\rVert}_{\mathcal{L}^1}$ has a huge impact on solutions of* .
\[thm:nn\] If ${\lVertA\rVert}_{\mathcal{L}^1} < 4/n$ for $n\in{\mathbb{N}}$, then all solutions $f$ of satisfy $f,f^2, \dotsc,f^n \in \mathcal{B}$.
For $1/2<\alpha<\infty$, the coefficient condition $A\in\mathcal{L}^\alpha$ places all solutions of in $\bigcap_{0<p<\infty} H^p$, see [@R:2007 Corollary 1.9]. This property is no longer true for $\alpha=1/2$ as certain solutions may lie outside the Nevanlinna class $\mathcal{N}$; apply [@P:1982 Theorem 4] to $Q(r)=(1-r)^{-2} ( \log(e/(1-r)))^{-1/2}$, $0\leq r<1$. It seems that non-Nevanlinna solutions produced in this manner do not belong to $\mathcal{B}$ as they are exponentials of very badly behaved Bloch functions themselves. The following result indicates that not all Bloch solutions of are smooth enough to be contained in $\mathcal{N}$. By the discussion above, these solutions cannot be detected by any coefficient condition which implies $A\in \mathcal{L}^1$.
As usual, the Hardy space $H^p$ for $0<p<\infty$ consists of $f\in \mathcal{H}({\mathbb{D}})$ for which $${\lVertf\rVert}_{H^p}^p = \lim_{r\to 1^-} \, \frac{1}{2\pi} \, \int_0^{2\pi} |f(re^{i\theta})|^p \, d\theta < \infty,$$ while the Nevanlinna class $\mathcal{N}$ contains $f\in\mathcal{H}({\mathbb{D}})$ such that $$\lim_{r\to 1^-} \, \frac{1}{2\pi} \, \int_0^{2\pi} \log^+ |f(re^{i\theta})| \, d\theta<\infty,
\quad \log^+ = \max \{ \log, 0\}.$$
\[thm:ex\] Let $0<C<\infty$. Then, there exists a coefficient $A\in\mathcal{H}({\mathbb{D}})$ with ${\lVertA\rVert}_{\mathcal{L}^0} < C$ such that admits a (zero-free) solution $f\in \mathcal{B} \setminus \mathcal{N}$.
The following result complements Theorem \[thm:ex\] by offering a condition under which non-Nevanlinna solutions do not appear.
\[thm:zfbase\] If ${\lVertA\rVert}_{\mathcal{L}^0}\leq 1$ and there exists one zero-free solution of which belongs to $\bigcup_{0<p<\infty} H^p$, then all solutions of are in $\bigcup_{0<p<\infty} H^p$.
The coefficient condition ${\lVertA\rVert}_{\mathcal{L}^0} \leq 1$ corresponds to the classical univalency criterion [@N:1949 Theorem I] due to Nehari, which implies that all non-trivial solutions of have at most one zero in ${\mathbb{D}}$. Theorem \[thm:zfbase\] should be compared to [@H:2013 Theorem 4] which holds in a more general setting.
Oscillation of solutions
------------------------
If $A\in\mathcal{H}({\mathbb{D}})$ and there exists $0<R<1$ such that $|A(z)| (1-|z|^2)^2 \leq 1$ for all $R<|z|<1$, then all non-trivial solutions of vanish at most finitely many times in ${\mathbb{D}}$ [@S:1955 Theorem 1]. This is the case, in particular, if $A\in\mathcal{L}^\alpha$ for any $0<\alpha<\infty$. The following example concerns a case when all solutions belong to $\mathcal{B}$ while one of them has infinitely many zeros. This is Hille’s example, see [@H:1949] and [@S:1955 p. 162].
\[ex:hille\] Let $0<\gamma<\infty$. On one hand, all solutions of the differential equation for $A(z)=(1+4\gamma^2)/(1-z^2)^2$, $z\in{\mathbb{D}}$, are bounded and hence in $\mathcal{B}$. This follows from the estimates in [@S:2012 p. 131], for example. On the other hand, the particular solution $$f(z) = \sqrt{1-z^2} \, \sin \!\left( \gamma \log\frac{1+z}{1-z} \right), \quad z\in{\mathbb{D}},$$ has infinitely many (real) zeros $z_n = (e^{\pi n/\gamma}-1)/(e^{\pi n/\gamma}+1)$, $n\in{\mathbb{Z}}$. $\diamond$
\[remark:defects\] By the discussion above, the coefficient condition $A\in\mathcal{L}^1$ implies that all non-trivial solutions of belong to $\bigcap_{0<p<\infty} H^p$ and have at most finitely many zeros. We have shown that neither of these properties is characteristic to Bloch solutions of under the restriction $A\in \mathcal{L}^0$.
We point out that, although $A\in\mathcal{L}^1$ is not sufficient to place all solutions of in $\mathcal{B}$, it guarantees that solutions are normal in the sense $$\sup_{z\in{\mathbb{D}}} \, f^{\#}(z) (1-|z|^2) = \sup_{z\in{\mathbb{D}}} \, \frac{|f'(z)|}{1+|f(z)|^2} \, (1-|z|^2) < \infty.$$ This follows from [@GNR:preprint Proposition 7] by using the fact that all non-trivial solutions have at most finitely many zeros provided that $A\in\mathcal{L}^1$.
Solutions of finite valance
---------------------------
Let $n(f,\zeta) = \# \{ z\in{\mathbb{D}}: f(z)=\zeta\}$ be the counting function for $\zeta$-points of $f\in\mathcal{H}({\mathbb{D}})$; let $D(z,r)$ denote the Euclidean disc of radius $0<r<\infty$ centered at $z\in{\mathbb{D}}$; and let $dm$ be the Lebesgue area measure. According to [@P:1977 Satz 1], if $f\in\mathcal{B}$ and $$\label{eq:intnumber}
V_f= \sup_{z\in{\mathbb{C}}} \, \int_{D(z,1)} n(f,\zeta)\, dm(\zeta) < \infty,$$ then $f\in{\rm BMOA}$. Hence, Bloch functions of finite valence belong to ${\rm BMOA}$. Recall that $f\in{\rm BMOA}$ if and only if ${\lVertf\rVert}_{{\rm BMOA}}^2 = \sup_{a\in{\mathbb{D}}}\, {\lVertg_a\rVert}_{H^2}^2<\infty$, where $g_a(z) = f(\varphi_a(z)) - f(a)$ and $\varphi_a(z)=(a-z)/(1-\overline{a}z)$ for $a,z\in{\mathbb{D}}$.
If ${\lVertA\rVert}_{\mathcal{L}^1}$ is sufficiently small, then all finitely valent solutions of are not only in ${\rm BMOA}$ but also possess a specific type of regularity.
\[thm:bmoa\_not\] Let $A\in \mathcal{L}^1$. If $f$ is a solution of which satisfies , then $$\label{eq:newest}
\int_{{\mathbb{D}}} |f'(z)|^2 \left( \log\frac{e}{1-|z|} \right)^{-\beta} dm(z)<\infty$$ for any ${\lVertA\rVert}_{\mathcal{L}^1}/2<\beta<\infty$.
Example \[ex:valent\] below shows that, regardless of the size of ${\lVertA\rVert}_{\mathcal{L}^1}$, both finitely and infinitely valent (non-trivial) solutions of are possible.
Converse problem {#sec:converse}
----------------
Before going any further, we discuss a problem converse to Theorem \[thm:imp\]: How is the growth of the coefficient $A\in\mathcal{H}({\mathbb{D}})$ restricted if all solutions of are in $\mathcal{B}$?
The argument in [@S:2012] reveals the following estimates. Let $f_1,f_2$ be linearly independent *bounded* solutions of for $A\in\mathcal{H}({\mathbb{D}})$. Without any loss of generality, we may assume that $f_1 f_2' - f_1'f_2 = 1$. By a straight-forward computation $A = f_1'f_2''- f_1''f_2'$, and therefore $\sup_{z\in{\mathbb{D}}} |A(z)| (1-|z|^2)^3 < \infty$. Moreover, the spherical derivative $w^{\#} = |w'|/(1+|w|^2)$ of $w=f_1/f_2$ satisfies $w^{\#} = 1/(|f_1|^2 + |f_2|^2) \leq |f_1'|^2 + |f_2'|^2$, and hence $\sup_{z\in{\mathbb{D}}} \, w^{\#}(z) (1-|z|^2)^2 < \infty$. It is clear that these estimates withstand the weaker assumption $f_1,f_2\in\mathcal{B}$. The following result improves the growth estimate for $A$ and is related to a problem mentioned in [@S:2012 p. 131].
\[thm:conv\_preli\] Let $f_1,f_2\in\mathcal{B}$ be linearly independent solutions of for $A\in\mathcal{H}({\mathbb{D}})$. Then, $\sup_{z\in{\mathbb{D}}} |A(z)|(1-|z|^2)^{5/2}\lesssim\max\{{\lVertf_1\rVert}_{\mathcal{B}},{\lVertf_2\rVert}_{\mathcal{B}}\}<\infty$.
Here $\lesssim$ denotes a one sided estimate up to a constant. The betting is that Theorem \[thm:conv\_preli\] is not sharp. It would be desirable to show $A\in\mathcal{L}^0$ if $f_1,f_2\in\mathcal{B}$. We do not know whether this is true (even for $f_1,f_2$ bounded), however. Theorem \[thm:conv\_preli\] fails to be true if we have information only on one non-trivial solution of . For example, $f(z)=\exp(-(1+z)/(1-z))$ is a bounded solution of for $A(z)=-4z/(1-z)^4$, $z\in{\mathbb{D}}$. In this case admits also non-Bloch solutions such as $$f(z) \int_0^z \frac{1}{f(\zeta)^2} \, d\zeta, \quad z\in{\mathbb{D}},$$ which is linearly independent to $f$ and grows too fast on the positive real axis to be included in $\mathcal{B}$ (by the Bernoulli-l’Hôpital theorem).
Let $A\in\mathcal{H}({\mathbb{D}})$. If there exist linearly independent *bounded* solutions $f_1, f_2$ of such that $\inf_{z\in{\mathbb{D}}} \, (|f_1(z)|+ |f_2(z)|) >0$, then $A\in \mathcal{L}^0$ by an argument based on the corona theorem [@GHR:preprint p. 3]. We extend this observation for $\mathcal{B}$ with an argument independent of the corona theorem.
\[thm:conv\] Let $f_1,f_2\in\mathcal{B}$ be linearly independent solutions of for $A\in\mathcal{H}({\mathbb{D}})$ such that $\inf_{z\in{\mathbb{D}}} \, (|f_1(z)|+ |f_2(z)|) >0$. Then, $A\in \mathcal{L}^0$ and $(f_1/f_2)^{\#}$ is bounded in ${\mathbb{D}}$.
Solutions of bounded and vanishing mean oscillation {#sec:25}
---------------------------------------------------
Coefficient conditions, which place all solutions of in ${\rm BMOA}$, are considered in [@GHR:preprint]. We derive a result similar to [@GHR:preprint Theorem 3] by using known growth estimates for solutions of . This method is somewhat surprising, since it was not known to work with slowly growing solutions. By the Carleson measure description in [@Z:2003 Theorem 1], Theorem \[thm:bmoa\] is weaker than [@GHR:preprint Theorem 3].
\[thm:bmoa\] Let $A\in\mathcal{H}({\mathbb{D}})$. If $$\label{eq:bmoaa}
\sup_{a\in{\mathbb{D}}} \, \left( \log\frac{e}{1-|a|} \right) \int_{{\mathbb{D}}} |A(z)| (1-|\varphi_a(z)|^2) \, dm(z)$$ is sufficiently small, then all solutions of belong to ${\rm BMOA}$.
Coefficient conditions, which place all solutions of in ${\rm VMOA}$, are also discussed in [@GHR:preprint]. We consider two related results which, as opposed to ones in [@GHR:preprint], are given in terms of the radial growth of the coefficient. Recall that $f\in{\rm VMOA}$ if and only if ${\lVertg_a\rVert}_{H^2}^2\to 0^+$ as $|a|\to 1^-$.
\[thm:vmoa\] Let $A\in\mathcal{H}({\mathbb{D}})$. If there exists $0\leq r_0<1$ such that $$\label{eq:vmoageneral}
\int_{r_0}^1 M_\infty(r,A)^2
\exp\!\left(\, 2 \int_{r_0}^r M_\infty(t,A) (1-t) \, dt \right) (1-r^2)^3\, dr<\infty,$$ then all solutions of belong to ${\rm VMOA}$.
Theorem \[thm:vmoa\] gives rise to the following corollary. The coefficient condition allows solutions of to be unbounded, see Example \[ex:unbounded\] below.
\[cor:vmoa\] Let $A\in\mathcal{H}({\mathbb{D}})$. If $$\label{eq:corvmoa}
\sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|^2)^2 \left( \log\frac{e}{1-|z|} \right) \log \log\frac{e}{1-|z|} < \infty,$$ then all solutions of belong to ${\rm VMOA}$.
Proof of Theorem \[thm:imp\] {#sec:f}
============================
The following proof is based on the growth estimate [@H:2000 Theorem 4.2] for solutions of . The known approaches to Bloch solutions of depend on other methods (duality relations [@GHR:preprint] and straight-forward integration [@HKR:2016]).
Let $f$ be a non-trivial solution of , and let $0\leq r_0<1$ be fixed. If $r_0<r<1$ and $e^{i\theta}\in\partial{\mathbb{D}}$, then $$f(re^{i\theta}) = f(r_0 e^{i\theta}) + f'(r_0e^{i\theta}) (re^{i\theta}-r_0 e^{i\theta})
- \int_{r_0 e^{i\theta}}^{re^{i\theta}} f(\zeta) A(\zeta) (re^{i\theta}-\zeta) \, d\zeta,$$ by the representation theorem [@H:2000 Theorem 4.1]. Therefore $$|f(r e^{i\theta})| \leq \Big( M_\infty(r_0,f) + M_\infty(r_0,f')(1-r_0) \Big)
\exp\!\left(\, \int_{r_0}^r |A(t e^{i\theta})| (1-t) \, dt \right)$$ by Gronwall’s lemma [@L:1993 Lemma 5.10]. This growth estimate, the assumption , and the identity $f''=-Af$ imply that $f''\in \mathcal{L}^0$. This completes the proof as $f\in\mathcal{B}$ by [@Z:2007 Theorem 5.4].
We proceed to show that Theorem \[thm:imp\] sharpens [@HKR:2016 Corollary 4(b)].
\[remark:imp\] Suppose that the coefficient condition in [@HKR:2016 Corollary 4(b)] holds, that is, $A\in\mathcal{H}({\mathbb{D}})$ and $$\label{eq:bz}
\sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|)^2 \int_0^{|z|} \frac{dr}{1-r} < 1.$$ Fix any $0<r_0<1$, and compute $$\begin{aligned}
& \sup_{r_0<r<1} \, M_\infty(r,A) (1-r)^2 \exp \!\left( \, \int_{r_0}^r M_\infty(t,A) (1-t) \, dt \right)\\
& \qquad \leq \sup_{r_0<r<1} \,\frac{1}{\log\frac{1}{1-r}} \, \exp\!\left( \log\log\frac{1}{1-r} - \log\log\frac{1}{1-r_0} \right)
= \frac{1}{\log\frac{1}{1-r_0}}.\end{aligned}$$ Therefore, the assumptions of Theorem \[thm:imp\] are satisfied. We point out that Theorem \[thm:imp\] applies also to cases such as $A(z) = (1-z)^{-2} ( \log (e/(1-z)))^{-1}$, $z\in{\mathbb{D}}$, for which [@HKR:2016 Corollary 4(b)] is inconclusive. In particular, Theorem \[thm:imp\] can be utilized even with equality in .
Under an additional smoothness assumption, the coefficient condition falls also into the category $A\in\mathcal{L}^1$. This is the case, for example, if $M_\infty(r,A)(1-r)^2(\log (e/(1-r))$, $r_0<r<1$, is increasing. $\diamond$
The following example shows that, even if ${\lVertA\rVert}_{\mathcal{L}^0}$ is arbitrarily small, it is possible that *all* non-trivial solutions of lie outside $\mathcal{B}$.
\[ex:nonbloch\] Let $1<\gamma<\infty$ be fixed. The differential equation for $A(z)=(1-\gamma^2)/(1-z^2)^2$, $z\in{\mathbb{D}}$, admits linearly independent solutions $$f_1(z) = \frac{(1+z)^{(\gamma+1)/2}}{(1-z)^{(\gamma-1)/2}}, \quad
f_2(z) = \frac{(1-z)^{(\gamma+1)/2}}{(1+z)^{(\gamma-1)/2}}, \quad z\in{\mathbb{D}},$$ which clearly satisfy $f_1,f_2\notin \mathcal{B}$. Since the singularities of $f_1,f_2$ are located at distinct points, we conclude that all linear combinations of $f_1,f_2$, and therefore all non-trivial solutions of , lie outside $\mathcal{B}$. $\diamond$
Proof of Theorem \[thm:nn\]
===========================
We begin with an auxiliary result, which shows that the coefficient condition $A\in\mathcal{L}^1$ is associated with solutions of at most logarithmic growth. This should be compared to the case of the coefficient condition $A\in\mathcal{L}^0$, which implies that all solutions of satisfy $\sup_{z\in{\mathbb{D}}} |f(z)| (1-|z|^2)^p < \infty$ for sufficiently large $p=p({\lVertA\rVert}_{\mathcal{L}^0})< \infty$, see [@P:1982 Example 1].
\[lemma:reprelog\] Let $A\in \mathcal{L}^1$.
1. All solutions $f$ of satisfy $$\label{eq:logest}
\sup_{z\in{\mathbb{D}}} \, |f(z)| \left( \log\frac{e}{1-|z|} \right)^{-\alpha} < \infty$$ for ${\lVertA\rVert}_{\mathcal{L}^1}/4 < \alpha<\infty$.
2. Any solution $f$ of , which satisfies for $\alpha=1$, belongs to $\mathcal{B}$.
The proof of Lemma \[lemma:reprelog\](i) resembles that of [@GR:2017 Theorem 2]; a similar estimate could also be obtained from [@H:2000 Theorem 4.2]. Lemma \[lemma:reprelog\](ii) is an immediate consequence of and [@Z:2007 Theorem 5.4], but plays an important role in the proof of Theorem \[thm:nn\].
\(i) Let $f$ be a solution of , and $0\leq \delta<R<1$. Since $$|f(z)|
\leq \int_\delta^{|z|} \!\!\!\int_\delta^t \, \big| f''(sz/|z|) \big| \, ds dt
+M_\infty(\delta,f') + M_\infty(\delta,f), \quad \delta<|z|<1,$$ we obtain $$\begin{aligned}
\sup_{\delta < |z| <R} \, \frac{|f(z)|}{\big( \log\frac{e}{1-|z|} \big)^\alpha}
& \leq \left( \sup_{\delta < |\zeta| <R} \, \frac{|f(\zeta)|}{\big( \log\frac{e}{1-|\zeta|} \big)^\alpha} \right) {\lVertA\rVert}_{\mathcal{L}^1}
\, \sup_{\delta < |z| <R} I_\alpha(z) \\
& \qquad + M_\infty(\delta,f') + M_\infty(\delta,f),\end{aligned}$$ where $I_\alpha(z)$ is as below. Since $$\lim_{|z|\to 1^-} I_\alpha(z)
= \lim_{|z|\to 1^-} \left( \log\frac{e}{1-|z|} \right)^{-\alpha}\int_0^{|z|}
\!\!\!\int_0^t \, \frac{\big( \log\frac{e}{1-s} \big)^{\alpha-1}}{(1-s^2)^2} \, ds dt = \frac{1}{4\alpha}$$ by the Bernoulli-l’Hôpital theorem, we deduce for ${\lVertA\rVert}_{\mathcal{L}^1}/4 < \alpha<\infty$ by choosing a sufficiently large $0\leq \delta<1$, reorganizing the terms and finally letting $R\to 1^-$.
If $n=1$, then $f\in\mathcal{B}$ follows directly from Lemma \[lemma:reprelog\]; first, apply part (i) and then (ii). If $n\geq 2$, then we may assume that $f\in\mathcal{B}$ by the first part of the proof. Since ${\lVertA\rVert}_{\mathcal{L}^1} < 2$ by the assumption, every solution $f$ of satisfies for $\alpha=1/2$ by Lemma \[lemma:reprelog\](i). Note that $(f^2)''=2(f')^2 - 2 f^2 A$ by . We deduce $(f^2)''\in \mathcal{L}^0$, which implies $f^2\in\mathcal{B}$.
We proceed by induction. Assume that $f^{k-1}\in \mathcal{B}$ for $2< k\leq n$. As above, we know that $f\in\mathcal{B}$. Since ${\lVertA\rVert}_{\mathcal{L}^1} < 4/n\leq 4/k$ by the assumption, every solution $f$ of satisfy for $\alpha=1/k$ by Lemma \[lemma:reprelog\](i). Now $$(f^k)''= k f' (f^{k-1})' - k f^k A$$ by . We deduce $(f^k)'' \in \mathcal{L}^0$, which gives $f^k\in\mathcal{B}$. The claim follows.
Proof of Theorem \[thm:ex\]
===========================
The following proof takes advantage of universal covering maps to create a Bloch function with special properties. Similar arguments appear in the literature several times. The idea for the following Bloch construction is borrowed from [@CCS:1980 p. 229].
Let $0<C<\infty$. By the proof of [@P:1982 Theorem 4], when applied to $Q(r)=C/(1-r)^2$, there exists $g\in \mathcal{B} \setminus \mathcal{N}$ with ${\lVertg\rVert}_{\mathcal{B}} \lesssim C$ such that $f=e^g \not\in\mathcal{N}$ is a solution of for $A=-g''-(g')^2$ with ${\lVertA\rVert}_{\mathcal{L}^0} \leq 4C$.
Let $\mathcal{Z}=\{ x+iy\in{\mathbb{C}}: x,y\in{\mathbb{Z}}\}$ be the set of integral lattice points, and let $E$ be its preimage $E = \{z\in{\mathbb{D}}: f(z) \in \mathcal{Z}\}$. Since $E\subset{\mathbb{D}}$ is a countable closed set, $E$ has capacity zero and therefore the universal covering map from ${\mathbb{D}}$ onto ${\mathbb{D}}\setminus E$ is an inner function [@F:1935]; see also [@S:1979 p. 261]. Let this inner function be denoted by $I$. The function $f\circ I$ belongs to $\mathcal{B}$ since its image, contained in ${\mathbb{C}}\setminus \mathcal{Z}$, does not contain (schlicht) discs of arbitrarily large radius; see [@C:1979 Theorem 2.6], for example. Note that $f\circ I$ is non-vanishing, and define $B\in\mathcal{H}({\mathbb{D}})$ by $$B = -\frac{(f \circ I)''}{f\circ I} = (A \circ I) (I')^2 - (g' \circ I) \, I''.$$ By the Schwarz-Pick lemma, and its extension [@R:1985 Theorem 2], we deduce $$\begin{split}
{\lVertB\rVert}_{\mathcal{L}^0} & \leq \sup_{z\in{\mathbb{D}}} \, (1-|z|^2)^2 \,
\frac{{\lVertA\rVert}_{\mathcal{L}^0}}{(1-|I(z)|^2)^2} \cdot \frac{(1-|I(z)|^2)^2}{(1-|z|^2)^2} \\
& \qquad + \sup_{z\in{\mathbb{D}}} \, (1-|z|^2)^2 \, \frac{{\lVertg\rVert}_{\mathcal{B}}}{1-|I(z)|^2} \cdot \frac{2! \, (1-|I(z)|^2)}{(1-|z|)^2(1+|z|)}\\
& \leq {\lVertA\rVert}_{\mathcal{L}^0} + 4 \, {\lVertg\rVert}_{\mathcal{B}}.
\end{split}$$ We conclude that $h = f\circ I \in \mathcal{B}$ is a zero-free solution of $h''+Bh=0$, where ${\lVertB\rVert}_{\mathcal{L}^0} \lesssim C$ with a comparison constant independent of $C$. Finally, [@S:1979 Proposition 3.3] implies that $f \circ I$ does not belong to $\mathcal{N}$.
Proof of Theorem \[thm:zfbase\]
===============================
The following result shows that slow growth of the coefficient ensures the existence of zero-free solution bases.
\[lemma:zfbase\] If ${\lVertA\rVert}_{\mathcal{L}^0}\leq 1$, then admits linearly independent zero-free solutions $f_1$ and $f_2$ such that $\log f_1 - \log f_2 \in{\rm BMOA}$.
If $A\in\mathcal{L}^0$, then any zero-free solution $f$ of satisfies $\log f\in\mathcal{B}$ by [@GNR:preprint Theorem 4(ii)]. The contribution of Lemma \[lemma:zfbase\] lies in the fact that linearly independent zero-free solutions are shown to be closely related to each other. If $A\in\mathcal{L}^1$, then any zero-free solution $f$ of satisfies $\log f\in{\rm BMOA}$ by [@GNR:preprint Theorem 4(i)], and therefore the $\mathcal{L}^1$-counterpart of Lemma \[lemma:zfbase\] is trivial.
Let $g_1$ and $g_2$ be linearly independent solutions of where ${\lVertA\rVert}_{\mathcal{L}^0}\leq 1$. It follows that $h=g_1/g_2$ is a locally univalent meromorphic (not necessarily analytic) function whose Schwarzian derivative $S_h = 2A$ satisfies ${\lVertS_h\rVert}_{\mathcal{L}^0}\leq 2$, and therefore $h$ is univalent in ${\mathbb{D}}$ by [@N:1949 Theorem I]. Consequently, there exist two distinct values $\zeta_1,\zeta_2\in{\mathbb{C}}\cup \{\infty\}$ which belong to the complement of $h({\mathbb{D}})$ with respect to the extended complex plane. If $\zeta_j\in{\mathbb{C}}$ then define $f_j = g_1 - \zeta_j g_2$, while otherwise let $f_j=g_2$. We conclude that $f_1$ and $f_2$ are linearly independent zero-free solutions of .
Finally, $w=f_1/f_2$ is a locally univalent analytic zero-free function, whose Schwarzian derivative agrees with $S_h$. It follows that $w$ is univalent, and therefore $\log w \in{\rm BMOA}$ by [@B:1980 Corollary 1, p. 21]. The claim follows.
Let $f_1$ and $f_2$ be linearly independent non-vanishing solutions of . Their existence follows from ${\lVertA\rVert}_{\mathcal{L}^0}\leq 1$ as in the proof of Lemma \[lemma:zfbase\]. Without loss of generality, we may assume that $f_2\in\bigcup_{0<p<\infty} H^p$ is the zero-free solution given by the hypothesis. Any solution $f$ of can be represented in the form $f=\alpha f_1 + \beta f_2= f_2 \, ( \alpha \, e^{\log f_1- \log f_2} + \beta ),$ where $\alpha,\beta\in{\mathbb{C}}$ are constants depending on $f$. Since $\log f_1 - \log f_2\in{\rm BMOA}$ by Lemma \[lemma:zfbase\], we deduce $\exp(\log f_1 - \log f_2)\in \bigcup_{0<p<\infty} H^p$ by [@CS:1976 Theorem 1]. This proves the assertion.
Proof of Theorem \[thm:bmoa\_not\]
==================================
Theorem \[thm:bmoa\_not\] reveals that finitely valent solutions possess a unique property, which is not even found from all bounded analytic functions. To construct a bounded function $f\in\mathcal{H}({\mathbb{D}})$ for which fails, consider a Blaschke sequence which is not a zero-sequence for the weighted Dirichlet space $\mathcal{D}_s$ for fixed $0<s<1$, and let $f$ be the corresponding Blaschke product. See [@PP:2011 p. 1981] for more details.
Let $\beta$ be any constant such that ${\lVertA\rVert}_{\mathcal{L}^1}/2<\beta<\infty$, and fix $\alpha$ such that ${\lVertA\rVert}_{\mathcal{L}^1}/4 < \alpha<\beta/2$. Since $f$ is a solution of for $A\in\mathcal{L}^1$, holds by Lemma \[lemma:reprelog\](i). As in [@P:1977 p. 593], we compute $$\begin{aligned}
\int_{D(0,r)} |f'(z)|^2 \, dm(z) & = \int_{f(D(0,r))} \Bigg( \sum_{z\in{\mathbb{D}}\, : f(z)=\zeta} 1 \Bigg) \, dm(\zeta)\\
& \leq \int_{D(0,M_\infty(r,f))} n(f,\zeta)\, dm(\zeta)
\leq 4 \big( M_\infty(r,f) + 1 \big)^2 \cdot V_f\\
& \lesssim \left( \log\frac{e}{1-r} \right)^{2\alpha}, \quad 0<r<1,\end{aligned}$$ by and the generic change of variable formula [@A:1992 Proposition 2.1]. Now $$\begin{aligned}
& \int_{{\mathbb{D}}} |f'(z)|^2 \left( \log\frac{e}{1-|z|} \right)^{-\beta} \, dm(z) \\
& \qquad = \int_0^1 \left( \int_{D(0,r)} |f'(z)|^2 \, dm(z) \right) \frac{\beta}{(1-r) \big( \log\frac{e}{1-r} \big)^{\beta+1}} \, dr \\
& \qquad \lesssim \int_0^1 \frac{dr}{(1-r) \big( \log\frac{e}{1-r} \big)^{1+ \beta-2\alpha}} < \infty\end{aligned}$$ by Fubini’s theorem.
The following example concerns the valence of solutions of .
\[ex:valent\] Let $0<\alpha<1$. As in [@HKR:2016 Example 5(b)], we conclude that $f(z)=(\log(e/(1-z)))^\alpha$ is a solution of for $$A(z)=\frac{-\alpha}{(1-z)^2} \bigg( (\alpha-1)\left(\log\frac{e}{1-z} \right)^{-2}
+\left(\log\frac{e}{1-z}\right)^{-1} \bigg), \quad z\in{\mathbb{D}},$$ where ${\lVertA\rVert}_{\mathcal{L}^1}\lesssim \alpha$. Since $z\mapsto \log(e/(1-z))$ is univalent in ${\mathbb{D}}$, we see that $f$ is finitely valent for $\alpha\in (0,1) \cap \mathbb{Q}$ and infinitely valent for $\alpha\in (0,1) \setminus \mathbb{Q}$. $\diamond$
Proofs of Theorems \[thm:conv\_preli\] and \[thm:conv\]
=======================================================
Let $f_1$ and $f_2$ be linearly independent solutions of for $A\in\mathcal{H}({\mathbb{D}})$. We may assume that the Wronskian determinant satisfies $f_1f_2'-f_1'f_2 = 1$. Differentiate this identity once to obtain $f_1f_2''-f_1''f_2 = 0$, and differentiate it twice to deduce $f_1'''f_2 - f_1 f_2''' = f_1'f_2'' - f_1''f_2' = A$, where the last equality follows from .
Since $f_1,f_2\in \mathcal{B}$, we conclude that $f_1'',f_2''\in \mathcal{L}^0$. Define $h(z) = |f_1(z)| + |f_2(z)|$ for $z\in{\mathbb{D}}$. Function $h$ is non-vanishing as the Wronskian determinant satisfies $f_1f_2'-f_1'f_2 = 1$. On one hand, $$\begin{split}
|A(z)| & = \frac{|f_1(z) A(z)| + |f_2(z) A(z)|}{|f_1(z)| + |f_2(z)|} = \frac{|f_1''(z)|+|f_2''(z)|}{|f_1(z)| + |f_2(z)|}\\
& \lesssim \frac{\max\{{\lVertf_1\rVert}_{\mathcal{B}},{\lVertf_2\rVert}_{\mathcal{B}}\}}{(1-|z|^2)^3} \cdot \frac{1-|z|^2}{h(z)}, \quad z\in{\mathbb{D}},
\end{split}$$ with an absolute comparison constant. On the other hand, $$|A(z)| \leq |f_1'''(z)| |f_2(z)| + |f_1(z)| |f_2'''(z)|
\lesssim \frac{\max\{{\lVertf_1\rVert}_{\mathcal{B}},{\lVertf_2\rVert}_{\mathcal{B}}\}}{(1-|z|^2)^3} \, h(z), \quad z\in{\mathbb{D}}.$$
Since $\min\{ x/y, y \} \leq \sqrt{x}$ for all $0<x,y<\infty$, we obtain $$\min\left\{ (1-|z|^2)/h(z), \, h(z)\right\} \leq \sqrt{1-|z|^2}, \quad z\in{\mathbb{D}}.$$ The assertion $\sup_{z\in{\mathbb{D}}} |A(z)|(1-|z|^2)^{5/2}\lesssim\max\{{\lVertf_1\rVert}_{\mathcal{B}},{\lVertf_2\rVert}_{\mathcal{B}}\}$ follows.
The proof of Theorem \[thm:conv\] is similar to the one above, with the difference that the auxiliary function $h$ in the proof of Theorem \[thm:conv\_preli\], is now uniformly bounded away from zero by the assumption.
Since $f_1,f_2\in \mathcal{B}$, we have $f_1'',f_2''\in \mathcal{L}^0$. By , $$\begin{split}
\sup_{z\in{\mathbb{D}}} \, |A(z)| (1-|z|^2)^2 & = \sup_{z\in{\mathbb{D}}}\, \frac{|f_1(z) A(z)| + |f_2(z) A(z)|}{|f_1(z)| + |f_2(z)|} \, (1-|z|^2)^2 \\
& \leq \left( \, \inf_{z\in{\mathbb{D}}} \big( |f_1(z)| + |f_2(z)| \big) \right)^{-1} \big( {\lVertf_1''\rVert}_{\mathcal{L}^0} + {\lVertf_2''\rVert}_{\mathcal{L}^0} \big).
\end{split}$$ Let $w=f_1/f_2$, which implies that $w'=-1/f_2^2$. To see that $w^{\#}$ is bounded in ${\mathbb{D}}$, it suffices to write $$\label{eq:mmest}
\begin{split}
\sup_{z\in{\mathbb{D}}} \, w^{\#}(z) & = \sup_{z\in{\mathbb{D}}} \, \frac{1}{|f_1(z)|^2 + |f_2(z)|^2}
\leq \sup_{z\in{\mathbb{D}}} \, \frac{2}{\big( |f_1(z)| + |f_2(z)| \big)^2} \\
& \leq 2 \left( \, \inf_{z\in{\mathbb{D}}} \big( |f_1(z)| + |f_2(z)| \big) \right)^{-2}.
\end{split}$$ This completes the proof.
We take the opportunity to mention an interesting application of . Let $f_1,f_2$ be linearly independent solutions of for $A\in\mathcal{H}({\mathbb{D}})$ such that $\inf_{z\in{\mathbb{D}}} \, (|f_1(z)|+ |f_2(z)|) >0$, and let $z_1,z_2\in{\mathbb{D}}$ be (necessarily distinct) points at which $f_1(z_1)=0=f_2(z_2)$. Let $\gamma(z_1,z_2)$ denote the straight line segment from $z_1$ to $z_2$. Since $z_1$ is a (simple) zero of $w=f_1/f_2$, and $z_2$ is a (simple) pole of $w$, we deduce $$1 \lesssim \int_{\gamma(z_1,z_2)} \frac{|w'(z)|}{1+|w(z)|^2} \, |dz|
\leq \left( \, \sup_{z\in{\mathbb{D}}} \, w^{\#}(z) \right) | z_1 - z_2 |$$ as the spherical length of $w(\gamma(z_1,z_2))$ is uniformly bounded from below. Therefore, implies that $|z_1-z_2|$ is uniformly bounded away from zero.
Proof of Theorem \[thm:bmoa\]
=============================
By [@Z:2003 Theorem 1] and the subharmonicity of $|A|$, we deduce $${\lVertA\rVert}_{\mathcal{L}^1} \lesssim \, \sup_{a\in{\mathbb{D}}} \left( \log\frac{e}{1-|a|} \right) \int_{{\mathbb{D}}} |A(z)| (1-|\varphi_a(z)|^2) \, dm(z).$$ Consequently, when proving Theorem \[thm:bmoa\], we may assume that all solutions of are in $\mathcal{B}$ by [@HKR:2016 Corollary 4(b)] or Theorem \[thm:imp\].
Let $f$ be a solution of and consider its normalized hyperbolic translates $g_a(z) = f(\varphi_a(z)) - f(a)$ for $a\in{\mathbb{D}}$. To prove $f\in{\rm BMOA}$ it suffices to show that $$\label{eq:need}
\sup_{a\in{\mathbb{D}}} \sup_{0<r<1} m(r,g_a)
= \sup_{a\in{\mathbb{D}}} \sup_{0<r<1} \, \frac{1}{2\pi} \int_0^{2\pi} \log^+ |g_a(re^{i\theta})| \, d\theta
< \infty$$ by [@B:1980 Corollary 2, p. 15]. We proceed to verify that the proximity functions $m(r,g_a)$ satisfy .
A straight-forward computation reveals that $g_a\in\mathcal{H}({\mathbb{D}})$ is a solution of the non-homogenous linear differential equation $$g_a'' + B_a \, g_a' + C_a \, g_a = - f(a) \, C_a,$$ where $B_a,C_a\in\mathcal{H}({\mathbb{D}})$ are given by $$B_a(z) = - \, \frac{\varphi_a''(z)}{\varphi_a'(z)},
\quad C_a(z) =A\big(\varphi_a(z)\big) \, \varphi_a'(z)^2,
\quad z\in{\mathbb{D}}.$$ By [@HKR:2009 Corollary 3(a)], we deduce $$\begin{aligned}
m(r,g_a) & \lesssim 1 + \log^+ \!\big( |f'(a)|(1-|a|^2) \big) \\
& \qquad + \int_0^{2\pi} \log^+ \!\left( \int_0^r |A(\varphi_a(se^{i\theta}))| |\varphi_a'(se^{i\theta})|^2 |f(a)| (1-s) \, ds \right) d\theta\\
& \qquad + \int_{D(0,r)} |A(\varphi_a(z))| |\varphi_a'(z)|^2 (1-|z|^2) \, dm(z)\\
& \qquad + \int_{D(0,r)} \left| \frac{\varphi_a''(z)}{\varphi_a'(z)} \right| dm(z)
+ \int_{D(0,r)} \left| \left( \frac{\varphi_a''}{\varphi_a'} \right)'\!(z)\right| (1-|z|^2) \, dm(z)\end{aligned}$$ for all $0<r<1$, where the comparison constant is independent of $a\in{\mathbb{D}}$. The area integrals involving $B_a$ and $B_a'$ are uniformly bounded for $0<r<1$ and $a\in{\mathbb{D}}$ by standard estimates, while $$\sup_{a\in{\mathbb{D}}} \sup_{0<r<1} \, \int_{D(0,r)} |A(\varphi_a(z))| |\varphi_a'(z)|^2 (1-|z|^2) \, dm(z)$$ is at most by a conformal change of variable. Recalling that $\log^+ x \leq x$ for all positive $x$, we conclude $$\begin{aligned}
& \sup_{a\in{\mathbb{D}}} \sup_{0<r<1} \, \int_0^{2\pi} \log^+
\left( \int_0^r |A(\varphi_a(se^{i\theta}))| |\varphi_a'(se^{i\theta})|^2 |f(a)| (1-s) \, ds \right) d\theta \notag\\
& \qquad \lesssim \sup_{a\in{\mathbb{D}}} \, M_\infty(|a|, f) \int_{{\mathbb{D}}} |A(z))| (1-|\varphi_a(z)|^2) \, dm(z) \label{eq:finn}.\end{aligned}$$ The quantity is finite by and the fact $f\in\mathcal{B}$. This proves , and hence Theorem \[thm:bmoa\].
Proofs of Theorem \[thm:vmoa\] and Corollary \[cor:vmoa\] {#sec:slast}
=========================================================
The proof of Theorem \[thm:vmoa\] is based on the following result [@R:2003 Corollary 5.3]: If $f\in\mathcal{H}({\mathbb{D}})$ and $$\label{eq:vmoan}
\int_0^1 M_\infty(r,f'')^2 (1-r^2)^3 \, dr<\infty,$$ then $f\in{\rm VMOA}$.
Let $f$ be a non-trivial solution of , and let $0<r_0<1$ be fixed. As in the proof of Theorem \[thm:imp\], we obtain $$\begin{aligned}
M_\infty(r,f'') \leq M_\infty(r,A) \, M_\infty(r,f)
\lesssim M_\infty(r,A) \exp\!\left(\, \int_{r_0}^r M_\infty(t,A) (1-t) \, dt \right)\end{aligned}$$ for $r_0<r<1$, where the comparison constant is independent of $r$. The assertion follows as $f\in{\rm VMOA}$ by and .
Fix any $0<r_0<1$. The coefficient condition implies that there exists an absolute constant $0<C<\infty$ such that $$\exp\!\left(\, 2 \int_{r_0}^r M_\infty(t,A) (1-t) \, dt \right)
\lesssim \left( \log\log\frac{e}{1-r} \right)^{2C}, \quad r_0<r<1,$$ where the comparison constant is independent of $r$. The condition is satisfied by a straight-forward computation, which concludes the proof.
The following example shows that the coefficient condition allows solutions of to be unbounded.
\[ex:unbounded\] Let $0<\alpha<\infty$. Note that $f(z) = (\log\log e^e/(1-z))^\alpha$, $z\in{\mathbb{D}}$, is a zero-free unbounded solution of for $$A(z) = - \alpha \, \frac{\alpha - 1 + \left( \log\frac{e^e}{1-z} - 1\right) \!\left( \log \log\frac{e^e}{1-z}\right)}
{(1-z)^2\left( \log\frac{e^e}{1-z} \right)^2\left( \log \log\frac{e^e}{1-z}\right)^2}, \quad z\in{\mathbb{D}}.$$ It is immediate that is satisfied. $\diamond$
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[^1]: The author is supported in part by the Academy of Finland \#286877.
|
June 2015\
[ **Dimensional reduction in causal set gravity**]{}\
[**Abstract**]{}
[Results from a number of different approaches to quantum gravity suggest that the effective dimension of spacetime may drop to $d=2$ at small scales. I show that two different dimensional estimators in causal set theory display the same behavior, and argue that a third, the spectral dimension, may exhibit a related phenomenon of “asymptotic silence.” ]{}
Introduction
============
In classical physics, the dimension of spacetime is a fixed parameter, specified from the outset. In quantum gravity, this may no longer be the case: dimension may be a quantum observable, taking different values under different circumstances. In particular, there are intriguing hints from a number of different approaches to quantum gravity that the effective dimension of spacetime drops to two at very small distances [@Carlip1; @Carlip2]. This “spontaneous dimensional reduction” was first noted in high temperature string theory [@Atick], and then a few years later in the discrete path integral approach of causal dynamical triangulations [@Ambjorn]. Since then, the same behavior has been seen in the asymptotic safety program [@Reuter; @Percacci], the short distance approximation to the Wheeler-DeWitt equation [@Carlip1], aspects of loop quantum gravity [@Modesto], some formulations of noncommutative geometry [@Benedetti; @Nozari; @Arzano] or minimum length [@Modestob], and perhaps Ho[ř]{}ava-Lifshitz gravity [@Horava]. The generality of these results suggests that dimensional reduction may be a fundamental feature of quantum gravity.
One important approach to quantum gravity, however, seems to be an exception. In [@Eichhorn], Eichhorn and Mizera show that the spectral dimension of a causal set *increases* at short distances. In this paper, I will show that two other dimensional estimators for causal sets—the Myrheim-Meyer dimension of a small causal set and the dimension determined by the causal set Laplacian—display the more standard drop to $d=2$ and short distances. I will argue that the Eichhorn–Mizera result may have a different interpretation, as an indication of short distance “asymptotic silence” [@Heinzle], a behavior that has also been associated with dimensional reduction [@Carlip1; @Carlip2; @Carlip3].
This paper should be read as a report on work in progress. As we shall see, a number of relevant concepts (e.g., Hadamard Greens functions on causal sets) and calculations (e.g., Myrheim-Meyer dimension for “small” causal sets with more than six elements) do not yet exist. But the preliminary results are promising, and this seems to be a program worthy of further study.
Causal sets
===========
A causal set [@Bombelli] is a discrete spacetime in which events have prescribed causal relations. Such a set is characterized by a partial order $\prec$ (where $x\prec y$ means “$x$ is to the past of $y$”) satisfying\
1. transitivity: $x\prec y$ and $y\prec z \Rightarrow x\prec z$;
2. acyclicity: $x \prec y$ and $y \prec x \Rightarrow x=y$;
3. local finiteness: for any $x$ and $y$, the number of elements $z$ such that $x\prec z\prec y$ is finite.
Mathematically, these conditions define a locally finite partially ordered set, or poset. Physically, the causal relations should be thought of as determining “most” of the metric. Indeed, in the continuum, the causal structure of a globally hyperbolic manifold determines the metric up to a conformal factor [@Malamet]; in causal set theory, the missing conformal factor is simply the number of points in a region.
Causal sets with clear physical meaning can be constructed by randomly “sprinkling” points on a fixed spacetime. Given a globally hyperbolic manifold $M$ with metric $g$, select a set of points by a Poisson process so that the probability of finding $m$ points in any region of volume $V$ is $$\begin{aligned}
P_V(m) = \frac{(\rho V)^m}{m!}e^{-\rho V}
\label{a1}\end{aligned}$$ for some discreteness scale $\rho^{-1}$. Assign to these points the causal relations determined by the metric $g$, and then “remove” the manifold $M$, leaving only the set of points and relations. At scales larger than $\rho^{-1}$, the resulting causal set is believed to approximate $M$ well. In particular, if $M$ is Minkowski space, the causal set preserves statistical Lorentz invariance [@LIV], a highly nontrivial characteristic for any discretization.
Myrheim-Meyer dimension of a small causal set
=============================================
As in other discrete approaches to quantum gravity, it is not obvious what one means by the “dimension” of a causal set. For a space with an analog of a Riemannian metric, a popular choice is the spectral dimension, but it is not obvious that this is appropriate to a Lorentzian spacetime; I will return to this issue later. For a causal set, the most common choice for a dimensional estimator is the Myrheim-Meyer dimension [@Myrheim; @Meyer], which is based on a count of the number of causally related points.
More precisely, let us start with a causal set derived from a Poisson sprinkling of points in $d$-dimensional Minkowski space. Choose an Alexandrov interval, or “causal diamond,” $\mathcal{A}$, that is, the intersection of the future of some point $p$ and the past of another point $q$. Let $\langle C_1\rangle$ be the average number of points in $\mathcal{A}$, and $\langle C_2\rangle$ be the average number of causal relations, that is, pairs $x,y$ such that $x\prec y$. $\langle C_1\rangle$ and $\langle C_2\rangle$ depend on the volume of $\mathcal{A}$ and the discreteness scale $\rho^{-1}$, but a suitable ratio depends only on the dimension: $$\begin{aligned}
\frac{\langle C_2\rangle\,}{\langle C_1\rangle^2} =
\frac{\Gamma(d+1)\Gamma(\frac{d}{2})}{4\Gamma(\frac{3d}{2})}
\label{b1}\end{aligned}$$ For an arbitrary causal set, the Myrheim-Meyer dimension $d_M$ is then defined as the value $d$ for which (\[b1\]) holds. One can also consider a sprinkling of points in a curved spacetime; if the curvature is small, a generalization of (\[b1\]) involving chains of three and four related points can eliminate distortions due to curvature [@Roy].
We are interested here in the dimension of “small” causal sets. There are several different things this might mean:\
1. One might simply take a random causal set with a small number $C_1$ of elements. As a practical matter, $C_1$ must be *very* small: the number of distinct causal sets with $C_1$ elements goes as $2^{C_1^2/4}$, and the causal sets have only been fully enumerated up to $C_1=16$ [@McKay].
2. For larger $C_1$, random causal sets are dominated by Kleitman-Rothschild, or KR, orders [@KR; @Henson]. These are three-layered posets with approximately $C_1/4$ elements in the first and third layers and $C_1/2$ elements in the second; an element in the first or third layer is causally related to about half of the elements in the second layer, and almost every element in the first layer is related to almost every element in the third. Numerical studies indicate that these sets become important at $C_1\sim 50$ [@Henson]. While KR orders must be dynamically suppressed at large scales if causal set theory is to reproduce anything like our universe, it is plausible that they remain important at reasonably small scales.
3. The preceding criteria do not include dynamics, in part because the dynamical behavior of causal set theory is not well understood. One might, however, consider random sprinklings of points in known spacetimes—Minkowski space, for instance—and look at their small scale behavior.
The first two of these approaches show clear signs of dimensional reduction. For example, suppose we start with a large causal set and chose a subset containing four elements. There are a total of 16 possible causal structures among those elements, having between zero and six causal relations. If these structures occur with equal probability, the average $\langle C_2\rangle$ is $\frac{13}{4}$, and the Myrheim-Meyer dimension (\[b1\]) is 2.27. For random causal sets with four, five, or six elements, as enumerated in the Chapel Hill poset atlas [@CH], the Myrheim-Meyer dimensions range from 2.15 to 2.27. Similarly, for a random KR order, the dimension is 2.38.
For the third approach, more numerical work is needed. But Reid has looked at random sprinklings in Minkowski space [@Reid], and the results show a decrease in the Myrheim-Meyer dimension to a bit less than 2 for small subintervals, as expected in short distance dimensional reduction.
Laplacians and Greens functions
===============================
Consider a massless field in a $d$-dimensional spacetime. At short distances, the Hadamard Greens function takes the form $$\begin{aligned}
G^{(1)}(x,x') \sim \left\{ \begin{array}{lc} \sigma(x,x')^{-(d-2)/2} \quad& d>2\\
\ln\sigma(x,x') & d=2 \end{array}\right.
\label{c1}\end{aligned}$$ where Synge’s world function $\sigma(x,x')$ is half the squared geodesic distance between $x$ and $x'$. The dimension is thus determined, in a manifestly physical way, by the rate at which the two-point function blows up at coincident points.
As usual, it is not immediately obvious how to extend this expression to a discrete spacetime. Recently, however, considerable progress has been made in defining Laplacians and retarded Greens functions on causal sets obtained by random sprinklings of points in Minkowski space in two [@Sorkina], four [@Benincasa], and arbitrary [@Dowker; @Aslanbeigi] dimensions. While the retarded Greens functions are not the same as the Hadamard functions (\[c1\]), they still provide use useful information.
Aslanbeigi et al. have examined the behavior of these quantities averaged over causal sets obtained by sprinklings on $d$-dimensional Minkowski space [@Aslanbeigi]. The averaged Laplacians have plane wave eigenfunctions $e^{ip\cdot x}$, as expected from Poincar[é]{} invariance, with calculable eigenvalues $g(p)$. Hence $$\begin{aligned}
G_R(x,x') = \int_{\mathcal{C}}\!d^dp\, g(p)^{-1} e^{ip\cdot(x-x')}
\label{c1a}\end{aligned}$$ In the IR limit relevant for long distance behavior, $g(p)^{-1}\sim 1/p^2$, confirming that the causal set Laplacians approximate the standard continuum operators. In the UV, though, one finds that $$\begin{aligned}
g(p)^{-1} \sim \alpha + \beta (p\cdot p)^{-d/2}
\label{c2}\end{aligned}$$ For the contour $\mathcal{C}$ appropriate for a retarded Greens function, the integral (\[c1a\]) near the coincidence limit $\sigma\rightarrow0$ gives a delta function plus a finite correction, the normal behavior for a retarded Greens function [@Aslanbeigi]. But if, as in the continuum case, the Hadamard function can be obtained by choosing a different contour in (\[c1a\]), then (\[c2\]) will lead to a Hadamard function $G^{(1)}\sim\ln\sigma$ at short distances, the standard form for a two-dimensional massless field theory,[^1] although one may worry whether this reduction occurs below the discreteness scale.
To be confident of this claim, one would have to construct the full analog of the Hadamard Greens function in causal set theory and examine its UV limit. Recent work on field theory on causal sets [@Johnston; @Belenchiab] suggests an approach to this problem, and work is in progress. But as in the preceding section, we already see strong hints of dimensional reduction.
Spectral dimension and asymptotic silence
=========================================
Consider a random walk on a $d$-dimensional manifold with a Riemannian metric. Diffusion from an initial position $x$ to a final position $x'$ in a time $s$ is described by a heat kernel $K(x,x';s)$, which behaves for small $s$ as [@Ambjorn] $$\begin{aligned}
K(x,x';s) \sim (4\pi s)^{-d/2} e^{-\sigma(x,x')/2s}
\left( 1 + \mathcal{O}(s)\right)
\label{d1}\end{aligned}$$ In particular, the return probability $K(x,x;s)$ is determined by the dimension. By generalizing (\[d1\]) to an arbitrary space, discrete or continuous, on which a random walk can be defined, one obtains an effective dimension, the spectral dimension.
For several approaches to quantum gravity, including causal dynamical triangulations [@Ambjorn] and asymptotic safety [@Reuter], the spectral dimension exhibits short distance dimensional reduction to $d=2$. For causal set theory, though, it does not. On the contrary, the spectral dimension increases at short distances [@Eichhorn]. What should one make of this?
Eichhorn and Mizera argue in [@Eichhorn] that the peculiar behavior of causal set theory comes from the Lorentzian signature of the metric, which in many cases leads to a “radical nonlocality”—a typical point can have infinitely many nearest neighbors, points connected by a single causal link. Now, as stressed in [@Carlip1; @Carlip2], the importance of spectral dimension comes in part from the fact that Greens functions can be obtained as Laplace transforms of the heat kernel: (\[c1\]) is a Laplace transform of (\[d1\]). But for causal sets, the Greens functions of [@Sorkina; @Benincasa; @Dowker; @Aslanbeigi] contain nonlocal corrections, and the direct connection to the heat kernel for a random walk may be broken.
The results of [@Eichhorn] could, however, have a different implication. In a Lorentzian setting, a high spectral dimension—especially a high value of the “causal spectral dimension” of [@Eichhorn]—implies a suppression of the probability that two random walkers will meet within a given diffusion time. The observed rapid rise in spectral dimension at very short distances thus suggests that “nearby” points are increasingly causally disconnected. A very similar behavior occurs in cosmology near a spacelike singularity, where it is known as “asymptotic silence” [@Heinzle]. As I first pointed out in [@Carlip1], this phenomenon, which leads to locally Kasner-like behavior of the metric, might explain dimensional reduction: at certain scales, $d$-dimensional Kasner space has an effective dimension of two [@Hu].
It should be possible to test this conjecture more directly. In the continuum, asymptotic silence is an “anti-Newtonian” limit, in which the speed of light goes to zero and nearby spacelike separated points become (nearly) causally disconnected. In the causal set context, defining “nearby” is nontrivial, but not impossible [@Rideout], and one can measure the minimum number of links $N$ required for two nearby points to share a common point in the future. The short distance asymptotic silence conjecture is that while $N$ should behave classically for pairs of points with large spatial separations, it should become much larger than its classical value as the spatial distance shrinks. If this is the case, the arguments of [@Carlip1] would again predict spontaneous dimensional reduction.
Conclusion
==========
While the evidence for short distance dimensional reduction in quantum gravity is far from conclusive, there are enough hints from enough different approaches to make the phenomenon at least plausible. But details remain elusive. We do not even know whether dimensional reduction is mainly kinematical or whether it depends sensitively on the dynamics: in the asymptotic safety scenario, for instance, the mere existence of a non-Gaussian UV fixed point is enough to indicate two-dimensional behavior [@Percacci], while in some approaches based on noncommutative geometry the nature of dimensional reduction depends sensitively on a choice of deformed Laplacian [@Arzano].
Causal set theory offers a promising avenue to explore these issues. Much of what we know about causal sets is nondynamical, and there are several approaches to the dynamics that may not be equivalent [@Wallden]. While this paper is a start, there is clearly much more to be done:\
- A systematic study of the Myrheim-Meyer dimension of small subsets of random sprinklings on various known manifolds could reveal more about the influence of large scale spacetime geometry, and thus dynamics, on small scale dimension. One might also look at the curvature-corrected dimensional estimator introduced in [@Roy].
- A construction of the causal set Hadamard function, perhaps following [@Johnston; @Belenchiab], and a study of its asymptotics in the manner of [@Aslanbeigi], would tell more reliably whether Greens functions exhibit dimensional reduction. One might also compute the heat kernels, and through that the spectral dimensions, of the Laplacians in [@Aslanbeigi].[^2]
- More direct tests of short distance asymptotic silence would certainly be illuminating.
It may be that different dimensional estimators give different answers, and the full picture might require a better understanding of the quantum dynamics of causal sets. But the preliminary indications of short distance dimensional reduction in causal set theory seem promising.
**Acknowledgments**
This work was supported in part by Department of Energy grant DE-FG02-91ER40674.
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[^1]: A similar phenomenon occurs in Ho[ř]{}ava-Lifshitz gravity [@Horava], but in contrast to that model, the causal set result does not require a violation of Lorentz invariance.
[^2]: Just after this preprint first appeared, a preprint by Belechia et al. [@Belenchia] answered this last question. The heat kernels of the nonlocal Laplacians [@Aslanbeigi] do, in fact, lead to a spectral dimension that falls to $d=2$ at short distances.
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abstract: |
Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds.
We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is a $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge-de Rham interpretation of the $L^p$ cohomology in degree $1$, and that the map from $L^2$ to $L^p$ cohomology in this degree is injective.
address:
- 'Département de Mathématiques, Université de Nantes, 44322, Nantes, FRANCE'
- 'Département de Mathématiques, Université de Cergy-Pontoise, 95302, Pontoise, FRANCE'
- 'Department of Mathematics, ANU, Canberra, ACT 0200, AUSTRALIA'
author:
- Gilles Carron
- Thierry Coulhon
- Andrew Hassell
title: Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends
---
[^1]
[^2]
Introduction {#sec:introduction}
============
Let $M$ be a complete Riemannian manifold with infinite measure. The Riesz transform $T$ on $M$ is the operator $$f \to d \Delta^{-1/2} f ,$$ where $\Delta$ is the positive Laplace operator on $M$. The Riesz transform is always a bounded map from $L^2(M)$ to $L^2(M; T^*M)$. It is of interest to figure out the range of $p$ for which $T$ extends to a bounded map $L^p(M) \to L^p(M; T^*M)$. Equivalently, we can ask whether $$\| \, |d f | \, \|_p \leq \| \Delta^{1/2} f \|_p \text{ for all } f \in C_c^\infty(M).$$It has been shown in [@CD] that the Riesz transform is bounded on $L^p$ for $1 < p < 2$ as soon as the manifold satisfies the doubling property as well as a natural heat kernel upper bound. The situation for $p > 2$ is more complicated: there is some understanding of what happens in the more restricted class of manifolds satisfying upper and lower Gaussian estimates for the heat kernel (see [@ACDH], [@AC]), and it is also known that the above more general assumptions do not imply the boundedness of the Riesz transform for all $p>2$. The counterexample is simply the connected sum of two copies of $\RR^n$, where one easily sees that the Riesz transform is unbounded for $p>n$ (see [@CD]), precisely because the boundedness for such $p$ would imply a lower Gaussian heat kernel estimate on this manifold, which is known to be false. The first aim of the present article is to find out what happens in the remaining range $2<p\le n$, and to treat more generally by the same token manifolds with a finite number of Euclidean ends. Our second aim is to give, in a somewhat more general class of manifolds, a cohomological consequence of the boundedness of the Riesz transform for some $p>2$, which explains the unboundedness in the range $p\ge n$ in the examples just mentioned.
On $\RR^n$, it is a classical result in harmonic analysis that the Riesz transform is bounded on $L^p$ for all $1 < p < \infty$. In this case, the kernel of the Riesz transform is given by $$c d \frac1{|z-z'|^{n-1}} = c' \frac{(z-z')_i dz^i}{|z-z'|^{n+1}}.$$ (We shall use $z$ for a Euclidean coordinate throughout this paper, while the prime denotes the ‘right variable’ of a kernel $K(z,z')$; also, $\hat z$ denotes $\frac{z}{|z|}$ and we write $r, r'$ for $|z|, |z'|$.) It is useful to compactify $\RR^n$ to a compact manifold with boundary $Z$ by adding a sphere at infinity, and using $|z|^{-1}$ as a boundary defining function[^3] for it. If we consider this kernel of $\Delta^{-1/2}$ near the ‘right boundary’ of $Z \times Z$, namely $Z \times \partial Z$, where $|z'| \to \infty$, we see that it has an expansion given by $$\frac1{|z-z'|^{n-1}} = \frac1{|z'|^{n-1}} \big( 1 - 2 \frac{z \cdot \hat z'}{|z'|} + \frac{|z|^2}{|z'|^2} \big)^{-\frac{n-1}{2}} \sim \frac1{|z'|^{n-1}} + (n-1) \frac{ z \cdot \hat z'}{|z'|^n} + \dots$$ The leading power in this expansion is $|z'|^{-n+1}$ and the coefficient multiplying it is $1$. This kernel, by itself, is not bounded on $L^p$ for $p \geq n$ because the decay of the kernel puts it only in $L^{n/(n-1) + \epsilon}$, $\epsilon > 0$, as a function of $z'$ with $z$ fixed, so it can only be boundedly paired with elements in $L^p$ for $p < n$. However, when we apply a $z$-derivative, the leading term is killed (since $d 1 = 0$) and the kernel of $T$ decays at one order better, namely $|z'|^{-n}$. This allows pairing with elements of $L^p$ for any $p < \infty$. Let $M$ be a manifold with Euclidean ends, and assume the number of ends is at least two.
It is now relatively easy to explain why the Riesz transform on $M$ is not bounded for $p \geq n$. For simplicity we shall assume here that $M$ has exactly two Euclidean ends. Let us compactify $M$ to a compact manifold ${\overline{M}}$ in the analogous way to $Z$ above. The boundary of ${\overline{M}}$ is then the disjoint union of two $n-1$-spheres, which we shall denote $\partial {\overline{M}}_+$ and $\partial {\overline{M}}_-$. It turns out that the kernel of $\Delta^{-1/2}$ has a similar expansion at ${\overline{M}}\times \partial {\overline{M}}$, of the form $$\sum_{j=n-1}^\infty |z'|^{-j} a_j(z), \quad a_j \in {{\mathcal{C}^\infty}}({\overline{M}}\times \partial M),$$ but it is no longer true that the leading term $a_{n-1} = 1$. Rather, $a_{n-1}$ is the harmonic function on ${\overline{M}}$ that equals $1$ on $\partial {\overline{M}}_+$, and zero on $\partial {\overline{M}}_-$. As a consequence, applying the derivative operator in the left variable $z$ does not make this leading term disappear, since $d a_{n-1} \neq 0$. Hence the kernel of $T$ only decays to order $n-1$ at the right boundary of ${\overline{M}}\times {\overline{M}}$, and therefore can only be paired boundedly with elements of $L^p$ for $p < n$.
In this paper we shall prove
\[main\] Let $M$ be a complete ${{\mathcal{C}^\infty}}$ Riemannian manifold of dimension $n \geq 3$ which is the union of a compact part and a finite number of Euclidean ends. Then the Riesz transform is bounded from $L^p(M)$ to $L^p(M; T^*M)$ for $1 < p < n$, and is unbounded on $L^p$ for all other values of $p$ if the number of ends is at least two.
Our method is to analyze the kernel $\Delta^{-1/2}$ based on the formula $$\Delta^{-1/2} = \frac2{\pi} \int_0^\infty (\Delta + k^2)^{-1} \, dk.
\label{sp-int}$$ Since $M$ is a manifold which is conic at infinity, the Laplacian $\Delta$ on $M$ lies in the class of scattering differential operators [@scatmet] and we can use methods from the scattering calculus to analyze the kernel of $\Delta^{-1/2}$. We shall analyze the kernel of $\Delta^{-1/2}$ rather precisely and work out the leading term in the expansion at the right boundary. From this, it will be straightforward to analyze the kernel of $T$ and to prove the theorem. The plan of the paper is as follows. We briefly describe the scattering calculus in section \[scatt\]. In sections \[para\] – \[RT\] we prove the theorem using the analysis of the resolvent of the Laplacian on asymptotically conic spaces in [@HV] as a model. We give some large time asymptotics on derivatives of the heat kernel on manifolds with Euclidean ends in section \[heat\].
In section \[coh\] we change point of view and consider a much more general class of manifolds, namely complete manifolds $M$ of dimension $n$ satisfying the Nash inequality and with a uniform upper bound $O(r^n)$ on the volume of geodesic balls of radius $r$. We *assume* that the Riesz transform on $M$ is bounded on $L^p$ for some $p > 2$ and give several geometric and topological consequences: a Hodge-de Rham interpretation of the $L^p$ cohomology of $M$ (Proposition \[HdR\]), injectivity of the map from $L^2$ to $L^p$ cohomology (Lemma \[inj\]) and derive a contradiction if $p > n$ and $M$ has at least two ends (Corollary \[nb\]), thus generalizing the unboundedness part of Theorem \[main\] for $p > n$ to this larger class of manifolds. In the final section we discuss our results in the context of previously known examples and pose some open problems.
Scattering Calculus {#scatt}
===================
As noted above, we shall use the scattering calculus [@scatmet] to analyze the kernel of $\Delta^{-1/2}$ on manifolds with several Euclidean ends. The scattering calculus is expressed in terms of compactifications on $M$ and, especially, of the double space $M^2$ which carries the kernel of the resolvent and of the operator $\Delta^{-1/2}$. The space $M$ is compactified by adding a sphere at infinity $S^{n-1}$ for each Euclidean end, and declaring $r^{-1}= 1/|z|$ and $\hat z = z/|z|$ to be local coordinates near a boundary point; in particular, $r^{-1}$ is taken to be a defining function for the boundary (which we shall sometimes refer to as ‘infinity’). We sometimes use $x = r^{-1}$ to denote this boundary defining function and $y = \hat z$ to denote boundary coordinates, extending to a collar neighbourhood of the boundary, as is customary when using the scattering calculus. The metric takes the form $$dr^2 + r^2 h(y, dy) = \frac{dx^2}{x^4} + \frac{h(y, dy)}{x^2}$$ at each end, and is therefore a *scattering metric* as defined in [@scatmet] (of a particularly simple form, being an exact conic metric near infinity).
We are mostly interested in the case when the number of ends is at least two. In fact, for the sake of clear exposition we shall assume from now on that the number of ends is exactly two, although all proofs in this paper generalize in an obvious way to any finite number of ends. We shall label these ends $+$ and $-$, thus for example we shall use $z_+$ as the Euclidean variable on the positive end, and $z_-$ for the variable on the negative end; when it is not necessary to stipulate which end is being considered, we shall just use $z$.
It is not so obvious which compactification of $M^2$ is most appropriate for dealing with the Schwartz kernels of operators such as the Laplacian, or functions of the Laplacian, on manifolds with Euclidean ends. There are several different asymptotic regimes of interest when dealing with such kernels. One regime, the ‘near-diagonal’ regime, is when the two variables $z, z'$ remain a finite distance apart as they both go to infinity. Another is when they both go to infinity with the ratio $r/r'$ approaching a limit and with $\hat z,\hat z'$ both approaching a limit. Finally there is the case that one variable approaches a limit, while the other remains fixed. The kernel has different behaviour in each of these asymptotic regimes, so they need to be represented by distinct parts of the boundary of the compactification. The space $({\overline{M}})^2$ is thus too ‘small’ a compactification of $M^2$ for our purposes, since it only has the third regime distinguished; the first two are squashed into the corner.
It turns out that there is a space denoted $\MMsc$, the ‘scattering double space’, which satisfies these criteria. It is obtained by performing two blowups on $M^2$. The first is blowing up the corner $(\partial {\overline{M}})^2$, creating the so-called b-double space, and the second is blowing up the boundary of the diagonal (which lifts to the b-double space to be transverse to the boundary, hence this blowup is well-defined). Each asymptotic regime is represented by a boundary hypersurface of $\MMsc$. The first, ‘near-diagonal’ regime is represented by the boundary hypersuface created by the second blowup, denoted $\sf$ for ‘scattering face’; the second is represented by the boundary hypersurface created by the first blowup, denoted $\bfc$ for ‘b-face’ (since it is present in the b-calculus) and the third regime is represented by the two boundary hypersurfaces $\partial {\overline{M}}\times {\overline{M}}$, ${\overline{M}}\times \partial {\overline{M}}$ of $({\overline{M}})^2$, denoted ${\operatorname{lb}}$ and ${\operatorname{rb}}$ for ‘left boundary’ a nd ‘right boundary’. Note that when $M$ has $k$ ends, then $\bfc$ has $k^2$ components and $\sf$, ${\operatorname{lb}}$ and ${\operatorname{rb}}$ each have $k$ components.
The structure of $\sf$ and $\bfc$ is as follows. Each component of $\sf$ is naturally diffeomorphic to $\overline{\RR^n} \times S^{n-1}$, and $z - z'$ and $\hat z$ are coordinates on the interior of each component of $\sf$. Each component of $\bfc$ is naturally isomorphic to a blowup of the space $S^{n-1} \times S^{n-1} \times [0,1]$, with coordinates $(\hat z, \hat z', |z|(|z| + |z'|)^{-1})$; the blowup is of the submanifold $\{ \hat z = \hat z', |z|(|z| + |z'|)^{-1} = 1/2 \}$ which corresponds to the boundary of the diagonal.
The *scattering calculus* is an algebra of pseudodifferential operators on $M$ which is defined by the properties of their Schwartz kernels. Namely, $A$ is a scattering psuedodifferential operator of order $(m,0)$ on $M$ iff the kernel of $A$, when lifted to $\MMsc$, is conormal[^4] of order $m$ at the diagonal of $\MMsc$ smoothly up to the boundary $\sf$, is smooth elsewhere at $\sf$, and is rapidly decreasing at $\bfc$, ${\operatorname{lb}}$ and ${\operatorname{rb}}$. The resolvent of the Laplacian $(\Delta - \lambda^2)^{-1}$ is a scattering pseudodifferential operator of order $(-2, 0)$ on $M$ for $\Real \lambda \neq 0$. In fact, the structure of the resolvent on the spectrum, i.e. the kernel of $(\Delta - (\lambda \pm i0)^{2})^{-1}$ for real $\lambda$, can also be described on $\MMsc$, although here the kernel is no longer rapidly decreasing at $\bfc$, ${\operatorname{lb}}$ and ${\operatorname{rb}}$, rather it is a ‘Legendrian distribution’ [@HV]. Our approach is partly modelled on the analysis in this paper. However, we take advantage of the assumption here that $M$ has exact Euclidean ends, which leads to great simplifications over the analysis of [@HV] since we can exploit the well-known explicit formulae for the resolvent of the Laplacian on $\RR^n$ and use these as ingredients for a parametrix of the resolvent kernel on $M$, thereby avoiding the need to use Legendrian distributions in this paper.
*Notation.* We write $z$ for a Euclidean variable $z = (z_1, z_2, \dots, z_n) \in \RR^n)$ and write $\ang{z} = \sqrt{1 + |z|^2}$, while $x$ is used for $|z|^{-1}$, or, sometimes, where more convenient, for $\ang{z}^{-1}$. For a manifold with corners $X$, we write ${\dot C^\infty}(X)$ for the space of smooth functions which vanish to infinite order at the boundary of $X$. We use notation $[X; S_1, S_2, \dots S_n]$ to denote the blowup of $X$ at the submanifolds $S_1$, $S_2$, …(in that order).
Parametrix construction {#para}
=======================
To analyze the operator $\Delta^{-1/2}$ we return to the formula . We first observe that the off-diagonal terms in the kernel of $\Delta^{-1/2}$ come from a neighbourhood of zero in the integral . Indeed, let $s_0(k)$ be a cutoff function equal to $1$ in a neighbhourhood of $k=0$ and equal to zero outside a compact set, and let $s_1(k) = 1 - s_0(k)$. Then we may insert the factor $1 = s_0(k) +s_1(k)$ into the integral . With the factor $s_i$ inserted, the integral gives a function $g_i(\Delta)$ of $\Delta$, $i = 0$ or $1$, where $$g_1(t) = \int_0^\infty s_1(k) \frac1{k^2 + t^2} \, dk$$ is easily checked to be a classical symbol of order $-1$. By the symbolic functional calculus [@HV:Symbolic], this term is a scattering pseudodifferential operator of order $-1$, hence $d g_1(\Delta)$ is a scattering pseudodifferential operator of order zero. It is therefore bounded on $L^p$ for all $1 < p < \infty$ [@Stein]. So we are reduced to studying $d g_0(\Delta)$, given by the integral with factor $s_0(k)$ inserted.
We shall write down a fairly explicit parametrix for $(\Delta + k^2)^{-1}$ for small $k$. In doing so, we need to consider the different asymptotics that this kernel takes when $k= 0$ and $k \neq 0$. Indeed, on $\RR^n$ the kernel decays as $|z-z'|^{-(n-1)/2}$ for $k \neq 0$ and $|z-z'|^{-n+2}$ for $k = 0$, as $|z-z'| \to \infty$, which (except when $n=3$) is a different rate. This can be encoded geometrically by blowing up at the boundary when $k = 0$. Consider the space $$\MMscsp = [\MMsc \times [0, k_0]; \bfc \times \{ 0 \}; {\operatorname{lb}}\times \{ 0 \}; {\operatorname{rb}}\times \{ 0 \}].
\label{scsp-blowup}$$ We shall denote the boundary hypersurfaces which are the lifts of $\bfc \times [0, k_0]$, ${\operatorname{lb}}\times [0, k_0]$ and ${\operatorname{rb}}\times [0, k_0]$ to $\MMscsp$ by $\bfc$, ${\operatorname{lb}}$ and ${\operatorname{rb}}$, and $\sf \times [0, k_0]$ by $\sf$; this is of course an abuse of notation, but in the context it will always be clear whether it is a boundary hypersurface of $\MMsc$ or $\MMscsp$ that is referred to. We shall denote the new boundary hypersurfaces corresponding to the three blowups by $\bfacez$, $\lbz$ and $\rbz$, according as they arise from the first, second or third blowups in respectively, and we shall denote $\MMsc \times \{ 0 \}$ by $\zf$, for ‘zero face’. We also define $\Diagscsp$ to be $\Diagsc \times [0, k_0] \subset \MMscsp$. Let $\chi$ be a smooth function on $\MMscsp$ which is equal to one in a neighbourhood of $\Diagscsp$, and whose support meets the boundary of $\MMscsp$ only at $\sf$ and $\zf$.
We recall the well-known expression for the resolvent kernel $(\Delta + k^2)^{-1}$ on $\RR^n$ for $n \geq 3$: $$(\Delta + k^2)^{-1} = \frac{e^{-k|z-z'|}}{|z-z'|^{-n+2}} f_n(k|z-z'|),
\label{res}$$ where $f_n(t)$ is symbolic of order $(n-3)/2$ as $t \to \infty$, while it is $O(1)$ and has a classical expansion in powers and logarithms as $t \to 0$. In fact, $f_n$ is a polynomial of order $(n-3)/2$ when $n \geq 3$ is odd. It is straightforward to check
\[conormal\] Let $Z$ be the compactification of $\RR^n$. Then the resolvent kernel $(\Delta + k^2)^{-1}$ is such that $\chi (\Delta + k^2)^{-1}$ is conormal at $\Diagscsp$, and $$(1 - \chi) (\Delta + k^2)^{-1} \in \rho_{\sf}^0 (\rho_{\bfc} \rho_{{\operatorname{lb}}} \rho_{{\operatorname{rb}}})^{\infty}(\rho_{\bfacez}\rho_{\lbz}\rho_{\rbz})^{n-2} {{\mathcal{C}^\infty}}(\MMscsp).
\label{conorm}$$ Here, ‘conormal to $\Diagscsp$’ means that the kernel is conormal in $z-z'$ which defines $\Diagscsp$ and smooth in the remaining variables $x = |z|^{-1}, \hat z, k$, uniformly up to the boundary.
For example, let us check the statement of the lemma near the triple intersection $\bfacez \cap \rbz \cap {\operatorname{rb}}$. Coordinates near this codimension three corner are $\hat z, \hat z'$ and boundary defining functions $\rho_{\rbz} = k/x = k|z|$ for $\rbz$, $\rho_{{\operatorname{rb}}} = x'/k = 1/(k|z'|)$ for ${\operatorname{rb}}$ and $\rho_{\bfacez} = x$ for $\bfacez$. Near this corner, $|z'|$ is much larger than $|z|$ so we may expand $$k|z-z'| = k|z'| \big( 1 - \frac{2z \cdot \hat z'}{|z'|} + \frac{|z|^2}{|z'|^2} \big)^{1/2} =
\frac1{\rho_{{\operatorname{rb}}}} \big( 1 - 2\hat z \cdot \hat z' \rho_{{\operatorname{rb}}} \rho_{\rbz} + (\rho_{{\operatorname{rb}}} \rho_{\rbz} )^2 \big)^{1/2},$$from which it is easy to check that holds.
We also need a single space version of this space. Let $$\Msp = [{\overline{M}}\times [0, k_0]; \partial M \times \{ 0 \}].$$ Denote the boundary hypersurfaces $\bbdy$, $\zf$ and $\ff$ which arise from $\partial {\overline{M}}\times [0, k_0]$, ${\overline{M}}\times \{ 0 \}$ and from the blowup, respectively, and denote corresponding boundary defining functions by $\rho_{\bbdy}$, $\rho_{\zf}$ and $\rho_{\ff}$. Again, it will always be clear in context whether $\zf$ refers to the zero-face of $\Msp$ or $\MMscsp$.
We have
\[const\] Let $v \in {\dot C^\infty}({\overline{M}})$. Then there is a function $u \in \rho_{\bbdy}^{\infty} \rho_{\ff}^{n-2} {{\mathcal{C}^\infty}}(\Msp)$, such that $(\Delta + k^2) u | \zf$ is equal to $v$, and $(\Delta + k^2) u$ vanishes to infinite order at both $\ff$ and $\bbdy$.
We first use results from [@tapsit] to show that we can solve $\Delta f = v$ on $M$. The Laplacian on an asymptotically Euclidean manifold $M$ may be written in the form $$\Delta = x^{n/2+1} P x^{n/2 - 1},$$where $P$ is an elliptic b-differential operator on $M$. A short computation shows that near infinity, $P$ takes the form $$-(x \partial_x)^2 + \big( \frac{n-2}{2} \big)^2 + x^2 \Delta_{S^{n-1}}, \quad x = |z|^{-1},$$where $\Delta_{S^{n-1}}$ is the standard Laplacian on the $(n-1)$-sphere. This is a strictly positive operator, so $P$ is ‘totally elliptic’, and hence is Fredholm acting between the b-Sobolev spaces[^5] $H^{2}_b(M) \to L^2_b(M)$. Thus $\Delta$ itself is Fredholm acting between $x^{n/2-1} H^{2}_b(M) \to x^{n/2+1} L^2(M)$. Also $P$ is self-adjoint with respect to the measure induced by $g_b$, so its index is equal to zero, hence it is invertible if and only if its null space is trivial. This is therefore also true for $\Delta : x^{n/2-1} H^{2}_b(M) \to x^{n/2+1} L^2(M)$.It is also shown in [@tapsit], section 5.25, that if $P f \in {\dot C^\infty}(M)$, with $f$ in $L^2(M)$, then $f$ has an asymptotic expansion of the form $$f \sim \sum_j x^{n-2+j} a_j \phi_j(\hat z), \quad
\label{f-exp}$$ where $\phi_j$ is a spherical harmonic with eigenvalue $j (j+n-2): \Delta_{S^{n-1}} \phi_j = j(j+n-2) \phi_j$. In particular, such a function tends to zero at infinity. It follows from this and from the maximum principle that there is no nontrivial solution to $\Delta f = 0$, with $f \in x^{n/2-1} H^{2}_b(M)$, because by $f$ would be a harmonic function tending to zero at infinity. Hence we can solve $\Delta f = v$, $v \in {\dot C^\infty}(M)$, where $f$ has an expansion .
Let $f$ be as in the previous paragraph. We first find a formal expansion for $u$ near the corner $\zf \cap \ff$ of $\Msp$. Coordinates near this corner are $x, y$ and $K = k/x$. Let us look for an expansion for $u$ of the form $$u = \sum_j x^{n-2+j} \phi_j a_j(K), \quad K = \frac{k}{x},$$ where $a_j(0)$ is given by the expansion for $f$, so that $u | \zf = f$. The operator $\Delta + k^2$ may be written $$(x^2 D_x)^2 + i(n-1) x^3 D_x + x^2 \Delta_{S^{n-1}} + k^2.$$ Acting on the $j$th term this gives $$x^2 \Big( (xD_x)^2 + i(n-2) xD_x + \Delta_{S^{n-1}} + K^2 \Big).$$ Here $D_x$ indicates the derivative keeping $k$ fixed. When we switch to using coordinates $(x, y, K)$, then we must replace $x D_x$ by $xD_x - K D_K$, getting $$x^2 \Big( (xD_x - K D_K)^2 + i(n-2) (xD_x - K D_K) + \Delta_{S^{n-1}} + K^2 \Big).$$ Acting on the $j$th term, we may replace $\Delta_{S^{n-1}}$ by $j(j+n-2)$ and $xD_x$ by $-i(j+n-2)$, getting the operator $$R_j \equiv x^2 \Big( ( K D_K)^2 -i(n-2+2j) (K D_K) + K^2 \Big).$$ The equation $R_j (a_j(K)) = 0$ has a smooth solution for every $j$, with initial condition $a_j(0)$ determined by the coefficient in . We may cut this off with a cutoff function in $K$ whose derivative is supported where $K \in [1,2]$. The error term is then of the form $$\sum_j x^{n+j} \phi_j \tilde b_j(K),$$ with $\tilde b_j$ supported in $[1,2]$.
We now change to variables which are smooth at the other corner, $\bbdy \cap \ff$, namely $k$ and $\rho = x/k = 1/K$. The error term above may be written $$\sum_j k^{n+j} \phi_j b_j(\rho),$$ where $b_j$ is supported in $[1/2, 1]$. Let us try to solve it away with a series of the form $$\sum_j k^{n-2+j} \phi_j c_j(\rho).$$
Writing the operator in these new variables we get $$\begin{gathered}
(k \rho^2 D_\rho)^2 + i(n-1)k^2 \rho^3 D_\rho + x^2 \Delta_{S^{n-1}} + k^2 \\
= k^2 \Big( (\rho^2 D_\rho)^2 + i(n-1) \rho^3 D_\rho + \rho^2 \Delta_{S^{n-1}} + 1 \Big).
\end{gathered}$$ Let $c_j = e^{-1/\rho} e_j$. Then $e_j$ satisfies the equation $$\begin{gathered}
\Big((\rho^2 D_\rho -i)^2 + i(n-1) \rho (\rho^2 D_\rho - i) + \rho^2 j(j+n-2) + 1 \Big) e_j = e^{1/\rho}b_j
\implies \\
\Big( -2 \rho^2 \partial_\rho +(n-1)\rho + \big( ( \rho^2 D_\rho)^2 + i(n-1) ( \rho^3 D_\rho) +
\rho^2 j(j+n-2) \big) \Big) e_j = e^{1/\rho} b_j
\end{gathered}$$ This is a regular singular ODE with a solution of the form $e_j = \rho^{(n-1)/2} \tilde e_j$ where $\tilde e_j$ is smooth down to $\rho = 0$. This gives us a formal series in powers of $\rho_{\ff}$ at $\ff$ in which each term is uniformly rapidly decreasing with all derivatives at $\bbdy$ (i.e., as $\rho \to 0$). Borel summing at $\ff$, we get a formal solution that matches with $f$ to infinite order at $\ff$. Making a correction that vanishes to infinite order at $\zf$, in order to make $u$ agree exactly with $f$ at $\zf$, we get a function $u$ which satisfies all conditions of the lemma.
We now use this lemma to define a harmonic function on $M$ which will be key to the parametrix construction. We begin by choosing a smooth function $\phi : \RR^n \to \RR$ which is equal to $1$ for $|z|$ large and is supported in $\{ |z| > 1 \}$. Using this we define functions $\phi_\pm$ on $M$, with $\phi_\pm$ supported on the $\pm$ end of $M$, in the obvious way. Then let $u_\pm$ be the function given by Lemma \[const\] from the function $v = -\phi_\pm$. It follows that $$(\Delta + k^2) (e^{k z \cdot \omega} \phi_\pm + u_\pm)
\in \rho_{\zf} \rho_{\bbdy}^\infty \rho_{\ff}^\infty {{\mathcal{C}^\infty}}({\overline{M}}).
\label{bdy-1}$$ Moreover, $$\Phi_\pm = \phi_\pm + u_\pm | \zf \text{ is a harmonic function equal to $1$ at $\partial {\overline{M}}_\pm$ and $0$ at $\partial {\overline{M}}_\mp$.}
\label{bdy-2}$$
We now define our parametrix. It is based on the resolvent kernel for $\RR^n$, but there is a crucial additional term ($G_3$ below) which corrects the leading order coefficient of the kernel at the face $\rbz$ (see the discussion of this coefficient in the Introduction). We now write $\phi_\pm$ to denote this function of the left variable o $M^2$ and $\phi_\pm'$ denote this function of the right variable on $M^2$. Let $G_{\interior}(k)$ be a parametrix, modulo smoothing operators, for $(\Delta +k^2)^{-1}$ in the interior of $M$. We may assume that it is localized sufficiently close to the diagonal. We recall that the resolvent of the Laplacian on $\RR^n$ has the form . Using this notation we define $$\begin{gathered}
\tilde G(k) = G_1(k) + G_2(k) + G_3(k), \text{ where } \\
G_1(k) =
\frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \Big( \phi_+ \phi_+' + \phi_- \phi_-' \Big) \\
G_2(k) = G_{\interior}(k) \Big( 1 - \phi_+ \phi_+' - \phi_- \phi_-' \Big) \\
G_3(k) = \frac{e^{-k |z'|}}{|z'|^{n-2}} f_n(k |z'|) \bigg( u_+ \phi_+'
+ u_- \phi_-' \bigg) .
\end{gathered}\label{G}$$
Error term and resolvent {#error}
========================
In this section, we correct the parametrix to the exact resolvent. The main point is that we obtain complete information about the regularity of the kernel of the error term on $\MMscsp$, and therefore of the resolvent itself on this space. This allows us to determine the regularity of $\Delta^{-1/2}$ and $d \Delta^{-1/2}$ and compute its behaviour to leading order at the boundary hypersurfaces of $\MMsc$.
Applying $\Delta + k^2$ to our parametrix (on the left), we get $$(\Delta + k^2) \tilde G(k) \equiv (\Delta + k^2) (G_1(k) + G_2(k) + G_3(k)) = \Id + \tilde E(k),$$ where this equation defines $\tilde E(k)$. We may think of $\tilde E(k)$ either as a kernel on $\MMscsp$ or, by restricting to $M^2 \times [0, k_0]$ which is the interior of $\MMscsp$ together with the interior of $\zf$, a family of kernels parametrized by $k$ acting on functions on $M$.
By construction, the complete symbol of the diagonal singularity of $\tilde G(k)$ is the inverse, modulo symbols of order $-\infty$, of the complete symbol of $\Delta + k^2$. Thus $\tilde E(k)$ is smooth at the diagonal. Also, we see due to the properties of $u_{\pm}$ that $\tilde E(k)$ vanishes to infinite order at $\bfc, \bfacez, {\operatorname{lb}}$, ${\operatorname{rb}}$ and $\lbz$. The crucial property of $\tilde E(k)$ is the order of vanishing at $\rbz$. To calculate this, we need to determine the leading coefficient of the expansion of $\tilde G(k)$ at $\rbz$. These terms come from $G_1(k)$ and $G_3(k)$. Since $|z-z'| = |z'| - z \cdot \hat z' + O(\rho_{\rbz}\rho_{\bfacez}^{-1})$ at $\rbz$, we have $k|z-z'| = k|z'| + O(\rho_{\rbz})$; note that $k|z|$ vanishes on $\rbz$, while $k|z'|$ is finite in the interior of $\rbz$. Hence $$G_1(k) = \frac{e^{-k |z'|}}{|z'|^{n-2}} f_n(k |z'|) \phi_\pm(z) + O(\rho_{\rbz}^{n-1}) \text{ at } \rbz.$$ If we combine this with $G_3(k)$, then using we see that the leading coefficient becomes $$\frac{e^{-k |z'|}}{|z'|^{n-2}} f_n(k |z'|) \Phi_\pm(z) + O(\rho_{\rbz}^{n-1})
= \rho_{\rbz}^{n-2} e^{-k |z'|} f_n(k|z'|) \Phi_\pm(z) + O(\rho_{\rbz}^{n-1}) \text{ at } \rbz.$$ The leading term annihilated by the operator $\Delta + k^2$ (since $\Phi_\pm$ is harmonic and $k=0$ at $\rbz$), so the error term $\tilde E(k)$ is $O(\rho_{\rbz}^{n-1})$ at $\rbz$ — an improvement of one order over what might be expected, and the main point of introducing the correction term $G_3(k)$. Thus, we have $$\label{Ek}
\tilde E(k) \in \rho_{\sf}^\infty \rho_{\bfc}^\infty \rho_{\bfacez}^\infty \rho_{{\operatorname{lb}}}^\infty \rho_{\lbz}^\infty \rho_{{\operatorname{rb}}}^{\infty} \rho_{\rbz}^{n-1} {{\mathcal{C}^\infty}}(\MMscsp).$$ Note that both $x = \ang{z}^{-1}$ and $x' = \ang{z'}^{-1}$ are smooth on $\MMscsp$, the former function vanishing simply at ${\operatorname{lb}}, \lbz, \bfc, \bfacez, \sf$ and the latter vanishing simply at ${\operatorname{rb}}, \rbz, \bfc, \bfacez, \sf$. Thus implies that the kernel of $\tilde E(k)$ is $(\ang{z} \ang{z'})^{1-n}$ times a bounded function on $\MMscsp$. This implies that $\tilde E(k)$ is Hilbert-Schmidt, uniformly for $k \in [0, k_0]$, hence compact for each $k \in [0, k_0]$. Therefore $\Id - \tilde E(0)$ has finite dimensional null space and cokernel of the same dimension on $L^2(M)$. We next show that we can modify our parametrix by the addition of a finite rank term so that the new error term is invertible for small $k$. The correction term will be $$G_4(k) = \sum_{i=1}^N \phi_i \langle \psi_i, \cdot \rangle \, \quad N = \dim \null (\Id - \tilde E(0))
\label{g4-def}$$ where the $\phi_i$, $\psi_i$ are in ${\dot C^\infty}(M)$ and independent of $k$. Since $\tilde E(0)$ maps into ${\dot C^\infty}(M)$, the null space is contained in ${\dot C^\infty}(M)$ and hence is independent of the choice of $l$. Thus we choose $\psi_i$ to span the null space of $\Id - \tilde E(0)$, and we would like to choose $\phi_i$ so that $\Delta \phi_i $ span a space supplementary to the range of $\Id + \tilde E(0)$. This is possible since $\Delta$ has trivial null space, and $\Delta$ is self-adjoint, hence the range of $\Delta$ on ${\dot C^\infty}(M)$ is dense in $L^2(M)$. Choosing such $\phi_i$, we define $G_4(k)$ (which is actually independent of $k$) by . We now define $$G(k) = G_1(k) + G_2(k) + G_3(k) + G_4(k) = \tilde G(k) + G_4(k)$$ and define $E(k)$ by setting $$(\Delta + k^2) G(k) = \Id + E(k) \implies E(k) = \tilde E(k) + (\Delta + k^2) G_4(k);$$ $E(k)$ enjoys all the properties of $\tilde E(k)$ listed above. In addition, since $E(0)$ is such that $\Id + E(0)$ is invertible, it follows that actually $\Id + E(k)$ is invertible for all sufficiently small $k$; we assume that $k_0$ is chosen so that $\Id + E(k)$ is invertible for all $k \in [0, k_0]$.
We now analyze the inverse of $\Id + E(k)$. Let us write $$(\Id + E(k))^{-1} = \Id + S(k),$$ where this equation defines $S(k)$. The decay of the kernel $E(k)$ at the boundary of $\MMscsp$ implies that $E(k)$ is Hilbert-Schmidt on $L^2(M)$. Hence $S(k)$ is also Hilbert-Schmidt. The regularity of $\tilde E(k)$ on $\MMscsp$, and the fact that $x'/(x' + k) \in {{\mathcal{C}^\infty}}(\MMscsp)$ vanishes simply at ${\operatorname{rb}}, \bfc, \sf$, imply that $$E(k) \in x^N (x')^{n-1} \big( \frac{x'}{x' + k} \big)^N L^\infty(M^2 \times [0, k_0]) \text{ for all }N.$$ Using this and the formula $$S(k ) = E(k) + E(k)^2 + E(k) S(k) E(k)$$ shows that $$S(k) \in x^N (x')^{n-1} \big( \frac{x'}{x' + k} \big)^N L^\infty(M^2 \times [0, k_0]) \text{ for all } N.
\label{Sk}$$
We are particularly interested in the kernel $G(k) S(k)$, which we shall call $G_5(k)$, since the addition of $G_5(k)$ will correct the parametrix $G(k)$ to the exact resolvent kernel.
\[g5\] Let $l = 0, 1, 2 \dots$. Then the kernel $$\int_0^{k_0} s_0(k) \nabla^{(l)} G(k) S(k) \, dk
\label{t5-int}$$ is in $$\ang{z}^{-(n-1+l)} \ang{z'}^{-(n-1)} L^\infty(M^2) \cap \ang{z}^{-(n-2+l)} \ang{z'}^{-n} L^\infty(M^2).$$
Much more precise statements can be made about the kernels and , for example by using Melrose’s Pushforward Theorem [@RBMCalcCon], which shows that these kernels are actually conormal, with respect to the boundary and the diagonal, on $\MMsc$. However, the $L^\infty$ statements will suffice for our purposes and are more straightforward to prove.
Let us break up $G(k)$ into two parts $G(k) = \chi G(k) + (1 - \chi) G(k)$, where $\chi$ is as in Lemma \[conormal\]. Thus $\chi G(k)$ is a smooth family of scattering pseudodifferential operators, while $(1 - \chi) G(k)$ has no singularity at the diagonal.
We first consider $(1 - \chi) G(k)$ which is localized away from the diagonal. Let $m_x$ denote the multiplication operator by $x=\ang{z}^{-1}$ on $M$. Then we have $$\nabla^{(l)} (1 - \chi) G(k)S(k) = \big( \nabla^{(l)} (1 - \chi) G(k)m_x^{n+l} \big) \big( m_x^{-(n+l)} S(k) \big).$$ The kernel $\nabla^{(l)} G(k)$ decays to order $n-2+l$ at $\lbz$ and to order $\infty$ at ${\operatorname{lb}}$ and $\bfc$. If we multiply this kernel by $\ang{z'}^{-(n+l)}$, which corresponds to composing with $m^{n+l}$ on the right, then it also decays to order $n+l$ at $\bfacez$ and $\sf$. This means that we can write $$\nabla^{(l)} (1 - \chi) G(k)m_x^{n+l} \in x^{n-2+l} \big( \frac{x}{x+k} \big)^2 L^\infty(M^2 \times [0, k_0]), \quad x = \ang{z}^{-1}$$ since $x$ is a product of boundary defining functions for ${\operatorname{lb}}, \lbz, \bfc, \bfacez, \sf$, and $x/x+k$ vanishes to first order at ${\operatorname{lb}}$ and $\bfc$. In a similar way, using , we find that $$m_x^{-(n+l)} S(k) \in (x')^{n-1} \big( \frac{x'}{x'+k} \big)^2 L^1(M; L^\infty(M \times [0, k_0]));$$ note that composing with $m_x^{-(n+l)}$ on the left is harmless here because the kernel $S(k)$ vanishes to infinite order on every boundary hypersurface where $x^{-(n+1)}$ blows up. For this same reason the kernel is $L^1$ in the left variable, uniformly in the right variable and in $k$. It follows that the composition $$\nabla^{(l)} (1 - \chi) G(k)S(k) \in x^{n-2+l} (x')^{n-1} \big( \frac{x}{x+k} \big)^2 \big( \frac{x'}{x'+k} \big)^2 L^\infty(M^2 \times [0, k_0]).
\label{Linfty}$$ Now we integrate in $k$. If we ignore the $(x/x+k)^2$ factor (which is bounded) then we find that $$\int_0^{k_0} s_0(k) \nabla^{(l)} (1 - \chi) G(k)S(k) \, dk \in x^{n-2+l} (x')^n L^\infty(M^2),$$ because $$\int_0^{k_0} \big( \frac{x'}{x'+k} \big)^2 \, dk \leq
\int_0^{\infty} \big( \frac{x'}{x'+k} \big)^2 \, dk = C x', \
C = \int_0^\infty \big( \frac{1}{1+\overline{k}} \big)^2 \, d\overline{k}.$$ In exactly the same way we show that $$\int_0^{k_0} s_0(k) \nabla^{(l)} (1 - \chi) G(k)S(k) \, dk \in x^{n-1+l} (x')^{n-1} L^\infty(M^2).$$
Finally we consider the integral with $G(k)$ replaced by $\chi G(k)$. We may regard $\nabla^{(l)} \chi G(k)$ as a smooth family of scattering pseudodifferential operators, and $S(k)$ as an element of $$\rho_{{\operatorname{rb}}}^{\infty} \rho_{\rbz}^{n-1} {\dot C^\infty}(M; {{\mathcal{C}^\infty}}(\Msp)).
\label{S-space}$$ Since scattering pseudodifferential operators map ${\dot C^\infty}(M)$ to itself continuously, it follows that $\nabla^{(l)} \chi G(k) S(k)$ is also an element of the space . Performing the $k$ integral we get an extra vanishing factor at ${\operatorname{rb}}\subset \MMsc$, yielding $x^\infty (x')^n L^\infty(M^2)$, which proves the Lemma for this piece. This completes the proof.
Riesz Transform {#RT}
===============
Recall that in Section \[para\] we split $\Delta^{-1/2} = g_0(\Delta) + g_1(\Delta)$, where $d g_1(\Delta)$ was bounded from $L^p$ to $L^p$ for all $1<p<\infty$, and $$g_0(\Delta) = \frac{2}{\pi} \int_0^\infty s_0(k) (\Delta + k^2)^{-1} \, dk.$$ Hence, it remains to analyze $d g_0(\Delta)$. Let us decompose $$(\Delta + k^2)^{-1} = G_1(k) + G_2(k) + G_3(k) + G_4(k) + G_5(k)$$ as in the previous section and write $d g_0(\Delta) = T_1 + T_2 + T_3 + T_4 + T_5$ correspondingly.
The easiest kernel to deal with is $T_4$; this kernel is in ${\dot C^\infty}(M^2)$, hence is bounded from $L^p$ to $L^p$ for $1 \leq p \leq \infty$. The kernel $T_2$ is bounded on $L^p$ for $1 < p < \infty$ because it is a classical zero order pseudodifferential operator with proper support; see Chapter VI, section 5 of [@Stein]. The kernel $T_1$ we decompose further as as $T_1 = T_{1,1} + T_{1,2} + T_{1,3}$, where $$\begin{gathered}T_{1,1} = \Big( d_z \frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \Big) \Big( \phi_+ \phi_+' + \phi_- \phi_-' \Big) , \\T_{1,2} = \chi \Big( \frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \Big) \Big( (d\phi_+) \phi_+' + (d\phi_-) \phi_-' \Big), \\T_{1,3} = (1 - \chi) \Big( \frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \Big) \Big( (d\phi_+) \phi_+' + (d\phi_-) \phi_-' \Big)\end{gathered}$$and $\chi$ is as defined above Lemma \[conormal\]. It is clear that $T_{1,1}$ is bounded on $L^p$ for $1 < p < \infty$, because t he Riesz kernel on $\RR^n$ has this property. Also, $T_{1,2}$, like $T_2$, it is a classical zero order pseudodifferential operator with proper support, hence bounded on all $L^p$.
We next consider $T_5$. Lemma \[g5\], with $l=1$, shows that $T_5$ is in $L^p(M; L^{p'}(M))$, where $p'^{-1} = 1 - p^{-1}$, for all $p \in (1, \infty)$, which implies that $T_5$ is bounded on $L^p$ for $1 < p < \infty$. Thus we are left with $T_{1,3} + T_3$.
The kernel of $T_{1,3} + T_3$ such that $$T_{1,3} + T_3 \in \rho_{{\operatorname{rb}}}^{n-1} \rho_{{\operatorname{lb}}}^n \big( \rho_{\sf} \rho_{\bfc} \big)^{2n-2} {{\mathcal{C}^\infty}}(\MMsc).
\label{T}$$ Moreover, the leading coefficient of $T_{1,3} + T_3$ at ${\operatorname{rb}}$ is a constant times $d \Phi_\pm$.
Let us first consider the kernel of $T_{1,3}$ near ${\operatorname{rb}}$ and away from $\bfc$. This given by $$\int_0^{k_0} \bigg(
\frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \Big( d \phi_+ \phi_+' + d \phi_- \phi_-' \Big)
\bigg) dk;$$ note that $T_{1,3}$ is supported away from ${\operatorname{lb}}$, $\bfc$ and $\sf$ since the support of $d \phi$ is compact. It is a smooth function of $z$ and $x', y', k/x'$ which is rapidly decreasing in $k/x'$. It vanishes to order $(x')^{n-2}$ at $x' = 0$; note that $x'$ is a boundary defining function for ${\operatorname{rb}}$ in this region. Moreover, it is given by $$(x')^{n-2} e^{-k/x'} f_n(k/x') \Big( d \phi_+ \phi_+' + d \phi_- \phi_-' \Big) + O((x')^{n-1}).$$ Changing variable of integration to $k/x'$ and taking into account $dk = x' d(k/x')$ we see that the integral is $$C_n (x')^{n-1} \Big( d \phi_+ \phi_+' + d \phi_- \phi_-' \Big) + O((x')^{n})$$ at ${\operatorname{rb}}$. If we do the analogous calculation for $T_3$ and add the results we find that the kernel of $T_{1,3} + T_3$ is given by $$C_n (x')^{n-1} \Big( d \Phi_+ \phi_+' + d \Phi_- \phi_-' \Big) + O((x')^{n})$$ at ${\operatorname{rb}}$. This proves the last statement of the lemma.
A similar computation can be done for $T_3$ at ${\operatorname{lb}}$, but now the result vanishes to order $n$ at the left boundary, because the derivative $d$, which is applied to the left variable of the kernel, increases the order of vanishing by $1$ at the left boundary.
Consider next the kernel $T_3$ near the triple intersection of ${\operatorname{rb}}$, $\rbz$ and $\bfacez$. In this case, local boundary defining functions are $\rho_{{\operatorname{rb}}} = x'/k$, $\rho_{\bfacez} = x$ and $\rho_{\rbz} = k/x$. We claim that the kernel is actually a smooth function of $x'/k$, $x'/x$, $x$, $y$ and $y'$ in this region, which is a stronger statement, since $x'/x = \rho_{{\operatorname{rb}}} \cdot \rho_{\rbz}$. To see this, note that the kernel of $T_3$ is equal to $e^{-1/\rho_{{\operatorname{rb}}}}$ times a ${{\mathcal{C}^\infty}}$ function on $\MMscsp$. Generally, if $h(u,v)$ is any smooth function of $u$ and $v$, $u, v \geq 0$, then $e^{-1/u} h(u,v)$ is a smooth function of $u$ and $v/u$. In other words, the function $\tilde h(u, w) = e^{-1/u} h(u, w/u)$ is smooth. (This is easily checked directly by differentiating $\tilde h$; inverse powers of $u$ are harmless due to the $e^{-1/u}$ factor.) Now let $u = \rho_{{\operatorname{rb}}}$ and $v = \rho_{\rbz}$, and treat the other coordinates as parameters, and the claim follows.
The kernel $T_3$ vanishes to order $n-2$ at $\rbz$ and $2n-3$ at $\bfacez$. We change variable of integration to $k/x'$ as before, and the change of measure $dk = x'd(k/x')$ gives us additional vanishing at *both* $\rbz$ and $\bfacez$, since $x' = (x'/x) x$ vanishes at both $\rbz$ and $\bfacez$. Thus the result is a smooth function of $(x'/x, x, y, y')$ which vanishes to order $n-1$ at ${\operatorname{rb}}= \{x'/x = 0\}$ and order $2n-2$ at $\bfc = \{x = 0\}$, which verifies the statement of the lemma near the corner ${\operatorname{rb}}\cap \bfc \subset \MMsc$. The other regions of $\MMsc$ are treated similarly.
This lemma implies that, for $1 < p < n$, $T_{1,3} + T_3$ is an element of $L^p(M; L^{p'}(M))$. Moreover, for $p \geq n$, this is not true since the function $(x')^{n-1}$ is not in $L^{p'}$ then, and the coefficient of $(x')^{n-1}$ is $d \Phi_\pm$ which does not vanish identically. Therefore $T_{1,3} + T_3$ cannot be applied to any bounded function equal to $x(\log x)^{-1}$ near infinity, which lies in $L^p$ for $p \geq n$. This completes the proof of Theorem \[main\].
If $M$ has one Euclidean end then the same argument shows that the Riesz transform is bounded on $L^p$ for all $1 < p < \infty$. In this case, the parametrix $\tilde G(k)$ can be taken to be (compare with ) $$\frac{e^{-k|z-z'|}}{|z-z'|^{n-2}} f_n(k |z-z'|) \phi \phi' +
G_{\interior}(k) \big( 1 - \phi \phi' \big) +
\frac{e^{-k |z'|}}{|z'|^{n-2}} f_n(k |z'|) (1-\phi) \phi' .$$ In this case the role of $\Phi_\pm$ in the computation above is played by the constant function $1$. The argument is the same as above, except that the gradient of $1$ vanishes so that we get $\rho_{{\operatorname{rb}}}^n$ instead of $\rho_{{\operatorname{rb}}}^{n-1}$ in (as outlined in the introduction), leading to the boundedness for all $p$ strictly between $1$ and $\infty$.
Heat kernel {#heat}
===========
As part of the analysis of the heat kernel we analyzed the structure of the resolvent $(\Delta + k^2)^{-1}$ for real $k$, including an analysis of the asymptotics of its kernel when $k \to 0$. This analysis remains valid for any cone $\{ k = i\lambda \mid \Imag \lambda \geq \epsilon \Real \lambda \}$ for any $\epsilon > 0$. We can use this to obtain information about the heat kernel $H(t, z, z')$ of $e^{-t \Delta}$ on $M$ via the contour integral $$e^{-t \Delta} = \frac1{2\pi i} \int_\Gamma e^{-t \lambda^2} \big( \Delta - \lambda^2 \big)^{-1} 2\lambda \, d\lambda
\label{contour-int}$$ where $\Gamma$ is the contour $\{ \lambda = s e^{-i\pi/12} \cup
\lambda = s e^{i\pi/12} \mid s \in \RR^+ \}$.
Let us focus on the heat kernel in the following asymptotic regime: We fix a point $z \in M$, which we think of as being in the ‘compact part’ of $M$ (where the metric is not flat), and fix an end of $M$ and a point $\omega \in S^{n-1}$ which we think of as a point at infinity for this end. Consider the behaviour of the heat kernel $H(t,z, z')$ where $z' = r' \omega$ and $t \to \infty$, $r' \to \infty$ so that $\sqrt{t}/r'$ approaches a finite positive limit $\sigma$.
Assume that $M$ has Euclidean ends, with the number of ends at least two. Under the limiting process described above, $t^{n/2} \nabla_{z}^{(l)} H(t, z, z')$ approaches a limit, for any value of $l$. Indeed $$\lim_{t \to \infty} t^{n/2} \nabla_{z}^{(l)} H(t, z, z') = (4\pi)^{-n/2} \ e^{-1/4\sigma^2} \nabla_{z}^{(l)} \Phi(z),\ \sigma = \frac{\sqrt{t}}{r'} > 0 \text{ fixed.}
\label{heat-limit}$$ where $\Phi$ is the harmonic function which tends to $1$ at the given end and tends to $0$ at all other ends. In particular, we have a lower bound on the derivatives of the heat kernel for large time: $$\sup_{z, z' \in M} \big| \nabla_z^{(l)} H(t, z, z') \big| \geq c_l t^{-n/2}, \, \text{ for some } \, c_l > 0, \quad t \geq 1.$$
For $l=0$ this result is not surprising. The point of this proposition is that *taking derivatives in the $z$ variable gives no additional decay in the heat kernel* (in this asymptotic regime). This contrasts with Euclidean space where each additional derivative gives additional decay of $t^{-1/2} = (\sigma r')^{-1}$.
The $k$th $z$-derivative of the heat kernel is given by the contour integral with the resolvent replaced by the $k$th $z$-derivative of the resolvent. Clearly, to prove the theorem we only have to consider the kernel of the resolvent in a neighbourhood of ${\operatorname{rb}}$ and $\rbz$.
Near the interior of $\rbz$, and away from ${\operatorname{rb}}$ the function $\Lambda = \lambda/x'$ is a smooth function, which goes to infinity at ${\operatorname{rb}}$; in fact, $\Lambda^{-1}$ is a boundary defining function for ${\operatorname{rb}}$. In this integral , the term $e^{-t\lambda^2} = e^{-\sigma^2 \Lambda^2}$ therefore vanishes together with all its derivatives at ${\operatorname{rb}}$, since $\sigma > 0$ by assumption, which means that we may ignore the expansion of the resolvent at ${\operatorname{rb}}$. Hence to find the asymptotics the heat kernel near ${\operatorname{rb}}$ in this regime we only need to consider the expansion of the resolvent at $\rbz$ (up to a correction that vanishes to infinite order as $r' \to \infty$).
Using the $L^\infty$ bounds , we may write the $k$th derivative of the resolvent kernel in the form $$\begin{gathered}
K_0(z, y', \Lambda) + K_1(z, y', \Lambda, r'), \\
K_0 = (r')^{-(n-2)} e^{i \Lambda} f_n(\Lambda) \nabla_{z}^{(l)} \Phi(z), \quad K_1 = O((r')^{-(n-1)})
\end{gathered}$$ in the region of interest. Let us first substitute $K_0$ for the resolvent into the integral . Thus we want to compute the limit $$\lim_{t \to \infty} t^{n/2} \frac1{\pi i} \int_\Gamma e^{-t \lambda^2} (r')^{-n+2} e^{i \Lambda} f_n(\Lambda) \nabla_{z}^{(l)} \Phi(z) \lambda \, d\lambda.
\label{contour-int-2}$$ Substituting $\lambda = (r')^{-1} \Lambda$ and $t = \sigma^2 (r')^2$, and using $\lambda d\lambda = (r')^{-2} \Lambda d\Lambda$, we get $$\lim_{t \to \infty} \frac1{\pi i} \sigma^{n} \nabla_{z}^{(l)} \Phi(z) \int_\Gamma e^{-\sigma^2 \Lambda^2} e^{i \Lambda} f_n(\Lambda) \Lambda \, d\Lambda.
\label{contour-int-3}$$ Taking the limit is trivial, since is independent of $t$. To perform the integral, consider the case of $\RR^n$, with kernel $(\Delta - \lambda^2)^{-1}(z,z')$ with $z$ fixed to be the origin. This gives rise to an integral $$\frac1{\pi i} (r')^{-n} \int_\Gamma e^{- \sigma^2 \Lambda^2} e^{i \Lambda} f_n(\Lambda) \Lambda \, d\Lambda
\label{contour-int-4}$$ which is equal to $$(4 \pi t)^{-n/2} e^{-(r')^2/4t} = (4 \pi t)^{-n/2} e^{-1/(4\sigma^2)}.$$ Multiplying through by $(r')^n$ gives $$\frac1{\pi i} \int_\Gamma e^{- \sigma^2 \Lambda^2} e^{i \Lambda} f_n(\Lambda) \Lambda \, d\Lambda = (4\pi)^{-n/2} \sigma^{-n} e^{-1/(4\sigma^2)}.$$ Hence, is equal to $$(4\pi)^{-n/2} e^{-1/(4\sigma^2)} \nabla_{z}^{(l)} \Phi(z) ,$$ which is the right hand side of . If we now substitute $K_1$ for the resolvent in , which vanishes to an additional order as $r' \to \infty$ as compared to $K_0$, then the integral also vanishes to an additional order, giving a zero contribution to the limit . This proves the proposition.
It is also of interest to compute the leading behaviour of the heat kernel $H(t, z, z')$ as $t \to \infty$ and as $z, z'$ both tend to infinity, but along different ends. Suppose that $z = r \omega$, where $\omega \in S^{n-1}_-$ is fixed and that $z' = r' \omega'$, $\omega' \in S^{n-1}_+$ is fixed, and suppose further that $\sqrt{t}/r \to \sigma, \sqrt{t}/r' \to \sigma'$ where $\sigma, \sigma' \in (0, \infty)$.
Under this asymptotic regime, the limit $$\lim_{t \to \infty} t^{n-1} H(t, z, z') = q(\sigma, \sigma'), \quad \frac{\sqrt{t}}{r} \to \sigma, \ \frac{\sqrt{t}}{r'} \to \sigma'$$exists and is finite. Hence in this asymptotic regime the heat kernel has $t^{-n+1}$ decay as $t \to \infty$.
For $n \geq 3$ this is faster than the usual $t^{-n/2}$ decay. Hence Gaussian lower bounds do not hold for the heat kernel on $M$. This was observed in [@BCF], and can be heuristically explained in terms of Bro wnian motion on $M$. Here we give an explicit quantitative description of the failure of this lower bound.
We shall perform a similar computation as in the proof of the previous proposition. Since $|z|/|z'| \to \sigma'/\sigma$ under this limiting regime, and $z, z'$ go to infinity along different ends, we end up at the ‘anti-diagonal’ part of $\bfc$. Hence we need to consider the resolvent kernel near the anti-diagonal part of $\bfc$ and $\bfacez$, where $y \in S^{n-1}_-$ and $y' \in S^{n-1}_+$. It is the $G_3(k)$ term which is important here; we need the leading behaviour of $u_+$ at the negative end. It is not hard to show that $$u_+ = A |z|^{-n+2} e^{-k|z|} f_n(k|z|) + O(|z|^{-n+1}) \text{ for some } A > 0.$$at this end. Indeed, the harmonic function $\Phi_+$ is equal to $A' |z|^{-n+2} + O(|z|^{-n+1})$ as $z \to \infty$ along this end, for some $A' > 0$. The leading coefficient $a_0(K)$ from Lemma \[const\] must then be equal to a constant times $
e^{-K} f_n(K)$, which follows readily from the fact that $|z|^{-n+2} e^{-k|z|} f_n(k|z|)$ satisfies the equation $(\Delta + k^2) u = 0$. The specific structure of the parametrix $G(k)$, together with the estimate with $l=0$, shows that in this region the resolvent kernel may be written as a sum$$\begin{gathered}
K_0(y, y', r', \Lambda) + K_1(y, y', r', \Lambda), \\
K_0 = A r^{-n+2} (r')^{-n+2} e^{i \lambda r} f_n(\lambda r)e^{i \lambda r'} f_n(\lambda r') , \quad K_1 = O((r')^{-2(n-1)})
\end{gathered}$$ in the region of interest. Substituting $K_0$ for the resolvent into the integral , we obtain$$\lim_{t \to \infty} t^{n-1} \frac1{\pi i} \int_\Gamma e^{-t \lambda^2} A r^{-n+2} (r')^{-n+2} e^{i \lambda r} f_n(\lambda r)e^{i \lambda r'} f_n(\lambda r') \lambda \, d\lambda.
\label{contour-int-22}$$Let $\alpha = \sigma'/\sigma = \lim r/r'$. Substituting $\lambda = (r')^{-1} \Lambda$ and $t = (\sigma')^2 (r')^2$, and using $\lambda d\lambda = (r')^{-2} \Lambda d\Lambda$, we get $$\lim_{t \to \infty} t^{n-1} (r')^{-2(n-1)} \alpha^{-n+2} \frac1{\pi i} \int_\Gamma e^{-(\sigma')^2 \Lambda^2} e^{i \Lambda} f_n(\Lambda) e^{i \alpha \Lambda} f_n(\alpha \Lambda) \Lambda \, d\Lambda = C(\alpha, \sigma').
\label{contour-int-33}$$Thus the limit exists and is finite, when $K_0$ is substituted for the resolvent. As in the previous proof, when $K_1$ is substituted for the resolvent the limit is zero, since $K_1$ decays to an additional order at infinity. This completes the proof.
Riesz transform and $L^p$ cohomology {#coh}
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Here $(M^n,g)$ is a complete Riemannian manifold of dimension $n$.
We want here to discuss some consequence of the boundedness of the Riesz transform on $L^p$ for some $p> 2$ for the $L^p$ cohomology. On $(M,g)$, the space of $L^2$ differential forms admits the Hodge decomposition $$L^2(T^*M)=\cH^1(M)\oplus \overline{d C^\infty_0(M)}\oplus \overline{d^*
C^\infty_0(\Lambda^2 T^*M)},$$ where $\cH^1(M)=\{\alpha\in L^2(T^*M), d \alpha=0=d^*\alpha\}$ (see [@DR]). Let us recall now the definition of reduced $L^p$-cohomology: for $p\ge 1$ , the first space of reduced $L^p$ cohomology of $(M,g)$ is $$H^1_p(M)=\frac{\{\alpha\in L^p(T^*M),
d\alpha=0\}}{\overline{dC^\infty_0(M)}} \ ,$$ where we take the closure in $L^p$. The first space of reduced $L^2$ cohomology can be identified with $\cH^1(M)$. As noticed in [@ACDH], if we assume that for some $p\ge 2$ the Riesz transform $T:=d\Delta^{-1/2}$ is bounded on $L^p$ and on $L^{p/(p-1)}$, then the Hodge projector $$P=d\Delta^{-1}d^*=TT^*:L^p(M; T^*M)\cap L^2(M; T^*M)\longrightarrow
L^2(M; T^*M)$$ i.e. the orthogonal projector of $L^2(M; T^*M)$ onto the space of ‘exact forms’ extends by continuity to a bounded operator $$P: L^p(M; T^*M)\longrightarrow L^p(M; T^*M).$$
We assume now that $(M^n,g)$ is a complete Riemannian manifold, $n \geq 2$, satisfying the Nash inequality $$\label{nash}\mu \left(\int_M f^2 d\vol\right)^{1+2/n}\le \left(\int_M |f|
d\vol\right)^{4/n} \int_M |df|^2 d\vol,$$ for all $f\in C^\infty_0(M)$ and some $\mu>0$, and that the volume growth of geodesic balls is uniformly bounded: $$\label{volumgro}
\forall x\in M,\, \forall r>0,\ \vol B(x,r)\le C r^n.$$ It follows from [@Ca] that (\[nash\]) implies a matching lower bound: $$\label{volumlo}
\forall x\in M,\, \forall r>0,\ \vol B(x,r)\ge c r^n.$$ Note that (\[nash\]) easily implies the Faber-Krahn inequality : $$\label{fk}
\ \lambda_1(\Omega) \ge \mu
\left( \vol \Omega\right)^{-2/n},$$ for all $\Omega\subset M$ with finite measure, where $$\lambda_1(\Omega)=\inf\left\{\frac{ \int_\Omega |df|^2d\vol}{\int_\Omega f^2d\vol} , f\in
C^\infty_0(\Omega)\setminus\{0\}\right \}$$ is the first eigenvalue for the Laplacian on $\Omega$ for the Dirichlet boundary conditions (in fact, (\[nash\]) and (\[fk\]) are equivalent, see [@G]). Also, if $n>2$, (\[nash\]) is equivalent to the Sobolev inequality : $$\label{sobo}
\nu \left(\int_M |f|^{\frac{2n}{n-2}} d\vol\right)^{1-2/n}\le \int_M
|df|^2 d\vol,\ \forall f\in C^\infty_0(M),$$ for some $\nu>0$ (see for instance [@BCLS]).
According to [@CD], we know that on $(M,g)$ the Riesz transform is bounded on $L^q$ for $q\in]1,2]$. Hence if we assume that for some $p\ge 2$ the Riesz transform is also bounded on $L^p$ then, according to the above remark, the Hodge projector $$P=:L^p(M; T^*M)\cap L^2(M; T^*M)\longrightarrow L^2(M; T^*M)$$ extends by continuity to a bounded operator $$P: L^p(M; T^*M)\longrightarrow L^p(M; T^*M).$$
\[last\] Under the hypotheses (\[nash\], \[volumgro\]), if the Riesz transform is bounded in $L^p$ for some $p>2$, then $P\left(L^p(M; T^*M)\right)$ is the closure in $L^p(M; T^*M)$ of $d C^\infty_0(M)$.
According to [@CKS], the Nash inequality implies that the semigroup $e^{-t\Delta}$ satisfies the bound $$\label{heatmapping}
\|e^{-t\Delta}\|_{L^1\to L^\infty}\le Ct^{-n/2}, \forall\,t>0.$$ A result of N. Varopoulos ([@V]) then implies the following mapping property for $q\in ]1,n[$: $$\label{mapping} \Delta^{-1/2}:L^q(M)\rightarrow
L^{qn/(n-q)}(M).$$ In order to prove the lemma, we have to show that if $\alpha\in C^\infty_0(M; T^*M)$ then $P\alpha$ can be approximated in $L^p$ by a sequence of elements of $d C^\infty_0(M)$. We seek a sequence $\chi_k$ of smooth functions with compact support such that $$L^p \operatorname{-}\lim_{k\to\infty} d(\chi_k\Delta^{-1} d^*\alpha)=P\alpha.$$ Since we assume that the Riesz transform is bounded in $L^p$, we know that its adjoint $$\Delta^{-1/2}d^*:L^{p/(p-1)}(M; T^*M)\rightarrow L^{p/(p-1)}(M)$$ is bounded. Hence we have $\Delta^{-1/2}d^*\alpha\in L^{p/(p-1)}(M)$. Note that the condition $p/(p-1)<n$ is satisfied since we are assuming that $n\ge 2$, in which case $p> 2\ge n/(n-1)$. Thus, by , $$\Delta^{-1/2}\Delta^{-1/2}d^*\alpha=\Delta^{-1}d^*\alpha\in
L^{\frac{pn}{n(p-1)-p}}(M).$$Choose a point $o \in M$ and choose $$\chi_k(x)=
\left\{\begin{array}{lll}
1& {\rm if} & x\in B(o,k)\\
0&{\rm if} & x\not \in B(o,2k)\\
\end{array}\right.$$ $${\rm with}\ \|d\chi_k\|_{L^\infty}\le C/k.$$ Since $$d(\chi_k\Delta^{-1} d^*\alpha)=\chi_k(d\Delta^{-1}
d^*\alpha)+d\chi_k(\Delta^{-1} d^*\alpha )$$ and $$\chi_k(d\Delta^{-1}
d^*\alpha)=\chi_kP\alpha$$ obviously tends to $P\alpha$ in $L^p$ as $k\to\infty$, we need only to show that $$\lim_{k\to\infty} \|d\chi_k(\Delta^{-1} d^*\alpha) \|_{L^p}=0.$$ We know that $\varphi=\Delta^{-1} d^*\alpha $ is harmonic outside a big ball $B(o,R_0)$ containing the support of $\alpha$. Now the Faber-Krahn inequality (\[fk\]) implies a mean value inequality for harmonic functions (see [@G1], Lemma 6.9) which yields: $$|\varphi(x)|\le
C(\mu,n,p)r^{-\frac{(p-1)n-p}{p}}\|\varphi\|_{L^{\frac{pn}{n(p-1)-p}}(B(x,r))},$$ provided that $B(x,r)\subset M\setminus B(o,R_0)$ . In particular if $\rho(x)=\dist(x,o)-R_0>0$ we obtain $$|\varphi(x)|^p \le \frac{C}{ \rho(x)^{(p-1)n-p}}.$$ Hence we finally obtain, if say $k\ge 2R_0$: $$\|d\chi_k(\Delta^{-1} d^*\alpha) \|_{L^p}^p\le C\frac{\vol
B(o,2k) }{k^{(p-1)n}} \le C k^{(2-p)n}$$ which indeed goes to zero when $k\to\infty$. We have proved that $$PL^p(M; T^*M))\subset \overline{dC^\infty_0(M)}.$$ The converse inclusion follows from the fact that $$dC^\infty_0(M)\subset PL^p(M; T^*M))$$ and that $P$, being a bounded projector, has a closed range.
As a consequence of Lemma \[last\], if the assumptions (\[nash\],\[volumgro\]) are satisfied and if the Riesz transform is bounded on $L^p$ for some $p>2$, then $H^1_p(M)$ can be identified with:$$\{\alpha \in L^p(M; T^*M) \mid
d\alpha=0\ {\rm and}\ P\alpha=0\}.$$ Moreover we also have
Under the hypotheses (\[nash\], \[volumgro\]), if the Riesz transform is bounded in $L^p$ for some $p>2$, then $$\{\alpha\in L^p(M; T^*M) \mid d^*\alpha=0\}=\{\alpha\in L^p(M; T^*M) \mid P\alpha=0\}.$$ \[coclosed\]
As a matter of fact, we have $\{\alpha\in L^p(M; T^*M), P\alpha=0\}=\Imag (\Id -P)$. The density of $C_0^\infty(M; T^*M)$ in $L^p(M; T^*M)$ and the boundedness of $P$ in $L^p$ imply that $(\Id
-P)(C_0^\infty(M; T^*M))$ is dense in $\Imag (\Id -P)$. But we have $$(\Id -P)(C_0^\infty(M; T^*M))\subset \{\alpha\in L^p(M; T^*M) \mid d^*\alpha=0\}.$$ The latter space is closed, hence we have the inclusion $$\{\alpha\in L^p(M; T^*M) \mid P\alpha=0\}\subset \{\alpha\in L^p(M; T^*M) \mid d^*\alpha=0\}.$$ Now assume that $\alpha\in L^p(M; T^*M)$ is coclosed. We define a sequence of cutoff functions by $$\chi_k(x)=
\left\{\begin{array}{lll}
1& {\rm if} & x\in B(o,k)\\
\frac{ \log\left(k^2/\dist(x,o)\right)}{\log k} & {\rm if} & x\in B(o,k^2)\setminus B(o,k)\\
0&{\rm if} & x\not \in B(o,k^2)\\
\end{array}\right.\label{cutoffk}$$ Then $L^p-\lim_{k\to\infty}\chi_k\alpha=\alpha$ but now $\chi_k \alpha\in
L^2$ and $P\chi_k\alpha=-T\Delta^{-1/2}({\rm int}_{\nabla\chi_k}\alpha)$, because $d^*(\chi_k \alpha)=-{\rm int}_{\nabla\chi_k}\alpha$ (here ${\rm int}_{\nabla\chi_k}\alpha$ denotes the contraction of $\alpha$ with the vector field $\nabla \chi_k$). Take $l\in ]1,n[ $ with $$\frac1l=\frac1p+\frac1n.$$ Then we have $$\left\|{\rm int}_{\nabla\chi_k}\alpha\right\|_{L^l}\le
\left\|\alpha\right\|_{L^p(M\setminus B(o,k))}\, \left\|\nabla\chi_k\right\|_{L^n}.$$ But an easy computation leads to $$\left\|\nabla\chi_k\right\|^n_{L^n}\le C (\log k)^{-(n-1)}.$$ With , we have $$\lim_{k\to \infty}
\left\|\Delta^{-1/2}({\rm int}_{\nabla\chi_k}\alpha)\right\|_{L^p}=0$$ hence by continuity of $T$ we obtain $P\alpha=0$. In particular we obtain a Hodge-de Rham interpretation of the $L^p$ cohomology:
\[HdR\] Under the hypotheses (\[nash\], \[volumgro\]), if the Riesz transform is bounded in $L^p$ for some $p>2$, then $$H^1_p(M)\simeq \{\alpha \in L^p(M; T^*M) \mid
d\alpha=0\ {\rm and}\ d^*\alpha=0\}.$$
In general one cannot compare $L^p$ cohomology for different values of $p$. However let us now assume that the Ricci curvature of $M$ is bounded from below: $$\label{ricci}
\Ricci\ge -(n-1)\kappa^2 g.$$ From the Bochner formula $$\Delta \alpha \equiv \nabla^*\nabla \alpha+\Ricci (\alpha, \cdot)$$ we see that if $\alpha$ is a harmonic $1$-form in $L^2$, then it satisfies the subelliptic estimate $$\Delta |\alpha|\le (n-1)\kappa^2 |\alpha| .
\label{subelliptic}$$ Indeed if for $\varepsilon>0$ we define $f_\varepsilon= \sqrt{|\alpha|^2+\varepsilon}$, then it is classical to show that the Bochner formula and the Kato inequality imply $$\Delta f_\varepsilon\le (n-1)\kappa^2 f_\varepsilon.$$ Passing to the limit $\varepsilon=0$, we get the desired subelliptic estimate (only in the distributional sense). Hence with the Nash inequality we can deduce that $\alpha$ is in fact bounded and $$\|\alpha\|_{L^\infty} \le C(n,\kappa,\mu) \|\alpha\|_{L^2};
\label{bounded}$$ this can be done using a Nash-Moser iteration scheme [@Berard], but we can also use our upper bound on the heat operator with the inequality (\[subelliptic\]), to assert that for every $x\in M$ $$t\mapsto \left(e^{-t(\Delta-(n-1)\kappa^2 )}
|\alpha|\right)(x)$$ is non-decreasing. With the mapping properties of the heat operator (\[heatmapping\]) we obtain $$|\alpha|(x)\le C t^{-n/2}e^{t(n-1)\kappa^2}
\|\alpha\|_{L^2};$$ with $1/t=(n-1)\kappa^2$, we obtain the desired bound. Hence there is a well defined map $$\cH^1(M)\rightarrow H^1_p(M).$$ Proposition \[HdR\] immediately implies
\[inj\] Assume that (\[nash\], \[volumgro\], \[ricci\]) hold, and that for some $p>2$, the Riesz transform is bounded on $L^p$ and on $L^{p/(p-1)}$. Then the natural map $$\cH^1(M)\rightarrow H^1_{p}(M)$$ is injective.
A corollary of this lemma is:
\[nb\]Assume that $n>2$, and that $(M,g)$ satisfies the assumptions (\[nash\], \[volumgro\], \[ricci\]) and that it has more than two ends, then for every $p\ge n$ the Riesz transform is not bounded on $L^p$.
In $n>2$, then $(M,g)$ satisfies the Sobolev inequality (\[sobo\]), and following [@CSZ], we know that if $M$ has more than two ends there exists a non-constant bounded harmonic function $h$ with finite Dirichlet energy[^6], hence $dh$ is a harmonic $L^2$ $1-$form. Take $\chi_k$ as in . We have $d\chi_kh=\chi_k dh+hd\chi_k$, but if $p\ge n$, then if $V(r):=\vol B(o,r)$ we have $$\| hd\chi_k\|^p_{L^p}\le C\|h\|^p_{L^\infty} \int_k^{k^2} ((\log k)r)^{-p}dV(r)$$ and integrating by parts we have : $$\int_k^{k^2} r^{-p}dV(r)= \frac{ \vol B(o,k^2)}{k^{2p}}-
\frac{ \vol B(o,k)}{k^{p}}+p\int_k^{k^2} \frac{ \vol B(o,r)}{r^{p+1}}dr.$$ This quantity is bounded for $p>n$ and grows as $C\log k$ if $p=n$. Hence we obtain that $dh=L^p-\lim_{k\to\infty} d(\chi_kh)$. Hence $dh$ is zero in reduced $L^p$ cohomology. Since $dh$ is non-zero in $\cH^1(M)$, and since under (\[nash\],\[volumgro\]) the Riesz transform is bounded on $L^p$ for $1<p<2$, Lemma (\[inj\]) says that it can not be bounded on $L^p$ if $p\ge n$.This generalizes the unboundedness part of Theorem \[main\] to the much larger class of manifolds satisfying , , .
Concluding remarks and open problems {#conc}
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In this final section we discuss some questions suggested by the work above, and pose some open problems.
For any complete Riemannian manifold $M$ of infinite measure there are numbers $\pmin \leq 2 \leq \pmax$ such that the Riesz transform is bounded on $L^p$ for all $p$ between $\pmin$ and $\pmax$ (it may or may not be bounded at $p = \pmin$ or $p = \pmax$). We may call these values the lower and upper thresholds for $M$.
There are a number of classes of manifolds on which the Riesz transform is known to be bounded on $L^p$ for all $p$ (in other words, $\pmin = 1$ and $\pmax = \infty$); for example, manifolds with nonnegative Ricci curvature [@Bakry], Cartan-Hadamard manifolds with a spectral gap [@Lo], noncompact symmetric spaces [@Anker] and Lie groups of polynomial growth [@Alex] (see [@ACDH] for more examples and references). On the other hand, Coulhon and Ledoux showed in [@CL] that for any $p_0>2$ there is a manifold $M$ with bounded geometry such that $\pmax \leq p_0$. Another example, with polynomial volume growth, was given in [@CDfull]. Since, as we mentioned earlier, it was shown by Coulhon and Duong that $\pmin = 1$ for a large class of complete manifolds, and that $\pmax = 2$ for certain simple surfaces, one could wonder whether $2$ is the upper threshold for a large class of manifolds. But H.-Q. Li [@Li] proved that for $n$-dimensional cones with compact basis, $$\pmax = \begin{cases} n \Big( \frac{n}{2} - \sqrt{\big(\frac{n-2}{2}
\big)^2 + \lambda_1} \Big)^{-1}, \quad \lambda_1 < n-1 \\+\infty, \qquad \lambda_1 \geq n-1, \end{cases}$$where $\lambda_1$ is the smallest nonzero eigenvalue of the Laplacian on the basis. Note that $\pmax>n$ here.
Is a result similar to H.-Q. Li’s valid for smooth manifolds with one conic or asymptotically conic end? what happens for several conic ends?
Manifolds with more than one Euclidean end satisfy the doubling condition but not the scaled $L^2$ Poincaré inequality. Our result sheds some light on the implications for the Riesz transform of these conditions: it follows from [@Li] and [@CouLi] that doubling together with Poincaré (equivalently, upper and lower Gaussian estimates of the heat kernel) are not sufficient for the Riesz transform to be bounded for [*all*]{} $p>2$. Theorem \[main\] shows that these conditions are not necessary for the Riesz transform to be bounded for [*some*]{} $p>2$.
The class of manifolds with Euclidean ends is of course extremely special. One can attempt to enlarge the class of known examples synthetically, i.e. by creating further examples from known examples by performing various operations. Our results may be seen as obstructions to the stability of the $L^p$ boundedness of the Riesz transform under gluing for $p$ above the dimension.
Under which conditions is boundedness of the Riesz transform on $L^p$ stable under the following operations on manifolds:
- gluing,
- compact metric perturbations,
- taking products, $((M_1, g_1), (M_2, g_2)) \to (M_1 \times M_2, g_1 \oplus g_2)$,
- warped products.
We only mention, without proof, one result along these lines. Namely, if the Riesz transform is bounded on $L^p$ on a complete Riemannian manifold $M$ of infinite measure, and with Ricci curvature bounded from below, then it is bounded on $L^p$ on $M \times N$ for any compact $N$.
We also mention a conjecture on manifolds obtained by gluing several copies of a simply connected nilpotent Lie group (endowed with a left invariant metric). According to [@Alex] we know that the Riesz transform is bounded for every $p$ on a simply connected nilpotent Lie group. Let $(N,g_0)$ be a simply connected nilpotent Lie group of dimension $n>2$ (endowed with a left invariant metric). According to [@Alex] we know that the Riesz transform on $(N,g_0)$ is bounded for every $p$. Let $\nu$ be the homogeneous dimension of $N$; for instance we can set $$\nu=\lim_{R\to \infty} \frac{\log \vol B(o,R)}{\log R},$$ $o\in N$ being a fixed point. Let $(M,g)$ be a manifold obtained by gluing $k>1$ copies of $(N,g_0)$. The manifold $M$ is diffeomorphic to the sphere with $k$ points removed. According to [@CD] we know that on $(M,g)$ the Riesz transform is bounded on $L^p$ for $p\in ]1,2]$. The argument of Section \[coh\] can be applied (changing the $n$ appearing in analytic inequalities by the homogeneous dimension $\nu$). Then (\[nb\]) implies that the Riesz transform is not bounded on $L^p$ for $p\ge \nu$. We can moreover compute the $L^p$ cohomology of $M$. The arguments of Proposition 3.3 of [@Canew] are given for the $L^2$ cohomology of such $M$ but they can be easily modified for $L^p$ cohomology. We find that for $p\in ]1,\nu[$, $$H^1_p(M)\simeq H^1_c(M)=\R^{k-1},$$ whereas for $p\ge \nu$ we find $$H^1_p(M)=\{0\}.$$ That is, the conclusion of Lemma \[inj\] is satisfied if and only if $p<\nu$. This gives another proof of the fact that the Riesz transform is not bounded on $L^p$ for $p\ge \nu$.
Show that the Riesz transform on $(M,g)$ is bounded on $L^p$ for $p\in]1,\nu[$.
Finally, it would be interesting to get results for differential forms. One can consider either $\nabla \Delta^{-1/2}$ or $(d + \delta) \Delta^{-1/2}$, where $\Delta = (d + \delta)^2$ is the Laplacian on forms and $\Delta^{-1/2}$ really means $f(\Delta)$ where $f(0) = 0$ and $f(x) = x^{-1/2}$ for $x > 0$ (this projects off the $L^2$ null space of $\Delta$, i.e. the $L^2$-cohomology, which is trivial in the case of $0$-forms when $M$ has infinite measure).
Determine the upper and lower thresholds for $(d + \delta) \Delta^{-1/2}$ or $\nabla \Delta^{-1/2}$ acting on $k$-forms on manifolds with Euclidean ends.
Extend the results of section \[coh\] to differential forms of all degrees.
[10]{}
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[^1]: T.C. is supported in part by the European Commission (IHP Network “Harmonic Analysis and Related Problems” 2002-2006, Contract HPRN-CT-2001-00273-HARP). He also acknowledges for this work the support of Macquarie University and Australian National University
[^2]: A.H. is supported in part by an Australian Research Council Fellowship and acknowledges the support of Université de Nantes.
[^3]: This means that smooth functions near the boundary of the compactification are given precisely by smooth functions of $\hat z$ and $1/|z|$.
[^4]: In other words, the kernel of $A$ has a singularity at the diagonal characteristic of pseudodifferential operators of order $m$, and this holds smoothly up to the boundary, in the sense that it could be extended across the boundary as a conormal distribution. See [@Ho], section 18.2, for the precise definition.
[^5]: Here $L^2_b(M)$ is the $L^2$-space with respect to the b-metric $g_b = x^2 g$; thus $L^2_b(M) = \ang{z}^{n/2} L^2(M)$. Also $H^2_b(M)$ is the b-Sobolev space of order $2$, defined as the set of functions $g \in L^2(M)$ such that $Qg \in L^2(M)$ for all b-differential operators of order $2$ on $M$ (near infinity, such operators take the f orm $\ang{z}^2 \sum_{i,j} a_{ij} \partial_{z_i} \partial_{z_j}$, with $a_{ij} \in {{\mathcal{C}^\infty}}({\overline{M}})$).
[^6]: In fact if $M\setminus K=U_+\cup U_-$ with $K$ compact and $U_\pm$ unbounded then $\lim_{x\to\infty,x\in U_\pm} h(x)=\pm 1$.
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abstract: 'Quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition in the Dirac-Feynmann path integrals. The quantum trajectories are the key concept to understand extreme nonlinear optical phenomena, such as high-order harmonic generation (HHG), above-threshold ionization (ATI), and high-order terahertz sideband generation (HSG). While HHG and ATI have been mostly studied in atoms and molecules, the HSG in semiconductors can have interesting effects due to possible nontrivial “vacuum" states of band materials. We find that in a semiconductor with non-vanishing Berry curvature in its energy bands, the cyclic quantum trajectories of an electron-hole pair under a strong terahertz field can accumulate Berry phases. Taking monolayer MoS$_2$ as a model system, we show that the Berry phases appear as the Faraday rotation angles of the pulse emission from the material under short-pulse excitation. This finding reveals an interesting transport effect in the extreme nonlinear optics regime.'
author:
- Fan Yang
- 'Ren-Bao Liu'
title: Berry phases of quantum trajectories in semiconductors under strong terahertz fields
---
After excitation by a weak near-resonant laser in semiconductors, an electron and a hole can be created and driven into large amplitude oscillations by an intense low-frequency ac electric field such as from a terahertz (THz) laser. The recollisions between the electron and hole during the oscillations will generate high-order sidebands relative to the excitation frequency [@HSG_RBL; @Zaks_Liu]. This high-order THz sideband generation (HSG) has potential electro-optical applications such as wide-band optical multiplexers, terabit/sec optical communications, and optical pulses with ultra-high repetition rate [@HSG_RBL; @Zaks_Liu]. The HSG is analogous to the high-order harmonic generation (HHG) in atoms and molecules [@HHG_Krause; @HHG_Corkum; @HHG_QT]. Both HSG and HHG spectra are characterized by a wide-band plateau with a sharp cut-off. The HSG spectrum has been well understood using the quantum trajectory theory developed in HHG, in which the quantum evolution of particles under a strong field is described by a few paths that satisfy the stationary phase condition (i.e., the saddle points) in the formalism of Dirac-Feynmann path integrals [@HSG_RBL; @HHG_QT]. The HSG or HHG cutoff is determined by the maximum energy the particles in the quantum trajectories can acquire from the driving field [@HSG_RBL; @HHG_QT].
A fundamental difference between HHG in atomic systems and HSG in semiconductors is that the “vacuum" state of a semiconductor can have non-trivial structures (such as in topological insulators [@SCZhang; @Kane]). When the eletron (or hole) in a spin-orbit-coupled semiconductor is accelerated by an ac electric field ${\mathbf F}\left(t\right)$, not only does the quasi-momentum evolve according to the semiclassical equation $\dot {{\mathbf k}}=-e{\mathbf F}\left(t\right)$ [@Solid_AM], but also its Bloch wavefunction (the direction of the spin) is changed. Thus the evolution driven by the electric field leads to a geometric phase in addition to the dynamical one, which is the famous Berry phase in a cyclic evolution [@Berry]. This geometric phase is of fundamental importance for a gauge-invariant description of the nonlinear optics in insulators [@Sipe2nd; @spin_current; @Sipe].
The Berry phase effect is clearly seen in the representation of the polarization operator in the basis of Bloch states, which is found by Blount [@Blount] $$\left\langle {\psi _{n,{\mathbf k}} } \right|{\mathbf r}\left| {\psi _{m,{\mathbf k}'} } \right\rangle = i\left[ {\delta_{nm} \nabla _{\mathbf k} + \left\langle {u_{n,{\mathbf k}} } \right|\nabla _{\mathbf k} \left| {u _{m,{\mathbf k}} } \right\rangle } \right]\delta \left( {{\mathbf k} - {\mathbf k}'} \right),$$ where the first term gives change of the quasi-momentum and the second term is the so-called Berry connection or Berry vector potential. Through the polarization operator, the Berry phase (or, more intrinsically, the Berry curvature) appears naturally in various optical effects in condensed matter systems as revealed by some recent works. For example, the presence of the Berry phase effect was noticed in the optical birefringence effects of a pure spin current [@spin_current1st] or the second-order non-linear spectroscopy of spin currents [@spin_current]. Also the photogalvanic effect in topological insulator surfaces can depend on the Berry curvature [@CPGE]. The interaction between an intense THz laser and semiconductors [@Zaks_Liu] provides a new opportunity to explore the Berry phase effect in the regime of extreme nonlinear optics.
In this Letter, we show that the optical response of a semiconductor under an intense THz field explicitly includes the Berry phase. We analyze the effect using the quantum trajectory theory and apply the theory to monolayer $\rm{MoS}_2$ as a model system. In the time-domain response, we find that the Faraday rotation angle of the emission delayed by integer multiples of the THz laser period is given by the Berry phase of a specific trajectory. The quantum trajectory approximation is verified by numerical simulations.
Let us consider a general semiconductor under a strong THz field ${\mathbf F}(t)$, which enters into the Hamiltonian through a uniform electromagnetism vector potential: ${\mathbf p} \to {\mathbf p} + e{\mathbf A}\left( t \right)$, with ${\mathbf F}= - \partial {\mathbf A}/\partial t$. Because ${\mathbf A}$ preserves the translational symmetry, Bloch’s theorem still applies and we write the Hamiltonian in the ${\mathbf k}$-space representation $H \left(\tilde {{\mathbf k}}\left( t \right)\right)$ [@QNiu2], with $\tilde {{\mathbf k}}\left( t \right) = {\mathbf k} + e{\mathbf A}\left( t \right)$. The instantaneous Bloch states of $H \left(\tilde {{\mathbf k}}\left( t \right)\right)$ are obtained from the original Bloch states by simply changing ${\mathbf k}$ to $\tilde{{\mathbf k}}$, $$H\left( {\tilde {{\mathbf k}}\left( t \right)} \right)\left| { \pm ,\mu ,\tilde {{\mathbf k}}\left( t \right)} \right\rangle = E^{\pm}_{ \tilde {{\mathbf k}}\left( t \right)}\left| { \pm ,\mu ,\tilde {{\mathbf k}}\left( t \right)} \right\rangle,$$ where $+$ and $-$ are the indices of the conduction and valence bands, respectively, and $\mu$ is the spin index introduced to indicate possible band degeneracy of the system.
Now let us calculate the linear response of this system to a near infrared (NIR) laser that creates electron-hole pairs at the band edge with interaction Hamiltonian $\hat H_{\text{NIR}}=-\hat{{\mathbf P}}\cdot {\mathbf E}_{\text{NIR}}e^{-i\Omega t}+\text{h.c.}$, where the interband polarization operator $\hat{{\mathbf P}}$ in the interaction picture is $$\hat {{\mathbf P}}\left(t\right) = \int {d{\mathbf k}} \hat e_{\mu, {\mathbf k}}^\dag \hat h_{\nu, -{\mathbf k}}^\dag {\mathscr{D}}_{\mu \nu,{\mathbf k}}\left(t\right).$$ Here $\hat e$ and $\hat h$ are electron and hole operators, respectively, and $\mathscr{D}_{\mu \nu ,{\mathbf k}}\left(t\right) = - ie \left\langle {\psi_{+,\mu,{\mathbf k}}\left(t\right)} \right|\nabla _{{\mathbf k}} \left| {\psi_{-,\nu,{\mathbf k}}\left(t\right)} \right\rangle$ is the interband dipole moment, with $| {\psi_{\pm,\mu,{\mathbf k}}\left(t\right)} \rangle$ denoting the adiabatic evolution of the instantaneous Bloch states under the driving of the THz field $$\left| {\psi_{\pm,\mu,{\mathbf k}}\left(t\right)} \right\rangle = \left| {\pm, \alpha ,\tilde {{\mathbf k}}\left( t \right)} \right\rangle \left[ {\hat T e^{ { -i\int_{ - \infty }^t E_{\tilde{{\mathbf k}} \left(\tau\right)}^{\pm}d\tau + i \int_{ - \infty }^t {\mathscr{A}_{\tilde {{\mathbf k}}\left( \tau \right)}^ {\pm} \cdot d\tilde {{\mathbf k}}\left( \tau \right)} } }} \right]_{\alpha \mu},$$ where $\hat T$ is the time-ordering operator, the Berry connection is defined as $\left({\mathscr{A}}_{\tilde {{\mathbf k}}}^{\pm}\right)_{\mu \nu} = i\left\langle {\pm ,\mu , \tilde {{\mathbf k}}} \right|\nabla _{{\mathbf k}} \left| {\pm,\nu , \tilde {{\mathbf k}}} \right\rangle$, and summation of repeated dummy indices is assumed. In general, the Berry connection can be non-Abelian. Assuming the initial state is the vacuum state $\left|G\right\rangle$ with empty conduction bands and filled valence bands, we obtain the linear response to the NIR optical field as $$\begin{aligned}
&\notag \left\langle {{\mathbf P}\left( t \right)} \right\rangle = \frac{{ - i}}{V}\int_{ - \infty }^t {dt'} \left\langle G \right|\hat{{\mathbf P}}\left( t \right)\hat H_{\text{NIR}} \left( {t'} \right)\left| G \right\rangle \\
&\notag = i\int_{ - \infty }^t {dt'} \int {\frac{{d {\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ - i\int_{t'}^t {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} - i\Omega t' }{{\mathbf d}}_{\nu \mu ,\tilde {{\mathbf k}}\left( t \right)}^\dag \left[ \hat Te^{ i \int_{t'}^t {{\mathscr{A}}_{\tilde{{\mathbf k}}\left( \tau \right)}^ + \cdot d\tilde {{\mathbf k}}\left( \tau \right)} } \right]_{\mu \mu '} \\
&\quad {{\mathbf d}}_{\mu '\nu ',\tilde {{\mathbf k}}\left( {t'} \right)} \cdot { {\mathbf E}}_{\text{NIR}}\left[ \hat T e^{ i \int_{ t' }^t { {\mathscr{A}}_{\tilde{{\mathbf k}}\left( \tau \right)}^ - \cdot d\tilde {{\mathbf k}}\left( \tau \right)} } \right]_{\nu '\nu }^\dag , \label{NAbelQT}\end{aligned}$$ where $d$ is the dimension of the system, $\varepsilon _{\tilde {{\mathbf k}}} = E^{+}_{\tilde {{\mathbf k}}} - E^{-}_{\tilde {{\mathbf k}}}$ is the energy of the electron-hole pair and ${{\mathbf d}}_{\nu \mu,\tilde {{\mathbf k}} } = - ie \left \langle +,\nu,\tilde{{\mathbf k}}\right|\nabla _{{\mathbf k}} \left| -,\mu,\tilde{{\mathbf k}} \right\rangle$ is the instantaneous dipole moment. The Berry phase enters Eq. (\[NAbelQT\]) due to the requirement of the gauge invariance of the physical result under the local gauge transformation $\left| {\pm , \mu , {\mathbf k}} \right\rangle \to \left| {\pm , \alpha , {\mathbf k}} \right\rangle U^{\pm}_{\alpha \mu} \left( {\mathbf k} \right)$, which introduces the gauge freedom of ${{\mathbf d}}_{\nu \mu,\tilde {{\mathbf k}} }$. For the case without band degeneracy, the Berry phase becomes Abelian, and the response is reduced to $$\begin{aligned}
\left\langle {{\mathbf P}\left( t \right)} \right\rangle = \frac{{i}}{{\left( {2\pi } \right)^d }}\int_{ - \infty }^t {dt'} \int {d {\mathbf k}} {{\mathbf d}}_{\tilde {{\mathbf k}} \left( t \right)}^* {{\mathbf d}}_{\tilde {{\mathbf k}} \left( {t'} \right)} \cdot {{\mathbf E}}_{\text{NIR}} \notag \\
e^{ - i\int_{t'}^t {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau + i \int_{t'}^t {{{\mathscr {A}}_{\tilde {{\mathbf k}} \left( \tau \right)} } \cdot d\tilde {{\mathbf k}} \left(\tau\right)} - i\Omega t'} } , \ \ \quad\qquad \label{AbelQT}\end{aligned}$$ where ${\mathscr{A}}_{{\mathbf k}}={\mathscr{A}}^+_{{\mathbf k}}- {\mathscr{A}}^-_{{\mathbf k}}$ is the combined Berry connection of the electron-hole pair. It is interesting to note that the Berry term $\mathscr{A}_{\tilde {{\mathbf k}} } \cdot {\dot {\tilde {{\mathbf k}}}}$ in the phase factor can be regarded as a generalization of the interaction energy ${\mathbf A} \cdot \dot{{\mathbf r}}$ in the electromagnetism theory in real space to the momentum space [@QNiu2].
Now let us focus on the geometric phase part of Eq. (\[AbelQT\]). Without loss of generality, we consider an elliptically polarized THz field in the $x$-$y$ plane $$\label{Ft}
{\mathbf F}\left( t \right) = F \left( \cos\left(\theta\right)\cos \left(\omega t\right),\sin\left(\theta\right)\sin \left(\omega t\right), 0 \right).$$ Then the electron-hole pair goes along an elliptical path in the ${\mathbf k}$-space under the driving of the THz field: $$\tilde {{\mathbf k}}\left( t \right) = \left( {k_x - k_0\cos\theta \sin \left(\omega t\right),k_y + k_0\sin\theta \cos \left(\omega t\right)},k_z \right), \label{path}$$ where $k_0 = {eF}/\omega$. After an integer multiple periods of the THz field $t-t'=nT=2n\pi/\omega$, the electron-hole pair completes a closed path in the ${\mathbf k}$-space. During this cyclic evolution, the geometric phase acquired by the electron-hole pair equals the Berry curvature times the area $S\left(\theta,k_0\right)=n\pi k_0^2 \sin\left(2\theta\right)/2$ enclosed by the path, which is further related to the polarization and strength of the THz light.
To measure the Berry phase effect of quantum trajectories, we can apply a short NIR laser pulse to the system at time $t'=t_0$, with the width of the pulse much smaller than $T$. Thus the pulse can be approximated by a $\delta$-pulse ${{\mathbf E}}_{\text{NIR}}={\mathbf E}\delta\left(t-t_0\right)$. Then response of the system at $t_n=t_0+nT$ explicitly contains the Berry phase of a closed path: $\phi_B^{\left(n\right)}\left({\mathbf k}\right)=\int_{t_0}^{t_n} { {\mathscr{A}}_{\tilde {{\mathbf k}}\left( \tau \right)} \cdot d\tilde {{\mathbf k}}\left( \tau \right)}$. In order to separate the Berry phase from the dynamical phase, we can introduce some specific interference that singles out the geometric phase part. We note that under the time-reversal transformation, the direction of the path in ${\mathbf k}$-space is reversed and the Berry phase becomes opposite, while the dynamical phase is unchanged. It leads us to consider the solid state systems that preserve the time-reversal and inversion symmetry and have nontrivial Berry phases, such as the topological insulators [@SCZhang; @Kane], monolayer $\rm{MoS}_2$ and other group-VI dichalcogenides [@MOS2_Mak1; @MOS2; @MOS2_Zeng; @MOS2_Mak; @MOS2_Cao] and bilayer graphene [@Graphene_RMP; @BilayerG]. We denote one state of the Kramers pair as the pseudospin state $\Uparrow$ and the other as $\Downarrow$. From the time-reversal and inversion symmetry, we obtain the following relations ${{\mathbf d}}_{ \Uparrow \Uparrow ,{\mathbf k}} = {{\mathbf d}}_{ \Downarrow \Downarrow , - {\mathbf k}}^* = - {{\mathbf d}}_{ \Downarrow \Downarrow ,{\mathbf k}}^*:={{\mathbf d}}_{{\mathbf k}}$, $\varepsilon _{ \Uparrow , {\mathbf k}} = \varepsilon _{ \Downarrow , - {\mathbf k}}: = \varepsilon _{{\mathbf k}}$ and $\left({\mathscr{A}}_{{\mathbf k}}\right) _{\Uparrow \Uparrow} = \left({\mathscr{A}}_{ - {\mathbf k}}\right)_{\Downarrow \Downarrow}^*:={\mathscr{A}}_{{\mathbf k}}$. Thus we get the key formula $$\phi_B^{\left(n\right)}\left({\mathbf k}\right)=\phi^{\left(n\right)}_{B,\Uparrow\Uparrow}\left({\mathbf k}\right)
=-\phi^{\left(n\right)}_{B,\Downarrow\Downarrow}\left({\mathbf k}\right)
=\int_{t_0}^{t_n} {{\mathscr{A}}_{\tilde {{\mathbf k}}} \cdot d\tilde {{\mathbf k}}},$$ i.e. the Berry phase of the two time-reversal related paths are opposite to each other. With these considerations, the response at $t_n$ is simplified as $$\begin{aligned}
\left\langle {{\mathbf P}\left( t_n \right)} \right\rangle = & i\int {\frac{d{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} +i \phi_B^{\left(n\right)}\left({\mathbf k}\right) } } {{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)}^* {{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)} \cdot { {\mathbf E}} \ \notag\\
+ & i\int {\frac{d{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} -i \phi_B^{\left(n\right)}\left({\mathbf k}\right) } } {{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)}{{\mathbf d}}^*_{\tilde {{\mathbf k}}\left( t_0 \right)} \cdot { {\mathbf E}}. \label{RespN}\end{aligned}$$ Here $\left\langle {{\mathbf P}\left( t_n \right)} \right\rangle$ is given by the interference between two kinds of responses with opposite Berry phases.
Equation (\[RespN\]) can be studied using the quantum trajectory theory, i.e. the stationary phase formalism [@HSG_RBL; @HHG_QT]. In the path integral, the electron-hole pairs move along all possible trajectories when driven by the THz field, with the phase given by the action $S_{\pm}\left({\mathbf k}\right)= \int_{t_0}^{t_n} \left(\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)}\pm{\mathscr{A}}_{\tilde {{\mathbf k}}} \cdot e{{\mathbf F}}\left(\tau\right) \right) d\tau$. As the THz field is strong, the motion amplitude of the electron-hole pair is much larger than the quantum fluctuation. Thus the response is dominated by the stationary phase points of the actions $$\nabla_{{\mathbf k}}S_{\pm}\left({\mathbf k}\right)= \int_{t_0}^{t_n} \left(\nabla_{{\mathbf k}}\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} \pm \nabla_{{\mathbf k}}{\mathscr{A}}_{\tilde {{\mathbf k}}} \cdot e{{\mathbf F}}\left(\tau\right) \right) d\tau=0 , \label{saddle}$$ plus the Gaussian quantum fluctuation around them. $\nabla_{{\mathbf k}}{\varepsilon}_{\tilde{{\mathbf k}}}={{\mathbf v}}_{\tilde{{\mathbf k}}}$ is the semiclassical velocity of the electron-hole pair. The Berry connection term gives a gauge dependent motion $\nabla _{{\mathbf k}} \left({\mathscr{A}}_{\tilde {{\mathbf k}}} \cdot e{{\mathbf F}}\right)= \nabla_{{\mathbf k}} \left({\mathscr{A}}_{\tilde {{\mathbf k}}} \cdot e{{\mathbf F}}\right) - \left(e{{\mathbf F}}\cdot \nabla_{{\mathbf k}}\right) {\mathscr{A}}_{\tilde {{\mathbf k}}} - \frac{d}{{d\tau }}{\mathscr{A}}_{\tilde {{\mathbf k}}}$. Because we are considering the cyclic evolution along a closed loop, the last gauge dependent term vanishes and the first two terms gives a gauge-invariant physical quantity $\Omega _{\tilde k_i \tilde k_j } eF_j {{\mathbf e}}_i$, with $\Omega _{\tilde k_i \tilde k_j }
=\partial_{k_i}{\mathscr{A}}_{\tilde {k_j}}-\partial_{k_j}{\mathscr{A}}_{\tilde {k_i}}$ being the Berry curvature. We note that this is the well-known anomalous velocity that is responsible for various Hall effects [@Chang_Niu1995; @Sundaram_Niu; @QNiu2]. The stationary phase condition in Eq. (\[saddle\]) therefore means the return of the electron to the hole after $nT$ under the acceleration by the THz field. The Berry phase $\phi_B^{\left(n\right)}\left({\mathbf k}\right)$ in the actions is generally a slowly-varying function of ${\mathbf k}$ and much smaller than the dynamical phase factor. Therefore the stationary phase points are determined by $\int_{t_0}^{t_n} {{{\mathbf v}}_{\tilde {{\mathbf k}}\left( \tau \right)} d\tau}=0$, which has a simple solution ${{\mathbf v}}_{{\mathbf k}}=0$ if the effective mass model is used. This means that the response is dominated by the trajectories of the electron-hole pairs whose paths in the ${\mathbf k}$-space are centered at the extreme points of the energy band (see Fig. \[Schem\]). Thus Eq. (\[RespN\]) is approximated by $$\begin{aligned}
& \notag \left\langle {{\mathbf P}\left( t_n \right)} \right\rangle \\
\approx & \cos\left(\phi_B^{\left(n\right)}\right)_{{{\mathbf v}}_{{\mathbf k}}=0}\int {\frac{2id{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} } } \Re\left[{{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)}^* {{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)} \cdot { {\mathbf E}}\right]\notag \\
\ - &\sin\left(\phi_B^{\left(n\right)}\right)_{{{\mathbf v}}_{{\mathbf k}}=0}\int {\frac{2id{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} } } \Im\left[{{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)}^* {{\mathbf d}}_{\tilde {{\mathbf k}}\left( t_0 \right)}\cdot { {\mathbf E}}\right] .
\label{RespQT}\end{aligned}$$
![(color online). Schematics of optical response to a short NIR pulse and quantum trajectories in semiconductors under strong THz fields. (a) An electron-hole pair is excited by an NIR pulse and then driven along quantum trajectories by an elliptically polarized THz field. At $t_n-t_0=nT$, the electron-hole pair completes an elliptical loop in ${\mathbf k}$-space (see (b)) and obtains a nonzero Berry phase. The polarization of optical emission at $t_n$ is rotated by an angle $n\phi$ that is equal to the Berry phase $\phi^{\left(n\right)}_B$. (b) Paths of the electron-hole pairs in ${\mathbf k}$-space, where the green solid curve is for the electron-hole pair satisfying the stationary phase condition in Eq. (\[saddle\]). The red dashed one does not satisfy Eq. (\[saddle\]). (c) The green solid curve gives the quantum trajectory of an electron-hole pair that recombines at $t_1$. The red dashed curve corresponds to the red dashed path in (b), in which the electron-hole pair does not make a close path in real space.[]{data-label="Schem"}](fig1.pdf){width="\columnwidth"}
To be specific, from now on we consider the monolayer $\rm{MoS}_2$ as the model system. This material has interesting spin-valley coupling and has potential applications for novel spin- and valley-based information processing [@MOS2_Mak1; @MOS2; @MOS2_Zeng; @MOS2_Mak; @MOS2_Cao]. The effective Hamiltonian describing the Bloch states at the band edges is given by [@MOS2] $$H\left( {\mathbf k} \right) = A\left( {\tau k_x \sigma _x + k_y \sigma _y } \right) + M\sigma _z,$$ where $2M=1.59 \ {\rm{eV}}$ is the band gap, $A=3.51 \ {\rm{eV}} \cdot \AA$ and $\tau=\pm 1$ is the index of the $\pm K$ valley. The energy spectrum is $\varepsilon_{{\mathbf k}}=2\sqrt{M^2+A^2 k^2}$ and the stationary phase point is ${\mathbf k}=0$. The two valleys are related by time-reversal transformation and we denote the state at the $-K$ valley as the pseudospin state $\Uparrow$. We choose the gauge such that the Berry connection ${\mathscr{A}}_{{\mathbf k}} = \frac{\varepsilon_{{\mathbf k}}-2M}{\varepsilon_{{\mathbf k}}k^2}\left( {k_x {{\mathbf e}}_y - k_y {{\mathbf e}}_x } \right)$ and the dipole moment ${{\mathbf d}}_{{\mathbf k}} \approx d_{cv,{\mathbf k}}\left( {{\mathbf e}}_x +i{{\mathbf e}}_y \right)$ for small ${\mathbf k}$ in the $-K$ valley. For the sake of simplicity we have not included the Coulomb interaction between electrons and holes. This is justified for the band edge excitation since the exciton binding energy (100s of meV [@note]) is much greater than the THz field and therefore the exciton bound states are far off-resonant from the NIR excitation.
We assume that the NIR field is linearly polarized in the $x$-$y$ plane with ${\mathbf E}=E{\mathbf e}_{\parallel}$. The dipole moment gives $${{\mathbf d}}_{{\mathbf k}}^* {{\mathbf d}}_{{\mathbf k}} = \left| {d_{cv,{\mathbf k}} } \right|^2 \left[ \left( {{\mathbf e}}_x {{\mathbf e}}_x + {{\mathbf e}}_y {{\mathbf e}}_y \right) + i\left( {{\mathbf e}}_x {{\mathbf e}}_y - {{\mathbf e}}_y {{\mathbf e}}_x \right) \right]. \label{Spe_dipole}$$ The real part of (\[Spe\_dipole\]) leads to the longitudinal response along ${\mathbf e}_{\parallel}$ while the imaginary part leads to the transverse response along ${\mathbf e}_{\bot}$ (which is related to the Faraday rotation of the emission) with ${{\mathbf e}}_\parallel \times {{\mathbf e}}_ \bot = {{\mathbf e}}_z$. Thus the longitudinal response of Eq. (\[RespQT\]) is $$\left\langle {{\mathbf P}\left( t_n \right)} \right\rangle_{\parallel}= \cos\left(\phi_B^{\left(n\right)}\right)_{{\mathbf k}=0} \int {\frac{2id{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} } } \left|d_{cv,\tilde {{\mathbf k}}\left( t_0 \right)}\right|^2 E ,$$ and the transverse response is $$\left\langle {{\mathbf P}\left( t_n \right)} \right\rangle_{\bot}= \sin\left(\phi_B^{\left(n\right)}\right)_{{\mathbf k}=0} \int {\frac{2id{{\mathbf k}}}{{\left( {2\pi } \right)^d }}} e^{ { - i\int_{t_0}^{t_n} {\varepsilon _{\tilde {{\mathbf k}}\left( \tau \right)} d\tau} } } \left|d_{cv,\tilde {{\mathbf k}}\left( t_0 \right)}\right|^2 E .$$ The Faraday rotation angle of the optical emission is exactly given by the Berry phase $${\phi}_{r}^{\left(n\right)}\left(\theta,k_0\right) = \phi_B^{\left(n\right)}\left({\mathbf k}=0\right) \approx \frac{{n\pi A^2 k_0^2 }}{{2M^2 }}\sin \left( {2\theta } \right), \label{Rotangle}$$ where $A^2/M^2$ is the Berry curvature for small ${\mathbf k}$ [@MOS2].
![(color online). Faraday rotation of the optical emission of monolayer $\rm{MoS}_2$ under a strong THz field. (a) Schematic for opposite Berry phases of quantum trajectories in different valleys. (b) and (c) plot Faraday rotation angle ${\phi}_{r}^{\left(n\right)}$ of the optical emission at $t=nT$ ($n=1,2,3$) as a function of the polarization ellipticity ($\theta$) and the strength ($F$) of the THz field, respectively. The symbols show the numerical integration (NI) results and the lines are the stationary phase approximation (SPA) in Eq. (\[Rotangle\]). In (b), the THz field strength $F=8\ {\rm{kV/cm}}$ (i.e., $k_0=0.02$). In (c), the THz field is circularly polarized (i.e., $\theta=\pi/4$).[]{data-label="Phir_fig"}](fig2.pdf){width="\columnwidth"}
The Faraday rotation effect of an elliptically polarized THz field can be intuitively understood as illustrated in Fig. \[Phir\_fig\](a). The electron-hole pair created by a linearly polarized short NIR pulse excitation is a superposition of the valley states $\left| \Uparrow \right\rangle + \left| \Downarrow \right\rangle$. After the cyclic evolution under the THz field, the states at $\mp K$ valleys (corresponding to $\tau=\pm 1$) obtain the same dynamical phase $\phi_D$ and opposite Berry phases $\pm\phi_B$, which are the Berry curvature fluxes through the area enclosed by the quantum trajectories. Thus the final state is $e^{i\phi _D } \left( {e^{i\phi _B} \left| \Uparrow \right\rangle + e^{ - i\phi _B } \left| \Downarrow \right\rangle } \right)$, which results in emission with linear polarization rotated by an angle $\phi_B$.
Equation (\[Rotangle\]) shows clearly the effect of the Berry phase on the optical response of the semiconductor in an intense THz field and provides a new method to directly measure the Berry phase of the energy bands in momentum space. Since the Faraday rotation angle is independent of the strength of the optical response, the measurement does not rely on the specific form of the energy spectrum, the value of $d_{cv,{\mathbf k}}$, or the dephasing of the electron-hole pair during the evolution. However, the result does depend on the optical selection rule of the dipole moment ${{\mathbf d}}_{{\mathbf k}}\sim \left( {{\mathbf e}}_x +i{{\mathbf e}}_y \right)$, which is due to the rotational symmetry of the system. Thus Eq. (\[Rotangle\]) may be applied to other two-dimensional spin-orbit coupled semiconductors with (approximate) rotational symmetry. For a semiconductor having a different form of the dipole moment, the Faraday rotation angle of the optical emission may have a more complex relation to the Berry phase.
To verify the validity of Eq. (\[Rotangle\]), we compare it with the numerical results obtained from the standard numerical integration of Eq. (\[AbelQT\]) for a $\rm{MoS}_2$ monolayer. The frequency of the THz field is $\omega=4 \ \text{meV}$ and the NIR pulse has the gaussian form $E{{\mathbf e}}_x \exp \left( { - i\Omega t - t^2 /\delta t ^2 } \right)$, where $\Omega=2M$ and the width of the pulse is such that $\omega \delta t = 0.2$ ($\ll 2\pi$). Some results calculated for different $n$, $\theta$ and field strength $F$ (i.e. $k_0$) are shown in Fig. \[Phir\_fig\], where the lines are the stationary phase approximation and the symbols give the numerical integration results. We can see that Eq. (\[Rotangle\]) is a good approximation.
In the discussions above, we only consider the emissions at $t=t_0 + nT$. However, in the case of $t-t_0 \ne nT$, the trajectories of the electron-hole pairs also obtain a geometric phase. For the monolayer $\rm{MoS}_2$, the dipole moment is nearly constant for small ${\mathbf k}$ in the gauge we chose above. Then based on the same reasoning, we see that the Faraday rotation angle of the emission at any time $t$ is given by the geometric phase of the trajectory that satisfies the stationary phase point equation $\int_{t_0}^{t} {{{\mathbf v}}_{\tilde {{\mathbf k}}\left( \tau \right)} d\tau}=0$.
In summary, we have obtained a Berry phase dependent theory of optical response in spin-orbit-coupled semiconductors under strong THz fields, where the Berry phase enters the formula as required by the gauge invariance of the optical response. This theory is investigated using the quantum trajectory theory and applied to the monolayer $\rm{MoS}_2$. The Faraday rotation angle of the optical emission is exactly equal to the Berry phase of the quantum trajectory that satisfies the stationary phase condition. This result can be generalized to semiconductors without time-reversal symmetry. Even more interesting, the theory can be applied to semiconductors with non-Abelian Berry connection such as the three-dimensional topological insulators [@SCZhang; @Kane]. The quantum trajectory will then be accompanied by a nontrivial (pseudo)spin rotation that is determined by the non-Abelian Berry phase.
This work is supported by Hong Kong RGC/GRF 401512 and the CUHK Focused Investments Scheme.
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---
abstract: 'We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a given neighbourhood. Our proof follows from a graph transform type method and is performed in the state space of the system. We do not require the map to be invertible. From our method follows also the smoothness of the established manifolds, which depends on the smoothness of the map, as well as rate conditions, which follow from bounds on the derivative of the map. Our method is tailor made for rigorous, interval arithmetic based, computer assisted validation of the needed assumptions.'
address:
- 'AGH University of Science and Technology, al. Mickiewicza 10, 30-059 Kraków, Poland'
- 'Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland'
author:
- 'Maciej J. Capiński[^1]'
- 'Piotr Zgliczyński[^2]'
title: Geometric proof for normally hyperbolic invariant manifolds
---
Invariant manifolds, normal hyperbolicity 34C45, 34D35, 37D10
[^1]: Research supported by the Polish National Science Center Grant 2012/05/B/ST1/00355
[^2]: Research supported by the Polish National Science Center Grant 2011/03/B/ST1/04780
|
---
abstract: 'Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling parameter inside a wedge with two coaxial cylindrical boundaries. It is assumed that the field obeys Dirichlet boundary condition on bounding surfaces. The application of a variant of the generalized Abel-Plana formula enables to extract from the expectation values the contribution corresponding to the geometry of a wedge with a single shell and to present the interference part in terms of exponentially convergent integrals. The local properties of the vacuum are investigated in various asymptotic regions of the parameters. The vacuum forces acting on the boundaries are presented as the sum of self-action and interaction terms. It is shown that the interaction forces between the separate parts of the boundary are always attractive. The generalization to the case of a scalar field with Neumann boundary condition is discussed.'
author:
- |
A. A. Saharian$^{1,2}$[^1] and A. S. Tarloyan$^{1,3}$\
\
*$^1$Department of Physics, Yerevan State University*\
*0025 Yerevan, Armenia*\
*$^2$The Abdus Salam International Centre for Theoretical Physics*\
*34014 Trieste, Italy*\
*$^3$ Yerevan Physics Institute, Yerevan, Armenia*\
*0036 Yerevan, Armenia*
title: Wightman function and scalar Casimir densities for a wedge with two cylindrical boundaries
---
PACS numbers: 11.10.Kk, 03.70.+k
Introduction
============
The nontrivial properties of the vacuum state are among the most important predictions in quantum field theory. These properties are manifested in the response of the vacuum to external influences such as external fields. A simple model of the influence is realized by imposing prescribed boundary conditions on the field operator. The distortion of the spectrum for the zero-point fluctuations of a quantum field by these conditions results in the shifts in the vacuum expectation values of physical observables, such as the vacuum energy density and stresses, and induces vacuum forces acting on constraining boundaries. This is the well known Casimir effect (see [Most97,Plun86,Bord01,Milt02]{} and references therein). The Casimir effect is common to all systems characterized by fluctuating quantities and has important implications on all scales, from cosmological to subnuclear. In addition to its fundamental interest this effect also plays an important role in the fabrication and operation of nano- and micro-scale mechanical systems and has become an increasingly popular topic in quantum field theory.
An interesting topic in the investigations of the Casimir effect has always been the dependence of the physical characteristics of the vacuum on the geometry of constraining boundaries. Analytic results can usually be found only for highly symmetric geometries including planar, spherically and cylindrically symmetric boundaries. Recently exact results for the Casimir force in geometries of a sphere and a cylinder above a plate are obtained in [@Bulg06; @Emig06] (see also [@Most07]). Aside from their own theoretical and experimental interest, the problems with this type of boundaries are useful for testing the validity of various approximations used to deal with more complicated geometries. In the present paper we consider a less symmetric exactly solvable geometry of boundaries which is a combination of a wedge with coaxial cylindrical shells. The Casimir effect for wedge-shaped regions is well investigated in literature [Most97,jphy,Deutsch,brevikI,brevikII,Nest02]{}. For a conformally coupled scalar and electromagnetic fields the vacuum expectation value of the energy-momentum tensor inside the wedge is azimuthal symmetric. In particular, the vacuum energy-momentum tensor is finite everywhere apart points on the edge. This property is a direct consequence of the conformal invariance in the corresponding problems and does not take place for a non-conformally coupled scalar field. For a scalar field with an arbitrary curvature coupling parameter satisfying Dirichlet boundary condition on the wedge sides the vacuum energy-momentum tensor is evaluated in [Reza02,Saha05cyl]{}. In addition to the azimuthal dependence this tensor, unlike to the case of conformally coupled fields, is also non-diagonal with nonzero azimuthal-radial off-diagonal component.
The investigations of quantum effects for cylindrical boundaries have received a great deal of attention. In addition to traditional problems of quantum electrodynamics under the presence of material boundaries, the Casimir effect for cylindrical geometries is also important in the flux tube models of confinement [@Fish87; @Barb90] and for determining the structure of the vacuum state in interacting field theories [@Ambj83]. The calculation of the vacuum energy for the electromagnetic field with boundary conditions defined on a cylinder turned out to be technically a more involved problem than the analogous one for a sphere. First the Casimir energy of an infinite perfectly conducting cylindrical shell has been calculated in Ref. [@Dera81] by introducing ultraviolet cutoff and later the corresponding result was derived by using other methods [Milt99,Gosd98,Lamb99]{}. The local characteristics of the corresponding electromagnetic vacuum such as energy density and vacuum stresses are considered in [@Sah1cyl] for the interior and exterior regions of a conducting cylindrical shell, and in [@Sah2cyl] for the region between two coaxial shells (see also [@Saha00rev]). The electromagnetic vacuum forces acting on the boundaries in the geometry of two cylinders are also considered in Refs. [@Mazz02]. In Ref. [@Rome01] scalar vacuum densities and the zero-point energy for general Robin boundary condition on a cylindrical surface in arbitrary number of spacetime dimensions are studied for massive scalar field with general curvature coupling parameter. The corresponding problem for the geometry of two coaxial cylindrical shells is considered in [@Saha06cyl]. A large number of papers is devoted to the investigation of the various aspects of the Casimir effect for a dielectric cylinder (see, for instance, [@Milt02; @Nest04] and references therein).
In the geometry of a wedge with coaxial cylindrical boundary the modes are still factorizable for both scalar and electromagnetic fields and the corresponding problems are exactly solvable. The total Casimir energy of a semi-circular infinite cylindrical shell with perfectly conducting walls is considered in [@Nest01] by using the zeta function technique. For a scalar field with an arbitrary curvature coupling parameter obeying Dirichlet boundary condition the Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor in the geometry of a wedge with an arbitrary opening angle and with a cylindrical boundary are investigated in [@Reza02; @Saha05cyl]. The corresponding Casimir densities for the electromagnetic field with perfect conductor boundary conditions on bounding surfaces are considered in [@Saha07El]. The closely related problem with a cylindrical shell in the geometry of a cosmic string is discussed in [@Beze06Sc; @Beze07El] for scalar and electromagnetic fields. In both scalar and electromagnetic cases the application of a variant of the generalized Abel-Plana formula [@Saha00rev] enables to extract from the vacuum expectation values the parts corresponding to the geometry of a wedge without the cylindrical shell and to present the shell induced parts in terms of rapidly converging integrals. This geometry is also interesting from the point of view of general analysis for surface divergences in the expectation values of local physical observables for boundaries with discontinuities. The nonsmoothness of the boundary generates additional contributions to the heat kernel coefficients (see, for instance, the discussion in [@Nest04; @Apps98; @Dowk00; @Nest03] and references therein).
In this paper we investigate one-loop vacuum quantum effects for a scalar field in the geometry of a wedge with two coaxial cylindrical shells assuming Dirichlet boundary condition on bounding surfaces. This geometry generalizes various special cases previously considered in literature for wedge-shaped and cylindrical boundaries. In addition, we also study the role of nonzero mass of the field quanta. The presence of boundaries eliminates the translational invariance and as a result the properties of the vacuum are nonuniform. The most important quantities characterizing the local properties of the vacuum are the expectation values of the field square and the energy-momentum tensor. In addition to describing the physical structure of the quantum field at a given point, the energy-momentum tensor acts as the source of gravity in the Einstein equations. It therefore plays an important role in modelling a self-consistent dynamics involving the gravitational field. As the first step for the investigation of vacuum densities we evaluate the positive frequency Wightman function. This function gives comprehensive insight into vacuum fluctuations and determines the response of a particle detector of the Unruh-DeWitt type. Having the vacuum energy-momentum tensor we can derive the vacuum forces acting on constraining boundaries evaluating the vacuum stresses at points on the bounding surfaces. As we will see below, in the geometry under consideration these forces are position dependent on the boundary and cannot be obtained by the global method using the total Casimir energy (on the advantages of the local method see also [@Acto96]). In the limiting case from the results of the present paper the local vacuum densities are obtained for the geometry of a rectangular waveguide (for the local analysis of quantum fields confined in rectangular cavities see [@Acto96; @Acto94; @Acto95]).
The paper is organized as follows. The next section is devoted to the evaluation of the Wightman function for a massive scalar field in the region bounded by two cylindrical shells and by the wedge walls. This function is decomposed into three parts: the first one corresponds to the geometry of a wedge without cylindrical shells, the second one is induced by a single cylindrical shell when the second shell is absent, and the third one is induced by the presence of the second shell. By using the formula for the Wightman function, in section \[sec:phi2EMT\] the vacuum expectation values of the field square and the energy-momentum tensor are evaluated and their behavior is investigated in various asymptotic regions of the parameters. In section \[sec:forces\] we consider the vacuum forces acting on bounding surfaces. For separate boundary elements these forces are decomposed into self-action and interaction parts. The interaction forces are investigated in detail and numerical examples are presented. On the example of interaction forces we also demonstrate the limiting transition to the geometry of a rectangular waveguide. Finally, the results are summarized and discussed in section \[sec:Conclusion\].
Wightman function {#sec:WF}
=================
We consider a real scalar field $\varphi $ inside a wedge with opening angle $\phi _{0}$ and with two coaxial cylindrical shells of radii $a$ and $b$, $%
a<b$ (see figure \[fig1\]). For the field with curvature coupling parameter $\xi $ the corresponding field equation has the form $$\left( \nabla ^{i}\nabla _{i}+\xi R+m^{2}\right) \varphi \left( x\right) =0,
\label{fieldeq}$$where $R$ is the curvature scalar for a $(D+1)$-dimensional background spacetime, $\nabla _{i}$ is the covariant derivative operator. For special cases of minimally and conformally coupled scalars one has $\xi =0$ and $\xi
=\xi _{D}\equiv (D-1)/4D$, respectively. Here we will assume that the background spacetime is flat and, hence, in Eq. (\[fieldeq\]) we have $R=0$. As a result the eigenmodes are independent of the curvature coupling parameter. However, the local characteristics of the vacuum such as the energy density and vacuum stresses depend on this parameter. In accordance with the problem symmetry we will use cylindrical coordinates $\left( r,\phi
,z_{1},...,z_{N}\right) $, $N=D-2$, and will assume that the field obeys Dirichlet boundary conditions on bounding surfaces: $$\varphi |_{r=j}=\varphi |_{\phi =0}=\varphi |_{\phi =\phi _{0}}=0,\;j=a,b.
\label{boundcondD}$$These boundary conditions modify the spectrum of the zero-point fluctuations compared with the case of free space and change the physical properties of the vacuum. Among the most important characteristics of the vacuum are the expectation values of the field square and the energy-momentum tensor. These expectation values can be obtained from two-point functions in the coincidence limit. As a two-point function here we will consider the positive frequency Wightman function $\left\langle 0|\varphi (x)\varphi
(x^{\prime })|0\right\rangle $, where $|0\rangle $ is the amplitude for the vacuum state. This function also determines the response of Unruh-DeWitt type particle detectors [@Birr82]. Here we consider the spatial region $%
0\leqslant \phi \leqslant \phi _{0}$, $a\leqslant r\leqslant b$. The formulae for the regions $r\leqslant a$ and $r\geqslant b$ are obtained in limiting cases. Note that by using the corresponding formulae we can discuss various combinations of boundaries in the regions $0\leqslant \phi \leqslant
\phi _{0}$ and $\phi _{0}\leqslant \phi \leqslant 2\pi $. For example, we can consider the situation with two cylindrical shells in the first region and without shells in the second one.
By expanding the field operator and using the standard commutation relations, the positive frequency Wightman function is presented as a sum over the eigenmodes:$$\left\langle 0|\varphi (x)\varphi (x^{\prime })|0\right\rangle =\sum_{\alpha
}\varphi _{\alpha }(x)\varphi _{\alpha }^{\ast }(x), \label{W1}$$where $\left\{ \varphi _{\alpha }(x),\varphi _{\alpha }^{\ast }(x)\right\} $ is a complete orthonormal set of positive and negative frequency solutions to the field equation satisfying boundary conditions ([boundcondD]{}). In the region between the cylindrical shells, $a\leqslant
r\leqslant b$, the eigenfunctions are specified by the set of quantum numbers $\alpha =(n,\gamma ,\mathbf{k})$,$~\ n=1,2,\cdots $, and have the form$$\varphi _{\alpha }(x)=\beta _{\alpha }g_{qn}(\gamma a,\gamma r)\sin (qn\phi
)\exp \left( i\mathbf{kr}_{\parallel }-i\omega t\right) , \label{eigfunc}$$where$$\omega =\sqrt{\gamma ^{2}+k_{m}^{2}},\;k_{m}^{2}=|\mathbf{k}|^{2}+m^{2},\
q=\pi /\phi _{0},$$and $\mathbf{r}_{\parallel }=\left( z_{1},...,z_{N}\right) $, $\mathbf{k}%
=(k_{1},\ldots ,k_{N})$, $-\infty <k_{j}<\infty $. In formula (\[eigfunc\]) we have introduced the notation $$g_{qn}(\gamma a,\gamma r)=Y_{qn}(\gamma a)J_{qn}(\gamma r)-J_{qn}(\gamma
a)Y_{qn}(\gamma r), \label{gn}$$with $J_{qn}(z)$ and $Y_{qn}(z)$ being the Bessel and Neumann functions. The eigenfunctions $\varphi _{\alpha }(x)$ defined by (\[eigfunc\]) satisfy the boundary conditions on the inner shell and on the wedge sides. The eigenvalues for the quantum number $\gamma $ are quantized by boundary condition (\[boundcondD\]) on the surface $r=b$ and are solutions of the equation$$J_{qn}(\gamma a)Y_{qn}(\gamma b)-Y_{qn}(\gamma a)J_{qn}(\gamma b)=0.
\label{modeeq}$$In the discussion below the corresponding positive roots we will denote by $%
\gamma a=\sigma _{qn,l}$, $l=1,2,\ldots $, assuming that they are arranged in the ascending order, $\sigma _{qn,l}<\sigma _{qn,l+1}$.
The normalization coefficient $\beta _{\alpha }$ in (\[eigfunc\]) is found from the standard orthonormality condition for the eigenfunctions: $$\int d^{N}\mathbf{r}_{\parallel }\int_{a}^{b}dr\,r\int_{0}^{\phi _{0}}d\phi
\,\varphi _{\alpha }(x)\varphi _{\alpha ^{\prime }}^{\ast }(x)=\frac{1}{%
2\omega }\delta _{nn^{\prime }}\delta _{ll^{\prime }}\delta (\mathbf{k-k}%
^{\prime }). \label{normcond}$$By making use of the standard integral for cylinder functions (see, for instance, [@Prud86]), one finds$$\beta _{\alpha }^{2}=\frac{\pi ^{2}q\gamma T_{qn}^{ab}(\gamma a)}{(2\pi
)^{D-1}\omega a}, \label{betalf}$$with the notation$$T_{\nu }^{ab}(z)=\frac{z}{J_{\nu }^{2}(z)/J_{\nu }^{2}(\eta z)-1},\;\eta
=b/a. \label{Tnab}$$The substitution of eigenfunctions (\[eigfunc\]) into mode-sum formula (\[W1\]) leads to the following expression for the positive frequency Wightman function$$\begin{aligned}
\left\langle 0|\varphi (x)\varphi (x^{\prime })|0\right\rangle &=&\frac{\pi
^{2}q}{a}\int d^{N}\mathbf{k}\sum_{n=1}^{\infty }\sum_{l=1}^{\infty }\frac{%
zg_{qn}(z,zr/a)g_{qn}(z,zr^{\prime }/a)}{(2\pi )^{D-1}\sqrt{z+k_{m}^{2}a^{2}}%
} \notag \\
&&\times \sin (qn\phi )\sin (qn\phi ^{\prime })\exp (i\mathbf{k}\Delta
\mathbf{r}_{\parallel }-i\omega \Delta t)T_{qn}^{ab}(z)\big|_{z=\sigma
_{n,l}}, \label{W2}\end{aligned}$$where $\Delta \mathbf{r}_{\parallel }=\mathbf{r}_{\parallel }-\mathbf{r}%
_{\parallel }^{\prime }$ and $\Delta t=t-t^{\prime }$. As the expressions for the eigenmodes $\sigma _{n,l}$ are not explicitly known, formula ([W2]{}) for the Wightman function is not convenient. In addition, the separate terms in the sum are highly oscillatory for large values of quantum numbers. For the further evaluation of the summation over $l$ we apply formula [Saha00rev]{} $$\begin{aligned}
\frac{\pi ^{2}}{2}\sum_{l=1}^{\infty }h(\sigma _{qn,l})T_{qn}^{ab}(\sigma
_{n,l}) &=&\int_{0}^{\infty }\frac{h(x)dx}{J_{qn}^{2}(x)+Y_{qn}^{2}(x)}
\notag \\
&&-\frac{\pi }{4}\int_{0}^{\infty }dx\,\Omega _{a,qn}(x,\eta x)\left[
h(xe^{\pi i/2})+h(xe^{-\pi i/2})\right] , \label{Abel}\end{aligned}$$which is a direct consequence of the generalized Abel-Plana formula (for applications of the generalized Abel-Plana formula in investigations of the vacuum densities in the Casimir effect see also [@Saha07rev]). In ([Abel]{})$$\Omega _{a,qn}(x,y)=\frac{K_{qn}(y)/K_{qn}(x)}{%
K_{qn}(x)I_{qn}(y)-K_{qn}(y)I_{qn}(x)}, \label{Oma}$$and $I_{qn}(x)$, $K_{qn}(x)$ are the modified Bessel functions.
As a function $h(x)$ in summation formula (\[Abel\]) we choose$$h(x)=\frac{xg_{qn}(x,xr/a)g_{qn}(x,xr^{\prime }/a)}{\sqrt{%
x^{2}+k_{m}^{2}a^{2}}}\exp (-i\Delta t\sqrt{x^{2}/a^{2}+k_{m}^{2}}).
\label{hx}$$The corresponding conditions for this formula to be valid are satisfied if $%
r+r^{\prime }+|\Delta t|<2b$. In particular, this is the case in the coincidence limit $t=t^{\prime }$ for the region under consideration. As a result, the Wightman function is presented in the form$$\begin{aligned}
\left\langle 0|\varphi (x)\varphi (x^{\prime })|0\right\rangle &=&\frac{q}{%
2^{D-2}\pi ^{D-1}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\int d^{N}\mathbf{k}\,e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel }} \notag
\\
&&\times \Bigg\{\int_{0}^{\infty }dx\frac{h(x)/a}{J_{qn}^{2}(x)+Y_{qn}^{2}(x)%
}-\frac{2}{\pi }\int_{k_{m}}^{\infty }dx\frac{x\Omega _{a,qn}(ax,bx)}{\sqrt{%
x^{2}-k_{m}^{2}}} \notag \\
&&\times G_{qn}(ax,rx)G_{qn}(ax,r^{\prime }x)\cosh (\Delta t\sqrt{%
x^{2}-k_{m}^{2}})\Bigg\}, \label{W3}\end{aligned}$$where $h(x)$ is defined by (\[hx\]) and we have introduced the notation $$G_{qn}(x,y)=K_{qn}(x)I_{qn}(y)-I_{qn}(x)K_{qn}(y). \label{Gnj}$$In the limit $b\rightarrow \infty $ the second term in figure braces on the right of (\[W3\]) vanishes, whereas the first term does not depend on $b$. It follows from here that the part with the first term presents the Wightman function for the geometry of a wedge with a single cylindrical shell of radius $a$. The corresponding problem for a massless scalar field is investigated in [@Saha05cyl]. For points $r<b$ the second term in figure braces on the right of (\[W3\]) is finite in the coincidence limit and, hence, the renormalization procedure for the VEVs of the field square and the energy-momentum tensor is reduced to the corresponding procedure for the geometry with a single shell. In addition, in the coincidence limit of the arguments the $x$-integral in (\[W3\]) is exponentially convergent in the upper limit.
In formula (\[W3\]), the part corresponding to the geometry with a single cylindrical shell with radius $a$ can be further transformed by using the identity$$\begin{aligned}
\frac{g_{qn}(x,xr/a)g_{qn}(x,xr^{\prime }/a)}{J_{qn}^{2}(x)+Y_{qn}^{2}(x)}
&=&J_{qn}(xr/a)J_{qn}(xr^{\prime }/a)-\frac{1}{2}\sum\limits_{\sigma =1}^{2}%
\frac{J_{qn}(x)}{H_{qn}^{(\sigma )}(x)} \notag \\
&&\times H_{qn}^{(\sigma )}(xr/a)H_{qn}^{(\sigma )}(xr^{\prime }/a),
\label{iden1}\end{aligned}$$where $H_{qn}^{(\sigma )}(x)$, $\sigma =1,2$, are the Hankel functions. In the corresponding integral over $x$ with the second term on the right of (\[iden1\]) we rotate the integration contour by the angle $\pi /2$ for $%
\sigma =1$ and by the angle $-\pi /2$ for $\sigma =2$. Due to the well known properties of the Hankel functions, under the condition $r+r^{\prime
}-|\Delta t|>2a$, the integrals over the arcs of the circle with large radius vanish, whereas the integrals over $(0,iak_{m})$ and $(0,-iak_{m})$ cancel out. Introducing the Bessel modified functions one obtains$$\begin{aligned}
\int_{0}^{\infty }dz\frac{h(x)/a}{J_{qn}^{2}(x)+Y_{qn}^{2}(x)}
&=&\int_{0}^{\infty }dx\,x\frac{J_{qn}(xr)J_{qn}(xr^{\prime })}{\sqrt{%
x^{2}+k_{m}^{2}}}\exp (-i\Delta t\sqrt{x^{2}+k_{m}^{2}}) \notag \\
&&-\frac{2}{\pi }\int_{k_{m}}^{\infty }dx\frac{xI_{qn}(ax)}{K_{qn}(ax)}\frac{%
K_{qn}(xr)K_{qn}(xr^{\prime })}{\sqrt{x^{2}-k_{m}^{2}}}\cosh (\Delta t\sqrt{%
x^{2}-k_{m}^{2}}). \label{int1}\end{aligned}$$By taking into account this relation, the Wightman function is presented in the form$$\begin{aligned}
\left\langle 0|\varphi (x)\varphi (x^{\prime })|0\right\rangle
&=&\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle
_{0}+\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{a}-\frac{q%
}{2^{D-3}\pi ^{D}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\, \notag \\
&&\times \int d^{N}\mathbf{k}e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel
}}\int_{k_{m}}^{\infty }dx\,x\frac{\Omega _{a,qn}(ax,bx)}{\sqrt{%
x^{2}-k_{m}^{2}}}G_{qn}(ax,rx)G_{qn}(ax,r^{\prime }x) \notag \\
&&\times \cosh (\Delta t\sqrt{x^{2}-k_{m}^{2}}). \label{W4}\end{aligned}$$In this formula,$$\begin{aligned}
\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{0} &=&\frac{q}{%
2^{D-2}\pi ^{D-1}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\int d^{N}\mathbf{k}\,e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel }} \notag
\\
&&\times \int_{0}^{\infty }dx\,x\frac{J_{qn}(xr)J_{qn}(xr^{\prime })}{\sqrt{%
x^{2}+k_{m}^{2}}}\exp (-i\Delta t\sqrt{x^{2}+k_{m}^{2}}), \label{W0}\end{aligned}$$is the Wightman function for the wedge without cylindrical boundaries, and $$\begin{aligned}
\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{a} &=&-\frac{q}{%
2^{D-3}\pi ^{D}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\int d^{N}\mathbf{k}\,e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel }} \notag
\\
&&\times \int_{k_{m}}^{\infty }dx\,x\frac{I_{qn}(ax)}{K_{qn}(ax)}\frac{%
K_{qn}(xr)K_{qn}(xr^{\prime })}{\sqrt{x^{2}-k_{m}^{2}}}\cosh (\Delta t\sqrt{%
x^{2}-k_{m}^{2}}), \label{Wa}\end{aligned}$$is the part of the Wightman function induced by a single cylindrical shell with radius $a$ in the region $r>a$. Hence, the last term on the right of (\[W4\]) is induced by the presence of the second shell with radius $b$.
An equivalent form for the Wightman function is obtained from (\[W4\]) by using the identity$$\begin{aligned}
&&\sum_{j=a,b}n_{j}\Omega _{j,qn}(ax,bx)G_{qn}(jx,xr)G_{qn}(jx,xr^{\prime })
\notag \\
&=&\frac{K_{qn}(bx)}{I_{qn}(bx)}I_{qn}(xr)I_{qn}(xr^{\prime })-\frac{%
I_{qn}(ax)}{K_{qn}(ax)}K_{qn}(xr)K_{qn}(xr^{\prime }), \label{iden2}\end{aligned}$$with the notations $n_{a}=1$, $n_{b}=-1$, and$$\Omega _{b,qn}(x,y)=\frac{I_{qn}(x)/I_{qn}(y)}{%
K_{qn}(x)I_{qn}(y)-K_{qn}(y)I_{qn}(x)}. \label{Omb}$$This leads to the following representation for the Wightman function$$\begin{aligned}
\left\langle 0|\varphi (x)\varphi (x^{\prime })|0\right\rangle
&=&\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle
_{0}+\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{b}-\frac{q%
}{2^{D-3}\pi ^{D}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\, \notag \\
&&\times \int d^{N}\mathbf{k}e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel
}}\int_{k_{m}}^{\infty }dx\,x\frac{\Omega _{b,qn}(ax,bx)}{\sqrt{%
x^{2}-k_{m}^{2}}}G_{qn}(bx,xr)G_{qn}(bx,xr^{\prime }) \notag \\
&&\times \cosh (\Delta t\sqrt{x^{2}-k_{m}^{2}}). \label{W5}\end{aligned}$$In this formula,$$\begin{aligned}
\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{b} &=&-\frac{q}{%
2^{D-3}\pi ^{D}}\sum_{n=1}^{\infty }\sin (qn\phi )\sin (qn\phi ^{\prime
})\int d^{N}\mathbf{k}\,e^{i\mathbf{k}\Delta \mathbf{r}_{\parallel }} \notag
\\
&&\times \int_{k_{m}}^{\infty }dx\,x\frac{K_{qn}(bx)}{I_{qn}(bx)}\frac{%
I_{qn}(xr)I_{qn}(xr^{\prime })}{\sqrt{x^{2}-k_{m}^{2}}}\cosh (\Delta t\sqrt{%
x^{2}-k_{m}^{2}}) \label{Wb}\end{aligned}$$is the part induced by a single cylindrical shell of radius $b$ in the region $r<b$ and the last term on the right is induced by the presence of the second shell. Note that formulae (\[Wa\]) and (\[Wb\]) are related by the interchange $a\rightleftarrows b$, $I_{n}\rightleftarrows K_{n}$. For a massless scalar field these formulae are derived in [@Saha05cyl].
VEVs of the field square and the energy-momentum tensor {#sec:phi2EMT}
========================================================
Field square
------------
In this section we consider the VEVs for the field square and the energy-momentum tensor in the region between the cylindrical shells. The VEV of the field square is obtained from the Wightman function in the coincidence limit of the arguments. In this limit and for points away from the boundaries the divergences are contained in the term $\left\langle
\varphi (x)\varphi (x^{\prime })\right\rangle _{0}$ only. The corresponding renormalization procedure is realized by subtracting the part for the Minkowskian spacetime without boundaries. By using decompositions (\[W4\]) and (\[W5\]) for the Wightman function and taking the coincidence limit of the arguments, for the renormalized VEV of the field square one finds$$\langle \varphi ^{2}\rangle _{\mathrm{ren}}=\langle \varphi ^{2}\rangle _{0,%
\mathrm{ren}}+\langle \varphi ^{2}\rangle _{j}+\langle \varphi ^{2}\rangle
_{jj^{\prime }}, \label{VEVphi2}$$where $j^{\prime }=a$ ($b$) for $j=b$ ($a$) and the last term on the right is given by the formula $$\begin{aligned}
\langle \varphi ^{2}\rangle _{jj^{\prime }} &=&-2qA_{D}\sum_{n=1}^{\infty
}\sin ^{2}(qn\phi ) \notag \\
&&\times \int_{m}^{\infty }dx\,x\left( x^{2}-m^{2}\right) ^{\frac{D-3}{2}%
}\Omega _{j,qn}(ax,bx)G_{qn}^{2}(jx,rx), \label{phi21}\end{aligned}$$with the notation$$A_{D}=\frac{2^{2-D}}{\pi ^{(D+1)/2}\Gamma ((D-1)/2)}. \label{AD}$$To obtain this result we have used the formula$$\int_{0}^{\infty }dk\int_{k_{m}}^{\infty }dx\,\frac{k^{s}f(x)}{\sqrt{%
x^{2}-k_{m}^{2}}}=\frac{\pi ^{N/2}}{\Gamma \left( N/2\right) }B\left( \frac{%
N+s}{2},\frac{1}{2}\right) \int_{m}^{\infty }dx\,\left( x^{2}-m^{2}\right) ^{%
\frac{D-3}{2}}f(x), \label{intform1}$$where $B(x,y)$ is the Euler beta function. In formula (\[phi21\]), the term $\langle \varphi ^{2}\rangle _{0,\mathrm{ren}}$ is the renormalized VEV for the geometry of a wedge without cylindrical shells and the term $\langle
\varphi ^{2}\rangle _{j}$ is induced by a single cylindrical shell of radius $j$ when the second shell is absent. Hence, the last term is induced by the second shell of radius $j^{\prime }$.
The formulae for single shell terms are directly obtained from (\[Wa\]) and (\[Wb\]) in the coincidence limit. By making use of formula ([intform1]{}), in the case $j=a$ one finds $$\langle \varphi ^{2}\rangle _{a}=-2qA_{D}\sum_{n=1}^{\infty }\sin
^{2}(qn\phi )\int_{m}^{\infty }dx\,x\left( x^{2}-m^{2}\right) ^{\frac{D-3}{2}%
}\frac{I_{qn}(ax)}{K_{qn}(ax)}K_{qn}^{2}(rx), \label{phi2a}$$and the formula for $\langle \varphi ^{2}\rangle _{b}$ is obtained from here by the replacements $a\rightarrow b$, $I\rightleftarrows K$. Note that, as $%
\Omega _{j,qn}(x,y)>0$ for $x<y$, the both terms $\langle \varphi
^{2}\rangle _{j}$ and $\langle \varphi ^{2}\rangle _{jj^{\prime }}$ are negative. For points away from the cylindrical shells the last two terms on the right of formula (\[phi21\]) are finite. Note that both single shell and the second shell induced parts vanish on the wedge sides $\phi =0,\phi
_{0}$, $a<r<b$. The part $\langle \varphi ^{2}\rangle _{j}$ diverges on the cylindrical surface $r=j$ with the leading term$$\langle \varphi ^{2}\rangle _{a}\approx -\frac{\Gamma \left( (D-1)/2\right)
}{(4\pi )^{(D+1)/2}|r-j|^{D-1}}. \label{Phi2neara2}$$for points with $|r/j-1|\ll |\sin \phi |,|\sin (\phi _{0}-\phi )|$. For points near the edges $(r=j,\phi =0,\phi _{0})$ the leading terms in the corresponding asymptotic expansions are the same as for the geometry of a wedge with the opening angle $\phi _{0}=\pi /2$. The surface divergences in the VEVs of local physical observables are well known in quantum field theory with boundaries and are investigated for various types of bulk and boundary geometries (see, for example, [Deutsch,Birr82,Full89,Cand82,Kennedy,Baac86]{}).
The term $\langle \varphi ^{2}\rangle _{jj^{\prime }}$ in (\[VEVphi2\]) vanishes on the shell $r=j$ and diverges on the shell $r=j^{\prime }$. The corresponding surface divergences are the same as those for a single cylindrical shell of radius $j^{\prime }$. It follows from here that if we present the VEV of the field square in the form$$\langle \varphi ^{2}\rangle _{\mathrm{ren}}=\langle \varphi ^{2}\rangle _{0,%
\mathrm{ren}}+\sum_{j=a,b}\langle \varphi ^{2}\rangle _{j}+\Delta \langle
\varphi ^{2}\rangle , \label{interphi2}$$then the interference term $\Delta \langle \varphi ^{2}\rangle $ is finite everywhere. Let us consider the behavior of the interference part in asymptotic regions of the parameters. In the limit $a\rightarrow 0$ for fixed values $r$ and $b$, this term vanishes as $a^{2q}$. In the limit $%
b\rightarrow \infty $ and for a massless field the interference part tends to zero like $1/b^{D+2q-1}$. In the same limit under the condition $mb\gg 1$ the interference part is suppressed by the factor $e^{-2mb}/b^{(D-1)/2}$. For small values of the wedge opening angle one has $q\gg 1$ and, hence, the order of the modified Bessel functions in the formulae for the VEVs is large. By using the corresponding uniform asymptotic expansions (see, for example, [@Abra64]) we can see that the main contribution comes from the term with $n=1$ and from the lower limit of the $x$-integral. To the leading order for the interference term we find$$\Delta \langle \varphi ^{2}\rangle \approx \frac{4q^{(D-1)/2}(a/b)^{2q}\sin
^{2}(q\phi )}{(2\pi )^{(D+1)/2}(b^{2}-a^{2})^{(D-1)/2}}. \label{Deltaphi2q}$$As we see, in this limit the interference part is exponentially suppressed. For points not too close to the cylindrical shells, similar suppression takes place for single shell induced parts.
Vacuum energy-momentum tensor
-----------------------------
The VEV for the energy-momentum tensor is obtained by using the formula$$\langle 0|T_{ik}|0\rangle =\lim_{x^{\prime }\rightarrow x}\partial
_{i}\partial _{k}^{\prime }\langle 0|\varphi (x)\varphi (x^{\prime
})|0\rangle +\left[ \left( \xi -\frac{1}{4}\right) g_{ik}\nabla _{l}\nabla
^{l}-\xi \nabla _{i}\nabla _{k}\right] \langle 0|\varphi ^{2}|0\rangle .
\label{emtvev1}$$Note that in this formula we have used the form of the metric energy-momentum tensor which differs from the standard one by the term which vanishes for the solutions of the field equation (see, for instance, [Saha04EMT]{}). As in the case of the field square, for points away from the boundaries the renormalization is realized by subtracting the part corresponding to the Minkowski spacetime without boundaries. By using the formulae for the Wightman function and for the VEV of the field square, for the renormalized VEV we obtain$$\langle T_{i}^{k}\rangle _{\mathrm{ren}}=\langle T_{i}^{k}\rangle _{0,%
\mathrm{ren}}+\langle T_{i}^{k}\rangle _{j}+\langle T_{i}^{k}\rangle
_{jj^{\prime }}, \label{VEVemt}$$where $j^{\prime }=a$ ($b$) for $j=b$ ($a$) and the non-zero components of the last term on the right are given by the formulae (no summation over $i$)$$\begin{aligned}
\langle T_{i}^{i}\rangle _{jj^{\prime }} &=&\frac{1}{2}qA_{D}\sum_{n=1}^{%
\infty }\int_{m}^{\infty }dx\,x^{3}\left( x^{2}-m^{2}\right) ^{\frac{D-3}{2}%
}\Omega _{j,qn}(ax,bx) \notag \\
&&\times \left\{
a_{i,qn}^{(+)}[G_{qn}(jx,rx)]-a_{i,qn}^{(-)}[G_{qn}(jx,rx)]\cos (2qn\phi
)\right\} , \label{vevemt_ii} \\
\langle T_{1}^{2}\rangle _{jj^{\prime }} &=&q^{2}A_{D}\sum_{n=1}^{\infty
}n\sin (2qn\phi )\int_{m}^{\infty }dx\,x^{2}(x^{2}-m^{2})^{\frac{D-3}{2}}
\notag \\
&&\times \Omega _{j,qn}(ax,bx)G_{qn}(jx,rx)\left[ \frac{2\xi }{rx}%
G_{qn}(jx,rx)+(1-4\xi )G_{qn}^{\prime }(jx,rx)\right] , \label{vevemt_12}\end{aligned}$$with $G_{\nu }^{\prime }(x,y)=\partial _{y}G_{\nu }(x,y)$. In formula ([vevemt\_ii]{}) we have introduced notations$$\begin{aligned}
a_{i,l}^{(\pm )}[g(y)] &=&(4\xi -1)\left[ g^{\prime 2}(y)+\left( 1\pm \frac{%
l^{2}}{y^{2}}\right) g^{2}(y)\right] +2g^{2}(y)\frac{1-m^{2}r^{2}/y^{2}}{D-1}%
, \label{ajpm} \\
a_{1,l}^{(\pm )}[g(y)] &=&g^{\prime 2}(y)+\frac{4\xi }{y}g(y)g^{\prime
}(y)-g^{2}(y)\left\{ 1\pm \left[ 1-4\xi (1\mp 1)\right] \frac{l^{2}}{y^{2}}%
\right\} , \label{ajpm1} \\
a_{2,l}^{(\pm )}[g(y)] &=&\left( 4\xi -1\right) \left[ g^{\prime
2}(y)+g^{2}(y)\right] -\frac{4\xi }{y}g(y)g^{\prime }(y)+\frac{l^{2}}{y^{2}}%
g^{2}(y)\left( 4\xi \pm 1\right) , \label{ajpm2}\end{aligned}$$with $g(y)=G_{qn}(jx,y)$ and in (\[ajpm\]) $i=0,3,\ldots ,D$. In particular, for the vacuum energy density and stresses along directions parallel to the cylinder axis we have the relations $\langle
T_{0}^{0}\rangle _{\mathrm{ren}}=\langle T_{3}^{3}\rangle _{\mathrm{ren}%
}=\ldots =\langle T_{D}^{D}\rangle _{\mathrm{ren}}$. This property is a direct consequence of translation invariance of the problem along these directions. In (\[VEVemt\]) the term $\langle T_{i}^{k}\rangle _{j}$ is induced by a single cylindrical surface with radius $j$ when the second shell is absent and the term $\langle T_{i}^{k}\rangle _{jj^{\prime }}$ is induced by the presence of the second shell. Note that the off-diagonal component $\langle T_{1}^{2}\rangle _{jj^{\prime }}$ vanishes on the wedge sides and on the cylindrical shell $r=j$. The formulae for the components $%
\langle T_{i}^{k}\rangle _{a}$ are obtained from (\[vevemt\_ii\]), ([vevemt\_12]{}) by the replacements$$\Omega _{j,qn}(ax,bx)\rightarrow
I_{qn}(ax)/K_{qn}(ax),\;G_{qn}(jx,rx)\rightarrow K_{qn}(rx). \label{VEVemta}$$The formulae for $\langle T_{i}^{k}\rangle _{b}$ are obtained from the corresponding expressions for $\langle T_{i}^{k}\rangle _{a}$ by the replacements $a\rightarrow b$, $I\rightleftarrows K$. Single shell parts in both interior and exterior regions are investigated in [Reza02,Saha05cyl]{} for a massless scalar field. These parts diverge on the shell and for $|r/j-1|\ll |\sin \phi |,|\sin (\phi _{0}-\phi )|$ the leading term in the corresponding asymptotic expansion is given by the formula (no summation over $i$) $$\langle T_{i}^{i}\rangle _{j}\approx \frac{D(\xi -\xi _{D})\Gamma \left(
(D+1)/2\right) }{2^{D}\pi ^{(D+1)/2}|r-j|^{D+1}},\quad i=0,2,\ldots ,D.
\label{T00asra2}$$For the other components to the leading order one has $\langle
T_{1}^{1}\rangle _{j}\sim \langle T_{2}^{1}\rangle _{j}\sim |r-j|^{-D}$.
As in the case of the field square, the VEV of the energy-momentum tensor can be presented in the form$$\langle 0|T_{i}^{k}|0\rangle =\langle T_{i}^{k}\rangle
_{0}+\sum_{j=a,b}\langle T_{i}^{k}\rangle _{j}+\Delta \langle
T_{i}^{k}\rangle , \label{Tikren}$$where the surface divergences are contained in the single shell parts only and the interference part is finite on the shells. The explicit formula for the latter is obtained by subtracting from the last term on the right ([vevemt\_ii]{}) and (\[vevemt\_12\]) the corresponding single shell part. It can be checked that the separate terms in formulae (\[interphi2\]) , ([Tikren]{}) satisfy the standard trace relation$$T_{i}^{i}=D(\xi -\xi _{D})\nabla _{i}\nabla ^{i}\varphi ^{2}+m^{2}\varphi
^{2}, \label{trrel}$$and the continuity equation $\nabla _{i}T_{k}^{i}=0$. For the geometry under consideration the latter takes the form$$\begin{aligned}
\partial _{r}\left( rT_{2}^{1}\right) +r\partial _{\phi }T_{2}^{2} &=&0,
\label{conteq1} \\
\partial _{r}\left( rT_{1}^{1}\right) +r\partial _{\phi }T_{1}^{2}
&=&T_{2}^{2}. \label{conteq2}\end{aligned}$$The behavior of the VEV for the energy-momentum tensor in the asymptotic regions of the parameters is investigated in the way similar to that used for the field square. In the limit $a\rightarrow 0$ the main contribution comes from the term with $n=1$ and the interference part behaves as $a^{2q}$. For large values of the radius of the exterior shell, $b\rightarrow \infty
$, this part vanishes as $e^{-2mb}/b^{(D-1)/2}$ for a massive field and like $1/b^{D+2q-1}$ for a massless one. For large values of the parameter $q$, the interference term in the VEV of the energy-momentum tensor is suppressed by the factor $(a/b)^{2q}$.
In the discussion above we have considered a model where the physical interactions are replaced by the imposition of boundary conditions on the field for all modes. Of course, this is an idealization as real physical interactions cannot constrain all the modes of a fluctuating quantum field [@Deutsch; @Cand82; @Grah02]. In general, the physical quantities in problems with boundary conditions can be classified into two main groups (see also [@Jaff06]). The first group includes quantities which do not contain surface divergences. For these quantities the renormalization procedure is the same as in quantum field theory without boundaries and they can be evaluated by boundary condition calculations. The contribution of the higher modes into the boundary induced effects in these quantities is suppressed by the parameters already present in the idealized model. Examples of such quantities are the vacuum densities away from boundaries and the interaction forces between disjoint bodies. For the quantities from the second group, such as the vacuum densities on the boundary and the total vacuum energy, the contribution of the arbitrary higher modes is dominant and they contain divergences which cannot be eliminated by the standard renormalization procedure of quantum field theory without boundaries. Of course, the model where the physical interaction is replaced by the imposition of boundary conditions on the field for all modes is an idealization. The appearance of divergences in the process of the evaluation of physical quantities of the second type indicates that more realistic physical model should be employed for their evaluation. In literature on the Casimir effect different field-theoretical approaches have been discussed to extract the finite parts from the diverging quantities. However, in the physical interpretation of these results it should be taken into account that these terms are only a part of the full expression of the physical quantity and the terms which are divergent in the idealized model can be physically essential and their evaluation needs a more realistic model. It seems plausible that such effects as surface roughness, or the microstructure of the boundary on small scales can introduce a physical cutoff needed to produce finite values for surface quantities. Another possibility, proposed in Refs. [@Grah02], is to replace a boundary condition by a renormalizable coupling between the fluctuating field and non-dynamical smooth background field representing the material (for the evaluation of the vacuum energy in smooth background fields see also [Bord96]{}). In this model the standard renormalization procedure of quantum field theory without boundaries provides the finite result for the quantities which are divergent in the boundary condition limit. An alternative mechanism for introducing a cutoff which removes singular behavior on boundaries is to allow the position of the boundary to undergo quantum fluctuations [@Ford98]. Such fluctuations smear out the contribution of the high frequency modes without the need to introduce an explicit high frequency cutoff.
The main subject of the present paper is the investigation of the VEVs for the field square and the energy-momentum tensor at points away from the boundaries and the vacuum interaction forces between separate parts of boundaries. In the scheme where a cutoff function is used instead of point-splitting, these quantities are cutoff independent and fall into the first group. They do not contain surface divergences and are completely determined within the framework of standard procedure of quantum field theory without boundaries. We expect that similar results would be obtained in the model where instead of externally imposed boundary condition the fluctuating field is coupled to a smooth background potential that implements the boundary condition in a certain limit [@Grah02].
Vacuum interaction forces {#sec:forces}
=========================
In this section we investigate the vacuum forces acting on the bounding surfaces due to the presence of the second cylindrical shell. First of all let us consider the forces acting on the wedge sides. These forces are determined by the $_{2}^{2}$-component of the energy-momentum tensor evaluated for $\phi =0,\phi _{0}$. Note that the off-diagonal components $%
\langle T_{1}^{2}\rangle _{j}$ and $\langle T_{1}^{2}\rangle _{jj^{\prime }}$ vanish on the wedge sides and, hence do not contribute to the force. The corresponding effective pressure is presented in the form$$p_{2}=p_{2,\mathrm{wedge}}+p_{2,\mathrm{cyl}}, \label{p2wcyl}$$where $p_{2,\mathrm{wedge}}$ is the vacuum effective pressure on the wedge side when the cylindrical shells are absent and the part $p_{2,\mathrm{cyl}}$ is induced by the shells. For a conformally coupled massless scalar in $D=3$ one has$$p_{2,\mathrm{wedge}}=-\frac{q^{4}-1}{480\pi ^{2}r^{4}}. \label{p2wedge}$$The corresponding force is attractive for $\phi _{0}<\pi $ and repulsive for $\phi _{0}>\pi $. The second term on the right of (\[p2wcyl\]) is decomposed as
$$p_{2,\mathrm{cyl}}=p_{2,\mathrm{cyl}}^{(j)}+p_{2,\mathrm{cyl}}^{(jj^{\prime
})}, \label{p2}$$
where $p_{2,\mathrm{cyl}}^{(j)}=-\langle T_{2}^{2}\rangle _{j}|_{\phi =0}$ is the effective azimuthal pressure on the wedges induced by a single cylindrical boundary with radius $j$, $j=a,b$, and $p_{2,\mathrm{cyl}%
}^{(jj^{\prime })}=-\langle T_{2}^{2}\rangle _{jj^{\prime }}|_{\phi =0}$ is induced by the presence of the second cylindrical boundary. Substituting $%
i=2 $ and $\phi =0,\phi _{0}$ in the formulae for the VEVs of the energy-momentum tensor from the previous section, for the forces induced by the shells we find $$\begin{aligned}
p_{2,\mathrm{cyl}}^{(a)} &=&-\frac{q^{3}A_{D}}{r^{2}}\sum_{n=1}^{\infty
}n^{2}\int_{m}^{\infty }dx\,x\left( x^{2}-m^{2}\right) ^{\frac{D-3}{2}}\frac{%
I_{qn}(ax)}{K_{qn}(ax)}K_{qn}^{2}(rx), \label{p2cyla} \\
p_{2,\mathrm{cyl}}^{(jj^{\prime })} &=&-\frac{q^{3}A_{D}}{r^{2}}%
\sum_{n=1}^{\infty }n^{2}\int_{m}^{\infty }dx\,x\left( x^{2}-m^{2}\right) ^{%
\frac{D-3}{2}}\Omega _{j,qn}(ax,bx)G_{qn}^{2}(jx,rx). \label{p2cyljjp}\end{aligned}$$The expression for $p_{2,\mathrm{cyl}}^{(b)}$ is obtained from (\[p2cyla\]) by the replacements $a\rightarrow b$, $I\rightleftarrows K$. Single shell parts in the forces acting on the wedge sides, $p_{2,\mathrm{cyl}}^{(j)}$, are finite for all values $r$ except the points on the edge $r=j$. The second shell-induced part, $p_{2,\mathrm{cyl}}^{(jj^{\prime })}$, is finite for all $r$ except the points on the edge $r=j^{\prime }$, $j^{\prime }=a,b$, $j^{\prime }\neq j$. Note that $p_{2,\mathrm{cyl}}^{(jj^{\prime })}=0$ for $r=j$. The integrands in (\[p2cyla\]) and (\[p2cyljjp\]) are positive and, hence, the corresponding vacuum forces are attractive. As before we can write$$p_{2,\mathrm{cyl}}=\sum_{j=a,b}p_{2,\mathrm{cyl}}^{(j)}+\Delta p_{2,\mathrm{%
cyl}}, \label{p2interf}$$where the interference part $\Delta p_{2,\mathrm{cyl}}$ is finite for all values $a\leqslant r\leqslant b$. As it follows from (\[p2cyla\]), ([p2cyljjp]{}), the corresponding forces do not depend on the curvature coupling parameter.
In the limit $a\rightarrow 0$ the main contribution into $p_{2,\mathrm{cyl}%
}^{(a)}$ and $\Delta p_{2,\mathrm{cyl}}$ comes from the term with $n=1$ and these quantities behave like $a^{2q}$. In the limit $b\rightarrow \infty $ and for a massive scalar field the parts $p_{2,\mathrm{cyl}}^{(b)}$ and $%
\Delta p_{2,\mathrm{cyl}}$ are exponentially suppressed by the factor $%
e^{-2mb}$. In the same limit and for a massless field the main contribution comes from the summand with $n=1$ and these parts behave as $1/b^{D+2q-1}$. Now we consider the forces acting on the wedge sides in the limit of small values of the opening angle when the parameter $q$ is large, $q\gg 1$. In this limit the order of the modified Bessel functions is large and we can use the uniform asymptotic expansions for these functions. By using these expansions, it can be seen that the main contribution comes from the $n=1$ term and from the lower limit of the integral. To the leading order we find$$p_{2,\mathrm{cyl}}^{(j)}\approx -\frac{q^{(D+3)/2}\exp [-2q|\ln (j/r)|]}{%
(2\pi )^{(D+1)/2}r^{2}|r^{2}-j^{2}|^{(D-1)/2}}. \label{p2jcylq}$$In the similar way, for the interference part of the force one has:$$\Delta p_{2,\mathrm{cyl}}\approx \frac{2q^{(D+3)/2}(a/b)^{2q}}{(2\pi
)^{(D+1)/2}r^{2}(b^{2}-a^{2})^{(D-1)/2}}. \label{Deltp2q}$$In figure \[fig2\] we have plotted the quantities $a^{4}p_{2,\mathrm{cyl}%
}^{(j)}$, $j=a,b$, and $a^{4}p_{2,\mathrm{cyl}}$ as functions of $r/a$ for $%
D=3$ massless scalar field. The graphs are given for the wedges with $\phi
_{0}=\pi /2$ (full curves) and $\phi _{0}=3\pi /2$ (dashed curves) and for $%
b/a=1.5$.
Now we turn to the interaction forces acting on the cylindrical boundaries. These forces are determined by the $_{1}^{1}$-component of the energy-momentum tensor evaluated on the corresponding surfaces. Similar to the previous case, the effective pressure on the cylindrical shell $r=j$ is presented as the sum $$p^{(j)}=p_{1}^{(j)}+p^{(jj^{\prime })}, \label{p1j}$$where $p_{1}^{(j)}=-(\langle T_{1}^{1}\rangle _{0}+\langle T_{1}^{1}\rangle
_{j})|_{r=j}$ is the radial vacuum stress on the cylinder with the radius $j$ when the second cylinder is absent and $p^{(jj^{\prime })}=-\langle
T_{1}^{1}\rangle _{jj^{\prime }}|_{r=j}$ is the additional stress on this cylindrical surface when the second cylinder is present. Note that the off-diagonal component $\langle T_{1}^{2}\rangle _{jj^{\prime }}$ vanishes on the shell $r=j$ and does not contribute to the force. The part $%
p_{1}^{(j)}$ includes the self-action force on the cylindrical shell and belongs to the second group of quantities in the classification given in the previous section. Its evaluation requires more realistic model for the interaction of the quantum field. Unlike to the self-action force, the interaction force given by the second term on the right of (\[p1j\]) is finite for all nonzero distances between the shells and can be evaluated by boundary condition calculations. From the last term on the right of ([vevemt\_ii]{}) taking $i=1$ and $r=j$ one finds:$$p^{(jj^{\prime })}=-\frac{qA_{D}}{j^{2}}\sum_{n=1}^{\infty }\sin ^{2}(qn\phi
)\int_{m}^{\infty }dx\,x\left( x^{2}-m^{2}\right) ^{\frac{D-3}{2}}\Omega
_{j,qn}(ax,bx). \label{Deltap1j}$$From this formula we see that $p^{(jj^{\prime })}<0$ and the corresponding forces are always attractive. The expression for the interaction forces between the cylindrical shells can also be written in the form$$p^{(jj^{\prime })}=\frac{qn_{j}A_{D}}{j}\frac{\partial }{\partial j}%
\sum_{n=1}^{\infty }\sin ^{2}(qn\phi )\int_{m}^{\infty }dx\,x\left(
x^{2}-m^{2}\right) ^{\frac{D-3}{2}}\ln \left[ 1-\frac{I_{qn}(ax)K_{qn}(bx)}{%
I_{qn}(bx)K_{qn}(ax)}\right] , \label{pjint1}$$where, as before, $n_{a}=1$, $n_{b}=-1$. As for the forces acting on the wedge sides, the interaction forces do not depend on the curvature coupling parameter.
Now we consider the behavior of the interaction forces in asymptotic regions of the parameters. In the limit $a\rightarrow 0$ the main contribution in the sum of formula (\[Deltap1j\]) comes from the $n=1$ term and $%
j^{2}p^{(jj^{\prime })}\sim a^{2q}$. For large values of the exterior shell radius, $b\rightarrow \infty $, and for a massive field the interaction forces $p^{(jj^{\prime })}$ are suppressed by the factor $e^{-2mb}$. In the same limit and for a massless field one has $j^{2}p^{(jj^{\prime })}\sim
1/b^{D+2q-1}$. For small values of the wedge opening angle, assuming that $%
q\gg 1$, in the way similar to that used for the estimation of the forces acting on the wedge sides, one finds$$j^{2}p^{(jj^{\prime })}\approx -\frac{4q^{(D+3)/2}(a/b)^{2q}\sin ^{2}(q\phi )%
}{(2\pi )^{(D+1)/2}r^{2}(b^{2}-a^{2})^{(D-1)/2}}. \label{pjjq}$$In figure \[fig3\] we have plotted the interaction forces acting on cylindrical shells, $a^{4}p^{(jj^{\prime })}$, as functions of $\phi /\phi
_{0}$ for wedges with $\phi _{0}=\pi /2$ (full curves) and $\phi _{0}=3\pi
/2 $ (dashed curves) and for $b/a=1.5$ in the case of $D=3$ massless scalar field. The curves a are for $p^{(ab)}$ and the curves b are for $p^{(ba)}$.
Note that in the geometry of two coaxial cylindrical shells without a wedge the corresponding interaction forces are given by the formula [Saha06cyl]{} $$p^{(jj^{\prime })}=-\frac{A_{D}}{2j^{2}}\sideset{}{'}{\sum}_{n=0}^{\infty
}\int_{m}^{\infty }du\,u\left( u^{2}-m^{2}\right) ^{\frac{D-3}{2}}\Omega
_{j,n}(au,bu), \label{pjj2cyl}$$where the prime on the sum sign means that the term $n=0$ should be halved. For $D=3$ massless scalar field and for $b/a=1.5$ from this formula we have $%
p^{(ab)}\approx -0.437/a^{4}$ and $p^{(ba)}\approx -0.254/a^{4}$. As it has been shown in [@Saha06cyl], the interaction forces (\[pjj2cyl\]) can also be obtained from the corresponding part in the total Casimir energy differentiating over the radii of cylindrical shells. In the geometry under consideration in the present paper the Casimir forces are position dependent on the boundary and cannot be obtained by global methods using the total Casimir energy.
In the limit $\phi _{0}\rightarrow 0$, $a,b\rightarrow \infty $, assuming that $b-a\equiv L_{1}$ and $a\phi _{0}\equiv L_{2}$ are fixed, from the formulae given above we obtain the corresponding results for the geometry of a rectangular waveguide with sides $L_{1}$ and $L_{2}$. Here we discuss this limiting transition for the case of the interaction forces $p^{(jj^{\prime
})}$. The consideration of the other quantities is done in the similar way. In the limit under consideration the parameter $q$ is large and we can replace the modified Bessel functions by the corresponding uniform asymptotic expansions. By using these expansions it can be seen that to the leading order we have$$\Omega _{j,\nu }(a\nu z,b\nu z)\approx \frac{2\nu \sqrt{1+a^{2}z^{2}}}{%
e^{2\nu \sqrt{1+a^{2}z^{2}}L_{1}/a}-1},\;\nu =qn. \label{Omlim}$$Introducing in (\[pjjq\]) a new integration variable $z=x/qn$ and by making use of (\[Omlim\]), after some transformations, to the leading order we find$$p^{(jj^{\prime })}\approx -\frac{2\pi A_{D}}{L_{1}^{D}L_{2}}%
\sum_{n=1}^{\infty }\sin ^{2}(\pi ny/L_{2})\int_{0}^{\infty }dx\,\frac{%
x^{D-2}\sqrt{x^{2}+c_{n}^{2}}}{e^{2\sqrt{x^{2}+c_{n}^{2}}}-1}%
,\;c_{n}^{2}=m^{2}L_{1}^{2}+(\pi nL_{1}/L_{2})^{2}, \label{pjjlim}$$where $y=a\phi $. The expression on the right of this formula is the vacuum interaction force per unit surface between the facets of the rectangular parallelepiped separated by the distance $L_{1}$ and $y$ is the Cartesian coordinate parallel to these facets. Other facets of the parallelepiped are located at $y=0$ and $y=L_{2}$. Introducing in (\[pjjlim\]) $y=y^{\prime
}+L_{2}/2$ and taking the limit $L_{2}\rightarrow \infty $ with fixed value $%
y^{\prime }$, from (\[pjjlim\]) the vacuum forces for two infinite parallel Dirichlet plates are obtained. Note that the local vacuum densities for a quantum field confined within rectangular cavities are investigated in [@Acto96; @Acto94; @Acto95] (for corresponding global quantities such as the total Casimir energy see [@Most97; @Milt02] and references therein).
Conclusion {#sec:Conclusion}
==========
In this paper we have considered one-loop quantum vacuum effects for a massive scalar field in the geometry of a wedge with two coaxial cylindrical shells. We have assumed that the field satisfies Dirichlet boundary condition on the bounding surfaces. This geometry generalizes various special cases previously discussed in literature, including wedge-shaped regions, cylindrical boundaries, and rectangular waveguides. The most important local characteristics of the quantum vacuum are the VEVs for the field square and the energy-momentum tensor. To evaluate these VEVs, as the first step we construct the positive frequency Wightman function. The corresponding eigensum contains a summation over the zeros of the combination of Bessel and Neumann functions. The application of the generalized Abel-Plana formula to the corresponding sum allows to present the Wightman function in decomposed form given by formulae (\[W4\]) and (\[W5\]). In this representations the first term on the right is the Wightman function for the wedge without cylindrical boundary, the term $%
\left\langle \varphi (x)\varphi (x^{\prime })\right\rangle _{j}$ is induced by a single shell with radius $j$ when the second shell is absent, and the last terms on the right are induced by the presence of the second shell. For points away from the shells the last two terms are finite in the coincidence limit and the renormalization is needed for the first term only. By taking the coincidence limit, we have obtained similar representations for the VEVs of the field square and the energy-momentum tensor, formulae (\[VEVphi2\]) and (\[VEVemt\]). More symmetric decompositions are given by formulae ([interphi2]{}) and (\[Tikren\]), where the last interference term is finite everywhere including points on the shells. In the limit $a\rightarrow 0$ the interference parts tends to zero like $a^{2q}$. For large values of the exterior shell radius, $b\rightarrow \infty $, the interference terms in the VEVs behave as $e^{-2mb}/b^{(D-1)/2}$ for a massive field and as $%
1/b^{D+2q-1}$ for a massless one. For a wedge with small opening angle, $%
q\gg 1$, the main contribution into the interference parts of the VEVs comes from the summands with $n=1$ and these parts are suppressed by the factor $%
(a/b)^{2q}$.
In section \[sec:forces\] we have considered the vacuum forces acting on constraining boundaries. In the geometry under consideration these forces are position dependent on the boundary and cannot be obtained by global methods using the total Casimir energy. The forces acting on the wedge sides are determined by the $_{2}^{2}$-component of the vacuum energy-momentum tensor and are presented in the decomposed form (\[p2wcyl\]). In this representation the first term on the right determines the force when the shells are absent and the second term is induced by the shells. In its turn the latter is decomposed into a single shell and second shell induced parts (see formula (\[p2\])) given by formulae (\[p2cyla\]), (\[p2cyljjp\]). Both these forces are always attractive and do not depend on the curvature coupling parameter. Further we consider the forces acting on the cylindrical shells. These force are presented in the form (\[p1j\]) where the first term on the right is the force acting on the cylindrical shell with radius $j
$ when the second shell is absent and the second term is induced by the presence of the second shell. The latter, given by formula (\[Deltap1j\]), is always attractive and does not depend on the curvature coupling parameter. For large values of the parameter $q$, this part is suppressed by the factor $(a/b)^{2q}$. In the limit $\phi _{0}\rightarrow 0$, $%
a,b\rightarrow \infty $, assuming that $b-a$ and $a\phi _{0}$ are fixed, from the results of the present paper we obtain the corresponding formulae for the VEVs in the geometry of a rectangular waveguide. We have demonstrated this on the example of the interaction force between the cylindrical shells.
Note that we have considered quantities which are well defined within the framework of standard renormalization procedure of quantum field theory without boundaries. We expect that similar results would be obtained from the model discussed in [@Grah02] where instead of externally imposed boundary condition the fluctuating field is coupled to a smooth background potential that reproduces the boundary condition in a limiting case. The generalization of the results in the present paper for a scalar field with Neumann boundary conditions is straightforward. For this case in the expressions (\[eigfunc\]) of the eigenfunctions the function $\cos (qn\phi )$ stands instead of $\sin (qn\phi )$ and the quantum number $n$ takes the values $0,1,2,\ldots $. The corresponding eigenvalues for $\gamma $ are zeros of the function $J_{qn}^{\prime }(\gamma a)Y_{qn}^{\prime
}(\gamma b)-Y_{qn}^{\prime }(\gamma a)J_{qn}^{\prime }(\gamma b)$. The formula for the summation over these zeros is given in [@Saha00rev]. The formulae for the Wightman function and the VEV of the field square in Neumann case are obtained from the corresponding formulae for Dirichlet scalar by the replacements $\sin (qn\phi )\rightarrow \cos (qn\phi )$, $%
I_{qn}(jx)\rightarrow I_{qn}^{\prime }(jx)$, $K_{qn}(jx)\rightarrow
K_{qn}^{\prime }(jx)$, $j=a,b$, and with the term $n=0$ included in the summation. In the expressions for the VEVs of the energy-momentum tensor this leads to the change of the sign for the second term in the figure braces on the right of (\[vevemt\_ii\]) and to the change of the sign for the off-diagonal component (\[vevemt\_12\]).
Acknowledgements {#acknowledgements .unnumbered}
================
AAS would like to acknowledge the hospitality of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The work was supported by the Armenian Ministry of Education and Science Grant No. 0124.
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[^1]: E-mail: [email protected]
|
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abstract: 'Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki’s toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system $E_6$.'
address: 'Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy'
author:
- Bert van Geemen
title: 'A linear system on Naruki’s moduli space of marked cubic surfaces'
---
-0.5truecm -0.5truecm 17truecm 22truecm -.1truecm
\#1\#2\#3\#4\#5\#6\#7
(124, 50) (5, 35) (7, 35)[(1, 0)[28]{}]{} (37, 35) (39, 35)[(1, 0)[28]{}]{} (69, 35) (71, 35)[(1, 0)[28]{}]{} (101, 35) (103, 35)[(1, 0)[28]{}]{} (133, 35) (69, 33)[(0, -11)[28]{}]{} (69, 3) (5, 41)[(0, 0)\[b\][$#1$]{}]{} (37, 41)[(0, 0)\[b\][$#3$]{}]{} (69, 41)[(0, 0)\[b\][$#4$]{}]{} (101, 41)[(0, 0)\[b\][$#5$]{}]{} (133, 41)[(0, 0)\[b\][$#6$]{}]{} (69, 0)[(0, 0)\[t\][$#2$]{}]{} (-60, 35)[(0, 0)\[lb\][$#7$]{}]{}
[**Introduction**]{}
Recently Allcock, Carlson and Toledo [@ACT2] studied the moduli space of smooth cubic surfaces using the intermediate jacobian of the cubic threefold which is the triple cover of projective three space branched along a cubic surface. They show that this moduli space, as well as the moduli space of marked cubic surfaces ${{\cal M}}^0$ (that is, cubic surfaces with an ordered set of six skew lines) are open subsets of certain 4-ball quotients. The Weyl group $W(E_6)$ of the root system $E_6$ acts on ${{\cal M}}^0$ by permuting the markings on any given cubic surface, the quotient variety is the moduli space of cubic surfaces. The quasi projective variety ${{\cal M}}^0$ has a natural compactification ${{\cal M}}$ given by geometrical invariant theory. The projective variety ${{\cal M}}$ coincides with the Baily-Borel compactification of the ball quotient. The action of $W(E_6)$ extends to ${{\cal M}}$.
Using Borcherds’ work on automorphic forms on ball quotients, Allcock and Freitag [@Fr] found a $W(E_6)$-equivariant embedding of ${{\cal M}}$ in a nine dimensional projective space. The action of $W(E_6)$ on the projective space is obtained from the unique ten dimensional irreducible linear representation of $W(E_6)$. This map actually already appears in a paper by A. B. Coble published in 1917 [@Co3] (and see also [@Y]) where ${{\cal M}}$ is identified with the moduli space of six points in the projective plane. The same embedding of ${{\cal M}}$ was also found by Matsumoto and Terasoma [@MT] who used the theta constants associated to the intermediate jacobians.
An explicit smooth projective compactification ${{\cal C}}$ (‘the cross ratio variety’) of the moduli space ${{\cal M}}$ with a biregular action of the Weyl group was constructed by Naruki [@Naruki]. It is a modification of a toric variety associated to the root system $D_4$. Naruki constructs and studies his model as a subvariety of the product of $270$ projective lines, each component of this map is given by a cross ratio (of certain tritangent planes containing a given line on the cubic surface). The Weyl group acts via permutations of these $270$ projective lines.
In this paper we explicitly identify the nine dimensional linear system on Naruki’s model ${{\cal C}}$ which defines the map $F$ to ${{\bf P}}^9$ discovered by Coble, Allcock and Freitag (see Theorem \[main\]) $$F:{{\cal C}}\longrightarrow {{\cal M}}\quad (\subset{{\bf P}}^9).$$ We also give explicit formulas for the $W(E_6)$-action on this linear system in section \[we6V\].
A tritangent plane of a cubic surface is a plane which cuts out three lines on the surface. If these three lines meet in a point, that point is called an Eckart point. We obtain a nice parametrization, equivariant for the Weyl group of the root system $F_4$, of the 45 divisors in ${{\cal M}}$ which parametrize marked cubic surfaces with an Eckart point, see Theorem \[tritpar\]. A study of the linear relations between tritangent planes leads to the discovery that ${{\cal M}}$ is the singular locus of a variety $X$ defined by six quintic polynomials, see \[quints\]. The group $W(E_6)$ acts on $X$ and it would be very interesting to have a moduli interpretation for $X$.
The Weyl group of $E_6$ is defined as a reflection group on a real six dimensional vector space. Complexifying and projectivizing this vector space one obtains a biregular action of $W(E_6)$ on a ${{\bf P}}^5$. In his book [@H], Bruce Hunt suggested an identification of the moduli space with the unique $W(E_6)$-invariant quintic hypersurface $I_5$ in ${{\bf P}}^5$. In section \[6dire\] we construct a dominant rational map $\Sigma:{{\cal M}}\longrightarrow {{\bf P}}^5$ which is equivariant for the action of $W(E_6)$ and we show that its image is $I_5$ (Thm. \[i5\]), but, unfortunately, this map has degree at least 10 (Thm \[thmdeg\]).
The results of this paper are obtained from computations with rational functions on the toric variety, many of them computer assisted. It does lead to very explicit formulas and parametrizations, somewhat in contrast to the ball quotient approach where the modular forms in question are hard to describe explicitly.
I’m indebted to E. Freitag for suggesting to undertake this study and for many discussions. I would also like to thank him and E. Carlini for assistance with the computations.
Cubic surfaces their moduli space
=================================
We briefly recall the basics on cubic surfaces and $E_6$, see [@H] and references given there for proofs. We relate this to the modular orthogonal geometry used by Allcock and Freitag.
The 27 lines.
-------------
Any smooth cubic surface $S$ has 27 lines and there are sets of six disjoint lines $\{a_1,\ldots,a_6\}$. Blowing down the lines $a_i$ to points $p_i$ defines a birational isomorphism $S\rightarrow {{\bf P}}^2$. The images of the other 21 lines on $S$ are the 15 lines $<p_i,p_j>$ and the 6 conics which pass through all six points except one of the $p_i$. The corresponding lines are denoted by $c_{ij}$ and $b_j$. The birational inverse ${{\bf P}}^2\rightarrow S$ is given by the linear system of all cubics passing through the points $p_1$, $\ldots$, $p_6$.
The root system $E_6$.
----------------------
The Picard group of $S$ is isomorphic to ${{\bf Z}}^7$ and a ${{\bf Z}}$-basis is given by the pull-back $l$ of (the divisor class of) a line in ${{\bf P}}^2$ and the classes of the lines $a_i$. The intersection form is determined by $$l^2=1,\quad a_i^2=-1, \quad l\cdot a_i=0,\quad a_i\cdot a_j=0$$ for $i\neq j$. The classes of the lines are $$c_{ij}=l-(a_i+a_j),\qquad b_i=2l-(a_1+\ldots +\hat{a}_i+\ldots +a_6).$$
The canonical class of $S$ is $K_S:=-3l+a_1+\ldots+a_6$ and $K_S^2=3$. The class of a hyperplane section of $S$ is $-K_S$. The primitive cohomology of $S$ is thus the orthogonal complement of $K_S$. This ${{\bf Z}}$-module, with the bilinear form $(x,y):=-x\cdot y$, is isomorphic to the root lattice $Q(E_6)$ of the root system $E_6$: $$Q(E_6)\cong K_S^\perp :=\{x\in Pic(S):\; x\cdot K_S=0\;\}.$$ A ${{\bf Z}}$-basis for $Q(E_6)$ is given by: $$\begin{array}{ccccccccc}\alpha_1=a_2-a_1,&\,& \alpha_3=a_3-a_2,&\,&
\alpha_4=a_4-a_3,&\,&
\alpha_5=a_5-a_4,&\, &\alpha_6=a_6-a_5,\\
&&&&\phantom{x}&&&&\\
&&&&\alpha_2=l-a_4-a_5-a_6.&&&&
\end{array}$$ This is a basis of simple roots of $E_6$: $$\DynkinEEE{\alpha_1}{\alpha_2}{\alpha_3}{\alpha_4}{\alpha_5}{\alpha_6}$$ The set $E_6^+$ of positive roots of $E_6$ consists of the 36 elements in $Q(E_6)$ given by $$h_{ij}:=-a_i+a_j,\quad(i<j)\qquad h_{ijk}:=l-a_p-a_q-a_r,\quad
h:=2l-a_1-\ldots-a_6,$$ where $\{i,j,k,l,p,q,r\}=\{1,2,\ldots,6\}$. In particular, $\alpha_2=h_{123}$ and with this convention our notation is compatible with that of [@H]. The root system $E_6:=E_6^+\cup (-E_6^+)$ $\subset Pic(S)$ contains $72$ vectors, called roots.
The Weyl group $W(E_6)$.
------------------------
The Weyl group $W(E_6)$ is the subgroup of $GL(Q(E_6))$ generated by reflections in the roots. We denote by $s_i$ the reflection in the hyperplane perpendicular to the root $\alpha_i$. More generally we write $s_\alpha$, with $\alpha\in E_6$, for the reflection in the hyperplane perpendicular to $\alpha$.
The orthogonal geometry.
------------------------
Allcock and Freitag use a non-degenerate quadratic form $Q$ on the vector space ${{\bf F}}_3^5$ and its orthogonal group $O(5,3)$ to describe the combinatorics of the lines on a cubic surfaces and of divisors on the moduli space ${{\cal M}}$. The basic facts are ([@Fr], section 2): $$O(5,3)\cong W(E_6)\times \{\pm 1\},$$ there are $72$ vectors with $Q(x)=-1$, these are called the short roots (note $Q(x)=Q(-x)$). There are $90$ vectors with $Q(x)=-2$, the long roots, and there are $80$ nonzero vectors with $Q(x)=0$, called isotropic vectors. (See also [@MT], $\S$3.)
Boundary divisors of ${{{\cal M}}}$ {#bdiv}
-----------------------------------
If a root is the class of an effective divisor on the blow up of ${{\bf P}}^2$, then this effective divisor is a ${{\bf P}}^1$ which is contracted to a node on the cubic surface. This sets up a correspondence between the set of irreducible divisors in ${{\cal M}}$ parametrizing nodal cubic surfaces and $E_6^+$. These divisors are labelled by pairs $\pm x$ of ‘short roots’ in [@Fr].
The divisor in ${{\cal M}}$ corresponding to $\alpha\in E_6^+$ is denoted by $D_\alpha$ (or by $D_{ij}$ if $\alpha=h_{ij}$ etc.). These divisors are the fixed point sets of the corresponding reflections $s_\alpha\in W(E_6)$ in ${{\cal M}}$. The reflection $s_\alpha\in Aut(Pic(S))$ may be identified with the Picard-Lefschetz transformation associated to the general nodal cubic surface $S_0$ in $D_\alpha$.
Lines and weights.
------------------
Let $P(E_6)\subset Q(E_6)\otimes{{\bf Q}}$ be the weight lattice of $E_6$: $$P(E_6):=\{x\in Q(E_6)\otimes_{{\bf Z}}{{\bf Q}}:\; (x,y)\in{{\bf Z}},\;\forall y\in Q(E_6)\,\}.$$ The intersection number of the class $c$ of a line on $S$ with a root is an integer, hence $c$ defines an element $x_c\in P(E_6)$. In this way one obtains a $W(E_6)$-orbit of $27$ weights (which are also denoted by $a_i$, $b_i$, $c_{ij}$ with $1\leq i\leq 6$, $1\leq i<j\leq 6$, cf. [@H], $\S$ 6.1.3). Note that $a_1$ is perpendicular to all simple roots except $\alpha_1$ and that $(a_1,\alpha_1)=-1$, thus $a_1$ is minus a fundamental root of $E_6$.
The tritangent planes and tritangent divisors. {#trite6}
----------------------------------------------
Since hyperplane sections of $S$ correspond to cubics on the $p_i$, it is easy to see that there are 45 planes, the tritangent planes, which intersect $S$ in three lines, in Schl[ä]{}fli’s notation these are denoted by: $$(ij)=\{ a_i,b_j,c_{ij}\},\qquad
(ij.kl.mn)=\{ c_{ij},c_{kl},c_{mn}\},$$ where$\{i,\ldots,n\}=\{1,\ldots,6\}$. Another labelling for the tritangents was given by Cayley and is used by Naruki. The dictionary between the labels is given in [@se2], p.371. The 45 tritangent divisors in ${{\cal M}}$ are written as $D_t$ where $t$ is one of Schläfli’s labels. The tritangent divisors correspond to pairs $\pm x$ of long roots of [@Fr].
Three lines lie in a tritangent plane iff the sum of their classes in $Pic(S)$ is $-K_S$ iff the corresponding weights are linearly dependent. The orthogonal complement in $E_6$ of the span of three such weights is a root system of type $D_4$. If the tritangent is labelled by $t$, we will denote this $D_4$ by $t^\perp$.
The subsystem $D_4$ {#subd4}
-------------------
An important example is the case that $t=(16)={\rm w}$. In that case $t^\perp$ is the $D_4\subset E_6$ spanned by the simple roots $\alpha_2,\;\alpha_3,\;\alpha_4$ and $\alpha_5$. This root system is discussed in section \[toric\].
The $W(F_4)$ and tritangents. {#F4}
-----------------------------
To a tritangent $t$ one associates an element $\gamma({\rm t})\in W(E_6)$ which is the product of the relections in 4 orthogonal roots in $t^\perp\cong D_4$. Thus $\gamma(t)$ is $-I$ on the span of $t^\perp$ and is $+I$ on the orthogonal complement which is the span on the subspace spanned by the weights corresponding to the lines in $t$. For $t=(16)={\rm w}$ one may take $\gamma({\rm w})= s_2s_5s_3(s_4s_5s_3s_4)s_2(s_4s_3s_5s_4)$. The $\gamma(t)$’s are a conjugacy class of 45 elements in $W(E_6)$ which correspond (via their $+1$-eigenspace) with the tritangents. The centralizer of a $\gamma(t)$ in $W(E_6)$ is isomorphic to the Weyl group $W(F_4)$. The fixed point set of a $\gamma(t)$ on ${{\cal C}}$ is the tritangent divisor $D_t$ which parametrises cubic surfaces for which the three lines in $t$ meet in one point, called an Eckart point ([@Naruki] §8).
The toric variety {#toric}
=================
For general facts on toroidal compactifications we refer to [@Fu], for root systems see [@Hu].
The torus.
----------
The $D_4$-adjoint torus $$T\stackrel{\cong}{\longrightarrow} ({{\bf C}}^*)^4,\qquad t\longmapsto
(\lambda(t),\mu(t),\nu(t),\rho(t))$$ comes with a natural identification of its character group ${\rm
Hom}(T,{{\bf C}}^*)\cong{{\bf Z}}^4$ with the sublattice $$M:=\langle \, e_1-e_2,\, e_2-e_3,\,e_3-e_4,\,e_3+e_4\rangle\subset
\oplus_{i=1}^4{{\bf Z}}e_i.$$ The lattice $M$, with the scalar product induced by the standard inner product on $\oplus{{\bf Z}}e_i$, is the root lattice $Q(D_4)$ of $D_4$. We often use: $${\rm Hom}(T,{{\bf C}}^*)\stackrel{\cong}{\longrightarrow} M,\qquad
\lambda\mapsto e_1-e_2,\;\mu\mapsto e_3+e_4,\;\nu\mapsto e_3-e_4,\;
\rho\mapsto e_2-e_3.$$ For $\alpha\in M$ we define a regular function on $T$ by: $$f_\alpha:=\lambda^a\mu^b\nu^c\rho^d\qquad{\rm with}\quad \alpha=a(e_1-e_2)+
b(e_2-e_3)+c(e_3-e_4)+d(e_3+e_4)\in M.$$
The root system.
----------------
The root system $D_4$ consists of the following 24 vectors in $M$: $$D_4=\{\,\pm e_i\pm e_j\;\in M:\quad 1\leq i<j\leq 4\,\}.$$ The set $$\Delta_0:=\{e_1-e_2,\, e_2-e_3,\,e_3-e_4,\,e_3+e_4\}\quad(\subset D_4)$$ is a fundamental system (or base of the root system), that is any root is a linear combination of these 4 vector with all coefficients either positive (such a root is called positive) or negative. Let $N=M^*$ be the dual lattice of $M$, $$N:={\rm Hom}_{{\bf Z}}(M,{{\bf Z}})=\{x\in \left(\oplus{{\bf Z}}e_i\right)^*\otimes_{{\bf Z}}{{\bf R}}:\;
\langle x,\alpha \rangle\in{{\bf Z}}\quad\forall \alpha\in M\,\},$$ here $\langle.,.\rangle$ is the pairing between $\left(\oplus{{\bf Z}}e_i\right)^*\otimes_{{\bf Z}}{{\bf R}}$ and its dual. Let $\{\epsilon_1,\ldots,\epsilon_4\}\subset (\oplus {{\bf Z}}e_i)^*\otimes_{{\bf Z}}{{\bf R}}$ be the dual basis of $\{e_1,\ldots,e_4\}$. Then the basis of $N$ which is dual to $\Delta_0$ is $$\epsilon_1,\quad (\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2,\quad
(\epsilon_1+\epsilon_2+\epsilon_3-\epsilon_4)/2, \quad \epsilon_1+\epsilon_2
\qquad (\in N).$$
The Weyl group. {#defSR}
---------------
The Weyl group $W(D_4)$ of the root system is the subgroup of $GL(M\otimes{{\bf R}})$ generated by the reflections in the roots (so $s_\alpha(\beta)=\beta-(\beta,\alpha)\alpha)$ and $(.,.)$ is the standard inner product on $\oplus{{\bf Z}}e_i$). This group has 192 elements and is a semidirect product of $S_4$ (permuting the $e_i$) and $({{\bf Z}}/2{{\bf Z}})^3$ (changes the sign of an even number of the $e_i$). The Weyl group acts simply transitively on the fundamental systems.
The Weyl group acts on $N$ and the 4 elements of the dual basis above are in distinct orbits of lengths 8, 8, 8 and 24 respectively. We define $$S:=\{\pm \epsilon_i\}\,\cup\,\{(\pm
\epsilon_1\pm\epsilon_2\pm\epsilon_3\pm\epsilon_4)/2\},\qquad
R:=\{\pm\epsilon_i\pm\epsilon_j\},$$ $S$ and $R$ each have 24 elements.
The Weyl chambers.
------------------
The (closed) Weyl chamber $C(\Delta)$ of a fundamental system $\Delta\;(\subset D_4)$ is the (maximal) cone in $N\otimes_{{\bf Z}}{{\bf R}}=\left(\oplus{{\bf Z}}e_i\right)^*\otimes_{{\bf Z}}{{\bf R}}$ defined by: $$C(\Delta):=\{x\in N\otimes_{{\bf Z}}{{\bf R}}\;:\quad \langle x,\alpha\rangle\geq 0\quad
\forall \alpha\in \Delta\}.$$ If $\Delta=\{\alpha_1,\ldots,\alpha_4\}$ then the edges (i.e. the one dimensional faces) of $C(\Delta)$ are the 4 half-lines ${{\bf R}}_{\geq 0}\tau_i$ with $\{\tau_1,\ldots,\tau_4\}$ the dual basis of $\Delta$. The decomposition $$N\otimes_{{\bf Z}}{{\bf R}}=\cup_{\Delta} C(\Delta)$$ is a regular cone decomposition of the vector space $N\otimes_{{\bf Z}}{{\bf R}}$, it defines in a fan in $N$ whose faces are the faces of the 192 Weyl chambers. This fan has 48 edges which correspond to the elements of $S\cup R$.
The toroidal compactification. {#torcom}
------------------------------
Associated to this fan is a toric variety $\tilde{T}$, $$\tilde{T}=\cup_{\Delta} A(\Delta),\qquad A(\Delta)\cong {{\bf C}}^4$$ and the inclusion $T\subset A(\Delta)$ is defined by the inclusion of the rings of regular functions $${{\bf C}}[A(\Delta)]:=
\langle \;f_\alpha\;:\alpha\in M,\quad
\langle x,\alpha\rangle\geq 0\quad\forall x\in C(\Delta)\,\rangle
\;\hookrightarrow\;
{{\bf C}}[T]:={{\bf C}}[\lambda^{\pm 1},\,\mu^{\pm 1},\,\nu^{\pm 1},\,\rho^{\pm 1}].$$ For example ${{\bf C}}[A(\Delta_0)]=
{{\bf C}}[\lambda,\,\mu,\,\nu,\,\rho]$. Each edge ${{\bf R}}_{\geq 0}\tau$, with $\tau\in S\cup R$, defines a divisor $V(\tau)$ in $\tilde{T}$ ([@Fu], §3.3) and these $48$ divisors are the complement of $T$ in $\tilde{T}$: $$\tilde{T}-T=\cup_{\tau\in S\cup R}\, V(\tau).$$
The regular functions $f_\alpha$, $\alpha\in M$, on $T$ extend to rational functions on $\tilde{T}$. The divisor of $f_\alpha$ is given by: $$(f_\alpha)=\sum_{\tau} n_\tau V(\tau)\qquad {\rm with}\quad
n_\tau:=\langle\tau,\alpha\rangle.$$
Example. {#exal}
--------
The divisor of $\lambda=f_{e_1-e_2}$ is given by: $$(\lambda)=D^+_\lambda-D^-_{\lambda}\quad{\rm with}\;
\left\{\begin{array}{lcl}
D^+_\lambda=V(\epsilon_1)+V(-\epsilon_2)+
\sum_{\pm,\pm}V((\epsilon_1-\epsilon_2
\pm\epsilon_3\pm\epsilon_4)/2)+D'\\
D^-_\lambda=V(-\epsilon_1)+V(\epsilon_2)+
\sum_{\pm,\pm}V((-\epsilon_1+\epsilon_2
\pm\epsilon_3\pm\epsilon_4)/2)+D''
\end{array} \right.$$ where $D'$ and $D''$ are combinations of the divisors $V(\tau)$ with $\tau\in
R$ with coefficients in $\{\,-2,\,-1,\,0,\,1,\,2\}$.
The cross ratio variety. {#crv}
------------------------
Naruki’s (smooth, projective) cross ratio variety ${{\cal C}}$ is obtained from the toric variety $\tilde{T}$ as follows ([@Naruki], §10-12): $$\begin{array}{ccccrcccccc}
{{\cal M}}&\longleftarrow&{{\cal C}}&\stackrel{r}{\longleftarrow}&
\hat{T}&\stackrel{\pi''}{\longrightarrow}&
{{\tilde T}}''&\stackrel{\pi'}{\longrightarrow}&
{{\tilde T}}'&\stackrel{\pi_e}{\longrightarrow}&{{\tilde T}}.\\
\end{array}$$
The map $\pi_e$ is the blow up of ${{\tilde T}}$ in the identity element $e\in T$. The exceptional divisor $\pi_e^{-1}(e)\cong {{\bf P}}^3$ is denoted by ${{\bf P}}^3_{\rm w}$. The image in ${{\cal M}}$ of its strict transform in ${{\tilde T}}''$ is the tritangent divisor $D_{\rm w}=D_{(16)}$.
The map $\pi'$ is the blow up of ${{\tilde T}}'$ in the strict transforms in ${{\tilde T}}'$ of the 12 curves in the $W(D_4)$-orbit of the curve in ${{\tilde T}}$ defined by $\lambda=\nu=\rho=1$. The morphism $r$ contracts the strict transforms in $\hat{T}$ of the 12 exceptional divisors in ${{\tilde T}}''$ to surfaces in ${{\cal C}}$ and is an isomorphism on the complement ([@Naruki], Prop. 11.3).
The map $\pi''$ is the blow up in the strict transform in ${{\tilde T}}''$ of the 16 surfaces in the $W(D_4)$-orbit of $\mu=\rho=1$. The 16 exceptional divisors in $\hat{T}$ map under $r$ to divisors in ${{\cal C}}$, their $W(E_6)$-orbit consists of $40$ divisors, the other 24 are the images under $r$ of the strict transforms of the $V(\tau)$’s with $\tau\in R$ ([@Naruki], Prop. 11.2). We call these 40 divisors the cusp divisors of $\hat{T}$.
There is a morphism ${{\cal C}}\rightarrow {{\cal M}}$, where ${{\cal M}}$ is the moduli space of semistable marked cubic surfaces, which contracts the $40$ cusp divisors to points (cf. [@Naruki], Introduction and §12), the cusps of ${{\cal M}}$. The Weyl group $W(E_6)$ acts biregularly on ${{\cal C}}$ and ${{\cal M}}$ and the morphism ${{\cal C}}\longrightarrow {{\cal M}}$ is $W(E_6)$-equivariant.
The $W(E_6)$-action on boundary divisors. {#e6bdiv}
=========================================
According to Naruki [@Naruki], Prop. 11.3’, the boundary ${{\cal C}}-{{\cal M}}^0$ consists of two $W(E_6)$-orbits of divisors, one orbit is formed by the 36 boundary divisors $D_\alpha$ with $\alpha\in E_6^+$. The other orbit consists of the 40 cusp divisors and will not be of interest for us. In Naruki’s toroidal construction, the 36 $D_\alpha$’s are parametrized by the 12 positive roots $D_4^+$ of $D_4$ and by the 24 elements of a set of $S$ (see \[defSR\]) of weights of $D_4$. In this section we determine the corresponding $W(D_4)$-equivariant bijection between $E_6/\{\pm 1\}$ and $(D_4/\{\pm 1\})\cup S$, see table \[tabwe6b\] for the final result.
{#blT}
To do the required computations, it is sufficient to work on the blow up of ${{\tilde T}}$ in the origin, rather then on ${{\cal C}}$ or ${{\cal M}}$, cf. \[crv\]. For each positive root $\alpha\in D_4$ the closure in $\tilde{T}$ of the subtorus defined by $f_\alpha=1$ in $T$ is an irreducible divisor. Since it contains $e$, its pull-back to $\tilde{T}'$ has two irreducible components, one is ${{\bf P}}^3_{\rm w}$ and the other is its strict transform which we will denote by $D^1_\alpha$. The image in ${{\cal M}}$ of the strict transform of $D^1_\alpha$ in $\hat{T}$ is $D_\alpha$, so these twelve divisors are labelled via $D_4={\rm w}^\perp\subset E_6$.
The other $24$ boundary divisors in ${{\cal M}}$ are the images in ${{\cal M}}$ of the strict transforms of the $V(\beta)$ with $\beta\in S$ ([@Naruki], Prop. 11.1). The Weyl group $W(D_4)$ has three orbits on $S$ and it suffices to identify one divisor from each orbit. That is done in the following lemma. The resulting labelling of all 36 divisors is given in table \[tabwe6b\].
Lemma. {#lemact}
------
Let $s_1$, $s_6$ be the reflections in $W(E_6)$ defined by the roots $\alpha_1=h_{12}$, $\alpha_6=h_{56}$ respectively. Then we have: $$s_1^*D^1_\lambda=V(-\epsilon_2)$$ hence $V(-\epsilon_2)=D_{13}$. Similarly we have: $$\begin{array}{lcl}
s_6^*D^1_{\lambda\nu\rho}&=&V((\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2),
\\
s_1^*V((\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2)&=&
V((\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2)
\end{array}$$ and thus $V((\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2)=D_{26}$, $V((\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2)=D_{16}$.
[[**Proof.**]{}$\;\;$]{}The divisor of the rational function $\lambda-1$ on $\tilde{T}'$ is $$(\lambda-1)=D^1_\lambda+{{\bf P}}^3_{\rm w}-D^-_\lambda,$$ where $D_\lambda^-$ is as in \[exal\]. Therefore $s_1^*(\lambda-1)$ will have exactly two effective components, one being $s_1^*D^1_\lambda$ which must be in the orbit of length $36$ and the other will be a tritangent divisor. From [@Naruki], p. 13 we have: $$s_1:\;\lambda\longmapsto
\frac{\lambda\mu\nu\rho^2(1-\lambda)}{\lambda\mu\nu\rho^2-1}$$ and hence that $$s_1:\;\lambda-1\longmapsto
f_1:=\frac{1-\lambda^2\mu\nu\rho^2}{\lambda\mu\nu\rho^2-1}.$$ Since $\lambda^2\mu\nu\rho^2=f_{2e_1}$ (note that $\lambda^2\mu\nu\rho^2$ is not a root) and $\lambda\mu\nu\rho^2=f_{e_1+e_2}$ we see that the denominator has a pole of order one on $V(-\epsilon_2)$ but the numerator has vanishing order zero on that divisor, hence $V(-\epsilon_2)$ must be one of the two effective components of $(f_1)$. The other effective component is defined by $1-\lambda^2\mu\nu\rho^2=0$, which is the local equation of the tritangent divisor $D_{\bar{\rm x}}$ ($\bar{\rm x}=(26)$, cf. Table 3 of [@Naruki]). Note that $\lambda=f_{e_1-e_2}$ and $e_1-e_2=h_{23}$, so $D^1_\lambda=D_{23}$ and that $s_1$ permutes the indices $1$ and $2$ of an $h_{ij}$, hence $s_1^*D_{23}=D_{13}$ and $s_1^*D_{(16)}=D_{(26)}$.
Using the formulas from [@Naruki], p. 13 again we get: $$s_6:\;1-\lambda\nu\rho\longmapsto
f_2:=\frac{1-\lambda\mu\nu^2\rho^2}{1-\mu\nu\rho}.$$ Since $\lambda\mu\nu^2\rho^2=f_{e_1+e_2+e_3-e_4}$ and $\mu\nu\rho=f_{e_2+e_3}$, we see that $V((\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2)$ is one of the two effective components of $(f_2)$. The other component corresponds to the tritangent divisor $D_{\bar{{\rm z}}}=D_{(15)}$ defined by $1-\lambda\mu\nu^2\rho^2=0$. Note that $\lambda\nu\rho=f_{e_1-e_4}$ and $e_1-e_4=h_{25}$, so $D^1_{\lambda\nu\rho}=D_{25}$ and that $s_6$ permutes the indices $5$ and $6$ of an $h_{ij}$, hence $s_6^*D_{25}=D_{26}$ and $s_6^*D_{(16)}=D_{(15)}$.
Next we apply $s_1$ to $f_2$ and obtain: $$s_6:f_2\longmapsto f_3:= \frac{-\mu\rho(\lambda + \nu - \lambda\nu -
\lambda\nu\rho - \lambda\mu\nu\rho + \lambda^2\mu\nu^2\rho^2)}
{(\lambda\mu\rho-1)(\lambda\nu\rho-1)}$$ In the open subset $U=A(\Delta_0)=Spec({{\bf C}}[\lambda,\mu,\nu,\rho])$, this function is zero on $\mu=0$, which is $V((\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2)\cap U$ (the zero locus in $U$ of the $i$-th element in $\{\lambda,\mu,\nu,\rho\}$ is the divisor corresponding to the $i$-th vector of the dual basis). Thus we found one of the two effective components of the divisor of $f_3$. Note that $(\rho=0)\cap U$ lies in $V(\epsilon_1+\epsilon_2)$, which is not in the orbit of the $36$ divisors and that the third factor of the numerator of $f_3$ defines the tritangent divisor labelled by $\bar{q}_1=(25)$. Since $s_1$ permutes the indices $1$ and $2$ of an $h_{ij}$, we get $s_1^*D_{26}=D_{16}$ and $s_1^*D_{(15)}=D_{(25)}$.
{#divld}
The labelling of these 36 divisors on $\tilde{T}'$ allows us to express various divisors in a convenient manner. For example (cf. \[exal\]): $$(\lambda)=D_{13}+D_{26}+D_{136}+D_{246}+D_{256}+D_{345}-
D_{12}-D_{36}-D_{126}-D_{346}-D_{356}-D_{245}$$ and similarly: $$(\lambda-1)=D_{\rm w}+D_{23}-D_{12}-D_{36}-D_{126}-D_{346}-D_{356}-D_{245}.$$
The CAF-linear system.
======================
{#section-3}
To identify the linear system on the moduli space ${{\cal M}}$ introduced by Coble, Allcock and Freitag and to describe the $W(E_6)$-action on it, we consider two divisors with support in the boundary of the toric variety $\tilde{T}$.
Definition. {#defs}
-----------
Let $R,\;S\subset N$ be as in section \[defSR\]. We define divisors in $\tilde{T}$ (cf. \[torcom\]) by: $$D_S:=\sum_{\tau\in S} V(\tau),\qquad D_R:=\sum_{\tau\in R} V(\tau).$$
Lemma. {#lemma.}
------
We have $$H^0(\tilde{T},{{\cal O}}(D_S+2D_R))=\langle\,f_0=1\,\rangle\oplus \langle\,
f_\alpha\,:
\,\alpha\in D_4\,\rangle,$$ in particular, $\dim H^0(\tilde{T},{{\cal O}}(D_S+2D_R))=25$. The divisor $D_S+2D_R$ is very ample on $\tilde{T}$.
[[**Proof.**]{}$\;\;$]{} The space of global sections of the line bundle associated to a divisor $\sum n_\tau V(\tau)$ is spanned by certain $f_\alpha$’s: $$H^0(\tilde{T},{{\cal O}}(\sum_\tau n_\tau V(\tau)))\,=\,\langle\,f_\alpha\,:\,
\alpha \in M\;{\rm and}\; \langle \tau , \alpha \rangle \geq -n_\tau\, \rangle
.$$ Thus we must find the $\alpha\in M$ with $\langle\tau,\alpha\rangle\geq -1$ for $\tau\in S$ and $\langle\tau,\alpha\rangle\geq -2$ for $\tau\in R$. Let $\alpha=\sum m_ie_i$ with $m_i\in{{\bf Z}}$. Taking $\tau=\pm\epsilon_i\in S$ we get $-1\leq m_i\leq 1$, taking $\tau=(\pm\epsilon_1\pm\ldots\pm\epsilon_4)/2$ $\in S$ we get $-2\leq \pm m_1\pm m_2\pm m_3\pm m_4\leq 2$, hence at most two of the $m_i$ are non zero and thus $\alpha=0,\,\pm e_i$ or $\pm e_i\pm e_j$ with $i\neq j$. However $\pm e_i\not\in M$ and therefore $\alpha$ is either zero or a root. All these $\alpha$ also satisfy $\langle\tau,\alpha\rangle\geq -2$ for $\tau\in
R$.
The proof of the very ampleness is standard, cf. [@Fu], and since we do not really need it, we omit the proof.
Divisors near the identity. {#dive}
---------------------------
The functions $x_r:=r-1$ with $r\in\{\lambda,\,\mu,\,\nu,\,\rho\}$ are local coordinates near the identity element $e=(1,1,1,1)\in T$. Any rational function $f$ on $T$ which is regular in $e$ can be developed in a Taylor series: $$f=f_d+f_{d+1}+\ldots,\qquad{\rm with}\quad f_k\in
{{\bf C}}[x_\lambda,\,x_\mu,\,x_\nu,\,x_\rho]$$ with $f_k$ homogeneous of degree $k$ and $d\geq 0$. If the polynomial $f_d$ is not identically zero we say that $f$ vanishes to order $d$ in $e\in T$ and we write $m_e(f)=d$, $f_d$ is called the leading term of $f$.
For $\alpha=a(e_1-e_2)+\ldots+d(e_2-e_3)\in M-\{0\}$ we have: $$f_\alpha-1=
(x_\lambda+1)^a(x_\mu+1)^b(x_\nu+1)^c(x_\rho+1)^d-1
=ax_\lambda+bx_\mu+cx_\nu+dx_\rho+\, H.O.T.$$ hence $f_\alpha-1$ vanishes to order 1 at $e$ and a product $\prod_{i=1}^m
(f_{\alpha_i}-1)$ of such functions vanishes to order $m$ at $e$.
Definition. {#defV}
-----------
We define the vector space $V$ of rational functions on $\tilde{T}$ to be the subspace of those global sections of ${{\cal O}}(D_S+2D_R)$ which vanish to order at least 3 at $e\in \tilde{T}$: $$V:=\{\,f\in\,H^0(\tilde{T},{{\cal O}}(D_S+2D_R)):\; m_e(f)\geq 3\,\}.$$
Lemma. {#lemV}
------
The dimension of $V$ is 10. A basis for $V$, multiplied by $\lambda\mu\nu\rho^2$, is given in table \[second\]. [[**Proof.**]{}$\;\;$]{}Note that 10 is the expected dimension of $V$ since the spaces of constant, linear and quadratic polynomials in 4 variables have dimension $1,\,4,\,10$ respectively. Thus we only have to show that each monomial $x_\lambda^ax_\mu^bx_\nu^cx_\rho^d$ with $a+b+c+d\leq 2$ is the leading term of a function in $H^0(\tilde{T},{{\cal O}}(D_S+2D_R))$. Obviously we can use $f_0=1$ to get leading term $1$ and the $r-1$ to get leading term $x_r$. For the roots $\alpha=\lambda\rho,\,\mu\rho,\,\nu\rho$ the leading term is a linear combination of the leading terms of the $r-1$’s which we already have. Subtracting these linear terms we get functions with the leading terms $x_\lambda x_\rho,\,x_\mu x_\rho,\,x_\nu x_\rho$. The Taylor series of $t-1$ with $t=\lambda\nu\rho,\,\mu\nu\rho,\,
\lambda\mu\rho$, give us, modulo the leading terms we already found, the leading terms $x_\lambda x_\nu,\,x_\mu x_\nu,\, x_\lambda x_\nu$. To get the $x_r^2$ use that $$r^{-1}=1-x_r+x_r^2-\ldots.$$ Thus we found all the 15 desired leading terms and we conclude that $V$ has codimension 15 in $H^0(\tilde{T},{{\cal O}}(D_S+2D_R))$.
Example. {#f12x34x56}
--------
The following function lies in $V$: $$\nu^{-1}\rho^{-1}(\rho-1)(\lambda\nu\rho-1)(\mu\nu\rho-1)\\
=
\lambda\mu\nu\rho^2-\lambda\mu\nu\rho-\lambda\rho-\mu\rho
+\lambda+\mu
+\nu^{-1}-(\nu\rho)^{-1}.$$ The first expression shows it vanishes to order three in $e$, the second that it is a linear combination of roots, hence it lies in $H^0(\tilde{T},{{\cal O}}(D_S+2D_R))$.
The action of $W(E_6)$ on the vector space $V$. {#we6V}
===============================================
{#section-4}
Naruki [@Naruki] defined a biregular action of $W(E_6)$ on ${{\cal C}}$ ([@Naruki], $\S5$, p. 13). We show that this induces an action of $W(E_6)$ on the vector space $V$ defined in \[defV\]. The vector space $V$ may be identified, via pull-back $$V\cong H^0(\tilde{T}',{{\cal O}}(2D_R+D_S-3{{\bf P}}^3_{\rm w})),$$ where $\tilde{T}'$ is the blow up of $\tilde{T}$ in the identity element $e$ and ${{\bf P}}^3_{\rm w}$ is the exceptional fiber.
The main problem is to find the images of the divisor $D_S-3{{\bf P}}^3_{\rm w}$ under $s_1,\,s_6\in W(E_6)$ and to show that the images are linearly equivalent to this divisor. For this we use the following rational function: $$C_1:=\frac{(\lambda^2\mu\nu\rho^2-1)^3}
{(\lambda-1)(\lambda\rho-1)(\lambda\nu\rho-1)
(\lambda\mu\nu\rho^2-1)(\lambda\mu\nu\rho-1)(\lambda\mu\rho-1)}.$$
Lemma. {#exadiv}
------
The rational function $C_1$ on $\tilde{T}'$ has divisor $$(C_1)=3D_{\bar{\rm x}}-3{{\bf P}}^3_{\rm w}+\sum_{\pm,i=2}^4 V(\pm \epsilon_i)
-D^1_{\lambda}-D^1_{\lambda\rho}-D^1_{\lambda\nu\rho}-D^1_{\lambda\mu\nu\rho^2}
-D^1_{\lambda\mu\nu\rho}-D^1_{\lambda\mu\rho}+D$$ for some divisor $D$ which is a combination of the divisors $V(\tau)$ with $\tau\in R$. Here $D_{\bar{\rm x}} $ is the tritangent divisor defined by the strict transform of the zero locus of $\lambda^2\mu\nu\rho^2-1$ in $\tilde{T}$.
[[**Proof.**]{}$\;\;$]{}The proof is straightforward using the formula from \[torcom\] and the examples in the proof of Lemma \[lemact\], for example $$(\lambda^2\mu\nu\rho^2-1)=D_{\bar{\rm x}}+{{\bf P}}^3_{\rm
w}-2V(-\epsilon_1)-\sum_{\pm,\pm,\pm}
V((-\epsilon_1\pm\epsilon_2\pm\epsilon_3\pm\epsilon_4)/2),$$ and the divisor of $\lambda-1$, in $\tilde{T}$, was determined in \[exal\].
{#section-5}
For $f\in V$, the composition $f\circ s_1$ does not lie in $V$. However, we will show that the quotient $(f\circ s_1)/C_1$ does lie in $V$. To get an action of all of $W(E_6)$ however, the correct definition for the action of $s_1$ on $V$ is $s_1(f)=-(f\circ s_1)/C_1$.
Theorem. {#actwe6}
--------
The action of $W(E_6)$ on ${{\cal C}}$ defines an action of $W(E_6)$ on $V$ by the following formulas: $$s_i(f):=\left\{\begin{array}{ccl}
-(f\circ s_1)/C_1&\quad& {\rm if}\;i=1,\\
f\circ s_i& &{\rm if}\;2\leq i\leq 5,\\
-(f\circ s_6)/C_6&& {\rm if}\;i=6,
\end{array}\right.$$ here the rational maps $s_i:T\rightarrow T$ are as defined by Naruki in [@Naruki], p. 13 and $C_6=C_1\circ \tau$ where $\tau(\lambda,\mu,\nu,\rho)=(\nu,\mu,\lambda,\rho)$.
The representation of $W(E_6)$ on $V$ is its unique 10 dimensional irreducible representation and is denoted by $10_s$ in [@Fra].
[[**Proof.**]{}$\;\;$]{}Recall that $D_S=\sum D_\alpha$ with $\alpha\in E_6-D_4$ a positive root. Write: $$D_S=D^{(0)}_S+D^{(1)}_S,\qquad D^{(0)}_S=\sum_\alpha D_\alpha$$ where we sum over the positive roots $\alpha\in E_6$, $\alpha\not\in D_4$ which are fixed under $s_1$. Then $s_1^*D_S=D^{(0)}_S+s_1^*D^{(1)}_S$. Since $s_1^*D_{\rm w}=D_{\bar{\rm x}}$ (cf. the proof of \[lemact\]), we get: $$s_1^*(D_S-3D_{\rm w})=D^{(0)}_S+s_1^*D^{(1)}_S-3D_{\bar{\rm x}}.$$ One verifies, using the tables \[tabwe6\] and \[tabwe6b\] and the lemma above, that $$(C_1)=3D_{\bar{\rm x}}-3D_{\rm w}+D^{(1)}_S-s_1^*D^{(1)}_S$$ hence $s_1^*(D_S-3D_{\rm w})+(C_1)=D_S-3D_{\rm w}$. This suggests that $f\mapsto \pm(f\circ s_1)/C_1$ defines an endomorphism of $V$. To check this and to get a $W(E_6)$ representation on $V$, one computes matrices and checks the defining relations for $W(E_6)$ (we used a computer, note this direct method avoids a detailed discussion of the divisor $D_R$ and verifies that one has to put a ‘$-$’ sign in the definition of $s_1$ and $s_6$). Since the only representations of $W(E_6)$ of dimension at most 10 are the trivial one, denoted by $1=1_p$, the 6 dimensional reflection representation $6_p$, their tensor products with the determinant representation $1_n$ and $6_n$, and $10_s$, it suffices to compute the traces of a reflection $s_i$ (which is 0) and of a product of two commuting reflections (which has trace 2) to prove that $V\cong 10_s$.
Table of a basis of $V$ {#second}
-----------------------
To obtain functions in $V$, all entries have to be divided by $\lambda\mu\nu\rho^2$. All ten functions are in one $W(E_6)$-orbit.
$$\begin{array}{rcl}
f_1&=&(\lambda\rho-1)(\mu\rho-1)(\nu\rho-1)(\lambda\mu\nu\rho-1),\\
g_1&=&(\rho-1)(\lambda\mu\rho-1)(\lambda\nu\rho-1)(\mu\nu\rho-1),\\
f_2&=&( \mu\rho-1)( \nu\rho-1)(1 - \lambda^2\mu\nu\rho^2), \\
g_2&=&( \rho-1)( \mu\nu\rho-1)(1 - \lambda^2\mu\nu\rho^2), \\
f_3&=&( \lambda\rho-1)( \mu\rho-1)(1 - \lambda\mu\nu^2\rho^2), \\
g_3&=&( \rho-1)( \lambda\mu\rho-1)(1 - \lambda\mu\nu^2\rho^2), \\
f_4&=&\rho( \mu\rho-1)(\lambda + \nu - \lambda\nu - \lambda\nu\rho -
\lambda\mu\nu\rho + \lambda^2\mu\nu^2\rho^2), \\
g_4&=&\rho( \mu-1)(\lambda + \nu - \lambda\nu - \lambda\nu\rho -
\lambda\mu\nu\rho + \lambda^2\mu\nu^2\rho^2), \\
f_5&=& ( \lambda\mu\rho-1)( \mu\nu\rho-1)(1 - \lambda\nu\rho^2), \\
g_5&=& ( \mu\rho-1)(\lambda\mu\nu\rho-1)(1- \lambda\nu\rho^2).
\end{array}$$
Crosses.
--------
Allcock and Freitag construct a 10 dimensional space $W$ of automorphic forms on the 4-ball ([@Fr], between 4.3 and 4.4) which defines the map ${{\cal M}}\hookrightarrow {{\bf P}}^9$. The vector space $W$ is spanned by certain automorphic forms which, up to a scalar multiple, can be characterized by the fact that their divisors in the ball-quotient ${{\cal M}}$ are crosses ([@Fr], Theorem 4.6). A cross is defined to be a divisor $$D_\alpha+D_\beta+D_\gamma+D_\delta+D_t$$ where $t$ is a tritangent, defining a subroot system $t^\perp$ of type $D_4$ in $E_6$ (as in \[trite6\]) and $\alpha$,$\ldots,\delta\in t^\perp\cap E_6^+$ are mutually perpendicular (cf. [@Fr], Definition 3.2). For each tritangent $t$, there are 3 crosses containing $D_t$, thus there are $45\cdot 3=135$ crosses. For example, the crosses associated to $t=(16)$ have $\{\alpha,\ldots,\delta\}$ equal to one of the three sets: $$\{h_{23},\,h_{45},\,h_{123},\,h_{145}\},
\qquad
\{h_{24},\,h_{35},\,h_{124},\,h_{135}\},
\qquad
\{h_{25},\,h_{34},\,h_{125},\,h_{134}\}.$$ The following theorem identifies $W$ with $V$ (as spaces of global sections of a line bundle on ${{\cal M}}$).
Theorem. {#main}
--------
The rational map $\tilde{F}:\tilde{T}\longrightarrow {{\bf P}}^9$ defined by a basis of the vector space $V$ defines a $W(E_6)$-equivariant morphism $$F:{{\cal C}}\longrightarrow {{\cal M}}\;\subset {{\bf P}}^9$$ which blows down the 40 cusp divisors to the 40 cusps. The image of $F$ is the moduli space ${{\cal M}}$ which is embedded into ${{\bf P}}^9$ via the map defined by Allcock and Freitag.
[[**Proof.**]{}$\;\;$]{}Using the results of $\cite{Fr}$ and the $W(E_6)$-action on $W$ and $V$, it suffices to show that there is a function $f\in V\cong H^0(\tilde{T}',{{\cal O}}(2D_R+D_S-3D_{\rm w}))$ such that the corresponding section has, modulo cusp divisors, a cross as zero divisor in $\tilde{T}'$. In fact, the exceptional divisors in the blow ups $\pi'$ and $\pi''$ get blown down in the composition $\hat{T}\rightarrow {{\cal C}}\rightarrow {{\cal M}}$ and under push-pull via ${{\cal M}}\leftarrow \hat{T}\rightarrow \tilde{T}'$ crosses in ${{\cal M}}$ correspond to crosses in $\tilde{T}'$ and cusp divisors in $\tilde{T}'$ get contracted to points in ${{\cal M}}$.
Let $f=f_1$ in table \[second\], then the divisor in $\tilde{T}'$ of the corresponding function in $V$ is: $$\left({f_1\over {\lambda\mu\nu\rho^2}}\right)=
4D_{\rm w}+D_{24}+D_{124}+D_{35}+D_{135}-D_S+D'$$ where $D'$ is a divisor with support in $D_R$.
Thus the zero divisor on $\tilde{T}'$ of the section corresponding to $f_1$ is $D_{\rm w}+D_{24}+D_{124}+D_{35}+D_{135}+2D_R-D'$. Note that ${\rm w}=(16)$ and that the four roots $h_{24}$, $h_{124}$, $h_{35}$, $h_{125}$ are in $D_4=(16)^\perp$ and are perpendicular. The remaining part, $2D_R-D'$, has support on cusp divisors. We observe that using the explicit bases of $V$ and the method of [@Fr] Corollary 7.3, one can also prove directly that $F$ factors over ${{\cal M}}$ and embeds ${{\cal M}}$ into ${{\bf P}}^9$.
Cross ratios. {#exptab}
-------------
The basis of $V$ given in \[second\] has the property that the quotients $f_i/g_i$ are double ratios associated to tritangents (see the table 2 of [@Naruki]), and we have in fact one double ratio from each $D_4$-orbit: $$r({\rm w})=\frac{f_1}{g_1},\quad
r(\bar{{\rm x}})=\frac{g_2}{f_2},\quad
r(\bar{{\rm z}})=\frac{g_3}{f_3},\quad
r(\bar{{\rm q}}_1)=\frac{g_4}{f_4},\quad
r({\rm y})=\frac{f_5}{g_5}.$$ (For completeness sake: ${\rm w}=(16)$, $\bar{{\rm x}}=(26)$, $\bar{{\rm z}}=(15)$, $\bar{{\rm q}}_1=(25)$, ${\rm y}=(16.23.45)$.) Note that the last factor in each function in \[second\] is the local equation of the associated tritangent.
The fact that we find one cross ratio from each $D_4$ orbit already implies that ${{\cal C}}$ is birationally isomorphic with $F({{\cal C}})$ (use the argument of [@Naruki], $\S$ 5.5).
The involution $\gamma(t)\in W(E_6)$ associated to a tritangent $t$, see \[F4\], has trace $-6$ on $V$ (cf. [@Fra], Table II), hence it has a 2 dimensional space of invariants $V_t$ in $V$. There are, upto scalar multiple, 3 functions in $V_t$ whose divisors are crosses (cf. [@Fr], Lemma 4.5). The pairs of functions $f_i$, $g_i$ span such $V_t$’s. The third function in $V_{(16)}$ is: $$h_1:=f_1-g_1=\rho(\lambda-1)(\mu-1)(\nu-1)(\lambda\mu\nu\rho^2-1).$$ The stabilizer $W(F_4)$ of $t$ acts on $V_t$ through the action of a dihedral group with $12$ elements; the subgroup $W(D_4)$ (generated by reflections in the long roots) acts a $S_3$ and the reflections in the short roots act as $-1$ on $V_t$. In fact, the elements $\sigma_1,\,\sigma_2\in
W(F_4)$ given by Naruki in [@Naruki], $\S 8$, p. 16 act as $-1$ on $V_{(16)}$.
Complex invariants. {#excom}
-------------------
In example \[f12x34x56\] we considered the following function from $V$: $$f=\nu^{-1}\rho^{-1}(\rho-1)(\lambda\nu\rho-1)(\mu\nu\rho-1).$$ Its divisor satisfies, modulo components with support in $D_R$: $$(f)+D_S-3D_{\rm
w}=D_{16}+D_{34}+D_{25}+D_{125}+D_{256}+D_{136}+D_{146}+D_{234}+D_{345}.$$ The effective divisor on the right is the sum of the $D_\alpha$ where $\alpha$ runs over the positive roots of three mutually perpendicular $A_2$’s: $$\{h_{16},\;h_{125},\;h_{256}\},\qquad
\{h_{25},\;h_{234},\;h_{345}\},\qquad
\{h_{34},\;h_{136},\;h_{146}\}.$$ There are 40 such triples of orthogonal $A_2$’s in $E_6$ which are permuted transitively by $W(E_6)$ ([@H], 6.1.5.3; this particular triple is denoted by $[16,25,34]$). The corresponding $40$ functions in $V$ were considered by Coble who called them complex invariants (cf. [@Co3], p. 340-341), see also [@Y]. There are $80=2\cdot 40$ functions in the $W(E_6)$-orbit of a complex invariant, the sign of a complex invariant is not well defined.
Images of divisors in ${{\cal C}}$
==================================
{#section-6}
We can use Naruki’s model ${{\cal C}}$ and the explicit basis of $V$ to study the moduli space ${{\cal M}}\subset {{\bf P}}^9$. Here we consider various divisors in ${{\cal M}}$ as subvarieties of ${{\bf P}}^9$, in particular we find a nice parametrization of a tritangent divisor.
The boundary divisors. {#short}
----------------------
We consider the image in ${{\bf P}}^9$ of one of the 36 boundary divisors $D_\alpha\subset {{\cal M}}$ (\[bdiv\] and section \[e6bdiv\]). These parametrize cubic surfaces with at least one node. The divisor $D_\alpha$ is the fixed point set of the involution $s_\alpha$. The trace of $s_\alpha$ on $W$ is zero, hence $W$ is the direct sum of two $5$-dimensional eigenspaces of $s_\alpha$. Since $F$ is equivariant for $W(E_6)$, $D_\alpha$ will lie in a ${{\bf P}}^4$. The centralizer in $W(E_6)$ of the reflection $s_\alpha$ acts on the divisor $D_\alpha$ and on the eigenspaces of $s_\alpha$. This subgroup is isomorphic to $S_6$. For example if $\alpha=h$, one obtains the ‘standard’ $S_6$ generated by all the $s_i$ except $s_2$.
In particular we consider the image of $D_{345}=V(\epsilon_1)$ under $F$. This divisor is defined by $\lambda=0$ on the open subset $A(\Delta_0)=Spec({{\bf C}}[\lambda,\mu,\nu,\rho])$ of $\tilde{T}$. Since the 10 functions listed in table \[second\] are regular on $A(\Delta_0)$ and do not vanish simultaneously, we can simply take $\lambda=0$ and determine (the closure of) the image. The image spans only a ${{\bf P}}^4$ since the following linear functions vanish on this divisor (in the notation of table \[second\]): $$f_1-f_2,\qquad g_1-g_2,\qquad f_3-g_5,\qquad f_2-f_4-g_5,
\qquad g_2-g_3-f_4+g_4.$$ The image of ${{\cal M}}$ in ${{\bf P}}^9$ is defined by cubics (see [@Fr]), and one can show that the image of a boundary divisor is the Segre cubic hypersurface in this ${{\bf P}}^4$ (cf. [@H], 3.2).
The cusp divisors.
------------------
We consider one of the 40 cusp divisors in ${{\cal C}}$ (cf. \[crv\]), for example $V(\epsilon_1+\epsilon_2)$, note that $\epsilon_1+\epsilon_2\in R$. This divisor is defined by $\rho=0$ in $Spec({{\bf C}}[\lambda,\mu,\nu,\rho])$. Putting $\rho=0$ in the 10 functions in table \[second\] one finds that the image of this divisor is the point $$(1: 1: 1: 1: 1: 1: 0: 0: 1: 1).$$
Tritangent divisors. {#trit}
--------------------
The tritangent divisor $D_t$ is the fixed point set of the involution $\gamma(t)\in W(E_6)$. Each $\gamma(t)$ has trace $-6$ on $V$ [@Fra], hence it has two eigenspaces, of dimension $2$ and $8$, in $V$. Since the dimension of the divisor $D_t$ is three we get $D_t\subset{{\bf P}}^7$.
The centralizer of $\gamma(t)$ is isomorphic to $W(F_4)$ and this group acts on both $D_t$ and ${{\bf P}}^7$. We consider the case $t=(16)={\rm w}$, hence $D_t$ is birationally isomorphic to the exceptional fiber ${{\bf P}}^3_{\rm w}$ of the blow up of the torus $T$ in the identity element $e$. Since $e$ is fixed by $W(D_4)$, we get an induced action of $W(D_4)$ on ${{\bf P}}^3_{\rm w}$, and we will see that this action extends to a linear action of $W(F_4)$.
The root lattice $Q(F_4)$ of $F_4$ is the lattice in ${{\bf R}}^4$ generated by the 48 roots of $F_4$ which are (cf. [@Hu], III 12.1) the 24 roots $\pm e_i\pm e_j$ of $D_4$ (these roots have length $2$ and are called the long roots of $F_4$) and the 24 vectors $\pm e_i$, $(\pm e_1\pm e_2\pm e_3\pm e_4)/2$ which have length 1, the short roots of $F_4$. $$Q(F_4)=\langle \pm e_i\pm e_j,\;(\pm e_1\pm e_2\pm e_3\pm e_4)/2\rangle_{{\bf Z}}\qquad(\subset{{\bf R}}^4).$$
Theorem {#tritpar}
-------
Any tritangent divisor $D_t$ is $W(F_4)$-equivariantly birationally isomorphic to ${{\bf P}}^3$ via the map $${{\bf P}}^3={{\bf P}}(Q(F_4)\otimes_{{\bf Z}}{{\bf C}})\longrightarrow D_t\hookrightarrow {{\bf P}}^7$$ given by the linear system of cubics which are zero in the short roots of $F_4$.
[[**Proof.**]{}$\;\;$]{}Since $W(E_6)$ acts transitively on the tritangent divisors, it is sufficient to consider the case $t=(16)$. We show that the functions from $V$ give the desired map ${{\bf P}}^3_{\rm w}\rightarrow D_{(16)}$.
The local coordinate functions $\lambda-1,\ldots,\rho-1$ near $e$ induce projective coordinates $x_\lambda,\ldots,x_\rho$ on ${{\bf P}}^3_{\rm w}$. Since $\lambda=e_1-e_2,\ldots,\rho=e_2-e_3$ it is more convenient to use coordinates $y_i$ with $$(x_\lambda:x_\mu:x_\nu:x_\rho)=(y_1-y_2:y_3+y_4:y_3-y_4,y_2-y_3).$$
The group $W(F_4)$ is generated by the subgroup $W(D_4)$ and $\sigma_1$, $\sigma_2$ given in [@Naruki], $\S$8. Using the explicit formulas for the $\sigma_i$ one finds that these act on ${{\bf P}}^3_{\rm w}$ as reflection in the planes $y_4=0$ and $y_1-y_2-y_3-y_4$ respectively. (For example $\sigma_1$ interchanges $\mu$ and $\nu$ and fixes the other roots, thus on ${{\bf P}}^3_{\rm w}$ it is the linear map which permutes $y_3+y_4$ and $y_3-y_4$ and fixes $y_1-y_2$ and $y_2-y_3$.) Thus these $\sigma_i$ are reflections in the short roots. This implies that we may identify ${{\bf P}}^3_{\rm w}$ with ${{\bf P}}(Q(F_4)\otimes_{{\bf Z}}{{\bf C}})$.
All functions in $V$ vanish to third order in $e$, but not all vanish to fourth order, hence restricted to ${{\bf P}}^3_{\rm w}$ the map $F$ is given by the leading terms of third order. Note that $f_1$ and $g_1$ from table \[second\] vanish to order four at $e$, hence the image of ${{\bf P}}^3_{\rm w}$ spans at most a ${{\bf P}}^7$, as we observed earlier. The leading terms of the other $8$ basis functions are cubics which all contain the 12 points: $$(1:0:0:0)_y,\ldots,(0:0:0:1)_y \qquad{\rm and}\quad (1:\pm 1:\pm 1:\pm 1)_y.$$ For example, $f_2$ from table \[second\] has leading term (up to sign): $$(x_\mu+x_\rho)( x_\nu+x_\rho)(2x_\lambda+x_\mu+x_\nu+2x_\rho)
=(y_2+y_4)(y_2-y_4)(2y_1).$$
One can verify that the $8$ leading cubics are independent and that these $12$ points impose independent conditions on the ($20$ dimensional) space of cubics. Thus the map ${{\bf P}}^3_{\rm w}\rightarrow D_{(16)}$ is given by the subspace of cubics vanishing in these points and the image of ${{\bf P}}^3_{\rm w}$ spans a ${{\bf P}}^7$.
We observe that the twelve basepoints are in three $D_4$-orbits of length four. Two points in two distinct orbits determine a line on which there is a unique line from the third orbit. For example the points $(0:0:0:1)_y,\;(1:1:1:1)_y,\;(1:1:1:-1)_y$ are on a line. In this way we get $16$ lines, on each of these there are 3 base points. Actually, the two fourth-order leading terms are zero exactly on these 16 lines.
Incidence of tritangent divisors. {#inctri}
---------------------------------
The tritangent divisor $D_{(16)}$ is one of $45$ such divisors, recall that $(16)={\rm w}=\{a_1,\,b_6,\,c_{16}\}$. The remaining $44$ tritangent divisors now divide into 4 groups, $44=4+4+4+2\cdot16$ as follows. For each $l\in (16)$, there are $4$ other tritangents containing $l$ (for example, the $\{a_1,\,b_i,\,c_{1i}\}$ for $2\leq i\leq 5$ are the other tritangents which also contain $a_1$).
The remaining $32$ tritangents do not have a line in common with $(16)$. These come in pairs as follows. Given one of these $32$, say $\{l_1,l_2,l_3\}$, after a permutation of the indices one has that $a_1$ and $l_1$ meet (and $a_1$ does not meet $l_2$ and $l_3$) and thus there is a line $m_1$ such that $\{a_1,l_1,m_1\}$ is a tritangent. Similarly $b_6$ and $l_2$ determine a line $m_2$ and $c_{16}$ and $l_3$ determine a line $m_3$. Now $\{m_1,m_2,m_3\}$ is another tritangent which has no line in common with $(16)$. For example, $\{b_5,a_2,c_{25}\}$ determines $\{c_{15},c_{26},c_{34}\}$. (To see all this, consider a general cubic surface and the planes $V_{(16)}$ and $V'$ spanned by the lines in $(16)$ and the $l_i$ respectively. These planes meet in a line which by assumption does not lie in the cubic surface. Thus this line meets the surface in $3$ points and through each of these points there passes exactly one line from $(16)$ and one line from the $l_i$.)
If the lines in $(16)$ all pass through an Eckart point $P_{\rm w}$ and similarly the lines $l_i$ all pass through an Eckart point $P'$, the line $L$ spanned by $P_{\rm w}$ and $P'$ meets the cubic surface in a third point $P''$ which is an Eckart point, being the intersection of the $m_i$. (To see that each $m_i$ passes through $P''$, consider the plane spanned by, say, $a_1$ and $l_1$; it cuts out $m_1$ and contains the line $L$, hence $m_1$ meets $L$ in $P''$, similarly for the other pairs of lines.)
As a consequence, a point in the intersection of two tritangent divisors without a common line will lie in a third tritangent divisor.
Tritangent divisors and ${{\bf P}}^3_{\rm w}$. {#leadtri}
----------------------------------------------
The intersections of ${{\bf P}}^3_{\rm w}$ with the other $44$ tritangent divisors are given by the leading terms of their equations. Those tritangent divisors which have a line in common with ${\rm w}=\{a_1,\,b_6,\,c_{16}\}$ have a linear leading term, in fact one finds the following $12$ linear terms: $$y_i\quad(1\leq i\leq 4),\qquad{\rm and}\quad
y_1\pm y_2\pm y_3\pm y_3$$ where the last $8$ come in two $W(D_4)$-orbits distinguished by the parity of the number of minus signs. For example, the tritangent $(15)=\bar{{\rm z}}$ defined by $\lambda\mu\nu^2\rho^2-1$ has leading term $y_1+y_2+y_3-y_4$.
The tritangents which do not have a line in common with ${\rm w}$ have a leading term of degree two, in fact the two tritangents in a pair have the same leading term (as they should, see the last part of \[inctri\]). These quadrics correspond to the 16 lines in ${{\bf P}}^3_{\rm w}$ containing $3$ of the $12$ base points of $F$ (see \[trit\]); each line determines a unique quadric by the condition that it contains the other $9$ base points (and this quadric will not contain any of the $3$ points on the line). For example, the tritangent divisor $\bar{q}_1=\{a_2,b_5,c_{25}\}$ is defined by $\lambda+\nu-\lambda\nu-\lambda\nu\rho-\lambda\mu\nu\rho+
\lambda^2\mu\nu^2\rho^2$ has leading term $$Q_{25}:=y_1^2+y_2y_3-y_2y_4-y_3y_4$$ which does not contain the $3$ colinear points $(1:0:0:0)_y$, $(1:-1:-1:1)_y$ and $(1:1:1:-1)_y$. The other tritangent divisor having the same leading term is $\bar{q}=\{c_{15},c_{26},c_{34}\}$ which is defined by $1-\lambda\nu\rho
-\lambda \mu \nu \rho- \lambda \mu \nu \rho^2+\lambda^2 \mu \nu \rho^2 +
\lambda \mu \nu^2 \rho^2$.
The other $15$ quadrics can be obtained from $Q_{25}$ by the action of $W(D_4)$, that is, by permuting the coordinates and changing the signs of an even number of the $y_i$. These quadrics are smooth and hence are isomorphic to ${{\bf P}}^1\times{{\bf P}}^1$.
Equations for the moduli space. {#unif}
===============================
{#section-7}
The universal marked cubic surface is embedded in a ${{\bf P}}^3$-bundle over ${{\cal C}}$. Over the moduli space ${{\cal M}}^0$ of smooth marked cubic surfaces, this bundle is the projectivization of the tangent bundle ([@ACT2] $\S$ 10). In Naruki’s paper [@Naruki] one finds an explicit cubic polynomial in $R[X,Y,Z,T]$, with $R:={{\bf C}}[\lambda,\mu,\nu,\rho]$, which defines the universal family over an open part of ${{\cal M}}$. He also gives 45 linear forms in $R[X,Y,Z,T]$ which define the tritangent planes.
We will verify that there are linear relations between these, suitably normalized, linear forms with coefficients which are elements from $V$ (note that elements from $V$ are rational functions on $T$ and thus are in the field of fractions of $R$). This allows us to recover the cubic equations found by Allcock and Freitag which define ${{\cal M}}$. We also find a six dimensional vector space of quintic polynomials, on which $W(E_6)$ acts via its standard representation, which define a variety $X\subset{{\bf P}}^9$ whose singular locus contains ${{\cal M}}$.
{#section-8}
Consider two tritangents which contain a common line. For any point in the interior of ${{\cal C}}$, the corresponding planes are distinct. However over one of the 36 boundary divisors the planes may coincide. Over a boundary divisor $D_\alpha$ the 6 pairs of lines in the double six corresponding to $\alpha$ (cf. [@H]) on the universal marked surface specialize to the six lines through the node of the universal surface over $D_\alpha$. The reflection $s_\alpha$ in $W(E_6)$ interchanges the lines in each of the six pairs and fixes the other 15 lines. Thus if $s_\alpha$ maps one tritangent set to another, then the lines in the planes and thus the planes themselves will coincide over $D_\alpha$.
Lemma. {#lemma.-1}
------
Let $t_1$, $t_2$ be two distinct tritangent sets which have a line in common. Then there are exactly two reflections in $W(E_6)$ which map $t_1$ to $t_2$. The corresponding roots in $E_6$ are perpendicular.
[[**Proof.**]{}$\;\;$]{}Since $W(E_6)$ acts transitively on the set of lines, we may assume that the common line is $b_6$. Then the $t_i$ are of type $\{a_i,b_6,c_{i6}\}$ with $1\leq i\leq 5$ and applying a suitable element of $W(E_6)$ we may assume that $t_1=\{a_1,b_6,c_{16}\}$, $t_2=\{a_2,b_6,c_{26}\}$. By inspection of the lists of double sixes in [@H] one finds exactly one double six which contains the pairs $(a_1,a_2)$ and $(c_{16}, c_{26})$ (it is $N_{12}$) and one which contains the pairs $(a_1,c_{26})$ and $(c_{16}, a_2)$ (it is $N_{345}$). Thus only reflections in $h_{12}$ (which permutes the indices $1$ and $2$) and in $h_{345}$ (which interchanges $a_1\leftrightarrow c_{26}$ and $a_2\leftrightarrow c_{16}$) permute these two tritangent sets. It is easy to verify that $h_{12}$ and $h_{345}$ are perpendicular.
{#section-9}
Given three linear forms $K,\,L,\,M\in R[X,Y,Z,T]$ which define tritangent planes to the universal cubic surface having a line in common, there is a linear relation, with coefficients in $R$, $$AK+BL+CM=0.$$ The next proposition shows that three tritangent planes with a line in common define three crosses. Recall that a cross is a divisor in ${{\cal M}}$ determined by the choice of a tritangent set $t$ and on 4 perpendicular roots in $t^\perp\cong D_4$. In the example below we then verify that these crosses are the divisors of the coefficients in the linear relation.
Proposition.
------------
Let $t_1$, $t_2$ and $t_3$ be tritangent sets with a line in common. Then there are crosses $X_i$ determined by the tritangent sets $t_i$, the pair of roots whose reflections interchange $t_j$ and $t_k$ (with $\{i,j,k\}=\{1,2,3\}$) and the pair of roots which is perpendicular to all the weights in the union of these three tritangent sets.
[[**Proof.**]{}$\;\;$]{}Again we use the $W(E_6)$ action, and so we may assume that $t_i=\{a_i,b_6,c_{i6}\}$. These span the subspace $\langle
x_1,\,x_2,\,x_5,\,x_6\rangle$ ([@H], 6.1.3) hence only the roots $h_{45}=-x_3+x_4$ and $h_{145}=x_3+x_4$ are perpendicular to this subspace. The two roots whose reflections interchange $t_1$ and $t_2$ are $h_{12}$, $h_{345}$. The roots $h_{12}$, $h_{345}$, $h_{45}$ and $h_{145}$ are orthogonal and lie in the $D_4$ perpendicular to the weights in $t_3$. Therefore there is a cross $X_3$ which is the sum of the tritangent divisor corresponding to $t_3$ and the four boundary divisors corresponding to these four roots. Similarly one finds crosses $X_1$ and $X_2$.
Example. {#exeqq}
--------
We consider the tritangents which contain the line $b_6$. They are: $$\begin{array}{cccrcr}
{\rm set}& {\rm label} &{\rm local\;equation}&&&{\rm linear\ \;form}\\
\{a_1,b_6,c_{16}\}& \;(16)={\rm w}\; &1& \qquad& &W\\
\{a_2,b_6,c_{26}\}& \;(26)=\bar{{\rm x}}\;&\lambda^2\mu\nu\rho^2 -1&
\qquad\lambda X&-&(\lambda\rho-1)(\lambda\mu\nu\rho-1)W\\
\{a_3,b_6,c_{36}\}& \;(36)={\rm x}\;&
\mu\nu\rho^2 -1&-X&+&(\rho-1)(\mu\nu\rho-1)W\\
\{a_4,b_6,c_{46}\}& \;(46)=x\;&-\rho(\mu\nu-1) &X&&\\
\{a_5,b_6,c_{56}\}& \;(56)=\xi\;&\mu-\nu &X&+&\rho(\mu-1)(\nu-1)W\\
\end{array}$$ The conversion of the labels is given in [@se2], the equation of the planes is given in [@Naruki], Table 1, but we changed the sign of $(36)$ and we multiplied the local equation of $(46)$ by a unit.
We write $t_i:=\{a_i,b_6,c_{16}\}$. Then $t_2=s_1(t_2)$, $t_3=s_3(t_2)$, $t_4=s_4(t_3)$ and $t_5=s_5(t_4)$ where $s_i$ is the reflection in $\alpha_i$. The two roots perpendicular to the span of the sets $t_1$, $t_2$ and $t_3$ are $h_{45}$ and $h_{123}$. The cross $X_1$ is then: $$X_1=D_{23}+D_{145}+D_{45}+D_{123}+D_{(16)}$$ and $X_2=s_1(X_1)$, $X_3=s_3(X_2)$.
Note that $X_1$ is the divisor of the section corresponding to $$A_1=\rho(-1 + \lambda)(-1 + \mu)(-1 + \nu)(-1 + \lambda\mu\nu\rho^2)
(\lambda\mu\nu\rho^2)^{-1}\qquad(\in V),$$ and that $A_1=h_1$ in \[exptab\]. Similarly we define $A_2=s_1(A_1)$, $A_3=s_3(A_2)\in V$.
We define $L_{i6}\in {{\bf C}}(\lambda,\ldots,\rho)[X,W]$ to be the quotient of the linear form defining the tritangent plane $(i6)$ by the local equation of the tritangent divisor $D_{(i6)}$ as listed in the table. One can then verify the following linear relation: $$A_1L_{16}+A_2L_{26}+A_3L_{36}=0.$$
Proposition. {#rank}
------------
Let $A_i$ and $L_{ij}$ be as in Example \[exeqq\]. Define functions $B_i$, …, $F_i\in V$ by: $$B_i=s_4(A_i),\quad C_i=s_3(B_i),\quad D_i=s_1(C_i),\quad E_i=s_5(D_i),
\quad F_i=s_5(B_i).$$ Then we have $Mv=0$ where $$M=\left(\begin{array}{ccccc}
A_1&A_2&A_3&0&0\\
B_1&B_2&0&B_3&0\\
C_1&0&-C_2&C_3&0\\
0&D_1&D_2& -D_3&0\\
0& E_1& E_2 & 0 & E_3\\
F_1& F_2&0&0&- F_3\\
\end{array}\right),\qquad
v=
\left(\begin{array}{c}L_{16}\\L_{26}\\L_{36}\\L_{46}\\L_{56}\end{array}\right).$$ In particular, $M$ has rank at most three.
[[**Proof.**]{}$\;\;$]{}Applying the reflection $s_4$ in $\alpha_4=h_{34}$ (which permutes the indices $3$ and $4$) to the linear relation from Example \[exeqq\] we obtain a relation between the linear forms defining the tritangents corresponding to $t_1=s_4(t_1)$, $t_2=s_4(t_2)$ and $t_4=s_4(t_3)$. One verifies that this is $B_1L_{16}+B_2L_{26}+B_3L_{46}=0$ with coefficients $B_i=s_4(A_i)$. Similarly, one verifies the other relations. Since each entry of $v$ is of the form $a_iX+b_iW$ we see that $\ker(M)$ contains the two vectors $a=(a_1,\ldots,a_5)$ and $b=(b_1,\ldots,b_5)$. Thus the rank of $M$ is at most $5-2=3$.
Equations.
----------
To obtain equations for ${{\cal M}}\subset{{\bf P}}^9$ from this proposition, one chooses a basis $X_0$,$\ldots$,$X_9$ of $V$. Then each function in $V$ is a linear form in the $X_i$ with coefficients in ${{\bf C}}$. Thus each entry of the matrix $M$ is a linear form in the $X_i$. Since the rank of $M$ is at most $3$, the determinant of each $4\times 4$ submatrix of $M$, which is a degree 4 polynomial in the $X_i$, is identically zero as function on ${{\cal M}}$. Therefore each such determinant gives a, possibly trivial, quartic polynomial in the ideal of ${{\cal M}}$.
Cubics. {#cubrel}
-------
To get cubic equations we consider the following submatrix of $M$: $$N=\left(\begin{array}{cccc}
A_1&A_2&A_3&0\\
B_1&B_2&0&B_4\\
C_1&0&-C_3&C_4\\
\end{array}\right).$$ The matrix $N$ has rank at most two since $Nw=0$, where $w=(L_{16},\ldots,L_{46})$, gives two vectors in $\ker{N}$ (put $X=1$, $W=0$ and $X=0$, $W=1$ in $w$). In particular, $$\det\left(\begin{array}{ccc}
A_2&A_3&0\\
B_2&0&B_4\\
0&C_3&-C_4\\
\end{array}\right)=-A_2B_4C_3+A_3B_2C_4=0.$$ The corresponding cubic polynomial in the $X_i$ is not identically zero in ${{\bf C}}[\ldots,X_i,\ldots]$ and is one of those found in [@Fr] Lemma 6.3. Theorem 6.4 of that paper implies that ${{\cal M}}$ is defined by the $W(E_6)$-orbit of this cubic equation.
Quintics. {#quints}
---------
One verifies that the determinant of the following submatrix of $M$ is a degree 5 polynomial in the $X_i$ which is not identically zero: $$M_2=\left(\begin{array}{ccccc}
A_1&A_2&A_3&0&0\\
C_1&0&-C_3&C_4&0\\
0&D_2&D_3& -D_4&0\\
0& E_2& E_3 & 0 & E_5\\
F_1& F_2&0&0&- F_5\\
\end{array}\right).$$ By Proposition \[rank\] the rank of $M_2$ is at most $3$. Therefore the determinant of any $4\times 4$ submatrix of $M_2$ is zero on ${{\cal M}}$. Since the partial deriviatives of $\det(M)$ with respect to the $X_i$ are linear combinations of determinants of such submatrices, we conclude that the quintic hypersurface $X$ in ${{\bf P}}V$ defined by $\det(M)$ is singular along moduli space of marked cubic surfaces ${\cal M}\subset {{\bf P}}^9$.
Using the $10\times 5$ matrix obtained from all ${5\choose 3}=10$ linear relations between 3 of the 5 tritangent planes containing the line $b_6$, we get ${10 \choose 5}$ quintics, but they are either $0$ or the same as $\det(M)$ up to sign. It can be checked that the $W(E_6)$-orbit of such a quintic has $27$ elements and that these quintics span a copy of the standard 6-dimensional representation $6_p$ of $W(E_6)$.
Hunt’s quintic. {#6dire}
===============
Supercrosses.
-------------
We show how to construct 27 quintic polynomials, which we call supercrosses, on $V$ which are permuted, up to sign, as the 27 lines on the cubic surface under the action of $W(E_6)$. We show that the supercrosses span a $6$-dimensional vector space on which $W(E_6)$ acts as $6_n$ and that they define a rational map $$\Sigma:{{\cal M}}\longrightarrow {{\bf P}}^5$$ which maps the moduli space onto the the unique $W(E_6)$-invariant hypersurface of degree $5$ in ${{\bf P}}^5$. This hypersurface was investigated by Hunt in [@H].
{#section-10}
The line $a_1$ on a marked cubic surface defines a weight of $E_6$. The roots $\alpha_2,\ldots,\alpha_6$ are perpendicular to this weight and span a root system, of type $D_5$, consisting of $2\cdot 20=40$ roots. In the notation of [@H], this system is ‘in standard form’ $$a_1^\perp=\{\pm x_j\pm x_k\,:\;1\leq j<k\leq 5\}\cong D_5.$$
Any line on a cubic surface lies in 5 tritangent planes. The tritangent planes containing $a_1$ are the $(1i)=\{a_1,b_j,c_{1j}\}$, $2\leq j\leq 6$. The three weights corresponding to the three lines in a tritangent are linearly dependent, hence span a line, and the orthogonal complement of the line is a root system of type $D_4$, in fact $a_1=-(2/3)x_6$, $b_j=x_{j-1}+(1/3)x_6$, thus $$\{a_1,b_j,c_{1j}\}^\perp=\{\,x_{j-1},\,x_6\}^\perp=\{\pm x_i\pm x_k:\;
i<k,\;i,\,k\in\{1,\ldots,\widehat{j-1},\ldots,5\}\,\}\cong D_4.$$
Now the main point is that the 20 positive roots which are perpendicular to $a_1$ split in 5 sets of 4 perpendicular roots such that each of the 5 sets is also perpendicular to the weights corresponding to the lines in a tritangent plane containing $a_1$. Thus each line $l$ determines $5$ crosses. In the notation of [@H]: $$\begin{array}{ccccc}
(12)=\langle a_1,\,b_2,\,c_{12}\rangle&=&\langle x_1,\,x_6\rangle
\quad&\qquad \{\pm x_2+ x_3,\;\pm
x_4+x_5\,\}=&\quad\{h_{34},\,h_{56},\,h_{134},\,h_{156}\,\}\\
(13)=\langle a_1,\,b_3,\,c_{13}\rangle&=&\langle x_2,\,x_6\rangle
\quad&\qquad \{\pm x_1+ x_4,\;\pm
x_3+x_5\,\}=&\quad\{h_{25},\,h_{46},\,h_{125},\,h_{146} \,\}\\
(14)=\langle a_1,\,b_4,\,c_{14}\rangle&=&\langle x_3,\,x_6\rangle
\quad&\qquad \{\pm x_1+ x_5,\;\pm
x_2+x_4\,\}=&\quad\{h_{26},\,h_{35},\,h_{126},\,h_{135}\,\}\\
(15)=\langle a_1,\,b_5,\,c_{15}\rangle&=&\langle x_4,\,x_6\rangle
\quad&\qquad \{\pm x_1+ x_3,\;\pm
x_2+x_5\,\}=&\quad\{h_{24},\,h_{36},\,h_{124},\,h_{136}\,\}\\
(16)=\langle a_1,\,b_6,\,c_{16}\rangle&=&\langle x_5,\,x_6\rangle
\quad&\qquad \{\pm x_1+ x_2,\;\pm
x_3+x_4\,\}=&\quad\{h_{23},\,h_{45},\,h_{123},\,h_{145}\,\}\\
\end{array}$$
The functions $F_l$.
--------------------
To each cross corresponds a function, up to scalar multiple, in $V$. Fixing one such function and applying $W(E_6)$ we find other fuctions, unique up to sign, whose divisors are crosses. Fix a line $l$, then we can associate to it the function $F_l$, unique up to sign, which is the product of the 5 functions in $V$ corresponding to the $5$ crosses associated to $l$. The divisor of $F_l$ is then essentially the sum of the $5$ tritangent divisors $D_t$ with $l\in t$ and the $20$ boundary divisors $D_\alpha$ with $\alpha\in l^\perp\cap E_6^+$. If $m$ is a line and $m=\sigma(l)$ for some $\sigma\in W(E_6)$, we define $F_m:=\det(\sigma)\sigma(F_l)$ where $\det(\sigma)$ is the determinant of $\sigma$ in the 6-dimensional reflection representation. The $F_m$’s will be called a supercrosses, they are uniquely determined by $F_l$.
Proposition. {#proposition.-1}
------------
The 27 functions $F_l$ on Naruki’s cross ratio variety span a 6 dimensional vector space. The Weylgroup $W(E_6)$ acts on this vector space as $6_n$, the tensor product of the standard 6 dimensional representation with its determinant.
[[**Proof.**]{}$\;\;$]{}The functions $F_l$, with scalar factors suitably normalized, satisfy the linear relations $F_l\pm F_m\pm F_n=0$ whenever the lines $l,\,m,\,n$ are in a tritangent plane. From this one concludes that they span a space of dimension 6 on which $W(E_6)$ acts (the relations $F_{a_i}\pm F_{b_j}\pm F_{c_{ij}}=0$ imply one can can express the $F_{c_{ij}}$ in terms of the $F_{a_i}$ and $F_{b_j}$, now use the relations $F_{c_{ij}}\pm F_{c_{kl}}\pm F_{c_{mn}}$ to eliminate the $F_{b_j}$).
Since reflections in the stabilizer of an $F_l$ act by as multiplication by $-1$ on $F_l$, the representation is the twist of the standard representation.
{#section-11}
The theorem provides us with a $W(E_6)$-equivariant rational map $$\Sigma:{{\cal M}}\longrightarrow {{\bf P}}^5.$$ By computing the differential of $\Sigma$ in some point of ${{\cal M}}$ we found that it has maximal rank. Hence the (closure of the) image of $\Sigma$ is a $W(E_6)$-invariant hypersurface in ${{\bf P}}^5$.
Theorem. {#i5}
--------
The hypersurface $\Sigma({{\cal M}})\subset{{\bf P}}^5$ is Hunt’s quintic, the unique quintic hypersurface which is $W(E_6)$-invariant. It is defined by: $$I_5:=\sum_l \lambda_l^5=0$$ where $\lambda_l$ is the linear form on ${{\bf P}}^5$ defined by the $E_6$-weight which corresponds to the line $l$.
[[**Proof.**]{}$\;\;$]{}We will show that the following sextic relation holds: $$\prod_{l\in A} F_l=\prod_{l\in B} F_l$$ where $A=\{a_1,\ldots, a_6\}$ and $B=\{b_1,\ldots,b_6\}$ form a double six of lines. As observed by Naruki (see [@H], p.235), this equation is reducible, being the product of $I_5$ and a linear factor which is the linear form defined by the root corresponding to the double six given by $A$ and $B$. The $W(E_6)$-invariance of the image implies that the image is defined by $I_5$.
The divisors of both sides of the equation are the sum of the $6\cdot 5=30$ tritangent divisors $D_{(ij)}$ as well as the sum of $6\cdot 20=120$ boundary divisors. We already determined the positive roots in $a_1^\perp$ above, those in $b_1^\perp$ are: $$b_1^\perp=\{h_{jk}=-x_{j-1}+x_{k-1}:\;2\leq j<k\leq 6\}\cup
\{h_{pqr}:\;2\leq p<q<r\leq 6\;\}.$$ Thus each $h_{ij}$ occurs 4 times whereas each $h_{pqr}$ occurs 3 times in both the left and the right hand side, note that $4\cdot 15+3\cdot 20=120$). Thus, upto scalar multiple, the left and right hand side coincide. Using the reflection $s$ in the root $h$ (note $s(a_i)=b_i$) one finds the equality.
{#section-12}
Direct computations show that the images of the 36 divisors are 36 points in ${{\bf P}}^5$, these are the roots of $E_6$. The images of the 45 tritangent divisors are the 45 ${{\bf P}}^3$’s in Hunt’s quintic (see the proof of the theorem below).
Theorem. {#thmdeg}
--------
The rational map $$\Sigma:{{\cal M}}\longrightarrow I_5$$ has generic degree at least 10.
[[**Proof.**]{}$\;\;$]{} We verified by machine computation that $\Sigma$ has maximal rank at the point $(\lambda,\mu,\nu,\rho)=(-1,-1,2,3)\in T$. This point lies in the intersection of the two tritangent divisors $(12)=\zeta$ defined by $\lambda=\mu$ and $(13)=z$ defined by $\lambda\mu=1$ (cf. [@Naruki] Table 3). These tritangents have the line $a_1$ in common. Since $\Sigma$ is $W(E_6)$-equivariant we conclude that $\Sigma$ has maximal rank at the general point in the intersection of any two tritangent divisors with a line in common.
We consider the restriction of $\Sigma$ to the intersection of the tritangent divisors $D_{\rm w}=D_{(16)}$ and $D_{(26)}$ which have the line $b_6$ in common. The divisor $D_{\rm w}$ is birationally isomorphic to ${{\bf P}}^3_{\rm w}$, the exceptional fiber of the blow up of $T$ in $e$, and we consider the map induced by $\Sigma$ on this ${{\bf P}}^3$. The local equation of $(26)=\bar{\rm x}$ is $\lambda^2\mu\nu\rho^2=1$ and its intersection with ${{\bf P}}^3_{\rm w}$ is given by $y_1=0$. (cf. \[inctri\], \[leadtri\]). Note that $\Sigma$ has maximal rank in a general point of ${{\bf P}}^3_{\rm w}\cap (y_1=0)$.
On ${{\bf P}}^3_{\rm w}$ the leading terms of the $F_l$ are of degree 15 or 16 (only for $F_{a_1}$, $F_{b_6}$ and $F_{c_{16}}$), hence the restriction of $\Sigma$ is given by homogeneous polynomials of degree 15 and the image of ${{\bf P}}^3_{\rm w}$ under $\Sigma$ lies in the intersection of the hyperplanes defined by $a_1,\,b_6$ and $c_{16}$ which is a ${{\bf P}}^3$. After omitting leading terms which are multiples of $y_1$ and dividing the remaining ones by their common factor $y_2y_3y_4$, we found that $\Sigma$ restricts to ${{\bf P}}^3_{\rm w}\cap (y_1=0)$ to give a map $$\Sigma_r:{{\bf P}}^2\longrightarrow {{\bf P}}^2$$ defined by homogeneous polynomials of degree $12$. One coordinate function is $$F_2:=y_3y_4(y_3-y_4)(y_3+y_4)(y_2^2-y_3y_4)^2(y_2^2+y_3y_4)^2,$$ the other two, $F_3$ and $F_4$, are obtained by permuting the coordinates cyclically. All these functions satisfy $$F(y_2,y_3,y_4)=-(y_2y_3y_4)^8F(y_2^{-1},y_3^{-1},y_4^{-1})$$ hence the map $\Sigma_r$ has degree at least 2.
The inverse image of a general point $(x_2:x_3:x_4)\in{{\bf P}}^2$ under $\Sigma_r$ is defined by the two equations, each homogeneous of degree $12$: $$G_1:=x_3F_2-x_2F_3=0,\qquad G_2:=x_4F_2-x_2F_4=0.$$ The 0-cycle defined by these equations has degree $12^2=144$, but the linear system defined by the $F_i$ has base points. Below we list the base points and their contribution to the intersection multiplicities (determined with computer). Here $\omega$ is a primitive cube root of unity. $$\begin{array}{ll}
(0:0:1),\quad (0:1:0),\quad (1:0:0),\qquad& m_P=20,\\
(0:1:\pm 1),\quad (1:0:\pm 1),\quad (1:\pm 1:0),&m_P=1,\\
(1:1:-1),\quad (1:-1:1),\quad (-1:1:1),&m_P=9,\\
(1:1:1),& m_P=9,\\
(1:\omega:\pm\omega^2),\quad (1:\omega^2:\pm \omega),\quad
(1:-\omega:\pm\omega^2),\quad (1:-\omega^2:\pm\omega)\qquad&m_P=4.
\end{array}$$ Thus we find that the base points contribute $$3\cdot 20+6\cdot 1+3\cdot 9+1\cdot 9+8\cdot 4=134$$ to the intersection, so there remain 10 points unaccounted for. Since $\Sigma$ has maximal rank in a general point of this ${{\bf P}}^2$, we conclude that the degree of $\Sigma$ is at least 10.
Tables.
=======
The following tables identify the $36$ positive roots of $E_6$, in the notation of Hunt [@H], with the $12$ positive $D_4$ roots, in the notation of Naruki [@Naruki], and $24$ $D_4$-weights. We also list the functions $f_\alpha$ on $T$ corresponding to the positive roots $\alpha\in D_4$.
{#tabwe6}
$$\begin{array}{cclccl}
\mbox{roots of $D_4$}&f_\alpha\;&
\mbox{roots of $E_6$}\;&
\mbox{roots of $D_4$}&f_\alpha\;&
\mbox{roots of $E_6$}\\
e_1-e_2&\lambda&h_{23}=-x_1+x_2\quad&
e_1+e_2&\lambda\mu\nu\rho^2&h_{145}=x_3+x_4\\
e_1-e_3&\lambda\rho&h_{24}=-x_1+x_3&
e_1+e_3&\lambda\mu\nu\rho&h_{135}=x_2+x_4\\
e_1-e_4&\lambda\nu\rho&h_{25}=-x_1+x_4&
e_1+e_4&\lambda\mu\rho&h_{134}=x_2+x_3\\
e_2-e_3&\rho&h_{34}=-x_2+x_3&
e_2+e_3&\mu\nu\rho&h_{125}=x_1+x_4\\
e_2-e_4&\nu\rho&h_{35}=-x_2+x_4&
e_2+e_4&\mu\rho& h_{124}=x_1+x_3\\
e_3-e_4&\nu&h_{45}=-x_3+x_4&
e_3+e_4&\mu&h_{123}=x_1+x_2\\
\end{array}$$
{#tabwe6b}
$$\begin{array}{rlrlrl}
D_4{\rm -weight}& E_6{\rm -root} & D_4{\rm -weight}& E_6{\rm -root}&D_4{-\rm
weight}& E_6{\rm -root}\\
\epsilon_1&h_{345}&(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2 &
h_{16}&(-\epsilon_1-\epsilon_2-\epsilon_3+\epsilon_4)/2 & h_{56} \\
\epsilon_2&h_{245} &(\epsilon_1+\epsilon_2-\epsilon_3-\epsilon_4)/2 &
h_{236}&(-\epsilon_1-\epsilon_2+\epsilon_3-\epsilon_4)/2 & h_{46} \\
\epsilon_3&h_{235}&(\epsilon_1-\epsilon_2+\epsilon_3-\epsilon_4)/2
&h_{246}&(-\epsilon_1+\epsilon_2-\epsilon_3-\epsilon_4)/2&h_{36}\\
\epsilon_4&h_{234}&(\epsilon_1-\epsilon_2-\epsilon_3+\epsilon_4)/2
&h_{256}&(\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2&h_{26}\\
-\epsilon_1&h_{12}&(-\epsilon_1+\epsilon_2+\epsilon_3-\epsilon_4)/2
&h_{346}&(-\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4)/2&h_{126}\\
-\epsilon_2&h_{13}&(-\epsilon_1+\epsilon_2-\epsilon_3+\epsilon_4)/2
&h_{356}&(\epsilon_1-\epsilon_2+\epsilon_3+\epsilon_4)/2&h_{136} \\
-\epsilon_3&h_{14}&(-\epsilon_1-\epsilon_2+\epsilon_3+\epsilon_4)/2
&h_{456}&(\epsilon_1+\epsilon_2-\epsilon_3+\epsilon_4)/2&h_{146} \\
-\epsilon_4&h_{15}&(-\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4)/2
&h&(\epsilon_1+\epsilon_2+\epsilon_3-\epsilon_4)/2&h_{156}\\
\end{array}$$
$W(E_6)$-representations.
-------------------------
In the notation of Frame [@Fra], the (unique) 10 dimensional representation $V$ of $W(E_6)$ is denoted by $10_s$. One has: $$\begin{array}{rcl}
Sym^2(10_s)&=&1+15_m+15_q+24_p,\\
Sym^3(10_s)&=&20_s+2\cdot 30_m+2\cdot 30_p+80_s,\\
Sym^4(10_s)&=&2\cdot 1+1_n+3\cdot 15_m+4 \cdot 15_q+20_p+20_s+\ldots,\\
Sym^5(10_s)&=&2\cdot 6_p+ 2\cdot 6_n+15_p+15_q+7\cdot 30_m+7\cdot 30_p+\ldots,\\
Sym^6(10_s)&=&5\cdot 1+3\cdot 1_n + 11\cdot 15_m+14\cdot 15_q+\ldots,
\end{array}$$ here $6_p$ is the standard $6$-dimensional representation and $6_n$ is the tensor product of $6_p$ with its determinant. On ${{\bf P}}^5$ the representations $6_p$ and $6_n$ are the same. In particular, there are two 1-dimensional families of $6$-dimensional representations in $S^5V$.
[AMR]{}
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---
author:
- 'W. Wang, M.G. Lang, R. Diehl, H. Halloin, P. Jean, J. Knödlseder, K. Kretschmer, P. Martin, P. Roques, A.W. Strong, C. Winkler, and X.L. Zhang'
date: Received
title: 'Spectral and intensity variations of Galactic [$^{26\!}$Al ]{}emission'
---
[Gamma-ray line emission from the radioactive decay of [$^{26\!}$Al ]{}reflects nucleosynthesis in massive stars and supernovae. We use INTEGRAL [$^{26\!}$Al ]{}measurements to characterize the distribution and characteristics of [$^{26\!}$Al ]{}source regions throughout the Galaxy.]{} [The spectrometer SPI aboard INTEGRAL has accumulated over five years of data on [$^{26\!}$Al ]{}gamma-ray emission from the Galactic plane. We analyzed these data using suitable instrumental-background models and adopted sky distribution models to produce high-resolution [$^{26\!}$Al ]{}spectra of Galactic emission, spatially resolved along the Galaxy plane.]{} [We detect the [$^{26\!}$Al ]{}line from the inner Galaxy at $\sim 28\sigma$ significance. The line appears narrow, and we constrain broadening in the source regions to $<1.3$ keV ($2\sigma$). Different sky distribution models do not significantly affect those large-scale results. The [$^{26\!}$Al ]{}intensity for the inner Galaxy is derived as $(2.9\pm 0.2)\times 10^{-4}\
\mathrm{ph\ cm^{-2}\ s^{-1}\ rad^{-1}}$, consistent with earlier results from COMPTEL and SPI data. This can be translated to an [$^{26\!}$Al ]{}mass of $2.7\pm 0.7$ [$M_\odot$ ]{}in the Galaxy as a whole. The [$^{26\!}$Al ]{}intensity is also confirmed to be somewhat brighter in the 4th than in the 1st quadrant (ratio $\sim 1.3\pm 0.2$). [$^{26\!}$Al ]{}spectra separately derived for regions along the Galactic plane show clear line centroid shifts, attributed largely to the Galaxy’s large-scale rotation. The [$^{26\!}$Al ]{}line toward the direction of the Aquila region ($20\degr < l < 40\degr$) appears somewhat broadened. Latitudinal variations of [$^{26\!}$Al ]{}emission towards the inner Galaxy are studied, finding a latitudinal scale height of $130^{+120}_{-70}$ pc $(1\sigma)$ for [$^{26\!}$Al ]{}in the inner Galaxy and a hint (3$\sigma$) of peculiar [$^{26\!}$Al ]{}emission towards the region $l<0^\circ,\ b>5^\circ$. ]{}
Introduction
============
The unstable isotope [$^{26\!}$Al ]{}has a mean lifetime of 1.04 Myr. [$^{26\!}$Al ]{}first decays into an excited state of $^{26}$Mg, which de-excites into the $^{26}$Mg ground state by emitting gamma-ray photons with the characteristic energy of 1809 keV . The 1809 keV gamma-ray line from radioactive [$^{26\!}$Al ]{}serves as a tracer of the recent nucleosynthesis activity in the Galaxy.
The 1809 keV $\gamma$-ray line emission from the Galaxy was first detected with the Ge spectrometer on the HEAO-C space experiment (Mahoney et al. 1982). The Compton Observatory sky survey 1991-2000 then with COMPTEL imaging of the [$^{26\!}$Al ]{}line across the sky showed that [$^{26\!}$Al ]{}emission extends along the Galactic plane, thus clearly establishing [$^{26\!}$Al ]{}nucleosynthesis as a widely-distributed Galactic phenomenon (Diehl et al. 1995, Plüschke et al. 2001). The structure of this emission, alignments of emission maxima with spiral-arm tangent, and comparisons with tracers of candidate [$^{26\!}$Al ]{}sources, all point to a predominant origin of [$^{26\!}$Al ]{}in massive stars (Prantzos and Diehl 1996, Chen et al. 1995, Diehl et al. 1995, Knödlseder 1999).
The detailed study of [$^{26\!}$Al ]{}line emission from the Galaxy is one of the main science goals of the INTEGRAL mission. SPI aboard INTEGRAL is a high-resolution spectrometer with energy resolution of 3 keV (FWHM) at 1809 keV, which therefore adds high-resolution spectroscopic information to [$^{26\!}$Al ]{}astronomy. The Compton Observatory in its early design phase included a fifth instrument ‘GRSE’, a high-resolution spectrometer later abandoned for cost and complexity reasons. COMPTEL as a scintillation-detector based instrument had modest spectral resolution of about 10% (FWHM). The detailed measurement of [$^{26\!}$Al ]{}line position and shape is expected to reveal more information beyond the COMPTEL imaging survey about the [$^{26\!}$Al ]{}sources and their location through the Doppler effect, induced from Galactic rotation and dynamics of the ejected [$^{26\!}$Al ]{}as it propagates in the interstellar medium around its stellar sources.
Earlier INTEGRAL analysis had used 1.5 years of SPI data to first explore the large-scale spectral characteristics of [$^{26\!}$Al ]{}emission in the inner Galaxy (Diehl et al. 2006a, 2006b). A detection of the [$^{26\!}$Al ]{}line from the inner Galaxy with a significance of $\sim
16\sigma$ confirmed the narrowness of the [$^{26\!}$Al ]{}line (FWHM $<2.8$ keV, $2\sigma$), which had already been seen by RHESSI (Smith et al. 2003) and HEAO-C (Mahoney et al. 1984) earlier. We use INTEGRAL/SPI data accumulated from five years to extend this study towards spatially-resolved details of [$^{26\!}$Al ]{}line spectroscopy across the inner regions of the Galaxy. In this paper, we concentrate on the spectral and intensity variations of [$^{26\!}$Al ]{}emission along the Galactic plane. The main goal is to explore if there are variations from the large-scale properties of [$^{26\!}$Al ]{}emission, such as bulk motion or enhanced turbulence in specific regions. Our study will be limited by the brightness of the signal per region, and hence proceed from broad to more constrained regions along the plane, as the current exposure and [$^{26\!}$Al ]{}brightness allows; with more exposure, in particular towards outer regions of the Galaxy at higher longitudes, the extended INTEGRAL mission is expected to eventually allow such studies up to the spatial resolution of about 2.8$^{\circ}$ (FWHM) of the SPI instrument. We also derive spectra for separate regions of Galactic latitudes, to probe the symmetry and the scale height of Galactic [$^{26\!}$Al ]{}emission.
In this paper, we first describe the SPI observations and methods of data analysis. Then we proceed with an update of the large-scale characteristics of Galactic [$^{26\!}$Al ]{}emission, before refining the spatial resolution of our study. We discuss the implications of our findings with respect to the large-scale distribution of [$^{26\!}$Al ]{}sources, the interstellar medium in their vicinity, and possible regional deviations. Studies of specific regions and their [$^{26\!}$Al ]{}emission are underway and will be reported in separate papers (Cygnus – Martin et al., in preparation; Orion – Lang et al., in preparation; Sco-Cen – Diehl et al., in preparation).
![Exposure of the sky in Galactic coordinates (the number at the color bar in units of ksec) for the data selected from 5-year SPI observations for our [$^{26\!}$Al ]{}study (INTEGRAL orbits 43 – 650). The database covers the whole sky, with 29736 individual pointings, equivalent to a total (deadtime-corrected) exposure time of 61 Ms. Observations emphasize the Galactic plane, specially in the inner Galaxy region, but also specific regions of interest such as Cas A, Cygnus, Carina-Vela, Orion, Virgo, and the Crab pulsar and nebula.](f1_exposure.eps){width="8.5cm"}
Observations and Data Analysis
==============================
SPI observations
----------------
The INTEGRAL spacecraft was launched on October 17, 2002, into a high-inclination, high-eccentricity orbit intended to avoid the increased background from the Earth’s trapped radiation belts. INTEGRAL’s orbital period is $\sim$ 3 days. The spectrometer SPI consists of 19 Ge detectors actively shielded by a BGO anti-coincidence shield. It has a tungsten coded mask in its aperture which allows imaging at $\sim 2.8^{\circ}$ resolution within a $16^{\circ} \times 16^{\circ}$ full coded field of view (imaging on INTEGRAL is mainly performed at lower energies by the IBIS telescope, with which SPI is co-aligned). The Ge detectors are sensitive to gamma-rays between 15 keV and 8 MeV, with a total effective area $\sim 70$ cm$^{2}$ at 1 MeV, and achieve an energy resolution of $\sim 2.5$ keV at 1 MeV (Roques et al. 2003, Attié et al. 2003). However, cosmic-ray (CR) impacts degrade this resolution over time, and the instrument SPI is periodically switched off for 14 days twice a year while annealing (by heating from cryogenic temperatures $\sim$ 80 K to 100$^{\circ}$C) is applied to the detectors to restore the energy resolution back to its pre-launch value by thermal curing heating of the CR-induced defects (Roques et al. 2003, Leleux et al. 2003).
In space operations, INTEGRAL with its IBIS and SPI telescopes is pointed with a fixed attitude for intervals of typically $\sim
2000$ s (referred to as [*pointings*]{}), which are successively arranged as a standard pattern of neighbouring pointings $\simeq
2^\circ$ apart ([*dithering*]{}), and covering target region of interest for improved imaging (Jensen et al. 2003; Courvoisier et al. 2003).
We obtain a database from the 5-year SPI observations, i.e. from the INTEGRAL orbits 43 – 650, which encompasses 29736 pointings of the spacecraft and its instruments across the sky, equivalent to a total deadtime-corrected exposure time of 61 Ms (see the exposure map in Figure 1).
The instrument operation has been interrupted from a few short anomalies and the occurrences of solar flares, and most significantly for the typically two-week annealing episodes described above, and by the regular perigee passages with switch-offs due to the high background intensity from the radiation belts. The sensitivity of our observations were further reduced by the failure of two of the 19 detectors (December 2003 and July 2004).
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Data Analysis
-------------
Our data analysis steps include: (1) selection and assembly of data which are free of contaminations by anomalies and, e.g., solar-flare events; (2) modelling the instrumental background; (3) deriving spectra by fitting the measured data in narrow energy bins with our background model and a model of the spatial distribution of celestial gamma-ray emission, folded through the instrumental response into the data space of the measurement. These steps have been described in several papers, e.g. Strong et al. (2005), Diehl et al. (2003), and as applied to gamma-ray lines specifically in Wang et al. (2007) and Diehl et al. (2006b). Here we give a concise summary and add comments as relevant for the studies of this paper.
We first apply a selection filter to reject corrupted or invalid data. This is done for data sections called ‘science window’, typically a time interval of $\sim 30$ minutes corresponding to one pointing. We apply selection windows to ‘science housekeeping’ parameters such as the count rates in several background tracers, proper instrument status codes, and orbit phase. For tracing background, we employ the SPI plastic scintillator anticoincidence counter (PSAC), and the rate of saturating events in SPI’s Ge detectors (i.e., events depositing $>$ 8 MeV in a single Ge detector; hereafter referred to as GEDSAT rates, see Figure 2). This leads to exclusions of solar-flare periods and other periods of clearly increased / abnormal backgrounds. Additionally, regular background increases during and after passages through the Earth’s radiation belts are eliminated through a 0.05–0.916 window on orbital phase.
From the selected events, spectra are accumulated per each detector and pointing in energy bins of 0.5 keV in the spectral range of 1785 – 1826 keV, and together with dead time and pointing information assembled into the analysis database. From identically-selected data, we also establish a database for the background modelling from the adjacent continuum, combining broader spectral ranges on both sides of the [$^{26\!}$Al ]{}line into single bins.
We select events which trigger one and only one of our 19 Ge detectors (’single events’, SE). We exclude the $\simeq$50% fraction of ’multiple’ events, because for events interacting in more than one detector different combinations of single-detector energies with their correspondingly-different resolutions would be superimposed for identical values of total energy, thus leading to an ill-determined spectral energy response. This reduces the total signal obtained, but guarantees that the spectral response of the instrument is well-defined for our dataset.
Instrumental background is dominated by a rather smooth spectral continuum, with instrumental lines superimposed - for our case, a broad instrumental-line feature centered at 1810 keV comprises 30% of the count rate (see Fig. 2 in Diehl et al. 2006b). For our model of the continuum background, we make use of the simultaneously-measured events in energy bands adjacent to the [$^{26\!}$Al ]{}line region, at 1785 – 1802 keV plus 1815 – 1826 keV, and use each detector’s count rates in this adjacent range to normalize the GEDSAT time series with their superior statistical precision at regular intervals of several days. In Figure 2 we show both the original GEDSAT count rate history and the background model for one of our detectors, as it varies over the dataset. It is seen that anomalous spikes of our background are excluded from our data, and that there still is significant temporal structure in our fitted background model. This ‘adjacent-energies’ background variation model per detector thus is based on high-statistics GEDSAT rates over short time scales of $\sim$ 100 pointings, as these have been found to trace the prompt CR activation rather well. Adjusting these ’templates’ to the actual counts per detector in the 0.5 keV bins used for spectral analysis of the sky signal then ensures that over longer time scales (typically 3 days, where counts in those narrow bins are statistically sufficient) any second-order deviations from GEDSAT tracing actual backgrounds are accounted for. Through this procedure we ensure that pointing-to-pointing variations are given by our background tracer, independent of the coded-mask orientation on the sky. The broad instrumental line feature centered at 1810 keV is actually found to also be rather well modelled by these 0.5 keV bin adjustments, indicating that the line and continuum backgrounds have similar temporal behaviour.
The cosmic-signal spectra are obtained from our measured database of detector spectra when we combine above background models with a spatial model for sky emission (Figure 3), allowing for adjustments of intensity parameters for background and sky intensities per energy bin:
D\_[e,d,p]{}=\_[m,n]{}\_[j=1]{}\^[k\_1]{} A\_[e,d,p]{}\^[j,m,n]{}\_s\^j I\_j\^[m,n]{} + \_t \_[i=1]{}\^[k\_2]{}\^i\_[b,t]{} B\^i\_[e,d,p]{} + \_[e,d,p]{}, where [*e,d,p*]{} are indices for data space dimensions: energy, detector, pointing; [*m,n*]{} indices for the sky dimensions (galactic longitude, latitude); $A$ is the instrument response matrix, $I$ is the intensity per pixel on the sky, $k_1$ is the number of independent sky intensity distribution maps; $k_2$ is the number of background components, $\delta$ is the count residue after the fitting. The coefficients $\beta_s$ for the sky map intensity are constant in time, while $\beta_{b,t}$ is allowed time dependent, to cater for different background normalizations for each camera configuration of 19/18/17 functional detector elements. The sky brightness amplitudes $\beta_s$ comprise the resultant spectra of the signal from the sky. We generally use a maximum-likelihood fitting method (implemented in a SPI standard tool called [*spimodfit*]{}, properly accounting for Poisson statistics in our spectra; for more details, see Strong et al. 2005).
We thus obtain per energy bin the fitted intensity parameter values with their uncertainties, their covariance matrices, and the fit residuals. Analysis of residuals, uncertainties, and covariance matrices are made to determine the validity of the fit of such a combined model to our measured set of spectra. Residuals after the model fitting are shown in Figure 4 (residuals with the time and energies). Reduced $\chi^2$ values around 1.0 confirm that both our background model and the sky distribution model(s) are adequate.
We use the sky intensity distribution of $^{26}$Al as derived from 9 years of COMPTEL observations (maximum entropy map, Plüschke et al. 2001) as the standard reference sky model. We compare this to different [$^{26\!}$Al ]{}emission tracer models or maps, to assess the impact of different sky distribution models on the [$^{26\!}$Al ]{}line shape and intensity (see §3). For simultaneous fits of sets of partial-sky distribution models (in order to derive spatially-resolved spectra), we use analytical forms of sky distribution models with prescribed symmetry to avoid biases to such analysis (see below).
After we obtain fitted-amplitude coefficient spectra for the sky components, we characterize these spectra and in particular the [$^{26\!}$Al ]{}line details therein through two different approaches. When the sky signal is weak, we fit the spectra with Gaussians plus a linear residual background, and use the Gaussian intensities, centroid energies, and FWHM widths for relative comparisons, such as trends along the plane of the Galaxy. When the signal is sufficiently strong so that we are sensitive to line width details, we describe the line component not by a single Gaussian any more, but rather by the convolution of a Gaussian with the asymmetric instrumental line response as it develops from degradation of our detectors’ resolution and their periodic restorations through annealings. This allows us to infer immediate information about the celestial [$^{26\!}$Al ]{}dynamics, which we identify with this Gaussian, and in particular its width, which arises from Doppler shifting of the line energies with motion of decaying [$^{26\!}$Al ]{}nuclei relative to the observer. In both cases, we fit these spectral models to the amplitude values and their uncertainties as determined from the model fit to the large set of spectra, performing a maximum-likelihood fit with the Levenberg-Marquardt algorithm. In the latter case, we derive the entire probability distribution of the fitted parameter values through a Monte Carlo Markov Chain (MCMC) method, which allows us in particular to determine probability constraints for the celestial [$^{26\!}$Al ]{}line broadening from kinematics. This parameter challenges the spectral resolution of our instrument (about 3 keV at the [$^{26\!}$Al ]{}line energy), hence so far is bounded from above only. Such asymmetric parameter probability distributions may be far from Gaussian, hence require such more sophisticated treatment (Kretschmer et al. 2009, in preparation).
[$^{26\!}$Al ]{}in the Inner Galaxy – An Update
===============================================
The spectral characteristics of [$^{26\!}$Al ]{}emission in the inner Galaxy serve to study the current nucleosynthesis activity and the properties of the interstellar medium near the [$^{26\!}$Al ]{}sources on a large-scale averaged scale. We define the “inner Galaxy” as the region $-30^\circ < l < 30^\circ, \ -10^\circ < b <10^\circ$), and may use this as a representative region for this purpose, since it coincides with the bright ridge of observed 1809 keV emission as observed along the plane of the Galaxy. We specifically exclude regions at higher longitudes such as Cygnus: This region in particular has been recognized as special, in its supernova to Wolf-Rayet ratio, for example (Knödlseder et al. 2004); other emission at large longitudes also may be deviant from the Galactic average and over-emphasized because nearby. We therefore believe that using the $-30^\circ < l < 30^\circ$ region will give us a more representative picture of [$^{26\!}$Al ]{}source environments in the Galaxy.
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.
Fitting our set of observations to the sky intensity distribution of the [$^{26\!}$Al ]{}maximum entropy image (MaxEnt) from COMPTEL (Figure 3, left), we obtain the updated inner-Galaxy [$^{26\!}$Al ]{}emission spectrum shown in Figure 5. MaxEnt is one possible sky intensity distribution compatible with the COMPTEL data. The [$^{26\!}$Al ]{}line is detected at $\sim 28\sigma$ significance.
The [**[$^{26\!}$Al ]{}gamma-ray flux**]{} from the inner Galaxy turns out as $(2.93\pm 0.15)\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$rad$^{-1}$. This is consistent with our earlier values $(3.3\pm 0.4)\times 10^{-4}\
\mathrm{ph\ cm^{-2}\ s^{-1}\ rad^{-1}}$ (Diehl et al. 2006a, 2006b), and also with the COMPTEL imaging-analysis value of $(2.8\pm 0.4)\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$rad$^{-1}$ (Plüschke et al. 2001). The above value is derived using an asymmetric line shape as best matching our expectations from SPI’s spectral response and eventual additional celestial line broadening the total [$^{26\!}$Al ]{}gamma-ray flux of the Gaussian fit as determined for the inner Galaxy region is $(2.73\pm 0.17)\times 10^{-4}\ \mathrm{ph\ cm^{-2}\ s^{-1}\
rad^{-1}}$.
In our model-fitting approach to derive [$^{26\!}$Al ]{}spectra from SPI spectra per pointing (Eq. 1), the adopted sky distribution models may affect the [$^{26\!}$Al ]{}flux and line shape results. We estimate the variations and potential systematic uncertainties introduced by the spatial distribution model of the [$^{26\!}$Al ]{}emission through variation of this sky distribution model within a plausible range.
Our “standard” is the measurement of [$^{26\!}$Al ]{}gamma-ray emission directly, from COMPTEL. Uncertainties from COMPTEL imaging analysis have however been shown to allow for a range of images which are consistent with the COMPTEL measurements (Knödlseder et al. 1999). We consider the MaxEnt image the compromise between making use of COMPTEL’s imaging resolution of about 3.8$^\circ$ (FWHM) and the attempts to suppress artifacts from statistical noise. A more conservative COMPTEL image is obtained from the Multi-resolution Expectation Maximization method (MREM), which carefully eliminates noise contributions at each iteration and builds up the image starting from large spatial scales, terminating once the image obtained does not need further refinement in this statistical sense (Knödlseder et al. 1999; Knödlseder 1999). The smooth MREM image from COMPTEL also shows the inner Galactic ridge as well as Cygnus being bright in [$^{26\!}$Al ]{}emission, yet does not show the few-degree scale features of the MaxEnt map; we consider these two maps as adequate tests for systematics from the range of direct [$^{26\!}$Al ]{}measurements.
Considering the limitations of gamma-ray telescopes, it has been plausible to alternatively use maps obtained in astronomically-more developed wavelength bands, once it it clear that those trace [$^{26\!}$Al ]{}sources in the Galaxy. Detailed studies have shown (Knödlseder et al. 1999; Diehl et al. 1996) that among the best tracers of [$^{26\!}$Al ]{}sources are (1) the infrared emission of warm dust grains, mapped with the COBE/DIRBE and arising from radiative heating of dust around clusters of massive stars (Bennett et al. 1996), and (2) radiation of free electrons (free-free emission, Bremsstrahlung) observed at radio frequencies with the WMAP satellite, arising from the ionizing massive-star radiation around massive star clusters. From astrophysical arguments, also maps of interstellar gas in different forms should trace the locations and space density of [$^{26\!}$Al ]{}sources. (3) Molecular gas is observed through CO line emission at radio frequencies, and has been mapped in rather fine resolution (Dame et al., 1987 and 2001), (4) atomic hydrogen (HI) sky surveys have been accumulated (e.g. Dickey and Lockman 1990), and (5) cosmic-ray interactions with interstellar gas produces penetrating continuum gamma-ray emission which has been mapped in the EGRET sky survey (Hunter et al. 1997).
Finally, analytical models for the distribution of [$^{26\!}$Al ]{}sources have been constructed, based on above knowledge of Galactic structure in its different components (Robin et al. 2003), properly weighted from astrophysical arguments. (6) Double-exponential functions (in galactocentric radius, and scale height above the Galactic plane) have been constructed, as well as more sophisticated models including (7) spiral structure components and building on the distribution of free electrons in the Galaxy as derived from pulsar dispersion measurements (Taylor & Cordes 1993, and Cordes & Lazio 2002). The scale height of [$^{26\!}$Al ]{}sources has been found to lie between the molecular disk (about 50 pc) and the thick disk (about 0.3-1 kpc), with plausible values around 200 pc.
We compare [$^{26\!}$Al ]{}line spectra determined from these different models and tracers of [$^{26\!}$Al ]{}sources in the Galaxy. Fifteen different sky distribution maps have been analyzed, and variations on [$^{26\!}$Al ]{}brightness, [$^{26\!}$Al ]{}line centroid and width parameters are shown in Figure 6. Systematic variations are within statistical uncertainties, when we vary the spatial models for [$^{26\!}$Al ]{}emission. We use their scatter to estimate a “systematic” uncertainty, which turns out as $(2.9\pm 0.2)\times 10^{-4}{\rm ph\ cm^{-2}\
s^{-1}\ rad^{-1}}$, $1809.0\pm 0.09$ keV, $0.5 \pm 0.45$ keV for the flux, centroid, and width parameters (with 1$\sigma$ error bars), separately.
As before (Diehl et al., 2006a), we convert our measured [$^{26\!}$Al ]{}intensity into an estimate of the [**total current [$^{26\!}$Al ]{}mass**]{} in the Galaxy, using an assumed geometrical source-distribution model to extrapolate across the entire Galaxy from our inner-Galaxy normalization of such a model, as discussed above. This is required because the flux to mass conversion relies on [$^{26\!}$Al ]{}source distances, not measured directly in projected sky brightness distribution maps. Several three-dimensional distribution models are applied and compared here, e.g., an exponential disk model, a “young-disk” model, a geometrical representation based on dust emission, and a multi-component model including spiral-arm structures and based on abundances of free electrons in interstellar space. We vary scale height parameters of the appropriate components to also study the latitude extent of the [$^{26\!}$Al ]{}emission. In all models, we have taken the distance of the Sun to the Galactic center as $R_0=$ 8.5 kpc ([$^{26\!}$Al ]{}mass sensitively depends on $R_0$, smaller $R_0$ will globally reduce the size of the Galaxy, and result in the smaller amount of [$^{26\!}$Al ]{}).
The average measured [$^{26\!}$Al ]{}flux of $(2.9\pm 0.2) \times 10^{-4}\
\mathrm{ph\ cm^{-2}\ s^{-1}\ rad^{-1}}$ for the inner Galaxy thus translates into a Galactic [$^{26\!}$Al ]{}mass of $(2.7\pm 0.7)$ [$M_\odot$ ]{}using a plausible scale height of 180 pc. If we ignored the ejection of [$^{26\!}$Al ]{}into surrounding cavities and corresponding champagne flows (see our own scale height determination below), and used the lower scale heights of O stars or of the molecular disk, we would obtain lower total amounts around or even below 2 [$M_\odot$ ]{}. This emphasizes the need for spatially-resolved [$^{26\!}$Al ]{}studies (see Sect 5 and Fig. 12 below). The quoted uncertainty here includes both the statistical uncertainty as propagated from the fitting method, and the systematical uncertainty derived from variations among plausible representations of the sky distribution of the emission plus alternative 3-dimensional [$^{26\!}$Al ]{}source distribution models (see also Figure A4.1 in Supplemental materials of Diehl et al. 2006a).
The [**line width**]{} of [$^{26\!}$Al ]{}conveys global information about the spread in projected velocities of [$^{26\!}$Al ]{}nuclei when they decay and emit characteristic 1809 keV gamma-rays. Astrophysical origins of [$^{26\!}$Al ]{}line width may come from two effects: random motions in the interstellar medium (Chen et al. 1997) and Galactic differential rotation (Kretschmer et al. 2003). A line broadening of $\sim$ 1 keV from our line-shape constraints corresponds to thermal Doppler velocities of $\sim$ 120 km s$^{-1}$.
The Gaussian used to represent the [$^{26\!}$Al ]{}line (Fig. 5 left) shows a width of $3.16\pm 0.15$ keV (FWHM), consistent with the instrumental width determined as $\sim 3.1$ keV near 1809 keV from nearby instrumental lines at 1764 and 1779 keV. This indicates that the cumulative [$^{26\!}$Al ]{}line emission in the inner Galaxy is intrinsically rather narrow in energy around the laboratory value, with little kinematic Doppler broadening, well below SPI’s energy resolution.
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For such a strong signal, we can be more ambitious, however, and attempt to quantitatively constrain the intrinsic (celestial) line broadening in a statistical probability analysis as described above (see also Diehl et al. 2006b and Kretschmer et al. 2004). We know the instrumental line response of our instrument at each epoch of the mission from detailed studies of a large number of instrumental lines. The response gradually degrades and develops an increasing tail component on the low-energy side of the photopeak energy, due to incomplete charge collection as the detector’s Ge lattice experiences damaging from cosmic-ray impacts. Annealings restore better charge collection and a symmetric instrumental-line response whose width is determined and dominated by statistics of the charge collection. We assemble the effective instrumental-line response for our study through appropriate weightings of these time-dependent responses. Then we fit the convolution of a parameterized Gaussian with this response shape function, and thus derive values for the celestial [$^{26\!}$Al ]{}line centroid energy and broadening. MCMC sampling of the probability distributions of these parameters then allows us to translate the parameter value uncertainties into probability constraints by integrating over these distributions. In particular, the probability distribution of the intrinsic width may not peak at zero, indicating a small but by itself non-detectable broadening of the line. In this case, our approach yields a reliable estimate of the upper bound on line broadening, which is the astrophysically-relevant quantity to constrain kinematics of decaying [$^{26\!}$Al ]{}nuclei in the interstellar medium. The fitted parameters are the line centroid, the intrinsic width of celestial [$^{26\!}$Al ]{}, the intensity of the line, and two parameters for the underlying continuum (see Figure 5 right). The intrinsic line width is now constrained to $<1.3$ keV (2 $\sigma$, which is consistent and yet significantly smaller than the earlier constraints based on fewer SPI data (Diehl et al. 2003, 2006b).
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In the inner Galaxy, Galactic differential rotation alone can lead to significant Doppler shifts towards specific longitudes where the projected-velocity differences with respect to the solar orbit reach maxima; line broadening results if we integrate over a larger longitude range with different bulk velocity differences. Kretschmer et al. (2003) have simulated the [$^{26\!}$Al ]{}line shape diagnostics in the inner Galaxy due to Galactic rotation and [$^{26\!}$Al ]{}ejection from sources, and find that line broadening of up to 1 keV is expected if the signal is integrated over the inner region of the Galaxy. Our present large-scale line-shape constraints are consistent with these expectations.
If we interpret line broadening of the [$^{26\!}$Al ]{}line from the inner Galaxy in terms of interstellar-medium characteristics, the intrinsic-width constraint of $<1.3$ keV corresponds to 160 km s$^{-1}$ as a corresponding 2$\sigma $ limit on ISM velocities. This is well within the plausible and acceptable range for the environment of normal interstellar-medium turbulence (Chen et al. 1997), leading us to conclude that, within uncertainties, the average velocities of decaying [$^{26\!}$Al ]{}in the Galaxy are not abnormally-high (compare discussion after Naya et al. 1996 in Chen et al. 1997).
In summary, the measured line width of [$^{26\!}$Al ]{}from the inner Galaxy is consistent with Galactic rotation and modest interstellar-medium turbulence around the sources of [$^{26\!}$Al ]{}. This confirms earlier results on the [$^{26\!}$Al ]{}line width from HEAO-C (Mahoney et al. 1984), RHESSI (Smith 2003), and SPI on INTEGRAL (Diehl et al. 2006b). The GRIS balloon experiment had reported a very broad line with a width $\sim 5.4$ keV (Naya 1996), which is inconsistent with these measurements and clearly ruled out. This is reassuring: within uncertainties, the average velocities of decaying [$^{26\!}$Al ]{}in the Galaxy are not abnormally-high (see discussion in Chen et al. 1997).
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The relative position of [**the [$^{26\!}$Al ]{}gamma-ray line centroid**]{} with respect to the laboratory-determined [$^{26\!}$Al ]{}decay energy value of 1808.65 $\pm 0.07$ keV (Firestone & Ekström 2004) yields information on potential bulk motion of [$^{26\!}$Al ]{}in the Galaxy. The [$^{26\!}$Al ]{}line centroid energy from the whole inner Galaxy is determined at $1808.92\pm 0.06$ keV from the Gaussian fit and $1809.03\pm 0.08$ keV from the instrumental-response-convolved fit, even after considering uncertainties, the measured centroid energy is higher than the laboratory value. The effect of this blueshift for the whole inner Galaxy will be discussed below.
In our determination of [$^{26\!}$Al ]{}line properties, we represent the spectra derived from detailed model fitting by a line component for [$^{26\!}$Al ]{}plus a linear component which could capture any underlying systematics in our background or sky modellings. We expect that our adjacent-energy background model component will eliminate Galactic continuum emission, at least to first order, if the spectral shape is rather flat across our 40 keV energy range (1785–1826 keV). The diffuse gamma-ray continuum in the inner Galaxy is $\sim 2\times 10^{-6}\ \mathrm{ph\ cm^{-2}\ s^{-1}
rad^{-1}\ keV^{-1}}$ in the energy band of 1 – 2 MeV (Strong et al. 1999). Indeed, we find that offsets above zero in our spectra are rather small and negligible, supporting this property of our background model.
[$^{26\!}$Al ]{}emission along Galactic longitudes
==================================================
The 9-year COMPTEL imaging of [$^{26\!}$Al ]{}line emission had already suggested some asymmetry in the inner Galaxy: the fourth Galactic quadrant appears somewhat brighter than the first quadrant (Plüschke (2000) finds a significance of 2.5$\sigma$ for a brightness difference). COMPTEL could provide the image details of [$^{26\!}$Al ]{}in the Galaxy, but no significant spectral information due to its spectral resolution of about 150 keV near the [$^{26\!}$Al ]{}line. SPI with its Ge detectors features sufficiently-high spectral resolution to allow astrophysical constraints from [$^{26\!}$Al ]{}line shapes, averaged over the Galaxy as discussed above, but also for different regions along the Galactic plane due to its imaging properties as a coded-mask telescope.
![[$^{26\!}$Al ]{}spectrum from the Galactic center ($-5^\circ<l<
5^\circ,\ -10^\circ<b< 10^\circ$). The determined line centroid energy ($1808.58\pm 0.27$ keV) is consistent with the laboratory value.](f9_HomoDisk_l-5.eps){width="8.5cm"}
In this section, we will proceed towards increasing spatial resolution along the Galactic plane, starting out from testing Galactic asymmetries between the first and fourth quadrant. [$^{26\!}$Al ]{}line parameters toward the different directions of the Galactic plane are determined using separate sky maps covering each sky region, simultaneously fitting these together with our background model to the entire sky survey database. [$^{26\!}$Al ]{}line fluxes, centroid energies, and line widths then are derived by a simple Gaussian fit to the [$^{26\!}$Al ]{}line in the resulting spectra, as we are interested in relative changes between different portions of the sky. This will allow us to identify line shifts from bulk motion such as expected from large-scale Galactic rotation, and hints for additional line broadenings in particular regions, which would reflect increased [$^{26\!}$Al ]{}velocities in such regions. A homogenous disk model (see Figure 2 right, $-60^\circ < l < 60^\circ, \
-10^\circ < b < 10^\circ$, scale height 200 pc ) is used here to avoid a bias of sky distribution models along the Galactic plane, for such relative comparison.
[$^{26\!}$Al ]{}spectra for the 1st ($0^\circ < l < 60^\circ$) and 4th quadrant ($-60^\circ < l < 0^\circ$ ) are presented in Figure 7. In the 4th quadrant we note a blueshift of $0.49\pm 0.07$ keV relative to the centroid energy of [$^{26\!}$Al ]{}line in the laboratory, but no significant redshift in the 1st quadrant is apparent ($\sim
0.04\pm 0.10$ keV). Both spectra have width values compatible with no significant [$^{26\!}$Al ]{}line broadenings. The indicated [$^{26\!}$Al ]{}asymmetry between the two inner Galactic quadrants appears again, with a flux ratio of $\sim 1.3\pm 0.2$.
We proceed further towards more confined Galactic regions using four sub-maps with 30 degree width along Galactic longitude, to obtain [$^{26\!}$Al ]{}spectra for these regions shown in Figure 8: (1) $0^\circ < l < 30^\circ$, (2) $30^\circ < l < 60^\circ$, (3) $-30^\circ < l < 0^\circ$, and (4) $-60^\circ < l < -30^\circ$. Centroid energy shifts of the [$^{26\!}$Al ]{}line are found, $+0.15\pm 0.12$ keV and $-0.61\pm 0.09$ keV in regions (1) and (3) respectively, as expected from large-scale Galactic rotation. The inner region ($-30^\circ < l < 30^\circ$) is seen to be much brighter than the two outer regions of the Galaxy, consistent with the COMPTEL [$^{26\!}$Al ]{}map (note that here we use a homogeneous sky distribution model as the model-fitting prior). The indicated [$^{26\!}$Al ]{}emission asymmetry for the 1st and 4th quadrants also shows up again between regions (3) and (1), with a flux ratio of $\sim 1.15\pm 0.18$. Even further out, region (4) may also be brighter than (2) by $\sim 1.6\pm
0.6$. Since smaller regions include less [$^{26\!}$Al ]{}signal, these differences are, however, not significant.
Challenging the imaging capability of SPI for diffuse and extended emission, we refine spatial structure even more towards smaller longitude intervals. In Figure 9, we show a spectrum representing the Galactic center region ($-5^\circ<l< 5^\circ, -10^\circ
<b<10^\circ$), where the [$^{26\!}$Al ]{}line is still significant ($>
4\sigma$). The line centroid energy is determined at $1808.58\pm
0.27$ keV, and is consistent with the laboratory value – no line shift from bulk motion is indicated.
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"} {width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
Then we also derive spectra for six smaller longitude intervals of 20 degree width along the Galactic plane ($-60^\circ < l <
60^\circ$, see Figure 10). The [$^{26\!}$Al ]{}line is still detected for the inner Galaxy ($-40^\circ < l < 40^\circ$, $> 6\sigma$ for each $20^\circ$ bin region), but only marginal for the two outer regions ($40^\circ < l < 60^\circ$ and $-60^\circ < l <
-40^\circ$, $< 4\sigma$). This may be attributed to both less [$^{26\!}$Al ]{}brightness and to less exposure in these regions, compared to the inner Galaxy.
For the four inner regions, [$^{26\!}$Al ]{}line centroid shifts are observed. When we use the [$^{26\!}$Al ]{}line centroid energy of 1808.65 keV from the laboratory as a reference (rather than our own determination, Fig. 9), we obtain [$^{26\!}$Al ]{}line centroid energy shifts along Galactic longitudes as shown in Figure 11. Evidently, towards positive longitudes of $\sim 0^\circ - 40^\circ$, the redshift in energy is minor, $\sim 0.1$ keV, while for negative longitudes, the blueshift is substantial, $\sim 0.4$ keV for $-20^\circ <l<
0^\circ$ and up to 0.8 keV for $-40^\circ <l< -20^\circ$. This asymmetry of [$^{26\!}$Al ]{}line energy shifts along the Galactic plane is inconsistent with simple azimuthally-symmetric Galactic rotation; it is clear evidence for pronounced spiral-arm structure in the inner Galaxy, and possibly peculiar bulk motion along the Galaxy’s bar (e.g. Englmaier & Gerhard 1999, Hammersley et al. 2007 and references therein).
Figure 11 shows the intensity distribution of [$^{26\!}$Al ]{}emission (from the same 20$^{\circ}$ regions, Figure 10) along the Galactic plane, adding the (longitude-range normalized) Galactic-Center region [$^{26\!}$Al ]{}intensity ($|l|< 5^\circ$, Figure 9) for comparison. The variability of [$^{26\!}$Al ]{}intensity along the Galactic plane again is evident.
We note that the [$^{26\!}$Al ]{}line in the region of $20^\circ < l <
40^\circ$ appears somewhat broadened, with a Gaussian width of FWHM$\sim 4.15\pm 0.75$ keV (also see Figure 11). This may hint towards a peculiar [$^{26\!}$Al ]{}source region towards this direction, which could be associated with the Aquila region (Rice et al. 2006). Broadening could result from higher turbulence if the [$^{26\!}$Al ]{}source region is younger than average and dominated by the [$^{26\!}$Al ]{}ejection from more massive stars (see Knödlseder et al. 2004). Further studies would be interesting, and have the potential to identify star formation otherwise occulted by foreground molecular clouds.
Latitudinal variations of [$^{26\!}$Al ]{}emission
==================================================
The interpretation of [$^{26\!}$Al ]{}imaging and spectral results relies on (uncertain) distances of [$^{26\!}$Al ]{}sources. Along the line-of-sight, the detected [$^{26\!}$Al ]{}signal could originate from local star-formation complexes ($\sim 100$ pc), or from the nearest part of the Sagittarius-Carina arm ($1-2$ kpc), or from the Galactic center region ($\sim 8$ kpc), or even from the distant side of the Galaxy ($> 10$ kpc).
![[**Top:**]{} [$^{26\!}$Al ]{}line energy shifts along the Galactic plane ( $-60^\circ <l<60^\circ$, with longitude bin widths of $20^\circ$, [$^{26\!}$Al ]{}spectra from Figure 10) relative to the line centroid of the [$^{26\!}$Al ]{}line (fitted energy 1808.65 keV). [**Center:**]{} [$^{26\!}$Al ]{}intensity distribution along the Galactic plane (from Figures 9, 10). For comparison, the COMPTEL-derived [$^{26\!}$Al ]{}intensity profile is shown (solid line, and dashed lines when integrated over the same longitude bins). [**Bottom:**]{} [$^{26\!}$Al ]{}FWHM (Gaussian fitting) variation along the Galactic longitudes, a broad [$^{26\!}$Al ]{}line feature is detected toward the longitudes $20^\circ < l <
40^\circ$. ](f11_l_eng.eps "fig:"){width="7.5cm"} ![[**Top:**]{} [$^{26\!}$Al ]{}line energy shifts along the Galactic plane ( $-60^\circ <l<60^\circ$, with longitude bin widths of $20^\circ$, [$^{26\!}$Al ]{}spectra from Figure 10) relative to the line centroid of the [$^{26\!}$Al ]{}line (fitted energy 1808.65 keV). [**Center:**]{} [$^{26\!}$Al ]{}intensity distribution along the Galactic plane (from Figures 9, 10). For comparison, the COMPTEL-derived [$^{26\!}$Al ]{}intensity profile is shown (solid line, and dashed lines when integrated over the same longitude bins). [**Bottom:**]{} [$^{26\!}$Al ]{}FWHM (Gaussian fitting) variation along the Galactic longitudes, a broad [$^{26\!}$Al ]{}line feature is detected toward the longitudes $20^\circ < l <
40^\circ$. ](f11_flux_cBin_c1deg.eps "fig:"){width="7.5cm"} ![[**Top:**]{} [$^{26\!}$Al ]{}line energy shifts along the Galactic plane ( $-60^\circ <l<60^\circ$, with longitude bin widths of $20^\circ$, [$^{26\!}$Al ]{}spectra from Figure 10) relative to the line centroid of the [$^{26\!}$Al ]{}line (fitted energy 1808.65 keV). [**Center:**]{} [$^{26\!}$Al ]{}intensity distribution along the Galactic plane (from Figures 9, 10). For comparison, the COMPTEL-derived [$^{26\!}$Al ]{}intensity profile is shown (solid line, and dashed lines when integrated over the same longitude bins). [**Bottom:**]{} [$^{26\!}$Al ]{}FWHM (Gaussian fitting) variation along the Galactic longitudes, a broad [$^{26\!}$Al ]{}line feature is detected toward the longitudes $20^\circ < l <
40^\circ$. ](f11_fwhm.eps "fig:"){width="7.5cm"}
In this section, we explore a possibility to resolve the [$^{26\!}$Al ]{}signals for local complexes from the large scales of the Galaxy through different latitudinal signatures. For such analysis, we split a homogenous-disk model (Figure 3 right) into sub-maps along Galactic latitudes, and derive separate [$^{26\!}$Al ]{}spectra for the different latitude ranges simultaneously through model fitting to our observations. We study the [$^{26\!}$Al ]{}emission for three intervals along latitudes, $-5^\circ<b<5^\circ $ (low latitudes), $5^\circ<b<20^\circ$ and $-20^\circ<b<-5^\circ$ (intermediate latitudes). [$^{26\!}$Al ]{}emission for low latitudes ($|b|<5^\circ$) should be dominated by a large-scale origin in the Galactic disk, while [$^{26\!}$Al ]{}emission at higher latitudes ($|b|>5^\circ$) could originate from more local star-formation systems such as OB associations in the Gould Belt. This definition is similar to the one used in pulsar population studies (Wang et al. 2005). The Gould Belt is an ellipsoidal shaped ring-like structure delineated by the groups of nearby stellar groups within 1 kpc, with semi-major and minor axes of $\sim 500$ pc and 340 pc, respectively (Perrot & Grenier 2003). The center of this structure is located towards $l=130^\circ$ , displaced from the Sun’s location by about 200 pc (Guillout et al. 1998). The Vela region is located near the outer boundary of the Gould Belt towards $l\sim -90^\circ$ . The nearby Sco-Cen region at about 140 pc distance towards $l\sim -10^\circ,
\ b\sim 10^\circ$ probably also belongs to the Gould Belt. The origin of this structure is debated, between triggered sequential star formation propagating outwards from its center, and an external triggering event such as a high-velocity cloud falling through the Galaxy’s plane (see Perrot & Grenier 2003 and Pöppel 1997).
Figure 12 displays the [$^{26\!}$Al ]{}intensity distribution along Galactic latitudes for $|l|<60^\circ$. No [$^{26\!}$Al ]{}signal is detected at negative latitudes ($-20^\circ<b<-5^\circ$), while weak [$^{26\!}$Al ]{}emission is detected at positive latitudes ($\sim 2\sigma,\
5^\circ<b<20^\circ$). We also derive latitudinally-separated [$^{26\!}$Al ]{}emission for the 1st and 4th quadrants separately again through model fitting, also shown in Figure 11. No [$^{26\!}$Al ]{}signal is detected for off-plane regions of the 1st quadrant, while in the 4th quadrant, [$^{26\!}$Al ]{}emission is still clearly detected (3$\sigma$) in the latitude region $5^\circ<b<20^\circ$. This [$^{26\!}$Al ]{}emission towards $b>5^\circ,\ l<0^\circ$ could be attributed to the nearby Sco-Cen star-formation complex at 140 pc distance.
We also determine the scale height of the Galactic plane in [$^{26\!}$Al ]{}emission by comparing the fit quality for sets of two different plausible geometrical models for the [$^{26\!}$Al ]{}source density distribution in the Galaxy, varying their scale height parameter (Fig. 12 right). We use a Galactocentric double-exponential disk model (ExpDisk) described by $$\rho(R,z)\propto\mathrm{e}^{-\left( \frac{R}{R_0} +
\frac{|z|}{z_0} \right)},$$ where $R$ is the Galactocentric distance within the plane, and $z$ is the height above the plane, with $R_0=3.5$ kpc and $z_0$ as the scale radius and height parameter. Alternatively, we use a model which includes the spiral-arm structure of the Galaxy as derived by Taylor & Cordes (1993); we use only their components for the inner Galaxy and the spiral arms, with identical scale height parameter (in this model, the density perpendicular to the disk is described as sech$\left(z/z_0\right)$). In order to avoid a bias from bright special regions such as Cygnus/Vela/Carina, we restrict this analysis to data within $|l|<60\degr$ and $|b|<30\degr$. Fig. 12 (right) shows the variation of log-likelihood values with different scale heights for both model types. Here, the values for the exponential-disk models have been shifted by $+16$, as the spiral-arm model systematically provides a better description of our data. With $-2\log$L being asymptotically $\chi^2$ distributed, we derive a scale height of $130^{+120}_{-70}$ pc $(1\sigma)$ for the [$^{26\!}$Al ]{}emission in the inner Galactic disk, from the spiral arm model constraints. This confirms previous such studies based on COMPTEL data (Diehl et al. 1998).
{width="8.5cm"} {width="8.5cm"}
Summary and Conclusions
=======================
The study of the [$^{26\!}$Al ]{}line and its details in different regions of the Galaxy is one of the main goals of the SPI spectrometer on INTEGRAL. Using five years of SPI data, we find the [$^{26\!}$Al ]{}signal from the inner Galaxy ($|l|<30^\circ,\ |b|<10^\circ $) with a high significance of $\sim 28\sigma$. The [$^{26\!}$Al ]{}flux integrated over the inner Galaxy of $(2.9\pm 0.2)\times 10^{-4}\ \mathrm{ph\ cm^{-2}\
s^{-1}\ rad^{-1}}$ is consistent with our own and other earlier measurements, though significantly lower than the ones from measurements with instruments which have even more modest spatial resolution on the sky (see Figure 2 of Diehl et al. 2004). Taking the distance of the Sun to the Galactic center as $R_0=$ 8.5 kpc, we convert the measured [$^{26\!}$Al ]{}flux to a Galactic [$^{26\!}$Al ]{}mass of $(2.7\pm 0.7)$ [$M_\odot$ ]{}. We compared different plausible models for the sky distribution of [$^{26\!}$Al ]{}emission and how it may affect the global [$^{26\!}$Al ]{}flux measurement. We find that different sky distribution models do not substantially affect [$^{26\!}$Al ]{}intensity and line shapes for the integrated inner Galaxy, and we have used the observed variations of [$^{26\!}$Al ]{}mass values to estimate a systematic uncertainty, added to the statistical uncertainty from the model fitting, to yield the $\pm 0.7$ [$M_\odot$ ]{}uncertainty quoted.
The [$^{26\!}$Al ]{}line centroid energy appears blue-shifted relative to the laboratory value of $1808.65\pm 0.07$ keV if integrated over the inner Galaxy ($1809.0\pm 0.1$ keV). With refined spatial resolution this turns out to be mostly due to asymmetric bulk motion which we find along the plane of the Galaxy, and attribute to asymmetries in inner spiral arms and to the Galaxy’s bar. In particular, for the central longitude bin towards the Galactic Center, the signal is strong enough to allow for a rather small longitude range integration, and we find a [$^{26\!}$Al ]{}line centroid consistent with the laboratory value and thus with the absence of bulk motion relative to the Sun. Also, here the [$^{26\!}$Al ]{}line appears as most-narrow with an upper limit of 1.3 keV (2$\sigma$). The measured line width of [$^{26\!}$Al ]{}from the large-scale integrated inner Galaxy with an upper limit of 1.3 keV (2$\sigma$) is consistent with expectations from both Galactic rotation and modest interstellar-medium turbulence around the sources of [$^{26\!}$Al ]{}(turbulent velocities constrained below 160 km s$^{-1}$ even when disregarding the effects of galactic rotation). Our line width results are consistent with previous reports by HEAO-C (Mahoney et al. 1984), RHESSI (Smith 2003), but the very broad line with a width $\sim 5.4$ keV reported by GRIS (Naya et al. 1996) is clearly ruled out by our SPI measurements ( $4\sigma$ significance level ).
From our study of [$^{26\!}$Al ]{}emission in spatially-restricted regions along the plane of the Galaxy, we find:
- [$^{26\!}$Al ]{}brightness appears asymmetric for the two inner quadrants, with a flux ratio of the 4th quadrant to the 1st of $\sim 1.3 \pm
0.2$ (Figure 7).
- The [$^{26\!}$Al ]{}line energy varies clearly along the Galactic plane (Figure 11): a minor redshift ($\sim 0.1$ keV) for positive longitudes, but significant blueshift ($\sim 0.4-0.8$ keV) for negative longitudes.
- The [$^{26\!}$Al ]{}line towards the direction of $20^\circ < l < 40^\circ$ shows a hint for additional line broadening.
- The scale height of Galactic-plane [$^{26\!}$Al ]{}emission is $130^{+120}_{-70}$ pc $(1\sigma)$, determined towards the inner Galaxy ($|l|<60\degr$).
- There is a strong hint for [$^{26\!}$Al ]{}emission in the fourth Galactic quadrant at intermediate latitudes $l<0^\circ,\ b>5^\circ$ (Fig.11).
This leads us to several astrophysical implications and the “bar” structures:
- The disk of our Galaxy apparently is not azimuthally-symmetric. The [$^{26\!}$Al ]{}brightnesses for the two inner quadrants appear different, both from our INTEGRAL and from earlier COMPTEL results, favoring the fourth quadrant. [$^{26\!}$Al ]{}line shifts towards the blue in such brighter regions are more pronounced than the redshifts on the fainter side of the disk, unexpected from a simple and symmetric Galaxy model and its large-scale rotation properties. These phenomena may reflect variations in the space distribution of young stars, possibly related to the inner parts of the spiral arms and their interfaces to the “molecular ring” and the “bar” structures. Large non-circular motions have been seen in HI and CO observations (Mulder & Liem 1986; Gerhard & Vietri 1986), and also from the NIR light distribution (Binney et al. 1997), in addition to source count asymmetries (Nikolaev & Weinberg 1997) and non-symmetric gas dynamics (Englmaier & Gerhard 1999). The bar is most clearly traced in infrared emission from dust (Marshall et al. 2008), which suggests a position angle of $\sim 25^\circ$ of the bar with the near end pointing towards us in the first quadrant, and a total length of about 4 kpc. The transition regions between spiral arms and bar are likely to incur star formation (Verley et al. 2007), and affect the dynamics of gas and stars. Simply superimposing a homogeneous star-forming bar to the rotating spiral would, however, not agree with our results - the inner Galaxy structure may be more complex. Localized star-forming regions could lead to peculiar motion of hot gas ejected from winds and supernovae, which might dominate over large-scale Galactic rotation. Such activity could be the cause of the observed asymmetry of the [$^{26\!}$Al ]{}line intensities and line energy shifts in the inner Galaxy. On the other hand, other more nearby [$^{26\!}$Al ]{}source regions could be responsible for these irregularities, such as the nearest part of the Sagittarius-Carina arm in the fourth quadrant, or regions/complexes attributed to the Gould Belt such as the Scorpius-Centaurus-Lupus groups. Refined [$^{26\!}$Al ]{}studies and their combination with astrometry from other tracers of Galactic structure could help to understand the role of our Galaxy’s bar.
- The [$^{26\!}$Al ]{}emission asymmetry ($1.3\pm 0.2$) between fourth and first quadrant of the Galaxy is lower than the intensity contrast of $1.8^{+0.5}_{-0.3}$ reported for positron annihilation emission from the disk of the Galaxy (Weidenspointner et al. 2008). [$^{26\!}$Al ]{}by itself releases a positron in 82% of its decays; it is uncertain, however, how this translates into annihilation photons, from the variety of slowing down and annihilation processes which determine the fate of positrons in interstellar space. Both spatial and temporal variations and non-linearities scaling with gas density may occur.
- The scale height of Galactic-plane [$^{26\!}$Al ]{}emission is significantly larger than the molecular-gas disk scale height of 50 pc, yet significantly smaller than the “thick disk” part of the Galaxy. It is consistent with [$^{26\!}$Al ]{}being ejected from star-forming regions, and partly extending more towards the Galactic halo where gas pressure is lower than within the plane of the Galaxy (“champagne flows”).
- Localized regions may deviate in interesting detail from the large-scale averaged properties of [$^{26\!}$Al ]{}source regions. The direction of $20^\circ < l < 40^\circ$ corresponds to the Aquila region, and our hint for additional line broadening may be due to increased interstellar turbulence from stellar-wind and supernova activity at the characteristic age of stellar groups of that region, which may have created a supershell of substantial size (320 $\times$ 550 pc, see Maciejewski et al. 1996). Similar arguments are being explored for the Cygnus region, based on earlier hints of peculiar [$^{26\!}$Al ]{}emission (Martin et al. in preparation). The hint for [$^{26\!}$Al ]{}emission at $l<0^\circ,\ b>5^\circ$ (Fig. 12) may be attributed to the relatively nearby star forming complexes of the Sco-Cen association (de Geus 1992). Sco-Cen and several nearby stellar groups are attributed to the larger structure of the “Gould Belt”, which is suggested to have been more actively forming stars during the last 30 Myr (Grenier 2000, Perrot & Grenier 2003). These examples indicate that spatially-resolved [$^{26\!}$Al ]{}emission properties may enable new diagnostics of the interactions of massive stars with their surroundings, when combined with other astronomical constraints on cold gas and stars of such regions.
INTEGRAL/SPI will continue to accumulate more data for several years, to cover more sky regions in the outer Galaxy and at intermediate latitudes. This will allow us to deepen the studies reported in this paper, and to refine spatial information on the [$^{26\!}$Al ]{}line and/or increase the significance of the results reported above, as signal statistics increases and instrumental-background models are tightened. These studies will help us to better understand the properties of ISM near groups of massive stars, and the bulk motion of gas in the inner Galaxy from Galactic rotation and other peculiar kinematics. Nearby [$^{26\!}$Al ]{}sources may be also discriminated with better exposure towards candidate regions, improving our determination of the [$^{26\!}$Al ]{}mass in the Galaxy and towards regions where the stellar census is known to a better degree. Studies of [$^{26\!}$Al ]{}from nearby star-formation regions (e.g., Cygnus, Vela, Sco-Cen, Orion) are a promising diagnostic for the massive star origin of Galactic [$^{26\!}$Al ]{}and kinematics of [$^{26\!}$Al ]{}ejecta in ISM. From this, important constraints on massive-star and supernova nucleosynthesis are obtained, refining our models of their complex interiors (see e.g. Woosley and Heger 2007, or Chieffi and Limongi 2006).
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the anonymous referee for the comments to improve the draft. The INTEGRAL project is supported by government grants in the member states of the hardware teams. The SPI project has been completed under responsibility and leadership of CNES. We are grateful to ASI, CEA, CNES, DLR, ESA, INTA, NASA, and OSTC for support. W. Wang is also supported by the National Natural Science Foundation of China under grant 10803009.
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|
---
abstract: 'Recent measurement on an LC resonator magnetically coupled to a superconducting qubit\[arXiv:1005.1559\] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes-Cummings model exactly without the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch-Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch-Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.'
address: |
$^{1}$ Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang Normal University, Jinhua 321004, P. R. China\
$^{2}$ Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China\
$^{3}$Department of Physics, Southwest University of Science and Technology, Mianyang 621010, P. R. China\
$^{4}$Department of Modern Physics, University of Science and Technology of China, Hefei 230026, P. R. China
author:
- 'Qing-Hu Chen$^{1,2}$, Lei Li$^{3}$, Tao Liu$^{3}$, and Ke-Lin Wang$^{4}$'
title: 'Theory of spectrum in qubit-Oscillator systems in the ultrastrong coupling regime'
---
Introduction
============
The Jaynes-Cummings (JC) model[@JC] describes the interaction of a two-level atom with a single bosonic mode, which is fundamental model in quantum optics. Recently, the JC model is also closely related to condensed matter physics. It can be realized in some solid-state systems, such as one Josephson charge qubit coupling to an electromagnetic resonator [@Wallraff], the superconducting quantum interference device coupled with a nanomechanical resonator[@squid], and the LC resonator magnetically coupled to a superconducting qubit[@exp]. In conventional quantum optics, the coupling between the “natural” two-level atom and the single bosonic mode is quite weak, the rotating-wave approximation (RWA) has been usually employed. With the advent of circuit quantum electrodynamics (QED), on-chip superconducting qubits (the “artificial ” two-level atoms) could be engineered to interact very strongly with oscillators (cavities)[@Wallraff; @squid; @Chiorescu; @exp; @Schuster; @Deppe; @Fink; @Hofheinz], RWA can not describe well the strong coupling regime[@liu], so the studies to the JC model without RWA is highly called for.
However, it is more difficult to solve the JC model without RWA than with RWA. In the absence of RWA, due to the presence of the counter-rotating terms, the photonic number is not conserved, so the photonic Fock space has infinite dimensions. The standard diagonalization procedure (see, for example, Ref. [@exp]) is the first candidate, which is to apply a truncation procedure considering only a truncated number of photons. Typically, the convergence is assumed to be achieved if the numerical results are determined within very small relative errors. Within this method, one has to diagonalize very large, sparse Hamiltonian in strong coupling regime. Furthermore, the calculation might become prohibitive for higher excited states where more photons should be involved.
Fortunately, several non-RWA approaches[@chenqh; @zheng; @liu; @Liutao; @Amico; @Yuyu] has been recently proposed in a few contexts. Especially, by using extended bosonic coherent states, three of the present authors and a collaborator have solved the Dicke model without RWA exactly in the numerical sense[@chenqh]. The JC model is just special Dicke model with only one two-level atom.
Recently, the spectrum for an LC resonator magnetically coupled to a superconducting qubit was measured experimentally. A 50 MHz Bloch-Siegert shift when the qubit is in its symmetry point was observed, which clearly shows that the system enter the ultra-strong coupling regime. Therefore JC model with RWA is invalid to describe this strong coupling system. In this paper, we numerically solve JC model without RWA exactly. Based on the some key data drew from the spectrum[@exp], we obtain a fit of the experimental parameters. All spectrum line can then be calculated. The Bloch-Siegert shifts in arbitrary level transitions and in a wide range of the coupling parameters can also be estimated.
The paper is organized as follows. In Sec.II, the numerically exact solution to the JC model is proposed in detail. The numerical results and discussions are given in Sec.III. The brief summary is presented finally in the last section.
Model
=====
The interaction between the flux qubit and the LC resonator in the experiment [@exp] is described by $$H_{int}=\hbar g(a^{\dagger }+a)\sigma _z$$ where $a^{\dagger }$, $a$ are the photon creation and annihilation operators in the basis of Fock states of the LC resonator, $g$ is the flux qubit-cavity coupling constant. The RWA has not been employed here. The effective Hamiltonian for the flux qubit can be written as the standard one for a two-level system $$H=-\left( \epsilon \sigma _z+\Delta \sigma _x\right) /2$$ where $\Delta $ and $\epsilon $ are is the tunneling coupling between the two persistent current states and the transition frequency of the flux qubit. $\epsilon=I_p (\Phi-\Phi_0/2) $ with $I_p$ the persistent current in the qubit loop, $\Phi$ an externally applied magnetic flux, and $\Phi_0$ the flux quantum. In the above two equations, the Pauli matrix notations $\sigma _k(k=x,y,z)$ $\ $ are used in the basis of the two persistent current states. Then the Hamiltonian for the whole system reads $$\begin{aligned}
H &=&-\left( \epsilon \sigma _z+\Delta \sigma _x\right) /2+\hbar
\omega
_r\left( a^{\dagger }a+\frac 12\right) \nonumber \\
&&+\hbar g(a^{\dagger }+a)\sigma _z\end{aligned}$$ where $\omega _r$ is the cavity frequency. For convenience, we denote $$\hbar \omega _q=\sqrt{\epsilon ^2+\Delta ^2},\tan \theta =\Delta /\epsilon$$ Then the final Hamiltonian is ( $\hbar \;$is set to unity) $$\begin{aligned}
H &=&-\frac{\omega _q}2\left[ \cos (\theta )\sigma _z+\sin (\theta
)\sigma
_x\right] +\omega _r\left( a^{\dagger }a+\frac 12\right) \nonumber \\
&&+g\left( a^{\dagger }+a\right) \sigma _z\end{aligned}$$
By introducing the new operators $$A=a+\alpha ,B=\alpha -a,\alpha =g/\omega _r$$ we have $$H=\left(
\begin{array}{cc}
\omega _r\left( A^{\dagger }A-\alpha ^2\right) +\epsilon _{-} &
-\omega
_q\sin (\theta )/2 \\
-\omega _q\sin (\theta )/2 & \omega _r\left( B^{\dagger }B-\alpha
^2\right) +\epsilon _{+}
\end{array}
\right)$$ where $\epsilon _{\pm }=\left( \omega _r\pm \omega _q\cos \theta
\right) /2$. Note that the linear term for the original bosonic operator $a^{\dagger }(a)$ is removed, and only the number operators $A^{+}A$ and $B^{+}B$ are left. Therefore the wavefunction can be expanded in terms of these new operators as $$\left| {}\right\rangle =\left(
\begin{array}{l}
\left| \varphi _1\right\rangle \\
\left| \varphi _2\right\rangle
\end{array}
\right) =\left(
\begin{array}{l}
\sum_{n=0}^{N_{tr}}c_n\left| n\right\rangle _A \\
\sum_{n=0}^{N_{tr}}d_n\left| n\right\rangle _B
\end{array}
\right)$$ For $A$ operator, we have $$\begin{aligned}
\left| n\right\rangle _A &=&\frac{A^n}{\sqrt{n!}}\left| 0\right\rangle _A=%
\frac{\left( a+\alpha \right) ^n}{\sqrt{n!}}\left| 0\right\rangle _A \\
\left| 0\right\rangle _A &=&e^{-\frac 12\alpha ^2-\alpha
a^{+})}\left| 0\right\rangle _a.\end{aligned}$$ $B$ operator has the same properties. Inserting Eqs. (6) and (7) into the Schr$\stackrel{..}{o}$ dinger equation, we have $$\begin{aligned}
\left[ \epsilon _{-}+\omega _r\left( m-\alpha ^2\right) \right] c_m
&&
\nonumber \\
-\frac{\omega _q\sin (\theta )}2\sum_nD_{mn}d_n &=&Ec_m \\
\left[ \epsilon _{+}+\omega _r\left( m-\alpha ^2\right) \right] d_m
&&
\nonumber \\
-\frac{\omega _q\sin (\theta )}2\sum_nD_{mn}c_n &=&Ed_m\end{aligned}$$ where $$D_{mn}=\exp (-2\alpha ^2)\sum_{k=0}^{\min
[m,n]}(-1)^{-k}\frac{\sqrt{m!n!} (2\alpha
)^{m+n-2k}}{(m-k)!(n-k)!k!}$$ In principle, all eigenvalues and eigenfunctions can be obtained in Eqs. (10) and (11). As before, to obtain the true exact results, the truncated number $N_{tr}$ should be taken to infinity. Fortunately, it is not necessary. It is found that finite terms in state (7) are sufficient to give very accurate results with a relative errors less than $10^{-5}$ in the whole parameter space. We believe that we have exactly solved the JC model numerically. The numerical results are given in the next section.
*Solutions in the symmetry point*.– The spectrum in the symmetry point ($\epsilon=0$ ) is particularly interesting in experiments. In this case, Eq. (3) becomes $$H_0=-\frac \Delta 2\sigma _x+\hbar \omega _r\left( a^{\dagger
}a+\frac 12\right) +g\left( a^{\dagger }+a\right) \sigma _z$$ Associated with this Hamiltonian is a conserved parity $\Pi$, such that $\left[ H_0,\Pi \right] =0$, which is given by $$\Pi =e^{i\pi \sigma _y/4}e^{i\pi \widehat{N}}e^{-i\pi \sigma _y/4},
\widehat{N}=a^{\dagger }a+\sigma _z/2+1/2,$$ where $\widehat{N}$ is the excitation number operator. $\Pi$ has two eigenvalues $\pm 1$, depending on whether the excitation number is even or odd. So the system has the corresponding even or odd parity. It is easily proven that the wavefunction (6) with even and odd parity is of the form $$\left| \Psi _{\pm }\right\rangle =\ \left(
\begin{array}{l}
\sum_{n=0}^{N_{tr}}f_n\left| n\right\rangle _A \\
\pm \sum_{n=0}^{N_{tr}}f_n\left| n\right\rangle _B
\end{array}
\right)$$ where $\Psi _{+}$ $\left( \Psi _{-}\right) $ is corresponding to wavefunction with even(odd) parity. Inserting Eq. (13) into Eq. (3) gives $$\left[ \frac{\omega _r}2+\omega _r\left( m-\alpha ^2\right) \right]
f_m\mp \frac \Delta 2\sum_nD_{mn}f_n=Ef_m$$ The level transition is only allowed between the even and odd parity, i.e. $%
E_i^{(\pm )}\Leftrightarrow E_j^{(\mp )}$. The transition between the levels with the same parity is forbidden, $E_i^{(\pm
)}\nLeftrightarrow \ E_j^{(\pm )}$. The optical selection rules related to the parity have been discussed in the microwave-assisted transitions of superconducting quantum circuits[@liuyx; @Deppe].
Results and discussions
=======================
Díaz et al diagonalize a restricted Hilbert space to a certain number of photon states (in the Fock basis) and obtained fitted parameters[@exp; @pol]. The optimum fit of the experimental results within the present theoretical scheme gives $I_p=515nA,
g/2\pi =0.82GHz, \omega _r/2\pi =8.13GHz, \Delta /h=4.25GHz$, very close to their values. The calculations in this paper are based on these parameters, unless specified.
![ (Color online) Theoretical spectrum of the flux qubit coupled to the LC resonator from numerical calculations. []{data-label="Spectrum"}](theory_spect.eps)
We plot the numerical results for the spectrum for $E_n\rightarrow
E_0$( $ n=1,2,$ and $3$) in Fig. \[Spectrum\]. The experimental three spectral lines are just corresponding to the transitions between a few low energy levels, such as $ E_3\rightarrow E_0$ (upper), $E_2\rightarrow E_0$ (middle), and $E_1\rightarrow E_0$ (down). It is very interesting that our theoretical results for the spectrum are in excellent agreement with the experimental ones in Fig. 3 of Ref. [@exp]. Using these fitted parameters, the energy splitting on resonance $(E_2-E_1)/h$ obtained within the present approach is around $ 0.957GHz$, just in the scope of the experimental observation.
In Fig. 3 of Ref. [@exp], a weakly visible spectrum line just below the middle spectrum line was attributed to the thermally excited qubit. We calculate the spectrum line for the transition $E_3\rightarrow E_1$, as also list in Fig. \[Spectrum\] with a yellow line. Interestingly, it is just in the location observed experimentally shown in their Fig. 3. We believe that the state with $E_1$ is just corresponding to the qubit excited thermally mentioned in Ref. [@exp].
We would like to mention here that the experimentally observed spectrum lines have been explicitly related to the specified energy level transitions in the JC model without RWA. Then the comparison are easily performed.
Next, we specially consider the case in the symmetry point . Fig. \[energylevel\] (a) presents the energy levels from the numerically exact calculations. It was suggested in Ref. [@exp] that in the blue sideband spectral line [@Chiorescu] the minimum vanishes since the qubit is in the symmetry point where it produces no net flux and the transition is forbidden. In the symmetry point, the transition from $E_3^{(+)}\rightarrow E_0^{(+)}$ is forbidden due to the same parity, as shown in Fig. \[energylevel\]. This is the reason that the upper spectrum line around the symmetry point of Fig. 3 in Ref. [@exp] is almost invisible. It is perhaps just the optical selection rules related to the parity makes the qubit to produce no net flux. As also indicated in Fig. \[energylevel\], the other two transitions between levels with the different parity are allowed, so the intensities in the middle and down spectrum lines in the symmetry point are nearly the same as in the whole spectrum line.
Fig. \[energylevel\](b) shows the first 10 spectrums $E_n^{(-)}\rightarrow
E_0^{(+)}$ theoretically. In the experimental accessible detection, one can check the existence of these spectrums.
![ (Color online) (a) The energy levels $E_n^{(\pm )}$ and (b) the first 10 spectrums $E_n^{(-)}\rightarrow E_0^{(+)}$ in the symmetry point. []{data-label="energylevel"}](energylevel.eps)
We then turn to the Bloch-Siegert shift, which is just energy shift of the level transition with the consideration of the counter-rotating terms in the ultrastrong coupling regime. The Bloch-Siegert shift of the level transitions $E_i\rightarrow E_0$ in the symmetry point are exhibited in Fig. \[B\_S\_shift\]. The smallest Bloch-Siegert shift is around $50MHz$, nearly the same as that measured in the experiment[@exp]. Note that some level transitions $E_i\rightarrow E_0$ are forbidden in the symmetry point due to the same parity, which are also presented here only for the estimation of the magnitude of the Bloch-Siegert shift in the corresponding spectrum. For the main spectrum $E_i\rightarrow E_0$, the Bloch-Siegert shift becomes larger as $i$ increases, and its sign changes alternatively with either $i$, as shown in Fig. \[B\_S\_shift\](b).
![ (Color online) (a) The energy levels $E_i^{(\pm )}$ for in the symmetry point obtained within non-RWA (left panels) and RWA (right panels). (b) Bloch-Siegert shift for the different level transitions $E_i\rightarrow E_0$ in the symmetry point. []{data-label="B_S_shift"}](B_S_shift.eps)
To show the effect of the qubit-cavity coupling strength on the Bloch-Siegert shift, we fix all parameters fitted from experiments except the coupling parameter $g$. The Bloch-Siegert shift of the level transitions $E_1\rightarrow E_0$ in the symmetry point as a function of the effecting coupling constant $\alpha=g/\omega_r$ defined in Eq. (5) are plotted in Fig. \[shift\_g\]. In the weak coupling regime, say $\alpha\le0.01$, the Bloch-Siegert shift is so small ( less than 1 $MHz$) that it could not be distinguished from the spectrum line. When $\alpha \ge 0.1$, the Bloch-Siegert shift increases considerably with the coupling constant, and can reach the regime of $GHz$. This observation is also of practical interest. Recently, the coupling could easily be further enhanced in the circuit QED [@Devoret; @Bourassa; @Niemczyk] where $g$ is comparable with $\omega_r$, i.e. $\alpha$ is in the order of magnitude of $1.0$. If $\alpha>0.15$, the calculated Bloch-Siegert shift is observed to exceed $80 MHz$, the qubit line width at the symmetry point around $4 GHz$, it is predicted that the Bloch-Siegert shift could be clearly resolved experimentally in this strong-coupling regime, like the Lamb shift [@Fragner].
![ (Color online) Bloch-Siegert shift of the level transitions $E_1\rightarrow E_0$ in the symmetry point versus $\alpha=g/\omega_r$. []{data-label="shift_g"}](shift_g.eps)
conclusions
===========
In summary, by using extended bosonic coherent states, we solve the Jaynes-Cummings model without RWA exactly in the numerical sense. Within this technique, we can reproduce excellently the spectrum measured in a recent experiments on an LC resonator magnetically coupled to a superconducting qubit[@exp], which was demonstrated in the ultra-strong coupling regime. The Bloch-Siegert shift $E_1\rightarrow E_0$ in the symmetry point is estimated to be $50MHz$, very close to the experimental value. For the transition between the higher excited state $i>1$ and the ground-state, the magnitude of the Bloch-Siegert shift monotonously increases, but the sign changes as $(-1)^{(i+1)}$. The considerable Bloch-Siegert shift in turn demonstrate that the counter-rotating terms should be considered. The effect of the qubit-cavity coupling strength on the Bloch-Siegert shift is also investigated. It is predicted that the Bloch-Siegert shift can be distinguished experimentally for $\alpha>0.15$. The present technique are more suited for the stronger coupling regime, which experimental realizations may appear in the near future[@Devoret; @Bourassa; @Niemczyk].
Acknowledgements
================
The authors acknowledges useful discussions with P. Forn-Díaz. This work was supported by National Natural Science Foundation of China under Grant Nos. 10974180, National Basic Research Program of China (Grant Nos. 2011CB605903 and 2009CB929104).
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|
---
abstract: 'A metastable lattice gas with nearest-neighbor interactions and continuous-time dynamics is studied using a generalized Becker-Döring approach in the multidimensional space of cluster configurations. The pre-exponential of the metastable state lifetime (inverse of nucleation rate) is found to exhibit distinct peaks at integer values of the inverse supersaturation. Peaks are unobservable (infinitely narrow) in the strict limit $T\rightarrow 0$, but become detectable and eventually dominate at higher temperatures.'
author:
- 'Vitaly A. Shneidman'
- 'Gelu M. Nita'
title: 'Modulation of the nucleation rate pre-exponential in a low-temperature Ising system'
---
In a general case, the nucleation-controlled lifetime of a metastable state can be written as $$\tau= A\exp\left(W_\ast/T\right)
\label{volmer}$$ with $W_\ast$ being the minimal work (free energy change) to form a critical nucleus and the temperature $T$ measured in units of Boltzmann constant. The exponential term was anticipated already in the earliest estimations [@class] of the nucleation rate $I\sim 1/\tau$, which further found an enormous amount of applications in systems ranging from vapors [@Abr74] to glass-forming [@Deb96] or quantum [@Leg84] liquids. The structure of the pre-exponential, $A$, however, is not known in a general case. The difficulty of its derivation, whether in an analytical, numerical or experimental study, stems from the dominant contribution of the exponential in eq.(\[volmer\]), with minor uncertainties in $W_\ast$ (say, due to inaccuracies in the measured interfacial tension) implying large, orders of magnitude, discrepancies in the values of $A$.
With this, much attention is devoted to models which can exhibit nucleation and which are close to exact solvability. Here one can obtain an accurate expression for $W_\ast$, subsequently focusing on the pre-factor issue.
One of the best known example is the two-dimensional nearest neighbor Ising model, where metastability is achieved by orienting initially all spins one way (down) while non-zero magnetic field $h$ prescribes an opposite (upward) orientation. Allowing spin flips of non-conserved [@MetRosRos53Gla63] or conserved [@Kaw72] type adds the required dynamics to the problem. In a closely related lattice gas model the role of $h$ is played by supersaturation.
The pre-exponential in such systems attracts much attention both for the high- [@Fis67Lan71GunNikWal80GunMigSah83; @StoBinSch72BinSta76; @RikTomMiy94RicSidNov95; @JCP99] and low-temperature [@Nov97; @Nov02ParNov02; @BovMan02] regions. For $h\rightarrow 0$ the nucleus is macroscopic and its shape, as well as the value of $W_\ast$ can be obtained from the Wulff droplet construction [@RotWor81ZiaAvr82Zia86PRB01]. Monte Carlo simulations are possible for $W_\ast\lesssim 10\;-\;15 \;T$ which, for small $h$, restricts such studies to the aforementioned high-temperature region; transfer-matrix approaches also are available here [@GunRikNov93GunRikNov94Rut01]. For larger fields a straightforward Wulff construction may be inadequate, [@KotOli94], but for $T\rightarrow 0$ analytical treatment becomes possible due to dominant contribution of low-energy configurations. Technique of absorbing Markov chains [@Nov97; @Nov02ParNov02; @BovMan02] also can be used for simulations in the low-temperature region.
Neves and Schonmann[@NevSch91] evaluated $W_\ast^0$, the zero-temperature limit of $W_\ast$, obtaining the exponential part of the metastable lifetime. Their result is insensitive to specifics of the dynamics. Novotny [@Nov97] further showed that for discrete-time dynamics and a relatively large field, the pre-exponential remains finite in the limit $T\rightarrow 0$, approaching a piece-wise constant function of $h$, pointing towards a discontinuity at an integer value of the inverse field. Similar features will be observed at weaker fields as well [@BovMan02]. Integer values of inverse field, however, were excluded from the aforementioned rigorous mathematical treatments, leaving open questions with regard to this intriguing effect, especially in the physically more realistic case of $T>0$.
The present Letter aims to evaluate the pre-exponential at higher temperatures and in a finite domain of fields, spanning several integer values of $1/h$. This will clarify the nature of the discontinuities and, together with the available $W_\ast^0$, will provide predictive expressions for $\tau(T,h)$ at $T>0$. We will show that in contrast to intuitive expectation of discontinuities spreading out in a standard, [*tanh*]{}-like fashion, they are replaced by sharp peaks which persist, with finite heights and self-similar shapes, up to $T=0$.
A lattice gas model with continuous time dynamics will be considered. Specifically, the probability of creation of a particle on an empty site in an infinitesimal time interval $dt$ is taken as $\beta dt$, regardless of the surrounding; without restrictions, the time scale $\beta^{-1}$ can be taken as 1. Alternatively, the annihilation probabilities are proportional to $dt\exp(-\Delta E/T)$, with $\Delta E$ being the energy change due to broken bonds and field (“supersaturation”) which increases the energy by $2h$ when a particle is removed. This model, with various generalizations, is popular, e.g. in Monte Carlo simulations of the dynamic interface in crystallization problems - see, e.g. [@JacGilTem95JCG00] and references therein. It is expected that qualitatively the model also remains similar to the discrete-time Glauber type dynamics of Refs.[@NevSch91; @Nov97] (and the generalized Becker-Döring approach employed below bears certain parallels with the technique of absorbing Markovs chains [@Nov97; @Nov02ParNov02]), although the dynamics-sensitive pre-exponentials will not be identical even at $T=0$.
For a long time of the order of $\tau$ the rare particles will form isolated clusters of various sizes and shapes (*classes*), which will be distinguished by a running index, $i$. An empty site corresponds to $i=0$. Cluster shapes will be considered identical (and thus belonging to the same class) if they can made such by rotation or reflection. The key characteristics of each class are the numbers of particles, $s(i)$, the number of bonds $b(i)$ and the statistical weight $w_i\leq 8$. One can define the (quasi) equilibrium distribution $$f_i^{eq}=w_i z^{2s(i)-b(i)}\delta^{s(i)}
\label{feq}$$ with $
z=e^{-\varphi/T}$ and $\delta=z^{-2h}$ describing the temperature and field dependencies, respectively. $\varphi$ is the bond energy, subsequently taken as $1$ for simplicity of notations. In the $s, b$ space the function $ f_i^{eq}$ has a saddle point (for non-special fields - a single one [@BovMan02]) and the corresponding value of $s$ determines the critical cluster number, $s_\ast$. In a general case computer assistance is required in order to characterize all classes. Consistency of such predictions can be checked, e.g., against standard tables [@Dom60] for smaller $s$.
Once equilibrium properties are specified, one can introduce kinetic fluxes as a multidimensional version of the classical approach [@class], since in a low-temperature Ising system growth or decay of a cluster predominantly proceeds via random gain or loss of a single particle [@MarMar84]. If $\beta_{ik}dt$ is the probability to transform a cluster from class $i$ to class $k>i$ by adding a particle \[with $\beta{ik}=0$ if $s(k)\ne s(i)+1$\], the corresponding flux is given by $$I_{ik}=\beta_{ik}f_i^{eq}(v_i-v_k)\;,\;\;i<k
\label{Iik}$$ with $v_i \equiv f_i/f_i^{eq}$ and $v_0 = 1$. The Master Equation for the kinetic distributions $f_i$ takes the form $$\frac{df_i}{dt}=\sum_{k=0}^{i-1}I_{ki}-\sum_{k=i+1}^{k_{\max}+1}I_{ik}
\label{dfdt}$$ which automatically satisfies detailed balance.
For closing conditions, absorbing states are placed at all classes $k$ with $s(k)= s_{\max} +1$. Equivalently, all those absorbing states can be combined in a single absorbing class $k_{\max}+1$.
Due to an exponentially long lifetime, one can neglect the depletion of empty sites (for which, otherwise, an integral conservation law [@Pen97] should be employed instead of $v_0\equiv 1$). With this, eqs. $(\ref{Iik})$, $(\ref{dfdt})$ can be solved in the steady-state approximation; transient effects [@gelu_rem] also can be neglected here.
Introducing $b_{ik}=\beta_{ik}f_i^{eq}+\beta_{ki}f_k^{eq}$ ($0\le i,k\le
k_{\max}+1$) and $$M_{ik}=b_{ik}
-\delta_{ik}\sum_{l=0}^{k_{max}+1}b_{il}\;,\;\;1\le i,k\le k_{\max}
\label{mik}$$ one can show that the steady-state distributions $(v_1,\;v_2,\;\ldots)$ correspond to the first column of the matrix $-\hat M^{-1}$ (since only class $1$, with single-particle clusters, is connected to empty sites). The total flux $I$ coincides with $I_{01}$ where branching of paths does not yet occur. This gives $$I=(\hat M^{-1})_{11}+1
\label{Imhat}$$
For a single nucleation path, which leads to a tri-diagonal structure of the matrix $\hat M$, one recovers the classical result by Farkas [@class] $I^{-1}=b_{01}^{-1}+b_{12}^{-1}+\ldots$. Otherwise, the actual evaluation of $I$ via eq. (\[Imhat\]) is limited by one’s ability to obtain all classes and transition rates $\beta_{ik}$ for a sufficiently large $s_{\max}$, and the ability to inverse analytically a large matrix $\hat{M}$. At present, we were able to proceed up to $s_{\max}=9$ (1818 classes representing a total of 13702 cluster configurations) which allows us to consider fields $h>1/6$ with the critical number $s_\ast \leq 7$. A full exact expression for $\tau$ can be surveyed by a human eye only for more modest values of $s_{max}$, which implies a relatively small critical cluster (larger fields). For example, for $s_{\max}= 4$ kinetics is determined by 9 distinct classes with a total of 28 shapes (see, e.g. Fig. $2$ in Ref.[@JCP99]). Transition rates are easy to obtain (say, there are two ways a 3-particle “minus” shaped cluster can turn into 4-particle “T” shaped one, four ways it can turn into an “L” shaped one, etc.). The result which follows from eq.(\[Imhat\]), is expressed as a rational function of $z$ and $\delta$ $$\tau_{4}={P(\delta,z)}/{Q(\delta,z)}
\label{4spin}$$ with the subscript indicating the value of $s_{\max}$, and polynomials $P$ and $Q$ given by $$\begin{aligned}
\label{4full}
P(\delta,z)&=& 384 + 210000 \delta^9 z^9 + 16 \delta (48 + 185 z) + \\
\nonumber
&& 4 \delta^2 z (1576 + 2655 z) +8 \delta^3 z^2 (3081 + 3230 z) +\\
\nonumber
&& 2500 \delta^8 z^7 (21 + 250 z + 36 z^2)+ \\
\nonumber
&& 250 \delta^7 z^6 (695 + 2650 z + 1036 z^2) +\\
\nonumber
&& \delta^4 z^3 (65740 + 57797 z +15360 z^2) +\\
\nonumber
&& 5 \delta^5 z^4 (28574 + 28155 z + 19680 z^2) + \\
\nonumber
&& 5 \delta^6 z^5 (43375 + 70546z + 50000 z^2)\\
\nonumber
Q(\delta,z)&=& 8 \delta^4 z^4 (384 + 80 (24 + 85 \delta) z +\\
\nonumber
&& 250 \delta^2 (25 + 64 \delta) z^3 +125 \delta^3 (259 + 625 \delta) z^4 + \\
\nonumber
&& 20 \delta (615 + 1753 \delta) z^2 + 3750 \delta^4 (3 + 7 \delta) z^5)\end{aligned}$$
Eq.(\[4spin\]) is expected to be accurate in strong fields, $h\gtrsim 1/2$, with rather relaxed restrictions on temperature since all cluster configurations at $s\le 4$ are taken into account (although, for higher $T$ eventual destruction of the steady-state due to neglected cluster interactions should be kept in mind [@PRB99]). More consistently, this result should be treated asymptotically for $z\rightarrow 0$ and $\delta\rightarrow\infty$ with certain combinations of powers of $z$ and $\delta$ remaining finite, depending on the interval of field.
In order to isolate the pre-exponential, eq.(\[4spin\]) should be multiplied by $\exp(-W_\ast/T)=z^{W_\ast}$. In principle, an “observable” is $\tau$ itself, rather than $A$ or $W_\ast$ taken separately. To avoid ambiguity, the value of $W_\ast$ will be associated with its zero-temperature limit, $W_\ast^0$ [@NevSch91], with all temperature-dependent corrections being in the pre-exponential; for $h> 1$ the barrier will be taken as zero. The function $W_\ast^0(h)$ has a piece-wise linear structure, and $\exp(-W_\ast/T)$ is reduced to a product of integer powers of $z$ and $\delta$: $1$ for $h\ge 1$, $z^2\delta$ for $1/2\le h<1$, $z^4\delta^3$ for $1/4 \le h< 1/2$, etc. The resulting $A(h)$ is shown by a dashed line in Fig. \[fig1\] where numerical results, given as filled circles, were obtained for a much larger $s_{\max}$ and can be treated as “exact” in the present context. The case $T=0$ would correspond to a piece-wise constant structure of $A$, similar to Ref. [@Nov97] but with different constants and an additional “excluded singularity” at $h=1/2$: $A=1$ for $h>1$, $A=1/4$ for $1>h>1/4$ ($h\ne 1/2$), $A=1/16$ for $1/4>h>1/6$, etc. (These numbers can be deduced from the lowest energy path -see below-, serving as a checkpoint for more more elaborate expressions). This limit, however, becomes apparent only at a very low temperature, $z= 10^{-7}$.
For a larger cut-off, simplifications of analytics can be achieved due to the dominant contribution of low-energy configurations. Among all classes $k$ of clusters with the same $s(k)$, one can select only those which have a sufficiently large number of bonds: $b_s(k)\geq b_{max,s(k)}-r$, where $b_{max,s}$ is the number of bonds in the most compact cluster for a given $s$. An integer parameter $r$ indicates how close a cluster should be to the most compact configuration in order to be included in the kinetics. For sufficiently large $r$ ($r=4$ for $s_{max}=9$) all configurations are recovered. Alternatively, $r=0$ corresponds to the lowest energy path description, which is the closest to the kinetic part of the conventional one-dimensional random walk approach to nucleation [@class], although with microscopic rather than phenomenological coefficients. In addition, branching of paths is added starting from $s=7$. Already in the $r=0$ approximation peaks at integer $1/2h$ will appear in the pre-exponential, although one needs to include $r\ge 1$ for correct evaluation of their heights.
For the case $r =1$ and $z\ll 1$ the pre-exponential $A(h)$ can be described analytically in restricted domains of fields, the most interesting being those near the peaks (general expressions for $A(h)$ are also available, but are useless due to their size).
Introducing a [*finite*]{} combination $$\label{y_z}
y=\delta z^{1/n}$$ with $n=1,2,\ldots$ determining a corresponding peak, one can perform analytical expansions of $1/I$ in fractional powers of $z$. Symbolic computations with *Mathematica* were used here.
For $n=2$ one obtains $$\label{tau9}
\tau_{9}=\frac{1}{z^{5/2}}\frac{T_1(y)}{8y^7 T(y)} -
\frac{1}{z^2}\frac{T_2(y)}{16y^8 T^2(y)} +
\frac{1}{z^{3/2}}\frac{T_3(y)}{672y^9 T^3(y)}+\ldots$$ with the coefficients in this $s_{\max}=9$ approximation given by $$\begin{aligned}
T(y)&=& 8 + 63y^2\\
\nonumber
T_1(y)&=& 4 + 29y^2 + 79y^4 + 126y^6\\
\nonumber
T_2(y)&=&-208 - 1432y^2 + 3461y^4 + 49855y^6 +\\
\nonumber
&& 89649y^8 + 87318y^{10}\\
\nonumber
T_3(y)&=& 89152 - 297840y^2 - 13174644y^4 - 62801445y^6 +\\
\nonumber
&& 146767614y^8 +1284356493y^{10} + \\
\nonumber
&& 957680010y^{12} + 556604622y^{14}
\label{T9}\end{aligned}$$ The approximation works accurately in the vicinity of $h=1/4$, describing the rather complex near- and off-peak behavior - see Fig. \[fig2\]. The coefficient of $z^{-5/2}$ in eq.(\[tau9\]) multiplied, respectively by $y^3$ at $h>1/4$ or by $y^7$ at $h<1/4$, determines the scaling structure of the peak in the limit $T\rightarrow 0$, if the difference $h-1/4$ scales together with temperature. The structure of the neighboring peak at $h=1/2$ ($n=1$) follows the 4-particle approximation, eq.(\[4spin\]), with $\delta=y/z$ and $z\rightarrow 0$.
An important question is the sensitivity of the results to variations in $r$ and $s_{max}$. A smaller $r=0$ can reproduce the coefficient of $z^{-5/2}$ in eq.(\[tau9\]) in the limits $y\rightarrow\infty$ or $y\rightarrow 0$ (i.e. on both sides of the peak for $T\rightarrow 0$), but will not give the proper peak height for $y=1$, or scaling for finite $y$, or the correct higher-order terms. Cases with larger $r\ge 2$ presently could be studied only numerically and are shown by symbols in Fig. \[fig2\]. At small $h$ scatter appears in data, indicating the limits of numerical accuracy for very small $z$. There is no detectable difference with the analytical approximations in the regions of their validity for fields up to $h\gtrsim 1/4$. An similar expansion in $z$ for $r=1$ and $s_{max}=8$ also was performed, leading to a rather different structure of the $y$-dependent polynomials. The first coefficients in the $z$-expansion are nevertheless numerically close for stronger fields $ h\ge 1/4$ in the vicinity of the peak. So are the heights of the peak given, respectively by $0.4167-2.92z^{1/2}+\ldots$ and $0.4190-2.83z^{1/2}+\ldots$ in the 8-and 9-particle approximations. On the other hand, unlike the 9-particle case, $s_{\max}=8$ does not yield a proper $T\rightarrow 0$ limit for weaker fields $h<1/4$ since the boundary here is too close to the critical size $s_\ast=7$.
In summary, for a moderate field (supersaturation) the metastable state lifetime of a supersaturated lattice gas has been evaluated for $T\ll T_c$. The main result is the pre-exponential which, for the first time, was evaluated analytically beyond the zero-temperature limit, and which exhibits distinct peaks as a function of field. One can anticipate that similar peaks (which appear due to competition of several “critical sizes”) also will be observed in systems other than nearest-neighbor Ising models with non-conserved dynamics, whenever the nucleation barrier has a well-defined zero-temperature limit and the critical nucleus contains a reasonably small number of particles.
Authors are grateful to Mark Novotny for useful correspondence and comments on the manuscript.
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---
abstract: 'Using a sample of 89 snapshots from 58 hydrodynamic binary galaxy major merger simulations, we find that stellar remnants are mostly oblate while dark matter halos are mostly prolate or triaxial. The stellar minor axis and the halo major axis are almost always nearly perpendicular. This can be understood by considering the influence of angular momentum and dissipation during the merger. If binary mergers of spiral galaxies are responsible for the formation of elliptical galaxies or some subpopulation thereof, these galaxies can be expected to be oblate and inhabit their halos with the predicted shapes and orientations. These predictions are relevant to observational studies of weak gravitational lensing, where one must stack many optically aligned galaxies in order to determine the shape of the resulting stacked mass distribution. The simple relationship between the dark and luminous matter presented here can be used to guide the stacking of galaxies to minimize the information lost.'
author:
- 'Gregory S. Novak, Thomas J. Cox, Joel R. Primack, Patrik Jonsson, Avishai Dekel'
bibliography:
- 'ms.bib'
title: Shapes of Stellar Systems and Dark Halos from Simulations of Galaxy Major Mergers
---
Introduction
============
The shapes and mass profiles of dark matter halos from cosmological $N$-body simulations have long been studied [@dubinski:91; @navarro:96; @allgood:06 and references therein]. Cosmological simulations still lack sufficient resolution to track the shape and orientation of galaxies within their dark matter halos. There is no reason to believe that the shapes of galaxies and dark matter halos should be similar. It has only recently become feasible to perform large suites of high-resolution binary galaxy merger simulations [@naab:03; @cox:04; @cox:05; @robertson:06], and we here use such simulations in order to study the shapes of the resulting galaxies and their host halos statistically.
Observationally, the intrinsic shapes of elliptical galaxies have remained elusive. It has long been known that there seem to be at least two classes of elliptical galaxies: massive, anisotropic galaxies and lower mass, oblate rotators [@bender:88; @bender:92]. However, allowing the possibility of triaxiality leads to degeneracies in deprojection [@franx:91]. @alam:02 and @vincent:05 have used Sloan Digital Sky Survey (SDSS) data to conclude that not all elliptical galaxies can be oblate.
The relative orientations of galaxies and their dark halos is relevant to studies of weak gravitational lensing. Observers stack many images of galaxies in order to use the average deformation of the shapes of background galaxies to infer properties of the foreground mass distribution. It is important to stack galaxies coherently in order to build up a detectable signal. The model presented here represents a physically well-motivated *Ansatz* to help interpret the results of weak lensing observations. Section \[sec:methods\] gives a description of the galaxy merger simulations and our method of determining the shape of merger remnants, §\[sec:results\] gives our results, and §\[sec:conclusions\] summarizes our conclusions.
Methods {#sec:methods}
=======
We analyze the shapes of 89 snapshots from 58 of these simulations. We study two samples of simulations. One is the “G” series, which consists of major and minor mergers with progenitor spiral galaxies typical of the nearby universe and spanning a factor of 40 in baryonic mass and 20 total mass. In order to reduce the dependence on the progenitor galaxy model, here we only consider major mergers with mass ratios of 1:1 (G3-G3, G2-G2, G1-G1, and G0-G0) and roughly 3:1 (G3-G2, G2-G1, and G1-G0). We also analyze the “Sbc” series of merger simulations, which are 1:1 major mergers of massive, gas-rich spirals using a variety of different orbits and orientations.
To calculate the shape of a merger remnant, we iteratively diagonalize a moment of inertia tensor using an ellipsoidal window [@dubinski:91]: $$M_{ij} = \Sigma_k m_k r_{i,k} r_{j,k}
\label{eq:moment-of-inertia}$$ where $r_{i,k}$ is the position vector, $i,j$ refer to coordinates, and $k$ refers to particle number. The triaxial radius is given by @franx:91: $$\zeta = \sqrt{x^2/a^2 + y^2/b^2 + z^2/c^2}
\label{eq:triaxial-radius}$$ where $a$, $b$, and $c$ are the major, intermediate, and minor axis lengths, respectively. The sum over $k$ includes all particles for which $r$ lies within the ellipsoid $\zeta = 1$. The iteration is started with a spherical window ($a=b=c=$ baryonic half-mass radius), and after each iteration $a$,$b$, and $c$ are scaled so that half of the baryonic mass is enclosed. The result does not appreciably change if equation (\[eq:moment-of-inertia\]) is modified to include $\zeta^2$ in the denominator. Using a spherical window rather than an ellipsoidal one results in systematically larger axial ratios but does not change the main result.
Three-dimensional shapes of galaxies can be quantified with the triaxiality parameter $T=(a^2-b^2)/(a^2-c^2)$. We call an object oblate, triaxial, or prolate if $T<0.25$, $0.25<T<0.75$, or $0.75<T$, respectively. Shapes of galaxies can also be quantified by ellipticity $\epsilon=1-b/a$. Ellipticities are most often used to describe two-dimensional shapes; we occasionally refer to the three-dimensional ellipticity of perfectly prolate or oblate ($T=0$ or 1) objects since there is no ambiguity about the use of the equation.
Simulations were performed using the entropy-conserving version of the SPH code GADGET [@springel:01; @springel:02] with a gravitational smoothing length of 100 pc. The progenitor galaxies have baryonic masses from $1.6\times10^{9}$ to $2\times10^{11} M_\odot$, gas fractions between 20% and 70%, consist of $\sim$100,000 particles, and use a parameterization of star formation feedback from supernovae tuned to match the empirical Schmidt law [@kennicutt:98]. @cox:04 and @cox:05 contain further information about the simulations.
Results {#sec:results}
=======
Figure \[fig:shapes\] illustrates that most stellar remnants are oblate, while the dark matter halos in which they reside are mostly prolate or triaxial. Figure \[fig:orientation\] shows that the short axis of the stellar system and the long axis of the dark matter halo are almost always nearly perpendicular. This can be understood simply in terms of angular momentum and dissipation, as shown in Figure \[fig:diagram\].
This model helps interpret the findings from studies of weak gravitational lensing. @hoekstra:04 find that the ellipticity of dark halos is $0.77^{+0.18}_{-0.21}$ times the ellipticity of the light (i.e., halos are somewhat less flattened than galaxies), assuming that the two are aligned. According to our result, elliptical galaxies would either show an elliptical halo (if the long axis of the prolate halo is in the plane of the sky) or a circular halo (if the long axis of the halo is pointed toward the observer). Thus, the flattening of the dark matter would follow that of the luminous matter, in agreement with these observations.
The interpretation of the @hoekstra:04 data is complicated by the inclusion of spiral as well as elliptical galaxies in the sample. @mandelbaum:06a [@mandelbaum:06b] have done a similar study using SDSS galaxies separated by Hubble type and found that the projected halo shapes for elliptical galaxies are aligned with the projected stellar shapes, in agreement with @hoekstra:04 Finally, the projected positions of satellite galaxies also seem to indicate that the projected shapes of elliptical galaxies and halos are aligned [@sales:04; @brainerd:05; @yang:06].
Weak lensing studies necessarily underestimate the flattening of dark matter halos. Figures \[fig:lensing-1\] and \[fig:lensing-2\] quantify this by simulating the weak gravitational lensing observations. Given assumptions about the three-dimensional shapes and mass profiles of galaxies and their halos and a scheme for combining many galaxies into a single mass surface density, these two figures show shapes of the projected halo mass surface densities. They allow observers to translate their two-dimensional measurements to a range of possibilities for the three-dimensional structure of dark matter halos.
The hydrodynamic simulations discussed here do not represent a cosmologically unbiased sample, so they are *not* used as input to the simulated lensing observations. Instead we adopt a slightly idealized version of the correlation between halos and galaxies noted in this Letter. Nearly all of the baryonic components of the simulated galaxies are close to 2:1 oblate spheroids, so we assume that all early-type galaxies are so described. Thus, there is a simple mapping between viewing angle and optical ellipticity. We assume all galaxies follow the correlation between halos and galaxies noted here and that the halo mass density is given by a triaxial Navarro-Frenk-White profile: $\rho=\rho_0/(\zeta/r_s)(1+\zeta/r_s)^2$, where $\rho$ is the mass density and $\rho_0$ is a constant [@navarro:96; @jing:02].
@contopoulos:56 showed that for a triaxial ellipsoid with constant three-dimensional axis ratios, the contours of constant projected surface density are ellipses with constant ellipticity and position angle, independent of the radial density profile. We only use the ellipticity and position angle of the baryonic component, so the radial profile of the baryons does not matter. The @contopoulos:56 analysis does not apply to the stacked dark matter halos, so Figures \[fig:lensing-1\] and \[fig:lensing-2\] depend on the radial density distribution of the halos. In practice the difference is not large.
To simulate weak lensing measurements, we align the projected mass distributions based on projected light distributions, stack the projected halo mass distributions, and fit an ellipse to the halo mass surface density distribution where the area of the ellipse is constrained to equal $\pi (3 r_s)^2$. This size for the ellipse is motivated by the approximate radius at which weak lensing observations are sensitive to the halo shape (M. J. Hudson 2006, personal communication). The stacking either assumes a given inclination of the optical galaxy, averaging over the azimuthal angle (as in Fig. \[fig:lensing-1\]), or assumes that some *minimum* optical ellipticity is required to be included in the stack, averaging over the portion of the unit sphere that gives rise to sufficient optical ellipticities (as in Fig. \[fig:lensing-2\]).
Figure \[fig:lensing-1\] shows that galaxies with low optical ellipticities will have low halo ellipticities because there is no preferred axis to use to stack galaxies. The only situation where the projected halo ellipticity equals the three-dimensional halo ellipticity is when all stacked galaxies are viewed edge-on and halos are intrinsically oblate. Flattening is underestimated in all other cases.
Figure \[fig:lensing-2\] shows the result of the more realistic scenario where all galaxies with optical ellipticities greater than some value are included in the stack. This allows one to transform projected ellipticities to three-dimensional ellipticities. For example, if an observer sets the minimum optical ellipticity to 0.2 and measures a stacked halo ellipticity of 0.25, one can conclude that the three-dimensional ellipticity of halos is either 0.3 (for oblate halos), 0.5 (for prolate halos), or somewhere in between.
As one enforces tighter constraints on the optical ellipticity, the halo ellipticity goes up, but the cost is that fewer galaxies will make it into the stack. Under simple assumptions, one can estimate the signal-to-noise ratio (S/N) of the halo ellipticity measurement to be $$(\mbox{S/N})_{\mbox{tot}} =
\epsilon_{\mbox{2D}} \sqrt{ \Omega N_{\mbox{tot}}} / \sigma_1
\label{eq:s/n}$$ where $\epsilon_{\mbox{2D}}$ is the apparent ellipticity of the stacked halo mass surface density, $\Omega$ is the solid angle of viewing angles for which a galaxy will be included in the stack, $N_{\mbox{tot}}$ is the total number of galaxies in the survey, and $\sigma_1$ is the error on the halo ellipticity when only *one* galaxy is used. We define $\Psi$ as the part of this expression which depends on the signal and the available solid angle: $$\Psi = \epsilon_{\mbox{2D}} \sqrt \Omega
\label{eq:psi}$$ Figure \[fig:lensing-2\] thus also allows observers to estimate the quality of their measurement given the size of their survey and an estimate of the one-galaxy error on the halo ellipticity. In reality, observers do not know $\Omega$, but they could estimate it from the from the minimum ellipticity of galaxies in their sample as long as our assumption that the stellar remnants are 2:1 oblate spheroids is not far wrong.
Discussion {#sec:conclusions}
==========
We have analyzed the three-dimensional shapes of galaxies and dark matter halos resulting from more than 100 simulations of gas-rich galaxy mergers. Stellar remnants are nearly all oblate, with a few examples of triaxiality in the most gas-poor mergers. Dark matter halos are either prolate or triaxial, and the short axis of the baryons is perpendicular to the long axis of the dark matter. All of these facts can be understood in terms of the effects of angular momentum and dissipation during the merger. If there is a class of elliptical galaxies that were formed by gas-rich binary galaxy mergers, they can be expected to display these characteristics.
Real galaxies in a $\Lambda$CDM universe are thought to have experienced many mergers over the course of their history, and these multiple mergers can be expected to weaken the relationship between the shapes of galaxies and their halos described here. The extent to which the effects of large-scale structure, such as mass accretion along filaments, tend to preserve the relationship between galaxies and their halos is an interesting and open question.
G. S. N. was supported by the Krell Institute through the Computational Science Graduate Fellowship Program. Computing resources were provided by the UCSC Beowulf cluster UpsAnd and NERSC. We thank Andreas Burkert, S. M. Faber, Mike Hudson, and Laura Parker for useful discussions.
|
---
abstract: |
We obtain new information about divisors on the $d-$th symmetric power $C_{d}$ of a general curve $C$ of genus $g \geq 4.$ This includes a complete description of the effective cone of $C_{g-1}$ and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of $C_{d}$ in the range $\frac{g+1}{2} \leq d \leq g-2.$ We also obtain, for each $g \geq 5,$ a divisor on $C_{g-1}$ with non-equidimensional stable base locus.
For a general hyperelliptic curve $C$ of genus $g,$ we obtain a complete description of the effective cone of $C_{d}$ for $2 \leq d \leq g$ and an integral divisor on $C_{g-1}$ which has non-integral volume whenever $g$ is not a power of 2.
author:
- Yusuf Mustopa
title: |
Residuation of Linear Series\
and The Effective Cone of $C_{d}$
---
Introduction
============
Let $C$ be a smooth complex projective algebraic curve, and let $d \geq 1$ be an integer. The *$d$-th symmetric power* $C_{d}:=C^{d}/{\mathcal{S}_{d}}$ is a smooth $d-$dimensional complex projective variety which is a fine moduli space parametrizing effective divisors of degree $d$ on $C.$ In this paper we study the cone of effective divisors on $C_{d}$ and its refinements.
There are two natural divisor classes on $C_{d}.$ For each $p \in C$ the image of the embedding $$i_{p}:C_{d-1} \hookrightarrow C_{d}, \hspace{0.1cm} D' \mapsto D'+p$$ is an ample divisor on $C_{d}$ whose numerical class (which is independent of $p$) will be denoted by $x.$ The pullback of a theta-divisor on $\textrm{Pic}^{d}(C)$ via the natural map $$a_{d}:C_{d} \rightarrow \textrm{Pic}^{d}(C), \hspace{0.1cm} D \mapsto \mathcal{O}_{C}(D)$$ is a nef divisor on $C_{d}$ whose numerical class will be denoted by $\theta.$ (When $2 \leq d \leq g,$ we also have that $\theta$ is big.) The classes $x$ and $\theta$ are linearly independent in the real Néron-Severi space $N^{1}_{\mathbb{R}}(C_{d}),$ and when $C$ is general $N^{1}_{\mathbb{R}}(C_{d})$ is generated by $x$ and $\theta.$
We prove the following:
\[main1\] Let $C$ be a general nonhyperelliptic curve of genus $g \geq 4.$ For each $d \leq g-1,$ define $$r_{g,d}=1+\displaystyle\frac{g-d}{g^{2}-dg+(d-2)}.$$
The class $\theta-r_{g,d}x$ on $C_{d}$ is $\mathbb{Q}-$effective for $d \leq g-1.$
The class $\theta-r_{g,g-1}x$ on $C_{g-1}$ spans a boundary ray of the effective cone of $C_{g-1}.$
This result gives a new bound for the effective cone of $C_{d}$ in the range $\frac{g}{2}+1 \leq d \leq g-1.$ We will say more about the case where $d<\frac{g}{2}+1$ later in this introduction.
Theorem 3 in [@Kou] says that the diagonal class $$\Delta=2(-\theta+(g+d-1)x)$$ (which parametrizes effective divisors of degree $d$ having multiplicity) spans a boundary ray of the effective cone of $C_{d}$ for all $d \geq 2.$ Combining (ii) of Theorem A with that result yields
\[effcone1\] If $C$ is a general nonhyperelliptic curve of genus $g \geq 4,$ the effective cone of $C_{g-1}$ is spanned by the half-diagonal class $-\theta+(2g-2)x$ and the class $\theta-(1+\frac{1}{2g-3})x.$
We obtain more refined information in two distinct (but related) directions. The *stable base locus* $\textbf{B}(D)$ of a $\mathbb{Q}-$Cartier divisor $D$ on a projective variety $X$ is the set-theoretic intersection of the base loci of the linear systems $|mD|$ (where $m$ varies over all positive integers for which $mD$ is Cartier). The codimension of $\textbf{B}(D)$ is a rough measure of the size of $D;$ for instance, $\textbf{B}(D)=\emptyset$ if $D$ is ample and $\textbf{B}(D)=X$ if $D$ is not pseudoeffective.
The stable base locus of a divisor is not a numerical invariant in general. However, the so-called *stable* divisors (see Section \[baseloc\] for the definition) have stable base loci which are numerical invariants, and as such we can speak of stable classes in $N^{1}_{\mathbb{Q}}(X)$. The *movable cone* of a smooth projective variety $X$ is the closure of the convex cone in $N^{1}_{\mathbb{R}}(X)$ spanned by classes of divisors on $X$ whose stable base locus has no divisorial component.
\[main2\] Let $C$ be a general nonhyperelliptic curve of genus $g \geq 4,$ and let $d \leq g-1.$ Then the class $\theta-x$ on $C_{d}$ is stable with stable base locus $$C^{1}_{d}=\{D \in C_{d}:\dim{|D|} \geq 1\}.$$ In particular, since $C^{1}_{d}$ is of codimension at least 2, the class $\theta-x$ lies in the interior of the movable cone of $C_{d}.$
This theorem follows from the more general Theorem \[stab\_classes\], which when combined with Lemma \[nef\_small\_diag\] yields
If $C$ is a general nonhyperelliptic curve of genus $g \geq 5,$ there exists $s_{g} > 1+\frac{1}{g^{2}-g-1}$ such that the the class $\theta-sx$ on $C_{g-1}$ is stable with a non-equidimensional stable base locus whenever $1+\frac{1}{g^{2}-g-1} < s < s_{g}$.
We now turn to a different way of measuring the size of a divisor. The *volume* of a $\mathbb{Q}-$Cartier divisor $D$ on an $n-$dimensional projective variety $X$ is $$\textnormal{vol}_{X}(D)=\limsup_{m}\frac{n! \cdot h^{0}(\mathcal{O}_{X}(mD))}{m^{n}}$$ where the limit superior is taken over all positive integers $m$ for which $mD$ is Cartier. When $D$ is big, this number may be thought of as the “moving self-intersection” of $D$ (see, for instance, Theorem 11.4.11 in [@Laz2]). Note that $\textnormal{vol}_{X}(D) > 0$ precisely when $D$ is big, and that when $D$ is ample, $\textnormal{vol}_{X}(D)=D^{n}$ by Serre vanishing and asymptotic Riemann-Roch.
It can be shown that $\textnormal{vol}_{X}(D)$ is independent of the numerical class of $D,$ and that the real-valued function $\textnormal{vol}_{X}$ on $N^{1}_{\mathbb{Q}}(X)$ extends to a continuous real-valued function on $N^{1}_{\mathbb{R}}(X).$ (More recently, $\textnormal{vol}_{X}$ has been shown to be $\mathcal{C}^{1}$ in [@BFJ]).
\[main3\] Let $C$ be a general nonhyperelliptic curve of genus $g \geq 4.$ Then for $t \in [0,1+\frac{1}{g^{2}-g-1}],$ $$\textnormal{vol}_{C_{g-1}}(\theta-tx)=\displaystyle\sum_{k=0}^{g-1}\binom{g-1}{k}\frac{g!}{(k+1)!}t^{k}(1-t)^{g-1-k}.$$
We pause to discuss the proofs of the theorems listed thus far. Recall that for a smooth projective curve $C$ and two positive integers $r$ and $d,$ there exists a fine moduli variety $G^{r}_{d}(C)$ parametrizing linear series (complete or otherwise) of degree $d$ and dimension $r$ on $C.$ In particular, $G^{0}_{d}(C)$ is canonically isomorphic to $C_{d}.$
When $2 \leq d \leq g-1,$ there is a birational map $$\widetilde{\tau}:G^{g-d-1}_{2g-2-d}(C) \dashrightarrow C_{d}$$ defined by taking a complete linear series $|\mathcal{L}|$ to the unique element of its residual series $|K_{C}\otimes\mathcal{L}^{-1}|.$ This map is the moral heart of the paper and the technical heart of the proofs of Theorems A,C, D, and E. It follows from Gieseker’s Theorem that $\widetilde{\tau}$ is an isomorphism in codimension 1 of smooth projective varieties when $C$ is a general curve of genus $g$, and as a result we can use $\widetilde{\tau}$ to transfer information about divisors from $G^{g-d-1}_{2g-2-d}(C)$ to $C_{d}$ and back via Hartogs’ theorem.
We now describe the divisor classes from Theorem A. For a line bundle $\mathcal{L}$ on $C$ satisfying $e:=\dim|\mathcal{L}| \leq d \leq \deg{\mathcal{L}},$ the $e-$dimensional cycle $\Gamma_{d}(\mathcal{L})$ on $C_{d}$ parametrizes all effective divisors $D$ of degree $d$ that are subordinate to $|\mathcal{L}|,$ i.e. those $D$ for which $|\mathcal{L}(-D)| \neq \emptyset.$ A natural inner bound for the effective cone of $C_{g-1}$ in the fourth quarter of the $(\theta,x)-$plane is furnished by the cycle $\Gamma_{g-1}(K_{C}(-p))$ (where $p$ is a given point in $C$). This is a divisor whose class is $\theta-x.$
In Theorem 5 of [@Kou], Kouvidakis obtains $\theta-2x$ as an outer bound for the effective cone of $C_{g-1}$ by degeneration to a hyperelliptic curve. The inner bound $\theta-x$ reflects the fact that the canonical series $|K_{C}|$ separates 0-jets (i.e. is basepoint free) and Kouvidakis’ outer bound is obtained by degeneration to the case in which $|K_{C}|$ fails to separate 1-jets (i.e. fails to be an immersion). The divisor we obtain which spans a boundary ray of the effective cone of $C_{g-1}$ (in the nonhyperelliptic case) is supported on the set $$\bigcup_{p \in C}\Gamma_{g-1}(K_{C}(-2p))$$ and so it reflects in a precise manner the fact that $K_{C}$ separates 1-jets when $C$ is nonhyperelliptic. Indeed, this divisor is the ramification locus of the Gauss map $\gamma: C_{g-1} \dashrightarrow (\mathbb{P}^{g-1})^{\ast}$ which assigns to a general $D \in C_{g-1}$ the hyperplane spanned by the image of $D$ under the canonical embedding.
More generally, our divisor on $C_{d}$ for $d \leq g-1$ is supported on the set $$\bigcup_{p \in C}\Gamma_{d}(K_{C}(-(g-d+1)p))$$ This is the pullback via $\widetilde{\tau}$ of the divisor on $G^{g-d-1}_{2g-2-d}(C)$ parametrizing linear series with ramification points of degree $g-d+1$ or higher.
The bound for the effective cone implied by (i) of Theorem A is not sharp in the range $3 \leq d \leq \frac{g}{2}$; Theorem 5 of [@Kou] implies that the class $\theta-2x$ is effective for all such $d$. Our next result gives a new bound for the effective cone of $C_{\frac{g+1}{2}}.$ Recall that $W^{r}_{d}(C)$ is the determinantal subvariety of $\textnormal{Pic}^{d}(C)$ parametrizing line bundles with at least $r+1$ global sections.
Let $k \geq 3$ be an integer and let $C$ be a general curve of genus $2k-1.$
The class $\theta-(2-\frac{1}{k})x$ on $C_{k}$ is $\mathbb{Q}-$effective.
For all $t > 2-\frac{1}{k},$ the stable base locus of any divisor with class proportional to $\theta-tx$ contains the surface $$Z_{k}:=\bigcup_{\mathcal{L} \in W^{1}_{k+1}(C)}\Gamma_{k}(\mathcal{L})$$
The class $\theta-(2-\frac{1}{k})x$ spans a boundary ray of the effective cone of $C_{k}$ when $k=3.$ That is, the class $\theta-\frac{5}{3}x$ spans a boundary ray of the effective cone of $C_{3}$ when $C$ is a general curve of genus 5.
The divisor we consider is supported on the set $$\bigcup_{\mathcal{L} \in W^{1}_{k+1}(C)}\Gamma_{k}(K_{C}\otimes\mathcal{L}^{-1})$$ This is a direct generalization of the divisor shown by Pacienza in [@Pac] to span a boundary ray of the *nef* cone of $C_{\frac{g}{2}}.$ However, it is *not* nef in general, since in the case $k=3$ its top self-intersection is negative. We plan to address the nef cone of $C_{\frac{g+1}{2}}$ further in future work.
Special linear series on an arbitrary curve are poorly understood, and this accounts for much of the difficulty in the study of their symmetric powers. However, special linear series on *hyperelliptic* curves are understood quite well, and thus their consideration is a natural step in our study.
\[hyperell\] Let $C$ be a hyperelliptic curve of genus $g$, and let $2 \leq d \leq g.$
The class $\theta-(g-d+1)x$ spans a boundary ray of the effective cone of $C_{d}.$
The class $\theta$ spans a common boundary ray of the nef and movable cones of $C_{d}.$
For all $t \in [0,g-d+1],$ $$\textnormal{vol}_{C_{d}}(\theta-tx)=\frac{g!}{(g-d)!} \cdot \bigg(1-\frac{t}{g-d+1}\bigg)^{d}$$
The proof of this result, which is given in Section \[hyp\], is based on the observation that if $C$ is hyperelliptic, then the Abel map $a_{d}:C_{d} \rightarrow \textrm{Pic}^{d}(C)$ is a divisorial contraction for $2 \leq d \leq g.$ (Note that the converse statement is also true by Martens’ Theorem.)
Using a result of Pirola (Proposition \[pirola\]) we are able to deduce that $N^{1}_{\mathbb{R}}(C_{d})$ is 2-dimensional when $C$ is a general hyperelliptic curve (Corollary \[n\_s\_rank\]). Combining (i) with Theorem 3 of [@Kou] then yields
If $C$ is a general hyperelliptic curve of genus $g,$ then for $2 \leq d \leq g$ the effective cone of $C_{d}$ is spanned by the half-diagonal class $-\theta+(g+d-1)x$ and the class $\theta-(g-d+1)x.$
Since the nef and movable cone of $C_{d}$ share the ray spanned by $\theta$ as a common boundary, every big divisor class in the fourth quarter of the $(\theta,x)-$plane admits an “honest” Zariski decomposition which can be used to compute its volume; this is essentially the proof of (iii) of Theorem G. Setting $t=1$ in both Theorem E and (iii) of Theorem G yields
Let $C$ be a curve of genus $g \geq 4.$ Then $$\textnormal{vol}_{C_{g-1}}(\theta-x)=\begin{cases}
1&\textnormal{if $C$ is general nonhyperelliptic}\\
\frac{g!}{2^{g-1}}&\textnormal{if $C$ is general hyperelliptic}\end{cases}$$
It is straightforward to check that when $C$ is nonhyperelliptic and $p_{1},...,p_{g-1}$ are general points on $C,$ the intersection of the divisors $\Gamma_{g-1}(K_{C}(-p_{1})), \dots \Gamma_{g-1}(K_{C}(-p_{g-1}))$ is the union of $C^{1}_{g-1}$ and the unique element of the linear system $|K_{C}(-p_{1} \dots -p_{g-1})|.$ As mentioned earlier, each of the divisors $\Gamma_{g-1}(K_{C}(-p_{i}))$ has numerical class $\theta-x.$ Consequently the formula $\textnormal{vol}_{C_{g-1}}(\theta-x)=1$ bears out the interpretation of the volume as “moving self-intersection.”
By a well-known elementary identity, $\frac{g!}{2^{g-1}}$ is an odd integer when $g$ is a power of $2$ and fails to be an integer otherwise, so that symmetric powers of hyperelliptic curves furnish a class of examples of integral divisors with non-integral volume. Another class of such examples is treated in Section 2.3B of [@Laz1].
We do not touch on the interesting issues concerning the effective cone of $C_{2};$ the curious reader is referred to [@Chan] and [@Ross].
**Acknowledgments:** The bulk of this paper is my Ph.D thesis at Stony Brook University. I thank my advisor, Jason Starr, for his encouragement and support, as well as Lawrence Ein, Rob Lazarsfeld, Julius Ross, and Dror Varolin for valuable discussions.
I would also like to thank Li Li, Aleksey Zinger, and the anonymous referee for useful comments on the manuscript, Gianluca Pacienza for inviting me to IRMA to give a talk on this material, and Olivier Debarre, whose unpublished note has helped inspire the direction of this work.
**Notation and Conventions:** We work over the field of complex numbers. $C$ will always denote a smooth projective curve. All cycle classes on smooth varieties lie in the algebraic cohomology ring with coefficients in $\mathbb{R}$. If $\mathcal{Z}$ is a subvariety of the moduli space $\mathcal{M}_{g}$ of smooth projective curves of genus $g$, we say that a property holds for *a general curve of $\mathcal{Z}$* if it holds on the complement of the union of countably many proper subvarieties of $\mathcal{Z}.$ If $\mathcal{Z}=\mathcal{M}_{g},$ we say that the property holds *for a general curve of genus $g$*.
Preliminaries on $C_{d}$
========================
The Néron-Severi group of $C_{d}$
---------------------------------
Recall that the Néron-Severi group $NS(X)$ of a smooth projective variety $X$ is the additive group of divisors on $X$ modulo algebraic equivalence.
**Definition:** The **(real) Néron-Severi space** $N^{1}_{\mathbb{R}}(X)$ of $X$ is the real vector space $NS(X) \otimes_{\mathbb{Z}} \mathbb{R}.$
The following result is well-known.
\[ns\_group\] For any $d \geq 2$ there is an isomorphism $N^{1}_{\mathbb{R}}(C_{d}) \simeq \mathbb{R} \oplus N^{1}_{\mathbb{R}}(J(C)).$
Under this isomorphism, we may think of the summand $\mathbb{R}$ as being generated by $x$, and of the class $\theta$ as (not surprisingly) being contained in $N^{1}_{\mathbb{R}}(J(C)).$ In particular, $x$ and $\theta$ are linearly independent.
We may identify $NS(J(C))$ with the group $\textnormal{End}^{s}(J(C))$ of endomorphisms preserving the principal polarization (see Proposition 5.2.1 in [@BirLan] for details). Since $NS(J(C))$ is torsion-free, we have that $N^{1}_{\mathbb{R}}(J(C))$ is 1-dimensional precisely when inversion is the only nontrivial automorphism in $\textnormal{End}^{s}(J(C)).$
The fact that this holds for a general Jacobian is due to Lefschetz. The following refinement is due to Pirola. (Recall that $\mathcal{J}_{g}$ is the Jacobian locus in the moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g.$)
\[pirola\] [((ii) of Proposition 3.4 in [@Pir])]{.nodecor} Let $g \geq 2$ be an integer, and let $Y$ be a subvariety of codimension $\leq g-2$ in $\mathcal{J}_{g}.$ Then the rank of the Néron-Severi group of an abelian variety corresponding to a general point of $Y$ is 1.
This result is sharp. As pointed out in Remark 3.5 of *loc. cit.*, its conclusion fails if we take $Y$ to be the $(2g-2)-$dimensional locus parametrizing Jacobians of bielliptic curves.
A curve $C$ is called *e*-gonal if $e=\min\{k : C \textnormal{ admits a \textit{k}-to-1 covering of }\mathbb{P}^{1}\}.$
\[n\_s\_rank\] If $C$ is a curve corresponding to a general point of the $e-$gonal locus of $\mathcal{M}_{g}$ (where $e \geq 2$) then $N^{1}_{\mathbb{R}}(C_{d})$ is 2-dimensional for all $d \geq 2.$ In particular, this is true of both the general curve of genus $g$ and the general hyperelliptic curve of genus $g.$
Since the dimension of the $e-$gonal locus in $\mathcal{M}_{g}$ is $\min\{3g-3,2g+2e-5\},$ its image under the Torelli embedding is of codimension $\max\{0,g-2e+2\}$ in $\mathcal{J}_{g}.$ So the result follows immediately from Propositions \[ns\_group\] and \[pirola\].
Intersection theory on $C_{d}$
------------------------------
The following formula, which is a consequence of the Poincaré formula (p.25 of [@ACGH]) will be used freely.
\[intersect\] For all $0 \leq k \leq d \leq g,$ $$x^{k}\theta^{d-k}=\displaystyle\frac{g!}{(g-d+k)!}$$
Subordinate Loci
----------------
We now describe one of the most important constructions in this paper. Let $d \geq 2$ be an integer, let $(\mathcal{L},V)$ be a linear series of degree $n$ and dimension $r$ on $C,$ and assume that $n \geq d \geq r.$ Recall from [@ACGH] that there is a natural rank-$d$ vector bundle $E_{\mathcal{L}}$ on $C_{d}$ whose fibre over $D \in C_{d}$ is $H^{0}(D,\mathcal{L}|_{D}).$ As such, there is a morphism $$\alpha_{V}:V \otimes {\mathcal{O}}_{C_{d}} \rightarrow E_{\mathcal{L}}$$ whose fibre over each $D \in C_{d}$ is the restriction map $V \rightarrow H^{0}(D,\mathcal{L}|_{D}).$ The latter fails to be injective precisely when $D$ is subordinate to $(\mathcal{L},V),$ i.e. when $V \cap H^{0}(\mathcal{L}(-D)) \neq 0.$
**Definition:** The cycle ${\Gamma}_{d}(\mathcal{L},V)$ is the degeneracy locus of $\alpha_{V}.$ If $V=H^{0}(\mathcal{L})$, we will write ${\Gamma}_{d}(\mathcal{L})$ instead.
Note that $\Gamma_{d}(\mathcal{L},V)$ is supported on the set $$\{D \in C_{d}:V \cap H^{0}(\mathcal{L}(-D)) \neq 0\}.$$
The following result computes the fundamental class of $\Gamma_{d}(\mathcal{L},V);$ we refer to p.342 of [@ACGH] for the proof.
\[subordinate\] (3.2 on p. 342 of [@ACGH]) Let $C$ be a curve of genus $g,$ and let $n,d,$ and $r$ be integers satisfying $n \geq d \geq r.$ Then ${\Gamma}_{d}(\mathcal{L},V)$ is $r-$dimensional, and its fundamental class is $$\displaystyle\sum_{k=0}^{d-r}\binom{n-g-r}{k}\frac{x^{k}\theta^{d-r-k}}{(d-r-k)!}.$$
Diagonal Calculations
---------------------
We collect here two special cases of the computation of diagonal classes in Proposition 5.1 on p.358 of [@ACGH] which are used in the proof of Theorem A. First, we define the diagonal loci.
**Definition:** Let $d \geq 2$ be an integer and let $a_{1}, \dots ,a_{k}$ be a sequence of positive integers which is a partition of $d.$ Then $\Delta_{a_{1}, \dots ,a_{k}}$ is the reduced subscheme of $C_{d}$ supported on the image of the morphism $$\phi_{a_{1}, \dots ,a_{k}}:C^{k} \rightarrow C_{d} \hspace{0.3cm} (p_{1}, \dots ,p_{k}) \mapsto \sum_{i=0}^{k}a_{i}p_{i}$$ $\Delta_{2,1, \dots 1}$ will be denoted by $\Delta.$
\[small\_diag\] Let $C$ be a curve of genus $g.$ The fundamental class of ${\Delta}_{d}$ in $C_{d}$ is $$dx^{d-2} \cdot \Bigl(((d-1)g+1)x-(d-1)\theta\Bigr)$$
By Proposition 5.1 on p.358 of [@ACGH], this class is $${\sum}_{0 \leq \beta \leq \alpha \leq d-1}\frac{(-1)^{\alpha + \beta}}{{\beta}!(\alpha - \beta)!}\Bigl(d(\beta + 1 - g)+d^{2}(g - \beta)\Bigr)x^{d-1-{\alpha}}{\theta}^{\alpha}.$$ The result is thus immediate when $d=2;$ when $d \geq 3,$ it follows from the fact that ${\sum}_{1 \leq \beta \leq \alpha}(-1)^{\beta}{\beta}\binom{\alpha}{\beta}=0$ for all $\alpha\geq 2.$
\[other\_diag\] The numerical class of $\Delta_{g-d+1,d}$ is $$(1-\frac{1}{2}\delta_{(d,\frac{g+1}{2})}) \cdot d(g-d+1)x^{g-3} \cdot \biggl\{\Bigl(d(g-d+1)(g^{2}-g)-(g^{2}+1)(g-2)\Bigr)x^{2}$$ $$+\Bigl((2-2d)g^{2}+(2d^{2}-3)g-(2d^{2}-d-2)\Bigr)x\theta+(d-1)(g-d)\theta^{2}\biggr\}$$ where $\delta_{(d,\frac{g+1}{2})}$ is the Kronecker delta.
As a set, $\Delta_{g-d+1,d}$ is the image of the morphism $$\phi_{g-d+1,d}:C \times C \rightarrow C_{g+1}, (p,q) \mapsto (g-d+1)p+dq.$$ When $d \neq \frac{g+1}{2},$ this morphism is injective, so that the class of $\Delta_{g-d+1,d}$ is the pushforward class $[\phi_{g-d+1,d}]_{\ast}(C \times C).$ When $d=\frac{g+1}{2},$ we have the factorization $$\xymatrix{
C \times C
\ar[rr]^{\phi_{\frac{g+1}{2},\frac{g+1}{2}}}
\ar[dr]_{\pi}
&& C_{g+1}\\
& C_{2}
\ar[ur]_{\phi_{\frac{g+1}{2}}}}\\$$ where $\pi:C \times C \rightarrow C_{2}$ is the canonical quotient map and $\phi_{\frac{g+1}{2}}:C_{2} \rightarrow C_{g+1}$ is defined by $p+q \mapsto (\frac{g+1}{2})p+(\frac{g+1}{2})q.$ Since $\phi_{\frac{g+1}{2}}$ is injective, the class of $\Delta_{\frac{g+1}{2},\frac{g+1}{2}}$ is the pushforward class $[\phi_{\frac{g+1}{2}}]_{\ast}(C_{2}),$ which by our factorization is equal to $\frac{1}{2} \cdot [\phi_{\frac{g+1}{2},\frac{g+1}{2}}]_{\ast}(C \times C).$ Therefore the class of $\Delta_{g-d+1,d}$ is equal to $$(1-\frac{1}{2}\delta_{(d,\frac{g+1}{2})}) \cdot [\phi_{g-d+1,d}]_{\ast}(C \times C).$$
By Proposition 5.1 on p.358 of [@ACGH], the class $[\phi_{g-d+1,d}]_{\ast}(C \times C)$ is the coefficient of $t_{1}t_{2}$ in the expression $$\displaystyle\sum_{0 \leq \beta \leq \alpha \leq g-1}\frac{(-1)^{\alpha+\beta}}{\beta!(\alpha-\beta)!}\Bigl(1+(g-d+1)t_{1}+dt_{2}\Bigr)^{2-g+\beta}\Bigl(1+(g-d+1)^{2}t_{1}+d^{2}t_{2}\Bigr)^{g-\beta}x^{g-1-\alpha}\theta^{\alpha}.$$ This coefficient is equal to $$d(g-d+1)\displaystyle\sum_{\alpha=0}^{g-1}\frac{(-1)^{\alpha}}{\alpha!}\biggl\{\Bigl((d-1)g-d(d-1)\Bigr)\displaystyle\sum_{\beta=0}^{\alpha}(-1)^{\beta}\binom{\alpha}{\beta}\beta^{2}$$ $$+\Bigl((2-2d)g^{2}+(2d^{2}-d-2)g-(d^{2}-d-1)\Bigr)\displaystyle\sum_{\beta=0}^{\alpha}(-1)^{\beta}\binom{\alpha}{\beta}\beta$$ $$+\Bigl((d-1)g^{3}-(d^{2}-2)g^{2}+(d^{2}-d-1)g+2\Bigr)\displaystyle\sum_{\beta=0}^{\alpha}(-1)^{\beta}\binom{\alpha}{\beta}\biggr\}x^{g-1-\alpha}\theta^{\alpha}$$ Since the three sums over $\beta$ are equal to 0 for all $\alpha \geq 3,$ we finally obtain that the class $[\phi_{g-d+1,d}]_{\ast}(C \times C)$ is equal to $$d(g-d+1)x^{g-3}\biggl\{\Bigl((d-1)g^{3}-(d^{2}-2)g^{2}+(d^{2}-d-1)g+2\Bigr)x^{2}$$ $$+\Bigl((2-2d)g^{2}+(2d^{2}-3)g-(2d^{2}-2d-1)\Bigr)x\theta+\Bigl((d-1)g-d(d-1)\Bigr)\theta^{2}\biggr\}.$$
Results from the asymptotic theory of linear series {#asymp_thy}
===================================================
We collect the results on stable base loci and the volume function that will be used in the sequel. We refer to [@ELMNP] and [@Laz1] for a thorough treatment.
Base Loci {#baseloc}
---------
Recall the following definition from the Introduction.
**Definition:** Let $X$ be an irreducible projective variety and let $D$ be a $\mathbb{Q}-$Cartier divisor on $X.$ Then the **stable base locus of** $D$ is the algebraic set $$\textbf{B}(D)=\bigcap_{m}\textnormal{Bs}|mD|$$ where the intersection is taken over all positive integers $m$ for which $mD$ is Cartier.
For the proof of Theorem \[stab\_classes\] we will need to know that the stable base locus of a Cartier divisor $D$ can be realized as the base locus of some multiple of $D.$ The relevant result, which we state below, follows immediately from Proposition 2.1.21 in [@Laz1].
\[multiple\] Let $D$ be a Cartier divisor on an irreducible projective variety $X.$
- [There exists a positive integer $m_{0}$ such that $\textnormal{\textbf{B}}(D)=\textnormal{Bs}|km_{0}D|$ for all $k >> 0.$]{}
- [$\textnormal{\textbf{B}}(D)=\textnormal{\textbf{B}}(mD)$ for all $m \geq 1.$]{}
In particular, we can always find a positive integer $m$ for which $\textnormal{\textbf{B}}(mD)=\textnormal{Bs}|mD|.$
As mentioned in the introduction, the stable base locus of a $\mathbb{Q}-$Cartier divisor is not a numerical invariant; this can be seen by considering any smooth projective $X$ with $H^{1}(X,\mathcal{O}_{X}) \neq 0$ and comparing the trivial line bundle on $X$ to a non-torsion line bundle of degree 0 on $X.$ We now introduce the “outer and inner approximations” of the stable base locus.
**Definition:** Let $X$ be an irreducible projective variety and let $D$ be an $\mathbb{R}-$Cartier divisor on $X.$
- [The **augmented base locus** of $D$ is $$\textbf{B}_{+}(D)=\bigcap_{A}\textbf{B}(D-A)$$ where the intersection is taken over all ample $\mathbb{R}-$Cartier divisors $A$ for which $D-A$ is $\mathbb{Q}-$Cartier.]{}
- [The **restricted base locus** of $D$ is $$\textbf{B}_{-}(D)=\bigcup_{A}\textbf{B}(D+A)$$ where the union is taken over all ample $\mathbb{R}-$Cartier divisors $A$ for which $D+A$ is $\mathbb{Q}-$Cartier.]{}
While $\textbf{B}_{+}(D)$ is known to be Zariski-closed, we only know at this point that $\textbf{B}_{-}(D)$ is at worst a countable union of subvarieties of $X$ (Proposition 1.19 in [@ELMNP]).
The following result is a straightforward consequence of the definitions.
\[basicloc\] For all $\mathbb{R}-$Cartier divisors $D$ on $X,$ the following statements hold:
- [$\textnormal{\textbf{B}}_{-}(D)$ and $\textnormal{\textbf{B}}_{+}(D)$ are numerical invariants of $D,$ so that they are both well-defined for any class $D \in N^{1}_{\mathbb{R}}(X).$]{}
- [If $D$ is a $\mathbb{Q}-$Cartier divisor, then $\textnormal{\textbf{B}}_{-}(D) \subseteq \textnormal{\textbf{B}}(D) \subseteq \textnormal{\textbf{B}}_{+}(D).$ ]{}
Ampleness, nefness, and bigness can all be characterized in terms of augmented and restricted base loci:
\[char\_cones\] For all $D \in N^{1}_{\mathbb{R}}(X),$ the following hold:
- $D$ is ample if and only if $\textnormal{\textbf{B}}_{+}(D)=\emptyset.$
- $D$ is nef if and only if $\textnormal{\textbf{B}}_{-}(D)=\emptyset.$
- $D$ is big if and only if $\textnormal{\textbf{B}}_{+}(D) \neq X.$
- $D$ is pseudoeffective if and only if $\textnormal{\textbf{B}}_{-}(D) \neq X.$
We refer to [@ELMNP] for the proofs. In *loc. cit.*,(i) and (iii) are Example 1.7, while (ii) and (iv) are Example 1.18.
**Definition:** A class $D \in N^{1}_{\mathbb{R}}(X)$ is **stable** if $\textbf{B}_{-}(D)=\textbf{B}_{+}(D).$
Note that if $D$ is stable, we may speak of its stable base locus $\textbf{B}(D),$ and that the stable classes with empty stable base locus are precisely the ample classes.
The next two results will be used in the proofs of Theorems C and D; we refer to [@ELMNP] for their proofs.
\[stab\_dense\] (1.26 in [@ELMNP]) The set of stable classes is open and dense in $N^{1}_{\mathbb{R}}(X).$ In fact, for every $D \in N^{1}_{\mathbb{R}}(X)$ there exists $\epsilon > 0$ such that for any ample class $A$ satisfying $\lVert A \rVert < \epsilon,$ $D-A$ is stable.
\[stab\_drop\] (1.21 in [@ELMNP]) For every $\mathbb{R}-$divisor $D$, there is an $\epsilon > 0$ such that $\textnormal{\textbf{B}}_{-}(D-A)=\textnormal{\textbf{B}}_{+}(D-A)=\textnormal{\textbf{B}}_{+}(D)$ for every ample $A$ with $\lVert A \rVert < \epsilon.$
The following theorem of Nakamaye gives a useful characterization of the augmented base locus of a nef and big divisor. We refer to [@Nak] or p.249-251 in [@Laz2] for the proof.
\[nak\_base\_loc\] (Theorem 0.3 in [@Nak]) If $D$ is a nef and big divisor on a smooth projective variety $X,$ then $\textbf{B}_{+}(D)$ is the union of all positive-dimensional subvarieties $V$ of $X$ for which $D^{\dim{V}} \cdot V = 0.$
**Remark:** This result, while entirely sufficient for our purposes, has been generalized to arbitrary big divisors; see Theorem C in [@ELMNP2].
The Volume Function
-------------------
First, we briefly outline how the definition of the volume given in the introduction gives rise to a continuous function on the Néron-Severi space. In this subsection, $X$ will always denote an irreducible projective variety.
We omit proofs, referring to Section 2.2C of [@Laz1].
[((i) of Proposition 2.2.35 in [@Laz1])]{.nodecor} If $k$ is a positive integer and $D$ is a Cartier divisor on $X,$ then $$\textnormal{vol}_{X}(kD)=k^{n} \cdot \textnormal{vol}_{X}(D).$$
As a result, for any $\mathbb{Q}-$Cartier divisor $D$ on $X,$ we may define $$\textnormal{vol}_{X}(D):=\frac{1}{k^{n}} \cdot \textnormal{vol}_{X}(kD),$$ where $k$ is a positive integer for which $mD$ is an integral Cartier divisor.
\[vol\_num\_inv\] [(Proposition 2.2.41 in [@Laz1])]{.nodecor} Let $D_{1}$ and $D_{2}$ be Cartier divisors on $X$ which are numerically equivalent. Then $$\textnormal{vol}_{X}(D_{1})=\textnormal{vol}_{X}(D_{2}).$$
It follows that $\textrm{vol}_{X}:N^{1}_{\mathbb{Q}}(X) \rightarrow [0,\infty)$ is a well-defined function. The next result ensures that $\textrm{vol}_{X}$ can be uniquely extended to a continuous real-valued function on $N^{1}_{\mathbb{R}}(X).$
\[vol\_cont\] [(Theorem 2.2.44 in [@Laz1])]{.nodecor} Let $\lVert \cdot \rVert$ be any norm on $N^{1}_{\mathbb{R}}(X).$ Then there exists a constant $C > 0$ such that $$|\textnormal{vol}_{X}(\eta)-\textnormal{vol}_{X}({\eta}')| \leq C \cdot (\max(\lVert \eta \rVert , \lVert {\eta}' \rVert))^{n-1} \cdot \lVert{\eta-{\eta}'}\rVert$$ for any two classes $\eta,{\eta}' \in N^{1}_{\mathbb{Q}}(X).$
As stated earlier, the volume of any ample divisor is its top self-intersection. The following generalization, which is an immediate corollary of Theorem 1.4.40 in [@Laz1], will be used in the proof of Theorems E and G.
\[vol\_nef\] If $D$ is a nef divisor on an irreducible projective variety $X,$ then $\textnormal{vol}_{X}(D)=D^{n}.$
The Residuation Map {#resid}
===================
We begin with the natural generalization of the Abel map (which was defined in the introduction).
Let $C$ be a curve, and let $r$ and $d$ be positive integers. Then there is a fine moduli variety $G^{r}_{d}(C)$ parametrizing linear series of degree $d$ and dimension $r$ on $C,$ a determinantal subvariety $W^{r}_{d}(C)$ of $\textnormal{Pic}^{d}(C)$ parametrizing line bundles on $C$ of degree $d$ with at least $r+1$ global sections (we will refer to $W^{0}_{d}(C)$ as $W_{d}(C)$), and a natural morphism $$a_{r,d}:G^{r}_{d}(C) \rightarrow W^{r}_{d}(C)$$ which takes each linear series to its associated line bundle (we refer to Section 3 of Chapter IV of [@ACGH] for details).
The exceptional locus of $a_{r,d}$ is the subvariety $\mathfrak{I}^{r}_{d}(C)$ of $G^{r}_{d}(C)$ which parametrizes incomplete linear series.
Since $G^{0}_{d}(C)$ is canonically isomorphic to $C_{d},$ we have that $$a_{0,d}:C_{d} \rightarrow W_{d}(C)$$ is the Abel map $a_{d}.$ The inverse image $a_{0,d}^{-1}(W^{r}_{d}(C)),$ which is denoted by $C^{r}_{d},$ parametrizes effective divisors of degree $d$ which move in a linear series of dimension at least $r.$ Note that the exceptional locus $\mathfrak{I}^{0}_{d}(C)$ of $a_{0,d}$ is simply $C^{1}_{d}.$
If $C$ is a curve of genus $g$ and $2 \leq d \leq g-1,$ then there is an isomorphism $$\xymatrix{\tau : \textnormal{Pic}^{2g-2-d}(C) \ar[r]^-{\simeq} &\textnormal{Pic}^{d}(C)\\}$$ defined by $\tau(\mathcal{L})=K_{C} \otimes \mathcal{L}^{-1}.$ By the Riemann-Roch theorem, this restricts to an isomorphism $$\xymatrix{\tau : W^{g-d-1}_{2g-2-d}(C) \ar[r]^-{\simeq} &W_{d}(C)\\}.$$ This in turn lifts via $a_{0,d}$ and $a_{g-d-1,2g-2-d}$ to a birational map $$\widetilde{\tau} : G^{g-d-1}_{2g-2-d}(C) \dashrightarrow C_{d}.$$ Note that $\widetilde{\tau}$ is a biregular isomorphism precisely when $C^{1}_{d}=\emptyset.$
The following theorem is due to Gieseker.
\[gieseker\] (1.6 on p. 214 of [@ACGH]) Let $C$ be a general curve of genus $g.$ Let $d,r$ be integers satisfying $d \geq 1$ and $r \geq 0.$ Then $G^{r}_{d}(C)$ is smooth of dimension $g-(r+1)(g-d+r).$
It is a straightforward consequence of Theorem \[gieseker\] that when $C$ is a general curve of genus $g$ the codimension of $C^{1}_{d}$ is $g-d+1$ and the codimension of $\mathfrak{I}^{g-d-1}_{2g-2-d}(C)$ is 2, so that $\widetilde{\tau}$ is an isomorphism of smooth varieties in codimension 1.
Another consequence is that a general curve of genus $g$ is $\lceil{\frac{g}{2}+1}\rceil-$gonal. For all such curves, then, $\widetilde{\tau}$ fails to be a biregular isomorphism whenever $\frac{g}{2}+1 \leq d \leq g-1.$
\[picard\_isom\] Let $C$ be a general curve of genus $g \geq 3$ and let $d \geq 3$ be an integer satisfying $\frac{g}{2}+1 \leq d \leq g-1$.
- [$\widetilde{\tau}$ induces an isomorphism $$\xymatrix{
\widetilde{\tau}^{\ast} : \textnormal{Pic}(C_{d}) \ar[r]^-{\simeq} &\textnormal{Pic}(G^{g-d-1}_{2g-2-d}(C))\\}.$$]{}
- [Let $\mathcal{L}$ be a line bundle on $C_{d}$ and $\mathcal{M}$ be a line bundle on $G^{g-d-1}_{2g-2-d}(C).$ Then $$\begin{aligned}
H^{0}(C_{d},(\tau^{-1})^{\ast}\mathcal{M}) &\simeq H^{0}(G^{g-d-1}_{2g-2-d}(C),\mathcal{M})\\
H^{0}(G^{g-d-1}_{2g-2-d}(C),\tau^{\ast}\mathcal{L}) &\simeq H^{0}(C_{d},\mathcal{L})\end{aligned}$$]{}
Let $U=C_{d}-C^{1}_{d}$ and $V=G^{g-d-1}_{2g-2-d}(C)-\mathfrak{I}^{g-d-1}_{2g-2-d}(C).$ Clearly ${\tau}|_{V}:V \rightarrow U$ is an isomorphism and ${\tau}^{-1}|_{U}:U \rightarrow V$ is its inverse. These furnish natural isomorphisms $$H^{0}(U,\mathcal{L}|_{U})\simeq H^{0}(V,{\tau}^{\ast}\mathcal{L}|_{V})$$ $$H^{0}(V,\mathcal{M}|_{V}) \simeq H^{0}(U,({\tau}^{-1})^{\ast}\mathcal{M}|_{U})$$
Since $C$ is general, it follows from Hartogs’ theorem that these isomorphisms extend to all of $C_{d}$ and $G^{g-d-1}_{2g-2-d}(C).$
**Remark:** It follows from the first part of this proof that for any curve $C$ the singular locus of $G^{g-d-1}_{2g-2-d}(C)$ is contained in $\mathfrak{I}^{g-d-1}_{2g-2-d}(C).$
**Remark:** It is possible for $\mathfrak{I}^{g-d-1}_{2g-2-d}(C)$ to be of codimension 1 in $G^{g-d-1}_{2g-2-d}(C)$ even if $C^{1}_{d}$ is of codimension 2 in $C_{d}.$ For example, if $C$ is a nonhyperelliptic trigonal curve of genus 5, then $C^{1}_{3} \simeq \mathbb{P}^{1}$ and $\mathfrak{I}^{1}_{5}(C) \simeq \mathbb{P}^{2}.$
The following corollary is immediate.
$\widetilde{\tau}$ induces an isomorphism $$\xymatrix{
\widetilde{\tau}^{\ast} : N^{1}_{\mathbb{R}}(C_{d}) \ar[r]^-{\simeq} &N^{1}_{\mathbb{R}}(G^{g-d-1}_{2g-2-d}(C))\\}$$ of Néron-Severi spaces.
Our next task is to determine $\widetilde{\tau}^{\ast}$ explicitly. The divisor classes $x$ and $\theta$ on $C_{d}$ can be generalized in a straightforward fashion to obtain divisor classes on $G^{g-d-1}_{2g-2-d}(C).$
Recall that a vector bundle $E$ is ample if the hyperplane class on the *subbundle* projectivization $\mathbb{P}_{sub}(E^{\ast})$ of $E^{\ast}$ is ample.
\[det\_ample\] Let $X$ be a smooth projective variety, and let $E$ be an ample vector bundle of rank $s$ on $X.$ Then for all $s' \leq s,$ the Plücker class on the associated Grassmann bundle $G(s',E^{\ast})$ of rank-$s'$ subbundles of $E^{\ast}$ is ample.
If $\nu:G(s',E^{\ast}) \rightarrow X$ is the structure map, then the determinant of the inclusion $S_{s',E^{\ast}} \hookrightarrow \nu^{\ast}(E)$ of the tautological subbundle induces the Plücker embedding $G(s',E^{\ast}) \hookrightarrow \mathbb{P}_{sub}({\wedge}^{s'}E^{\ast}).$ The result then follows from the fact that the amplitude of $E$ implies the amplitude of its exterior powers (part (ii) of Corollary 6.1.16 on p.15 of [@Laz2]).
While we feel that the following result should be well known, we include its proof for lack of a reference.
\[plucker\_ample\] For any effective divisor $D$ of degree $r+1$ on $C,$ the set $$\widehat{X}_{D}:=\{(V,\mathcal{M}) \in G^{r}_{d}(C):V \cap H^{0}(\mathcal{M}(-D)) \neq 0\}$$ has the natural structure of an ample divisor on $G^{r}_{d}(C).$
We first recall some aspects of the construction of $G^{r}_{d}(C)$ in Section 3 of Chapter IV of [@ACGH]. Fix a Poincaré bundle $\mathcal{L}$ on $C \times \textrm{Pic}^{d}(C),$ and let $D'$ be an effective divisor on $C$ of degree $2g-d-1.$
If $\eta:C \times \textrm{Pic}^{d}(C) \rightarrow \textrm{Pic}^{d}(C)$ is projection onto the second factor, and $\Gamma$ is the product divisor $(D+D') \times \textrm{Pic}^{d}(C),$ then the direct image sheaf $\eta_{\ast}\mathcal{L}(\Gamma)$ is locally free of rank $g+r+1$ and its fibre over a line bundle $\mathcal{M}$ of degree $d$ on $C$ is $H^{0}(\mathcal{M}(D+D')).$ Indeed, $h^{1}(\mathcal{M}(D+D'))=0$ by Serre duality since the degree of $\mathcal{M}(D+D')$ is $2g+r$, so this follows from Riemann-Roch and base change in cohomology.
If ${\Gamma}'$ is the product divisor $D' \times \textrm{Pic}^{d}(C),$ then an entirely analogous argument tells us that $\eta_{\ast}\mathcal{L}({\Gamma}')$ is a rank-$g$ subbundle of $\eta_{\ast}\mathcal{L}(\Gamma).$ Since the dual of $\eta_{\ast}\mathcal{L}(\Gamma)$ is ample by Proposition 2.2 on p.310 of [@ACGH], the Plücker divisor ${\sigma}'$ on the Grassmann bundle $G(r+1,\eta_{\ast}\mathcal{L}(\Gamma))$ associated to $\eta_{\ast}\mathcal{L}({\Gamma}')$ is ample by Lemma \[det\_ample\].
For each line bundle $\mathcal{M}$ of degree $d$ on $C,$ there is a commutative diagram $$\xymatrix{
0 \ar[r] &H^{0}(\mathcal{M}) \ar[r]^{f_{1}} &H^{0}(\mathcal{M}(D+D')) \ar[r]^{g_{1}} &H^{0}(\mathcal{M}(D+D')|_{D+D'})\\
0 \ar[r] &H^{0}(\mathcal{M}(-D)) \ar[u]^{i} \ar[r]^{f_{2}} &H^{0}(\mathcal{M}(D')) \ar[u]^{i'} \ar[r]^{g_{2}} &H^{0}(\mathcal{M}(D')|_{D+D'}) \ar[u]^{i''}}$$ in which both rows are exact and all vertical arrows are injective, so that a diagram chase gives the equality $$(f_{1} \circ i)\bigl(H^{0}(\mathcal{M}(-D))\bigr)=f_{1}\bigl(H^{0}(\mathcal{M})\bigr) \cap i'\bigl(H^{0}(\mathcal{M}(D'))\bigr).$$
Therefore if $V$ is a subspace of $H^{0}(\mathcal{M}),$ we have $$f_{1}(V) \cap (f_{1} \circ i)\bigl(H^{0}(\mathcal{M}(-D))\bigr)=f_{1}(V) \cap i'\bigl(H^{0}(\mathcal{M}(D'))\bigr).$$
It follows at once from the definitions that $\widehat{X}_{D}=G^{r}_{d}(C) \cap {\sigma}'.$ Consequently $\widehat{X}_{D},$ which has a cycle structure induced by its being an intersection of cycles and is the restriction of an ample divisor, is ample.
As $D$ varies over $C_{r+1},$ the divisors $\widehat{X}_{D}$ sweep out an algebraic family whose common numerical class we will denote by $\widehat{X}.$ Also, we will denote by $\widehat{\theta}$ the numerical class of the pullback to $G^{r}_{d}(C)$ of a theta-divisor on $\textrm{Pic}^{d}(C).$
\[explicit\_isom\] Under the isomorphism ${\widetilde{\tau}}^{\ast}:N^{1}_{\mathbb{R}}(C_{d}) \rightarrow N^{1}_{\mathbb{R}}(G^{g-d-1}_{2g-2-d}(C)),$ $${\widetilde{\tau}}^{\ast}(\theta)=\widehat{\theta}, \hspace{0.2cm} {\widetilde{\tau}}^{\ast}(\theta-x)=\widehat{X}.$$
If $\widehat{\tau}: \textrm{Pic}^{2g-2-d}(C) \rightarrow \textrm{Pic}^{d}(C)$ is the morphism induced by taking Serre duals and $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are line bundles on $C$ of respective degrees $2g-2-d$ and $d$ satisfying $\mathcal{L}_{1} \otimes \mathcal{L}_{2} \simeq K_{C},$ we have the commutative diagram $$\xymatrix{
\textrm{Pic}^{0}(C) \ar[d]^{(-1)} \ar[r]^{\cdot \otimes \mathcal{L}_{1}} &\textrm{Pic}^{2g-2-d}(C) \ar[d]^{\widehat{\tau}} &G^{g-d-1}_{2g-2-d}(C) \ar[l] \ar@{-->}[d]^{\tau}\\
\textrm{Pic}^{0}(C) \ar[r]^{\cdot \otimes \mathcal{L}_{2}} &\textrm{Pic}^{d}(C) &C_{d} \ar[l]}$$ where the leftmost vertical arrow is multiplication by $-1$ and the left horizontal arrows on the top and bottom are multiplication by $\mathcal{L}_{1}$ and $\mathcal{L}_{2},$ respectively. Since multiplication by $-1$ induces the identity on the cohomology of $\textrm{Pic}^{0}(C)$, it induces the identity on the Néron-Severi group of $\textrm{Pic}^{0}(C).$ Therefore ${\tau}^{\ast}$ takes the theta class on $C_{d}$ to $\widehat{\theta}$ on $G^{g-d-1}_{2g-2-d}(C).$
If $D$ is an effective divisor of degree $g-d$ on $C,$ then it follows immediately from Riemann-Roch that $D' \in C_{d}$ satisfying $h^{0}(D')=1$ is subordinate to $|K_{C}(-D)|$ precisely when $D$ is subordinate to $|K_{C}(-D')|.$ This can be rephrased as saying that away from the loci of indeterminacy of $\tau$ and ${\tau}^{-1},$ the divisor $\widehat{X}_{D}$ is isomorphic to ${\Gamma}_{d}(K_{C}(-D))$ via $\tau.$ Since the fundamental class of ${\Gamma}_{d}(K_{C}(-D))$ is $\theta-x$ by Lemma \[subordinate\], we have that $\tau^{\ast}(\theta-x)=\widehat{X}.$
Since $\widetilde{\tau}$ is a biregular isomorphism if $C^{1}_{d}=\emptyset$, an immediate application of Proposition \[explicit\_isom\] yields the following result. Note that $C$ is not assumed to be general.
\[subnef\] If $C$ is any curve and $d$ is a positive integer for which $C^{1}_{d}=\emptyset,$ then the class $\theta-x$ on $C_{d}$ is ample.
In the case where $C$ is a general curve of even genus $g,$ Corollary \[subnef\] is subsumed by Pacienza’s computation of the nef cone of $C_{\frac{g}{2}}$ in [@Pac]. However, it yields the best inner bound for the nef cone of $C_{d}$ for $3 \leq d \leq \frac{g+1}{2}$ currently available when $C$ is a general curve of odd genus $g \geq 5.$
Proofs of The Main Results {#proofs}
==========================
The Nonhyperelliptic Case
-------------------------
In this section we prove Theorems A,C,D,E, and F. \[nonhyp\]
### The Effective Cone {#seceff}
The following calculation establishes (i) of Theorem A.
Let $C$ be a very general curve of genus $g \geq 4.$ For $2 \leq d \leq g$ let $D_{d}$ be the reduced divisor on $C_{d}$ supported on the set $$\bigcup_{p \in C}{\Gamma}_{d}\bigl(K_{C}(-(g-d+1)p)\bigr)$$ Then the numerical class of $D_{d}$ is $$(g-d+1)\bigl((g^{2}-dg+(d-2))\theta-(g^{2}-(d-1)g-2)x\bigr).$$
Fix $d-1$ general points $q_{1},...q_{d-1}$ on C. The two test curves in $C_{d}$ that we will use to compute the numerical class of $D_{d}$ are $\Delta_{d}$ and the curve $${\chi}_{d}:=\bigcap_{j=1}^{d-1}X_{q_{j}}=\{p+q_{1}+...+q_{d-1}:p \in C\}$$ The numerical class of ${\chi}_{d}$ is $x^{d-1},$ and by Proposition \[other\_diag\], the numerical class of $\Delta_{d}$ is $dx^{d-2}(((d-1)g+1)x-(d-1)\theta)$.
The intersection number ${\chi}_{d} \cdot D_{d}$ is the cardinality of the set $$\biggl\{q \in C : \exists p \in C \ni (g-d+1)p+q \leq |K_{C}(-q_{1}-...-q_{d-1})| \biggr\}$$ If $(g-d+1)p \leq |K_{C}(-q_{1}-...-q_{d-1})|,$ then there are $$((2g-2)-(d-1))-(g-d+1)=g-2$$ points $q$ (counting multiplicity) such that $$(g-d+1)p+q \leq |K_{C}(-q_{1}-...-q_{d-1})|,$$ and we may conclude that $${\chi}_{d} \cdot D_{d}=(g-2) \cdot {\Delta}_{g-d+1} \cdot {\Gamma}_{g-d+1}(K_{C}(-q_{1}-...-q_{d-1})).$$
When $d \neq \frac{g+1}{2},$ the intersection number $\Delta_{d} \cdot D_{d}$ is the cardinality of the set $$\mathfrak{K}_{g,d}:=\biggl\{q \in C: \exists p \in C \ni (g-d+1)p + dq \leq |K_{C}| \biggr\}$$ and when $d=\frac{g+1}{2},$ we have that $\Delta_{d} \cdot D_{d}$ is twice the cardinality of $\mathfrak{K}_{g,d}.$ Therefore our system of equations is $$\begin{aligned}
{\chi}_{d} \cdot D_{d} &= (g-2) \cdot {\Delta}_{g-d+1} \cdot {\Gamma}_{g-d+1}(K_{C}(-q_{1}-...-q_{d-1}))\\
\Delta_{d} \cdot D_{d} &= (1+\delta_{(d,\frac{g+1}{2})}) \cdot \Delta_{g-d+1,d} \cdot \Gamma_{g+1}(K_{C})\end{aligned}$$ If the numerical class of $D_{d}$ is $a\theta-bx$ for $a,b \in \mathbb{R},$ this becomes $$\begin{aligned}
ag-b &= g^{4}-2dg^{3}+(d^{2}+2d-4)g^{2}-(2d^{2}-5d+1)g-(2d-2)\\
adg-b &= dg^{4}-(2d^{2}-d+1)g^{3}+(d^{3}-2)g^{2}-(d^{3}-2d^{2}-1)g-(2d-2)\end{aligned}$$ after applying the class computations in Lemma \[subordinate\] and Propositions \[small\_diag\] and \[other\_diag\], and we have the solution $$\begin{aligned}
a &= g^{3}-(2d-1)g^{2}+(d^{2}-2)g-(d-1)(d-2)=(g-d+1)(g^{2}-dg+(d-2))\\
b &= g^{3}+(2-2d)g^{2}+(d^{2}-2d-1)g+(2d-2)=(g-d+1)(g^{2}-(d-1)g-2).\end{aligned}$$
We now turn to the proof of Theorem F. Before introducing the relevant divisor, we prove two preliminary lemmas.
\[combsum\] For all $m \geq 1,$ $$(2m+3)\cdot\displaystyle\sum_{l=0}^{m}(-1)^{l}(l+1)\displaystyle\binom{2m-l}{m}\displaystyle\binom{2m+2}{l+3}=-(m+2)\cdot\displaystyle\sum_{l=0}^{m}(-1)^{l}l(l+1)\displaystyle\binom{2m-l}{m}\displaystyle\binom{2m+3}{l+3}$$
It suffices to show that for all $m \geq 1,$ $$\displaystyle\sum_{l=0}^{m}(-1)^{l}(l+1)\displaystyle\binom{2m-l}{m}\Bigl(l(m+2)\displaystyle\binom{2m+3}{l+3}+(2m+3)\binom{2m+2}{l+3}\Bigr)=0.$$ First note that $$l(m+2)\displaystyle\binom{2m+3}{l+3}+(2m+3)\displaystyle\binom{2m+2}{l+3}=((m+1)l+2m)\displaystyle\binom{2m+3}{2m-l}$$ and $$\displaystyle\binom{2m-l}{m}\displaystyle\binom{2m+3}{2m-l}=\displaystyle\binom{2m+3}{m}\displaystyle\binom{m+3}{m-l}$$ for all $l,$ so that the left-hand side is proportional to the sum $$\displaystyle\sum_{l=0}^{m}(-1)^{l}\bigl((m+1)l^{2}+(3m+1)l+2m\bigr)\displaystyle\binom{m+3}{m-l}.$$ This is the $m$-th convolution of the sequences $\bigl\{(-1)^{l}((m+1)l^{2}+(3m+1)l+2m)\bigr\}_{l}$ and $\bigl\{\binom{m+3}{l}\bigr\}_{l}$ whose respective generating functions are $\frac{2m-2t}{(1+t)^{3}}$ and $(1+t)^{m+3}.$ Therefore it is the coefficient of $t^{m}$ in $(2m-2t)(1+t)^{m},$ which is zero.
\[orth\] If $|\mathcal{L}|$ is any pencil of degree $k+1$ on $C,$ then $\Gamma_{k}(\mathcal{L}) \cdot (\theta-(2-\frac{1}{k})x)=0.$
Fix a pencil $|\mathcal{L}|$ of degree $k+1$ on $C.$ It suffices to show that $\Gamma_{k}(\mathcal{L}) \cdot \theta=2k-1$ and $\Gamma_{k}(\mathcal{L}) \cdot x = k.$ By Lemmas \[intersect\] and \[subordinate\], we have that $$\begin{aligned}
\Gamma_{k}(\mathcal{L}) \cdot \theta &= (2k-1) \cdot \displaystyle\sum_{j=0}^{k-1}(-1)^{j}\displaystyle\binom{k-2+j}{j}\displaystyle\binom{2k-2}{k-1-j}\\
\Gamma_{k}(\mathcal{L}) \cdot x &= \displaystyle\sum_{j=0}^{k-1}(-1)^{j}\displaystyle\binom{k-2+j}{j}\displaystyle\binom{2k-1}{k-1-j}\end{aligned}$$ Since the sequences $\bigl\{(-1)^{j}\binom{k-2+j}{j}\bigr\}_{j},$ $\bigl\{\binom{2k-2}{j}\bigr\}_{j},$ and $\bigl\{\binom{2k-1}{j}\bigr\}_{j}$ have respective generating functions $(1+t)^{-(k-1)}$, $(1+t)^{2k-2}$, and $(1+t)^{2k-1}$, arguing as in the proof of Lemma \[combsum\] establishes the desired equalities.
Let $C$ be a general curve of genus $2k-1 \geq 5.$ Consider the diagram $$\xymatrix{
C_{k} \times C_{2k-5} \ar[d]^{\pi} \ar[r]^{\sigma} &C_{3k-5}\\
C_{k}}$$ where $\sigma$ is defined by $\sigma(D,E)=D+E$ and $\pi$ is projection. Since $C$ is general, we may deduce from Theorem \[gieseker\] that $C^{k-2}_{3k-5}$ is $(k-1)-$ dimensional. Furthermore, for each $D \in C_{k}$ we have that $\pi^{-1}(D) \cap \sigma^{-1}(C^{k-2}_{3k-5})$ is at most finite. It follows that the cycle $E_{(k)}:=\pi_{\ast}\sigma^{\ast}(C^{k-2}_{3k-5})$ is an effective divisor on $C_{k}.$
*Proof of Theorem F:* (i) By the push-pull formulas (e.g. Exercise D-8 in [@ACGH]) the class of $E_{(k)}$ is $$\displaystyle\frac{1}{k-1} \cdot \Bigl(\Bigl(\displaystyle\sum_{l=0}^{k-2}(-1)^{l}(l+1)\binom{2k-4-l}{k-2}\binom{2k-2}{l+3}\Bigr)\theta$$ $$+\Bigl(\displaystyle\sum_{l=0}^{k-2}(-1)^{l}l(l+1)\binom{2k-4-l}{k-2}\binom{2k-1}{l+3}\Bigr)x\Bigr)$$ Setting $m=k-2$ in Lemma \[combsum\], we see that this class is proportional to $\theta-(2-\frac{1}{k})x.$
\(ii) Let $D$ be an effective divisor on $C_{k}$ whose class is proportional to $\theta-tx$ for some $t > 2-\frac{1}{k}.$ By Lemma \[orth\], $D \cdot \Gamma_{k}(\mathcal{L})<0$ for all $\mathcal{L} \in W^{1}_{k+1}(C),$ so that $Z_{k} \subseteq \textbf{B}(D).$
\(iii) We employ a slight variation on an argument used in the proof of Theorem 5 of [@Kou]. First we show that $E_{(3)}$ is irreducible. We have a morphism $\mu: C \times W^{1}_{4}(C) \rightarrow C_{3}$ which takes each pair $(p,\mathcal{L})$ to the unique element of the linear system $|\mathcal{L}(-p)|.$ Since $C$ is general, the Brill-Noether locus $W^{1}_{4}(C)$ is a smooth irreducible curve, which implies the irreducibility of $C \times W^{1}_{4}(C).$ The irreducibility of $E_{(3)}$ then follows at once from the observation that $E_{(3)}$ is the image of $\mu.$
It follows from (i) that the class of $E_{(3)}$ is proportional to $\theta-\frac{5}{3}x.$ Suppose that there is an irreducible effective divisor $D'$ whose class is proportional to $\theta-tx$ for $t>\frac{5}{3}.$ Then for any $\mathcal{L} \in W^{1}_{4}(C)$ we have $\Gamma_{3}(\mathcal{L}) \cdot D' < 0,$ so that all such curves are contained in $D'.$ But this means $D'=E_{(3)},$ which is absurd.
### Results on Base Loci and Volume {#secbas}
The next theorem implies (ii) of Theorem A when combined with Lemma \[char\_cones\] and Theorem 3 in [@Kou], and it implies Theorem C when combined with Proposition \[plucker\_ample\].
\[stab\_classes\] Let $\mathcal{L}$ be a stable line bundle on $G^{g-d-1}_{2g-2-d}(C)$ with stable base locus $Z$ and numerical class $a\widehat{X}+b\widehat{\theta}$ satisfying $a>0$ and $a+b>0.$ Then the line bundle $({\tau}^{-1})^{\ast}\mathcal{L}$ on $C_{d}$ is stable with stable base locus $C^{1}_{d} \cup \tau^{-1}(Z).$
In particular, a line bundle $\mathcal{L}$ on $G^{g-d-1}_{2g-2-d}(C)$ with numerical class in the aforementioned range is stable precisely when the pullback bundle $({\tau}^{-1})^{\ast}\mathcal{L}$ on $C_{d}$ is stable.
Let $\mathcal{L}$ be a stable line bundle on $G^{g-d-1}_{2g-2-d}(C)$ satisfying the hypotheses, and let $\mathcal{M}:=(\tau^{-1})^{\ast}\mathcal{L}.$ By Proposition \[multiple\], we may assume without loss of generality that $\textrm{Bs}(|\mathcal{L}|)=\textbf{B}(\mathcal{L})=Z$ and $\textrm{Bs}(|\mathcal{M}|)=\textbf{B}(\mathcal{M}).$
The hypothesis on the coefficients $a$ and $b$ guarantees that the numerical class of $\mathcal{M},$ which is $(a+b)\theta-ax,$ lies in the fourth quarter of the $\theta,x-$plane, so that the stable base locus of $\mathcal{M}$ must contain $C^{1}_{d}.$ By Proposition \[picard\_isom\], pullback via ${\tau}^{-1}$ gives a natural isomorphism between $H^{0}(G^{g-d-1}_{2g-2-d}(C),\mathcal{L})$ and $H^{0}(C_{d},\mathcal{M}),$ so $\textbf{B}(\mathcal{M})=\textrm{Bs}(|\mathcal{M}|)=C^{1}_{d} \cup \tau^{-1}(Z).$ Indeed, if $x \in C_{d}-C^{1}_{d}$ is a basepoint of $|\mathcal{M}|,$ then $\tau(x)$ is a basepoint of $|\mathcal{L}|.$
The set of stable classes in in $N^{1}_{\mathbb{R}}(G^{g-d-1}_{2g-2-d}(C))$ having $Z$ as its stable base locus is open, so its image under $(\tau^{-1})^{\ast}$ is open as well. If $t_{0}:=\frac{a}{a+b},$ then by our previous calculation and Propositions \[stab\_dense\] and \[stab\_drop\], we have that for some $\epsilon > 0,$ $\theta-tx$ is stable with stable base locus $C^{1}_{d} \cup \tau^{-1}(Z)$ for all $t$ satisfying $0<|t-t_{0}|<\epsilon.$ We then have by the definitions of the augmented and restricted base loci that $\textbf{B}_{-}( \mathcal{M})=C^{1}_{d} \cup \tau^{-1}(Z)=\textbf{B}_{+}(\mathcal{M}).$
We now prove Theorems D and E. The following result is a rephrasing of Lemma 2.2 in [@Pac]; we refer to *loc. cit.* for its proof.
\[nef\_small\_diag\] If $C$ is any curve of genus $g$ and $d \geq 3,$ the numerical class $-\theta+dgx$ in $N^{1}_{\mathbb{R}}(C_{d})$ is nef and big, and its augmented base locus is $\Delta_{d}.$ In particular, $-\theta+dgx$ spans a boundary ray of the nef cone of $C_{d}.$
*Proof of Theorem D:* Let $C$ be a general curve of genus $g \geq 5.$ By Propositions \[stab\_dense\] and \[stab\_drop\] and Lemma \[nef\_small\_diag\], there exists a positive $t_{0}<g^{2}-g$ such that the class $-\theta+tx$ on $C_{g-1}$ is stable with stable base locus $\Delta_{g-1}$ whenever $t_{0}<t<g^{2}-g.$ It then follows from Theorem \[stab\_classes\] and Proposition \[explicit\_isom\] that $\widetilde{\tau}^{\ast}(-\theta+tx)=(t-1)\theta-tx$ is stable with stable base locus $C^{1}_{g-1} \cup \widetilde{\tau}^{-1}(\Delta_{g-1}).$ Since a given curve has only finitely many Weierstrass points, $\widetilde{\tau}^{-1}(\Delta_{g-1}) \nsubseteq C^{1}_{g-1}.$ Finally, since $g \geq 5,$ the dimension of $C^{1}_{g-1}$ is at least 2, so that $C^{1}_{g-1} \cup \widetilde{\tau}^{-1}(\Delta_{g-1})$ is non-equidimensional.
When $C$ is general, all its Weierstrass points are ordinary (e.g. [@EisHar]), so that $\widetilde{\tau}^{-1}(\Delta_{g-1})$ and $C^{1}_{g-1}$ are disjoint. However, it is possible for the stable base locus in the previous proof to be connected. The locus $\mathfrak{W}$ in $\mathcal{M}_{g}$ parametrizing curves with a Weierstrass point having gap sequence $\{1, \cdots ,g-2,g,g+1\}$ is of codimension 1, and since we are assuming $g \geq 5$ the hyperelliptic locus in $\mathcal{M}_{g}$ has codimension at least 3. Thus if the isomorphism class of $C$ is a general point in $\mathfrak{W},$ the above proof is valid for $C,$ and $\widetilde{\tau}^{-1}(\Delta_{g-1}) \cap C^{1}_{g-1} \neq \emptyset.$
*Proof of Theorem E:* It is an immediate consequence of (ii) in Proposition \[picard\_isom\] that for any line bundle $\mathcal{L}$ on $C_{g-1}$ we have $\textnormal{vol}_{C_{g-1}}(\mathcal{L})=\textnormal{vol}_{C_{g-1}}(\widetilde{\tau}^{\ast}\mathcal{L}).$ The result then follows from applying Lemmas \[intersect\], \[vol\_nef\], and \[nef\_small\_diag\].
The Hyperelliptic Case {#hyp}
----------------------
Let $C$ be a general hyperelliptic curve of genus $g \geq 2.$ Note that since $C^{1}_{d}$ is neither empty nor of the expected dimension $$(g-2(g-d+1))+1=2d-(g+1)$$ the computation of the class of $C^{1}_{d}$ given on p.326 of [@ACGH] does not apply.
*Proof of Theorem G:* *(i)* We denote the hyperelliptic pencil on $C$ by $|\mathcal{L}|.$ By Clifford’s Theorem, the dimension of the linear series $|{\mathcal{L}}^{\otimes (d-1)}|$ is $d-1.$ We have from Lemma \[subordinate\] that $\Gamma_{d}(\mathcal{L}^{\otimes (d-1)})$ is a divisor on $C_{d}$ and that its class is $\theta-(g-d+1)x.$ For any $D \in C_{d},$ it follows from Riemann-Roch that $\dim{|D|} \geq 1$ if any only if $$\dim|\mathcal{L}^{\otimes (d-1)}(-D)| \geq 0.$$ This equivalence of algebraic conditions gives the equality of cycles $$C^{1}_{d}=\Gamma_{d}(\mathcal{L}^{\otimes (d-1)}).$$ Since $C^{1}_{d}$ is the exceptional locus of the divisorial contraction $a_{0,d},$ it is not big; therefore its class spans a boundary of the effective cone of $C_{d}.$
*(ii):* Since $\theta$ is nef and big for $2 \leq d \leq g-1,$ this is a consequence of Theorem \[nak\_base\_loc\].
*(iii):* For $t \in (0,g-d+1],$ $$\textrm{vol}_{C_{d}}(\theta-tx)=\textrm{vol}_{C_{d}}{\Bigl(}(1-\frac{t}{g-d+1})\theta+(\frac{t}{g-d+1})(\theta-(g-d+1)x){\Bigr)}$$ $$=\Bigl(\frac{t}{g-d+1}\Bigr)^{d} \cdot \textrm{vol}_{C_{d}}\Bigl((\frac{g-d+1}{t}-1)\theta+(\theta-(g-d+1)x)\Bigr).$$
By (ii), $\theta$ spans a boundary of the movable cone, so that $(\frac{g-d+1}{t}-1)\theta$ is the positive part of a Zariski decomposition $(\frac{g-d+1}{t}-1)\theta+(\theta-(g-d+1)x).$ It then follows from Proposition 3.20 in [@Bou] and Lemma \[vol\_nef\] that $$\Bigl(\frac{t}{g-d+1}\Bigr)^{d} \cdot \textrm{vol}_{C_{d}}\Bigl((\frac{g-d+1}{t}-1){\theta}+(\theta-(g-d+1)x)\Bigr)=$$ $$\Bigl(\frac{t}{g-d+1}\Bigr)^{d} \cdot \Bigl(\frac{g-d+1}{t}-1\Bigr)^{d} \cdot \frac{g!}{(g-d)!}=\frac{g!}{(g-d)!}\Bigl(1-\frac{t}{g-d+1}\Bigr)^{d}.$$
Department of Mathematics, Stony Brook University\
Stony Brook, NY,11794-3651\
current address: Department of Mathematics, University of Michigan\
2074 East Hall, 530 Church Street\
Ann Arbor, MI 48109-1043\
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abstract: |
We model the shape and density profile of the dark matter halo of the low surface brightness, superthin galaxy UGC 7321, using the observed rotation curve and the scale height data as simultaneous constraints. We treat the galaxy as a gravitationally coupled system of stars and gas, responding to the gravitational potential of the dark matter halo. An isothermal halo of spherical shape with a core density in the range of 0.039 - 0.057 $M$$_{\odot}$ $pc$$^{-3}$ and a core radius between 2.5 - 2.9 $kpc$, gives the best fit to the observations for a range of realistic gas parameters assumed. We find that the best-fit core radius is only slightly higher than the stellar disc scale length (2.1 $kpc$), unlike the case of the high surface brightness galaxies where the halo core radius is typically 3-4 times the disc scale length of the stars. Thus our model shows that the dark matter halo dominates the dynamics of the low surface brightness, superthin galaxy UGC 7321 at all radii, including the inner parts of the galaxy.\
address:
- 'Department of Physics,Indian Institute of Science, Bangalore 560012, India'
- 'Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS-42, Cambridge, MA 02138, USA'
- 'Department of Physics,Indian Institute of Science, Bangalore 560012, India'
author:
- Arunima Banerjee
- 'Lynn D. Matthews'
- 'Chanda J. Jog'
title: Dark matter dominance at all radii in the superthin galaxy UGC 7321
---
galaxies: ISM ,galaxies: kinematics and dynamics ,galaxies: spiral ,galaxies: structure ,galaxies: halos ,galaxies: individual: UGC 7321
Introduction {#sec:intro}
============
Since spiral galaxies are rotationally supported systems, disc rotation curves generally serve as valuable tracers of the gravitational potential in the galactic plane. Through traditional mass-modelling, the observed curve is routinely used to infer the mass distribution of galaxies and hence their dark matter contents (e.g. Begeman 1987; Kent 1987; Geehan et al. 2006). In contrast, the thickness of the gas layer depends on the vertical gravitational force and thus traces the potential perpendicular to the plane e.g.,(Narayan & Jog 2002a).
Recently, the rotation curve and the outer galactic flaring data have been used together to probe the dark matter halos of a few galaxies. The rotation curve mainly determines the mass enclosed within a given radius, and therefore the power-law index of the density profile of the halo. The flaring curve, on the other hand, determines its shape uniquely. So, both the constraints have to be used on an equal footing to correctly determine the parameters of the dark matter halo of any galaxy.
The scale height data coupled with the rotation curve has been used to study the dark matter halos of NGC 4244 (Olling 1996) , NGC 891 (Becquaert & Combes 1997) and the Galaxy (Olling & Merrifield 2000, 2001) in the past. Narayan et al. (2005) studied the Galactic dark matter halo by rigorously incorporating the self-gravity of the gas into their model for the Galaxy unlike some of the previous studies given in the literature. They concluded that a steeper-than-isothermal, spherical halo best fits the observations, the scale height data at that time being available up to galactocentric distances of 24 $kpc$. These results were confirmed by Kalberla et al. (2007), who, however, included a dark matter ring in their model to explain their extended scale height data available till 40 $kpc$. In our previous work (Banerjee & Jog 2008), we studied the dark matter halo of M31, where we developed a model similar to the Galaxy (Narayan et al. 2005) . However, in addition, we included the bulge into the model, and also varied the shape of the halo as a free parameter, unlike the Galaxy case. Further, we fitted the rotation curve over the entire radial range instead of pinning it at a single point like the Galaxy case. We scanned the four dimensional grid of the four free parameters characterizing the halo, in a systematic manner, and found that an isothermal halo of an oblate shape of axis ratio $q$ = 0.4 gives the best fit to the available data.
In this paper, we apply for the first time a similar approach to study the dark matter halo properties of a low surface brightness (LSB) “superthin” galaxy: UGC 7321. UGC 7321 is a bulgeless, pure disc galaxy of Hubble Type Sd, and has a highly flattened stellar disc with a planar-to-vertical axis ratio of 10.3. A few of its key properties are summarized in Table 1. The galaxy has an extended disc, and the scale height data are available up to 6-7 disc scale lengths (Matthews & Wood 2003). So it is highly suitable for the application of the above method to probe its dark matter halo properties.
[^1] \[tab:gmrt\] 0.1in
Parameters Value
------------------------------------------------------------------------------------- ------------------------------
$A_{opt}$($kpc$) $16.3$ [^2]
$L_{B}$($L_{\odot}$) $1.0$ $\times$ $10^{9}$ [^3]
$M_{{\mbox{H\,{\sc i}}}}$($M_{\odot}$) $1.1$ $\times$ $10^{9}$ [^4]
$h_{R}$($kpc$) $2.1$ [^5]
$z_{0}$($pc$) $150$ [^6]
$\mu_{B,i}(0)$($mag$ $arcsec^{-2}$) $23.5$ [^7]
$v_{rot}$($km$$s^{-1}$) $105$
$Star$ $formation$ $rate$ ($M_{\odot}$ per year for massive stars $\ge 5 M_{\odot}$ $\sim 0.02$
Based on traditional mass-modelling which only uses the observed rotation curve as the constraint, it has been found that the late-type, low surface brightness galaxies are generally dark matter dominated, often within the inner portions of their stellar discs (de Blok & McGaugh 1997) . In the case of UGC 7321, other lines of evidence have already suggested that it, too, is a highly dark matter-dominated galaxy. It has large ratios of its dynamical mass to its mass and blue luminosity, ($M_{\rm dyn}/M_{HI}$ = 31 and $M_{\rm dyn}/L_{B}$ = 29, respectively; (cf. Roberts & Haynes 1994) , and an extraordinarily small stellar disc scale height ($\sim$150 $pc$ for a distance of 10 $Mpc$ based on an exponential fit; Matthews 2000). These properties suggest the need for a massive dark halo to stabilize the disc against vertical bending instabilities (Zasov et al. 1991) .
UGC 7321 is devoid of a central bulge component (Matthews et al. 1999) and its molecular gas content appears to be dynamically insignificant (Matthews & Gao 2001; Matthews & Wood 2001). We, therefore, model the galaxy as a gravitationally coupled, two-component system of stars and atomic hydrogen gas with the dark matter halo acting as a source of external force to this system. We use a four-parameter density profile for the dark matter halo (de Zeeuw & Pfenniger 1988; Bequaert & Combes 1997): the core density, the core radius, the power-law density index and the axis ratio of the halo being the four free parameters characterizing it. We methodically vary the four parameters within their respective feasible ranges, and try to obtain an optimum fit to both the observed rotation curve and the vertical scale height data at the same time. As we shall see, this method predicts a spherical, isothermal halo with a core density of about 0.039- 0.057 $M_{\odot}$ $pc$$^{-3}$ and core radius of 2.5 - 2.9 $kpc$ for this galaxy.
The layout of the present paper is as follows. We briefly discuss the model in §2, and in §3 the method of solving the equations and the input parameters used is discussed. In §4, we present the results, followed by the discussion and conclusions in §§5 and 6, respectively.
Description of the model used {#sec:des}
=============================
Gravitationally coupled, two-component, galactic disc model {#ssec:grav_coup}
-----------------------------------------------------------
The galaxy is modelled as a gravitationally-coupled, two-component system of stars and atomic hydrogen gas embedded in the dark matter halo, which exerts an external force on the system while remaining rigid and non-responsive itself. This is a simplified version of the Galaxy case (Narayan & Jog 2002b) , where a gravitationally-coupled, three-component system of stars, atomic and molecular hydrogen was considered. Here, the two components, present in the form of discs, are assumed to be axisymmetric and coplanar with each other for the sake of simplicity. Also, it is assumed that the components are in a hydrostatic equilibrium in the vertical direction. Therefore, the density distribution of each component will be jointly determined by the Poisson equation, and the corresponding equation for pressure equilibrium perpendicular to the midplane.
In terms of the galactic cylindrical co-ordinates ($R$, $\phi$, $z$), the Poisson equation for an azimuthally symmetric system is given by\
$$\frac{{\partial}^2{\Phi}_{total}}{{\partial}z^2} + \frac{1}{R}\frac{{\partial}}{{\partial}R}(R \frac{{\partial}\Phi_{total}}{{\partial}R})
= 4\pi G(\sum_{i=1}^{2} \rho_i + \rho_{h})
\eqno(1)$$\
where $\rho_i$ with i = 1 to 2 denotes the mass density for each disc component while $\rho_h$ denotes the mass density of the halo. $\Phi_{total}$ denotes the total potential due to the disc and the halo. For a nearly constant rotation curve as is the case here, the radial term can be neglected as its contribution to the determination of the scale height is less than ten percent as was noted by earlier calculations Narayan et al. (2005). So, the above equation reduces to\
$$\frac{{\partial}^2\Phi_{total}}{{\partial}z^2}
= 4\pi G(\sum_{i=1}^{2} \rho_i + \rho_{h})
\eqno(2)$$ The equation for hydrostatic equilibrium in the z direction is given by Rohlfs (1977) $$\frac{\partial}{{\partial}z}(\rho_{i}\langle(v_{z}^{2})_{i}\rangle) + \rho_{i}\frac{{\partial}\Phi_{total}}{{\partial}z} = 0 \eqno(3)$$\
where $\langle(v_{z}^{2})_{i}\rangle$ is the mean square random velocity along the $z$ direction for the $i^{th}$ component. Further we assume that each component is isothermal i.e., the random velocity $v_{z}$ remains constant with $z$.
Combining eq. (2) and eq. (3), we get $$\langle(v_{z}^{2})_{i}\rangle \frac{\partial}{{\partial}z}[\frac{1}{\rho_{i}}\frac{{\partial}\rho_{i}}{{\partial}z}] = -4\pi G(\sum_{i=1}^{2} \rho_i + \rho_{h})
\eqno(4)$$
This represents a set of two coupled, second-order, ordinary differential equations which needs to be solved to obtain the vertical density distribution of each of the two components. Although the net gravitational potential acting on each component is the same, the response will be different due to the different velocity dispersions of the two components.
Dark Matter Halo {#ssec:DM halo}
------------------
We use the four-parameter dark matter halo model (de Zeeuw & Pfenniger 1988; Bequaert & Combes 1997) with the density profile given by
$$\rho(R,z) = \frac{\rho_0}{\large [ 1+\frac{m^{2}}{{{R_c}}^{2}}\large]^p} \eqno(5)$$\
where $ m^{2}$=$R^{2} + ({z^{2}}/{q^{2}})$, $\rho_0$ is the central core density of the halo, ${R_c}$ is the core radius, $p$ is the power-law density index, and $q$ is the vertical-to-planar axis ratio of the halo (spherical: $q$ = 1; oblate: $q$ $<$ 1; prolate: $q$ $>$ 1).
Numerical Solution of the Equations & Input Parameters {#sec: Num}
======================================================
Solution of equations {#ssec: Sol}
---------------------
For a given halo density profile, eq. (4) is solved in an iterative fashion, as an initial value problem, using the fourth-order, Runge-Kutta method of integration, with the following two initial conditions at the mid-plane (i.e., $z$ = 0) for each component: $$\rho_i = (\rho_0)_i, \qquad \frac{d\rho_i}{dz} = 0 \eqno(6)$$\
As the modified mid-plane density $(\rho_0)_i $ for each component is not known a priori, the net surface density $\Sigma_i(R)$, given by twice the area under the curve of $\rho_i(z)$ versus z, is used as the secondary boundary condition, as this quantity is known from observations (see §3.2). The required value of $(\rho_i)_0$ is thus determined by a trial and error method, which gives the required $\rho_i(z)$ distribution after four iterations with an accuracy to the second decimal place. Existing theoretical models suggest a sech$^2$ profile for an isothermal density distribution. But for a three-component disc, the vertical distribution is shown to be steeper than a sech$^2$ function close to the mid-plane (Banerjee & Jog 2007).However, at large $z$ values, it is close to a sech$^2$ distribution. Hence we use the half-width-at-half-maximum of the resulting model vertical distribution to define the scale height as was done in Narayan & Jog (2002a,b).
Input Parameters {#ssec: Input}
----------------
We require the vertical velocity dispersion and the surface density of each of the two galactic disc components to solve the coupled set of equations at a given radius. The central stellar surface density is derived directly from the optical surface photometry (Matthews et al. 1999) by assuming a reasonable stellar mass-to-light ratio. The deprojected $B$-band central surface brightness of UGC 7321 (corrected for extinction) translates to a central luminosity density of 26.4 $M_{\odot}$ pc$^{-2}$. Using the $B-R$ color of the central regions ($\sim$1.2; Matthews et al. (1999) ) and the “formation epoch: bursts” models from Bell & de Jong (2001) predicts $(M/L)_{\star}$ = 1.9, which we adopt here. (Other models by Bell & de Jong give values of $(M/L)_{\star}$ ranging from 1.7 to 2.1). This in turn yields a central stellar surface density of 50.2 $M_{\odot}$ pc$^{-2}$ for UGC 7321.
The stellar velocity dispersion of this galaxy has been indirectly estimated to be 14.3 $km$$s^{-1}$ at the centre of the galaxy ($R$ = 0) (Matthews 2000). This is very close to the value of the central (vertical) stellar velocity dispersion (16 $km$$s^{-1}$) for the dwarf spiral galaxy UGC 4325 measured by Swaters (1999), and to the value (20 $km$$s^{-1}$) estimated analytically for the superthin galaxy IC 5249 by van der Kruit et al. (2001). We assume the central value of velocity dispersion to fall off exponentially with radius with a scale length of 2 $R_{d}$ (which is equal to 4.2 $kpc$ for UGC 7321) as is seen in the Galaxy (Lewis & Freeman 1989). Uson & Matthews (2003) give the deprojected surface density for UGC 7321 as a function of radius. The velocity dispersion of is obtained from the Gaussian fits to the edges of position-velocity cuts on the observed data. This gives a value between 7-9 $km$$s^{-1}$. The data are consistent to the typical value of the dispersion in other galaxies (See §5.2 for a detailed discussion).
The molecular hydrogen gas, H$_{2}$, has not been taken into account, as it appears to be dynamically insignificant compared to the other components of the disc. Matthews & Gao (2001) detected a weak CO signal from the central $\sim$2.7 $kpc$ of UGC 7321, which translates to a total molecular hydrogen mass of H$_{2}\approx2.3\times10^{7}~M_{\odot}$ (although this value is uncertain by at least a factor of 2-3 as a result of uncertainties in optical depth effects and the appropriate value of the CO-to-H$_{2}$ conversion factor). This corresponds to a mean H$_{2}$ surface density of $\Sigma_{H2}\approx 1~M_{\odot}$ in the inner galaxy, which agrees fairly well with independent estimates from the dust models of Matthews & Wood (2001) and from a study of the distribution of dark clouds from [*Hubble Space Telescope*]{} images (J. S. Gallagher & L. D. Matthews, unpublished). Therefore, the presence of H$_{2}$ has been ignored in subsequent calculations.
Results and analysis {#sec: Results}
====================
We perform an exhaustive scanning of the grid of parameters characterizing the dark matter density profile to obtain an optimum fit to both the observed rotation curve and the scale height data. To start with, we consider a spherical halo ($q$ = 1) for simplicity.
0.1in
$Parameter$ $Range$ $Resolution$
--------------------------------------- ---------------- --------------
$\rho_{0}$($M$$_{\odot}$ $p$c$^{-3}$) $0.0001 - 0.1$ $0.0001$
$0.001 - 0.5$ $0.001$
$R_{c}$($kpc$) $1.5 - 12$ $0.1$
$p$ $1 - 2$ $0.5$
We vary the remaining three free parameters characterizing the density profile of the halo (see eq. (5)) within their respective feasible ranges (as summarized in Table 2), and obtain the contribution of the halo to the rotation curve for each such grid point in this three-dimensional grid. The power-law density index $p$ is allowed to take the values 1, 1.5 and 2 successively. Here, a value of $p$ = 1 corresponds to the standard isothermal case used routinely for simplicity and also because it corresponds to the flat rotation curve. The value of $p$ = 1.5 refers to the NFW profile Navarro et al. (1996) at large radii, whereas $p$ = 2 gives an even steeper dark-matter halo profile, as was found for the Galaxy case Narayan et al. (2005). For each value of $p$, the core density $\rho_{0}$ and the core radius $R_{c}$ are varied as given in Table 2 to ensure an exhaustive scanning for the dark matter halo parameters since we have little prior knowledge of the plausible values these parameters can take in a superthin galaxy.
The rotation curve constraint {#sec: rotcurve}
-----------------------------
The total rotational velocity at each radius is obtained by adding the contribution from the stars, the gas and halo in quadrature as
$${v^{2}(R)} = v_{star}^{2}(R) + v_{gas}^{2}(R) + v_{halo}^{2}(R) \eqno(7)$$\
Here the way to obtain the different terms is discussed below. This result is matched with the observed rotational velocity at all radii.
The deprojected gas surface density versus radius data for UGC 7321 (Uson & Matthews 2003) can be modelled as one which remains constant at 5 $M$$_{\odot}$ $pc$$^{-2}$ at galactocentric radii less than 4 $kpc$, and which then falls off exponentially with a scale length of 2.8 $kpc$. The gas surface density does not include a correction for He. For this radial distribution, we calculated the contribution of the gas to the rotation curve (using eq. (2-158) & (2-160) of Binney & Tremaine (1987)), and found it to be negligible compared to that of the stellar component. However, it was included in the calculations for the sake of completeness.
The rotational velocity at any radius $R$ for a thin exponential stellar disc is given by Binney & Tremaine (1987)
$$v_{star}^{2}(R) = 4\pi G \Sigma_{0} R_{d} y^{2} [I_0(y)K_0(y) - I_1(y)K_1(y)] \eqno(8)$$\
where $\Sigma_{0}$ is the disc central surface density, $R_{d}$ the disc scale length and $y$ = $R$/[2$R$$_{d}$]{}, $R$ being the galactocentric radius. The functions $I_{n}$ and $K_{n}$ (where n=0 and 1) are the modified Bessel functions of the first and second kind, respectively.
For the spherical halo, the rotational velocity, $v_{halo}(R)$, is given by\
$$v_{halo}^{2}(R) = \frac{G M_{halo} (R)}{R} \eqno(9)$$\
where $M$$_{halo} (R)$, the mass enclosed within a sphere of radius $R$ for a the given halo density profile, and is obtained from the density as given by the right-hand side of eq. (5).
For an oblate halo of axis ratio $q$ and density index $p$, the circular speed $v_{halo}(R)$ is obtained by differentiating the expression for the potential from Sackett & Sparke (1990), and Becquaert & Combes (1997) to be:\
$$v_{halo}^{2}(R) = 4 \pi G \rho_{0} q \int_{0}^{1/q} \frac{R^2 x^2 [ 1 + \frac{R^2 x^2}{R_{c}^2 ( 1 + \epsilon^2 x^2)} ]^{-p}}{( 1 + \epsilon^2 x^2)^2} dx \eqno(10)$$\
where $\epsilon = (1- q^2)^{1/2}$. We obtain the value of the integral numerically in each case.
Thus upon obtaining the rotation curve corresponding to each grid point, we perform the ${\chi}^{2}$ analysis comparing computed to the observed rotation curve. The observed rotation curve is taken from Uson & Matthews (2003) and has 30 data points with very small error-bars (typically a few percent of the observed velocity amplitudes even after accounting for systematic uncertainties). It was derived by implicitly assuming a constant (Gaussian) velocity dispersion of 7 $kms^{-1}$. Ideally, we should have considered only those grid points which give ${\chi}^{2}$ values of the order of 30 (i.e., the number of data points) as those giving appreciably good fits to the observed curve Bevington (1969). But we relax this criterion and choose a larger range of grid points around the minimum i.e grid points which give ${\chi}^{2}$ values less than 300 for applying the next constraint i.e the vertical scale height data. This allows us to impose the simulataneous constraints (planar + vertical) on our model. (See §4.3 for a discussion). So finally we get 36 grid points for $p$ = 1, 80 for $p$ = 1.5 and 69 for $p$ = 2 case. As we shall see later, the final set of best-fit parameters obtained give reasonably good fits to both the observed rotation curve and the scale height data.
The scale height constraint {#sec: HI scaleheight}
---------------------------
For each value of $p$, we obtain the scale height distribution beyond 3 disc scale lengths, for each of the grid points filtered out by the first constraint as discussed in the previous section. Next we perform the ${\chi}^{2}$ analysis of our model scale height versus radius curves with respect to the observed one and try to fit our model to the observed data only beyond 3 disc scale lengths in keeping with the earlier studies in the literature (Narayan & Jog 2005; Banerjee & Jog 2008). For M31, the surface-density and therefore the vertical gravitational force due to the dark matter halo exceeds that of the disc only in the outer regions (See Fig.6 of Banerjee & Jog 2008). As the disc dynamics in this region are controlled by the halo alone, the above method helps us in studying the effect of the halo on the scale height distribution, decoupled from that of the other components. For the case of UGC 7321, at first we take the gas velocity dispersion to be equal to 7 $km$$s^{-1}$ . However it fails to give a good fit to the observed data. Next we try both 8 $km$$s^{-1}$ and 9 $km$$s^{-1}$ successively, but choose the latter for subsequent calculations as it gives much better fit to the observed data as compared to the 8 $km$$s^{-1}$ case.
For the choice of $v_{z}$ = 9 $kms^{-1}$, the best-fit core density is 0.041 $M$$_{\odot}$$pc$$^{-3}$ and a core radius is 2.9 $kpc$, as indicated by the smallest ${\chi}^{2}$ value. The small value of the best-fit halo core radius thus obtained indicates that the halo becomes important already at small radii. This suggests that the fitting of the theoretical curve with the observed one should not be restricted only to regions beyond 3 $R_{d}$ for an LSB galaxy like UGC 7321 as the halo is already important at small radii. Hence, we next fit the scale height data over entire radial range (i.e., 2-12 $kpc$) with the same constant $v_{z}$ value of 9 $kms^{-1}$. The best-fit values change by less than a few percent compared to the above case where the fit was done only beyond 3 $R_{d}$. The best-fit core density now becomes 0.039 $M$$_{\odot}$$pc$$^{-3}$ wheras the best-fit core radius continues to be 2.9 $kpc$.
Since the disagreement of the observed rotation curve with the predicted one is mostly in the inner galaxy, we check if the fit can be improved by reducing the central value of the stellar surface density by twenty percent or so, keeping the $v_{z}$ value contant at 9 $km$$s^{-1}$. This is reasonable as there are uncertainties of at least that order in evaluating both the $M$/$L$ ratio and the deprojected surface brightness of the stellar disc. However, this variation fails to improve the results significantly.
We then take a cue from the nature of the mismatch of our model curve with the observed one, which clearly shows the need to use a higher value of gas velocity dispersion in the inner parts, while a slightly lower value is required in the outer regions. Also, the nature of the mismatch rules out an oblate halo as a possible choice as that will lower the scale heights throughout the entire radial range, thus making the fits worse in the inner regions. To account for this, we then repeat the whole procedure by imposing a small gradient in the gas velocity dispersion by letting it vary linearly between 9.5 $km$$s^{-1}$ at $R$ = 7 $kpc$ and 8 $km$$s^{-1}$ at $R$ = 12.6 $kpc$. Although such a variation is ad-hoc, the observational constraints on this value are weak enough to allow for a small variation with galactocentric radius, with 9.5 $km$$s^{-1}$ approaching the upper limit allowed by the data. Using the same gradient in the inner regions, we get a gas velocity dispersion of 10.8 $km$$s^{-1}$ at $R$ = 2 $kpc$. We may note here that a similar gradient in the velocity dispersion was obtained in the case of the Galaxy (Narayan & Jog 2002b) and led to a better fit to the observed scale height in the inner Galaxy (See §5 for a detailed discussion). A fit to the whole range of observations (2 - 12 $kpc$) gives an isothermal halo of spherical shape with a core density of 0.043 $M$$_{\odot}$ pc$^{-3}$ and a core radius of 2.6 $kpc$ best fits the observations. These values are only slightly different (within 10 percent) from the values obtained with a constant velocity of $9 kms^{-1} $.
In Fig.1, we give our best-fit for the case of constant $v_{z}$ = 9 $km$$s^{-1}$, and the case with a $v_{z}$ slightly falling with radius as compared to the fit to the rotation curve alone, superimposed on the observed one. Our model curves follow the trend of the observed data well throughout the entire radial range.
In Fig.2, we compare the best-fit scale height distributions for the above two cases with the observed one. Clearly, the case with a gradient in gas velocity dispersion gives a remarkably better fit (${\chi}^{2}$ value 2.8), although as far as ${\chi}^{2}$ values are concerned, the case of constant $v_{z}$ = 9 $km$$s^{-1}$ cannot be ruled out altogether (${\chi}^{2}$ value 14.7) (This is because basic statistics suggests that the fit to the model is considered to be reasonably good if the ${\chi}^{2}$ value is of the order of the number of data points in the fit as discussed earlier at the end of §4.1. Here the total number of data points in the scale height data is 11.)
Quality of individual fits as a result of imposing two simultaneous constraints
-------------------------------------------------------------------------------
We reiterate the fact that our method is aimed at obtaining an optimum fit to both the observational constraints, namely the rotation curve and the HI scale height data. This evidently results in a compromise in the quality of individual fits to either of the observed curves (See Fig.1 & 2). Traditional mass modelling techniques resort to the rotation curve constraint alone, and therefore the fit is much better. However imposing two simultaneous constraints on the theoretical model gives a more realistic picture than the case in which best-fit is sought to a single constraint alone. It is noteworthy that even when the fit is sought to the rotation curve alone, the best-fit $R_{c}$ continues to be of the order of $R_{D}$ which is tha main result of this work. However the $\rho_{0}$ value obtained is different in the two cases.
Discussion {#sec:dis}
==========
The dark halo properties and overall stability of superthin galaxies like UGC 7321 are of considerable interest in the context of galaxy formation and evolution. In particular, such galaxies seem to pose a significant challenge to hierarchical models of galaxy formation, whereby galaxies are built-up through violent mergers of subgalactic clumps since such mergers may result in significant disc heating and trigger instabilities (e.g., D’Onghia & Burkert 2004, Eliche-Moral et al. 2006, Kormendy & Fisher 2005). While theorists have predicted that the thinnest galaxy disks must require massive dark halos for stabilization (Zasov et al. 1991; Gerritsen & de Blok 1999), little information has been available on the dark halo properties of individual superthin galaxies until now.
UGC 7321 is the first superthin galaxy for which both a detailed rotation curve and the gas layer thickness were derived Uson & Matthews (2003). This has allowed us to use both these constraints simultaneously to characterize its dark halo properties, as well as to obtain new insight into the stability of its disc against star formation. Below we comment further on the implications of several of our key findings.
The small core radius of the dark matter halo {#ssec: core radii}
---------------------------------------------
The core radii of the dark matter halos of massive high surface brightness galaxies studied so far are usually found to be comparable to their optical size, or equivalently, 3-4 times larger than the exponential stellar disc length Gentile et al. (2004). The Galaxy has a core radius of 8-9.5 $kpc$ which is equal 3$R_{d}$ (Narayan et al. 2005) while M31 has a core radius equal to 21 $kpc$ which is almost equal to 4$R_{d}$ (Banerjee & Jog 2008). For UGC 7321, we find a very small core radius of 2.5-2.9 $kpc$, which is just slightly greater than its disc scale length ($R_{d}$ = 2.1 $kpc$). This shows that the dark matter becomes important at small radii consistent with previous mass-modelling of LSB spirals, based on other techniques (de Blok & McGaugh 1997; de Blok et al. 2001). This is illustrated in another way in Fig.3, which gives a comparative plot of the surface-density of the stars, gas and the halo with radius in this galaxy. The halo surface density was calculated within the total gas scale height as was done for M31 (Banerjee & Jog 2008). It clearly shows that the surface-density and hence the gravitational potential of the halo becomes comparable to that of the disc already at R = 2$R_{d}$. This behaviour is quite different from that of a high surface density galaxy like M31 (cf Fig.6, Banerjee & Jog 2008), where the halo contribution starts to dominate at much larger radii (5$R_{d}$). Our results support the idea that superthin disks like UGC 7321 are among the most dark matter-dominated of disc galaxies.
Dependence on gas parameters
----------------------------
$\bullet$ **Gradient in gas velocity dispersion** As noted earlier, if we impose a constant velocity dispersion, we require a value of 9 $km$$s^{-1}$ to get a reasonably good fit to the observed scale height data, while an even better fit requires a velocity gradient implying even larger dispersion in the inner region (Fig.2). In the earlier work for the Galaxy (Narayan et al. 2005), a slope of -0.8 $km$$s^{-1}$ $kpc^{-1}$ for the gas velocity dispersion was obtained for the inner Galaxy between 2-12 $kpc$ (pinned at 8 $km$$s^{-1}$ at 8.5 $kpc$) as it gave the best-fit to the nearly constant scaleheights. Oort (1962) had tried the same idea but had needed a higher gradient of -2 $km$$s^{-1}$ $kpc^{-1}$ since he did not include the gas gravity and therefore needed a larger variation to account for the constant scaleheight in the inner Galaxy. Narayan et al. (2005) tried to constrain the halo properties using the outer galaxy data, where they had used gas velocity gradient of -0.2 $km$$s^{-1}$ $kpc^{-1}$. This is similar to the value we have for UGC 7321. This was based on the fact that some galaxies show a falling velocity dispersion which then saturates to 7 $km$$s^{-1}$ (See Narayan et al. 2005 for a discussion). Recently, Petric & Rupen (2007) have measured the velocity dispersion across the disc of the face-on galaxy NGC 1058. The authors find the velocity dispersion to have a fairly complex distribution, but nonetheless show a clear fall-off with radius (see Fig.8 of their paper). Using this figure, one can estimate a gradient of roughly -0.1 $km$$s^{-1}$$kpc^{-1}$ in the outer disc, which is consistent with values observed for other galaxies. A similar fall-off has also been seen in NGC 6946 (Boomsma et al. 2008) as well as in several other galaxies (Bottema et al. 1986; Dickey et al. 1990; Kamphius 1993). So this gives some observational support to our assumption.\
$\bullet$ **Superposition of two HI phases**
A more realistic case would be to treat the HI as consisting of two phases or components, characterized by a warm ($v_{z}$ = 11 $kms^{-1}$) and a cold medium ($v_{z}$= 7 $kms^{-1}$) respectively. These values match the range seen in the above fits and represent the two phases as observed in the Galaxy (Kulkarni & Heiles 1988). However,observationally the fraction of mass in these two phases as a function of radius is not known. Assuming that this fraction is constant with radius, we let its value vary as a free parameter.
The best-fit ${\chi}^{2}$ in this case is 13.7 as compared to 2.8 for the case with a velocity gradient treated earlier. Although we do not get as good a fit as was obtained in the case where there is a gradient in the velocity, the best-fit core radius $R_{c}$ still comes out to be 2.5 kpc which is again of the order of $R_{D}$. That the dark matter dominates at small radii therefore still remains a robust result irrespective of the input gas parameters used. The best-fit case gives the fraction of HI in the cold medium to be 0.2.\
We had taken this ratio to be constant for simplicity. Interestingly, this assumption is justified by the recent detailed study by Dickey et al. (2009) involving absorption and emission spectra in $21$ cm in the outer Galaxy. They use this to map the distribution of the cold and warm phases of the HI medium, and surprisingly find this ratio to be a robust quantity in the radial range of $R_{sun}$ to $3$ $R_{sun}$. They find this ratio is $\sim 0.15 - 0.2$, which agrees well with the best-fit ratio 0.2 that we obtain. It is interesting that this ratio obtained by two different techniques is similar in the two galaxies.
The case with a gradient with a higher velocity dispersion within the optical radius gives the lowest ${\chi}^{2}$ value (Fig.2), which we adopt as our best case. We note that this choice is not inconsistent with the constant phase ratio measured by Dickey et al. (2009) which was for the outer Galaxy.
$\bullet$ **High value of the gas velocity dispersion**
The high gas velocity dispersion required to get an improved fit to the scaleheight data is surprising given the superthin nature of the galaxy, whose small stellar scale height implies that it is among the dynamically coldest of galactic disks (e.g., Matthews 2000).
The origin of this high gas velocity is beyond the scope of this paper. However, independent of its origin, this high value of the gas velocity dispersion can partly explain why star formation is inefficent in UGC 7321. This is because, to first order, a higher gas dispersion will tend to suppress star formation since Toomre Q criterion ( Q $<$ 1) is less likely to be satisfied, hence the disc is less likely to be unstable to star formation.\
Conclusions
===========
We have modelled the LSB superthin galaxy UGC 7321 as a gravitationally-coupled system of stars and gas, responding to the gravitational potential of the dark-matter halo, and used the observed rotation curve and the vertical scale heights as simultaneous constraints to determine the dark halo parameters. We find that the best-fit gives a spherical, isothermal halo with a central density in the range of 0.039-0.057 $M$$_{\odot}$ $pc$$^{-3}$ and core radius of 2.5-2.9 $kpc$. The value of the best-fit core density is comparable to values obtained for HSB galaxies. The core radius is comparable to that of the disc scale length unlike HSB galaxies studied by this method, implying the importance of the dark-matter halo at small radii in UGC 7321. Thus we find that UGC 7321 is dark matter dominated at all radii, and the results of our analysis support the idea that the thinnest of the galaxies are the most dark matter dominated.\
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[^1]: All quantities are taken from Matthews et al. (1999) and Uson & Matthews (2003) which assume d = 10 $Mpc$
[^2]: Linear diameter at limiting observed $B$-band isophote of 25.5 $mag$ $arcsec^{-2}$
[^3]: Blue luminosity
[^4]: ${\mbox{H\,{\sc i}}}$ mass
[^5]: disc scale length measured from $R$-band image
[^6]: stellar scale height obtained from an exponential fit
[^7]: Deprojected (face-on) central disc surface brightness in the $B$ band, corrected for internal and Galactic extinction
|
---
abstract: 'We present Monte Carlo simulations on Eley-Rideal abstraction reactions of atomic hydrogen chemisorbed on graphite. The results are obtained via a hybrid approach using energy barriers derived from DFT calculations as input to Monte Carlo simulations. By comparing with experimental data we discriminate between contributions from different Eley-Rideal mechanisms. A combination of two different mechanisms yields, good quantitative and qualitative agreement between the experimentally derived and the simulated Eley-Rideal abstraction cross sections and surface configurations. These two mechanisms include a direct Eley-Rideal reaction with fast diffusing H atoms and a dimer mediated Eley-Rideal mechanism with increased cross section at low coverage. Such a dimer mediated Eley-Rideal mechanism has not previously been proposed and serves as an alternative explanation to the steering behavior often given as the cause of the coverage dependence observed in Eley-Rideal reaction cross sections.'
author:
- 'H.M. Cuppen'
- 'L. Hornek[æ]{}r'
title: Kinetic Monte Carlo Studies of Hydrogen Abstraction from Graphite
---
Introduction
============
Molecular hydrogen is the most abundant molecule in the interstellar medium (ISM), where it serves as an important coolant and as a precursor for the formation of more complex molecules. In the cold and dilute interstellar medium no efficient gas phase routes exist for the formation of H$_2$. Hence, the most likely route is to form the molecule on the surfaces of small dust particles. This process has been intensely studied over the past years [@Pirronello:1997; @Pirronello:1997a; @Pirronello:1999; @Hornekaer:2003; @Hornekaer:2005; @Perry:2003; @Cazaux:2004; @Chang:2005; @Cuppen:heating; @Perets:2005; @Katz:1999; @Dulieu:2005]. The general consensus is that molecular hydrogen can be formed by a diffusive Langmuir-Hinshelwood mechanism, in the very cold regions below $\sim$20 K [@Hollenbach:1971]. At these low temperatures the hydrogen atoms are physisorbed on the dust particles, but can still easily move on the surface. Once two atoms meet, they will react to form molecular hydrogen. If the surface temperature becomes too high, the residence time of atoms on the surface becomes so short that, at the low fluxes in the ISM, the chance of two atoms meeting becomes negligible. Observations show that molecular hydrogen is also formed in warmer areas ($>$ 20 K) like Photon Dominated Regions (PDRs) and post-shock regions. A possible mechanism at these conditions would be through an Eley-Rideal reaction where an incoming hydrogen atom reacts with another H atom that is chemically bound to the surface. Since a considerable fraction of the interstellar grains is expected to consist of carbonaceous material, a popular model system for this process is hydrogen abstraction from graphite. This Eley-Rideal reaction has been studied by Density Functional Theory (DFT) and Quantum Wave packet calculations [@Farebrother:2000; @Meijer:2001; @Sha:2002; @Sha:2002a; @Morisset:2004; @Martinazzo:2006]. Based on these calculations several different mechanisms have been proposed to contribute to the Eley-Rideal abstraction process. These include: Direct Eley-Rideal [@Farebrother:2000; @Meijer:2001; @Sha:2002; @Sha:2002a; @Morisset:2004; @Martinazzo:2006], barrier-less abstraction of one hydrogen atom forming part of a para-dimer configuration [@Bachellerie:2007] and abstraction by rapidly diffusing H atoms in physisorbed states [@Bonfanti:2007].
However, due to computational limitations such detailed calculations are unable to take the full complexity of the H-graphite system into account. Hence, in the calculations the flat surface approximation is often used and generally either none or only a single C atom on the simulated graphite surface were allowed to relax during the Eley-Rideal reaction. These simplifications are potentially problematic. By not allowing for relaxation of the C atoms on the surface zero barrier reaction channels, such as sticking of hydrogen atoms into specific dimer configuration [@Hornekaer:2006II; @Rougeau:2006], are in some cases artificially closed, making it impossible to evaluate the contributions from different proposed mechanisms to the Eley-Rideal abstraction process. Eg. calculations on the Eley-Rideal reaction by Martinazzo and Tantardini [@Martinazzo:2006] show a higher probability for reaction if the incoming H atom is 0.5 - 1.5 Å away from the target H atom instead of a direct hit on. This distance corresponds to the H-H distance in an ortho-dimer configuration (see Fig. 1) on the graphite surface (1.42 Å). Hence, surface corrugation and competing sticking reactions must be expected to influence the abstraction behavior. In these calculations, however, dimer formation was not a possible route since the flat surface approximation was used and only the carbon atom underneath the original hydrogen atom was allowed to relax. Impact parameters corresponding to the para-dimer (see Fig. 1) distance (2.84 Å) were not included in the study.
Both experimental observations and theoretical calculations show that hydrogen atoms preferentially form adsorbate clusters on the graphite surface. Scanning Tunneling Microscopy (STM) investigations show that at a hydrogen atom coverage of $\sim$ 0.5 $ \%$ more than 75 $\%$ of all surface configurations are clusters [@Hornekaer:2006II]. This indicates that $\sim$ 85 $\%$ of the hydrogen atoms are part of larger clusters. In particular, the existence of hydrogen dimer configurations has been studied theoretically [@Ferro:2003; @Miura:2003; @Hornekaer:2006I; @Hornekaer:2006II; @Rougeau:2006] and experimentally [@Hornekaer:2006I; @Andree2006]. Hornek[æ]{}r et al. [@Hornekaer:2006I] identified two stable hydrogen dimer configurations on the graphite surface, an ortho-dimer and a para-dimer (see Fig. \[dimers\]). DFT calculations show that formation of the para-dimer is barrierless and that formation of the ortho-dimer has a reduced barrier [@Hornekaer:2006II], which makes dimer formation a competing channel to Eley-Rideal abstraction at non-zero impact parameter.
One set of experiments on the Eley-Rideal abstraction reaction forming HD has been reported [@Zecho:2002; @Zecho:2002a]. In these experiments a high cross section for the abstraction reaction was observed which varied from 17 Å$^2$ at low coverage to 4 Å$^2$ at high coverage. This variation in the cross section with coverage has been ascribed to a steering effect [@Sha:2002; @Sha:2002a]. However, other mechanisms could also contribute to the high cross section at low coverage. Options include the barrier-less abstraction of one of the H atoms in a para-dimer configuration [@Bachellerie:2007] and abstraction by fast diffusing H atoms [@Bonfanti:2007]. We propose a further possibility, namely abstraction via a hydrogen dimer state, where the incoming atom is not immediately thermalized and some of its excess initial energy is used to overcome the barrier to H$_2$ formation and desorption. The finding that adsorption into the hydrogen para-dimer state is barrierless [@Hornekaer:2006II] make this a strong competing reaction channel.
Hence, in total 5 different abstraction mechanisms have been proposed to contribute to Eley-Rideal abstraction of hydrogen on graphite:
1. direct Eley-Rideal [@Farebrother:2000; @Morisset:2004].
2. direct Eley-Rideal with steering [@Zecho:2002; @Sha:2002; @Sha:2002a].
3. preferred direct Eley-Rideal with hydrogen atoms part of a para-dimer [@Bachellerie:2007].
4. a dimer mediated reaction where the incoming atom is first adsorbed into a dimer configuration and, before thermalizing to the substrate temperature, reacts with the other atom in the dimer to form H$_2$ and desorb.
5. Eley-Rideal reactions by fast diffusing H atoms in the physisorption state [@Bonfanti:2007].
However, due to the simplifications needed in the complex DFT and quantum wave packet calculations a quantitative comparison between experimental data and theory has not been possible. In this paper we employ a hybrid approach in which we include the findings of ab-initio DFT calculations in a Monte Carlo simulation program and then simulate Eley-Rideal abstraction experiments on a more realistic surface area with more complex hydrogen adsorbate configurations than what is possible in the DFT and wave packet calculations. Through this quantitative approach we aim to discriminate between the contributions of the different underlying mechanisms for hydrogen abstraction.
![The two stable dimer configurations found by Hornek[æ]{}r et al. [@Hornekaer:2006I].[]{data-label="dimers"}](802819jcp1.eps)
Monte Carlo model
=================
The Monte Carlo simulation program is a so-called lattice-gas simulation program, where the atoms are confined to an adaptive grid depending on the initial coverage. The hydrogen atoms can chemisorb to the graphite surface at the sites directly on top of the carbon atoms as indicated in Figure \[system\]. The puckering of the carbon atom upon chemisorption is included indirectly by using the barriers for chemisorption and binding energies from DFT calculations that allow this motion. Interaction between two hydrogen atoms is accounted for in a similar way as will be discussed in the following sections. The hydrogen atoms can physisorb to the surface without a barrier. Since the potential energy surface for physisorption is rather flat, there are probably no specific physisorption site, however since the lattice-gas model forces us to choose specific sites, we confine physisorption to the sites directly above the carbon atoms and an additional site at the center of the ring (Figure \[system\]).
![The adsorption sites used in the Monte Carlo simulations.[]{data-label="system"}](802819jcp2.eps)
The simulation starts with a clean graphite surface. The first event will be a deposition attempt of the first atom. The time at which this attempt will occur is $$t_{\rm dep} = -\frac{\ln\left(X\right)\sigma}{f} + t
\label{tdep}$$ where $\sigma$ is the density of sites, $X$ is a random number between 0 and 1, $f$ is the hydrogen flux in atoms per time per area, and $t$ is the current time. Deposition times for subsequent sticking events are determined by the same expression (Eq. \[tdep\]). How the atoms will bind to the surface is determined by another random number and depends on the barrier for chemisorption in that specific position. If the site is already occupied by another hydrogen atom, an abstraction reaction is considered by comparing a Boltzmann factor including the abstraction barrier against a random number. Upon reaction both atoms will leave the surface in the form of H$_2$, else the incoming atom is deflected. The abstraction channel can be closed by making the barrier infinitely large.
Once hydrogen atoms populate the surface, they can diffuse, desorb, or recombine with other atoms to form H$_2$. For each surface hydrogen the time at which they will undergo one of these events is determined by $$t^{i} = -\frac{\ln\left(X\right)}{R_{\rm dif}^i + R_{\rm des}^i + R_{\rm rec}^i} + t\label{t^i}$$ with $R_{\rm dif}^i$, $R_{\rm des}^i$, and $R_{\rm rec}^i$ the rate for diffusion, desorption, and reaction of atom $i$, respectively. Another random number determines which of the three events occur according to their relative probability of occurrence. Reaction between two chemisorbed atoms can only occur if they form a dimer configuration. The rates are given by $$R = \nu \exp\left(-\frac{E}{k T'}\right),
\label{R}$$ where $\nu$ is the attempt frequency which is assumed to be 10$^{12}$ Hz for physisorbed atoms and 10$^{13}$ Hz for chemisorbed atoms [@Hornekaer:2006I] and $T'$ is the ‘temperature’ of the atom that is involved.
In the experiments that we aim to reproduce, the atoms arrive at the surface at normal incidence and at a temperature around 2000 K which is much higher than the surface temperature. Furthermore, the atoms will gain energy due to the high binding energy if they chemisorb. The atoms will not be thermalized instantaneously, but will most likely gradually lose their energy to the substrate. The dissipation of the excess energy into the substrate is expected to be exponential [@Shalashilin:1998]. To reduce the computation complexity of the model we emulate this exponential energy loss by a simpler expression and use the following function to describe the temperature $$T'(t) = \max\left(T_{\rm s}, \frac{T_{\rm start}}{\left(1+B \left(t-t_{\rm a}\right)\right)^2}\right) \label{T'}$$ where $T_{\rm s}$ and $T_{\rm start}$ are the surface and starting temperature respectively, $t_{\rm a}$ is the time at which the atom has adsorbed on the surface. The sensitivity of the model to the exact functional form was checked and found to be small. For most cases we use $T_{\rm start} = 2000$ K, but also higher values of 7000 K and 10,000 K are considered, matching the binding energy of a monomer. The parameter $B$ can be chosen freely. Fig. \[therma\] studies the effect of this parameter. It displays the result of series of 10,000 simulations of one deposition event for a particular value of B. The top panel indicates the percentage of deposition attempts that resulted in sticking. The sticking fraction slowly approaches the monomer sticking barrier probability of 41 % for increasing $B$. A value of $B$ above $\sim 10^{9}$ s$^{-1}$ for $T_{\rm start} = 2000$ K is needed to get an initial sticking co-efficient in agreement with the experimental findings [@Zecho:2002]. Finally, the bottom panel gives the relaxation time. For comparison Shalashilin and Jackson [@Shalashilin:1998] found for a hydrogen atom on a Cu(111) surface that the thermal relaxation time is around 4 ps. This corresponds to a $B$ of $8\times 10^{10}$ s$^{-1}$ for $T_{\rm start} = 2000$ K and $2\times 10^{11}$ s$^{-1}$ for $T_{\rm start} =
10,000$ K.
![Influence of the $B$ parameter on the sticking fraction (top) and the relaxation time (bottom) of the hot atom. For detailed explanation see text. []{data-label="therma"}](802819jcp3.eps)
In the brief moment after a deposition, the rates in Eq. \[R\] are time dependent due to the decreasing temperature and Eq. \[t\^i\] cannot be used. Instead we use the method by Jansen et al. [@Jansen:1995] to determine $t^i$. This method makes use of $$-\ln\left(X\right) = \int_t^{t^i}R_{\rm dif}^i(t) {\rm d}t + \int_t^{t^i}R_{\rm des}^i(t) {\rm d}t +\int_t^{t^i}R_{\rm
rec}^i(t){\rm d}t,
\label{integral}$$ that transforms to Eq. \[t\^i\] if all rates are time dependent. To obtain $t_i$ the expression has to be solved. Using $$\begin{aligned}
\Omega(t^i) &=& \int_{t}^{t^i} \nu \exp\left( -\frac{E(1+B(t-t_{\rm a}))^2}{kT_{\rm start}}\right) {\rm d} t\\
& =& \frac{\nu}{2B} \sqrt{\frac{\pi T_{\rm start}}{E}} {\rm erf}\left((1+B(t-t_{\rm a})) \sqrt{\frac{E}{T_{\rm
start}}}\right)\end{aligned}$$ with ${\rm erf}$ the error function, Eq. \[integral\] becomes $$-\ln\left(X\right) = \Omega_{\rm dif}(t^i) + \Omega_{\rm des}(t^i) + \Omega_{\rm
rec}(t^i).$$ This can be solved numerically to $t_i$ using the Newton-Raphson method [@NumRec], since both $\Omega$ and $\frac{{\rm d}\Omega}{ {\rm d}t} $ decrease monotonically. Notice that different functions for $T'$ will result in a different expression for $\Omega(t^i)$. As the order in $t$ increases, solving $\Omega(t^i)$ becomes more computationally expensive.
Included reactions and energy barriers
======================================
The previous section described the general Monte Carlo algorithm. For all processes energy barriers are needed to determine the corresponding transition probabilities. These barriers are taken from independent DFT calculations [@Sljivancanin:2007] of the binding energies of the different configurations that are considered, sticking trajectories and diffusion trajectories. Since many possible configurations are formed during the simulations, especially for high coverages, including all these different possibilities explicitly would make the Monte Carlo program very slow and would require a huge set of barriers that all have to be calculated independently. To overcome this problem a number of simplifications are introduced resulting in the following sets of energy barriers:
Physisorbed atoms
-----------------
For the physisorption binding energy, the value of $\sim$ 40 meV is used based on results from selective adsorption experiments [@Ghio:1980]. For diffusion an activation energy of 4 meV taken from [@Bonfanti:2007] was used. We will come back to this diffusion rate at the discussion of mechanism IV. For physisorbed atoms $T'(t) = T_{\rm s}$ was used.
Sticking
--------
Numerous different barriers for sticking of an isolated hydrogen atom (a monomer) into the chemisorption site on the graphite surface have been given. Jeloaica and Sidis found a barrier of $\sim$0.2 eV using the coronene molecule as a model of a graphite surface [@Jeloaica:1999]. Sha et al. obtained a barrier slightly above 0.2 eV using a slab super cell with 4 layers each containing 8 carbon atoms [@Sha:2002; @Sha:2002a]. Hornek[æ]{}r et al. [@Hornekaer:2006II] found a barrier for chemisorption into a monomer of 0.15 eV using a single layer super cell containing 32 carbon atoms. The influence of adding a second carbon layer was investigated and found to be negligible. Using the same model surface a sticking barrier of 0.1 eV into the ortho-dimer and 0 eV into the para-dimer configurations were found. All possibilities for sticking into a trimer state, starting from the para-dimer, were seen to have non-zero sticking barriers between 0.1 and 0.15 eV. Adding a fourth atom resulting in a triple para-dimer configuration again did not exhibit a barrier [@Hornekaer:2006II]. Barrierless sticking into the para-dimer state was also found by Rougeau et al. [@Rougeau:2006].
Even though there is a clear dependence on the local configuration of the impact site, we decided only to include variations in sticking barriers for dimers in the simulation. The program determines if the incoming atoms can form a dimer ignoring the larger configuration it might be part of and determines the barrier for sticking accordingly. This assumption will cause deviations between simulations and experimental results at high coverage, since it overestimates the formation of trimer configurations that contain para-dimers. No sticking barrier was used for physisorption of H atoms. The barriers for sticking used in the simulation are summarized in Table \[Echem\].
Section \[Sense\] tests the different mechanisms to their sensitivity to various input parameters. The monomer sticking barrier is one of them.
Configuration
-------------------------- --- ----
para-dimer 0 0
ortho-dimer 0 1
other chemisorption site 0 15
physisorption 0 0
: The sticking barrier for different configurations[]{data-label="Echem"}
Configuration$^1$
-------------------- ---- ---
monomer -0 8
para-dimer (G) -2 9
ortho-dimer (A) -2 8
meta-dimer (E) -0 8
trimer (A, G) -3 9
trimer (G, I) -3 7
trimer (D, G) -3 5
trimer (B, G) -3 5
trimer (F, G) -3 3
tetramer (A, G, H) -5 9
tetramer (A, B, E) -5 9
tetramer (A, D, G) -5 7
tetramer (A, B, D) -5 5
tetramer (B, E, G) -5 1
tetramer (A, B, C) -5 0
tetramer (A, B, G) -4 8
tetramer (A, B, F) -4 5
: The total binding energy for different configurations[]{data-label="Ebin"}
[$^1$ See Fig. \[nummers\].]{}
![Schematic guide to obtain the configurations used in Table \[Ebin\]. A configuration is made up from hydrogen atoms positioned on top of the black carbon atom and the atoms indicated by the characters given in Table \[Ebin\]. In this way, an ortho-dimer is represented by the character (A). []{data-label="nummers"}](802819jcp4.eps)
Binding energies
----------------
The binding energies for different configurations of chemisorbed H atoms are displayed in table II. The different configurations are displayed in Fig. \[nummers\]. These values are based on DFT calculations reported in [@Hornekaer:2006II; @Sljivancanin:2007].
Diffusion
---------
![The diffusion barrier as a function of the energy difference between the initial and final configuration. The diffusion barriers between dimer and the tetramer configurations follow the same linear dependences.[]{data-label="Eb"}](802819jcp5.eps)
For several dimer and tetramer configurations the individual binding energies and the transition barriers between some of the configurations were determined via DFT calculations. The binding energy of the individual dimer and tetramer configurations are given in Table \[Ebin\]. The tetramer binding values and diffusion barriers are taken from [@Sljivancanin:2007]. Figure \[Eb\] plots the diffusion barrier as a function of the energy difference of the configurations. As the figure clearly shows there is a linear relation between the two quantities and dimers and tetramers follow the same relation. A least-squares fit resulted in $$E_{\rm diff} = 0.5 \Delta E + 1.04 \textrm{ eV}.
\label{E_diff}$$ In order to obey detailed balance, or microscopic reversibility, transition probabilities for diffusion should fulfill $$\frac{P_{ij}}{P_{ji}} = \exp\left(-\frac{\Delta E_{ij}}{kT}\right)$$ Using Eq. \[R\], it can be shown that the empirically found relation, Eq. \[E\_diff\], follows this requirement.
Desorption
----------
Finding a general expression for the desorption energy of the individual atoms from a configuration is less straightforward. The binding energies are determined for the configuration and not for the individual atoms. Again calculating all possible desorption pathways would not be feasible. We base the desorption on the total binding energy of the configuration of $n$ atoms, $E_{\rm bind}$. The desorption energy is then $$E_{\rm des} = \frac{E_{\rm bind}}{n} + E_{\rm stick}$$ where $E_{\rm stick}$ is the barrier for sticking in the same position. $E_{\rm stick}$ and $E_{\rm bind}$ can be found in Tables \[Echem\] and \[Ebin\].
Thermally activated H$_2$ formation
-----------------------------------
Formation of H$_2$ via reaction between two chemisorbed hydrogen atoms to gaseous molecular hydrogen occurs from the dimer states and has to overcome barriers of 2.49 eV for the ortho-dimer and 1.4 eV for the para-dimer state [@Hornekaer:2006I]. Reaction barriers from the isolated dimer states are used for all configurations. This approximation will again lead to inaccuracies for simulations at high coverage.
Abstraction
-----------
Five different Eley-Rideal abstraction mechanisms are considered in the model:
1. direct Eley-Rideal with all hydrogen atoms regardless of their local configuration.
2. a simple version of Eley-Rideal with steering where a direct Eley-Rideal reaction is allowed not just for H atoms impinging on an already occupied site but also for H atoms impinging on adjacent sites.
3. direct Eley-Rideal with only hydrogen atoms part of a para-dimer.
4. a dimer mediated reaction where the incoming atom is first adsorbed into a dimer configuration and, before thermalizing to the substrate temperature, forms H$_2$ and desorbs.
5. direct Eley-Rideal together with a high diffusion rate of the atoms in the physisorption state
The influence of the different Eley-Rideal mechanisms can be controlled by using different barriers for the direct Eley-Rideal reaction and different values of the thermalization parameter B. Furthermore, a simple version of steering is implemented with an adjustable parameter $s$ as will be described below. The values of the parameters used for simulating the different mechanisms are given in Table \[Abs\].
The height of the abstraction barrier is somewhat uncertain due to the limitations in the accuracy of DFT calculations. Morisset et al. [@Morisset:2004] found a barrier for direct abstraction of a monomer just below 10 meV. Others also found low barriers to abstraction [@Sha:2002a; @Martinazzo:2006]. Following Morisset et al. we employ a barrier of 9 meV but later investigate the effect of changing the value of the barrier. The same group also found that the abstraction reaction with one of the hydrogen atoms in a para-dimer can proceed without barrier [@Bachellerie:2007].
Mechanism [$E_{\rm ER,mono}$ \[meV\]]{} [$E_{\rm ER,dimer}$ \[meV\]]{} [$B$ \[$s^{-1}$\]]{} [$T_{\rm start}$ \[K\]]{} [$s$]{} $R_{\rm dif}$\[$s^{-1}]$
-------------------------- ------------------------------- -------------------------------- ---------------------- --------------------------- --------- --------------------------
I (Fig. \[dHDdt\]-I) 9 9 $10^{12}$ 2000 0 $7.2 \times 10^{12}$
II (Fig. \[dHDdt\]-II) 9 9 $10^{12}$ 2000 1 $7.2 \times 10^{12}$
III (Fig. \[dHDdt\]-III) $\infty$ 0 $10^{12}$ 2000 0 $7.2 \times 10^{12}$
IV (Fig. \[dHDdt\]-IV) $\infty$ $\infty$ $10^9$ 2000 0 $7.2 \times 10^{12}$
V (Fig. \[dHDdt\]-V) 9 9 $10^{12}$ 2000 0 $5.0 \times 10^{13}$
Comparison with experiments
===========================
The Monte Carlo simulation results are compared with the experimental data on the Eley-Rideal reaction presented by Zecho et al. [@Zecho:2002]. These experiments consist of two phases. First, a graphite substrate was exposed to a normal incidence deuterium beam with a flux of $3.8 \times 10^{15}$ atoms cm$^{-2}$s$^{-1}$ at 150 K. The exposure time was varied to give a range of initial coverages. The temperature of the atom beam was 2000 K. During the second phase the pre-exposed substrate was exposed to a normal incidence hydrogen beam and the formed HD molecules were measured using a mass spectrometer. The cross section of the deuterium abstraction with hydrogen, $\sigma$, was determined from these spectra using $$\frac{{\rm d[HD]_{\rm g}}}{{\rm d}t} = \sigma \Phi {\rm[D_{\rm ad,0}]} \exp\left(-\sigma \Phi t\right),\label{dHD/dt}$$ with $\Phi$ the H flux and ${\rm[D_{\rm ad,0}]}$ the initial D coverage. For the derivation of this equation, we refer to [@Zecho:2002]. The expression is obtained assuming only direct Eley-Rideal. Since the reaction mechanism is implicitly included in the cross section, it is hard to directly interpret the results obtained in this way, but we will use the method to compare the experimental data with our simulation results.
The hydrogen adsorbate configurations found in the Monte Carlo simulations are also compared to the adsorbate configurations observed in STM experiments. In particular the fraction of hydrogen atoms in dimer configurations or larger clusters in simulation and experiment are compared.
Results
=======
Several simulation runs were performed using a similar set of pre-exposures as used by Zecho et al. [@Zecho:2002]. The H and D fluxes were $4.8 \times 10^{13}$ cm$^{-2}$s$^{-1}$ at a temperature of 150 K. $250
\times 125$, $500 \times 250$, and $750 \times 500$ chemisorption sites were used depending on the initial D coverage. This corresponds to an array of $250 \times 250$, $500 \times 500$, and $750 \times 750$ to accommodate the extra physisorption sites. If the noise was still considerable for arrays of $750 \times 500$ multiple runs with different random seeds were made. In order to test to what extent the five proposed abstraction mechanisms contribute to the Eley-Rideal abstraction process comparisons to the experimental results, with different parameter sets (see Table \[Abs\]) were performed.
Figure \[dHDdt\] shows ${\rm d[HD]_{\rm g}}/{{\rm d}t}$ for the five different abstraction mechanisms. Panel (I) only includes the direct Eley-Rideal with a barrier of 9 meV independent of the configuration on the surface, panel (II) investigates the influence of steering, panel (III) only allows an Eley-Rideal reaction, with a zero barrier, if the surface atom is part of a para-dimer configuration, panel (IV) uses the slower thermalization with $B = 1\times 10^{8}$ s$^{-1}$ to test the dimer mediated mechanism, and panel (V) uses very fast diffusion. As ${\rm d[HD]_{\rm g}}/{{\rm
d}t}$ is plotted on a logarithmic scale, the curves should be linear according to Eq. \[dHD/dt\] with the slope the cross section times the flux. The flux has the same value throughout the simulations.
The key results of the simulations are summarized in Table \[Sumtable\] and compared to the experimental findings. Each entry consists of nine individual simulations. The top twelve rows represent the runs that allow only a single mechanism. The second section summarizes simulation series of combinations of mechanisms and the third part gives the results with a different sticking barrier and will be discussed in Section \[Sense\]. The final entries give the experimental results for comparison. The difference in cross section between high and low coverage, the saturation coverage, and the linearity of the initial signal in initial coverage are indicators of the agreement with the experiment. The latter linearity is measured by the Pearson correlation coefficient that gives one for perfect correlation and zero for no correlation. For a better comparison between the different simulation runs all cross sections at low coverage are given for $\sim0.013$ ML which is generally achieved after an exposure of 0.03 ML (the lowest considered pre-exposure). Simulations including mechanism IV (dimer mediated reaction) need a longer exposure to reach 0.013 ML whereas simulations with mechanism V (fast diffusion) reach it faster than 0.03 ML. The cross sections at high coverage are taken at the highest considered pre-exposure of 2.9 ML. The tenth column gives the corresponding coverages. In almost all cases the saturation coverage has been reached at this point. The last column of the table indicates the fraction of atoms that is part of a dimer configuration or larger cluster at a coverage of 0.5 % at conditions similar to the ones used by Hornek[æ]{}r at al. [@Hornekaer:2006I].
### Mechanism I: Direct Eley-Rideal {#mechanism-i-direct-eley-rideal .unnumbered}
The five plots in Fig. \[dHDdt\] all show a very different behavior. Panel (I), which tests the direct Eley-Rideal mechanism, shows a linear relation between $\ln\left({\rm d[HD]_{\rm g}}/{{\rm d}t}\right)$ and the deposition time over the whole time range. The slope appears to have some dependence of the initial D coverage. This would result in a cross section that is dependent of initial coverage. Figure \[cs\_1\] plots, among other quantities, the cross section which indeed shows some coverage dependence. It is however much less than seen in the experiments [@Zecho:2002]. The cross section and the value of the initial signal is obtained by fitting the curves shown in Figure \[dHDdt\]-I to Eq. \[dHD/dt\]. Since for some parameter choices only the first section is linear, only these points are included in the fit. The number of points included in the fit can have a strong effect on the final result, especially if the linear part is very short. This is reflected by the error bars. Various runs with different initial seed showed that this error should at least be 0.5 Å$^2$. The initial signal depends linearly on the deuterium coverage, which agrees with Figure 3c in [@Zecho:2002]. Figure \[cs\_1\] also displays the total HD yield. Since this value is comparable with the initial D coverage it is clear that only a negligible fraction of D atoms remains on the surface after hydrogen exposure. This is in good agreement with the experimental findings [@Zecho:2002]. The dimer fraction is very low in contrast with the experiments as can be seen in the first row in Table \[Sumtable\].
![Analysis of Figure \[dHDdt\]-I ($E_{\rm ER, mono} = E_{\rm ER, dimer} = 9$ meV). (a) Initial coverage and yield as a function the D pre-exposure. (b) The cross section and initial signal versus the initial D coverage.[]{data-label="cs_1"}](802819jcp7.eps)
### Mechanism II: Eley-Rideal with steering {#mechanism-ii-eley-rideal-with-steering .unnumbered}
A simple version of Eley-Rideal abstraction with steering is implemented by allowing atoms that land on empty sites close to chemisorbed hydrogen atoms to attempt a reaction with these atoms. For the reaction barrier the same $E_{\rm ER,mono}$ is used as for the direct hit. Eley-Rideal via a direct hit (mechanism I) is also allowed. If an atom lands on an empty site, the possibility of steering is considered by comparing a random number between 0 and 1 against parameter $s$. If the random number is smaller than $s$, then steering is activated. In the case of steering, the three neighbouring sites are checked for reacting species. If two or more different atoms are found, another random number determines with which of the atoms the incoming H atom will undergo the reaction attempt. Simulations for $s=1$ and 0.5 were performed. The results of these two simulation runs are summarized in Fig. \[cs\_4\] and the rows indicated by II in Table \[Sumtable\] and the $s=1$ simulation run is shown in Figure \[dHDdt\]-II. Fig. \[cs\_4\] shows the initial coverage and yield after 200 seconds of H exposure as function of the initial D pre-exposure. Notice that the yield is given in ML in these graphs and not as a percentage. It further gives the cross section and initial signal as a function of the initial coverage. For $s=1$, a reasonably high cross section at low coverage is obtained ($14.7 \pm 1.4$ Å$^2$), but the saturation coverage is very low (0.10 ML) and also the cross section at this coverage is too high. The series which simulates steering with $s=0.5$ gives somewhat lower cross section resulting in better agreement with the experimental findings at high coverage and too low a cross section at low coverage. Furthermore, an almost linear decrease in cross section is found as a function of coverage in disagreement with the experimental findings. Again the dimer ratio is too low.
### Mechanism III: Dimer Eley-Rideal {#mechanism-iii-dimer-eley-rideal .unnumbered}
The curves in panel (III) are not linear and show a maximum at later times for the low coverages. Since this panel presents the results with the dimer Eley-Rideal mechanism, this maximum indicates the time at which a maximum of D containing dimers is formed. This behavior contradicts the experiments [@Zecho:2002] where the \[HD\] signal decreases over time and is therefore not likely to be the primary mechanism involved in the abstraction.
![Analysis of Figure \[dHDdt\]-II ($E_{\rm
ER, mono} = E_{\rm ER, dimer} = 9$ meV and steering). (a) Initial coverage and yield as a function the D pre-exposure. (b) The cross section and initial signal versus the initial D coverage.[]{data-label="cs_4"}](802819jcp8.eps)
![Analysis of Figure \[dHDdt\]-IV ($B = 1 \times 10^8$ s$^{-1}$). (a) Initial coverage and yield as a function the D pre-exposure. (b) The cross section and initial signal versus the initial D coverage.[]{data-label="cs_3"}](802819jcp9.eps)
### Mechanism IV: Dimer mediated abstraction {#mechanism-iv-dimer-mediated-abstraction .unnumbered}
Panel (IV) presents the results for the dimer mediated mechanism. Here a clear linear dependence can be observed for very early times with a decreasing slope for increasing initial coverage. To obtain this graph large arrays upto $1500 \times 3000$ sites were used due to the low sticking rate at low coverage. Notice that the y-axis range of this panel is different from the others. Figure \[cs\_3\] studies this set of simulations more closely. The two upper panels show that not all D is converted into HD, but that a substantial amount of deuterium still resides at the surface after the 200 seconds exposure. This is in contrast with the experimental finding that all deuterium is abstracted after a 3 ML dose. Figure \[cs\_3\] also shows the cross section determined from Fig. \[dHDdt\]-II. Again the bars indicate the uncertainties in obtaining the slope. Since the linearity is much less than for mechanism I, the error bars are larger than in Fig. \[cs\_1\]. In contrast with this latter figure, Figure \[cs\_3\] shows a strong dependence of the cross section on the initial deuterium coverage. This corresponds with the trends observed by Zecho et al. [@Zecho:2002]. The values are also in the correct range, although the cross section is too high at low coverage and too low for high coverage. Both the cross section and the initial signal indicate that the dimer mediated mechanism becomes very inefficient for high coverages. The initial signal even decreases for increasing surface coverage indicating that at high coverage it is hard to make dimers due to the decreasing number of available sites. As the coverage increases, either by H or by D atoms, the mechanism becomes less efficient and the deuterium atoms will not be abstracted. This also results in a very high saturation coverage.
The dimer fraction is higher as compared to the other mechanisms, although still much too low. The increase in dimer fraction is due to the fact that many of the chemisorbed atoms desorb due to the initial energy. Since atoms in dimer position are less likely to desorb because of their higher binding energy, this results in an elevated dimer ratio.
![Simulations including mechanism V ($R_{\rm dif} = 5.0
\times 10^{13}$ s$^{-1}$). (a) Initial coverage and yield as a function the D pre-exposure. (b) The cross section and initial signal versus the initial D coverage.[]{data-label="cs_5"}](802819jcp10.eps)
### Mechanism V: Fast diffusion {#mechanism-v-fast-diffusion .unnumbered}
As discussed above, the barriers to diffusion in the chemisorbed state are quite high making this process essentially negligible at least at low coverage. Diffusion in the physisorbed state is, however, a completely different matter. If a simple expression for an activated process is assumed, then an atom thermalized to a surface temperature of 150 K will make around 20 hops, which corresponds to a distance of 5 [Å]{} before desorbing. However, as Bonfanti et al. [@Bonfanti:2007] pointed out diffusion is not an activated process, since the activation barrier of 4 meV is negligible at these temperatures. This asks for a different estimation of the moving rate. As a first approach we took the diffusion coefficient obtained by Bonfanti et al. [@Bonfanti:2007]. This results in a moving rate of $1.3 \times 10^{13}$ s$^{-1}$. The upper limit for diffusion will be free movement in two dimensions according to the average gas phase velocity. This results in a diffusion rate of $5.0 \times 10^{13}$ s$^{-1}$ at 150 K. We will perform simulations using both rates. The results for the highest rate and $E_{\rm ER,mono} = 9$ meV are shown in Figures \[dHDdt\]-V and \[cs\_5\]. Results of other simulation series with different combinations of $E_{\rm ER,mono}$ and the diffusion rate are summarized in Table \[Sumtable\]. A clear increase in the dimer fraction can be observed for increasing diffusion rate, although still slightly too low compared with experiments. The cross section curve as a function of the coverage has the correct dependence although it is still not high enough at low coverage. Furthermore the initial signal as a function of coverage does not have a linear dependence but exhibits a kink around 10 %. At this coverage the number of trimers starts declining in favour of the number of tetramers. Most of the configurations (60 %) are now a configuration of three or higher. This non-linearity could well be an artifact of the simplified treatment of reaction and sticking for these configurations which makes the model inaccurate at high coverage as discussed above.
### Combined mechanisms {#combined-mechanisms .unnumbered}
The experiments cannot be explained by one of the five mechanisms separately. Mechanism I is not efficient enough at low coverages, mechanism II leads to very low saturation coverages, mechanism III shows a maximum at $t>0$, mechanism IV is not efficient at high coverages. Moreover, all these four mechanisms show very low dimer fractions compared with experiments. Mechanism V has an increased number of dimers and a higher cross section, but again this is not high enough to match the experimental observations. We therefore investigate different combinations of the five mechanisms.
Again the results are summarized in Table \[Sumtable\]. The table clearly shows that a combination of mechanisms IV and V gives the best results, both in terms of dimer ratios and cross section. The best agreement is obtained for the high diffusion rate and $E_{\rm ER,mono} = 9$ meV (in bold face). The detailed analysis of these simulations is given in Figure \[cs\_3+5\]. The cross section has the correct coverage dependence, both in trend and in value. Also the saturation coverage of $0.4 \pm 0.2$ ML found experimentally is reproduced. The major discrepancy is again the initial HD signal which does not dependent linearly on the initial cross section, but as mentioned earlier this can be an artifact due to the simplified assumptions made for sticking and reaction for complex configurations, which make high coverage simulations inaccurate. The dimer ratio is still below the experimental values, but much higher than all previous series. The dimer ratio includes both ortho and para dimers, but the main contribution is from the para dimers (40 times more at 0.1 %), since these sites have no barrier for sticking. The residence time of the physisorbed atoms at ortho sites is not long enough for the atoms to easily enter the chemisorbed state. As the dimers have been observed in more equal ratios, the rate for crossing this barrier is probably higher, possibly due to a lower barrier or higher pre-exponential factor than used in the present model or due to a longer residence time at the ortho site because of the irregularity in the lattice caused by the first hydrogen atom. The reason that this combination of mechanisms give higher cross section at low coverage is because more reaction events go via a dimer state. In this state an atoms gains some energy due to the strong binding energy and this extra binding energy is used to overcome the barrier for reaction.
Other combinations show less agreement with the experiments on at least two points of comparison.
--------------------------------------------------------------------------------------------------------------------------------------------------- ---- ------------------- ----------------- -------------------- ---------- ---- --- --- --- -------- --- --- --- --- ----- --- ---- ----
Mechanism $T_{\rm start}$ steering Dimers
$(K)$ $s$ %
I (Fig. \[dHDdt\]-I) 9 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0 8 9 0 7 2 5 0 5 0 987 0 46 13
18 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0 7 5 0 5 2 6 0 5 0 995 0 48 13
0 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0 9 9 0 8 3 0 0 5 0 985 0 44 13
II (Fig. \[dHDdt\]-II) 9 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 1 13 1 1 1 11 4 0 5 0 997 0 10 11
9 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0.5 11 2 0 7 5 7 0 5 0 993 0 20 12
III (Fig. \[dHDdt\]-III) $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0 0 0 0 0 2 5 0 5 0 998 0 52 21
IV (Fig. \[dHDdt\]-IV) $1 \cdot 10^{9}$ 2000 $7.2\cdot 10^{11}$ 0 13 8 1 2 2 4 0 7 0 302 0 88 29
$4 \cdot 10^{11}$ 10000 $7.2\cdot 10^{11}$ 0 6 4 1 0 2 2 0 5 0 856 0 54 72
V 9 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 0 11 8 2 0 2 7 0 5 0 977 0 46 32
18 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 0 6 5 0 6 2 4 0 5 0 989 0 47 32
(Fig. \[dHDdt\]-V) 9 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 0 10 1 0 5 2 8 0 5 0 984 0 46 52
18 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 0 5 8 0 5 2 6 0 5 0 982 0 47 53
I + III 9 $2 \cdot 10^{12}$ 2000 $7.2\cdot 10^{11}$ 0 9 5 0 8 2 8 0 5 0 986 0 45 21
I + IV 9 $1 \cdot 10^{9}$ 2000 $7.2\cdot 10^{11}$ 0 10 0 1 0 3 1 0 5 0 984 0 42 30
II + IV 9 $1 \cdot 10^{9}$ 2000 $7.2\cdot 10^{11}$ 1 14 2 0 8 11 0 0 5 0 994 0 08 27
III + V 9 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{11}$ 0 8 3 0 8 2 7 0 5 0 983 0 45 52
II + V 9 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 1 14 3 0 9 10 5 0 5 0 998 0 12 30
18 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 1 11 9 0 5 9 1 0 5 0 996 0 13 30
9 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 1 13 6 1 0 10 4 0 5 0 997 0 13 50
18 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 1 11 4 0 5 9 1 0 5 0 997 0 14 51
9 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 0.5 10 9 0 7 6 6 0 5 0 986 0 20 33
18 $2 \cdot 10^{12}$ 2000 $1.3\cdot 10^{13}$ 0.5 10 5 1 5 5 6 0 5 0 992 0 22 33
9 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 0.5 10 9 0 5 5 7 0 5 0 987 0 21 52
18 $2 \cdot 10^{12}$ 2000 $5.0\cdot 10^{13}$ 0.5 9 6 0 5 5 3 0 5 0 996 0 23 52
IV + V 9 $1 \cdot 10^{9}$ 2000 $1.3\cdot 10^{13}$ 0 12 8 0 7 3 0 0 5 0 971 0 40 55
18 $1 \cdot 10^{9}$ 2000 $1.3\cdot 10^{13}$ 0 11 9 1 2 2 7 0 5 0 987 0 43 53
**(Fig. \[cs\_3+5\]) &**9 & $1 \cdot 10^{9}$ &**2000 & $5.0\cdot 10^{13}$ &**0 &**15&**6 &**1&**5 &**3&**2 &**0&**5 &**0&**980 & **0&**40 & **74\
& 18 & $1 \cdot 10^{9}$ & 2000 & $5.0\cdot 10^{13}$ & 0 & 10&8 & 0&7 & 2&8 & 0&5 & 0&985 & 0&43 & 76\
& 9 & $4 \cdot 10^{8}$ & 2000 & $5.0\cdot 10^{13}$ & 0 & 15&9 & 0&5 & 3&9 & 0&5 & 0&959 & 0&30 & 93\
& 9 & $3\cdot 10^{11}$ & 7000 & $5.0\cdot 10^{13}$ & 0 & 14&8 & 1&4 & 4&7 & 0&5 & 0&981 & 0&28 & 65\
& 9 & $4\cdot 10^{11}$ & 10000 & $5.0\cdot 10^{13}$ & 0 & 14&4 & 1&5 & 5&8 & 0&5 & 0&991 & 0&18 & 71\
III + IV + V & 9 & $1 \cdot 10^{9}$ & 2000 & $5.0\cdot 10^{13}$ & 0 & 16&0 & 0&6 & 2&8 & 0&5 & 0&970 & 0&41 & 75\
\
V & 9 & $2 \cdot 10^{12}$& 2000 & $1.3\cdot 10^{13}$ & 0 & 14&9 & 0&7 & 2&7 & 0&5 & 0&960 & 0&43 & 51\
& 18 & $2 \cdot 10^{12}$& 2000 & $1.3\cdot 10^{13}$ & 0 & 10&3 & 1&0 & 2&8 & 0&5 & 0&985 & 0&46 & 51\
& 9 & $2 \cdot 10^{12}$& 2000 & $5.0\cdot 10^{13}$ & 0 & 13&2 & 1&3 & 2&9 & 0&5 & 0&964 & 0&43 & 71\
& 18 & $2 \cdot 10^{12}$& 2000 & $5.0\cdot 10^{13}$ & 0 & 11&1 & 1&1 & 2&6 & 0&5 & 0&978 & 0&46 & 71\
IV + V & 9 & $1 \cdot 10^{9}$ & 2000 & $1.3\cdot 10^{13}$ & 0 & 10&4 & 0&6 & 3&2 & 0&5 & 0&957 & 0&37 & 64\
& 18 & $1 \cdot 10^{9}$ & 2000 & $1.3\cdot 10^{13}$ & 0 & 10&1 & 0&5 & 2&8 & 0&5 & 0&974 & 0&40 & 64\
& 9 & $1 \cdot 10^{9}$ & 2000 & $5.0\cdot 10^{13}$ & 0 & 18&1 & 0&8 & 3&2 & 0&5 & 0&962 & 0&38 & 83\
& 18 & $1 \cdot 10^{9}$ & 2000 & $5.0\cdot 10^{13}$ & 0 & 11&9 & 1&0 & 2&8 & 0&5 & 0&975 & 0&42 & 84\
Exp. [@Zecho:2002] && & & & & & & & &\
Exp. [@Hornekaer:2006I] & && & & & & & & & $\sim$85\
**********************************
--------------------------------------------------------------------------------------------------------------------------------------------------- ---- ------------------- ----------------- -------------------- ---------- ---- --- --- --- -------- --- --- --- --- ----- --- ---- ----
Sensitivity of the model on parameters \[Sense\]
================================================
By comparing the results listed in Table \[Sumtable\] it is clear that a combination of mechanisms IV and V reproduces the experimental findings quite well. This section will discuss the influence of the parameter settings on the final result. The table already showed four different parameter choices varying the diffusion rate and the barrier for abstraction. The diffusion rate has, as mentioned before, a clear influence on the number of formed dimers, but it appears to have very little effect on the obtained cross sections for the mechanisms IV and V. Lowering it to a very low value of $7.2\cdot 10^{11}$ s$^{-1}$, which turns it into a combination of mechanisms I and IV, results in a lower cross section at low coverage. We found that reducing the diffusion rate even further eventually leads to a completely flat cross section and a very high Pearson correlation coefficient. Also for a pure mechanism V a higher diffusion rate gives a stronger coverage dependence of the cross section as was suggested by Bonfanti et al. [@Bonfanti:2007].
The abstraction barrier has no effect on the dimer ratio, but it influences the cross section at low coverage. It generally gives a lower cross section for a higher barrier. For a pure mechanism I also a barrierless abstraction was considered (see Table \[Sumtable\]) . The cross section at low coverage did not raise to the values found in the experiments in this case.
The parameter $B$, which controls the thermalization of the sticking atoms, has a very narrow parameter range if only mechanism IV is considered. Lower values than $1 \cdot 10^{9}$ s$^{-1}$ result in very low sticking at low coverage and higher values will have only a limited effect on the cross section. If mechanism IV is used in combination with mechanism V the sticking will be increased because of the diffusion of the physisorbed atoms and also lower values than $1 \cdot 10^{9}$ s$^{-1}$ can be used leading to a higher cross section for low deuterium exposure and high dimer ratios. The linearity in the initial HD mass spectrometer signal is however decreased.
Adding mechanism III as a third mechanism in the simulations had a negligible effect on the results. The cross section and initial signal curves remained unchanged within their uncertainties.
The parameter, $T_{\rm start}$, which controls the initial energy of hydrogen atoms, has a different effect on the final results for the pure mechanism IV and the combination of IV and V. This parameter cannot be controlled independently of $B$, since the sticking probability is constrained within a certain window (see Fig. \[therma\]). For a pure mechanism IV a increased value of $T_{\rm start}$ results in a cross section which is too low and only weakly dependent on coverage. The correlation coefficient for the initial HD signal is higher as compared to $T_{\rm start}$ = 2000 K. This parameter combination further shows an extremely high dimer ratio. Only combinations of mechanism were found to yield similar high values. It appear that the high initial energy is only used to remove monomers and not for dimers to react or to desorb. The relaxation time is probably to fast for these events to occur. If a more elaborate implementation of this mechanism was considered, where the initial energy of atoms in dimers is high, because of their higher binding energy, the reaction from dimers would go up and the dimer ratio go down. The results would then probably closer resemble the $T_{\rm start}$ = 2000 K results. For a combination of mechanisms $T_{\rm start}$ has less of an effect. Here only the saturation coverage shows a large change.
Finally we checked the influence of increasing the monomer sticking barrier to the value of 0.25 eV which is an upper value for the sticking barrier reported by other authors [@Sha:2002; @Sha:2002a; @Morisset:2004]. This results in a higher cross section at low coverage and a higher dimer ratio in all simulated cases. However, the correlation coefficient of the initial mass spectrometer signal as a function of coverage goes down. In general this increase in the sticking barrier results in a better agreement with the experiments. Although the agreement of the pure mechanism V is improved much, the combination of mechanism IV and V remains the best option for both low and high values of $E_{\rm stick}$.
![Simulations including mechanism IV ($B = 1
\times 10^9$ s$^{-1}$) and mechanism V ($E_{\rm ER, mono} = E_{\rm ER, dimer} = 9$ meV and $R_{\rm dif} = 5.0\times 10^{13}$ s$^{-1}$). (a) Initial coverage and yield as a function the D pre-exposure. (b) The cross section and initial signal versus the initial D coverage.[]{data-label="cs_3+5"}](802819jcp11.eps)
Conclusion
==========
The Eley-Rideal abstraction of atomic hydrogen chemisorbed on the graphite surface has been studied via a hybrid approach using energy barriers derived from DFT calculations as input to Monte Carlo simulations. Through comparison with experimental data we discriminate between the contributions from different proposed Eley-Rideal mechanisms. Good quantitative and qualitative agreement between the experimentally derived and the simulated Eley-Rideal abstraction cross sections are found if two different Eley-Rideal abstraction mechanisms are included. One is a direct Eley-Rideal reaction with very fast diffusion of physisorbed H atoms leading to the formation of hydrogen dimer configurations, while the other is a dimer mediated Eley-Rideal mechanism with increased cross section at low coverage. Such a dimer mediated Eley-Rideal mechanism has not previously been proposed.
The effect on abstraction of fast diffusing physisorbed H atoms was first considered by Bonfanti et al. [@Bonfanti:2007], who suggested that diffusion of the physisorbed atoms could explain the high coverage dependence of the cross section. We tested this mechanism and it was indeed found that diffusion plays an important role, not only for the abstraction reaction but also to explain the high occurrence of dimers found by STM measurements. However, very little is known about the exact energetic landscape in the vicinity of a chemisorbed hydrogen atom, especially when the second atom is only weakly physisorbed. Since these weak interactions appear to be very important, further study would be desirable.
Furthermore, we investigated the effect of steering [@Sha:2002; @Sha:2002a] as a possibly alternative route to reproduce the experimental findings. The results presented here show that a simple steering mechanism which just results in increased cross sections for abstraction reactions is not sufficient to reproduce the experimental results. This finding does, however, not rule out steering as an important mechanism. It has for example been suggested that a coverage dependent steering effect could result from the creation of and interaction with electron-hole pairs on the surface [@Hammer-Private-communications]. Further investigations of steering should, however, consider the presence of preferred binding sites in the vicinity of adsorbed hydrogen atoms. In the present case, sticking in a dimer position with low or no barrier is a competing reaction to the steered Eley-Rideal reaction. Hence, the described dimer mediated Eley-Rideal mechanism offers a competing mechanism to steering. Full quantum mechanical calculations where both channels, dimer formation and hydrogen abstraction, are accessible, are needed to settle this case.
Acknowledgments
===============
HC is supported by the Netherlands Organization for Scientific Research (NWO) and the Leiden Observatory. LH acknowledges support from the Danish Natural Science Research Foundation.
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author:
- |
J. Madore\
Laboratoire de Physique Théorique et Hautes Energies[^1]\
Université de Paris-Sud, Bât. 211, F-91405 Orsay\
date: 'June, 1997'
title: 'Gravity on Fuzzy Space-Time'
---
\#1[\#1 ]{} @th \#1
*Dedicated to Walter Thirring on the occasion of his 70th birthday*
ESI Preprint 478 (1997). Lecture given at the International Workshop “Mathematical Physics - today, Priority Technologies - for tomorrow”, Kyiv, Ukraine, May 1997.
plus2pt minus2pt
Introduction and Motivation
===========================
Simply stated, ‘fuzzy space-time’ is a space-time in which the ‘coordinates’ do not commute. One typically replaces the four Minkowski coordinates $x^\mu$ by four generators $q^\mu$ of a noncommutative algebra which satisfy commutation relations of the form $$[q^\mu, q^\nu] = i \kbar q^{\mu\nu}. \eqno(1.1)$$ The parameter $\kbar$ is a fundamental area scale which we shall suppose to be of the order of the Planck area: $$\kbar \simeq \mu_P^{-2} = G\hbar.$$ Equation (1.1) contains in fact little information about the algebra. If the right-hand side does not vanish it states that at least some of the $q^\mu$ do not commute. It states also that it is possible to identify the original coordinates with the generators $q^\mu$ in the limit where the Planck mass $\mu_P$ tends to infinity: $$x^\mu = \lim_{\kbar \rightarrow 0} q^\mu. \eqno(1.2)$$ For mathematical simplicity we shall suppose this to be the case although one could include a singular ‘renormalization constant’ $Z$ and replace (1.2) by an equation of the form $$Z \, x^\mu = \lim_{\kbar \rightarrow 0} q^\mu. \eqno(1.3)$$ If, as we shall argue, gravity acts as a universal regulator for ultraviolet divergences then one could reasonably expect the limit $\kbar \rightarrow 0$ to be a singular limit. An argument in this sense has been given by Mangano [@Man97].
Perhaps not the simplest but certainly the most familiar example of a ‘fuzzy space’ is the quantized version of a 2-dimensional phase space, described by the ‘coordinates’ $p$ and $q$. This example has the advantage of illustrating what is for us the essential interest of the relation of the form (1.1) as expressed in the Heisenberg uncertainty relations. Since one cannot measure simultaneously $p$ and $q$ to arbitrary precision quantum phase space has no longer a notion of a point. It can however be thought of as divided into cells of volume $2\pi\hbar$. If the classical phase space is of finite total volume there will be a finite number of cells and the quantum system will have a finite number of possible states. A ‘function’ then on quantum phase space will be defined by a finite number of values and can be represented by a matrix.
By analogy with quantum mechanics we shall suppose that the generators $q^\mu$ can be represented as hermitian operators on some (complex) Hilbert space. The presence of the factor $i$ in (1.1) implies that the $q^{\mu\nu}$ are also hermitian operators. The $q^\mu$ have real eigenvalues but because of the relations (1.1) they cannot be simultaneously diagonalized; points are ill-defined and space-time consists of elementary cells of volume $(2 \pi \kbar)^2$. Now when a physicist calculates a Feynman diagram he is forced to place a cut-off $\Lambda$ on the momentum variables in the integrands. This means that he renounces any interest in regions of space-time of volume less than $\Lambda^{-4}$. As $\Lambda$ becomes larger and larger the forbidden region becomes smaller and smaller but it can never be made to vanish. There is a fundamental length scale, much larger than the Planck length, below which the notion of a point is of no practical importance. The simplest and most elegant, if certainly not the only, way of introducing such a scale in a Lorentz-invariant way is through the introduction of the ‘coordinates’ $q^\mu$. The analogs of the Heisenberg uncertainty relations imply then that $$\Lambda^2 \kbar \lesssim 1.$$ The existence of a forbidden region around each point in space-time means that the standard description of Minkowski space as a 4-dimensional continuum is redundant. There are too many points. Heisenberg already in the early days of quantum field theory proposed to replace the continuum by a lattice structure. A lattice however breaks Poincaré invariance and can hardly be considered as fundamental. It was Snyder [@Sny47a] who first had the idea of using non-commuting coordinates to mimic a discrete structure in a covariant way although something similar had previously been proposed by Markov [@Mar40]. In his article [@Fin69] on the subject Finkelstein cites Riemann as the first person to be concerned with the existence of discrete objects within a continuum space.
As a simple illustration of how a ‘space’ can be ‘discrete’ in some sense and still covariant under the action of a continuous symmetry group one can consider the ordinary round 2-sphere, which has acting on it the rotational group $SO_3$. As a simple example of a lattice structure one can consider two points on the sphere, for example the north and south poles. One immediately notices of course that by choosing the two points one has broken the rotational invariance. It can be restored at the expense of commutativity. The set of functions on the two points can be identified with the algebra of diagonal $2 \times 2$ matrices, each of the two entries on the diagonal corresponding to a possible value of a function at one of the two points. Now an action of a group on the lattice is equivalent to an action of the group on the matrices and there can obviously be no non-trivial action of the group $SO_3$ on the algebra of diagonal $2 \times 2$ matrices. However if one extends the algebra to the noncommutative algebra of all $2 \times 2$ matrices one recovers the invariance. The two points, so to speak, have been smeared out over the surface of a sphere; they are replaced by two cells. An ‘observable’ is an hermitian $2 \times 2$ matrix and has therefore two real eigenvalues, which are its values on the two cells. Although what we have just done has nothing to do with Planck’s constant it is similar to the procedure of replacing a classical spin which can take two values by a quantum spin of total spin 1/2. Only the latter is invariant under the rotation group. By replacing the spin 1/2 by arbitrary spin $s$ one can describe a ‘lattice structure’ of $n = 2s+1$ points in an $SO_3$-invariant manner. The algebra becomes then the algebra $M_n$ of $n \times n$ complex matrices. We shall discuss this example in more detail in Section 5.3.
It is to be stressed that we modify the structure of Minkowski space-time but maintain covariance under the action of the Poincaré group. A fuzzy space-time looks then like a solid which has a homogeneous distribution of dislocations but no disclinations. We can pursue this solid-state analogy and think of the ordinary Minkowski coordinates as macroscopic order parameters obtained by coarse-graining over scales less than the fundamental scale. They break down and must be replaced by elements of some noncommutative algebra when one considers phenomena on these scales. It might be argued that since we have made space-time ‘noncommutative’ we ought to do the same with the Poincaré group. This logic leads naturally to the notion of a $q$-deformed Poincaré (or Lorentz) group which act on a very particular noncommutative version of Minkowski space called $q$-Minkowski space. We discuss $q$-deformations in Section 5.2. It has also been argued, for conceptual as well as practical, numerical reasons, that the lattice version of space-time or of space is quite satisfactory if one uses a random lattice structure or graph. From this point of view the Lorentz group is a classical invariance group and is not valid at the microscopic level. We shall briefly mention this possibility in Section 5.1.
Let ${\cal A}_\kbar$ be the algebra generated in some sense by the elements $q^\mu$. We shall be here working on a formal level so that one can think of ${\cal A}_\kbar$ as an algebra of polynomials in the $q^\mu$ although we shall explicitly suppose that there are enough elements to generate smooth functions on space-time in the commutative limit. Since we have identified the generators as hermitian operators on some Hilbert space we can identify ${\cal A}_\kbar$ as a subalgebra of the algebra of all operators on the Hilbert space. We have added the subscript $\kbar$ to underline the dependence on this parameter but of course the commutation relations (1.1) do not determine the structure of ${\cal A}_\kbar$, We in fact conjecture that every possible gravitational field can be considered as the commutative limit of a noncommutative equivalent and that the latter is strongly restricted if not determined by the structure of the algebra ${\cal A}_\kbar$. We must have then a large number of algebras ${\cal A}_\kbar$ for each value of $\kbar$.
We argued above that the noncommutative structure gives rise to an ultraviolet cut-off. This idea has been developed by several authors [@HelTan54; @DopFreRob95; @KemManMan95; @KemMan96] since the original work of Snyder [@Sny47a; @Sny47b]. It is the right-hand arrow of the diagram $$\def\normalbaselines{\baselineskip=18pt}
\matrix{
{\cal A}_\kbar &\Longleftarrow &\Omega^*({\cal A}_\kbar)\cr
\Downarrow && \Uparrow \cr
\hbox{Cut-off} &&\hbox{Gravity}
}
\def\normalbaselines{\baselineskip=12pt} \eqno(1.4)$$ The top arrow is a mathematical triviality; the $\Omega^*({\cal
A}_\kbar)$ is what gives a differential structure to the algebra. We shall define and discuss it in Section 2.2. The main section is Section 4. In it we shall attempt, not completely successfully, to argue that each gravitational field is the unique ‘shadow’ in the limit $\kbar
\rightarrow 0$ of some differential structure over some noncommutative algebra. This would define the left-hand arrow of the diagram.
The composition of the three arrows is an expression of an old idea, due to Pauli and developed by Deser [@Des57] and others [@IshSalStr71], that perturbative ultraviolet divergences will one day be regularized by the gravitational field. The possibility which we shall consider here is that the mechanism by which this works is through the introduction of noncommuting ‘coordinates’ such as the $q^\mu$. A hand-waving argument can be given [@MadMou95] which allows one to think of the noncommutative structure of space-time as being due to quantum fluctuations of the light-cone in ordinary 4-dimensional space-time. This relies on the existence of quantum gravitational fluctuations. A purely classical argument based on the formation of black-holes has been also given [@DopFreRob95]. In both cases the classical gravitational field is to be considered as regularizing the ultraviolet divergences through the introduction of the noncommutative structure of space-time. This can be strengthened as the conjecture that the classical gravitational field and the noncommutative nature of space-time are two aspects of the same thing. It is our purpose here to explore in some detail this relation.
For an sampling of the early history of ideas on the microtexture of space-time we refer to Section 1.3 of the book by Prugovečki [@Pru95] as well as to the review articles by Kragh & Carazza [@KraCar94] and Gibbs [@Gib95]. When referring to the version of space-time which we describe here we use the adjective ‘fuzzy’ to underline the fact that points are ill-defined. Since the algebraic structure is described by commutation relations the qualifier ‘quantum’ has also been used [@Sny47a; @DopFreRob95; @MadMou96b]. This latter expression is unfortunate since the structure has no immediate relation to quantum mechanics and also it leads to confusion with ‘spaces’ on which ‘quantum groups’ act. To add to the confusion the word ‘quantum’ has also been used [@GreYau97] to designate equivalence classes of ordinary differential geometries which yield isomorphic string theories and the word ‘lattice’ has been used [@'tH96] to designate what we here qualify as ‘fuzzy’. The idea of a $q$-deformation goes back to the dawn of time. Almost immediately after Clifford introduced his algebras they were $q$-deformed with $q$ a root of unity by Sylvester [@Syl84] and by Cartan [@Car98]. This idea was taken up later in a special case by Weyl [@Wey50] and Schwinger [@Sch60] to produce a finite version of quantum mechanics.
Fuzzy space-time
================
Space-time as an algebraic structure
------------------------------------
We saw in the Introduction that by making the coordinates noncommutative we lost the space-time but retained an equivalent of the algebra of functions on it. The purpose of noncommutative geometry is to reformulate as much as possible the geometry of a space in terms of its algebra of functions and then generalize the corresponding results of differential geometry to the case of a noncommutative algebra. We have noticed that the main notion which is lost when passing from the commutative to the noncommutative case is that of a point. ‘Noncommutative geometry is pointless geometry.’ The original noncommutative geometry is based on the quantized phase space of non-relativistic quantum mechanics. In fact Dirac in his historical papers in 1926 [@Dir26a; @Dir26b] was aware of the possibility of describing phase-space physics in terms of the quantum analog of the algebra of functions, which he called the quantum algebra and he was aware of the absence of localization, expressed by the Heisenberg uncertainty relation, as a central feature of these geometries. Inspired by work by von Neumann, for several decades physicists studied quantum mechanics and quantum field theory as well as classical and quantum statistical physics giving prime importance to the algebra of observables and considering the state vector as a secondary derived object. This work has much in common with noncommutative geometry. The notion of a pure state replaces that of a point.
The details of the structure of the algebra ${\cal A}_\kbar$ will be contained, for example, in the commutation relations $[q^\lambda,
q^{\mu\nu}]$. The $q^{\mu\nu}$ can be also considered as extra generators and the Equations (1.1) as extra relations. In this case the $q^{\mu\nu}$ cannot be chosen arbitrarily. They must satisfy the four Jacobi identities: $$[q^\lambda, q^{\mu\nu}] + [q^\mu, q^{\nu\lambda}]
+ [q^\nu, q^{\lambda\mu}] = 0. \eqno(2.1)$$ One can define recursively an infinite sequence of elements by setting, for $p \geq 1$, $$[q^\lambda, q^{\mu_1 \cdots \mu_p}] = i \kbar q^{\mu_1 \cdots
\mu_{(p+1)}}. \eqno(2.2)$$ Several structures have been considered in the past [@Sny47a; @Mad89a; @DopFreRob95; @Mad95]. With our choice of normalization $q^{\mu_1 \cdots
\mu_p}$ has units of mass to the power $p-2$. We shall assume that for the description of a generic gravitational field the appropriate algebra ${\cal A}_\kbar$ has a trivial center, that the only elements which commute with all other elements are the constant multiples of the identity element. The only argument we have in favour of this assumption is the fact that it could be argued that if the center is not trivial then the ‘quantization’ has been only partial. It implies of course that the sequence of $q^{\mu_1 \cdots \mu_p}$ never ends, although all these elements need not be independent. The observables will be some subset of the hermitian elements of ${\cal A}_\kbar$. We shall not discuss this problem here; we shall implicitly suppose that all hermitian elements of ${\cal A}_\kbar$ are observables, including the ‘coordinates’. We shall not however have occasion to use explicitly this fact.
Consider the structure of the ‘classical’ limit ${\cal A}_0$ of ${\cal
A}_\kbar$ obtained by letting $\kbar \rightarrow 0$. For this we must suppose that $Z = 1$ in Equation (1.3). Assume that one can identify ${\cal A}_0 = {\cal C}(V_0)$ as the algebra of smooth, complex-valued functions on a real extension $V_0$ of space-time of dimension $\geq 4$. and that there is a projection of $V_0$ onto ordinary space-time. We set $$x^{\mu_1 \cdots \mu_p} =
\lim_{\kbar \rightarrow 0}q^{\mu_1 \cdots \mu_p}.$$ A set of independent elements of the complete set of the $x^{\mu_1
\cdots \mu_p}$ are local coordinates of $V_0$. The dimension of $V_0$ will depend on how many there are. If space-time is Minkowski space-time then the condition of Lorentz invariance in the commutative limit forces $x^\lambda$ and at least 4 of the 6 coordinates $x^{\mu\nu}$ to be independent [@DopFreRob95; @DubKerMad97]. In general the set $x^{\mu_1 \cdots \mu_p}$ for $p \geq 3$ can at least in part be functions of $x^\lambda$ and $x^{\mu\nu}$. One can consider $V_0$ as a Kaluza-Klein extension of space-time by a space which is perhaps of infinite dimension and in general not compact. It should be stressed however that $V_0$ is a mathematical fiction. The ‘real’ world is described by the algebra ${\cal A}_\kbar$; it is this algebra which we consider to be the correct Kaluza-Klein extension of space-time [@Mad89b; @MadMou95]. The difference in dimension between $V_0$ and space-time is one of the measures of the extent to which the verb ‘to quantize’ as applied to the coordinates of space-time is a misnomer; one could [*in extremis*]{} ‘quantize’ the coordinates of $V_0$. Even here the verb should be restricted to cases in which the right-hand side of (1.1) lies in the center of the algebra.
Quite generally the commutator of ${\cal A}_\kbar$ defines a Poisson structure on $V_0$. We have given arguments [@MadMou96b], based on simple models, that a differential calculus over ${\cal A}_\kbar$ should determine a metric-compatible torsion-free linear connection on $V_0$. It is natural then that there should be a relation between the Poisson structure and the curvature of the connection. One can show [@Mad97a] that certain natural hypotheses on the differential calculus yield in fact relations between the two. We have been however unable so far to present a realistic gravitational field explicitly as the ‘shadow’ of a differential calculus over a noncommutative algebra.
If there is a gravitational field then there must be some source, of characteristic mass $\mu$. If $\mu^2 \kbar$ tends to zero with $\kbar$ then $V_0$ will be without curvature. This case has been considered previously [@DopFreRob95; @DubKerMad97]. We are interested here in the case in which $\mu^2 \kbar$ tends to some finite non-vanishing value as $\kbar \rightarrow 0$.
Space-time as a differential structure
--------------------------------------
But space-time is more than just an algebra of functions; it has a differential structure. Although it was von Neumann who introduced the expression ‘noncommutative geometry’ it was only recently that mathematicians, notably Connes [@Con86; @Con94], have developed the theory of ‘differential noncommutative geometry’ which we shall use here. The central notion is that of a differential form. We shall define a differential by a set of simple rules which makes it obvious that it is equivalent to a derivative and ask the reader to believe that the rules have a rigorous and natural mathematical foundation. He will see that they are quite easy to manipulate in the simple noncommutative geometries we consider.
We first recall the commutative case. The set of smooth functions ${\cal A}$ on space-time is a commutative algebra, which it is convenient to consider over the complex numbers. A 1-form is a covariant vector field $A_\mu$, which we shall write as $A = A_\mu
dx^\mu$ using a set of basis elements $dx^\mu$. A 2-form is an antisymmetric 2-index covariant tensor $F_{\mu\nu}$ which we shall write as $$F = {1\over 2} F_{\mu\nu} dx^\mu dx^\nu$$ using the product of the basis elements. This product is antisymmetric $$dx^\mu dx^\nu = - dx^\nu dx^\mu \eqno(2.3)$$ but otherwise has no relations. Higher-order forms can be defined as arbitrary linear combination of products of 1-forms. A $p$-form can be thus written as $$\alpha = {1\over p!} \alpha_{\lambda_1 \cdots \lambda_p}
dx^{\lambda_1} \cdots dx^{\lambda_p}.$$ The coefficients $\alpha_{\lambda_1 \cdots \lambda_p}$ are smooth functions and completely antisymmetric in the $p$ indices.
We define $\Omega^0({\cal A}) = {\cal A}$ and for each $p$ we write the vector space of $p$-forms as $\Omega^p({\cal A})$. Each $\Omega^p({\cal A})$ depends obviously on the algebra ${\cal A}$ and, what is also obvious and very important, it can be multiplied both from the left and the right by the elements of ${\cal A}$; each $\Omega^p({\cal A})$ is an ${\cal A}$-bimodule. It is easy to see that $\Omega^p({\cal A}) = 0$ for all $p \geq 5$. We define $\Omega^*({\cal A})$ to be the set of all $\Omega^p({\cal A})$. It has a product $\pi$ induced by (2.3); it is a graded commutative algebra. The product defines a projection $$\Omega^1({\cal A}) \otimes_{\cal A} \Omega^1({\cal A})
\buildrel \pi \over \longrightarrow \Omega^2({\cal A}).$$ The algebra $\Omega^*({\cal A})$ can be written as a sum $$\Omega^*({\cal A}) = \Omega^+({\cal A}) \oplus \Omega^-({\cal A}) \eqno(2.4)$$ of even forms and odd forms. The $A$ is an odd form and $F$ is even. The algebra ${\cal A}$ is a subalgebra of $\Omega^+({\cal A})$.
Let $f$ be a function, an element of the algebra ${\cal A} = \Omega^0({\cal A})$. We define a map $d$ from $\Omega^p({\cal A})$ into $\Omega^{p+1}({\cal A})$ by the rules $$df = \partial_\mu f dx^\mu, \qquad d^2 = 0. \eqno(2.5)$$ It takes odd (even) forms into even (odd) ones. From the rules we find that $$d A = d(A_\mu dx^\mu)
= {1\over 2}(\partial_\mu A_\nu - \partial_\nu A_\mu) dx^\mu dx^\nu = F$$ if we set $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$ From the second rule we have $$d F = 0.$$ It is easy to see that if $\xi$ is a $p$-form and $\eta$ is a $q$-form then $$\xi \eta = (-1)^{pq} \eta \xi, \qquad
d(\xi \eta) = (d \xi) \eta + (-1)^p \xi d\eta.$$
The couple $(\Omega^*({\cal A}), d)$ is called the de Rham differential algebra or the de Rham differential calculus over ${\cal A}$. What distinguishes it is the fact that the 1-forms are dual to the Lie algebra of derivations of ${\cal A}$. The derivative $\partial_\mu f$ of a smooth function $f$ is a smooth function. We use the word derivation to distinguish the map $\partial_\mu$ from the result of the map $\partial_\mu f$. A general derivation is a linear map $X$ from the algebra into itself which satisfies the Leibniz rule: $X(fg) = (Xf)g + f(Xg)$. In the case we are presently considering a derivation can always be written in terms of the basis $\partial_\mu$ as $X = X^\mu \partial_\mu$. Such is not always the case. The relation between $d$ and $\partial_\mu$ is given by $$df (\partial_\mu) = \partial_\mu f.$$ This equation has the same content as the first of the relations (2.5). One passes from one to the other by using the particular case $$dx^\mu (\partial_\nu) = \delta^\mu_\nu. \eqno(2.6)$$ The basis $dx^\mu$ is said to be dual to the basis $\partial_\mu$. The derivations form a vector space (the tangent space) at each point, and (2.6) defines $df$ as an element of the dual vector space (the cotangent space) at the same point. Over an arbitrary algebra which has derivations one can always define in exactly the same manner a differential calculus based on derivations. These algebras have thus at least two, quite different, differential calculi, the universal one and the one based on the set of all derivations.
Over each algebra ${\cal A}$, be it commutative or not, there can exist a multitude of differential calculi. This fact makes the noncommutative version of geometry richer than the commutative version. As a simple example we define what is known as the universal calculus $(\Omega_u^*({\cal A}), d_u)$ over the commutative algebra of functions ${\cal A}$. We set, as always, $\Omega_u^0({\cal A}) = {\cal A}$ and for each $p \geq 1$ we define $\Omega_u^p({\cal A})$ to be the set of $(p+1)$-point functions which vanish when any two points coincide. It is obvious that $\Omega_u^p({\cal A}) \neq 0$ for all $p$. There is a map $d_u$ from $\Omega_u^p({\cal A})$ into $\Omega_u^{p+1}({\cal A})$ given by $(d_uf)(x^\mu, y^\mu) = f(y^\mu) - f(x^\mu)$ for $p = 0$. This can also be written without reference to points as $$d_uf = 1 \otimes f - f \otimes 1.$$ For $p \geq 1$, $d_u$ is given by a similar sort of alternating sum defined so that $d_u^2 = 0$. The algebra $\Omega_u^*({\cal A})$ is not graded commutative. It is defined for arbitrary functions, not necessarily smooth, and it has a straightforward generalization to arbitrary algebras, not necessarily commutative.
To explain the qualifier ‘universal’ let $(\Omega^*({\cal A}), d)$ be any other differential calculus over ${\cal A}$, for example the usual de Rham differential calculus. Then there is a unique $d_u$-homomorphism $\phi$ $$\Omega^*_u({\cal A}) \buildrel \phi \over \longrightarrow \Omega^*({\cal A})$$ of $\Omega^*_u({\cal A})$ onto $\Omega^*({\cal A})$. It is given by $$\phi (f) = f, \qquad \phi (d_u f) = d f.$$ If we choose a coordinate system and expand the function $f(y^\mu)$ about the point $x^\mu$, $$f(y^\mu) = f(x^\mu) + (y^\nu - x^\nu) \partial_\nu f(x^\mu) + \cdots,$$ we see that the map $\phi$ is given by $$\phi (y^\mu - x^\mu) = dx^\mu$$ and that it annihilates any 1-form $f(x^\mu,y^\mu) \in \Omega^1_u({\cal A})$ which is second order in $y^\mu- x^\mu$. One such form is $fd_ug - d_ugf$, given by $$(fd_ug -d_ugf)(x^\mu, y^\mu) =
- (f(y^\mu) - f(x^\mu))(g(y^\mu) - g(x^\mu)).$$ It does not vanish in $\Omega^1_u({\cal A})$ but its image in $\Omega^1({\cal A})$ under $\phi$ is equal to zero.
To form tensors one must be able to define tensor products, for example the tensor product $\Omega^1({\cal A}) \otimes_{\cal A} \Omega^1({\cal A})$ of $\Omega^1({\cal A})$ with itself. We have here written in subscript the algebra ${\cal A}$. This piece of notation indicates the fact that we identify $\xi f \otimes \eta$ with $\xi \otimes f \eta$ for every element $f$ of the algebra, a technical detail which is important. It means also that one must be able to multiply the elements of $\Omega^1({\cal A})$ on the left and on the right by the elements of the algebra ${\cal A}$. Since ${\cal A}$ is commutative of course these two operations are equivalent and this left (right) linearity is equivalent to the property of locality. It means that the product of a function with a 1-form at a point is again a 1-form at the same point, a property which distinguishes the ordinary product from other, non-local, products such as the convolution. In the noncommutative case there are no points and locality can not be defined; it is replaced by the property of left and right linearity with respect to the algebra.
There is an interesting relation between the differential $d$ and the Dirac operator $\Dirac$. Let $\psi$ be a Dirac spinor and $f$ a smooth function. It is straightforward to see that $$\partial_\lambda f \gamma^\lambda\psi = - [i\Dirac, f] \psi.$$ If we make the replacement $\gamma^\lambda \mapsto dx^\lambda$ the left-hand side becomes equal to $df \psi$ and we can write the differential as a commutator: $$df = - [i\Dirac, f].$$ It would be natural to try to generalize this relation to higher-order forms by using a graded commutator on the right-hand side. Because $dx^\mu dx^\nu + dx^\nu dx^\mu = 0$ whereas $\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu \neq 0$ one would find that $d^2 \neq 0$. This problem is connected with the fact that the square of the Dirac operator is not proportional to the identity. We shall see below a simpler example of this and mention how one solves the problem.
Consider now the noncommutative algebra ${\cal A}_\kbar$. First we note that if $X$ is a derivation of ${\cal A}_\kbar$ and $f$ an arbitrary element then in general $fX$ is no longer a derivation. The simplest examples to see this are the matrix algebras which we shall mention below. Also it is sometimes of interest to consider algebras which have no derivations. It is for these reasons that derivations do not play the same role in noncommutative geometry which vector fields play in ordinary geometry and it is much more convenient to use a differential calculus to describe the differential structure.
Suppose that a set of 1-forms $\Omega^1({\cal A}_\kbar)$ has been constructed and that there is a map $d$ of $\Omega^0({\cal A}_\kbar) = {\cal A}_\kbar$ into $\Omega^1({\cal A}_\kbar)$: $$\Omega^0({\cal A}_\kbar) \buildrel d \over \longrightarrow
\Omega^1({\cal A}_\kbar). \eqno(2.7)$$ We shall construct a differential calculus over ${\cal A}_\kbar$ using a procedure due to Connes and Lott [@ConLot92]. In the form which we shall use it the procedure has been described in detail elsewhere [@DimMad96] but the idea is simple. As in the commutative case we suppose that the $\Omega^1({\cal A}_\kbar)$ has the structure of an ${\cal A}_\kbar$-bimodule, that an element of $\Omega^1({\cal A}_\kbar)$ can be multiplied from the right and from the left by an arbitrary element of ${\cal A}_\kbar$ and the result is still an element of $\Omega^1({\cal A}_\kbar)$. We define the bimodule of 2-forms to be the largest set of elements of the form $dfdg$ with a product between $df$ and $dg$ subject only to the condition that it be consistent with the bimodule structure of $\Omega^1({\cal A}_\kbar)$ and with the condition $d^2 = 0$. We have then $$\Omega^0({\cal A}_\kbar) \buildrel d \over \longrightarrow
\Omega^1({\cal A}_\kbar) \buildrel d \over \longrightarrow
\Omega^2({\cal A}_\kbar).$$ If, for example, $f dg - dg f = 0$ as in the commutative case then we must have $d(f dg - dg f) = df dg + dg df = 0$. This construction can be continued to arbitrary $p$-forms [@DimMad96]. It is of course perfectly consistent to choose a smaller algebra of forms. One could set, for example, $\Omega^p({\cal A}_\kbar) = 0$ for all $p \geq 2$.
We shall find it convenient to define $\Omega^1({\cal A}_\kbar)$ in terms of a set of derivations of ${\cal A}_\kbar$. For each integer $n$ let $\lambda_i$ be a set of $n$ linearly independent antihermitian elements of ${\cal A}_\kbar$ and introduce the derivations $e_i$ defined by $$e_i f = [\lambda_i, f].$$ The Leibniz rule follows from the Jacobi identity for the bracket. In general the $e_i$ do not form a Lie algebra but they do however satisfy commutation relations as a consequence of the commutation relations of ${\cal A}$. In order for them to have the correct dimensions one must introduce a mass parameter $\mu$ and replace $\lambda_i$ by $\mu \lambda_i$. We shall set $\mu = 1$. We shall suppose that if an element of ${\cal A}_\kbar$ commutes with all of the $\lambda_i$ then it is a constant multiple of the unit element. This is the noncommutative equivalent of the statement that a function is a constant if all of its partial derivatives vanish. Define $\Omega^1({\cal A}_\kbar)$ and the map (2.7) by $$df (e_i) = e_i \, f. \eqno(2.8)$$
We shall suppose that there exists a set of $n$ elements $\theta^i$ of $\Omega^1({\cal A}_\kbar)$ such that $$\theta^i (e_j) = \delta^i_j. \eqno(2.9)$$ In the examples which we consider we shall show that the $\theta^i$ exist by explicit construction. We shall refer to the set of $\theta^i$ as a frame or Stehbein. It commutes with all the elements $f$ of ${\cal A}_\kbar$: $$f \theta^i = \theta^i f. \eqno(2.10)$$ This follows directly from (2.9) and from the definition of the module structure: $$f dg (e_i) = f e_i \, g, \qquad (dg) f (e_i) = (e_i \, g) f.$$ The ${\cal A}$-bimodule $\Omega^1({\cal A})$ is generated by all elements of the form $f dg$ or of the form $df g$. Because of the Leibniz rule these conditions are equivalent. Using the frame we can write $$f dg = (f e_i g) \theta^i, \qquad (dg) f = (e_i g) f \theta^i. \eqno(2.11)$$ The commutation relations of the algebra constrain then the relations between $f dg$ and $dg f$ for all $f$ and $g$.
Because of the commutation relations of the algebra the $\theta^i$ satisfy in general commutation relations. Since $\Omega^*({\cal A}_\kbar)$ is an algebra there is a natural product map $$\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar}
\Omega^1({\cal A}_\kbar) \buildrel \pi \over \longrightarrow
\Omega^2({\cal A}_\kbar)$$ We shall suppose that $\Omega^2({\cal A}_\kbar)$ is a submodule of $\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar)$ and that $\pi$ is a projection. We can write therefore $$\pi(\theta^i \otimes \theta^j) =
P^{ij}{}_{kl} \theta^k \otimes \theta^l \eqno(2.12)$$ where, because of (2.10), the $P^{ij}{}_{kl}$ are complex numbers with $$P^{ij}{}_{mn} P^{mn}{}_{kl} = P^{ij}{}_{kl}. \eqno(2.13)$$ The product $\theta^i \theta^j$ satisfies then the relations $$\theta^i \theta^j = P^{ij}{}_{kl} \theta^k \theta^l. \eqno(2.14)$$ In several important cases which we shall consider the $\theta^i$ anticommute. This corresponds to the expression $$P^{ij}{}_{kl} =
{1\over 2} (\delta^i_k \delta^j_l - \delta^j_k \delta^i_l) \eqno(2.15)$$ for the $P^{ij}{}_{kl}$.
Define $\theta = - \lambda_i \theta^i$. Then one sees that $$df = e_i f \theta^i = - [\theta, f] \eqno(2.16)$$ and it follows that as a bimodule $\Omega^1({\cal A})$ is generated by one element. We see also that $\theta$ plays a role in these differential calculi that the Dirac operator does in the commutative case. The $\theta$ is here however itself an element of $\Omega^1({\cal A})$ whereas $i \Dirac$ cannot be considered as a 1-form. Under the condition (2.9) the $\Omega^1({\cal A})$ is free of rank $n$ as a left or right module. It can therefore be identified with the direct sum of $n$ copies of ${\cal A}_\kbar$: $$\Omega^1({\cal A}_\kbar) = \bigoplus_1^n {\cal A}_\kbar. \eqno(2.17)$$ This equation states that a 1-form can be described by its components. It implies that the fuzzy space-times which we consider are the noncommutative equivalents of parallelizable manifolds. We see that the rank of $\Omega^1({\cal A})$ can be an arbitrary integer.
One can show [@MadMou96b] that $\theta$ satisfies the equation $$d\theta + \theta^2 = - {1\over 2} K_{ij} \theta^i \theta^j \eqno(2.18)$$ where the $K_{ij}$ are complex numbers. One can show further [@DimMad96; @MadMou96b] that the $\lambda_i$ must satisfy the consistency equations $$2 \lambda_l \lambda_m P^{lm}{}_{jk} -
\lambda_i F^i{}_{jk} - K_{jk} = 0 \eqno(2.19)$$ where the $F^i{}_{jk}$ are complex numbers. The structure elements $C^i{}_{jk}$ are defined by the equation $$d\theta^i =
- {1\over 2} C^i{}_{jk} \theta^j \theta^k. \eqno(2.20)$$ From general arguments [@DimMad96; @MadMou96b] it follows that $$C^i{}_{jk} = F^i{}_{jk} - 2 \lambda_l P^{(li)}{}_{jk}. \eqno(2.21)$$ The structure elements are not therefore in general complex numbers.
The simplest noncommutative algebras are the algebras $M_n$ of $n \times n$ complex matrices. Let $\lambda_i$ in $M_n$, for $1 \leq i \leq n^2-1$, be an antihermitian basis of the Lie algebra of the special unitary group $SU_n$. The product $\lambda_i \lambda_j$ can be written in the form $$\lambda_i \lambda_j = {1\over 2} C^k{}_{ij} \lambda_k +
{1\over 2} D^k{}_{ij} \lambda_k - {1 \over n} g_{ij}.$$ The $g_{ij}$ are the components of the Killing metric; we shall use it to raise and lower indices. The $C^k{}_{ij}$ here are the structure constants of the group $SU_n$ and $g_{kl}D^l{}_{ij}$ is trace-free and symmetric in all pairs of indices. For each $\lambda_i$ we introduce derivations $e_i$ as above in Equation (2.8). In this case the $e_i$ span the vector space of all derivations [@Dub88] of the algebra and form a Lie algebra with commutation relations $$[e_i, e_j] = C^k{}_{ij} e_k.$$ It is an elementary fact of algebra that any derivation $X$ of $M_n$ can be written as a linear combination $X = X^i e_i$ of the $e_i$ with the $X^i$ complex numbers. We have now $$d\lambda^i(e_j) = [\lambda_j, \lambda^i ] = - C^i{}_{jk}\lambda^k.$$ The frame [@DubKerMad89] is given by $$\theta^i = \lambda_j \lambda^i d\lambda^j.$$ The corresponding $P^{ij}{}_{kl}$ is given by (2.15) and $\theta$ satisfies $$d \theta + \theta^2 = 0. \eqno(2.22)$$ This is a particular case of (2.18) with $K_{ij} = 0$. We have seen that as a bimodule $\Omega^1(M_n)$ is generated by $\theta$ alone. For dimensional reasons $\Omega^1(M_n)$ cannot be of rank one. In fact the free $M_n$-bimodule of rank one is of dimension $n^4$ and the dimension of $\Omega^1(M_n)$ is equal to $(n^2-1)n^2 < n^4$. With the normalization which we have used for the generators $\lambda_i$ the element $$\zeta = {1\over n^2} 1 \otimes 1 - {1\over n} \lambda_i \otimes \lambda^i$$ is a projector in $M_n \otimes M_n$ which commutes with the elements of $M_n$. This can be written [@DubMadMasMou96] as $d(M_n) \zeta = 0$. We have the direct-sum decomposition $$M_n \otimes M_n = \Omega^1(M_n) \oplus M_n \,\zeta.$$
One can use matrix algebras to construct examples of differential calculi which have nothing to do with derivations. Consider the algebra $M_n$ graded as in supersymmetry with even and odd elements and introduce a graded commutator between two matrices $\alpha$ and $\beta$ as $$[\alpha, \beta] = \alpha \beta -
(-1)^{\vert \alpha \vert \vert \beta \vert} \beta \alpha$$ where $\vert \alpha \vert$ is equal to 0 or 1 depending on whether $\alpha$ is even or odd. One can define on $M_n$ a graded derivation $\hat d$ by the formula $$\hat d \alpha = - [\eta , \alpha], \eqno(2.23)$$ where $\eta$ is an arbitrary antihermitian odd element. Since $\eta$ anti-commutes with itself we find that $\hat d\eta = -2\eta^2$ and for any $\alpha$ in $M_n$ $$\hat d^2 \alpha = [\eta^2, \alpha]. \eqno(2.24)$$ The grading can be expressed as the direct sum $M_n = M_n^+ \oplus M_n^-$ of the even and odd elements of $M_n$. This decomposition is the analogue of (2.4). If $n$ is even it is possible to impose the condition $$\eta^2 = - 1. \eqno(2.25)$$ From (2.24) we see that $\hat d^2 = 0$ and $\hat d$ is a differential. In this case we shall write $\hat d = d$. We see that $\eta$ must satisfy $$d\eta + \eta^2 = 1, \eqno(2.26)$$ an equation which is to be compared with (2.18) and (2.22). If we define for all $p \geq 0$ $$\Omega^{2p}(M^+_n) = M^+_n, \qquad
\Omega^{2p+1}(M^+_n) = M^-_n \eqno(2.27)$$ then we have defined a differential calculus over $M^+_n$. The differential algebra based on derivations can be embedded in a larger algebra such that a graded extension of (2.16) exists for all elements [@Mad95]. In fact any differential calculus can be so extended.
As an example let $n=2$. To within a normalization the matrices $\lambda_i$ can be chosen to be the Pauli matrices. We define $\lambda_1$ and $\lambda_2$ to be odd and $\lambda_3$ and the identity even. The most general possible form for $\eta$ is a linear combination of $\lambda_1$ and $\lambda_2$ and it can be normalized so that (2.25) is satisfied. Using $\Omega^*(M_2^+)$ one can construct a differential calculus over the algebra of functions on a double-sheeted space-time [@ConLot90; @Coq89]. This doubled-sheeted structure permits one to introduce a description of parity breaking in the weak interactions.
If $n$ is not even or, in general, if $\eta^2$ is not proportional to the unit element of $M_n$ then $\hat d^2$ given by (2.24) will not vanish and $M_n$ will not be a differential algebra. It is still possible however to construct over $M_n^+$ a differential calculus $\Omega^*(M_n^+)$ based on (2.23). Essentially what one does is just eliminate the elements which are the image of $\hat
d^2$ [@ConLot92]. The problem here is an analog of the problem we mentioned in the commutative case where the square of the Dirac operator is not proportional to the identity.
As an example let $n=3$. There is a grading defined by the decomposition $3 = 2 + 1$ The most general possible form for $\eta$ is $$\eta = \left(
\begin{array}{ccc}
0 & 0 & a_1 \\
0 & 0 & a_2 \\
-a^*_1 & -a^*_2 & 0
\end{array}\right). \eqno(2.28)$$ For no values of the $a_i$ is it possible to impose the condition (2.25). The general construction yields $\Omega^0(M_3^+) = M_3^+ = M_2 \times M_1$ and $\Omega^1(M_3^+) = M_3^-$ as in the previous example but after that the elimination of elements which are the image of $\hat d^2$ reduces the dimensions. One finds $\Omega^2(M_3^+) = M_1$ and $\Omega^p(M_3^+) = 0$ for $p\geq 3$ [@ConLot92; @Mad95].
The noncommutative equivalent of a coordinate transformation of space-time could reasonably be considered to be an automorphism of the algebra ${\cal A}_\kbar$; if ${\cal A}_\kbar$ corresponds to the algebra of smooth functions then the corresponding limit coordinate transformation would then be considered as smooth. There are however problems with this identification. It can be seen from the “fuzzy-sphere” example to be described in Section 5.3 that the algebra of morphisms is sometimes to small. In this example one does not obtain as limit a general coordinate transformation [@Mad97b]. At the same time the algebra of morphisms is too big since some of them in the commutative limit change even the topology of the limit manifold [@MadSae97]. The important question is how the automorphisms are to be extended to the algebra $\Omega^*({\cal A}_\kbar)$. Consider as example the differential calculus defined in Section 2.7 by the set of $\lambda_i$ and let $\lambda_i \mapsto \lambda^\prime_i = u^{-1} \lambda_i u$ be an inner automorphism of the algebra. Then it is easy to see that $$d\lambda^\prime_i = - [\theta, \lambda^\prime_i].$$ A sufficient condition then for $u$ to respect the differential structure would be $$\theta^\prime = \theta.$$
Classical gravity
=================
There are several ways to introduce a gravitational field on ordinary, commutative space-time. Our main constraint is that we would like it to be expressed entirely in terms of the algebra of functions ${\cal A}$ and of a differential calculus $\Omega^*({\cal A})$ over ${\cal A}$ since this is what can be generalized to the noncommutative case. The geometry of ordinary smooth spaces was written from the point of view of the algebra of smooth functions by Koszul [@Kos60] in his lectures at the Tata Institute. We shall use the moving frame formalism since it is most convenient for what we believe to be the correct generalization to the noncommutative case. A moving frame is a nonsingular set of four 1-forms $\theta^\alpha$. Since we are especially interested in space-times which are near to Minkowski space we can suppose that the moving frame can be globally defined, an assumption which is a topological restriction. The $\theta^\alpha$ are of course dual to a set $e_\beta$ of derivations of ${\cal A}$: $$\theta^\alpha (e_\beta) = \delta^\alpha_\beta.$$ Equation (2.6) is a particular case of this with $\theta^\alpha = dx^\alpha$.
Let $g_{\alpha\beta}$ be the standard components of the Minkowski metric. We define a metric $g$ by the condition that the moving frame be orthonormal: $$g(\theta^\alpha \otimes \theta^\beta) = g^{\alpha\beta}. \eqno(3.1)$$ If we write in coordinates $\theta^\alpha = \theta^\alpha_\lambda dx^\lambda$ then (3.1) is equivalent to defining the metric by the line element $ds^2 = g_{\alpha\beta} \, \theta^\alpha_\mu \, \theta^\beta_\nu \,
dx^\mu \otimes dx^\nu$. We extend the metric to all tensor products by the requirement of linearity: $$f g(\theta^\alpha \otimes \theta^\beta) =
g(f \theta^\alpha \otimes \theta^\beta), \qquad
g(\theta^\alpha \otimes \theta^\beta) f =
g(\theta^\alpha \otimes \theta^\beta f). \eqno(3.2)$$ These linearity conditions are equivalent to a locality condition for the metric; the length of a vector at a given point depends only on the value of the metric and the vector field at that point. We shall return to this in Section 7. The second rule is here a triviality but in the noncommutative case this will not be so. A metric can be therefore considered as a bimodule map $$\Omega^1({\cal A}) \otimes_{{\cal A}} \Omega^1({\cal A})
\buildrel g \over \rightarrow {\cal A}. \eqno(3.3)$$ The structure functions $C^\alpha{}_{\beta\gamma}$ are defined as in the previous section by the equations analog to (2.20).
A covariant derivative or linear connection can be defined a rule which associates to each covariant vector $\xi$ a 2-index covariant tensor $D\xi$. It can be defined on the basis $dx^\lambda$ by $$D (dx^\lambda) = - \Gamma^\lambda_{\mu\nu} dx^\mu \otimes dx^\nu$$ and extended to an arbitrary 1-form $\xi = \xi_\lambda dx^\lambda$ by the Leibniz rule: $$D \xi = d\xi_\lambda \otimes dx^\lambda + \xi_\lambda D(dx^\lambda)
= d\xi_\lambda \otimes dx^\lambda
- \xi_\lambda \Gamma^\lambda_{\mu\nu} dx^\mu \otimes dx^\nu$$ It can also be written in terms of the moving frame. We define the Ricci rotation coefficients $\omega^\alpha{}_{\beta\gamma}$ by the equation $$D \theta^\alpha = - \omega^\alpha{}_{\beta\gamma}
\theta^\beta \otimes \theta^\gamma.$$ A covariant derivative can be defined as a map $$\Omega^1({\cal A}) \buildrel D \over \rightarrow
\Omega^1({\cal A}) \otimes_{\cal A} \Omega^1({\cal A}) \eqno(3.4)$$ which satisfies the Leibniz rules $$D (f \xi) = df \otimes \xi + f D\xi, \qquad
D (\xi f) = D (f \xi). \eqno(3,5)$$ The second rule is here a triviality but in the noncommutative case it will have to be modified.
Using a graded Leibniz rule, $D$ can be extended to higher-order forms and the curvature 2-form $\Omega^\alpha{}_\beta$ defined by the equation $$D^2 \xi = - \xi_\alpha \Omega^\alpha{}_\beta \otimes \theta^\beta.$$ The curvature is the field strength of the gravitational field. The minus sign is an historical convention. One can be write $\Omega^\alpha{}_\beta$ in terms of the basis as $$\Omega^\alpha{}_\beta =
{1\over 2} R^\alpha{}_{\beta\gamma\delta} \theta^\gamma \theta^\delta$$ an equation which defines the components $R^\alpha{}_{\beta\gamma\delta}$ of the Riemann tensor.
We have two maps of $\Omega^1({\cal A})$ into $\Omega^1({\cal A})$, the composite map $\pi \circ D$ as well as the exterior derivative. The difference between the two is the torsion: $$T = d - \pi \circ D. \eqno(3.6)$$ In particular $$T(dx^\lambda) = {1\over 2} \Gamma^\lambda_{[\mu\nu]} dx^\mu dx^\nu.$$ It vanishes with the antisymmetric part of $\Gamma^\lambda_{\mu\nu}$.
Let $\xi$ and $\eta$ be 1-forms and introduce a flip $\sigma$ in the tensor product: $\sigma(\xi \otimes \eta) = \eta \otimes \xi$. Then the covariant derivative can be extended to arbitrary tensors by the twisted Leibniz rule. $$D(\xi \otimes \eta) = D\xi \otimes \eta +
(\sigma \otimes 1) (\xi \otimes D\eta).$$ A covariant derivative is said to be compatible with the metric $g$ if $$(1 \otimes g)(D(\theta^\alpha \otimes \theta^\beta)) \equiv
- \omega^\alpha{}_{\gamma\delta} \, \theta^\gamma \,
g(\theta^\delta \otimes \theta^\beta) - \omega^\beta{}_{\gamma\delta}
\, \theta^\gamma \,
g(\theta^\alpha \otimes \theta^\delta) = 0. \eqno(3.7)$$ This will be the case if and only if $$\omega_{\alpha\beta\gamma} + \omega_{\gamma\beta\alpha} = 0. \eqno(3.8)$$ A metric-compatible $D$ without torsion is completely determined by the structure functions: $$\omega^\alpha{}_{\beta\gamma} = {1\over 2}
(C^\alpha{}_{\beta\gamma} - C_{\beta\gamma}{}^\alpha +
C_\gamma{}^\alpha{}_\beta). \eqno(3.9)$$
The classical theory of gravity involves also a set field equations for the metric, which are normally supposed to be derived from an action principle. We shall return to this in Section 7.
Noncommutative gravity
======================
In formulating a noncommutative theory of gravity we shall be as conservative as possible and change the definitions of the previous section only where absolutely necessary. Instead of the commutative algebra ${\cal A}$ we have now the noncommutative algebra ${\cal A}_\kbar$. We have also [*ipso facto*]{} the Kaluza-Klein extension $V_0$ of space-time. Let $e_i$ be a set of derivations and $\Omega^*({\cal A}_\kbar)$ the corresponding differential calculus as defined in Section 2. The frame $\theta^i$ plays then the role of the moving frame in the commutative case. One might think of $n$ as the ‘dimension’ but this is a delicate issue. One can only say that $$n \geq \hbox{dim} V_0.$$ Let $g_{ij}$ be the standard components of the Minkowski metric on a $d$-dimensional extension of space-time. We define again a metric $g$ by the condition that the moving frame be orthonormal: $$g(\theta^i \otimes \theta^j) = g^{ij}. \eqno(4.1)$$ We require as before that $$f g(\theta^i \otimes \theta^j) =
g(f \theta^i \otimes \theta^j), \qquad
g(\theta^i \otimes \theta^j) f =
g(\theta^i \otimes \theta^j f) \eqno(4.2)$$ and therefore a metric can be defined as a bimodule map $$\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar)
\buildrel g \over \rightarrow {\cal A}_\kbar. \eqno(4.3)$$ Because of the bilinearity and because of the relation (2.10) the coefficients $g^{ij}$ are here necessarily real numbers. In the commutative case they could have been chosen as arbitrary functions and using this freedom one can construct an arbitrary metric $g^\prime$ using the moving frame $\theta^\alpha$. This is a very important difference between the commutative and the noncommutative case. It is the reason why there is essentially only one metric associated to each differential calculus. The structure elements $C^i{}_{jk}$ are given by (2.21). It follows that they will be necessarily real numbers if the elements of the frame anticommute.
Let $\sigma$ be an ${\cal A}_\kbar$-bilinear map $$\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar)
\buildrel \sigma \over \longrightarrow
\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar).
\eqno(4.4)$$ We shall define a linear connection as a covariant derivative $D$ $$\Omega^1({\cal A}_\kbar) \buildrel D \over \rightarrow
\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar)
\eqno(4.5)$$ and a map $\sigma$ such that the following Leibniz rules [@DubMadMasMou95; @Mou95; @DubMadMasMou96] are satisfied: $$D (f \xi) = df \otimes \xi + f D\xi, \qquad
D (\xi f) = \sigma (\xi \otimes df) + (D\xi) f. \eqno(4.6)$$
The purpose of the $\sigma$ in the second equation is to place the differential in the first term to the left where it belongs while respecting the order of the various terms. A description of the bimodule structure of the module of 1-forms can be given in terms of a left-module structure with respect to a larger algebra and this leads to a natural splitting of the covariant derivative $D$ as the sum of two terms [@DubMadMasMou96]. In the commutative case it is easy to see that necessarily $\sigma$ is the flip of the previous section. As in the commutative case we introduce the elements $\omega^i{}_{jk}$ by the equation $$D \theta^i = - \omega^i{}_{jk} \theta^j \otimes \theta^k. \eqno(4.7)$$
There is no consensus at the moment concerning the necessity of two Leibniz rules. There are authors who maintain [@ChaFelFro93; @LanNguWal94; @Sit94; @ChaFroGra95; @FroGraRec97; @HecSch97] that it suffices to require that the covariant derivative satisfy a left (or right) Leibniz rule. There are others who propose [@CunQui95; @DabHajLanSin96; @Haj96] introducing both a left and right covariant derivative depending on which (left or right) Leibniz rule one chooses to enforce. There are interesting noncommutative cases [@Mad89b] where the two Leibniz rules are equivalent, We maintain [@Mou95; @MadMasMou95; @DubMadMasMou95; @GeoMadMasMou97; @DubMadMasMou96; @DimMad96] that without both rules it is not possible to correctly impose a reality condition on the linear connection [@KasMadTes97] nor will it be possible to construct nontrivial invariants to serve, for example as lagrangians. We shall discuss this second point in Section 7.
The torsion is defined exactly as in the commutative case (3.6). It is straightforward [@DubMadMasMou95; @Mou95; @DubMadMasMou96] to see that if the torsion is to be a bilinear map then the $\sigma$ must satisfy the condition $$\pi \circ (1 + \sigma) = 0. \eqno(4.8)$$ This condition is trivially satisfied in the commutative case. The most general such $\sigma$ is of the form [@MadMou96b] $$\sigma = (1 - \pi) \circ \tau - 1$$ where $\tau$ is an arbitrary ${\cal A}_\kbar$-bilinear map of $\Omega^1({\cal A}_\kbar) \otimes_{{\cal A}_\kbar} \Omega^1({\cal A}_\kbar)$ into itself. If $\tau = 2$ then $\sigma^2 = 1$. The condition that the connection be metric-compatible can be formulated exactly as in the commutative case: $$(1 \otimes g) D (\theta^i \otimes \theta^j) = 0.$$
If we define the complex numbers $S^{ij}{}_{kl}$ by the equation $$\sigma (\theta^i \otimes \theta^j))
=
S^{ij}{}_{kl} \theta^k \otimes \theta^l \eqno(4.9)$$ then the condition of metric compatibility becomes [@MadMou96b] $$\omega^i{}_{jk} + \omega_{kl}{}^m S^{il}{}_{jm} = 0.$$ This is analogous to the condition (3.8) of the commutative case but twisted by $\sigma$. We shall discuss the noncommutative generalization of curvature in Section 7.
For each differential calculus there is a linear connection defined in term of the form $\theta$ [@DubMadMasMou96] given by $$D\theta^i = - \theta \otimes \theta^i + \sigma(\theta^i \otimes \theta).
\eqno(4.10)$$ One verifies immediately that it satisfies the two Leibniz rules. In general it is neither torsion-free nor compatible with the metric.
Suppose that the algebra ${\cal A}_\kbar$ is such that $[q^\lambda, q^{\mu\nu}] = 0$ and that the matrix $q^{\mu\nu}$ has an inverse $q^{-1}_{\mu\nu}$. Define the differential structure by choosing $n=4$ and $$\lambda_\mu = {1\over i\kbar} q^{-1}_{\mu\nu} q^\nu.$$ It follows that the frame is given by $$\theta^\mu = dq^\mu.$$ The unique torsion-free linear connection compatible with the corresponding metric is the trivial connection given by [@MadMou96b] $D\theta^\mu = 0$. Apart from this example there are no non-trivial examples of linear connections on differential calculi over algebras which have anything to do with space-time. For this reason we must in the following section consider simpler models. It is hoped that there will eventually be a relation between noncommutative gravity and the quantum theory of gravity, whatever that may be. Speculations have been made [@MadMou95; @Bal97] along these lines.
Models
======
Lattice models
--------------
One of the advantages of noncommutative geometry is that it gives a prescription of how one can construct differential calculi over discrete structures, a construction which involves essentially using the universal calculus or some quotient of it over the algebra of functions on a finite set of points. This has been studied from a variety of points of view [@DimMul93; @DimMul94; @BimLizSpa94; @BalBimLanLizTeo96; @BimErcLanLizSpa96] and recently a book [@Lan97] has appeared to which we refer for further details. A comparison has yet to be made with the classical Regge calculus. One of the reasons for this is the difficulty in defining linear curvature within the context of noncommutative geometry. We shall return to this problem in Section 7.
$q$-models
----------
The quantum plane is the algebra ${\cal A}$ generated by ‘variables’ $x$ and $y$ which satisfy the relation $$xy = qyx, \eqno(5.1)$$ where $q$ is an arbitrary complex number. As usual it has over it many differential calculi $\Omega^*({\cal A})$. The commutation relations in $\Omega^1({\cal A})$ must be consistent with (5.1) but this condition is not enough to uniquely define the calculus. There is however a particularly interesting calculus known as the Wess-Zumino calculus [@PusWor89; @WesZum90] which is covariant under the co-action of a quantum group [@Wor87]. Since the elements of ${\cal A}$ do not in general commute the elements of $\Omega^1({\cal A})$ will not in general anti-commute. It has been shown [@DubMadMasMou95] that consistent with the Wess-Zumino calculus there is a 1-parameter family of linear connections which is without torsion but not compatible with a metric. The earliest example of a classical field theory on a noncommutative structure was furnished [@ConRie87; @Con88] by the electromagnetic field on a particular version of the quantum plane known as the noncommutative torus. In this case a careful analysis of the problem posed by the definition of the action was made. We shall return to this problem in Section 7.
One can extend the previous algebra by adding the inverses $x^{-1}$ and $y^{-1}$. For each integer $n$ and each set of $n$ linear-independent elements $\lambda_i$ of ${\cal A}$, there exists then a differential calculus $\Omega^*({\cal A})$ based on the derivations $e_i f = [\lambda_i, f]$ as in Section 2.2. Unless however $n = 2$ the frame has a singular limit as $q \rightarrow 1$. For $n=2$ and a special choice of the elements $\lambda_i$, given for $q^4 \neq 1$ by $$\lambda_1 = {1 \over q^4 - 1} x^{-2} y^2,\qquad
\lambda_2 = {1 \over q^4 - 1} x^{-2}, \eqno(5.2)$$ the resulting differential calculus is an extension of the Wess-Zumino calculus. The normalization has been chosen so that the structure elements $C^i{}_{jk}$ contain no factors $q$. The corresponding frame (2.9) is given by $$\theta^1 = - q^4 (q^2 + 1) x y^{-2} dx,\qquad
\theta^2 = - q^2 (q^2 + 1) x (x y^{-1} dy - dx). \eqno(5.3)$$ It satisfies the commutation relations $$(\theta^1)^2=0,\qquad (\theta^2)^2=0, \qquad
q^4 \theta^1\theta^2 + \theta^2\theta^1 = 0.$$ These relations determine the structure of the algebra $\Omega^*({\cal A})$. The corresponding $\theta$ cannot be considered as an element of the Wess-Zumino calculus since the $\theta^i$ are constructed using the inverses of $x$ and $y$. The $\lambda_i$ satisfy an equation of the form (2.19) with $F^i{}_{jk} = 0$, $K_{ij} = 0$ and $P^{lm}{}_{jk}$ defined in terms of the $R$-matrix of the associated quantum group.
Consider the quantum groups $GL_q(n)$ with generators $T^i_j$ and antipode $\kappa$. The left-invariant 1-forms $$\omega^i_j = \kappa(T^i_k) dT^k_j$$ generate [@Wor87] a bicovariant differential calculus $\Omega^*(GL_q(n))$. The exterior derivative is defined with the help of the right and left-invariant 1-form $\theta$. If $\sigma$ is any generalized permutation then the map $$\nabla^\sigma: \Omega^1(GL_q(n)) \rightarrow
\Omega^1(GL_q(n)) \otimes_{\cal A} \Omega^1(GL_q(n))$$ defined by $$\nabla^\sigma(\omega) = - \theta \otimes \omega +
\sigma(\omega \otimes \theta) \eqno(5.4)$$ defines a linear connection [@GeoMadMasMou97] associated to $\sigma$. This is to be compared with (4.10). It can be shown that for each $\sigma$, the only linear connection for generic $q$ is the one defined by (5.4). Further it can be shown that it has necessarily vanishing torsion. This is in contrast to the commutative case where there are an infinite number of linear connections not necessarily bicovariant nor torsion-free and where the generalized permutation is constrained to be the ordinary permutation. It is also in contrast to the cases with $q$ a root of unity. The arbitrariness in the deformed case lies merely in the generalized permutation for which it can be shown that there is at least a 2-parameter family, functions of $q$. The commutative limit is non-singular for a class of such functions which tend to the identity when $q \rightarrow 1$. More details are to be found in the article by Georgelin [*et al.*]{} [@GeoMadMasMou97]. See also the article by Heckenberger & Schmüdgen [@HecSch97].
For more details of $q$-deformed spaces and the groups which act on them we refer to the Cargèse lectures of Zumino [@ChuHoZum96]. For a discussion of the possible $q$ deformations of Minkowski space we refer to the literature [@FicLorWes95; @KehMeeZou95; @AscCas96; @Pod96; @PodWor96]. These $q$ deformations has been considered [@Maj97] as possible regulators of ultraviolet divergences in the sense of Snyder but as yet no linear connections have been constructed over them. The differential geometry of the $h$-deformed quantum plane [@Kup92; @Agh93] has been recently studied [@ChoMadPar97]; it is a noncommutative version of the Poincaré half-plane.
Finite models
-------------
Linear connections have been constructed [@MadMasMou95] over the finite differential calculi defined at the end of Section 2.2. The calculi based on derivations can be shown to have many linear connections but only one which is torsion-free and metric compatible. For $n=2$ the calculus (2.27) is the universal calculus over the algebra $M^+_2$ and it admits only the trivial connection. This model has been shown [@MadMouSit97] to be a singular contraction of the $n=2$ model based on derivations. The example with $n=3$ and differential calculus defined by (2.28) possesses a 1-parameter family of generalized permutations $\sigma$ (4.4) and for each of these there is a unique covariant derivative (4.5). All of these linear connections are torsion-free. For a special value of the parameter of $\sigma$ the connection is compatible with a metric.
The $SO_3$-invariant ‘lattice structure’ referred to in the Introduction can be constructed using the formalism of Section 2.2. For this we let $\lambda_i$ be a set of three antihermitian generators of the irreducible $n$-dimensional representation of the Lie algebra of $SU_2$. To discuss the commutative limit it is convenient to change the normalization of the generators $\lambda_i$. We introduce the parameter $\kbar$ with the dimensions of $(\hbox{length})^2$ and define ‘coordinates’ $x_i$ by $$x_i = i \kbar \lambda_i.$$ The $x_i$ satisfy therefore the commutation relations $$[x_i , x_j ] = i \kbar x_k \, C^k{}_{ij}. \eqno(5.5)$$ We choose the $\lambda_i$ so that $C_{ijk} = r^{-1} \epsilon_{ijk}$ where $r$ is a length parameter. These structure constants are in general independent from the structure elements defined by (2.21) but in the present case they are equal. Introduce the $SU_2$-Casimir metric $g_{ij}$. The matrix $g^{ij} x_i x_j$ is the Casimir operator. We choose $\kbar$ so that $g^{ij} x_i x_j = r^2$. We have then from (5.5) the relation $$4 r^4 = (n^2-1) \kbar^2. \eqno(5.6)$$ The commutative limit is the limit $\kbar \rightarrow 0$. Were we considering a noncommutative model of space-time then we would be tempted to identify $\kbar$ with the inverse of the square of the Planck mass, $\kbar = \mu^{-2}_P$, and consider space-time as fundamentally noncommutative in the presence of gravity.
The differential calculus has a basis [@Mad92a; @Mad92b]
$$\theta^i = - C^i{}_{jk} x^j dx^k - i\kbar r^{-2} \theta x^i. \eqno(5.7)$$ The 1-form $\theta$ can be written $$\theta = i \kbar^{-1} x_i \theta^i = r^2 \kbar^{-2} x_i dx^i. \eqno(5.8)$$ In the commutative limit $\theta$ diverges but $\kbar \theta \rightarrow
r^2 A$ where $A$ is the Dirac-monopole potential of unit magnetic charge. The commutative limit of the frame $\theta^i$ is a moving frame on a $U_1$-bundle over $S^2$. In this case it is not the frame bundle. A standard Kaluza-Klein reduction gives rise to the potential $A$ as well as the geometry of the sphere. We refer to this structure as the ‘fuzzy sphere’. Various field theories have been studied [@GroMad92; @GroPre95] on it and it has been generalized [@GroPre93; @GroKliPre96; @GroKliPre97a; @GroKliPre97c; @CarWat97] in several ways.
The linear connection on the fuzzy sphere is the same [@MadMasMou95] as that of the sphere. Recently over the same matrix algebra but with another differential calculus, obtained by using another solution to Equation (2.19) a less trivial connection has been constructed [@Mad97b] whose curvature is not invariant under the action of the rotation group.
Although we are primarily interested in the matrix version of surfaces as an model of an eventual noncommutative theory of gravity they have a certain interest in other, closely related, domains of physics. Without the differential calculus the fuzzy sphere is basically just an approximation to a classical spin $r$ by a quantum spin $r$ given by (5.6) with $\hbar$ in lieu of $\kbar$. It has been extended in various directions under various names and for various reasons [@Ber75; @deWHopNic88; @Hop89; @FaiFletZac89; @CahGutRaw90; @BorHopSchSch91]. In order to explain the finite entropy of a black hole it has been conjectured, for example by ’t Hooft [@'tH96], that the horizon has a structure of a fuzzy 2-sphere since the latter has a finite number of ‘points’ and yet has an $SO_3$-invariant geometry. The horizon of a black hole might be a unique situation in which one can actually ‘see’ the cellular structure of space. Matrices can also be used to give a finite ‘fuzzy’ description of the space complementary to a Dirichlet $p$-brane, a description which will allow one perhaps to include the reasonable property that points should be intrinsically ‘fuzzy’ at the Planck scale. This has much in common with the noncommutative version of Kaluza-Klein theory which we shall describe in the next section. Strings naturally play a special role here since they have a world surface of dimension two and an arbitrary matrix can always be written as a polynomial in two given matrices. We refer to the literature for a description of Dirichlet branes in general [@Pol96; @BonChu97; @Dij97] and within the context of $M$(atrix)-theory [@BanFisSheSus96; @GanRamTay96; @HoWu96; @Ban97]. The action of the matrix description of the complementary space is conjectured [@deWHopNic88] to be associated to the action in the infinite-momentum frame of a super-membrane of dimension $p$. Since quite generally the compactified factors of the surfaces normal to the $p$-branes are of the Planck scale we conclude [@MadSae97] that they have ill-defined topology and that a matrix description will include a sum over many topologies. Attempts have been made to endow them with a smooth differential structure [@Mad96; @GroKliPre97b]. Speculations have also been made [@AreVol97] concerning their relation with knots.
We have already mentioned that several models for the algebraic structure of space-time have been proposed [@Sny47a; @Mad89a; @DopFreRob95; @DubKerMad97] but there have been few discussions [@MadMou96b; @Mad97a] of associated differential structures and at present no interesting examples [@MadMou96b] of linear connections.
Kaluza-Klein theory
===================
Although the ultimate ambition of noncommutative geometry (in physics) is is to introduce a noncommutative version of space-time and to use it to describe quantum gravity, one can consider the much more modest task of introducing a modified version of Kaluza-Klein theory in which the hidden ‘internal’ space alone is described by a noncommutative geometry. In traditional Kaluza-Klein theory [@App87; @BaiLov87; @CoqJad88] the higher-order modes in the mode expansion of the field variables in the coordinates of the internal space are neglected, with the justification that they have all masses of the order of the Planck mass and would not be of interest in conventional physics. The alternative theory we here propose possesses [*ab initio*]{} only a finite number of modes; there are no extraneous modes to truncate. We would like to suggest also that the noncommutative version of Kaluza-Klein theory is more natural than the traditional one in that a hand-waving argument [@MadMou95] can be given which allows one to think of the extra algebraic structure as being due to quantum fluctuations of the light-cone in ordinary 4-dimensional space-time. We already suggested in the Introduction that this might be the origin of the noncommutative structure of space-time itself.
We suppose then that the algebra ${\cal A}_\kbar$ has the structure of a tensor product $${\cal A}_\kbar = {\cal C}(V) \otimes M_n \eqno(6.1)$$ of an algebra of smooth functions on space-time $V$ and a matrix algebra $M_n$. We introduce a differential calculus $\Omega^*({\cal A}_\kbar)$ over ${\cal A}_\kbar$ which is a tensor product of the de Rham differential calculus $\Omega^*(V)$ over $V$ and a differential calculus $\Omega^*(M_n)$ over the matrix factor. If we define $$\Omega^1_h = \Omega^1(V) \otimes M_n, \qquad
\Omega^1_v = {\cal C}(V) \otimes \Omega^1(M_n),$$ we can write $\Omega^1({\cal A})$ as a direct sum: $$\Omega^1({\cal A}) = \Omega^1_h \oplus \Omega^1_v. \eqno(6.2)$$ of two terms, the horizontal and vertical parts, using notation from traditional Kaluza-Klein theory. The exterior derivative $df$ of an element $f$ of ${\cal A}_\kbar$ has a similar decomposition $$df = d_hf + d_vf. \eqno(6.3)$$ We can choose $\theta^i = (\theta^\alpha, \theta^a)$ as a basis for $\Omega^1({\cal A})$ where $\theta^\alpha$ is a moving frame on $V$, supposed for convenience to be parallelizable and $\theta^a$ is a frame of the sort introduced in Section 2.2
Using the above differential structure one can study electromagnetism as well as gravity. We consider first the former. Most of the efforts to introduce noncommutative geometry into particle physics have been in fact directed towards trying to find an appropriate noncommutative generalization of an old idea [@ForMan80; @Man79; @ChaMan80; @Fai79] to try to unify Yang-Mills and Higgs fields by studying electromagnetism in higher dimensions. We write the electromagnetic field strength $F$ as $$F = {1 \over 2} F_{ij} \theta^i \theta^j.$$ Then the electromagnetic action on ${\cal A}_\kbar$ takes the form $$S = {1 \over 4} \int_V \tr (F_{ij} F^{ij}). \eqno(6.4)$$ The trace over the matrix factor is the equivalent of the integral over space-time.
Let $\omega$ be the electromagnetic potential, which we decompose $\omega = \omega_h + \omega_v$ in typical Kaluza-Klein fashion as the sum of a horizontal component and a vertical component. The gauge transformations are the unitary elements ${\cal U}_n$ of ${\cal A}_\kbar$. We notice that the form $\theta$ which we introduced in Section 2.2 is gauge-invariant [@DubKerMad89]. It is natural then to decompose $\omega_v$ as a sum $\omega_v = \theta + \phi$ where $\phi$ is the difference between two gauge potentials and so transforms covariantly under a gauge transformation. After a short calculation one arrives [@DubKerMad89] at a unification of Yang-Mills and Higgs fields with the potential of the Higgs particle given by the curvature of the covariant derivative in the algebraic ‘directions’. One calculates how the particle and mass spectra vary as one varies the extra noncommutative algebra and the associated differential calculi. Much ingenuity has gone into these calculations which often involve very sophisticated mathematics but which ultimately reduce to simple manipulations with matrices.
The simplest and most intuitive models are those which use differential calculi based on derivations [@DubKerMad89; @BalGurWal91], More general calculi constructed directly from a Dirac operator without the use of derivations, are less rigid and can be chosen so that the resulting action coincides with that of the Standard Model. The first example [@ConLot90; @Coq89] was based on the differential calculus defined by Equation (2.23) for $n=2$. The extension [@ConLot92] to $n=3$ and higher [@CoqHauSch95; @LizManMieSpa1996; @PriSch97] soon followed. There exist several reviews [@Kas93; @VarGra93] of these models. A comparison of the two approaches has been given [@MadMouSit97] in a simple case. The weak interactions violate parity and this fact must be included in a realistic model. No derivation-based model with explicit parity violation has been developed; the models mentioned above rely implicitly on spontaneous parity-breaking mechanisms like the ‘see-saw’ mechanism. The double-sheeted structure of the Dirac-based models lends itself more readily to the introduction of explicit parity violation.
Very few of the results of the preceding subsection can be developed within the context of the theory of gravity and none of them have as yet any significance for particle physics. We refer simply to the original literature. Gravity was included [@Mad89b] in the first noncommutative version of Kaluza-Klein theory and developed [@Mad90; @MadMou93] in subsequent articles. Recent reviews [@Mad95; @MadMou96a; @FroGraRec97] are to be found. We have already mentioned in Section 4 that there is no consensus concerning the definition of a linear connection and we mention in the next section the problems concerning the definition of curvature and the choice of action functional.
Open Problems
=============
The fundamental open problem of the noncommutative theory of gravity concerns of course the relation it might have to a future quantum theory of gravity either directly or via the theory of strings and membranes as mentioned at the end of Section 6. But there are more immediate technical problems which have not received a satisfactory answer. The most important ones concern the definition of the curvature. It is not certain that the ordinary definition of curvature taken directly from differential geometry is the quantity which is most useful in the noncommutative theory. The main interest of curvature in the case of a smooth manifold definition of space-time is the fact that it is local. Riemann curvature can be defined as a map $$\Omega^1({\cal C}(V)) {\buildrel R \over \longrightarrow}
\Omega^2({\cal C}(V)) \otimes_{{\cal C}(V)} \Omega^1({\cal C}(V)).$$ If $\xi \in \Omega^1({\cal C}(V))$ then $R(\xi)$ at a given point it depends only on the value of $\xi$ at that point. This can be expressed as a bilinearity condition; the above map is a ${\cal C}(V)$-bimodule map. If $f \in {\cal C}(V)$ then $$f R(\xi) = R(f\xi), \qquad R(\xi f) = R(\xi) f. \eqno(7.1)$$ In the noncommutative case bilinearity is the natural (and only possible) expression of locality. It has not yet been possible to enforce it in a satisfactory manner [@DubMadMasMou96].
In the noncommutative case considered in Section 2.2, where the module of 1-forms is free one, can formally identify the curvature as usual with the operator $D^2$ and set $$D^2 \theta^i = - {1 \over 2} R^i{}_{jkl}
\theta^k \theta^l \otimes \theta^j. \eqno(7.2)$$ Since $D^2$ is not necessarily right-linear as an operator we cannot conclude that the coefficients $R^a{}_{bcd}$ necessarily lie in the center of the algebra.
We define the Ricci map $$\Omega^1 \buildrel \mbox{Ric} \over \longrightarrow \Omega^1$$ by $\mbox{Ric} = - (1\otimes g) \circ D^2$. In terms of the frame we have $$\mbox{Ric} \, (\theta^i) =
{1 \over 2} R^i{}_{jkl} \theta^k g(\theta^l \otimes \theta^j).$$ It is given by $$\mbox{Ric} \, (\theta^i) = R^i{}_j \theta^k. \eqno(7.3)$$ Formally then one can write vacuum field equations as $$\mbox{Ric} = 0.$$
We are unable at the moment to propose a satisfactory definition of an action which would yield as field equations the vanishing of the Ricci map. and indeed we are not in a position to argue that there is even a valid action principle. A discussion of this point has been made by Connes and coworkers in a series of articles [@Con88; @KalWal95; @AckTol96; @Con96; @FroGraRec97] based on an idea of Sakarov [@Sak75] applied to a Kaluza-Klein theory similar to the one described in the previous section. The definition which these authors propose is valid only on the noncommutative generalizations of compact spaces with euclidean-signature metrics. Cyclic homology groups have been proposed [@Con86] as the appropriate generalization to noncommutative geometry of topological invariants; the appropriate definition of other, non-topological, invariants in not clear.
Mathematics
===========
At a more sophisticated level one would have to add a topology to ${\cal
A}_\kbar$ and consider a closed algebra. Since we have identified the generators as hermitian operators on a Hilbert space, the most obvious structure would be that of a von Neumann algebra. We refer to Connes [@Con94] for a description of these algebras within the context of noncommutative geometry. A large part of the interest of mathematicians in noncommutative geometry has been concerned with the generalization of topological invariants [@Con86; @CunQui95; @Mos97] to the noncommutative case. It was indeed this which lead Connes to introduce and develop cyclic cohomology. Another interest has been the generalization of the idea introduced by Atiyah of an homology theory dual to the $K$-theory of vector bundles. The fundamental object here is a $K$-cycle or spectral triple, a set $({\cal A}, D, {\cal H})$ consisting of an associative algebra ${\cal A}$ with a representation on a Hilbert space ${\cal H}$ and a ‘Dirac operator’ $D$ to define a differential calculus. All the examples of differential calculi which we have considered here can be formulated as spectral triplets; the simplest was given in Section 2.2 with ${\cal A} = M_2^+$, $D = \eta$ and ${\cal H} = {\mathbb C}^2$. Connes [@ConRie87; @Con88] has also developed and extended the notion of a Dixmier trace on certain types of algebras as a possible generalization of the notion of an integral. The mathematics of quantum groups and quantum spaces has also been considerably studied. We refer, for example, to the book by Majid [@Maj95].
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of this research was done while the author was visiting the Erwin Schrödinger Institute, Vienna. The author would like to thank W. Thirring for his hospitality. He would also like to thank G. Goldin and D. Lambert for interesting conversations.
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[^1]: Laboratoire associé au CNRS,
|
---
abstract: |
Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$\left\{
\begin{array}{l}
\Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N},\\
u \in H^{2}(\mathbb{R}^{N}),
\end{array}
\right.$$ where $N \geq 1$, $\Delta^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.
[**Mathematics Subject Classifications (2010):**]{} 35J20, 35J65
[**Keywords:**]{} Biharmonic operator, Multi-bump solution, Variational methods.
author:
- 'Claudianor O. Alves[^1] , Alânnio B. Nóbrega [^2]'
- |
[Universidade Federal de Campina Grande]{}\
[Unidade Acadêmica de Matemática]{}\
[CEP: 58429-900, Campina Grande - Pb, Brazil]{}\
title: '**Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator**'
---
Introduction
============
In this paper, we are concerned with the existence of multi-bump solutions for the following class of problems $$\label{1}
\left\{ \begin{array}{cc}
\Delta^2 u +(\lambda V(x)+1)u & = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, \\
u \in H^2(\mathbb{R}^N);\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&
\end{array}
\right.$$ where $N \geq 1$, $\Delta^{2}$ denotes the biharmonic operator, $\lambda >0$ is a positive parameter and $f:\mathbb{R} \rightarrow \mathbb{R}$ is a $C^1$ function verifying the following hypotheses:
$(f_1)$
: $f(0)=f'(0)=0.$
$(f_2)$
: $\liminf_{t \rightarrow +\infty} \frac{\left|f'(t)\right|}{\left|t\right|^{q-2}}<+\infty,
$ for $q \in (2,2_*)$ where $$2_*=
\left\{ \begin{array}{c}
\frac{2N}{N-4},\quad N\geq 5 \\
+\infty, \quad 1\leq N \leq 4 .
\end{array}
\right.$$
$(f_3)$
: There is $\theta >2$ such that $$0<\theta F(t) \leq f(t) t,\quad \mbox{for} \quad t\neq 0.$$
$(f_4)$
: $\frac{f(t)}{\left|t\right|}$ is an increasing function for $t\neq 0$.
Related to the potential $V:\mathbb{R}^N \rightarrow
\mathbb{R}$, we assume the following assumptions :
$(V_1)$
: $V(x)\geq 0, \ \forall \ x\in \mathbb{R}^N$;
$(V_2)$
: $\Omega= int V^{-1}(\{0\})$ is a non-empty bounded open set with smooth boundary $\partial \Omega$. Moreover, $\Omega$ has $k$ connected components, more precisely,
- $\Omega=\bigcup_{j=1}^{k}\Omega_j;$
- $dist(\Omega_i,\Omega_j)>0,\ i\neq j.$
$(V_3)$
: There is $M_0>0$ such that $ |\{ x\in \mathbb{R}^N ;\, V(x)\leq
M_0\}|<+\infty.$
Hereafter, if $A \subset \mathbb{R}^{N}$ is a mensurable set, $|A|$ denotes its Lebesgue’s measure.
In the last years, problems involving the biharmonic operator have been studied by many researchers, in part because this operator helps to describe the mechanical vibrations of an elastic plate, which among other things describes the traveling waves in a suspension bridge, see [@BDF; @FG; @G; @GK; @LM]. On the other hand, the biharmonic operator draws attention by the difficulties encountered when trying to adapt known results for the Laplacian, for example, we cannot always rely on a maximum principle, and also, if $u$ belongs $ H ^ 2 (A)$, we cannot claim that $u ^ \pm$ belong to $ H ^ 2 (A)$. Recently, many authors have studied various problems with the biharmonic operator, see for example, [@BG; @JQ; @P1; @P2; @YT; @ZWZ]. However, related to the existence of multi-bump solutions for an equation as (\[1\]), as far as we know, there is no results in this direction.
In , Ding and Tanaka have considered the problem $$\label{2}
\left\{ \begin{array}{l}
-\Delta u +(\lambda V(x)+Z(x))u = u^p, \quad \mbox{in} \quad \mathbb{R}^{N}, \\
u>0,\ \mbox{in}\ \mathbb{R}^N,
\end{array}
\right.$$ with $p \in \left(1,\frac{N+2}{N-2}\right)$ and $ N \geq 3$. In that paper, it was showed that the problem (\[2\]) has at least $2^k-1$ solutions for $\lambda$ large enough, which are called multi-bump solutions. These solutions have the following characteristics :\
For each non-empty subset $\Gamma \subset \{1,2,\cdots,k\}$ and $\varepsilon>0$ fixed, there is a $\lambda^*>0$ such that, (\[2\]) possesses a solution $u_{\lambda}$, for $\lambda \geq \lambda^*=\lambda^*(\varepsilon)$, satisfying: $$\left| \int_{\Omega_j}\left[\left|\nabla u_{\lambda}\right|^2+(\lambda V(x)+Z(x))u_{\lambda}^2\right]-\left( \frac{1}{2}-\frac{1}{p+1}\right)^{-1}c_j\right|< \varepsilon,\ \forall j \in \Gamma$$ and $$\int_{\mathbb{R}^N\setminus \Omega_{\Gamma}}\left[ \left| \nabla u_{\lambda}\right|^2+ u_{\lambda}^2\right]dx< \varepsilon,$$ where $\Omega_{\Gamma}= \bigcup_{j \in \Gamma}\Omega_j$ and $c_j$ is the minimax level of the energy functional related to the problem $$\label{3}
\left\{ \begin{array}{c}
- \Delta u +Z(x)u = u^p, \,\, \mbox{in} \,\, \Omega_j, \\
u>0, \,\, \mbox{in} \,\, \Omega_j, \\
u=0,\ \mbox{on}\ \partial\Omega_j .
\end{array}
\right.$$
We also highlight the papers due to Alves, de Morais Filho and Souto in [@A-M-S], Alves and Souto in [@AS], where the authors have considered a problem of type (\[2\]), assuming that $f$ has a critical growth for the case $N \ge 3$ and exponential critical growth when $N = 2$, respectively. We emphasize that in the above mentioned papers, the assumption $(V_3)$ was not assumed.
In all the above mentioned papers, it was essential the method developed in , which consists in modifying the nonlinearity to obtain a new problem, whose energy functional associated satisfies the $(PS)$ condition. After that, making some estimates, it is possible to prove that the solutions obtained for the modified problem are also solutions for the original problem when $\lambda$ is large enough. However, in our opinion, it is not clear that the method developed in can be used for our problem, because we are working with biharmonic operator. To overcome this difficulty, we have developed a new approach to get multi-bump avoiding the penalization on the nonlinearity. Our inspiration comes from an approach used in Bartsch & Wang . Here, we modify the sets where we will apply the Deformation Lemma, see Sections 4 and 5 for more details.
Our main result is the following
\[T1\] Suppose that $(f_1)-(f_4)$ and $(V_1)-(V_3)$ hold. Then, for each non-empty subset $\Gamma \subset \{1,\cdots,k\}$and $\varepsilon>0$ fixed, there is a $\lambda^*=\lambda^*(\varepsilon)>0$ such that, (\[1\]) possesses a solution $u_{\lambda}$, for $\lambda \geq \lambda^*$, satisfying: $$\left| \frac{1}{2}\int_{\mathbb{R}^N}\left[\left|\Delta u_{\lambda}\right|^2+(\lambda V(x)+1)\left|u_{\lambda}\right|^2\right]dx-\int_{\mathbb{R}^N}F(u_{\lambda})dx-c_j\right|< \varepsilon, \forall j \in \Gamma$$ and $$\int_{\mathbb{R}^N\setminus \Omega_{\Gamma}}\left[\left|\Delta u_{\lambda}\right|^2+\left|u_{\lambda}\right|^2\right]dx < \varepsilon,$$ where $\Omega_{\Gamma}=\cup_{j\in \Gamma}\Omega_j$ and $c_j$ is the minimax level of the energy functional related to the problem: $$\label{4}
\left\{ \begin{array}{c}
\Delta^2 u +u = f(u), \quad \mbox{in} \quad \Omega_j \\
u=\frac{\partial u}{\partial \eta}=0,\quad \mbox{on} \quad \partial\Omega_j .
\end{array}
\right.$$
The $(PS)_c$ Condition
======================
In this section, we fix some notations and show some properties of the energy functional associated with (\[1\]), for example, we will show that for each $c\geq 0$, the functional $I_{\lambda}$ satisfies the $(PS)_c$ condition, since that $\lambda$ is suitably chosen.
To begin with, we recall that the energy functional $I_{\lambda}:E_{\lambda}\rightarrow \mathbb{R}$ associated with the problem $(\ref{1})$ is given by $$I_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^N}\left[\left|\Delta u\right|^2+(\lambda V(x)+1)\left|u\right|^2\right]dx-\int_{\mathbb{R}^N}F(u)dx,$$ where $$E_{\lambda}=\left\{ u\in H^2(\mathbb{R}^N);\ \int_{\mathbb{R}^N}V(x)\left|u\right|^2 dx <+\infty\right\}.$$ The subspace $E_{\lambda}$ endowed with the inner product $$(u,v)_{\lambda}= \int_{\mathbb{R}^N} \left[\Delta u\Delta v+(\lambda V(x)+1)uv\right]dx,$$ is a Hilbert space and the norm generated by this inner product will be denoted by $\|\cdot\|_{\lambda}$.
Hereafter, if $\Theta \subset \mathbb{R}^{N}$ is a mensurable set, we denote by $E_\lambda(\Theta)$ the space $H^{2}(\Theta)$ endowed with the the inner product $$(u,v)_{\lambda,\Theta}= \int_{\Theta} \left[\Delta u\Delta v+(\lambda V(x)+1)uv\right]dx.$$ The norm associated with this inner product will be denoted by $\|\cdot\|_{\lambda,\Theta}$.
Next, we will show some technical lemmas, whose proofs follow with the same type of arguments found in . However for the readers’ convenience we will write their proofs.
\[l1\] Let $\{u_n\} \subset E_{\lambda}$ be a $(PS)_c$ sequence for $I_{\lambda},$ then $\{u_n\}$ is bounded in $E_{\lambda}$. Furthermore, $c\geq 0.$
Since $\{u_n\}$ is a $(PS)_c$ sequence, we have that $$I_{\lambda}(u_n)\rightarrow c\ \mbox{and}\ I'_{\lambda}(u_n) \rightarrow 0.$$ Thereby, for $n$ large enough, $$\label{5}
I_{\lambda}(u_n)-\frac{1}{\theta}I'_{\lambda}(u_n)u_n \leq c+1+\left|\left|u_n\right|\right|_{\lambda}.$$ On the other hand, $$I_{\lambda}(u_n)-\frac{1}{\theta}I'_{\lambda}(u_n)u_n=\left( \frac{1}{2}-\frac{1}{\theta}\right)\|u_n\|^2_{\lambda}+\int_{\mathbb{R}^N}\left[ \frac{1}{\theta}f(u_n)u_n-F(u_n)\right]\,dx.$$ Then, by $(f_3)$, $$\label{6}I_{\lambda}(u_n)-\frac{1}{\theta}I'_{\lambda}(u_n)u_n\geq \left( \frac{1}{2}-\frac{1}{\theta}\right)\|u_n\|^2_{\lambda}.$$ Gathering $ (\ref{5}) $ and $ (\ref{6}) $, we get $$\left( \frac{1}{2}-\frac{1}{\theta}\right)\|u_n\|^2 _{\lambda} \leq c+1+\|u_n\|_{\lambda},$$ showing that $\{u_n\}$ is bounded. Using the boundedness of $\{u_n\}$ and (\[5\]), we see that $$\label{7}0\leq\left(
\frac{1}{2}-\frac{1}{\theta}\right)\|u_n\|^2 _{\lambda} \leq
c+o_n(1).$$ Taking the limit $n \rightarrow +\infty$, it follows that $c \geq 0.$
\[c1\] Let $\{u_n\} \subset E_{\lambda}$ be a $(PS)_0$ sequence for $I_{\lambda}.$ Then, $u_n\rightarrow 0$ in $E_\lambda$.
This corollary is an immediate consequence of the arguments used in the proof of Lemma $\ref{l1}$.
\[l2\] Let $\{ u_n\}$ be a $(PS)_c$ sequence for $I_{\lambda}$ with $c \geq 0$. If $u_n \rightharpoonup u$ in $E_{\lambda},$ then $$\begin{aligned}
I_{\lambda}(v_n)-I_{\lambda}(u_n)+I_{\lambda}(u) &=& o_n(1) \nonumber\\
I'_{\lambda}(v_n)-I'_{\lambda}(u_n)+I' _{\lambda}(u) &= & o_n(1)\nonumber,
\end{aligned}$$ where $v_n=u_n-u.$ Hence, $\{v_n\}$ is a $(PS)_{c-I_{\lambda}(u)}$ sequence.
As the first step, note that $$\begin{aligned}
I_{\lambda}(v_n)-I_{\lambda}(u_n)+I_{\lambda}(u) &=\frac{1}{2}\left( \|v_n\|^2_{\lambda}-\|u_n\|^2_{\lambda}+\|u\|^2_{\lambda}\right)&\\
&\ \ -\int_{\mathbb{R}^N}\left( F(v_n)-F(u_n)+F(u)\right)dx &\\
&=o_n(1)-\int_{B_R(0)}\left( F(v_n)-F(u_n)+F(u)\right)dx&\\
&\ \ -\int_{\mathbb{R}^N \setminus B_R(0)}\left( F(v_n)-F(u_n)+F(u)\right)dx ,&
\end{aligned}$$ where $ R> $ 0 will be fixed later on. Once $u_n \rightharpoonup u$ in $E_{\lambda}$, we have
- $u_n \rightarrow u\ \mbox{in}\ L^p(B_R(0)) \ \mbox{for}\ 1 \leq p <2_*;$
- $u_n(x) \rightarrow u(x)\ a.e. \ \mbox{in}\ \mathbb{R}^N.$
Moreover, there are $h_1 \in L^2(B_R(0))$ and $ h_2 \in L^q(B_R(0))$ such that $$\left|u_n(x)\right|\leq h_1(x), h_2(x) \quad \mbox{a.e. in } \quad \mathbb{R}^{N}.$$ By Lebesgue’s Theorem, $$\label{8}
\int_{B_R(0)} \left| F(v_n)-F(u_n)+F(u)\right|\,dx \rightarrow 0.$$
On the other hand, from $(f_1)-(f_2)$, given $\epsilon >0$, there is $C_\epsilon>0$ satisfying $$\left|F\left(v_n\right)-F\left(u_n\right)\right| \leq \epsilon \left(\left|u_n\right|+\left|u\right|\right)\left|u\right|+C_{\epsilon}\left(\left|u_n\right|+\left|u\right|\right)^{q-1}\left|u\right|.$$ The above estimate combined with the boundedness of $\{u_n\}$ and Sobolev embeddings gives $$\begin{aligned}
\int_{\mathbb{R}^N \setminus B_R(0)}\left|F \left(v_n\right)- F\left(u_n\right) \right|dx \leq &\,\,\,\, \epsilon C_1 \left( \|u\|_{L^2(\mathbb{R}^N \setminus
B_R(0))}+\|u\|_{L^2(\mathbb{R}^N \setminus
B_R(0))}^2\right)&\\
& C_\epsilon\left(\|u\|_{L^q(\mathbb{R}^N \setminus
B_R(0))}+\|u\|_{L^q(\mathbb{R}^N
\setminus B_R(0))}^q \right).\\
\end{aligned}$$ The above estimate permits to fix $R>0$ large enough verifying $$\int_{\mathbb{R}^N
\setminus B_R(0)}\left|F \left(v_n\right)-
F\left(u_n\right) \right|dx \leq \epsilon.$$ By $(f_2)$, $$\int_{\mathbb{R}^N \setminus B_R(0)}\left|
F\left(u\right) \right|dx \leq \epsilon
\left|\left|u\right|\right|_{L^2(\mathbb{R}^N \setminus
B_R(0))}^2+C_{\epsilon}\|u\|_{L^q(\mathbb{R}^N \setminus
B_R(0))}^q.$$ Then, increasing $R$ if necessary, we can assume that $$\int_{\mathbb{R}^N \setminus B_R(0)}\left| F\left(u\right)
\right|dx \leq \epsilon.$$ Hence, $$\int_{\mathbb{R}^N
\setminus B_R(0)}\left|F \left(v_n\right)-
F\left(u_n\right) +F\left(u\right)\right|dx \leq \epsilon, \quad \forall n \in \mathbb{N}.$$ By arbitrariness of $ \epsilon, $ it follows that $$\label{9}
\limsup_{n \rightarrow +\infty} \int_{\mathbb{R}^N
\setminus B_R(0)}\left|F \left(v_n\right)-
F\left(u_n\right) +F\left(u\right)\right|dx=0.$$ From (\[8\]) and (\[9\]), we get the first limit. The second one follows by exploring the same type of arguments and the growth of $f'$.
\[l3\] Let $\{u_n\}$ be a $(PS)_c$ sequence for $I_{\lambda}$. Then $c=0$, or there exists $c_*>0$ independent of $\lambda,$ such that $c \geq c_*$ for all $\lambda >0.$
By Lemma $\ref{l1}$, $c \geq 0$. Supposing $c>0$, we get the inequality $$\begin{aligned}
c+o_n(1)\left|\left|u_n\right|\right|_{\lambda}& \geq I_{\lambda}(u_n)-\frac{1}{\theta}I'_{\lambda}(u_n)u_n
\geq \left( \frac{\theta-2}{2\theta}\right)\left|\left|u_n\right|\right|^{2}_{\lambda},&
\end{aligned}$$ which leads to $$\label{10}
\limsup_{n \rightarrow + \infty} \left|\left|u_n\right|\right|_{\lambda}^2 \leq \frac{2c\theta}{\theta-2}.$$ On the other hand, the growth of $f$ together with the Sobolev embedding gives $$I'_{\lambda}(u)u \geq \frac{1}{2}\left|\left|u \right|\right|_{\lambda}^2-K\left|\left|u \right|\right|_{\lambda}^q,$$ for some positive constant $K$. Thus, there exists $\delta>0$ such that $$\label{11}
I'_{\lambda}(u)u \geq
\frac{1}{4}\left|\left|u\right|\right|_{\lambda}^2,\ \mbox{for}\
\left|\left|u\right|\right|_{\lambda} < \delta.$$ Setting $c_*= \delta^2\frac{\theta -2}{2\theta}$ and $c<c_*$, $(\ref{10})$ yields $$\limsup_{n \rightarrow + \infty} \left|\left|u_n\right|\right|_{\lambda}^2 < \delta^2,$$ implying that for $n $ large enough, $$\label{12}
\left|\left|u_n\right|\right|_{\lambda}\leq \delta.$$ Hence, (\[11\]) and (\[12\]) combine to give $$I'_{\lambda}(u_n)u_n \geq
\frac{1}{4}\left|\left|u_n\right|\right|_{\lambda}^2,$$ leading to $$\left|\left|u_n\right|\right|_{\lambda}^2 \rightarrow 0.$$ Thus $$I_{\lambda}(u_n) \rightarrow I_{\lambda}(0)=0,$$ which contradicts the hypothesis that $ \{u_n \} $ is a $ (PS) _c $ sequence with $c>0$. Therefore, $c \geq c_*.$
\[l4\] Let $\{u_n\}$ be a $(PS)_c$ sequence for $I_{\lambda}.$ Then, there exists $\delta_0 >0$ independent of $\lambda,$ such that $$\liminf_{n
\rightarrow + \infty}
\left|\left|u_n\right|\right|_{L^{q}(\mathbb{R}^N)}^q \geq
\delta_0c.$$
By $(f_1)$ and $(f_2),$ given $\epsilon >0$, there is $C_\epsilon>0$ such that $$\frac{1}{2}f(t)t-F(t) \leq \epsilon\left|t\right|^2+C_{\epsilon}\left|t\right|^{q},\ \forall t \in \mathbb{R}.$$ Then, $$\label{13}
c \leq \liminf_{n \rightarrow
+\infty}\left(\epsilon\left|\left|u_n\right|\right|_{\lambda}^2+C_{\epsilon}\left|\left|u_n\right|\right|_{L^q(\mathbb{R}^N)}^{q}\right).$$ On the other hand, by $(f_3)$, $$\label{14}
I_{\lambda}(u_n)-\frac{1}{\theta}I'_{\lambda}(u_n)u_n\geq
\left(\frac{1}{2}-\frac{1}{\theta}\right)\left|\left|u_n\right|\right|_{\lambda}^2.$$ Combining $(\ref{13})$ with $(\ref{14})$, we get $$c \leq \frac{2
\epsilon c \theta}{\theta-2}+C_{\epsilon}\liminf_{n \rightarrow
+\infty}\left|\left|u_n\right|\right|_{L^q(\mathbb{R}^N)}^{q}.$$ Thereby, for $\epsilon$ small enough, $$\liminf_{n \rightarrow
+\infty}\left|\left|u_n\right|\right|_{L^q(\mathbb{R}^N)}^{q} \geq
\frac{c}{C_\epsilon}\left(1-\frac{2\epsilon
\theta}{\theta-2}\right)>0.$$ Now, the lemma follows fixing $$\delta_0=\frac{1}{C_\epsilon}\left(1-\frac{2\epsilon \theta}{\theta-2}\right).$$
\[l5\] Let $c_1>0$ be a constant independent of $\lambda$. Given $\epsilon >0$, there exist $\Lambda=\Lambda(\epsilon)>0$ and $R=R(\epsilon,c_1)>0$ such that, if $\{u_n\}$ is a $(PS)_c$ sequence for $I_{\lambda}$ with $c \in [0,c_1],$ then $$\limsup_{n \rightarrow +\infty}\left|\left|u_n\right|\right|_{L^q(\mathbb{R}^N \setminus B_R(0))}^{q} \leq \epsilon,\quad \forall \lambda \geq \Lambda.$$
For each $R >0$, fix $$A(R)=\{x \in \mathbb{R}^N/ \left|x\right|> R\ \mbox{and}\ V(x) \geq M_0\}$$ and $$B(R)=\{x \in \mathbb{R}^N/ \left|x\right|> R\ \mbox{and}\ V(x) < M_0\}.$$ Then, $$\begin{aligned}
\label{15}
\int_{A(R)}|u_n|^2dx &\leq \frac{1}{(\lambda M_0 +1)}\int_{\mathbb{R}^N}(\lambda V(x) +1)|u_n|^2dx&\nonumber\\
&\leq \frac{1}{(\lambda M_0 +1)}\left|\left|u_n\right|\right|_{\lambda}^2&\\
&\leq \frac{1}{(\lambda M_0 +1)}\left[ \left( \frac{1}{2}-\frac{1}{\theta}\right)^{-1}c+o_n(1)\right]\nonumber&\\
&\leq \frac{1}{(\lambda M_0 +1)}\left[ \left( \frac{1}{2}-\frac{1}{\theta}\right)^{-1}c_1+o_n(1)\right].\nonumber&
\end{aligned}$$ As $ c_1 $ is independent of $ \lambda, $ by $(\ref{15})$ there is $\Lambda>0$ such that $$\label{16}
\limsup_{n \to +\infty}\int_{A(R)}|u_n|^2dx < \frac{\epsilon}{2}, \quad \forall \lambda \geq \Lambda .$$ On the other hand, using the Hölder inequality for $p\in \left[1,2_*/2\right]$ , we obtain $$\begin{aligned}
\label{17}
\int_{B(R)}|u_n|^2dx &\leq \left(\int_{B(R)}\left|u_n\right|^{2p}dx\right)^{\frac{1}{p}}\left| B(R)\right|^{\frac{1}{p'}}\nonumber . &
\end{aligned}$$ Now, using the continuous embedding $E_{\lambda}\hookrightarrow L^{2p}(\Omega),$ it follows that $$\begin{aligned}
\int_{B(R)}|u_n|^2dx&\leq \beta \left|\left|u_n\right|\right|_{\lambda}^2\left| B(R)\right|^{\frac{1}{p'}},&
\end{aligned}$$ where $\beta$ is a positive constant. From (\[14\]), $$\begin{aligned}
\int_{B(R)}|u_n|^2dx
&\leq \beta c_1\left( \frac{1}{2}-\frac{1}{\theta}\right)^{-1}\left|
B(R)\right|^{\frac{1}{p'}}+o_n(1).\nonumber&
\end{aligned}$$ Now, by $(V_3)$, we know that $$\left|B(R)\right|\rightarrow 0 \quad \mbox{when} \quad R \rightarrow +\infty.$$ Therefore, we can choose $R$ large enough, such that $$\label{17}
\limsup_{n \to +\infty}\int_{B(R)}|u_n|^2dx < \frac{\epsilon}{2}.$$ Gathering (\[16\]) and (\[17\]), we find $$\limsup_{n \to +\infty}\int_{\mathbb{R}^N \setminus B_R(0)}|u_n|^2dx < \epsilon.$$ The last inequality combined with interpolation leads to $$\limsup_{n \to +\infty}\int_{\mathbb{R}^N \setminus B_R(0)}\left|u_n\right|^q dx < \epsilon, \lambda \geq \Lambda$$ increasing $R$ and $\Lambda$ if necessary.
\[p1\] Given $c_1>0,$ there exists $\Lambda=\Lambda(c_1)$ such that $I_{\lambda}$ verifies the $(PS)_c$ condition for all $c \in [0,c_1]$ and $\lambda
\geq \Lambda.$
Let $\{u_n\}$ be a $(PS)_c$ sequence. By Lemma \[l1\], $\{u_n\}$ is bounded and consequently, passing to a subsequence if necessary, $$\left\{ \begin{array}{c}
u_n\rightharpoonup u\ \ \mbox{in}\ E_{\lambda};\\
u_n(x)\rightarrow u(x) a.e. \ \mbox{in}\ \mathbb{R}^N;\\
u_n\rightarrow u \ \mbox{in}\ L^{s}_{loc}(\mathbb{R}^N) \quad \mbox{for} \quad 1 \leq s <2_*.
\end{array}
\right.$$ Then $I_{\lambda}'(u)=0$ and $I_{\lambda}(u)\geq 0$, because $$I_{\lambda}(u)=I_{\lambda}(u)-\frac{1}{\theta}I'_{\lambda}(u)u\geq \left( \frac{1}{2}-\frac{1}{\theta}\right)\|u\|^{2}_{\lambda} \geq 0.$$ Taking $v_n=u_n-u,$ we have by Lemma \[l2\] that $\{v_n\}$ is a $(PS)_{d}$ sequence, with $d=c-I_{\lambda}(u)$. Furthermore, $$0\leq d=c-I_{\lambda}(u)\leq c \leq c_1.$$ We claim that $d=0$. Indeed, otherwise $d>0$. Thereby, by Lemmas \[l3\] and \[l4\], $d \geq c_*$ and $$\label{18}
\liminf_{n \rightarrow +\infty} \| v_n\|_{L^q(\mathbb{R}^N)}^q \geq
\delta_0c_*>0.$$ Applying the Lemma \[l5\] with $\epsilon=\frac{\delta_0c_*}{2}>0,$ there exist $\Lambda, R>0$ such that $$\label{19}
\limsup_{n \rightarrow +\infty} \| v_n\|_{L^q(\mathbb{R}^N)\setminus
B_R(0)}^q \leq \frac{\delta_0c_*}{2}, \quad \mbox{for} \quad \lambda \geq \Lambda.$$ Combining $(\ref{18})$ with $(\ref{19})$, we obtain $$\liminf_{n \rightarrow
+\infty} \left|\left| v_n\right|\right|_{L^q(B_R(0))}^q \geq
\frac{\delta_0c_*}{2}>0,$$ which is an absurd, because as $v_n \rightharpoonup 0$ in $E_\lambda$, and the compact embedding $E_{\lambda}\hookrightarrow L^q(B_R(0))$ ensures that $$\liminf_{n
\rightarrow +\infty} \| v_n\|_{L^q(B_R(0))}^q=0.$$ Therefore $d=0$ and $\{v_n\}$ is a $(PS)_0$ sequence. Then, by Corollary \[c1\], $v_n \rightarrow 0$ in $E_\lambda$, or equivalently, $u_n \rightarrow u$ in $E_\lambda$, showing that for $\lambda$ large enough, $I_{\lambda}$ satisfies the $(PS)_c$ condition for all $c \in [0,c_1].$
The $(PS)_{\infty}$ Condition
=============================
In this section, we will study the behavior of a $(PS)_{\infty}$ sequence, that is, a sequence $\{u_n\} \subset H^2(\mathbb{R}^N)$ satisfying: $$\begin{aligned}
&u_n \in E_{\lambda_n}\ \mbox{and}\ \lambda_{n} \rightarrow +\infty;&\\
&I_{\lambda_n}(u_n)\rightarrow c,\quad \mbox{for some} \quad c\in [0,c_{\Gamma}];&\\
&\|I_{\lambda_n}'(u_n)\|_{E'_{\lambda_n}} \rightarrow 0,&\end{aligned}$$ where $c_{\Gamma}$ is a positive constant, which will be defined in the next section and it is independent of $\lambda$.
\[p2\] Let $\{u_n\}$ be a $(PS)_{\infty}$ sequence for $I_{\lambda}$. Then, there is a subsequence of $\{u_n\}$ , still denoted by itself, and $u \in H^2(\mathbb{R}^N)$ such that $$u_n \rightharpoonup u\ \mbox{in}\ H^2(\mathbb{R}^N).$$ Moreover,
i)
: $u\equiv 0$ in $\mathbb{R}^N \setminus \Omega_{\Gamma}$ and $u$ is a solution of $$\label{20}
\left\{ \begin{array}{c}
\Delta^2 u +u = f(u),\mbox{in}\ \Omega_j, \ \\
u=\dfrac{\partial u}{\partial \eta} =0,\ \mbox{on}\ \partial\Omega_j,
\end{array}
\right.$$ for all $j \in \Gamma;$
ii)
: $\left|\left| u_n-u\right|\right|^{2}_{\lambda_{n}} \rightarrow 0.$
iii)
: $\left\lbrace u_n\right\rbrace $ also satisfies $$\begin{aligned}
&\lambda_n \int_{\mathbb{R}^N}V(x)\left|u_n\right|^2dx \rightarrow 0,\ n \rightarrow +\infty&\\
&\left|\left|u_n\right|\right|^2_{\lambda_n,\mathbb{R}^N\setminus \Omega_{\Gamma}}\rightarrow 0 &\\
&\left|\left|u_n\right|\right|^2_{\lambda_n,\Omega'_j}\rightarrow \int_{\Omega_j}\left[ \left|\Delta u\right|^2+\left|u\right|^2\right]dx,\ \forall j\in \Gamma.&
\end{aligned}$$
In what follows, we fix $c \in [0,c_\Gamma]$ verifying $$I_{\lambda_n}(u_n)\rightarrow c\ \mbox{and}\ \|I'_{\lambda_n}(u_n)\|_{E'_{\lambda_n}} \rightarrow 0.$$ Then, there exists $n_0 \in \mathbb{N}$ such that, $$I_{\lambda_n}(u_n)-\frac{1}{\theta}I'_{\lambda_n}(u_n)u_n \leq c+1+\left|\left|u_n\right|\right|_{\lambda_n}, \quad \forall n \ge n_0.$$ On the other hand, from $f_3)$, $$I_{\lambda_n}(u_n)-\frac{1}{\theta}I'_{\lambda_n}(u_n)u_n \geq \left(\frac{1}{2}-\frac{1}{\theta}\right)\left|\left|u_n\right|\right|^2_{\lambda_n},\ \forall n \in \mathbb{N}.$$ So, for $n\ge n_0$, $$\left(\frac{1}{2}-\frac{1}{\theta}\right)\left|\left|u_n\right|\right|^2_{\lambda_n} \leq c+1+\left|\left|u_n\right|\right|_{\lambda_n},$$ implying that $\{\left|\left|u_n\right|\right|_{\lambda_n}\}$ is bounded in $\mathbb{R}$. As $$\left|\left|u_n\right|\right|_{\lambda_n} \geq \left|\left|u_n\right|\right|_{H^2(\mathbb{R}^N)}, \ \forall n \in \mathbb{N},$$ $\{u_n\}$ is also bounded in $H^2(\mathbb{R}^N)$, and so, there exists a subsequence of $\{u_n\}$, still denoted by itself, and $u \in H^{2}(\mathbb{R}^{N})$ such that $$u_n \rightharpoonup u\ \mbox{in}\ H^2(\mathbb{R}^N).$$ To show $(i)$, we fix for each $m \in \mathbb{N}^*$ the set $$C_m=\left\{x\in \mathbb{R}^N/ V(x) > \frac{1}{m}\right\}.$$ Hence $$\mathbb{R}^N\setminus \overline{\Omega}=\bigcup_{m=1}^{+\infty}C_m.$$ Note that, $$\begin{aligned}
\int_{C_m}\left|u_n\right|^2dx&=\int_{C_m}\frac{\lambda_n V(x)+1}{\lambda_n V(x)+1}\left|u_n\right|^2dx&\\
&\leq \frac{1}{\frac{\lambda_n}{m}+1}\left|\left|u_n\right|\right|_{\lambda_n}^2&\\
&\leq \frac{mM}{{\lambda_n}+m},&
\end{aligned}$$ where $M=\sup_{n \in \mathbb{N}}\|u_n\|_{\lambda_n}^2.$ By Fatou’s Lemma $$\begin{aligned}
\int_{C_m}\left|u\right|^2dx&\leq \liminf_{n \rightarrow +\infty}\int_{C_m}\left|u_n\right|^2dx&\\
&\leq \liminf_{n \rightarrow +\infty}\frac{mM}{{\lambda_n}+m}=0.&
\end{aligned}$$ Therefore, $u=0$ almost everywhere in $C_m$, and consequently, $u=0$ almost everywhere in $\mathbb{R}^N\setminus \overline{\Omega}.$ Besides, fixing $\varphi \in C_{0}^{\infty}(\mathbb{R}^N\setminus \overline{\Omega})$, we have $$\int_{\mathbb{R}^N\setminus \overline{\Omega}}\nabla u(x)\varphi(x)dx=-\int_{\mathbb{R}^N\setminus \overline{\Omega}} u(x)\nabla\varphi(x)dx=0,$$ from where it follows that $$\nabla u(x)=0,\ a.e. \ \mbox{in}\ \mathbb{R}^N\setminus \overline{\Omega}.$$ Since $\partial \Omega$ is smooth , $u \in H^2(\mathbb{R}^N\setminus \overline{\Omega})$ and $\nabla u \in H^1(\mathbb{R}^N\setminus \overline{\Omega}),$ by Trace Theorem , there are constants $K_1,K_2>0$ satisfying $$\left|\left|u\right|\right|_{L^2(\partial \Omega)} \leq K_1\left|\left|u\right|\right|_{H^2(\mathbb{R}^N\setminus \overline{\Omega})}=0,$$ and $$\left|\left|\nabla u\right|\right|_{L^2(\partial \Omega)} \leq K_2 \left|\left|\nabla u\right|\right|_{H^1(\mathbb{R}^N\setminus \overline{\Omega})}=0,$$ showing that $u \in H_0^2(\Omega).$ To complete the proof of $i)$, consider a test function $\varphi \in C_{0}^{\infty}(\Omega)$ and note that $$\label{21}
I'_{\lambda_n}(u_n)\varphi=\int_{\Omega}\left[\Delta u_n \Delta \phi + u_n \varphi\right]dx-\int_{\Omega}f(u_n)\varphi dx.$$ Since $\left\lbrace u_n\right\rbrace $ is a $(PS)_{\infty}$ sequence, we derive that $$\label{22}
I'_{\lambda_n}(u_n)\varphi \rightarrow 0.$$ Recalling that $u_n \rightharpoonup u$ in $H^2(\mathbb{R}^{N})$, we must have $$\label{23}
\int_{\Omega}\left[\Delta u_n \Delta \varphi + u_n \varphi\right]dx \rightarrow \int_{\Omega}\left[\Delta u \Delta \varphi + u \varphi\right]dx$$ and $$\label{24}
\int_{\Omega}f(u_n)\varphi dx \rightarrow \int_{\Omega}f(u)\varphi dx.$$ Therefore, from (\[21\])-(\[24\]), $$\int_{\Omega}\left[\Delta u \Delta \varphi +u \phi\right]dx=\int_{\Omega}f(u)\varphi dx,\ \forall \varphi \in C_0^{\infty}(\Omega).$$ As $C_{0}^{\infty}(\Omega)$ is dense in $H_0^2(\Omega)$, the above equality gives $$\int_{\Omega}\left[\Delta u \Delta v +uv\right]dx=\int_{\Omega}f(u)v dx,\ \forall v \in H_0^2(\Omega),$$ showing that $u$ is a weak solution of the problem $$\label{25}
\left\{ \begin{array}{c}
\Delta^2 u +u = f(u),\, \mbox{in} \, \Omega_j, \ \\
u=\dfrac{\partial u}{\partial \eta} =0,\ \mbox{on}\ \partial\Omega_j.
\end{array}
\right.$$ For $ii)$, note that $$\begin{aligned}
\label{26}
\left|\left|u_n-u\right|\right|_{\lambda_n}^2&= \left|\left|u_n\right|\right|_{\lambda_n}^2+\left|\left|u\right|\right|_{\lambda_n}^2-2\int_{\mathbb{R}^N}\left[\Delta u_n\Delta u+ (\lambda_n V(x)+1)u_nu\right]dx.&
\end{aligned}$$ From $i)$, $$\|u\|_{\lambda_n}^2=\|u\|^2_{H_0^2(\Omega)},$$ and so, $$\int_{\mathbb{R}^N}\left[\Delta u_n\Delta u+ (\lambda_n V(x)+1)u_nu\right]dx=\|u\|_{H_0^2(\Omega)}^2+o_n(1).$$ Thus, we can rewrite (\[26\]) as $$\begin{aligned}
\label{27}
\left|\left|u_n-u\right|\right|_{\lambda_n}^2&= \left|\left|u_n\right|\right|_{\lambda_n}^2-\left|\left|u\right|\right|_{H_0^2(\Omega)}^2+o_n(1).&
\end{aligned}$$ Gathering the boundedness of $\{\|u_n\|_{\lambda_n}\}$ with the limit $ \|I'_{\lambda_n}(u_n)\|_{E'_{\lambda_n}}\rightarrow 0,$ we find the limit $$I'_{\lambda_n}(u_n)u_n \rightarrow 0.$$ Hence, $$\label{28}
\left|\left|u_n\right|\right|_{\lambda_n}^2= I'_{\lambda_n}(u_n)u_n+\int_{\mathbb{R}^N}f(u_n)u_ndx =\int_{\mathbb{R}^N}f(u_n)u_ndx+o_n(1).$$ On the other hand, we know that the limit $ I'_{\lambda_n}(u_n)u \to 0 $ is equivalent to $$\int_{\Omega}\left[ \Delta u_n \Delta u + u_nu\right]dx-\int_{\Omega}f(u_n)udx=o_n(1),$$ which leads to $$\label{29}
\int_{\mathbb{R}^N}\left[ \left|\Delta u\right|^2 +
\left|u\right|^2\right]dx=\int_{\mathbb{R}^N}f(u)udx.$$ Combining (\[27\]) with (\[28\]) and (\[29\]), we see that $$\left|\left|u_n-u\right|\right|_{\lambda_n}^2=\int_{\mathbb{R}^N}f(u_n)u_ndx-\int_{\mathbb{R}^N}f(u)udx+o_n(1).$$ The same arguments used in the proof of Lemma \[l5\] gives $$\int_{\mathbb{R}^N}f(u_n)u_ndx\rightarrow \int_{\mathbb{R}^N}f(u)udx,$$ finishing the proof of $ii)$. Finally, to prove $iii),$ it is enough to use the inequality below $$\lambda_n\int_{\mathbb{R}^N}V(x)\left|u_n\right|^2dx=\lambda_n\int_{\mathbb{R}^N}V(x)\left|u_n-u\right|^2dx
\leq \|u_n-u\|_{\lambda_n}^2 \rightarrow 0.$$
A special minimax level
=======================
In this section, we denote by $I_j:H_0^2(\Omega_j) \rightarrow
\mathbb{R} $ and $I_{\lambda,j}:H^2(\Omega'_j) \rightarrow \mathbb{R}
$ the functionals given by $$I_j(u)=\frac{1}{2}\int_{\Omega_j}\left[\left|\Delta u\right|^2+\left| u\right|^2\right]dx-\int_{\Omega_j}F(u)dx$$ and $$I_{\lambda,j}(u)=\frac{1}{2}\int_{\Omega'_j}\left[\left|\Delta u\right|^2+(\lambda V(x)+1)\left| u\right|^2\right]dx-\int_{\Omega'_j}F(u)dx.$$
It is easy to show that $I_j$ and $I_{\lambda,j}$ satisfy the mountain pass geometry. Hereafter, we denote by $c_j$ and $c_{\lambda,j}$ the mountain pass levels related to the functionals $I_j$ and $I_{\lambda,j}$ respectively.
Since $I_j$ and $I_{\lambda,j}$ satisfy the Palais-Smale condition, from Mountain Pass Theorem due to Ambrosetti-Rabinowitz, there exist $w_j \in H_0^2(\Omega_j)$ and $v_j
\in H^2(\Omega'_j)$ satisfying $$I_j(w_j)=c_j,\ I_{\lambda,j}(v_j)=c_{\lambda,j}\ \mbox{and}\ I'_j(w_j)=I'_{\lambda,j}(v_j)=0.$$
In what follows, $c_{\Gamma}=\sum_{j=1}^{l}c_j$ and $R>0$ is a constant large enough verifying $$0
< I_j(\frac{1}{R}w_j), I_j(Rw_j)< c_j, \forall j \in \Gamma.$$ Hence, by definition of $c_j$, $$\max_{s \in
[1/R^2,1]}I_j(sRw_j)=c_j,\ \forall j \in \Gamma.$$ Consider $\Gamma=\{1,2,\cdots,l\}$, with $l \leq k$ and fix $$\gamma_0(s_1,s_2,\cdots,s_l)(x)=\sum_{j=1}^{l}s_jRw_j(x),\ \forall (s_1,\cdots,s_l)\in [1/R^2,1]^l.$$ From now on, we denote by $\Gamma_*$ the class of continuous path $\gamma \in C([1/R^2,1],E_{\lambda}\setminus \{0\})$ satisfying the following conditions: $$\gamma=\gamma_0\ \mbox{on}\ \partial([1/R^2,1]^l) \leqno{(a)}$$ and $$I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}(\gamma(s_1,\cdots,s_l))\ge 0, \leqno{(b)}$$ where $I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}: H^{2}(\mathbb{R}^N\setminus \Omega'_{\Gamma}) \to \mathbb{R}$ is the functional defined by $$I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}(u)=\frac{1}{2}\int_{\mathbb{R}^N\setminus \Omega'_{\Gamma}}\left[\left|\Delta u\right|^2+(\lambda V(x)+1)\left| u\right|^2\right]dx-\int_{\mathbb{R}^N\setminus \Omega'_{\Gamma}}F(u)dx.$$
Using the class $\Gamma_*$, we define the following minimax level $$b_{\lambda,\Gamma}=\inf_{\gamma \in \Gamma_*}\max_{(s_1,\cdots,s_l) \in [1/R^2,1]^l}I_{\lambda}(\gamma(s_1,\cdots,s_l)).$$
Notice that $\Gamma_* \neq \emptyset$, because $\gamma_0 \in \Gamma_*$.
\[l6\] For each $\gamma \in \Gamma_*,$ there is $(t_1,\cdots,t_l)\in[1/R^2,1]^l$ verifying $$I'_{\lambda,j}(\gamma(t_1,\cdots,t_l))\gamma(t_1,\cdots,t_l)=0,\ \mbox{for}\ j\in\{1,\cdots,l\}.$$
Given $\gamma \in \Gamma_*$, consider the map $\widetilde{\gamma}:[1/R^2,1]^l \rightarrow \mathbb{R}^l$ defined by $$\widetilde{\gamma}(s_1,\cdots,s_l)=\left(I'_{\lambda,1}(\gamma(s_1,\cdots,s_l))\gamma(s_1,\cdots,s_l), \cdots, I'_{\lambda,l}(\gamma(s_1,\cdots,s_l))\gamma(s_1,\cdots,s_l) \right).$$ For $(s_1,\cdots,s_l) \in \partial([1/R^2,1]^l),$ we know that $$\gamma(s_1,\cdots,s_l)=\gamma_0(s_1,\cdots,s_l).$$ Then, $$I'_{\lambda,j}(\gamma_0(s_1,\cdots,s_l))(\gamma_0(s_1,\cdots,s_l))=0 \Rightarrow s_j \not \in \{1/R^2,1\}, \forall j \in \Gamma,$$ otherwise, $$I'_{\lambda,j}(\gamma_0(s_1,\cdots,s_l))(\gamma_0(s_1,\cdots,s_l))=0$$ for $s_{j}=\frac{1}{R^2}$ or $s_{j}=1$, that is, $$I'_{j}(\frac{1}{R}w_j)(\frac{1}{R}w_j)=0 \quad \mbox{or} \quad I'_{j}(Rw_j)(Rw_j)=0$$ implying that $$I_{j}(\frac{1}{R}w_j)\geq c_j \quad \mbox{or} \quad I_{j}(Rw_j)\geq c_j,$$ which contradicts the choice of $R$. Hence, $$(0,0,\cdots,0) \not \in\widetilde{\gamma}(\partial([1/R^2,1]^l)).$$ Then, by Topological Degree $$deg(\widetilde{\gamma},(1/R^2,1)^l,(0,0,\cdots,0))=(-1)^l\not= 0,$$ from where it follows that there exists $(t_1,t_2,\cdots,t_l)\in (1/R^2,1)^l$ satisfying $$I'_{\lambda,j}(\gamma(t_1,t_2,\cdots,t_l))(\gamma(t_1,t_2,\cdots,t_l))=0,\ \mbox{for}\ j \in \{1,2,\cdots,l\}.$$
\[p3\]\
$a) \,\, \sum_{j=1}^{l}c_{\lambda,j}\leq b_{\lambda, \Gamma} \leq c_{\Gamma},\, \forall \lambda \geq 1.$\
$b)$ For $\gamma \in \Gamma_*\ \mbox{and}\ (s_1,\cdots,s_l)\in \partial([1/R^2,1]^l)$, we have $$I_{\lambda}(\gamma(s_1, \cdots,s_l))< c_{\Gamma},\, \forall \lambda \geq 1.$$
a)
: Since $\gamma_0 \in \Gamma_*$, $$\begin{aligned}
b_{\lambda,\Gamma} &\leq \max_{(s_1,\cdots,s_l)\in [1/R^{2},1]^l}I_{\lambda,j}(\gamma_0(s_1,\cdots,s_l))&\\
&\leq \max_{(s_1,\cdots,s_l)\in [1/R^{2},1]^l}I_{\lambda,j}(\sum_{i=1}^{l}s_iRw_i(x))&\\
&\leq \sum_{j=1}^{l}\max_{s_j \in [1/R^{2},1]}I_{j}(s_jRw_j(x))&\\
&\leq \sum_{j=1}^{l}c_j=c_{\Gamma.}&
\end{aligned}$$ For each $\gamma \in \Gamma_*$ and $(t_1,\cdots,t_l)\in [1/R^2,1]^{l}$ as in Lemma \[l6\], we find $$I_{\lambda,j}(\gamma(t_1,\cdots, t_l)) \geq c_{\lambda,j}, \forall j \in \Gamma.$$ In the last inequality we have used the following the equality below $$c_{\lambda,j}=\inf\{I_{\lambda,j}(u);\ u \in E_{\lambda}\setminus \{0\};\, I'_{\lambda,j}(u)u=0\}.$$ On the other hand, recalling that $I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}(\gamma(s_1,\cdots,s_l))\geq 0$, we derive that $$I_{\lambda}(\gamma(s_1, \cdots, s_l)) \geq
\sum_{j=1}^{l}I_{\lambda,j}(\gamma(s_1, \cdots, s_l)),$$ and so, $$\max_{(s_1,\cdots, s_l)\in [1/R^2,1]^l}I_{\lambda}(\gamma(s_1, \cdots, s_l)) \geq I_{\lambda}(\gamma(t_1, \cdots, t_l)) \geq \sum^{l}_{j=1}c_{\lambda,j}.$$ The last inequality combined with the definition of $b_{\lambda, \Gamma}$ gives $$b_{\lambda, \Gamma}\geq \sum_{j=1}^{l}c_{\lambda, j},$$ This completes the proof of $a).$
b)
: As $\gamma(s_1,\cdots,s_l)=\gamma_0(s_1,\cdots,s_l)$ on $\partial([1/R^2,1]^l),$ we derive that $$I_{\lambda}(\gamma_0(s_1,\cdots,s_l))=\sum_{j=1}^{l}I_j(s_jRw_j), \forall (s_1,\cdots,s_l) \in \partial([1/R^2,1]^l).$$ Since $$I_j(s_jRw_j) \leq c_j, \quad \forall j \in \Gamma$$ and there is $j_0 \in \Gamma$, such that $s_{j_0} \in \{1/R^2,1 \}$, we have $$I_{\lambda}(\gamma_0(s_1,\cdots,s_l)) < c_{\Gamma}.$$
$b_{\lambda, \Gamma} \rightarrow c_{\Gamma},$ when $\lambda
\rightarrow +\infty.$
Using the same arguments found in , it is possible to prove that $c_{\lambda, j} \rightarrow c_j$ for each $j \in \Gamma$. Therefore, by Proposition $\ref{p3}$, $b_{\lambda,\Gamma} \rightarrow
c_{\Gamma}$ when $\lambda \rightarrow +\infty.$
Proof of the Main Theorem
=========================
Hereafter, we consider $$M=1+\sum_{j=1}^{l}\sqrt{\left(\frac{1}{2}-\frac{1}{\theta}\right)c_j},$$ $$\overline{B}_{M+1}(0)=\{ u \in E_{\lambda};\|u\|_{\lambda}\leq M+1\},$$ and for small $\mu>0$ $$A_{\mu}^{\lambda}=\left\{ u \in \overline{B}_{M+1}; \left|\left|u\right|\right|_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}\leq \mu,\ I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}(u)\geq 0\ \mbox{and}\ \left|I_{\lambda,j}(u)-c_j\right|\leq \mu, \forall j \in \Gamma \right\},$$ and $$I_{\lambda}^{c_{\Gamma}}=\left\{ u \in E_{\lambda}/ I_{\lambda}(u)\leq c_{\Gamma}\right\}.$$ Note that $A_{\mu}^{\lambda}\cap I_{\lambda}^{c_{\Gamma}} \neq \emptyset,$ because $w= \sum_{j=1}^{l}w_j \in A_{\mu}^{\lambda}\cap I_{\lambda}^{c_{\Gamma}} .$ Fixing $$\label{30}
0< \mu < \frac{1}{4}\min\{c_j;j \in \Gamma\},$$ we have the following uniform estimate from below for $\|I'_{\lambda}(u)\|$ in the set $\left(A_{2\mu}^{\lambda}\setminus
A_{\mu}^{\lambda}\right)\cap I_{\lambda}^{c_{\Gamma}}.$
\[p4\] Let $\mu >0$ satisfy $(\ref{30})$. Then, there exist $\sigma_0>0$ independent of $\lambda$ and $\Lambda_* \geq 1$ such that $$\label{31}
\ \|I'_{\lambda}(u)\| \geq \sigma_0\ \mbox{for}\ \lambda \geq
\Lambda_*\ \mbox{and for all}\ u \in
\left(A_{2\mu}^{\lambda}\setminus A_{\mu}^{\lambda}\right)\cap
I_{\lambda}^{c_{\Gamma}}.$$
Arguing by contradiction, suppose that there are $\lambda_n \rightarrow +\infty$ and $u_n \in E_{\lambda_n}$, with $$u_n \in \left(A_{2\mu}^{\lambda_n}\setminus A_{\mu}^{\lambda_n}\right)\cap I_{\lambda}^{c_{\Gamma}} \quad \mbox{and} \quad \|I'_{\lambda_n}(u_n)\| \rightarrow 0.$$ Since $u_n \in A_{2\mu}^{\lambda_n}$, the sequence $\left\{\|u_n\|_{\lambda_n}\right\}$ is bounded. Consequently $\left\{I_{\lambda_n}(u_n)\right\}$ is also bounded. Then, passing to a subsequence if necessary, $$I_{\lambda_n}(u_n) \rightarrow c \in (-\infty,c_{\Gamma}].$$ By Proposition \[p2\], passing to a subsequence if necessary, $u_n
\rightarrow u$ in $H^2(\mathbb{R}^N)$ and $u \in
H_{0}^{2}(\Omega_{\Gamma})$ is a solution of the problem (\[20\]). Moreover, $$\begin{aligned}
&\lambda_n \int_{\mathbb{R}^N}V(x)\left|u_n\right|^2dx \rightarrow 0,& \label{32}\\
&\left|\left|u_n\right|\right|^2_{\lambda_n,\mathbb{R}^N\setminus \Omega_{\Gamma}}\rightarrow 0 & \label{33}\\
&\left|\left|u_n\right|\right|^2_{\lambda_n,\Omega'_j}\rightarrow \int_{\Omega_j}\left[ \left|\Delta u\right|^2+\left|u\right|^2\right]dx,\ \forall j\in \Gamma.& \label{34}
\end{aligned}$$ Since $c_{\Gamma}=\sum_{j=1}^{l}c_j$ and $c_j$ is the least energy level for $I_j$, one of the following cases occurs:
i)
: $u\left|_{\Omega_j}\neq 0\right.,\ \forall j \in \Gamma,$ or
ii)
: $u\left|_{\Omega_{j_0}}=0\right.,$ for some $j_0 \in \Gamma.$
If $i)$ happens, from $(\ref{32})-(\ref{34})$ $$I_j(u)=c_j, \quad \forall j \in \Gamma.$$ Hence $u_n \in A_{\mu}^{\lambda_n}$ for $n$ large enough, which is a contradiction.
If $ii)$ happens, from $(\ref{32})\ \mbox{and}\ (\ref{33})$ $$\left|I_{\lambda_{n},j_0}(u_n)-c_{j_0})\right| \rightarrow c_{j_0} \geq 4\mu,$$ which contradicts the hypothesis $u_n \in A_{2 \mu}^{\lambda_n}\setminus A_{\mu}^{\lambda_n}$ for all $n \in \mathbb{N}$. Since $i)$ or $ii)$ cannot happen, we get an absurd, finishing the proof.
\[p5\] Let $\mu$ satisfy $(\ref{30})$ and $\Lambda_* \geq 1$ the constant given in the Proposition \[p3\]. Then for $\lambda \geq
\Lambda_*$, there exists $u_{\lambda}$ a solution of $(\ref{1})$ satisfying $u_{\lambda}\in A_{\mu}^{\lambda}\cap
I_{\lambda}^{c_{\Gamma}}.$
We will suppose, by contradiction, that there are no critical points of $I_{\lambda}$ in $A_{\mu}^{\lambda}\cap I_{\lambda}^{c_{\Gamma}}.$ By Proposition $\ref{p1}$, $I_{\lambda}$ satisfies the $(PS)_d$ condition for $d \in [0,c_{\Gamma}] $ and $\lambda$ large enough. Thereby, there exists $d_{\lambda} >0$ such that $$\left|\left|I'_{\lambda}(u)\right|\right| \geq d_{\lambda},\
\forall u \in A_{\mu}^{\lambda}\cap I_{\lambda}^{c_{\Gamma}}.$$ On the other hand, by Proposition \[p4\], $$\|I'_{\lambda}(u)\|
\geq \sigma_0,\ \forall u \in (A_{2\mu}^{\lambda} \setminus
A_{\mu}^{\lambda})\cap I_{\lambda}^{c_{\Gamma}},$$ where $\sigma_0$ is independent of $\lambda.$ Now, we define the continuous functions $\Psi:E_{\lambda}\rightarrow \mathbb{R}$ and $H:I_{\lambda}^{c_{\Gamma}}\rightarrow \mathbb{R}$ by $$\begin{aligned}
\Psi(u)= 1,& \quad u \in A_{3\mu/2}^{\lambda},\\
\Psi(u)=0, & \quad u \not \in A_{2\mu}^{\lambda}, \\
0 \leq \Psi(u)\leq 1,&\quad \mbox{for}\ u \in E_{\lambda},
\end{aligned}$$ and $$H(u)=\left\{ \begin{array}{cc}
-\Psi(u)\left|\left|Y(u)\right|\right|^{-1}Y(u),&\ u \in A_{2\mu}^{\lambda},\\
0,& u\not \in A_{2\mu}^{\lambda},
\end{array}
\right.$$ where $Y$ is a pseudogradient vector field for $I_{\lambda}$ on $$X=\{ u \in E_{\lambda}; I_{\lambda}(u) \neq 0\}.$$ Notice that $$\left|\left|H(u)\right|\right| \leq 1 \ \mbox{for all}\ \lambda \geq \Lambda_*\ \mbox{and}\ u \in I_{\lambda}^{c_{\Gamma}}.$$ The above information ensures the existence of a flow $\eta: [0,+\infty) \times I_{\lambda}^{c_{\Gamma}} \rightarrow I_{\lambda}^{c_{\Gamma}}$ defined by $$\left\{ \begin{array}{ccc}
\dfrac{d\eta(t,u)}{dt}&=& H(\eta(t,u))\\
\eta(0,u)&=&u\in I_{\lambda}^{c_{\Gamma}},
\end{array}
\right.$$ verifying $$\label{35}
\dfrac{d I_{\lambda}(\eta(t,u))}{dt}\leq -\Psi(\eta(t,u))\left|\left|I'_{\lambda}(\eta(t,u))\right|\right|\leq 0,$$ $$\label{36}
\left|\left|\dfrac{d\eta}{dt}\right|\right|= \left|\left|H(\eta)\right|\right|\leq 1,$$ and $$\label{37}
\eta(t,u)=u,\ \forall\ t\geq 0\ \mbox{and}\ u \in I_{\lambda}^{c_{\Gamma}}\setminus A_{2 \mu}^{\lambda}.$$ In what follows, we set $$\beta(s_1,\cdots,s_l)=\eta(T,\gamma_0(s_1,\cdots,s_l)),\
\forall (s_1,\cdots,s_l)\in [1/R^2,1]^l,$$ where $T>0$ will be fixed later on.
Once $$\gamma_0(s_1,\cdots,s_l)\not \in A_{2 \mu}^{\lambda},\ \forall
(s_1,\cdots,s_l)\in \partial ([1/R^2,1]^l),$$ we deduce that $$\beta(s_1,\cdots,s_l)=\gamma_0(s_1,\cdots,s_l),\
\forall (s_1,\cdots,s_l)\in \partial([1/R^2,1]^l).$$ Moreover, it is easy to check that $$I_{\lambda,\mathbb{R}^N\setminus \Omega'_{\Gamma}}(\beta(s_1,\cdots,s_l))\geq 0, \quad \forall (s_1, \cdots,s_l)\in [1/R^2,1]^l,$$ showing that $\beta \in \Gamma_*$.
Note that $supp(\gamma_0(s_1,\cdots,s_l)) \subset
\overline{\Omega}_{\Gamma}$ for all $(s_1,\cdots,s_l) \in
[1/R^2,1]^l$ and $I_{\lambda}(\gamma_0(s_1,\cdots,s_l))$ independent of $\lambda \geq \Lambda.$ Furthermore, $$I_{\lambda}(\gamma_0(s_1, \cdots,s_l))\leq c_{\Gamma}, \forall (s_1, \cdots,s_l)\in [1/R^2,1]^l$$ and $$I_{\lambda}(\gamma_0(s_1, \cdots,s_l))= c_{\Gamma},\ \mbox{if}\ s_j=1/R, \forall j \in \Gamma.$$ Therefore, $$m_0=max\left\{ I_{\lambda}(u);u \in \gamma_0([1/R^2,1]^l)\setminus A_{\mu}^{\lambda}\right\}< c_{\Gamma},$$ and $m_0$ is independent of $\lambda$.
Since there is $K_*$ such that $$\left|I_{\lambda}(u)-I_{\lambda}(v)\right|\leq K_*\|u-v\|_{\lambda,\Omega'_j},\ \forall u,v \in \overline{B}_{M+1}\ \mbox{and}\ \forall j \in \Gamma,$$ we claim that if $T$ is large enough, the estimate below holds $$\label{38}
\max_{(s_1,\cdots,s_l \in
[1/R^2,1]^l)}I_{\lambda}\left(
\beta(s_1,\cdots,s_l)\right)\leq
\max\{m_0,c_{\Gamma}-\frac{1}{2K_*}\sigma_0\mu\}.$$ Indeed, fix $u=\gamma_0(s_1,\cdots,s_l) \in E_{\lambda}.$ If $u
\not \in A_{\mu}^{\lambda},$ $$I_{\lambda}\left(
\eta(t,u)\right)) \leq I_{\lambda}\left(
\eta(0,u)\right))=I_{\lambda}(u)\leq m_0, \quad \forall t \geq 0.$$ On the other hand, if $u \in A_{\mu}^{\lambda}$, by setting $\tilde{\eta}(t)=\eta(t,u),$ $\tilde{d}_{\lambda}=\min\{d_{\lambda},\sigma_0\}$ and $T=\frac{\sigma_0 \mu}{2 K_*d_{\lambda}}>0$, we analyze the following cases:\
**Case 1:** $\tilde{\eta}(t) \in A_{3\mu/2}^{\lambda}, \forall
t \in [0,T].$\
**Case 2:** $\tilde{\eta}(t_0) \in \partial
A_{3\mu/2}^{\lambda},\ \mbox{for some}\ t_0 \in [0,T].$\
[**Analysis of the Case 1:**]{} In this case, $$\Psi(\tilde{\eta}(t))\equiv 1, \quad \forall t \in [0,T]$$ and $$\| I'_{\lambda}(\tilde{\eta}(t))\| \geq \tilde{d}_{\lambda}, \forall t \in [0,T].$$ Hence, $$I_{\lambda}(\tilde{\eta}(T))=I_{\lambda}(u)+\int_{0}^{T}\frac{d}{ds}I_{\lambda}(\tilde{\eta}(s))ds \leq c_{\Gamma}-\int_{0}^{T}\tilde{d}_{\lambda}ds,$$ it follows that $$I_{\lambda}(\tilde{\eta}(T))\leq c_{\Gamma}-\tilde{d}_{\lambda}T=c_{\Gamma}-\frac{1}{2K_*}\sigma_0\mu.$$
[**Analysis of the Case 2:**]{} Let $0 \leq t_1 \leq t_2 \leq T$ satisfy $\tilde{\eta}(t_1)\in \partial A_{\mu}^{\lambda},$ $\tilde{\eta}(t_2)\in \partial A_{3\mu/2}^{\lambda}$ and $\tilde{\eta}(t)\in A_{3\mu/2}^{\lambda}\setminus
A_{\mu}^{\lambda}, \forall t\in [t_1,t_2].$ Then $$\label{39}
\|\tilde{\eta}(t_1)-\tilde{\eta}(t_2)\|\geq \frac{1}{2K_*}\mu.$$ Indeed, denoting $w_1=\tilde{\eta}(t)$ and $w_2=\tilde{\eta}(t_2),$ it follows that $$\|w_2\|_{\lambda,\mathbb{R^N}\setminus
\Omega'_{\Gamma}}=\frac{3}{2} \mu \quad \mbox{or} \quad
\left|I_{\lambda,j_0}(w_2)-c_{j_0}\right|=\frac{3}{2} \mu.$$ From definition of $A_{\mu}^{\lambda},$ we have $\|w_2\|_{\lambda,\mathbb{R^N}\setminus\Omega'_{\Gamma}}\leq \mu. $ Thus, $$\|w_2-w_1\|_{\lambda} \geq \frac{1}{K_*}\left|I_{\lambda,j_0}(w_2)-I_{\lambda,j_0}(w_1)\right|\geq \frac{1}{2K_*}\mu.$$ By Mean Value Theorem $$\label{40}
\|\tilde{\eta}(t_1)-\tilde{\eta}(t_2)\|_{\lambda}\le\left|\left|\dfrac{d\eta}{dt}\right|\right|\left| t_1-t_2\right|.$$ As $\left|\left|\dfrac{d\eta}{dt}\right|\right| \le 1$, from (\[39\]) and (\[40\]), $$\left| t_1-t_2\right| \ge \frac{1}{2K_*}\mu.$$ Hence $$I_{\lambda}(\tilde{\eta}(T))\leq I_{\lambda}(u)-\int_{0}^{T}\Psi(\tilde{\eta}(s))\|I'_{\lambda}(\tilde{\eta}(s))\|ds,$$ and so, $$I_{\lambda}(\tilde{\eta}(T))\leq c_{\Gamma}-\int_{t_1}^{t_2}\sigma_0ds \leq c_{\Gamma}-\frac{\sigma_0}{2K_*}\mu,$$ proving (\[38\]).
Thereby, $$b_{\lambda,\Gamma}\leq \max_{[1/R^2,1]^l}I_{\lambda}(\widehat{\eta}(s_1,\cdots,s_l))\leq \max\{m_0,c_{\Gamma}-\frac{1}{2K*}\sigma_0 \mu\} < c_{\Gamma},$$ which is an absurd, because $b_{\lambda, \Gamma}\rightarrow c_{\Gamma},$ when $\lambda \rightarrow \infty.$
Thus, we can conclude that $I_{\lambda}$ has a critical point $u_{\lambda}$ in $A_{\mu}^{\lambda}$ for $\lambda$ large enough.
**Completion of the Proof of Theorem \[T1\]:**
From the last proposition there exists $\{u_{\lambda_n}\}$ with $\lambda_n \rightarrow +\infty$ satisfying: $$I'_{\lambda_n}(u_{\lambda_n})=0,$$ $$\|u_{\lambda_n}\|_{\lambda_n,\mathbb{R}^N\setminus \Omega'_{\Gamma}} \rightarrow 0$$ and $$I_{\lambda_n,j}(u_{\lambda_n})\rightarrow c_j, \forall j \in \Gamma.$$ Therefore, from Proposition \[p2\], $$u_{\lambda_n} \rightarrow u \quad \mbox{in} \quad H^2(\mathbb{R}^N)\ \mbox{with}\ u\ \in H_{0}^{2}(\Omega_{\Gamma}).$$ Moreover, $u$ is a nontrivial solution of $$\label{PF}
\left\{ \begin{array}{c}
\Delta^2 u +u = f(u),\mbox{in}\ \Omega_j \ \\
u=\dfrac{\partial u}{\partial \eta} =0,\ \mbox{on}\ \partial\Omega_j,
\end{array}
\right.$$ with $I_j(u)=c_j$ for all $i \in \Gamma$. Now, we claim that $u=0$ in $\Omega_j$, for all $j \notin \Gamma$. Indeed, it is possible to prove that there is $\sigma_1>0$, which is independent of $j$, such that if $v$ is a nontrivial solution of (\[PF\]), then $$\|v\|_{H_0^{2}(\Omega_j)} \geq \sigma_1.$$ However, the solution $u$ verifies $$\|u\|_{H^{2}(\mathbb{R}^{N} \setminus \Omega'_\Gamma)}=0,$$ showing that $u=0$ in $\Omega_j$, for all $j \notin \Gamma$. This finishes the proof of Theorem \[T1\].
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[^1]: C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2 and INCT-MAT, [email protected]
[^2]: [email protected]
|
---
abstract: 'We make a review of the main nuclear effects that affect neutrino-nucleus cross sections. We discuss how the different models in the literature try to describe these different effects, and thus try to compare between them. We focus on the quasi-elastic reaction in the neutrino energy region of around 1 GeV, where recent data from MiniBoone are available. Among the issues discussed are the different treatment of medium corrections to initial and final state nucleon wave functions and the problem of the rescattering of ejected nucleons.'
author:
- 'M. Valverde'
- 'J. Nieves'
- 'J. E. Amaro'
- 'M. J. Vicente-Vacas'
title: Nucleon Emision Off Nuclei Induced By Neutrino Interactions
---
[ address=[Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan]{} ]{}
[ address=[Instituto de Física Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investigación de Paterna, Aptd. 22085, E-46071 Valencia, Spain]{}]{}
[ address=[Dpto. de Física Atómica, Molecular y Nuclear, Universidad de Granada,Facultad de Ciencias, Campus Fuentenueva S/N, E-18071 Granada, Spain]{}]{}
[ address=[Dpto. de Física Teórica e Instituto de Física Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investigación de Paterna, Aptd. 22085, E-46071 Valencia, Spain]{}]{}
Introduction
============
The study of neutrinos is playing a very relevant role in current research in nuclear, astro and particle physics. One of these major topics is neutrino oscillations, that since its discovery 10 years ago by the Super Kamiokande collaboration [@Kamiokande; @kamiokande2] have evolved and is now reaching the realm of precision experiments [@Hagiwara:2005pe]. This new generation of precision experiments is no longer hindered by statistical error, but is dominated by systematic uncertainties, one of the most important of these systematic errors being the neutrino-nucleus cross section. However neutrinos are a neutral, weakly interacting particle, thus its detection must rely on the observation of secondary particles that appear after the scattering of the incoming neutrino with one of the nuclei of the passive part of the detector set up. The study of neutrino oscillation physics requires a good determination of the incoming neutrino energy. However accelerator neutrino beams are produced by the muon decay, thus being far from monochromatic. The determination of the energy of a detected neutrino and the nature of the collision thus can be only done through kinematic reconstruction from the produced particles.
A proper knowledge of the neutrino-nucleus cross sections is therefore required for the experiment analysis. Nevertheless most of the codes and models used for the modelling of these processes in the analysis of experimental data are based on Fermi Gas models, namely the famous Marteau model [@Marteau:1999kt]. However most of these models are known from the experience on electron scattering physics to fail to describe the existing experimental data.
These reasons have motivated the interest of the theoretical nuclear physics community on the subject of neutrino-nucleus scattering. There is a general consensus among the community that a simple Fermi Gas model, widely used in the analysis of neutrino oscillation experiments, is no longer good enough for the level of precision required for neutrino experiments. Thus many different approaches has been proposed to model these reactions. In this talk I will describe the main nuclear physics effects that, based on the experience from electron scattering physics, are expected to arise. We will also try to review a few of the proposed nuclear models. For the sake of simplicity we will focus on the quasi-elastic reactions at neutrino energies from a few hundred MeV to around 1 GeV which are of interest to MiniBoone and T2K. Of course at these experiments pion production processes play an important role, however the pion physics involved in these processes is somewhat beyond the scope of this contribution. For a more general look at this neutrino interactions issues please look at the contribution of L. Alvarez Ruso in this same proceedings [@LuisARuso].
Nuclear effects in inclusive processes
======================================
In this section we introduce a model for the quasi-elastic inclusive charged current process $$\nu_l + \, ^Z\! A \to l^- + X$$ where the only detected particle being the outgoing lepton and therefore one must sum over all possible final hadron states, here denoted by $X$. In the case of neutral current processes the outgoing lepton part is a neutrino. So in order to get information about the process some of the hadronic products must be detected, be it a nucleon (usually proton) or pion. The residual nucleus is not detected in the Cerenkov radiation experiments so a sum over nuclear states is also needed, what is usually called semi-inclusive observables. In this talk we will focus in the situation where the only detected hadron is a nucleon. If we were to include also pions we should take into account the production of pions in the primary $\nu$-nucleon vertex. We will follow the model of references [@Nieves:2004wx] and [@Nieves:2005rq] which start from a local density Fermi Gas model, but in top of it they include a whole lot of nuclear effects. We will refer as this model as the Valencia-Granada model. Actually this model is actually based on a previous work dealing with electron scattering [@Gil:1997bm] that was able to reasonably describe the available experimental data.
Due to the difficult nature of neutrino cross sections experiments (that sometimes rely on some neutrino interaction model), it is very important to validate any neutrino-nucleus interaction model against electron-nucleus scattering experiments. As can be seen in Fig.\[fig:gil\] the electron version of the Valencia-Granada model describes rather well the existing experimental data. In this figure we can observe the main features of this kind of reactions. A large broad peak at low transfered energy $\omega$ (the quasi-elastic peak), and a second lower peak that is associated with the $\Delta(1232)$ production.
![Double differential cross section for the inclusive process $e^- + C^{12}\to e^- + X$. Picture taken from [@Gil:1997bm][]{data-label="fig:gil"}](eecarbon){width=".4\textwidth"}
A major feature of this model is that it is able to correctly describe the region between the two major peaks. This [*gap*]{} region is underestimated by most models, as they usually do not take into account any process beyond the dominant absorption by one nucleon (quasi-elastic) and delta production. However one must notice the existence of additional processes like non-resonant pion production and boson absorption by two nucleons. Thus in order to properly describe inclusive processes it is clearly needed the inclusion of additional non-resonant mechanisms. The first step towards the inclusion this processes in the framework of a model of neutrino-nucleus scattering is having an adequate model for neutrino-nucleon scattering in free space (that is, with no nuclear medium effects). Many interesting approaches are being developed to tackle this problem, see [*e.g.*]{} Ref.[@Hernandez:2007qq].
The differential cross section for the neutrino collision can be written $$\frac{d^2\sigma_{\nu l}}{d\Omega(\hat{k^\prime})dE^\prime_l} =
\frac{\lvert\vec{k}^\prime\rvert}{\lvert\vec{k}\rvert}\frac{G^2}{4\pi^2}
L_{\mu\sigma}W^{\mu\sigma} \label{eq:sec}$$ with $L$ and $W$ the leptonic and hadronic tensors, respectively. The leptonic tensor is well known and is obtained from the weak interaction in the Fermi contact approximation. On the other hand, the inclusive CC nuclear cross section is related to the imaginary part of the neutrino self-energy in the medium by: $$\sigma = - \frac{1}{\lvert\vec{k}\rvert} \int\,d^3\vec{r} \; {\rm Im}\Sigma_\nu (k;\rho(r))$$ We obtain the imaginary part of the neutrino self-energy in the medium ${\rm Im}\Sigma_\nu$ by means of the Cutkosky’s rules. We obtain for $k^0 > 0$ $${\rm Im} \Sigma_\nu(k) = \frac{8G}{\sqrt 2 M^2_W}\int \frac{d^3
k^\prime}{(2\pi)^3 }\frac{\Theta(q^0) }{2E^{\prime}_l}
~ {\rm Im}\left\{ \Pi^{\mu\nu}_W(q;\rho) L_{\nu\mu} \right\}
\label{eq:ims}$$ and thus, the hadronic tensor is basically an integral over the nuclear volume of the $W$-boson self-energy $\Pi_W^{\mu\nu}\left(q\,;\rho\right)$ inside the nuclear medium. In general we can then take into account the different in-medium effects and reaction mechanism modes ($W$ absorption by one nucleon or by a pair of nucleons, pion production, resonance excitation…) by including the correspondent diagrams in the $W$ self-energy diagram. We will focus in the charged current quasi-elastic process, that corresponds to the $W$ absorption by one nucleon. In general we obtain that the hadron tensor can be expressed (up to some constant) as: $$\begin{gathered}
W^{\mu\nu} = \int d^3\vec{r}\int \frac{d^3\vec{p}}{(2\pi)^3}\int_{\mu-q^0}^{\mu} d\omega \,
A^{\nu\mu}(p,q)|_{p^0=\bar{E}(\vec{p})} \\
S_h\left(\omega,\vec{p};\rho\right) S_p\left(q^0+\omega,\vec{p}+\vec{q};\rho\right) \, .\end{gathered}$$ In this expression the tensor $A^{\nu\mu}$ contains all the information related to the neutrino-nucleon interaction. The $S_p$ and $S_h$ are the particle and hole nucleon spectral functions and contain the information on the nucleon wave functions in the final and initial nuclear state respectively. $\mu$ is the chemical potential. In a simple local density Fermi Gas the nucleons are on mass shell and thus the hole spectral functions take the very simple form: $$S_h\left(\omega,\vec{p};\rho\right) = \delta(\omega - E(\vec{p}))\Omega(E_F - E(\vec{p}))$$ and an analogue expression for the particle one. This expression leads to a description that is completely equivalent to the usual Fermi gas model used in the literature. The only nuclear physics effects that are taken into account are the Pauli blocking effect (see the $\Theta$ function) and the Fermi motion of the nucleons, whose momenta are approximated to be distributed uniformly.
However, this is well known to be an oversimplificated model for the electron scattering process, and we expect it to be so also for the neutrino process. Thus we improve our model by including realistic spectral functions, $$\begin{gathered}
S_{p,h}(\omega,\vec{p}\,;\rho) = \\
\mp\frac{1}{\pi}\frac{{\rm Im}\Sigma(\omega,\vec{p}\,;\rho)}
{\left[\omega-{\bar E}(\vec{p}\,)-{\rm Re}\Sigma(\omega,\vec{p}\,;\rho) \right]^2 + \left[{\rm Im}\Sigma(\omega,\vec{p}\,;\rho)\right]^2}\end{gathered}$$ with $\omega\ge \mu$ or $\omega\le \mu$ for $S_p$ and $S_h$, respectively. The chemical potential $\mu$ is determined by $$\mu = M + \frac{k_F^2}{2M} + {\rm Re}\Sigma(\mu, k_F)$$ where in Valencia-Granada model the reference [@FO92] was followed for the nucleon self-energy $\Sigma(\mu,k_F)$. Notice that both particle and hole self-energies are included in this approach, in contrast with other models in the literature that only include the effect of nucleon wave functions in the hole states. It is also very important to notice how in the limit $\Sigma \to 0$ we recover the expressions of a non-interacting Fermi Gas model. This is an obvious point that should be tested in all models for lepton scattering off nuclei. The effect of particle (that is final state nucleons) spectral functions is often defined in the literature as final state interactions (FSI). This effect (see Fig.\[fig:fsi\]) usually produces a broadening of the nuclear response and a reduction of the response at the peak, however the total response is not much affected, specially when RPA corrections (see next paragraph) are also taken into account.
![$\nu_e$ and $\bar{\nu}_e$ inclusive quasi-elastic cross sections in oxygen as a function of the transferred energy, at two values of the transferred momentum. We show results for relativistic (REL) and non-relativistic nucleon kinematics. In this latter case, we present results with (FSI) and without (NOREL) FSI effects. For the three cases, we also show the effect of taking into account RPA correlations (lower lines at the peak). See Ref. [@Nieves:2004wx] for further details.[]{data-label="fig:fsi"}](ang-fsi_rev){width=".4\textwidth"}
Furthermore the excited nuclear states are expected to be correlated by means of the nucleon-nucleon interaction. We model this effect by including a series of particle-hole excitations, see Fig.\[fig:rpa\], of the RPA type. The inclusion of this diagrams modify the expression for the tensor $A^{\nu\mu}$ and induces a reduction of the cross section, specially at low $Q^2$ kinematics. We use an effective Landau-Migdal $ph-ph$ interaction where in the vector-isovector channel ($\vec{\sigma}\cdot\vec{\sigma}\vec{\tau}\cdot\vec{\tau}$ operator) we use an interaction with explicit $\pi$ meson (longitudinal) and $\rho$ meson (transverse) exchanges that also includes $\Delta(1232)$ degrees of freedom.
This point has been applied in [@AlvarezRuso:2009ad] to the MiniBoone experiment [@Katori:2009du] following the prescriptions of our model. In this reference it was found that the inclusion of this RPA correlations improves the description of the cross section measurements in the MiniBoone experiment, without including unphysical parameters, like effective Fermi momentum…
![Set of irreducible diagrams responsible for the polarization (RPA) effects in the $1p1h$ contribution to the $W$ self-energy.[]{data-label="fig:rpa"}](rpa){width=".4\textwidth"}
![Taken from Ref. [@Leitner:2008fg].[]{data-label="fig:leitner"}](leitner_rpa){width=".4\textwidth"}
Semi-inclusive observables: Nucleon rescattering
================================================
In the previous section we have focused in processes where the only detected particle is the outgoing (charged) lepton. However sometimes more information from the process is obtained from hadrons. Actually these are the only possible particles to be detected in neutral current processes. In this process a new effect must be taken into account in top of the ones described in the previous section. This is the rescattering of outgoing hadrons in its way out of the nucleus. Actually in the previous model the nucleon that interacts with the neutrino is put on mass shell and goes out of the nucleus. However it is well known that this nucleon strongly interacts with the other nucleons in the medium and can be deflected, inducing the emission of secondary nucleons or, given enough energy, pions. Of course this new processes do not change the total inclusive cross section as described in the previous section. However in the case of neutral currents it is necessary to properly describe this processes as secondary particle emission processes can result in background events, [*e.g.*]{} $\pi^0$ decay photons can mimic Cerenkov radiation from electrons, or in other processes energy can be transfered to undetected neutrons thus introducing problems in the incoming neutrino kinematics reconstruction. For that reason it is very important to properly model this rescattering processes. A few different approaches have been proposed to describe this:
1. Distorted wave impulse approximation. In this models the outgoing nucleon wave function is calculated using a wave equation complex potential. The imaginary part of this potential removes all the events where the outgoing nucleon collides. This approach is fully quantum mechanical, however it has the major disadvantage that is only suitable to deal with fully exclusive observables where the final nuclear state is also observed. These is because the optical potential is not unitary and thus it does not shuffle events from one channel to another (as should be done when dealing with semi-inclusive observables) but just remove those events where the nucleon undergoes a collision,changing its kinematics. For that reason it is well known that this approach underestimates cross sections in semi-inclusive reactions.
2. Monte Carlo cascade models. This is the usual approach in which the trajectory of the ejected hadrons is simulated via a semi-classical Monte Carlo algorithm that takes into account changes of energy and momentum of the emitted nucleon, as well as the possibility of having secondary hadrons.
3. Transport model. Recently a new approach has been proposed by the Giessen group [@Leitner:2006ww]. In this approach the semi-classical transport equations are explicitly solved for all ejected hadrons, thus allowing for a rigorous tracking of all particles.
In the following we shall focus on the model by the Valencia-Granada group [@Nieves:2005rq], which is a Monte Carlo cascade like model. We shall use a simplified version of the model where pion production processes are not included. In this model for any given leptonic kinematic $q^\mu$, a point ($\vec{r}$) in the nucleus is randomly selected where the gauge boson absorption takes place according to the profile $d^5\sigma / d\Omega'dE'd^3\vec{r}$. Then a nucleon with a random momentum is picked up from the Fermi sea with a given momentum $\vec{p}$. Its kinematics is determined via energy conservation $$E = q^0 + \sqrt{\vec{p}^2 + M^2}$$ and Pauli blocking effects are explicitly included. The nucleon is assumed to be in an average nucleon potential $V(r) =
k_F^2(\vec{r})/2M$ and then it is moved in discrete steps until it leaves the nucleus. At each of these steps the possibility of producing a secondary nucleon is explicitly taken into account by means of the cross section $$\hat{\sigma}^{N_{1}N_{2}} = \int d\Omega_{CM}
\frac{d\sigma^{N_{1}N_{2}}}{d\Omega_{CM}}C_T(q,\rho)
\Theta\left(\kappa-\frac{\lvert\vec{p}\cdot\vec{p}_{CM}\rvert} {\lvert\vec{p}\rvert\lvert\vec{p}_{CM}\rvert}\right)$$ where in-medium renormalization of the nucleon-nucleon interaction ($C_T(q,\rho)$) and Pauli blocking effects $\Theta\left(\kappa-
\lvert\vec{p}\cdot\vec{p}_{CM}\rvert
\ \lvert\vec{p}\rvert\lvert\vec{p}_{CM}\rvert\right)$ are explicitly included.
The effect of this cascade algorithm in the spectra of outgoing nucleons can be easily appreciated in Fig.\[fig:mcharged\]. The nucleons spectra produced by CC processes induced by muon neutrinos are shown in Fig. \[fig:mcharged\] for Argon. Of course neutrinos only interact via CC with neutrons and would emit protons, but these primary protons interact strongly with the medium and collide with other nucleons which are also ejected. As a consequence there is a reduction of the flux of high energy protons but a large number of secondary nucleons, many of them neutrons, of lower energies appear.
![Charged current $^{40}Ar(\nu,\mu^-+N)$ (upper panels) and $^{40}Ar(\bar{\nu},\mu^++N)$ (lower panels) cross sections as a function of the kinetic energy of the final nucleon. Left and right panels correspond to the emission of protons and neutrons respectively. The solid histogram shows results without FSI and the dashed one the full model. Please look Ref. [@Nieves:2005rq] for further details.[]{data-label="fig:mcharged"}](charged){width="45.00000%"}
This research was supported by spanish DGI and european FEDER funds, under contracts FIS2005-01143/FIS, FIS2006-3048, FPA2007-65748, CDS2007-00042, by Junta de Castilla y León (Spain) under contracts SA016A07 and GR12 and by the EU HadronPhysics2 project. M. Valverde wishes to acknowledge a fellowship from the Japanese Society for the Promotion of Science.
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|
---
author:
- 'S. Catalán , J. Isern , E. García–Berro , I. Ribas , C. Allende Prieto'
- 'A. Z. Bonanos'
title: 'The Initial-Final Mass Relationship from white dwarfs in common proper motion pairs$^\star$'
---
Introduction
============
White dwarfs are the final remnants of low- and intermediate-mass stars. About 95% of main-sequence stars will end their evolutionary pathways as white dwarfs and, hence, the study of the white dwarf population provides details about the late stages of the life of the vast majority of stars. Since white dwarfs are long-lived objects, they also constitute useful objects to study the structure and evolution of our Galaxy (Liebert et al. 2005a; Isern et al. 2001). For instance, the initial-final mass relationship (IFMR), which connects the properties of a white dwarf with those of its main-sequence progenitor, is of paramount importance for different aspects in modern astrophysics. It is required as an input for determining the ages of globular clusters and their distances, for studying the chemical evolution of galaxies, and also to understand the properties of the Galactic population of white dwarfs. Despite its relevance, this relationship is still poorly constrained, both from the theoretical and the observational points of view.
The first attempt to empirically determine the initial-final mass relationship was undertaken by [@wei77], who also provides a recent review on this subject (Weidemann 2000). It is still not clear how this function depends on the mass and metallicity of the progenitor, its angular momentum, or the presence of a strong magnetic field. The total age of a white dwarf can be expressed as the sum of its cooling time and the main-sequence lifetime of its progenitor. The latter depends on the metallicity of the progenitor of the white dwarf, but it cannot be determined from observations of single white dwarfs. This is because white dwarfs have such strong surface gravities that gravitational settling operates very efficiently in their atmospheres, and any information about their progenitors (e.g. metallicity) is lost in the very early evolutionary stages of the cooling track. Moreover, the evolution during the AGB phase of the progenitors is essential in determining the size and composition of the atmospheres of the resulting white dwarfs, since the burning processes that take place in H and He shells determine their respective thicknesses and their detailed chemical compositions, which are crucial ingredients for determining the evolutionary cooling times.
A promising approach to circumvent the problem, and also to directly test the initial-final mass relationship, is to study white dwarfs for which external constraints are available. This is the case of white dwarfs in open and globular clusters (Ferrario et al. 2005, Dobbie et al. 2006) or in non-interacting binaries, for instance, common proper motion pairs (Wegner 1973, Oswalt et al. 1988). Focusing on the latter, it is sound to assume that the members of a common proper motion pair were born simultaneously and with the same chemical composition. Since the components are well separated (100 to 1000 AU), mass exchange between them is unlikely and it can be considered that they have evolved as isolated stars. Thus, important information of the white dwarf, such as its total age or the metallicity of the progenitor, can be inferred from the study of the companion. In particular, if the companion is an F, G or K type star the metallicity can be derived with high accuracy from detailed spectral analysis. On the other hand, the age can be obtained using different methods. In particular, we will use stellar isochrones when the star is moderately evolved, or the X-ray luminosity if the star is very close to the ZAMS.
The purpose of this work is to present our spectroscopic analysis of both members of some common proper motion pairs containing a white dwarf, and the semi-empirical intial-final mass relationship that we have derived from this study. The paper is organized as follows. In §2 we present the observations done so far and describe the data reduction. Section 3 is devoted to discuss the classification and the analysis of the observed white dwarfs, whereas in §4 we present the analysis of the companions. This is followed by §5 where we present our main results and finally in §6 we elaborate our conclusions.
Observations and data reduction
===============================
The sample of common proper motion pairs to be observed was chosen from the available literature, mainly from the papers of [@sil01] and [@weg91], and from a cross-correlation of the SIMBAD database and the Villanova White Dwarf Catalog. We selected the pairs taking into account different requirements. Firstly, the white dwarf component should be classified as a DA (i.e., with the unique presence of Balmer lines), so that the fitting procedure is sufficiently accurate to derive realistic values for the effective temperature and surface gravity. Secondly, the other component of the pair should be a star of spectral type F, G or K for an accurate determination of the metallicity, and moderately evolved or very close to the ZAMS in order to be able to estimate its age. The complete list of targets is given in Table \[tab:1\].
The observations were carried out during different campaigns between the summer of 2005 and the spring of 2007. In Table \[tab:2\] we give details of the telescope-instrument configurations employed, as well as the resolution and spectral coverage of each setup.
For the white dwarf members we performed long-slit low-resolution spectroscopic observations covering some of the main Balmer lines (from H$\beta$ to H$8$). WD0315$-$011 was kindly observed for us by T. Oswalt with the RC spectrograph at the 4 m telescope at Kitt Peak National Observatory with a resoltuion of about $1.5$ Å FWHM. We performed as many exposures as necessary to guarantee a high signal-to-noise ratio final spectrum for each object (after the corresponding reduction). Spectra of high quality are essential to derive the atmospheric parameters with accuracy. We co-added individual 1800 s exposures to minimize the effects of cosmic ray impacts on the CCD.
The white dwarf spectra were reduced using the standard procedures within the single-slit tasks in IRAF[^1]. First, the images were bias- and flatfield-corrected, and then, the spectra were extracted and wavelength calibrated using arc lamp observations. We combined multiple spectra of the same star to achieve a final spectrum of high signal-to-noise ratio (S/N $>$ 100). Before this step, we applied the heliocentric correction of each spectrum, since we were co-adding spectra secured in different days. Finally, they were normalized to the continuum.
The FGK companions were observed with echelle spectrographs, obtaining high signal-to-noise high-resolution spectra (S/N $>$ 150), which are necessary to derive the metallicity with accuracy. For the reduction of the FGK stars spectra the procedure followed was similar to the case of white dwarfs but we used the corresponding echelle tasks in IRAF. In this case, we used the task [apscatter]{} in order to model and subtract the scattered light.
[lcll]{} System & White Dwarf & Companion & Sp. Type$^1$\
& &\
G 158$-$78/77 & WD0023$-$109 & G158$-$77 & K\
LP 592$-$80/ & WD0315$-$011 & BD $-$01 469A & K1IV\
LTT 1560 & & &\
GJ 166 A/B & WD0413$-$077 & HD 26965 & K1V\
G 116$-$16/14 & WD0913$+$442 & BD $+$44 1847 & G0\
G 163$-$B9B/A & WD1043$-$034 & G163$-$B9A & F9V\
LP 378$-$537 & WD1304$+$227 & BD $+$23 2539 & K0\
G 165$-$B5B/A & WD1354$+$340 & BD $+$34 2473 & F8\
G 66$-$36/35 & WD1449$+$003 & G66$-$35 & G5V\
EGGR 113/ & WD1544$+$008 & BD $+$01 3129 & G0\
BD $+$01 3129 & & &\
GJ 599 A/B & WD1544$-$377 & HD 140901 & G6V\
GJ 620.1 B/A & WD1620$-$391 & HD 147513 & G5V\
GJ 2125 / GJ 3985 & WD1659$-$531 & HD 153580 & F6V\
G 140$-$B1B/ & WD1750$+$098 & BD $+$09 3501 & K0\
BD $+$09 3501 & & &\
G 156$-$64/65 & WD2253$-$081 & BD $-$08 5980 & G6V\
\[tab:1\]
\
[lccccc]{} Observatory & Telescope & Spectrograph & R & Spectral\
& & & & Coverage\
& & White Dwarfs & &\
McDonald & 2.7 m HJS & LCS & 1,000 & 3885$-$5267 Å\
CAHA & 3.5 m & TWIN & 1,250 & 3570$-$5750 Å\
LCO & 6.5 m Clay & LDSS3 & 1,650 & 3600$-$6000 Å\
& & Low-mass Companions & &\
McDonald & 2.7 m HJS & 2dcoudé & 60,000 & 3400$-$10900 Å\
CAHA & 2.2 m & FOCES & 47,000 & 3600$-$9400 Å\
ORM & 3.5 m TNG & SARG & 57,000 & 4960$-$10110 Å\
LCO & 6.5 m Clay & MIKE & 65,000 & 4900$-$10000 Å\
\[tab:2\]
White dwarf analysis
====================
Classification
--------------
[lclc]{} Name & This Work & Previous & Reference\
WD0023$-$109 & DA & DA & EG65, WR91, OS94, MS99\
WD0315$-$011 & DA & DA & OS94, MS99, SOW01\
WD0413$-$077 & DA & DA & EG65, FKB97, MS99, HOS02, HBB03, KNH05, HB06\
WD0913$+$442 & DA & DA & EG65, WR91, BLF95, BLR01, ZKR03, KNH05, LBH05, HB06\
WD1043$-$034 & sdB$^1$ & DA/sd & WR91\
& & DAB & OS94, MS99\
WD1304$+$227 & DA & DA & O81, OS94, MS99, SOW01\
WD1354$+$340 & DA & DA & EG67, WR91, BLF95, MS99, SOW01\
WD1449$+$003 & M & DA & O81, WR91, OS94\
& & M & FBZ05\
WD1544$+$008 & sdO$^1$ & DA/sdO & W91\
& & DA & EG65, MS99\
& & DAB & SOW01\
WD1544$-$377 & DA & DA & EG65, W73, OS94, PSH98, BLR01, KNC01, SOW01\
& & DA & HOS02, HBB03, ZKR03, KNH05, KVS07\
WD1620$-$391 & DA & DA & W73, OS94, HBS98, PSH98, SOW01, HOS02, HBB03, HB06, KVS07\
WD1659$-$531 & DA & DA & W73, OS94, PSH98, SOW01, KVS07\
WD1750$+$098 & DC & DA & WR91, SOW01\
& & DC & EG65, OS94, MS99\
WD2253$-$081 & DA & DA & OS94, BLF95, MS99, BLR01, KNC01, SOW01, KNH05\
\[tab:3\]
\
\
References. (BLF95) [@ber95a]; (BLR01) [@ber01a]; (EG65) [@egg65]; (EG67) [@egg67]; (FBZ05) [@far05a]; (FKB97) [@fin97]; (HOS02) [@hol02]; (HBB03) [@hol03]; (HB06) [@hol06]; (KNC01) [@koe01]; (KNH05) [@kar05a]; (KVS07) [@kaw07a]; (MS99) [@mcc99a]; (O81) [@osw81]; (OS94) [@osw94]; (PSH98) [@pro98]; (SOW01) [@sil01a]; (W73) [@weg73a]; (WR91) [@weg91a]; (ZKR03) [@zuc03]
After the corresponding reduction, we carried out a first inspection of the spectra. All the objects in Table \[tab:3\] were previously classified as DA white dwarfs. However, we found that four of them are not of DA type. Particularly, WD1750$+$098 turned out to be of type DC although in the most recent reference (Silvestri et al. 2001) it appears classified as a DA. We believe that WD1544$+$008 is the same star as WD1544$+$009, which was classified as a DAB white dwarf by [@sil01]. However, it was identified as a sdO star by [@weg91]. The same authors also studied WD1043$-$034 and classified it as a sdB star, although [@mcc99] considered it as a DAB white dwarf. Taking into account the different inconsistencies in the literature, we decided to reobserve these objects in order to revise their spectral classifications, if necessary. The reduced spectra of these two stars were kindly analysed by P. Bergeron, who performed the corresponding fits and derived their temperatures and surface gravities, which turned out to be too low to be white dwarfs. As can be seen in Table \[tab:3\], WD1449$+$003 is an M star. This classification was also recently indicated by [@far05]. These authors also reported that WD0913$+$442 and BD $+$44 1847 are not a physical pair according to their parallaxes. It is worth mentioning that some of the previous misclassifications are probably due to the fact that the signal-to-noise ratio of the spectra used was low.
Atmospheric parameters
----------------------
[lcc]{} Name & $T_{\rm eff}$ (K) & $\log g$ (dex)\
WD0023$+$109 &$10380\pm230$ &$7.92\pm0.08$\
WD0315$-$011 &$7520\pm260$ &$8.01\pm0.45$\
WD0413$-$077$^1$ &$16570\pm350$ &$7.86\pm0.05$\
WD0913$+$442 &$8920\pm110$ &$8.29\pm0.10$\
WD1304$+$227 &$10800\pm120$ &$8.21\pm0.05$\
WD1354$+$340 &$13650\pm420$ &$7.80\pm0.15$\
WD1544$-$377 &$10600\pm250$ &$8.29\pm0.05$\
WD1620$-$391 &$24900\pm130$ &$7.99\pm0.03$\
WD1659$-$531 &$14510\pm250$ &$8.08\pm0.03$\
WD2253$-$081 &$7220\pm140$ &$8.25\pm0.20$\
\
Before calculating the atmospheric parameters of the white dwarfs ($T_{\rm eff}$ and $\log g$) we determined the radial velocities of each star using the IRAF task [fxcor]{}. Each spectrum was cross-correlated with a reference model from a grid computed by D. Koester (private communication). The obtained radial velocities, estimated with large error bars, were generally small (ranging from 10 to 50 km/s) compared with the resolution element (300 km/s) of our observations. In only one case (WD0023$+$109), the radial velocity measured turned out to be relevant (150 km/s). However, all radial velocities were taken into account for consistency.
After this previous step, we derived the atmospheric parameters of these stars performing a fit of the observed Balmer lines to white dwarf models following the procedure described in [@ber92]. The models had been previously normalized to the continuum and convolved with a Gaussian instrumental profile with the proper FWHM in order to have the same resolution as the observed spectra. The fit of the line profiles was then carried out using the task [specfit]{} of the IRAF package, which is based on $\chi^2$ minimization with the Levenberg-Marquardt method. We used [ specfit]{} for different $\log g$ values (7.0, 7.5, 8.0, 8.5 and 9.0) with $T_{\rm eff}$ as a free parameter, obtaining different $\chi^2$ for each fit. In each case, the initial estimate for $T_{\rm eff}$ obtained from the spectral energy distribution (photometry in the $BV$ and $JHK$ bands, 2MASS) was used as a starting guess. The uncertainties in the derived $T_{\rm eff}$ were estimated from the perturbations required to increase the value of the reduced $\chi^2$ by one.
The determination of $\log g$ was performed in an analogous way but to calculate the errors we took into account the prescription of [@ber92], who derive them from the independent fits of the individual exposures for any given star (before the combination). The results are given in Table \[tab:4\]. In Fig. 1 we show the fits for some of the DA white dwarfs in our sample.
       
\[fig:fitswd\]
Some of these white dwarfs had been the subject of previous analyses which allow us to perform a comparison with our results. For instance, WD0913$+$442 was also studied by [@ber95], who obtained atmospheric parameters compatible with the ones derived here. They also studied WD1354$+$340 and WD2253$-$081, but in these cases the effective temperatures obtained are compatible with ours while the surface gravities are not, although just outside the $1\sigma$ error bar. We have obtained lower values of $\log g$ in both cases, which could be due to the different resolution of the spectra ($\sim6$ FHWM in their case). This latter object, WD2253$-$081, is of particular interest since an accurate fit of its line profiles posed many problems to previous analyses because the lines seemed to be broader than the models predicted. This led different authors to consider the possibility of this star to be a magnetic white dwarf or to have its lines rotationally broadened. Both options were considered by [@kar05], who discarded the former possibility. With the purpose of solving the fitting problem of this star, in this work we have used updated models for DA white dwarfs with effective temperatures between 6000 and 10000 K. These models were kindly provided by D. Koester, who calculated them considering collision-induced absorption due to the presence of molecular hydrogen. This effect is very significant at low temperatures and it should be taken into account for an accurate determination of the atmospheric parameters. Contrarily to the results obtained by [@kar05] we did not need to consider rotational broadening to achieve a good fit. On the other hand, the southern hemisphere targets had been also studied by different authors. Recently, [@kaw07] derived the atmospheric parameters for WD1544$-$377, WD1620$-$391 and WD1659$-$531, which are in good agreement with our results.
Masses and cooling times
------------------------
Once we have derived the $T_{\rm eff}$ and $\log g$ of each star, we can obtain its mass ($M_{\rm
WD}$) and cooling time ($t_{\rm cool}$) from appropriate cooling sequences. We have used the cooling tracks of [@sal00] — model S0 — which consider a carbon-oxygen (C/O) core white dwarf (with a higher abundance of O at the center of the core) with a thick hydrogen envelope on top of a helium buffer, $q({\rm H})=M_{\rm H}/M=10^{-4}$ and $q({\rm He})=M_{\rm He}/M=10^{-2}$. These improved cooling sequences include an accurate treatment of the crystallization process of the C/O core, including phase separation upon crystallization, together with up-to-date input physics suitable for computing white dwarf evolution. In order to check the sensitivity of our results to the adopted cooling tracks, we also used the sequences of [@fon01] with different core compositions. In a first series of calculations, C/O cores with a composition of 50/50 by mass with thick H envelopes, $q({\rm H})=10^{-4}$, on top of a He buffer, $q({\rm He})=10^{-2}$, were adopted. We refer to these models as F0. In the second series of calculations, cooling sequences with a pure C core and the same envelope characteristics — model F1 — were used. As can be seen in Table \[tab:5\], the derived masses do not change appreciably when adopting different cooling sequences. On the contrary, small differences can be noted in the cooling times obtained, depending on the evolutionary tracks used. This stems naturally from the different core compositions of the cooling sequences adopted here. As can be noted by examining Table 5, considering a C/O core with equal carbon-oxygen mass fractions with thick envelopes (model F0) is quite similar to considering a C/O core with more O concentrated in the center of the core (model S0) in terms of the cooling time. Also, and as it should be expected, we obtain larger values for the cooling times when considering the pure C core sequences (model F1), since a white dwarf with a pure C core cools slower than a white dwarf with a C/O core because of the higher heat capacity of C in comparison with that of O, implying a larger amount of energy necessary to change the temperature of the core.
Some of these white dwarfs have mass estimates from previous investigations. [@sil01] calculated masses from gravitational redshifts for WD0315$-$011, WD1354$+$340, WD1544$-$377, WD1620$-$391, WD1659$-$531 and WD2253$-$081. The results of that study are compatible with the masses derived in this work except for WD1544$-$377, whose mass is 25% smaller when calculated from its gravitational redshift. However, [@kaw07] inferred the spectroscopic mass of this star, together with those of WD1620$-$391 and WD1659$-$531, that are in good agreement with our results. WD0913$+$442 was studied by [@kar05] and [@ber01]. The former inferred the spectroscopic mass of the white dwarf and the latter used photometry and the trigonometric parallax to estimate the mass. In both cases, the results are compatible with the value derived here.
[ccccccc]{} & & &\
Name & $M_{\rm WD}$ & $t_{\rm cool}$ & $M_{\rm WD}$ & $t_{\rm cool}$ & $M_{\rm WD}$ & $t_{\rm cool}$\
& $(\rm M_{\sun})$ & (Gyr) & $(\rm M_{\sun})$ & (Gyr) & $(\rm M_{\sun})$ & (Gyr)\
WD0023$-$109 & $0.56\pm0.03$ & $0.47\pm0.03$ & $0.56\pm0.03$ & $0.50\pm0.03$ & $0.56\pm0.03$ & $0.53\pm0.03$\
WD0315$-$011 & $0.60\pm0.20$ & $1.20\pm0.56$ & $0.60\pm0.18$ & $1.28\pm0.45$ & $0.60\pm0.18$ & $1.37\pm0.42$\
WD0413$-$077 & $0.54\pm0.02$ & $0.112\pm0.008$ & $0.54\pm0.02$ & $0.11\pm0.01$ & $0.54\pm0.02$ & $0.12\pm0.01$\
WD0913$+$442 & $0.78\pm0.01$ & $1.24\pm0.05$ & $0.78\pm0.05$ & $1.24\pm0.15$ & $0.78\pm0.05$ & $1.35\pm0.12$\
WD1304$+$227 & $0.73\pm0.02$ & $0.62\pm0.03$ & $0.73\pm0.02$ & $0.68\pm0.03$ & $0.73\pm0.02$ & $0.71\pm0.03$\
WD1354$+$340 & $0.50\pm0.04$ & $0.20\pm0.02$ & $0.50\pm0.04$ & $0.19\pm0.02$ & $0.50\pm0.03$ & $0.21\pm0.02$\
WD1544$-$377 & $0.78\pm0.02$ & $0.76\pm0.05$ & $0.78\pm0.02$ & $0.81\pm0.04$ & $0.78\pm0.02$ & $0.86\pm0.05$\
WD1620$-$391 & $0.63\pm0.01$ & $0.026\pm0.001$ & $0.63\pm0.01$ & $0.022\pm0.001$ & $0.63\pm0.01$ & $0.025\pm0.001$\
WD1659$-$531 & $0.66\pm0.01$ & $0.24\pm0.01$ & $0.66\pm0.01$ & $0.25\pm0.01$ & $0.66\pm0.01$ & $0.26\pm0.01$\
WD2253$-$081 & $0.75\pm0.09$ & $2.32\pm0.72$ & $0.75\pm0.09$ & $2.20\pm0.44$ & $0.75\pm0.09$ & $2.27\pm0.53$\
\[tab:5\]
S0 [@sal00]
F0, F1 [@fon01]
Low-mass companion analysis
===========================
Determination of $T_{\rm eff}$.
-------------------------------
\[tab:ms1\]
\
\
\
We have used the available photometry — $V$ from SIMBAD and $JHK$ from 2MASS (Table \[tab:ms1\]) — to derive the effective temperatures of these stars, $T_{\rm eff}$, following the method of [@mas06]. This procedure consists on calculating synthetic photometry using the non-overshoot Kurucz atmosphere model grid (Kurucz 1979)[^2]. Then, we developed a fitting algorithm that is based on the minimization of the $\chi^2$ parameter using the Levenberg-Marquardt method. $\chi^2$ is defined from the differences between the observed and synthetic $VHJK$ magnitudes. This function depends indirectly on $T_{\rm eff}$, $\log g$, \[Fe/H\] and a magnitude difference $\Re$, which is the ratio between the synthetic (star’s surface) and the observed flux at Earth, $\Re=-2.5\log(F_{\rm
star}/F_{\rm earth})$. Tests show that the spectral energy distribution in the optical/IR for the range of temperatures that corresponds to FGK stars is only weakly dependent on gravity and metallicity, which makes it possible to derive accurate temperatures for stars with poor determinations of $\log g$ and \[Fe/H\]. Taking this into account, we assume initial values of $\log
g=4.50$ and \[Fe/H\]=0.0 to estimate the effective temperatures. We did not consider interstellar extinction corrections, since they have negligible effects considering the nearby distances of the stars under study. The results are given in Table \[tab:ms1\].
[lccccccc]{} Name & $(R-I)_K$ & $(V-R)_K$ & $(V-I)_K$ & $(R-I)_C$ & $(V-R)_C$ & $(V-I)_C$\
BD $-$01 469A & 0.375 & 0.540 & 0.915 & 0.500 & 0.626 & 1.126\
HD 26965 & 0.305 & 0.340 & 0.645 & 0.432 & 0.441 & 0.873\
\[tab:ms2\]
It can be seen from Table \[tab:ms1\] that the $JHK$ 2MASS magnitudes for BD $-$01 469A and HD 26965 are saturated. Thus, in order to derive accurate effective temperatures for these stars, we considered $RI$ photometry (Eggen 1971) available from The Lausanne Photometric Database (GCPD) (Table \[tab:ms2\]). We used the relations of [@bes79] to transform between Cousins and the Kron-Eggen system in order to obtain $(R-I)_C$ and $(V-I)_C$. Then, we consider the suitable color-temperature relations derived by [@hou00] to infer their effective temperatures.
In the case of BD $-$01 469A we obtained $T_{\rm eff}=4525$ K and $T_{\rm eff}=4425$ K, for $(V-I)_C$ and $(V-R)_C$, respectively. Using the Stromgren $b-y$ index of $0.633$ also present at the same database we obtain $T_{\rm eff}=4500$ K considering the calibration of Olsen (1984). We consider as the final result the mean value of these three temperatures, $T_{\rm eff}=4480\pm50$ K. The effective temperature obtained for BD $-$01 469A is 230 K lower than the one reported in [@mcw90], who used $(B-V)$ from the Bright Star Catalog (BSC) and the corresponding calibration of color-temperature. These authors derived the effective temperature from an extrapolation, since their calibration did not cover stars with such low temperatures. Thus, we consider that the value that we have obtained is more reliable.
Regarding HD 26965 we used also the relations of [@hou00] obtaining $T_{\rm eff}=5200$ K and $T_{\rm eff}=5345$ K, for $(V-I)_C$ and $(V-R)_C$ respectively. Besides the Eggen $RI$ photometry, the $J$ and $K$ Johnson magnitudes (Johnson et al. 1968) of HD 26965 are also available at The Lausanne Photometric Database (GCPD). These photometric data are given in Table \[tab:ms3\]. We used the relation of [@bes88] to transform $V-K$ from the Johnson to the Johnson-Glass system. Then, we used the $(V-K)$-temperature calibration of [@hou00] obtaining $T_{\rm
eff}=5135$ K. To compare this result we can use also the $(V-K)$-temperature relation from [@mas06] that gives $T_{\rm eff}=5185$ K. We deem the values derived from the $(V-K)$ color are more accurate, so, our final value should be the mean of them, $T_{\rm eff}=5160\pm35$ K. This value is in reasonable agreement with $T_{\rm eff}=5090$ K, which is the result obtained by [@ste83] using also the available (Johnson 1966) $V-R$, $V-I$, $V-J$, $V-K$, and $V-L$ colors and the [@joh66] color calibrations.
\[tab:ms3\]
References. (RAL) Ram[í]{}rez et al. 2007; (RTLA) Reddy et al. 2003; (AMMSF) Affer et al. 2005\
Determination of \[Fe/H\]
-------------------------
To derive the metallicity of the stars we fitted the observed absorption lines with synthetic spectra computed with SYNSPEC (Hubeny & Lanz 1995)[^3] and Kurucz’s model atmospheres (Kurucz 1993). For each star, we used the model corresponding to the derived $T_{\rm eff}$ and assumed a value for $\log g$. SYNSPEC is a program for calculating the spectrum emergent from a given model atmosphere. SYNSPEC was originally designed to synthesize spectra from atmospheres calculated using TLUSTY (Lanz & Hubeny 1995), but may also be used with other model atmospheres as input (e.g. LTE Kurucz’s ATLAS models, as in our case). The program is complemented by the routine ROTINS that calculates the rotational and instrumental convolutions for the net spectrum produced by SYNSPEC.
Line selection and atomic data calibration is a crucial step to derive the metallicity of a star. We selected the lines from two sources: [@red03] and [@ram07] taking into account different requirements. The suitable stellar lines should have a relatively small equivalent width, i.e., $\Delta W_\lambda <50 \;{\rm m}\AA$ approximately. We discarded also the lines which fell in the spectral gaps between the spectral orders or those that appeared asymmetric, which were assumed to be blended with unidentified lines. It is very important also to consider lines for the same species but corresponding to different transitions and ionization states, since this can provide useful cross-checks to test if the derived effective temperature is correct. This is particularly interesting when stars are cooler, since it is more difficult to derive the temperature with accuracy. We selected also some stellar lines farther in the red part of the spectrum from the linelist of [@aff05].
![Fits of the observed spectra for HD 140901. The solid line is the fit corresponding to the derived $Z$ and the dotted and dashed lines are spectra computed for $+1 \sigma$ and $-1
\sigma$ from the average.[]{data-label="fig:fitms"}](fig02.ps){width="0.9\columnwidth"}
The first step of this procedure is to calibrate the atomic data list using the Kurucz’s solar spectrum[^4] and the corresponding solar atmosphere, which has $T_{\rm eff}=5777$ K, $\log g=4.437$ and $\xi=1.5\; {\rm km\; s^{-1}}$. For each selected line we changed the oscillator strength ($\log gf$) in the Kurucz’s atomic linelist until it reproduced the observed solar spectrum. In Table \[tab:ms4\] we give the values of [@ram07], [@red03] and [@aff05], and the adopted values that we have used in our analysis. The equivalent widths of the fitted solar lines measured with the IRAF task [splot]{} are given as well. Therefore, the oscillator strengths will be fixed when fitting the spectra of the FGK stars that we have observed. After discerning which lines were suitable for the fitting procedure we selected the value of $\log g$ (3.5, 4.0 or 4.5) that gave the same abundances for different species and different ionization states. We estimated the value of the microturbulence, $\xi$, using the relationship derived by [@all04] as function of $T_{\rm eff}$ and $\log g$ — see Table \[tab:ms5\]). After obtaining the metallicity considering the proper $\log g$ and $\xi$, we recalculated the $T_{\rm eff}$ performing again the corresponding fit to synthetic photometry, which led to negligible adjustments. Another parameter that could affect the determination of metallicity is the macroturbulence. We adjusted this parameter using a rotational profile and a Gaussian broadening function independently. Both approximations led to the same metallicities. In Fig. \[fig:fitms\] we show the spectral fits for one of the companions of the DA white dwarfs (HD 140901). We have chosen to plot the fits corresponding to Fe [i]{} and Fe [ii]{}, to show how the method works for different ionization states.
Age determination
-----------------
\
\
\
\
[lccccc]{} Name & $HR$ & Count Rate & $\log(L_x)$ & Age\
& & (c/s) & & (Gyr)\
HD 26965 & $-0.28\pm0.06$ & $0.796\pm0.052$ & $28.22\pm0.12$ & $1.07\pm0.37$\
HD 140901 & $-0.73\pm0.11$ & $0.150\pm0.023$ & $28.27\pm0.31$ & $0.94\pm0.50$\
HD 147513 & $-0.25\pm0.06$ & $0.650\pm0.045$ & $28.95\pm0.14$ & $0.33\pm0.12$\
\[tab:ms6\]
![Hertzsprung-Russell diagram for the companions. The isochrones of [@sch92] for different ages (ZAMS, 2, 3 and 7 Gyr, from left to right) and solar metallicity are also plotted.[]{data-label="fig:lumteff"}](fig03.ps){width="0.9\columnwidth"}
![X-ray luminosity versus age for stars with different spectral types according to [@rib07].[]{data-label="fig:xlum"}](fig04.ps){width="0.9\columnwidth"}
For most of our stars in our sample the parallax is known (from the Hipparcos Catalogue), thus, the calculation of the luminosity, $L$, is straightforward using the apparent magnitude after estimating the bolometric magnitude, $M_{\rm bol}$. For best accuracy we have used the $K$ band magnitude and the bolometric corrections of [@mas06]. In Fig. \[fig:lumteff\] we show the Hertzsprung-Russell diagram for the FGK stars in our list with known distances. The isochrones of [@sch92] for different ages and solar metallicity have been also plotted to show at which evolutionary state these stars are. As can be seen, the isochrone fitting technique is suitable for BD $+$34 2473, HD 153580 (both F stars) and for BD $-$01 469A (K subgiant). The rest of stars are too close to the ZAMS and hence the use of isochrones does not provide accurate values for their ages. When the isochrone fitting is appropriate, we have performed an interpolation in the grid of stellar models of [@sch92] considering the derived $T_{\rm eff}$, $Z$ and $L$ to obtain the ages of these stars, i.e., the total ages of the white dwarfs in the common proper motion pairs. Our results are given in Table \[tab:ms5\].
Another age indicator which could be used is X-ray luminosity. For some of these objects there are data available from the ROSAT All-Sky Bright Source Catalogue — 1RXS (Voges 1999) — which gives the count rate (number of detected counts per second) and the hardness ratio, $HR$. The hardness ratio is defined $HR=(H-S)/(H+S)$, where $H$ and $S$ are respectively the counts recorded in the hard and soft PSPC pulse height channels. To obtain the X-ray flux of a given star, we considered the calibrations of [@sch95]. In particular, we used the conversion factor to obtain the energy flux from the measured count rate, which depends on $HR$:
$$CF=(5.30HR+8.31)\times10^{-12}\;{\rm ergs}\;{\rm cm}^{-2}\;{\rm counts}^{-1}$$
[@rib07] calculated a relationship between the age and X-ray luminosity for stars of different spectral types (Fig. \[fig:xlum\]) using both cluster data and stars belonging to wide binaries, or using kinematic criteria. In Table \[tab:ms6\] we give the ROSAT information regarding these objects, the X-ray luminosity and the ages derived for the FGK companions with X-ray emission. The errors of the ages have been calculated considering the errors in the X-ray luminosity and an assumed cosmic dispersion for each relation (8 and 20% for G and K stars, respectively). There is also ROSAT information available for HD 153580, but since it is a member of a spectroscopic binary these relations cannot be applied.
[lccccccc]{} WD & Age & $t_{\rm cool}$ & $t_{\rm MS}$ & $M_{\rm F}$ & $M_{\rm I}$ & $Z$\
& (Gyr) & (Gyr) & (Gyr) & ($\rm M_{\sun}$) & ($\rm M_{\sun}$) &\
WD0315$-$011 & $4.17^{+3.04}_{-2.05}$ & $1.20\pm0.56$ & $2.97^{+3.09}_{-2.12}$ & $0.60\pm0.20$ & $1.48^{+0.87}_{-0.28}$ & $0.016\pm0.003$\
WD0413$-$017 & $1.07\pm0.37$ & $0.112\pm0.008$ & $0.96\pm0.37$ & $0.54\pm0.02$ & $2.07^{+0.53}_{-0.27}$ & $0.008\pm0.001$\
WD1354$+$340 & $3.26^{+0.74}_{-1.46}$ & $0.20\pm0.02$ & $3.06^{+0.74}_{-1.46}$ & $0.50\pm0.04$ & $1.46^{+0.31}_{-0.09}$ & $0.015\pm0.002$\
WD1544$-$377 & $0.94\pm0.50$ & $0.76\pm0.05$ & $0.18\pm0.50$ & $0.78\pm0.02$ & $4.13^{+?}_{-1.49}$ & $0.021\pm0.003$\
WD1620$-$391 & $0.33\pm0.12$ & $0.026\pm0.001$ & $0.30\pm0.12$ & $0.63\pm0.01$ & $3.45^{+0.65}_{-0.35}$ & $0.020\pm0.003$\
WD1659$-$531 & $2.51^{+0.34}_{-0.32}$ & $0.24\pm0.01$ & $2.27^{+0.34}_{-0.32}$ & $0.66\pm0.01$ & $1.58^{+0.08}_{-0.05}$ & $0.019\pm0.004$\
\[tab:mif\]
The initial-final mass relationship
===================================
![Final masses versus initial masses for the common proper motion pairs studied here and some precise available data.[]{data-label="fig:mif"}](fig05.ps)
Once we know the total age of the white dwarfs and the metallicity of their progenitors, the initial masses can be derived considering suitable stellar models. In our case we have used the stellar tracks of [@dom99]. The initial and final masses obtained are detailed in Table \[tab:mif\]. Other parameters, such as overall ages, cooling times, main-sequence lifetimes of the progenitors and metallicities are also given. As can be noted all the total ages exceed the cooling times, as expected.
In Fig. \[fig:mif\] we represent the final masses versus the initial masses obtained for the white dwarfs in our sample for which the age and metallicity have been derived. The lines correspond to the theoretical initial-final mass relationships of [@dom99] for different metallicities. For the sake of comparison we have also included the most precise data that are currently being used to define the semi-empirical initial-final mass relationship. For the Hyades and Praesepe, we plot the results obtained by [@cla01], and some recent results from Dobbie et al. (2004, 2006). We also used the results of [@dob06] for the only known Pleiades white dwarf. In the case of Sirius, we have used the initial and final masses derived by [@lie05b].
From an inspection of Fig. \[fig:mif\] it can be noted that the observational data present large dispersion, which is higher than the uncertainties, in comparison with the theoretical initial-final mass relationships of [@dom99]. According to our results, a main-sequence star of $1.5\, \rm M_{\sun}$ with approximately solar metallicity could end up as white dwarfs with masses that differ by $\sim25\%$ (cf. WD1354$+$340 and WD1659$-$531). Moreover, two white dwarfs of nearly the same masses could come from main-sequence stars with masses different by a factor of 2 (cf. WD1620$-$391 and WD1659$-$531). Apparently, this difference is not a consequence of metallicity, since it is practically the same for these objects (Table \[tab:mif\]). However, it is also interesting to note that the influence of metallicity on the theoretical initial-final mass relationship seems to be almost negligible below $2\, \rm M_{\sun}$. Other factors, such as magnetic fields or rotation (Dom[í]{}nguez et al. 1996) should be studied in detail in order to discern their effect on this relation.
The ages of star clusters are usually calculated to a higher accuracy than in the case of the individual low-mass stars considered in this work, which should allow to obtain the initial masses with better accuracy. However, from Fig. \[fig:mif\] it can be noted that white dwarfs in clusters display a large dispersion, especially between $3$ and $4\, \rm M_{\sun}$. Thus, this scatter in the observational data seems to be a real effect, rather than a consequence of the uncertainties in the mass estimates. Hence, there is no apparent reason for which the initial-final mass relationship should be considered a single-valued function. A thorough complete comparison of our results based in common proper motion pairs with cluster data will be discussed in a forthcoming paper (Catalán et al. 2008).
One of the most important contributions of our work is the study of the range of initial masses corresponding to $1.5-2\, \rm M_{\sun}$, which was not covered by the research based on open cluster data (Ferrario et al. 2005, Dobbie et al. 2006). The recent study of [@kal07] using old open clusters has also provided some new data in the low-mass domain. It is worth to mention that 5 of the 6 white dwarfs of our final sample have masses near the typical values derived by, e.g., Kepler et al. (2007), $M\sim0.6\,\rm M_{\sun}$, which represent 90% of the white dwarfs found in the SDSS. This stems from the fact that the progenitors of white dwarfs in open clusters were usually more massive ($M>2\,\rm M_{\sun}$) since clusters are relatively young and the low-mass stars, which would produce the typical white dwarfs, are still on the main sequence. Since some of the pairs that we have studied have larger ages than the typical values for open clusters, the white dwarfs that belong to these pairs can be less massive. Thus, we consider that white dwarfs in common proper motion pairs are more representative of the Galactic white dwarf field population than white dwarfs in open clusters.
Summary and Conclusions
=======================
We have studied a sample of common proper motion pairs comprised of a white dwarf and a FGK star. We have performed high signal-to-noise low resolution spectroscopy of the white dwarf members, which led us to carry out a full analysis of their spectra and to make a re-classification when necessary. From the fit of their spectra to white dwarf models we have derived their atmospheric parameters. Then, using different cooling sequences — namely those of [@sal00] and [@fon01] — their masses and cooling times were obtained. Simultaneously, we have performed independent high resolution spectroscopic observations of their companions. Using the available photometry we have obtained their effective temperatures. Then, from a detailed analysis of their spectra and using either isochrones or X-ray luminosities, we have derived their metallicities and ages (i.e., the metallicities of the progenitors of the white dwarfs and their total ages).
These observations allowed us to obtain the initial and final masses of six white dwarfs in common proper motion pairs, four of them corresponding to initial masses below $2\,\rm M_{\sun}$, a range which has not been previously covered by the open cluster data. Our semi-empirical relation shows significant scatter, compatible with the results obtained by [@fer05] and [@dob06], which are mainly based on open cluster data. However, the dispersion of the results is higher than the error bars, which leaves some open questions that should be studied in detail (e.g., rotation or magnetic fields).
We have shown that common proper motion pairs containing white dwarfs can be useful to improve the initial-final mass relationship, since they cover a wide range of ages, masses and metallicities, and they are also representative of the disk white dwarf population. We have seen that the accuracy in the total ages depends almost exclusively on the evolutionary state of the low-mass companions. Such relative accuracy becomes poor when the star is close to the ZAMS. However, this limitation may not be critical to many common proper motion pairs. Planned deep surveys like GAIA, LSST or the Alhambra Survey will discover thousands of new white dwarfs, some of them belonging to wide binaries. In the meantime, our most immediate priority is to further extend the sample of wide binaries valid for this study. We are working in the search for more wide binaries of our interest in the NLTT catalog (Gould & Chanamé 2004) and also in the LSPM-north catalog (Lépine & Bongiorno 2007). Detailed study of the current and future common proper motion pairs of this type should help to explain the scatter in the semi-empirical initial-final mass relationship and to discern whether this is a single-valued function. If consistency between observations and theoretical calculations is found, this would have a strong impact on stellar astrophysics, since this relationship is used in many different areas, such as chemical evolution of galaxies, the determination of supernova rates or star formation and feedback processes in galaxies.
We thank D. Koester for his useful comments in the fitting procedure and for providing us with his white dwarf models. We also wish to thank P. Bergeron for kindly sharing his fitting routines that were very useful to compare with our results. Finally, we are grateful to T. Oswalt & M. Rudkin for the observations of WD0315$-$011. S. C. would like to acknowledge support from MEC through a FPU grant. C. A. P ackowledges support from NASA (NAG5-13057, NAG5-13147). This research was supported in part by the MEC grants AYA05–08013–C03–01 and 02, by the European Union FEDER funds and by the AGAUR.
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[^1]: IRAF is distributed by the National Optical Astronomy observatories, which are operated by the Association of Universitites for Research in Astronomy, Inc., under cooperative agreement with the national Science Foundation ([http://iraf.noao.edu]{}).
[^2]: [ http://kurucz.harvard.edu/grids.html]{}
[^3]: [ http://nova.astro.umd.edu/Synspec43/synspec.html]{}
[^4]: [http://kurucz.harvard.edu/sun.html]{}
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---
abstract: 'Deep learning has produced state-of-the-art results for a variety of tasks. While such approaches for supervised learning have performed well, they assume that training and testing data are drawn from the same distribution, which may not always be the case. As a complement to this challenge, unsupervised domain adaptation can handle situations where a network is trained on labeled data from a source domain and unlabeled data from a related but different target domain with the goal of performing well at test-time on the target domain. Many unsupervised deep domain adaptation approaches have thus been developed. This survey will compare these approaches by examining alternative methods, the unique and common elements, results, and theoretical insights. We follow this with a look at application areas and open research directions.'
author:
- Garrett Wilson
- 'Diane J. Cook'
bibliography:
- 'bibliography.bib'
title: A Survey of Unsupervised Deep Domain Adaptation
---
Introduction
============
Supervised learning is arguably the most prevalent type of machine learning and has enjoyed much success across diverse application areas. However, many supervised learning methods make a common assumption: the training and testing data are drawn from the same distribution. When this constraint is violated, a classifier trained on the *source domain* will likely experience a drop in performance when tested on the *target domain* due to the differences between domains [@patel2015ieee]. *Domain adaptation* refers to the goal of learning a concept from labeled data in a source domain that performs well on a different but related target domain [@pan2010tkde; @goodfellow2016deep; @ganin2016jmlr]. *Unsupervised* domain adaptation specifically addresses the situation where there is labeled source data and only unlabeled target data available for use during training [@long2015icml; @ganin2016jmlr].
Because of its ability to adapt labeled data for use in a new application, domain adaptation can reduce the need for costly labeled data in the target domain. As an example, consider the problem of semantically segmenting images. Each real image in the Cityscapes dataset required approximately 1.5 hours to annotate for semantic segmentation [@cordts2016cvpr]. In this case, human annotation time could be spared by training an image semantic segmentation model on synthetic street view images (the source domain) since these can be cheaply generated, then adapting and testing for real street view images (the target domain, here the Cityscapes dataset).
An undeniable trend in machine learning is the increased usage of deep neural networks. Deep networks have produced many state-of-the-art results for a variety of machine learning tasks [@ganin2016jmlr; @goodfellow2016deep] such as image classification, speech recognition, machine translation, and image generation [@goodfellow2016deep; @goodfellow2016survey]. When trained on large amounts of data, these many-layer neural networks can learn powerful, hierarchical representations [@sun2016; @long2015icml; @goodfellow2016deep] and can be highly scalable [@ghifary2016]. At the same time, these networks can also experience performance drops due to domain shifts [@sun2016; @ganin2015icml]. Thus, much research has gone into adapting such networks from large labeled datasets to domains where little (or possibly no) labeled training data is available (for a list, see [@dapapers]). These unsupervised deep domain adaptation approaches, which combine the benefit of deep learning with the very practical use of domain adaptation to remove the reliance on potentially costly target data labels, will be the focus of this survey.
A number of surveys have been created on the topic of domain adaptation [@margolis2011literature; @beijbom2012domain; @patel2015ieee; @csurka2017domain; @csurka2017comprehensive; @wang2018deepsurvey; @zhao2018unsupervised; @kouw2018introduction; @bungum2011survey; @chu2018survey; @sun2015multisourcesurvey] and more generally transfer learning [@pan2010tkde; @lu2015tlsurvey; @shao2015tlsurvey; @weiss2016survey; @zhang2017transfer; @tan2018survey; @cook2013survey; @taylor2009transfer; @lazaric2012rlsurvey], of which domain adaptation can be viewed as a special case [@patel2015ieee]. Previous domain adaptation surveys lack depth of coverage and comparison of unsupervised deep domain adaptation approaches. In some cases, prior surveys do not discuss domain mapping [@kouw2018introduction; @csurka2017domain; @csurka2017comprehensive], normalization statistic-based [@kouw2018introduction; @zhao2018unsupervised; @csurka2017domain; @csurka2017comprehensive], or ensemble-based [@kouw2018introduction; @zhao2018unsupervised; @csurka2017domain; @csurka2017comprehensive; @wang2018deepsurvey] methods. In other cases, they do not survey deep learning approaches [@margolis2011literature; @beijbom2012domain; @patel2015ieee]. Still others are application-centric, focusing on a single use case such as machine translation [@bungum2011survey; @chu2018survey]. One earlier survey focuses on the multi-source scenario [@sun2015multisourcesurvey], while we focus on the more prevalent single-source scenario. Transfer learning is a broader topic to cover, thus surveys provide minimal coverage and comparison of the deep learning methods that have been designed for unsupervised domain adaptation [@pan2010tkde; @lu2015tlsurvey; @shao2015tlsurvey; @weiss2016survey; @zhang2017transfer; @tan2018survey], or they focus on tasks such as activity recognition [@cook2013survey] or reinforcement learning [@taylor2009transfer; @lazaric2012rlsurvey]. The goal of this survey is to discuss, highlight unique components, and compare approaches to unsupervised deep domain adaptation.
We first provide background on where domain adaptation fits into the more general problem of transfer learning. We follow this with an overview of generative adversarial networks (GANs) to provide background for the increasingly widespread use of adversarial techniques in domain adaptation. Next, we investigate the various domain adaptation methods, the components of those methods, and the results. Then, we overview domain adaptation theory and discuss what we can learn from the theoretical results. Finally, we look at application areas and identify future research directions for domain adaptation.
Background
==========
Transfer Learning {#transferlearning}
-----------------
The focus of this survey is domain adaptation. Because domain adaptation can be viewed as a special case of transfer learning [@patel2015ieee], we first review transfer learning to highlight the role of domain adaptation within this topic. Transfer learning is defined as the learning scenario where a model is trained on a source domain or task and evaluated on a different but related target domain or task, where either the tasks or domains (or both) differ [@pan2010tkde; @dredze2010multi; @weiss2016survey; @goodfellow2016deep]. For instance, we may wish to learn a model on a handwritten digit dataset (e.g., MNIST [@lecun1998mnist]) with the goal of using it to recognize house numbers (e.g., SVHN [@netzer2011reading]). Or, we may wish to learn a model on a synthetic, cheap-to-generate traffic sign dataset [@moiseev2013evaluation] with the goal of using it to classify real traffic signs (e.g., GTSRB [@Stallkamp-IJCNN-2011]). In these examples, the source dataset used to train the model is related but different from the target dataset used to test the model – both are digits and signs respectively, but each dataset looks significantly different. When the source and target differ but are related, then transfer learning can be applied to obtain higher accuracy on the target data.
### Categorizing Methods
In a transfer learning survey paper, Pan et al. [@pan2010tkde] defined two terms to help classify various transfer learning techniques: “domain” and “task.” A domain consists of a feature space and a marginal probability distribution (i.e., the features of the data and the distribution of those features in the dataset). A task consists of a label space and an objective predictive function (i.e., the set of labels and a predictive function that is learned from the training data). Thus, a transfer learning problem might be either transferring knowledge from a source domain to a different target domain or transferring knowledge from a source task to a different target task (or a combination of the two) [@pan2010tkde; @dredze2010multi; @weiss2016survey].
By this definition, a change in domain may result from either a change in feature space or a change in the marginal probability distribution. When classifying documents using text mining, a change in the feature space may result from a change in language (e.g., English to Spanish), whereas a change in the marginal probability distribution may result from a change in document topics (e.g., computer science to English literature) [@pan2010tkde]. Similarly, a change in task may result from either a change in the label space or a change in the objective predictive function. In the case of document classification, a change in the label space may result from a change in the number of classes (e.g., from a set of 10 topic labels to a set of 100 topic labels). Similarly, a change in the objective predictive function may result from a substantial change in the distribution of the labels (e.g., the source domain has 100 instances of class A and 10,000 of class B, whereas the target has 10,000 instances of A and 100 of B) [@pan2010tkde].
To classify transfer learning algorithms based on whether the task or domain differs between source and target, Pan et al. [@pan2010tkde] introduced three terms: “inductive”, “transductive”, and “unsupervised” transfer learning. In inductive transfer learning, the target and source tasks are different, the domains may or may not differ, and some labeled target data is required. In transductive transfer learning, the tasks remain the same while the domains are different, and both labeled source data and unlabeled target data are required. Finally, in unsupervised transfer learning, the tasks differ as in the inductive case, but there is no requirement of labeled data in either the source domain or the target domain.
### Domain Adaptation
One popular type of transfer learning is *domain adaptation*, which will be the focus of our survey. Domain adaptation is a type of transductive transfer learning. Here, the target task remains the same as the source, as well as the domain feature space, but the domain marginal probability distributions differ [@pan2010tkde; @purushotham2017variational]. Only part of the domain changes since the feature space is required to remain fixed between source and target.
In addition to the previous terminology, machine learning techniques are often categorized based on whether or not labeled training data is available. Supervised learning assumes labeled data is available, semi-supervised learning uses both labeled data and unlabeled data, and unsupervised learning uses only unlabeled data. However, domain adaptation assumes data comes from both a source domain and a target domain. Thus, prepending one of these three terms to “domain adaptation” is ambiguous since it may refer to labeled data being available in the source or target domains.
Authors apply these terms in various ways to domain adaptation [@jiang2008domain; @pan2010tkde; @saito2017icml; @daume2007acl; @weiss2016survey]. In this paper, we will refer to “unsupervised” domain adaptation as the case in which both labeled source data and unlabeled target data are available, “semi-supervised” domain adaptation as the case in which labeled source data in addition to some labeled target data are available, and “supervised” domain adaptation as the case in which both labeled source and target data are available [@beijbom2012domain]. The distinction between these categories describes the target domain, but only describe situations in which labeled data is available for the source domain. These definitions are commonly used in the methods surveyed in this paper as well as others [@sun2016; @saito2017icml; @long2015icml; @ganin2016jmlr; @ghifary2016; @carlucci2017autodial].
### Related Problems
Multi-domain learning [@dredze2010multi; @joshi2012multi] and multi-task learning [@caruana1997multitask] are related to transfer learning and domain adaptation. In contrast to transfer learning, the goal of these learning approaches is obtaining high performance on all specified domains (or tasks) rather than just on a single target domain (or task) [@pan2010tkde; @yang2015iclr]. For example, often it is assumed that the training data are drawn in an independent and identically distributed (i.i.d.) fashion, which may not be the case [@joshi2012multi]. One such example is the task of developing a spam filter for users who disagree on what is considered spam. If all the users’ data are combined, the training data will be drawn from multiple domains. While each individual domain may be i.i.d., the aggregated dataset may not be. If the data is split by user, then there may be too little data to learn a model for each user. Multi-domain learning can take advantage of the entire dataset to learn individual user preferences [@dredze2010multi; @joshi2012multi]. Some researchers have developed adversarial strategies to tackle this multi-domain learning challenge [@sebag2019multi; @hassan2018unsupervised].
When working with multiple tasks, instead of training models separately for different tasks (e.g., one model for detecting shapes in an image and one model for detecting text in an image), multi-task learning will learn these separate but related tasks simultaneously so that they can mutually benefit from the training data of other tasks through a (partially) shared representation [@caruana1997multitask]. If there are both multiple tasks and domains, then these approaches can be combined into multi-domain multi-task learning, as is described by Yang et al. [@yang2015iclr].
Another related problem is domain generalization, in which a model is trained on multiple source domains with labeled data and then tested on a separate target domain that was not seen during training [@muandet2013domain]. This contrasts with domain adaptation where target examples (possibly unlabeled) are available during training. Some approaches related to those surveyed in this paper have been designed to address this situation. Examples include an adversarial method introduced by Zhao et al. [@zhao2017icml] and an autoencoder approach by Ghifary et al. [@ghifary2015iccv] discussed in Section \[domainGeneralization\].
Generative Adversarial Networks {#gan}
-------------------------------
Many deep domain adaptation methods that we will discuss in the next section incorporate adversarial training. One popular use of adversarial training is generative adversarial networks (GANs). GANs have been directly incorporated into adversarial domain mapping methods (Section \[domainMapping\]) and have inspired similar adversarial training setups in adversarial domain-invariant feature learning methods (Section \[featureLevelAdaptation\]). Thus, to provide background for these techniques, we will first discuss GANs.
In recent years there has been a large and growing interest in GANs. Pitting two well-matched neural networks against each other (hence “adversarial”), playing the roles of a data discriminator and a data generator, the pair is able to refine each player’s abilities in order to perform functions such as synthetic data generation. Goodfellow et al. [@goodfellow2014nips] proposed this technique in 2014. Since that time, hundreds of papers have been published on the topic [@ganzoo; @adversarialnetspapers]. GANs have traditionally been applied to synthetic image generation, but recently researchers have been exploring other novel use cases such as domain adaptation.
![Realistic but entirely synthetic images of human faces generated by a GAN trained on the CelebA-HQ dataset [@karras2018progressive].[]{data-label="fig:Karras"}](files/Karras_CelebA_small.png){width="0.75\linewidth"}
GANs are a type of deep generative model [@goodfellow2014nips]. For synthetic image generation, a training dataset of images must be available. Popular datasets include human faces (CelebA [@Liu_2015_ICCV]), handwritten digits (MNIST [@lecun1998mnist]), bedrooms (LSUN [@Yu2015LSUNCO]), and sets of other objects (CIFAR-10 [@krizhevsky2009learning] and ImageNet [@5206848; @ILSVRC15]). After training, the generative model will be able to generate synthetic images that resemble those in the training data. For example, a generator trained with CelebA will generate images of human faces that look realistic but are not images of real people, as shown in Figure \[fig:Karras\]. To learn to do this, GANs utilize two neural networks competing against each other [@goodfellow2014nips]. One network represents a generator. The generator accepts a noise vector as input, which contains random values drawn from some distribution such as normal or uniform. The goal of the generator network is to output a vector that is indistinguishable from the real training data. The other network represents a discriminator, which accepts as input either a real sample from the training data or a fake sample from the generator. The goal of the discriminator is to determine the probability that the input sample is real. During training, these two networks play a minimax game, where the generator tries to fool the discriminator and the discriminator attempts to not be fooled.
Using the notation from Goodfellow et al. [@goodfellow2014nips], we define a value function $V(G,D)$ employed by the minimax game between the two networks: $$\begin{aligned}
\text{min}_G \text{max}_D V(D,G) =
\mathbb{E}_{x \sim p_\text{data}(x)} \left[ \log D(x) \right]
+ \mathbb{E}_{z \sim p_z(z)} \left[ \log(1 - D(G(z))) \right] \label{GANeq}\end{aligned}$$
Here, $x \sim p_{data}(x)$ draws a sample from the real data distribution, $z \sim p_z(z)$ draws a sample from the input noise, $D(x;\theta_d)$ is the discriminator, and $G(z;\theta_g)$ is the generator. As shown in the equation, the goal is to find the parameters $\theta_d$ that maximize the log probability of correctly discriminating between real ($x$) and fake ($G(z)$) samples while at the same time finding the parameters $\theta_g$ that minimize the log probability of $1-D(G(z))$. The term $D(G(z))$ represents the probability that generated data $G(z)$ is real. If the discriminator correctly classifies a fake input then $D(G(z))=0$. Equation \[GANeq\] minimizes the quantity $1-D(G(z))$. This occurs when $D(G(z))=1$, or when the discriminator misclassifies the generator’s output as a real sample. Thus the discriminator’s mission is to learn to correctly classify the input as real or fake while the generator tries to fool the discriminator into thinking that its generated output is real. This process is illustrated in Figure \[fig:GAN\].
=\[draw, minimum size=3em\] = \[pin edge=[to-,thin,black]{}\]
\(a) [z]{}; (b) \[right of=a\] [G]{}; (c) \[right of=b\] ; (d) \[right of=c\] [D]{}; (e) \[right of=d\] ; (a) edge node (b); (b) edge node (c); (c) edge node (d); (d) edge node (e);
(c2) \[above of=c\] ; (d2) \[right of=c2\] [D]{}; (e2) \[right of=d2\] ; (c2) edge node (d2); (d2) edge node (e2); (d) edge node (d2);
### Training
In recent years there have been impressive results from GANs. At the same time, this research faces numerous challenges. Training a GAN can encounter problems such as difficulty converging, mode collapse, and vanishing gradients.
GAN training may fail to converge. Because there are two players in the GAN game, each player’s move (i.e., update its neural network via gradient descent) toward a lower loss may undo the other player’s progress toward reaching its lower loss [@goodfellow2016survey]. For example, GANs have been observed to oscillate without making progress toward an equilibrium [@goodfellow2016survey]. In general, an equilibrium to a game may not even exist (e.g., rock-paper-scissors [@arora2017icml]), but Arora et al. [@arora2017icml] show that an approximate pure equilibrium does exist for a Wasserstein training objective if the generator wins the game. However, while an approximate equilibrium does exist, that does not mean that backpropagation will find it when training the GAN [@arora2017icml].
A common type of non-convergence that GANs may suffer is *mode collapse*, where the generator only learns to generate realistic samples for a few specialized modes of the data distribution [@goodfellow2016survey]. For example, a generator may learn to only generate images of a particular type of dog when the dataset contains images of many different types of animals [@goodfellow2016survey].
Another problem is *vanishing gradients*. A solution to the minimax game is found through iterative optimization: alternating between optimizing the discriminator objective and the generator objective. When the generated samples are initially very poor, however, the discriminator will be confident in whether the generated image is real or fake. Thus, $D(G(z))$, which is the probability of the generated sample being real, will be close to zero, causing the gradient of $\log(1-D(G(z))$ to be small [@goodfellow2014nips].
Many methods have been proposed to resolve these training challenges. Even in the original GAN paper, Goodfellow et al. [@goodfellow2014nips] proposed a variation of the objective in Equation \[GANeq\], replacing minimizing $\log(1-D(\tilde{\textbf{x}}))$ with maximizing $\log(D(\tilde{\textbf{x}}))$ to reduce problems from vanishing gradients (referred to as a non-saturating GAN). Since then, there has been a large amount of work proposing improvements over the original GAN using a variety of tricks [@salimans2016nips; @szegedy2016cvpr; @pmlr-v70-odena17a; @shrivastava2017cvpr; @heusel2017nips], network architecture choices [@radford2015; @salimans2016nips; @karras2018progressive], objective modifications [@zhao2016iclr; @berthelot2017; @metz2016; @mao2017least; @nowozin2016nips; @arjovsky2017icml; @gulrajani2017nips; @kodali2017; @fedus2017many; @jolicoeur2018relativistic; @miyato2018spectral; @odena2018generator; @nguyen2017cvpr], mixtures or ensembles [@ghosh2018multi; @hoang2018mgan; @park2018megan; @khayatkhoei2018disconnected; @mordido2018dropout; @durugkar2017generative; @arora2017icml; @zhang2018generative; @tolstikhin2017adagan], maximum mean discrepancy (MMD) [@dziugaite2015training; @li2015generative; @sutherland2016generative; @li2017mmd; @binkowski2018demystifying], making a connection to reinforcement learning [@finn2016; @pfau2016], or a combination of these modifications [@miyato2018cgans; @heusel2017nips; @zhang2018self]. For a more in-depth discussion of these methods, there are a number of survey papers directed at GAN variants that include a discussion of training challenges and work [@hong2017generative; @manisha2018; @hitawala2018].
Some GANs have been specifically designed with domain adaptation in mind [@liu2016nips; @mao2018unpaired; @shrivastava2017cvpr; @bousmalis2017cvpr; @hoffman2018icml; @bousmalis2018roboticgrasping; @sankaranarayanan2018cvpr; @wang2018domain; @wei2018generative]. As a result, the above training stabilization methods can be employed [@mao2018unpaired; @choi2018cvpr; @shrivastava2017cvpr; @sankaranarayanan2018cvpr; @wang2018domain]. While these training stability methods could similarly be applied to other adversarial training approaches, they are not typically needed in the non-GAN methods surveyed here.
### Evaluation
Once successfully trained, a GAN model can be difficult to evaluate and compare with other models. Multiple approaches and measures have been introduced to evaluate GAN performance. Often researchers have evaluated their models through visual inspection [@pmlr-v80-santurkar18a] such as performing user studies where participants mark which images they think look more realistic [@salimans2016nips]. However, ideally a more automated metric could be found. Past generative models were evaluated by computing log-likelihood [@theis2016iclr], but this is not necessarily tractable in GANs [@goodfellow2016survey]. A proxy for log-likelihood is a Parzen window estimate, which was used for early GAN evaluation [@theis2016iclr; @goodfellow2014nips; @makhzani2015; @nowozin2016nips], but in high dimensions (such as images), this could be far from the actual log-likelihood and not even rank models correctly [@theis2016iclr; @grover2017flow]. Thus, there has been much work proposing various evaluation methods for GANs: methods for detecting memorization [@goodfellow2014nips; @makhzani2015; @donahue2017iclr; @theis2016iclr; @radford2015; @berthelot2017], determining diversity [@arora2018; @pmlr-v80-santurkar18a; @pmlr-v70-odena17a; @heusel2017nips], measuring realism [@salimans2016nips; @heusel2017nips; @liu2018; @binkowski2018demystifying], and approximating log-likelihood [@wu2017iclr]. Xu et al. [@xu2018empirical] and Borji [@borji2018] survey and compare many of these GAN evaluation methods.
These techniques can be used for evaluating domain adaptation methods used for image translation (a form of image generation but conditioned on an input image) from one domain to another [@yoo2016pixel; @zhu2017iccv; @yi2017iccv; @choi2018cvpr; @royer2017xgan; @benaim2017nips]. However, many domain adaptation methods (even those that are adversarial such as those using GANs) are not used for generation but rather for tasks with more easily-defined loss functions, making these techniques largely not needed for adversarial domain adaptation methods. For example, accuracy [@liu2016nips; @ganin2016jmlr; @tzeng2017cvpr; @bousmalis2016nips; @bousmalis2017cvpr; @hoffman2018icml; @choi2018cvpr; @benaim2017nips; @fu2018geometry] or AUC scores [@purushotham2017variational] can be used to evaluate classification, intersection over union or pixel accuracy can be used to evaluate image segmentation [@hoffman2018icml; @benaim2017nips; @fu2018geometry; @li2018semantic; @perone2018unsupervised], and absolute difference can be used to evaluate regression [@shrivastava2017cvpr].
Methods
=======
In recent years, numerous new unsupervised domain adaptation methods have been proposed, with a growing emphasis on neural network-based approaches. Distinct lines of research have emerged. These include aligning the source domain and target domain distributions, mapping between domains, separating normalization statistics, designing ensemble-based methods, or focusing on making the model target discriminative by moving the decision boundary into regions of lower data density. In addition, others have explored combinations of these approaches. We will describe each of these categories together with recent methods that fall into these categories.
In this survey, we will focus on domain adaptation consisting of one source and one target domain, as is most commonly studied. Another case is multi-source domain adaptation, where there are multiple source domains but still only one target domain. Sun et al. [@sun2015multisourcesurvey] survey multi-source domain adaptation, and since then a number of other methods [@guo2018multi; @hoffman2018multisource; @zhao2018multisource; @peng2018moment; @carlucci2018agnostic; @mancini2018cvpr] have been developed for this case. It is also possible to perform multi-target domain adaptation [@gholami2018unsupervised], though this case is even more rarely studied.
Domain-Invariant Feature Learning {#domainInvariance}
---------------------------------
Most recent domain adaptation methods align source and target domains by creating a domain-invariant feature representation, typically in the form of a feature extractor neural network. A feature representation is domain-invariant if the features follow the same distribution regardless of whether the input data is from the source or target domain [@zhao2019learning]. If a classifier can be trained to perform well on the source data using domain-invariant features, then the classifier may generalize well to the target domain since the features of the target data match those on which the classifier was trained. However, these methods assume that such a feature representation exists and the marginal label distributions do not differ significantly (Section \[theory\]).
The general training and testing setup of these methods is illustrated in Figure \[fig:alignment\]. Methods differ in how they align the domains (the Alignment Component in the figure). Some minimize divergence, some perform reconstruction, and some employ adversarial training. In addition, they differ in weight sharing choices, which will be discussed in Section \[weightSharing\]. We discuss the various alignment methods below.
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### Divergence
One method of aligning distributions is through minimizing a divergence that measures the distance between the distributions. Four choices used in various domain adaptation approaches are maximum mean discrepancy, correlation alignment, contrastive domain discrepancy, and the Wasserstein metric.
Maximum mean discrepancy (MMD) [@gretton2007kernel; @gretton2012kernel] is a two-sample statistical test of the hypothesis that two distributions are equal based on observed samples from the two distributions. The test is computed from the difference between the mean values of a smooth function on the two domains’ samples. If the means are different, then the samples are likely not from the same distribution. The smooth functions chosen for MMD are unit balls in characteristic reproducing kernel Hilbert spaces (RKHS) since it can be proven that the population MMD is zero if and only if the two distributions are equal [@gretton2012kernel].
To use MMD for domain adaptation, the alignment component can be another classifier similar to the task classifier. MMD can then be computed and minimized between the outputs of these classifiers’ corresponding layers (a slightly different setup than that in Figure \[fig:alignment\]). Rozantsev et al. [@rozantsev2018ieee] use MMD, Long et al. [@long2015icml] use a multiple kernel variant of MMD (MK-MMD), and later Long et al. [@long2015icml] develop joint MMD (JMMD) [@long2017jmmd]. Bousmalis et al. [@bousmalis2016nips] also tried MMD but found using an adversarial objective performed better in their experiments.
Correlation alignment (CORAL) [@sun2016aaai] is similar to MMD with a polynomial kernel, computed from the distance between second-order statistics (covariances) of the source and target features. For domain adaptation, the alignment component consists of computing the CORAL loss between the two feature extractors’ outputs (in order to minimize the distance). A variety of distances have been used: Sun et al. [@sun2016] use a squared matrix Frobenius norm in Deep CORAL, Zhang et al. [@zhang2018mca] use a Euclidean distance in mapped correlation alignment (MCA), others have used log-Euclidean distances in LogCORAL [@wang2017iccv] and Log D-CORAL[@morerio2017correlation], and Morerio et al. [@morerio2018minimalentropy] use geodesic distances. Zhang et al. [@zhang2018aligning] generalize correlation alignment to possibly infinite-dimensional covariance matrices in RKHS.
Contrastive domain discrepancy (CCD) [@kang2019contrastive] is based on MMD but looks at the conditional distributions in order to incorporate label information (unlike CORAL or ordinary MMD). When minimizing CCD, intra-class discrepancy is minimized while inter-class margin is maximized. This has the problem of requiring target labels though, so Kang et al. [@kang2019contrastive] propose contrastive adaptation networks (CAN) that minimize cross-entropy loss on the labeled target data while alternating between estimating labels for target samples (via clustering) with adapting the feature extractor with the now-computable CCD (using the clusters). This approach outperforms the other methods on the Office dataset as shown in Table \[comparePerformance2\].
A problem known as “optimal transport” was originally proposed for studying resource allocation such as finding an optimal way to move material from mines to factories [@monge1781memoire; @redko2017theoretical], but it can also be used to measure the distances between distributions. If the cost of moving each point is a norm (e.g., Euclidean), then the solution to a discrete optimal transport problem can be viewed as a distance: the Wasserstein distance [@damodaran2018deepjdot] (also known as the earth mover’s distance). To align feature and label distributions with this distance, Courty et al. [@courty2017nips] propose joint distribution optimal transport (JDOT). To incorporate this into a neural network, Damodaran et al. [@damodaran2018deepjdot] propose DeepJDOT.
### Reconstruction
Rather than minimizing a divergence, Ghifary et al. [@ghifary2016] and Bousmalis et al. [@bousmalis2016nips] hypothesize that alignment can be accomplished by learning a representation that both classifies the labeled source domain data well and can be used to reconstruct either the target domain data (Ghifary et al.) or both the source and target domain data (Bousmalis et al.). The alignment component in these setups is a reconstruction network – the opposite of the feature extractor network – that takes the feature extractor output and recreates the feature extractor’s input (in this case, an image). Ghifary et al. [@ghifary2016] propose deep reconstruction-classification networks (DRCN), using a pair-wise squared reconstruction loss. Bousmalis et al. [@bousmalis2016nips] propose domain separation networks (DSN), using a scale-invariant mean squared error reconstruction loss.
### Adversarial {#featureLevelAdaptation}
Several varieties of feature-level adversarial domain adaptation methods have been introduced in the literature. In most the alignment component consists of a domain classifier. In one paper this component is instead represented by a network learning an approximate Wasserstein distance, and in another paper the component is a GAN.
A domain classifier is a classifier that outputs whether the feature representation was generated from source or target data. Recall that GANs include a discriminator that tries to accurately predict whether a sample is from the real data distribution or from the generator. In other words, the discriminator differentiates between two distributions, one real and one fake. A discriminator could similarly be designed to differentiate two distributions which instead represent a source distribution and a target distribution, as is done with a domain classifier. Note though that an adversarial domain classifier is used for adaptation, whereas a GAN is used for data generation. The domain classifier is trained to correctly classify the domain (source or target). In this scenario, the feature extractor is trained such that the domain classifier is unable to classify from which domain the feature representation originated. This is a type of zero-sum two-player game [@zhao2019learning] as in a GAN (Section \[gan\]). Typically, these networks are adversarially trained by alternating between these two steps. The feature extractor can be trained to make the domain classifier perform poorly by negating the gradient from the domain classifier with a *gradient reversal layer* [@ganin2015icml] when performing back propagation to update the feature extractor weights (e.g., in DANN [@ajakan2014domain; @ganin2015icml; @ganin2016jmlr] and VRADA [@purushotham2017variational]), maximally confusing the domain classifier (when it outputs a uniform distribution over binary labels [@tzeng2015iccv]), or inverting the labels (in ADDA [@tzeng2017cvpr]). The domain classifier may also be conditioned on the task classifier predictions when adapting between multimodal distributions [@long2018nips].
Shen et al. [@shen2018wasserstein] created WDGRL, a modification of DANN, by replacing the domain classifier with a network that learns an approximate Wasserstein distance. This distance is then minimized between source and target domains, which they found to yield an improvement. This method is similar to the divergence methods except here the divergence is learned with a network rather than computed based on statistics (e.g., using mean in MMD or covariance in CORAL). This method outperforms the other methods on the Amazon review dataset as shown in Table \[compareSentimentPerformance\].
Sankaranarayanan et al. [@sankaranarayanan2018cvpr] propose Generate to Adapt that uses a GAN as the alignment component. The feature extractor output is both fed to a classifier trained to predict the label (if the input is from the source domain) and also to a GAN trained to generate source-like images (regardless of if the input is source or target). For training stability, they use an AC-GAN [@pmlr-v70-odena17a]. They note one downside of using a GAN for adaptation is that it requires a large training dataset, but a common strategy is to use a pretrained network on a large dataset such as ImageNet. Using this pretraining, even on small datasets (e.g., Office) where the generated images are poor, the network still learns adaptation satisfactorily. Sankaranarayanan et al. [@sankaranarayanan2018cvprsemantic] similarly develop a similar approach for semantic segmentation.
Domain Mapping {#domainMapping}
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\[pixelLevelAdaptation\] An alternative to creating a domain-invariant feature representation is mapping from one domain to another. The mapping is typically created adversarially and at the pixel level (i.e., pixel-level adversarial domain adaptation), but not always, as discussed at the end of this section. This mapping can be accomplished with a conditional GAN. The generator performs adaptation at the pixel level by translating a source input image to an image that closely resembles the target distribution. For example, the GAN could change from a synthetic vehicle driving image to one that looks realistic as shown in Figure \[fig:driving\] [@yoo2016pixel; @zhu2017iccv; @royer2017xgan; @choi2018cvpr; @hoffman2018icml]. A classifier can then be trained on the source data mapped to the target domain using the known source labels [@shrivastava2017cvpr] or jointly trained with the GAN [@bousmalis2017cvpr; @hoffman2018icml]. We will first discuss how a conditional GAN works followed by the ways it can be employed for domain adaptation.
![Synthetic vehicle driving image (left) adapted to look realistic (right) [@hoffman2018icml].[]{data-label="fig:driving"}](files/synthetic_gta5.png "fig:"){width="0.4\linewidth"} ![Synthetic vehicle driving image (left) adapted to look realistic (right) [@hoffman2018icml].[]{data-label="fig:driving"}](files/real_cityscapes.png "fig:"){width="0.4\linewidth"}
### Conditional GAN for Image-to-Image Translation
The original formulation of a GAN was unconditional, where a GAN only accepted a noise vector as input. Conditional GANs, on the other hand, accept as input other information such as a class label, image, or other data [@goodfellow2014nips; @gauthier2014conditional; @mirza2014conditional; @denton2015deep]. In the case of image generation, this means that a particular type of image to generate can be specified. One such example is to generate an image of a particular class within an image dataset such as “cat” rather than a random object from the dataset. Another example is conditioning on an input image such as in Figure \[fig:driving\], mapping an input driving image from one domain (synthetic) to an output image in another domain (realistic). Other uses include: transferring style (e.g., make a photo look like a Van Gogh painting) [@zhu2017iccv; @yi2017iccv; @kim2017discogan], colorizing images [@isola2017cvpr], generating satellite images from Google Maps data (or vice versa) [@isola2017cvpr; @zhu2017iccv; @yi2017iccv], generating images of clothing from images of people wearing the clothing [@yoo2016pixel], generating cartoon faces from real faces [@taigman2016dtn; @royer2017xgan], converting labels to photos (e.g., semantic segmentation output to a photo) [@isola2017cvpr; @zhu2017iccv; @yi2017iccv], learning disentangled representations [@chen2016nips], improving GAN training stability [@pmlr-v70-odena17a], and domain adaptation, which will be discussed in Section \[i2iforda\].
GANs conditioned on an input image can be used to perform image-to-image translation. These networks can be trained with varying levels of supervision: the dataset may contain corresponding images in the domains (supervised [@yoo2016pixel; @isola2017cvpr]), only a few corresponding images (semi-supervised [@gan2017triangle]), or no corresponding images (unsupervised [@zhu2017iccv; @yi2017iccv; @kim2017discogan]). A popular and general-purpose supervised method is pix2pix, developed by Isola et al. [@isola2017cvpr]. A commonly used unsupervised method is CycleGAN [@zhu2017iccv], which is based on pix2pix, or methods similar to CycleGAN including DualGAN [@yi2017iccv] and DiscoGAN [@kim2017discogan].
Numerous modifications to these approaches have been proposed: one that is multimodal is MUNIT, a multimodal unsupervised image-to-image translator [@huang2018multimodal]. By assuming a decomposition into style (domain-specific) and content (domain-invariant) codes, MUNIT can generate diverse outputs for a given input image (e.g., multiple possible output images corresponding to the same input image). A modification to CycleGAN explored by Li et al. [@li2018twin] uses separate batch normalization for each domain (an idea similar to AdaBN discussed in Section \[normalizationStats\]). Mejjati et al. [@mejjati2018nips] and Chen et al. [@chen2018eccv] improve results with attention, learning which areas of the images on which to focus. While CycleGAN and similar approaches use two generators, one for each mapping direction, Benaim et al. [@benaim2017nips] developed a method for one-sided mapping that maintains distances between pairs of samples when mapped from the source to the target domain rather than (or in addition to) using a cycle consistency loss, and Fu et al. [@fu2018geometry] developed an alternative one-sided mapping using a geometric constraint (e.g., vertical flipping or 90 degree rotation). Royer et al. [@royer2017xgan] propose XGAN, a dual adversarial autoencoder capable of handling large domain shifts, where possibly an image in the source domain may correspond to multiple images in the target domain or vice versa. They tested mapping human faces to cartoon faces, which was a shift larger than CycleGAN could adequately handle. Choi et al. [@choi2018cvpr] propose StarGAN, a method for handling multiple domains with a single GAN. Approaches like CycleGAN need a separate generator (or two, one for each direction) for each pair of domains, which is not a scalable solution to many domains. StarGAN, on the other hand, only needs a single generator. This has the added benefit of allowing the generator to learn using all the available data rather than only the data in a specific pair of domains. During training they randomly pick a target domain at each iteration so the generator learns to generate images in all the domains. Anoosheh et al. [@anoosheh2018cvpr] propose an approach designed for the same purpose as StarGAN but using one generator per domain.
### Image-to-Image Translation for Domain Adaptation {#i2iforda}
While the above approaches map images from one domain to another without the explicit purpose of performing domain adaptation, they can also be used for domain adaptation. For example, the original CycleGAN paper was application agnostic, but others have experimented with applying CycleGAN to domain adaptation [@hoffman2018icml; @benaim2017nips; @fu2018geometry]. It is important to note though that these image-to-image translation approaches assume that the domain differences are primarily low-level [@bousmalis2017cvpr; @bousmalis2018roboticgrasping; @tzeng2017cvpr].
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If unsupervised domain adaptation is performed for classification, adaptation can be accomplished by training an image-to-image translation GAN to map data from source to target, training a classifier on the mapped source images with known labels, and then subsequently testing by feeding unlabeled target through this target-domain classifier [@shrivastava2017cvpr; @bousmalis2018roboticgrasping; @li2018semantic], as done in SimGAN [@shrivastava2017cvpr] and illustrated in Figure \[fig:mapping1\]. Alternatively, rather than learning a mapping from source to target, the opposite could be done: learn a mapping from target to source, train a classifier on the source images with known labels, and test by feeding target images to the image-to-image translation model (to make them look like source images) followed by the source-domain classifier [@chen2018seuda], as illustrated in Figure \[fig:mapping2\].
In either of these approaches, if the mapping and the classification models are learned independently, the class assignments may not be preserved. For instance, class 1 may end up being “renamed” to class 2 after the mapping since the mapping was learned ignoring the class labels. This issue can be resolved by incorporating a semantic consistency loss (see Section \[losses\]) and training the mapping and classification models jointly [@bousmalis2016nips; @hoffman2018icml], as done in PixelDA [@bousmalis2017cvpr].
If there is a way to perform hyperparameter tuning, a third option is possible (combination of Figure \[fig:mapping1\] and \[fig:mapping2\]): train a target-domain classifier on the source-to-target GAN (for which the GAN is not used during testing) and a source-domain classifier on the target-to-source GAN (for which the GAN is used during testing). The algorithm may then output a linear combination of the prediction results from the two classifiers [@russo2018cvpr]. While this approach does improve results, it requires a method of hyperparameter training (see Section \[hyperparameterTuningExisting\]).
All of the above approaches perform pixel-level mapping. An alternative approach is to perform feature-level mapping. Hong et al. [@hong2018cvpr] use a conditional GAN to learn to make the source features look more like the target features (a distinctly different idea than making the features domain invariant, which was discussed in Section \[domainInvariance\]). They found this particularly helpful for structured domain adaptation (e.g., semantic segmentation, in their case).
Up to this point, these domain mapping methods have used image-to-image translation to map images (or in one case features) from one domain to another and thereby improve domain adaptation performance. Another line of research using pixel-level image generation for domain adaptation is to use a GAN to generate corresponding images in multiple domains and then employ all but the last layer of the discriminator as a feature extractor for a classifier [@liu2016nips; @mao2018unpaired]. Liu et al. [@liu2016nips] train a pair of GANs called CoGAN on two domains of images. Mao et al. [@mao2018unpaired] propose RegCGAN using only one generator and discriminator but including a domain label prepended to the input noise vector.
Normalization Statistics {#normalizationStats}
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Normalization layers such as batch norm [@ioffe2015batchnorm] are used in most neural networks [@santurkar2018nips]. These have benefits including allowing for higher learning rates and thus faster training [@ioffe2015batchnorm], reducing initialization sensitivity [@ioffe2015batchnorm], smoothing the optimization landscape and making the gradients more Lipschitz [@santurkar2018nips], and allowing for deeper networks to converge [@wu2018groupnorm; @goodfellow2016deep]. Each batch norm layer normalizes its input to have zero mean and unit variance. At test time, running averages of the batch norm parameters can be used. Alternatives have been developed including instance norm allowing use in recurrent neural networks [@ba2016layer] and group norm removing the dependence on batch size [@wu2018groupnorm]. However, none of these normalization techniques were developed with domain adaptation in mind. In the case of domain adaptation, the normalization statistics for each domain likely differ. Another line of domain adaptation research involves using per-domain batch normalization statistics.
Li et al. [@li2018] assume that the neural net layer weights learn task knowledge and the batch norm statistics learn domain knowledge. If this is the case, then domain adaptation can be performed by modulating all the batch norm layers’ statistics from the source to target domain, a technique they call AdaBN. This has the benefit of being simple, parameter free, and complementary to other adaptation methods.
Carlucci et al. [@carlucci2017autodial] propose AutoDIAL, a generalization of AdaBN. In AdaBN, the target data is not used to learn the network weights but only for adjusting the batch norm statistics. AutoDIAL can utilize the target data for learning the network weights by coupling network parameters between source and target domains. They do this through adding domain alignment layers (DA-layers) that differ for source and target input data before each of the batch norm layers. Generally, batch norm computes a moving average of the statistics on a batch of the layer’s input data. However, in AutoDIAL, source and target input data to DA-layers are mixed by a learnable amount before feeding this to batch norm (meaning that the batch norm statistics are now computed over some source and some target data rather than just source data or just target data). This allows the network to automatically learn how much alignment is needed at various points in the network.
Ensemble Methods
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Given a base model such as a neural network or decision tree, an ensemble consisting of multiple models can often outperform a single model by averaging together the models’ outputs (e.g., regression) or taking a vote (e.g., classification) [@daho2014randomforest; @goodfellow2016deep]. This is because if the models are diverse then each individual model will likely make different mistakes [@goodfellow2016deep]. However, this performance gain corresponds with an increase in computation cost due to the large number of models to evaluate for each ensemble prediction, making ensembles common for some use cases such as competitions but uncommon when comparing models [@goodfellow2016deep]. Despite the incurred cost, several ensemble-based methods have been developed for domain adaptation either using the ensemble predictions to guide learning or using the ensemble to measure prediction confidence for pseudo-labeling target data.
### Self-Ensembling
An alternative to using multiple instances of a base model as the ensemble is using only a single model but “evaluating” (via a history or average) the models in the ensemble at multiple points in time during training – a technique called *self-ensembling*. This can be done by averaging over past predictions for each example (by recording previous predictions) [@laine2017iclr] or past network weights (by maintaining a running average) [@tarvainen2017nips]. Since an ensemble requires diverse models, these self-ensembling approaches require high stochasticity in the networks, which is provided by extensive data augmentation, varying the augmentation parameters, and including dropout. These methods were originally developed for semi-supervised learning.
French et al. [@french2018iclr] modify and extend these prior self-ensembling methods for unsupervised domain adaptation. They use two networks: a student network and a teacher network. Input images are fed first to stochastic data augmentation (Gaussian noise, translations, horizontal flips, affine transforms, etc.) before being input to both networks. Because the method is stochastic, the augmented images fed to the networks will differ. The student network is trained with gradient descent while the teacher network weights are an exponential moving average (EMA) of the student network’s weights. This method outperforms the other methods on the datasets in Table \[comparePerformance1\]. Athiwaratkun et al. [@athiwaratkun2018there] show that in at least one experiment stochastic weight averaging [@izmailov2018averaging] can further improve these results.
### Pseudo-Labeling
Rather than voting or averaging the outputs of the models in an ensemble, the individual model predictions could be compared to determine the ensemble’s confidence in that prediction. The more models in the ensemble that agree, the higher the ensemble’s confidence in that prediction. In addition, if performing classification on a particular example, an individual model’s confidence can be determined by looking at the last layer’s softmax distribution: uniform indicates uncertainty whereas one class’s probability much higher than the rest indicates higher confidence. Applying this to domain adaptation, a diverse ensemble trained on source data may be used to label target data. Then, if the ensemble is highly confident, those now-labeled target examples can be used to train a classifier for target data.
This is the method Saito et al. [@saito2017icml] developed called asymmetric tri-training (ATT). Two networks sharing a feature extractor are trained on the labeled source data (i.e., the ensemble in this case is of size two). Those two networks then predict the labels for the unlabeled target data, and if the two agree on the label and have high enough confidence on a particular instance, then the predicted label for that example is assumed to be the true label. After the target data is labeled by the first two networks, the third network (also sharing the same feature extractor) can be trained using the assumed-true labels (pseudo-labels). Diversity in the ensemble is handled with an additional loss (see Section \[losses\]).
Instead of using an ensemble, Zou et al. [@zou2018eccv] rely on just the softmax distribution for the confidence measure. When working with semantic segmentation, they found relying on the prediction confidence for pseudo-labeling results in transferring primarily easy classes while ignoring harder classes. Thus, they additionally propose adding a class-wise weighting term when pseudo-labeling to normalize the class-wise confidence levels and thus balance out the class distribution.
Target Discriminative Methods {#clusterAssumption}
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One assumption that has led to successes in semi-supervised learning algorithms is the *cluster assumption* [@chapelle2005semi]: that data points are distributed in separate clusters and the samples in each cluster have a common label [@shu2018vada]. If this is the case, then decision boundaries should lie in low density regions (i.e., should not pass through regions where there are many data points) [@chapelle2005semi]. A variety of domain adaptation methods have been explored to move decision boundaries into density regions of lower density. These have typically been trained adversarially.
Shu et al. [@shu2018vada] in virtual adversarial domain adaptation (VADA) and Kumar et al. [@kumar2018nips] in co-regularized alignment (Co-DA) both use a combination of variational adversarial training (VAT) developed by Miyato et al. [@miyato2018virtual] and conditional entropy loss. They are used in combination because VAT without the entropy loss may result in overfitting to the unlabeled data points [@kumar2018nips] and the entropy loss without VAT may result in the network not being locally-Lipschitz and thus not resulting in moving the decision boundary away from the data points [@shu2018vada]. Shu et al. [@shu2018vada] additionally propose a decision-boundary iterative refinement step with a teacher (DIRT-T) for use after training to further refine the decision boundaries on the target data, allowing for a slight improvement over VADA. An entropy loss was also used in AutoDIAL [@carlucci2017autodial] but without VAT.
In generative adversarial guided learning (GAGL), Wei et al. [@wei2018generative] propose to let a GAN move decision boundaries into lower-density regions. Using domain alignment methods that learn domain-invariant features like DANN (Section \[domainInvariance\]), typically the data fed to the feature extractor is either source or target data. However, Wei et al. propose to alternate this with feeding generated (fake) images and appending a “fake” label to the task classifier, thus repurposing the task classifier as a GAN discriminator. They found this to have the effect of moving the decision boundaries in the target domain into areas of lower density with a GAN, promoting target-discriminative features as a result.
Saito et al. [@saito2018adversarial] propose adversarial dropout regularization. Since dropout is stochastic, when they create two instances of the task classifier containing dropout, the resulting networks may produce different predictions. The difference between these predictions can be viewed as a discriminator. Using this discriminator to adversarially train the feature extractor has the effect of producing target discriminative features.
Combinations
------------
In recent work, researchers have proposed various combinations of the above methods. Domain mapping has been combined with domain-invariant feature learning methods either trained separately (in GraspGAN [@bousmalis2018roboticgrasping]) or jointly (in CyCADA [@hoffman2018icml]). Following AdaBN, many researchers started employing domain-specific batch normalization [@bousmalis2018roboticgrasping; @french2018iclr; @li2018twin; @kumar2018nips; @kang2019contrastive]. Kumar et al. [@kumar2018nips] propose co-regularized alignment (Co-DA), an approach in which two separate adversarial domain-invariant feature networks are learned with different feature spaces, drawing on ensemble-based methods. Kang et al. [@kang2018eccv] combine domain mapping with aligning the models’ attention by minimizing an attention-based discrepancy. Deng et al. [@deng2019cluster] combine target discriminative methods with self-ensembling. Lee et al. [@lee2019sliced] combine target discriminative methods and domain-invariant feature learning with a sliced Wasserstein metric.
Multi-adversarial domain adaptation (MADA) [@pei2018multi] combines adversarial domain-invariant feature learning with ensemble methods for the purpose of better handling multi-modal data. This is accomplished by incorporating a separate discriminator for each class and using the task classifier’s softmax probability to weight the loss from each discriminator for unlabeled target samples.
Saito et al. [@saito2018cvpr] combine elements of adversarial domain-invariant feature learning, ensemble methods, and target discriminative features in their maximum classifier discrepancy (MCD) method. They propose using a shared feature extractor followed by an ensemble (of size two) of task-specific classifiers, where the discrepancy between predictions measures how far outside the support of the source domain the target samples lie. The discriminator in this setup is the combination of the two classifiers. The feature extractor is trained to minimize the discrepancy (i.e., fool the classifiers that the samples are from the source domain) while the classifiers are trained to maximize the discrepancy on the target samples.
Components
==========
Table \[compare\] summarizes the neural network-based domain adaptation methods we discuss showing components each method uses including what type of adaptation, which loss functions, whether the method uses a generator, and which weights are shared. Below we discuss each of these aspects followed by how the networks are trained, what types of networks can be used, multi-level adaptation techniques, and how to tune the hyperparameters of these methods.
Losses
------
### Distance
Distance functions play a variety of roles in domain adaptation losses. A distance loss can be used to align two distributions by minimizing a distance function (e.g., MMD) as explained in Section \[domainInvariance\]. If using an ensemble, minimizing a distance function can align the outputs of the ensemble’s models: an L1 loss of the difference in predicted target class probabilities from two networks in Co-DA [@kumar2018nips] or a squared difference between the predictions of the student and teacher networks in self-ensembling [@french2018iclr]. (Note the squared difference loss is confidence thresholded, i.e., if the max predicted output is below a certain threshold then the squared difference loss is set to zero.)
Some of the described methods have been altered replacing the task loss with one of similarity. Laradji et al. [@laradji2018m] propose M-ADDA, a metric-learning modification to ADDA but with the goal of maximizing the margin between clusters of data points’ embeddings. Based on DANN, Pinheiro [@pinheiro2018cvpr] proposes SimNet, classifying based on how close an embedding is to the embeddings of a random subset of source images for each class. Hsu et al. [@hsu2018learning] propose $\text{CCN}^{++}$ incorporating a pairwise similarity network (trained with the same class is similar and different classes are dissimilar).
### Promote Differences
Methods that rely on multiple networks learning different features (such as to make an ensemble diverse) do so by promoting differences between the networks. Saito et al. [@saito2017icml] train the two classifiers labeling unlabeled data to use different features by adding a norm of the product of the two classifiers’ weights. Bousmalis et al. [@bousmalis2016nips] promote different features between two private feature extractors with a soft subspace orthogonality constraint, which is similarly used by Liu et al. [@liu2017adversarial] for text classification. Kumar et al. [@kumar2018nips] train the feature extractors to be different by pushing minibatch means apart. Saito et al. [@saito2018cvpr] maximize the discrepancy between two classifiers using a fixed, shared feature extractor to promote using different features.
### Cycle Consistency / Reconstruction
A cycle consistency loss or reconstruction loss is commonly used in domain mapping methods to avoid requiring a dataset of corresponding images to be available in both domains. This is how CycleGAN [@zhu2017iccv], DualGAN [@yi2017iccv], and DiscoGAN [@kim2017discogan] can be unsupervised. This means that after translating an image from one domain (e.g., horses) to another (e.g., zebras), the new image can be translated back to reconstruct the original image, as illustrated in Figure \[fig:CycleConsistent\]. Some variants of this have been proposed such as an L1 loss with a transformation function (e.g., identity, image derivatives, mean of color channels) [@shrivastava2017cvpr], a feature-level cycle-consistency loss (mapping from source to embedding to target then back to embedding resulting in the same embeddings) [@royer2017xgan], or using the loss in one [@choi2018cvpr] or both directions [@royer2017xgan; @hoffman2018icml]. Sener et al. [@sener2016nips] enforce cycle consistency in their $k$-nearest neighbors ($k$-NN) approach by requiring the distance between any source and target point labeled the same to be less than the distance between any source and target point labeled differently and derive a rule they can solve with stochastic gradient descent.
### Semantic Consistency
A semantic consistency loss can be used to preserve class assignments as illustrated in Figure \[fig:SemanticConsistency\] (a segmentation example). The semantic consistency loss requires that a classifier output (or semantic segmentation labeling) from the original source image is the same as the same classifier’s output on the pixel-level mapped target output.
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### Task
Nearly all of the domain adaptation methods include some form of task loss that helps the network learn to perform the desired task. For example, for classification, the goal is to output the ground truth source label, or for semantic segmentation, to label each pixel with the correct ground truth source label. The task loss used is generally a cross-entropy loss, or more specifically the negative log likelihood of a softmax distribution [@goodfellow2016deep] when using a softmax output layer. The exceptions not including a task loss are SimNet [@pinheiro2018cvpr] that classify based on distance to prototypes of each class, the work by Sener et al. [@sener2016nips] that uses $k$ nearest neighbors, and AdaBN [@li2018] that only adjusts the batch norm layers to the target domain. In addition, the image-to-image translation methods are application agnostic unless trained jointly for domain adaptation.
### Adversarial {#adversarial}
A variety of methods use a discriminator (or critic) for learning domain-invariant features, realistic image generation, or promoting target discriminative features by forcing a network (either a feature extractor or generator) to produce outputs indistinguishable between two domains (source and target or real and fake). This loss is different than the other losses discussed in this section because this *adversarial loss* is learned [@goodfellow2016survey; @isola2017cvpr] (where learning is more than a hyperparameter search) rather than being provided as a predefined function. During training, gradients from the discriminator are used to train the feature extractor or generator (e.g., negated by a gradient reversal layer, Section \[featureLevelAdaptation\]). This alternates with updating the discriminator itself to make the correct domain classification.
### Additions for Specific Problems
Some research focusing on specific problems has resulted in additional losses. For semantic segmentation, Li et al. [@li2018semantic] develop a loss making segmentation boundaries sharper to help when the mapped image-to-image translation images will be used for segmentation, Chen et al. [@chen2018cvprroad] develop a distillation loss in addition to performing location-aware alignment (e.g., “road” is usually at the bottom of each image), Hoffman et al. [@hoffman2016fcns] develop a class-aware constrained multiple instance loss, Zhang et al. [@zhang2017iccv] develop a curriculum where after learning some high-level properties on easy tasks the segmentation network is forced to follow those properties (interpretations include student-teacher setup or posterior regularization), and Perone et al. [@perone2018unsupervised] apply the self-ensembling method [@french2018iclr] replacing the cross-entropy loss with a consistency loss. For object detection, Chen et al. [@chen2018cvpr] use two domain classifiers (one on an image-level representation and the other on an instance-level representation) with a consistency regularization between them. For adaptation from synthetic images where it is known which pixels are foreground in the source images, Bousmalis et al. [@bousmalis2017cvpr] and Bak et al. [@bak2018eccv] mask certain losses to only penalize foreground pixel differences. For person re-identification, Wei et al. [@wei2018cvpr] include a person identity-keeping constraint in their domain mapping GAN.
Low-Confidence or Low-Relevance Rejection {#rejection}
-----------------------------------------
Given a measure of confidence, performance may increase if we can reject data points for training the target classifier that are not of sufficient confidence. This, of course, assumes our confidence measurement is accurate enough. Saito et al. [@saito2017icml] used the label agreement of an ensemble combined with the softmax distribution output (uniform is not confident, one probability much higher than the rest is confident). Sener et al. [@sener2016nips] used the label agreement of the $k$ nearest source data points. If the confidence is to low, then the example is rejected and not used in training until if later on when re-evaluated it is determined to be sufficiently confident. Inoue et al. [@inoue2018cvpr] used an object detector’s prediction probability as a measure of confidence, only using high-confidence detections for fine-tuning an object detection network. Similarly, a rejection approach could be used if we have a measure of relevance. For text classification, Zhang et al. [@zhang2017aspect] weight examples by their relevance to their target aspect based on a small set of positive and negative keywords (a form of weak supervision).
Weight Sharing {#weightSharing}
--------------
Methods employ different amounts of sharing network weights between domains or regularizing the weights to be similar. Most methods completely share weights between the feature extractors used on the source and target domains (as shown in Table \[compare\]). However, some techniques do not. Since deep networks consist of many layers, allowing them to represent hierarchical features, Long et al. [@long2015icml] propose copying the lower layers from a network trained on the source domain and adapting higher layers to the target domain with MK-MMD since higher layers do not transfer well between domains. In CoGAN, Liu et al. [@liu2016nips] share the first few layers of the generators and the last few layers of the discriminators, making the assumption that the domains share high-level representations. In AdaBN, Li et al. [@li2018] assume domain knowledge is stored in the batch norm statistics, so they share all weights except for the batch norm statistics. French et al. [@french2018iclr] define the teacher network as an exponential moving average of the student network’s weights (a type of ensemble). Instead of sharing weights, Rozantsev et al. [@rozantsev2018ieee; @rozantsev2018cvpr] propose two variants: regularizing weights to be similar but not penalizing linear transformations and transforming the weights from the source network to the target network with small residual networks. Bousmalis et al. [@bousmalis2016nips] propose domain separation networks (DSN): learning source-specific, target-specific, and shared features where the “shared” source domain encoder and “shared” target domain encoder do share weights, but the “private” source domain encoder and “private” target domain encoders do not. Others have similarly explored this idea of shared vs. specific features [@liu2017adversarial; @ren2018factorized; @cao2018dida].
Training Stages {#stages}
---------------
Some have trained networks for domain adaptation in stages. Tzeng et al. [@tzeng2017cvpr] train a source classifier first followed by adaptation. Taigman et al. [@taigman2016dtn] use a pre-trained encoder during adaptation. Bousmalis et al. [@bousmalis2018roboticgrasping] in GraspGAN first train the domain-mapping network followed by the domain-adversarial network. Hoffman et al. [@hoffman2018icml] in CyCADA train their many components in stages because it would not all fit into GPU memory at once.
Other methods train the domain adaptation networks jointly, which using an adversarial approach is done by alternating between training the discriminator and the rest of the networks (Sections \[gan\] and \[featureLevelAdaptation\]). However, variations exist for some other methods. Saito et al. [@saito2017icml] in ATT cycle through generating training the source networks, generating pseudo-labels, and training the target network. Zou et al. [@zou2018eccv] alternate between pseudo-labeling the target data and re-training the model using the labels (a form of self-training). Wei et al. [@wei2018generative] in GAGL alternate between feeding in real source and target data and the fake images generated by a GAN. Sener et al. [@sener2016nips] alternate between $k$-nearest neighbors and performing gradient descent.
Multi-Level
-----------
Some adaptation methods perform adaptation at more than one level. As discussed in Section \[combinations\], GraspGAN [@bousmalis2018roboticgrasping] and CyCADA [@hoffman2018icml] perform pixel-level adaptation with domain mapping and feature-level adaptation with domain-invariant feature learning. Hoffman et al. [@hoffman2018icml] found that performing both levels of adaptation significantly improves accuracy: using domain mapping to capture low-level image domain shifts and learning domain-invariant features to handle larger domain shifts than what pure domain mapping methods can support. Following this idea, Tsai et al. [@tsai2018cvpr] make semantic segmentation predictions and perform domain-invariant feature learning at multiple levels in their semantic segmentation network, and Zhang et al. [@zhang2018cvpr] perform domain-invariant feature learning at multiple levels while automatically learning how much to align to each level. Chen et al. [@chen2018cvpr] perform domain-invariant feature learning at both image and instance levels for object detection but also include a consistency regularization between the two domain classifiers.
Types of Networks {#networkTypes}
-----------------
Nearly all of the surveyed approaches focus on learning from image data and use convolutional neural networks (CNNs) such as ResNet-50 or Inception (Table \[comparePerformance2\]). Wang et al. [@wang2019transferable] explore the use of attention networks and Kang et al. [@kang2018eccv] a combination of CNNs and attention. In the case of time-series data, Purushotham et al. [@purushotham2017variational] propose instead using a variational recurrent neural network (RNN) [@NIPS2015_5653] or LSTM (a type of RNN) [@sepp1997lstm] rather than a CNN. The RNN learns the temporal relationships while adversarial training is used to achieve domain adaptation. For text classification (a type of natural language processing), Liu et al. [@liu2017adversarial] also use LSTMs while Zhang et al. [@zhang2017aspect] found a CNN to work just as well as RNNs or bi-LSTMs in their experiments. For relation extraction (another type of natural language processing), Fu et al. [@fu2017domain] also use a CNN. For time-series speech recognition, Zhao et al. [@zhao2017principled] use bi-LSTMs while Hosseini-Asl et al. [@hosseiniasl2019augmented] used a combination of CNNs and RNNs. In the related problem of domain generalization, a combination of CNNs and RNNs have been used for handling a radio spectrogram changing through time to identify sleep stages [@zhao2017icml].
------------------------------------------------------------- ------------ -- -------------- ----------- ----------- ---------- ---------- -- ------------- ----------- -- -- ------------ -- --
**Distance** **Diff.** **Cycle** **Sem.** **Task** **Feature** **Pixel**
**CAN**[@kang2019contrastive] DI,N CCD not BN
**French et al.**[@french2018iclr] En,N sq. diff. EMA
**Co-DA**[@kumar2018nips][^1] DI,En,N,TD L1 optional
**VADA**[@shu2018vada] DI,TD
**DeepJDOT**[@damodaran2018deepjdot] DI JDOT
**CyCADA**[@hoffman2018icml] DI,DM
**Gen. to Adapt**[@sankaranarayanan2018cvpr] DI
**SimNet**[@pinheiro2018cvpr] DI prototypes
**MADA**[@pei2018multi] DI,En
**MCD**[@saito2018cvpr] DI,En,TD
**GAGL**[@wei2018generative] DI,TD
**SBADA-GAN**[@russo2018cvpr][^2] DM
**MCA**[@zhang2018mca] DI MCA
**$\text{CCN}^{++}$**[@hsu2018learning] DI clusters
**M-ADDA**[@laradji2018m] DI clusters
**Rozant. et al.**[@rozantsev2018ieee] DI MMD regularize
**XGAN**[@royer2017xgan] DM some
**StarGAN**[@choi2018cvpr] DM
**PixelDA**[@bousmalis2017cvpr] DM
**AutoDIAL**[@carlucci2017autodial] N,TD not BN
**AdaBN**[@liu2018] N not BN
**JAN-A**[@long2017jmmd] DI JMMD
**LogCORAL**[@wang2017iccv] DI logCOR, mean
**Log D-CORAL**[@morerio2017correlation] DI logDCOR
**VRADA**[@purushotham2017variational] DI
**ATT**[@saito2017icml] En
**SimGAN**[@shrivastava2017cvpr] DM N/A[^3]
**ADDA**[@tzeng2017cvpr] DI
**CycleGAN**[@zhu2017iccv] DM [^4]
**RegCGAN**[@mao2018unpaired] DM
**Sener et al.**[@sener2016nips] DI $k$-NN
**DSN**[@bousmalis2016nips] DI some
**DRCN**[@ghifary2016] DI
**CoGAN**[@liu2016nips] DM some
**Deep CORAL**[@sun2016] DI CORAL
**DANN**[@ajakan2014domain; @ganin2015icml; @ganin2016jmlr] DI
**DAN**[@long2015icml] DI MK-MMD low
**Tzeng et al.**[@tzeng2015iccv][^5] DI
------------------------------------------------------------- ------------ -- -------------- ----------- ----------- ---------- ---------- -- ------------- ----------- -- -- ------------ -- --
Hyperparameter Tuning {#hyperparameterTuningExisting}
---------------------
Normal supervised learning-based hyperparamenter tuning methods do not carry over to unsupervised domain adaptation [@long2013iccv; @long2016nips; @ganin2016jmlr; @bousmalis2016nips; @wang2018domain; @perone2018unsupervised; @morerio2018minimalentropy]. A common supervised learning approach is to split the training data into a smaller training set and a validation set. After repeatedly altering the hyperparameters, retraining the model, and testing on this validation set for each set of hyperparameters, the model yielding the highest validation set accuracy is selected. Another option is cross validation. However, in unsupervised domain adaptation, there are now two domains, and the data for the target domain may not include any labels. When evaluating domain adaptation approaches on common datasets, generally the target data does contain labels, so work by some groups [@bousmalis2016nips; @russo2018cvpr; @wang2018domain; @wei2018generative; @carlucci2017autodial; @kumar2018nips; @shu2018vada] do use some labeled target data (or all of it [@long2013iccv; @shen2018wasserstein]) for hyperparameter tuning, which can be interpreted as an upper bound on how well the method could perform [@wang2018domain]. For example, some [@long2016nips; @carlucci2017autodial] tuned for Office on one $W$ labeled example per class on the $A\rightarrow$W task, while others [@russo2018cvpr; @wei2018generative] tuned with a validation set of 1000 randomly sampled target examples. Using any labeled target data is not ideal because real-world testing will not include labels for tuning (unless it is semi-supervised, in which case semi-supervised learning is recommended in Section \[theory\]).
One tuning method not requiring labeled target data is *reverse validation* [@ganin2016jmlr], which is a variant of *reverse cross validation* [@zhong2010crossval]. For a set of hyperparameters, the *reverse validation risk* can be estimated by first splitting source (labeled) and target (unlabeled) data into training and validation sets. Then, the labeled source and unlabeled target data is used to learn a classifier (as is normally done). Next, this forward classifier is used to label the target data and a new reverse classifier is learned (with the same algorithm) using the pseudo-labeled target data (as “source”) and unlabeled source data (as “target”, i.e., ignoring the known labels). This reverse classifier is evaluated on the source validation data to measure the reverse validation risk. Ganin et al. [@ganin2016jmlr] found this method works better if the reverse classifier is initialized with the weights of the forward classifier and if using early stopping on the source validation set and a pseudo-labeled target validation set. Finally, hyperparameters are selected (e.g., grid search, random search, Bayesian optimization, or other gradient-free optimization methods such as those implemented in Nevergrad [@nevergrad]) that minimize this reverse validation risk.
Alternatively, given some domain knowledge, one may devise relevant measures of similarity between the domains and tune parameters to increase the similarity. For example, French et al. [@french2018iclr] were able to improve performance on the challenging problem of MNIST $\rightarrow$ SVHN by tuning data augmentation hyperparameters for MNIST to match pixel intensities apparent in the SVHN dataset. By doing this, they were able to improve the state-of-the-art to 97.0% (Table \[comparePerformance1\]).
Results
=======
Tables \[comparePerformance1\] through \[comparePerformanceDatasets\] summarize the results of evaluating many of these methods on datasets used for image classification as well as sentiment analysis. Care must be taken in the extent to which conclusions are drawn from comparing published numbers in different papers since the provided accuracies are for different network architectures, hyperparameters, amount of data augmentation, random initializations (or averages over a number of them), etc. and the methods may perform differently in other application areas. However, interestingly, at least one method in each of the categories of surveyed gives promising results on at least one of the datasets.
With domain-invariant feature learning with the contrastive domain discrepancy, CAN [@kang2019contrastive] has the highest performance on the Office dataset (Table \[comparePerformance2\]). By using adversarial domain-invariant feature learning, WDGRL generally outperforms the other methods on the Amazon review dataset (Table \[compareSentimentPerformance\]) and Generate to Adapt is second highest of the methods evaluated on the Office dataset. By using adversarial pixel-level domain mapping, SBADA-GAN [@russo2018cvpr] obtains the highest accuracy on MNIST$\rightarrow$MNIST-M (Table \[comparePerformance1\]). AutoDIAL [@carlucci2017autodial], a normalization statistics method, does on-par with CAN and Generate to Adapt in two of Office adaptation tasks. The self-ensembling method by French et al. [@french2018iclr] outperforms all other methods on the datasets in Table \[comparePerformance1\], and Co-DA [@kumar2018nips] comes close using an ensemble (of size two) of adversarial domain-invariant feature networks. CyCADA increases accuracy from 54% to 82% for a synthetic season adaptation dataset [@hoffman2018icml] by combining both adversarial domain-invariant feature learning and domain mapping.
A number of these promising methods use adversarial techniques, which may be a key ingredient in solving domain adaptation problems. Adversarial approaches may be helpful on certain datasets (e.g., WDGRL on the Amazon review dataset on Office), certain types of data (e.g., VRADA was developed for time series data rather than image data), or may not require as extensive of tuning (e.g., Co-DA on MNIST$\rightarrow$SVHN). Or adversarial training may be an additional tool to incorporate into existing non-adversarial methods. For instance, promising non-adversarial methods such as AutoDIAL and by French et al. could be combined with adversarial methods (see Section \[combinePromisingMethods\]). In fact, Long et al. [@long2017jmmd] develop both JAN and then the adversarial version JAN-A, and JAN-A on average outperformed JAN on the Office dataset. CAN [@kang2019contrastive], which presently is the highest on the Office dataset, might also be improved by incorporating an adversarial component to it as in Long et al. [@long2017jmmd].
Interestingly, French et al. by far outperform all other methods on MNIST$\rightarrow$SVHN, though this requires a problem-specific data augmentation and hyperparameter tuning. This may indicate that for some problems, maybe in particular the more challenging domain adaptation problems, hyperparameter tuning for a specific dataset may be of utmost importance. Possibly if other domain adaptation methods similarly were tuned appropriately, they would also experience large improvements. This is an area of research requiring further work (see Section \[hyperparameterTuningFuture\]). However, Co-DA [@kumar2018nips] is not far behind on SVHN$\rightarrow$MNIST and MNIST$\rightarrow$MNIST-M and is the closest on MNIST$\rightarrow$SVHN, achieving 81.7% compared with 97.0%. A great advantage of Co-DA is that it does not require highly-problem-specific tuning on MNIST$\rightarrow$SVHN as required by French et al. (without they only achieved 37.5%). Possibly some components of Co-DA such as the adversarial domain adaptation or virtual adversarial training may be partially responsible for the decrease in hyperparameter sensitivity.
-------------------------------------------------- --------------------------- ------------------------- -- ------------------------- ------------------------- -- --------------------------- -- -------------------------------------------- -----------------------------------------------
**MN $\rightarrow$ US** **US $\rightarrow$ MN** **SV $\rightarrow$ MN** **MN $\rightarrow$ SV** **MN $\rightarrow$ MN-M** **$\text{SYN}_\text{N}$ $\rightarrow$ SV** **$\text{SYN}_\text{S}$ $\rightarrow$ GTSRB**
**Co-DA**[@kumar2018nips][^6]$^*$ 98.6 81.7 97.5 96.0
**DIRT-T**[@shu2018vada]$^*$ 99.4 76.5 98.7 96.2 99.6
**VADA**[@shu2018vada]$^*$ 94.5 73.3 95.7 94.9 99.2
**DeepJDOT**[@damodaran2018deepjdot] 95.7 96.4 96.7 92.4
**CyCADA**[@hoffman2018icml]$^*$ 95.6 $\pm$ 0.2 96.5 $\pm$ 0.1 90.4 $\pm$ 0.4
**Gen. to Adapt**[@sankaranarayanan2018cvpr]$^*$ 92.8 $\pm$ 0.9 90.8 $\pm$ 1.3 92.4 $\pm$ 0.9
**SimNet**[@pinheiro2018cvpr]$^*$ 96.4 95.6 90.5
**MCD**[@saito2018cvpr]$^*$ 96.5 $\pm$ 0.3 94.1 $\pm$ 0.3 96.2 $\pm$ 0.4 94.4 $\pm$ 0.3
**GAGL**[@wei2018generative]$^*$ 96.7 74.6 94.9 93.1 97.6
**SBADA-GAN**[@russo2018cvpr]$^*$ 97.6 95.0 76.1 61.1 99.4 96.7
**MCA**[@zhang2018mca] 96.6 96.8 89.0
**$\text{CCN}^{++}$**[@hsu2018learning]$^*$ 89.1
**M-ADDA**[@laradji2018m]$^*$ 98 97
**Rozantsev et al.**[@rozantsev2018ieee] 60.7 67.3
**PixelDA**[@bousmalis2017cvpr]$^*$ 95.9 98.2
**ATT**[@saito2017icml] 85.0 52.8 94.0 92.9 96.2
**ADDA**[@tzeng2017cvpr]$^*$ 89.4 $\pm$ 0.2 90.1 $\pm$ 0.8 76.0 $\pm$ 1.8
**RegCGAN**[@mao2018unpaired]$^*$ 93.1 $\pm$ 0.7 89.5 $\pm$ 0.9
**DTN**[@taigman2016dtn]$^*$ 84.4
**Sener et al.**[@sener2016nips] 78.8 40.3 86.7
**DSN**[@bousmalis2016nips]$^*$ 91.3 [@bousmalis2017cvpr] 82.7 83.2 91.2 93.1
**DRCN**[@ghifary2016] 91.80 $\pm$ 0.09 73.67 $\pm$ 0.04 81.97 $\pm$ 0.16 40.05 $\pm$ 0.07
**CoGAN**[@liu2016nips]$^*$ 91.2 $\pm$ 0.8 89.1 $\pm$ 0.8 62.0 [@bousmalis2017cvpr]
-------------------------------------------------- --------------------------- ------------------------- -- ------------------------- ------------------------- -- --------------------------- -- -------------------------------------------- -----------------------------------------------
-------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
**A $\rightarrow$ W** **D $\rightarrow$ W** **W $\rightarrow$ D** **A $\rightarrow$ D** **D $\rightarrow$ A** **W $\rightarrow$ A**
**CAN**[@kang2019contrastive][^7] 94.5 $\pm$ 0.3 99.1 $\pm$ 0.2 99.8 $\pm$ 0.2 95.0 $\pm$ 0.3 78.0 $\pm$ 0.3 77.0 $\pm$ 0.3
**Gen. to Adapt**[@sankaranarayanan2018cvpr]$^*$ 89.5 $\pm$ 0.5 97.9 $\pm$ 0.3 99.8 $\pm$ 0.4 87.7 $\pm$ 0.5 72.8 $\pm$ 0.3 71.4 $\pm$ 0.4
**SimNet**[@pinheiro2018cvpr]$^*$ 88.6 $\pm$ 0.5 98.2 $\pm$ 0.2 99.7 $\pm$ 0.2 85.3 $\pm$ 0.3 73.4 $\pm$ 0.8 71.8 $\pm$ 0.6
**MADA**[@pei2018multi]$^*$ 90.0 $\pm$ 0.1 97.4 $\pm$ 0.1 99.6 $\pm$ 0.1 87.8 $\pm$ 0.2 70.3 $\pm$ 0.3 66.4 $\pm$ 0.3
**AutoDIAL**[@carlucci2017autodial][^8][^9] 84.2 97.9 99.9 82.3 64.6 64.2
**$\text{CCN}^{++}$**[@hsu2018learning][^10]$^*$ 78.2 97.4 98.6 73.5 62.8 60.6
**Rozantsev et al.**[@rozantsev2018ieee] 76.0 96.7 99.6
**AdaBN**[@liu2018] 74.2 95.7 99.8 73.1 59.8 57.4
**JAN-A**[@long2017jmmd]$^*$ 86.0 $\pm$ 0.4 96.7 $\pm$ 0.3 99.7 $\pm$ 0.1 85.1 $\pm$ 0.4 69.2 $\pm$ 0.4 70.7 $\pm$ 0.5
**LogCORAL**[@wang2017iccv] 70.2 $\pm$ 0.6 95.5 $\pm$ 0.1 99.5 $\pm$ 0.3 69.4 $\pm$ 0.5 51.2 $\pm$ 0.3 51.6 $\pm$ 0.5
**Log D-CORAL**[@morerio2017correlation] 68.5 95.3 98.7 62.0 40.6 40.6
**ADDA**[@tzeng2017cvpr]$^*$ 75.1 97.0 99.6
**Sener et al.**[@sener2016nips] 81.1 96.4 99.2 84.1 58.3 63.8
**DRCN**[@ghifary2016] 68.7 $\pm$ 0.3 96.4 $\pm$ 0.3 99.0 $\pm$ 0.2 66.8 $\pm$ 0.5 56.0 $\pm$ 0.5 54.9 $\pm$ 0.5
**Deep CORAL**[@sun2016] 66.4 $\pm$ 0.4 95.7 $\pm$ 0.3 99.2 $\pm$ 0.1 66.8 $\pm$ 0.6 52.8 $\pm$ 0.2 51.5 $\pm$ 0.3
**Tzeng et al.**[@tzeng2015iccv] [^11]$^*$ 59.3 $\pm$ 0.6 90.0 $\pm$ 0.2 97.5 $\pm$ 0.1 68.0 $\pm$ 0.5 43.1 $\pm$ 0.2 40.5 $\pm$ 0.2
-------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
**Source $\rightarrow$ Target** **DANN**[@ganin2016jmlr][^12]$^*$ **DANN**[@ganin2016jmlr][^13]$^*$ **CORAL**[@sun2016aaai][^14] **ATT**[@saito2017icml] **WDGRL**[@shen2018wasserstein][^15]$^*$ **No Adapt.**[@sun2016aaai][^16]
--------------------------------- -- ----------------------------------- ----------------------------------- ------------------------------ ------------------------- ------------------------------------------ ----------------------------------
**B $\rightarrow$ D** 82.9 78.4 80.7 83.1
**B $\rightarrow$ E** 80.4 73.3 76.3 79.8 83.3 74.7
**B $\rightarrow$ K** 84.3 77.9 82.5 85.5
**D $\rightarrow$ B** 82.5 72.3 78.3 73.2 80.7 76.9
**D $\rightarrow$ E** 80.9 75.4 77.0 83.6
**D $\rightarrow$ K** 84.9 78.3 82.5 86.2
**E $\rightarrow$ B** 77.4 71.3 73.2 77.2
**E $\rightarrow$ D** 78.1 73.8 72.9 78.3
**E $\rightarrow$ K** 88.1 85.4 83.6 86.9 88.2 82.8
**K $\rightarrow$ B** 71.8 70.9 72.5 77.2
**K $\rightarrow$ D** 78.9 74.0 73.9 74.9 79.9 72.2
**K $\rightarrow$ E** 85.6 84.3 84.6 86.3
[@ll@]{}\
**MNIST**[@lecun1998mnist][^17] &\
**MNIST-M**[@ganin2016jmlr][^18] &\
**USPS**[@le1990handwritten][^19] &\
**SVHN**[@netzer2011reading][^20] &\
**$\text{SYN}_\text{N}$**[@ganin2016jmlr] &\
**$\text{SYN}_\text{S}$**[@moiseev2013evaluation][^21] &\
**GTSRB**[@Stallkamp-IJCNN-2011][^22] &\
**Office**[@saenko2010adapting][^23] &\
Theory
======
Having surveyed domain adaptation methods, we now address the question of when adaptation may be beneficial. Ben-David et al. [@ben2010ml] develop a theory answering this in terms of an ideal predictor on both domains, Zhao et al. [@zhao2019learning] further this theory by removing the dependence on a joint ideal predictor while focusing on domain-invariant feature learning methods, and Le et al. [@le2018theoretical] develop theory looking beyond domain-invariant methods. These theoretical results can help answer two questions: (1) when will a classifier (or other predictor) trained on the source data perform well on the target data, and (2) given a small number of labeled target examples, how can they best be used during training to minimize target test error?
Answering the first question, labeled source data and unlabeled target data are both required (unsupervised). Answering the second question, additionally some labeled target data are required (semi-supervised). We will first review the theoretical bounds followed by a discussion of what insights these bounds provide into answering the above two questions. Ben-David et al. [@ben2010ml] also address the case of multiple source domains, as do Mansour et al. [@mansour2009nips]. In this paper, we have focused on the cases containing only one source and one target (as is common in the methods we survey).
Unsupervised
------------
### Shared Hypothesis Space
Ben-David et al. [@ben2010ml] propose setting a bound on the target error based on the source error and the divergence between the source and target domains. The empirical source error is easy to obtain by first training and then testing a classifier. However, the divergence between the domains cannot be directly obtained with standard methods like Kullback-Leibler divergence due to only having a finite number of samples from the domains and not assuming any particular distribution. Thus, an alternative is to measure it using a classifier-induced divergence called $\mathcal{H} \Delta \mathcal{H}$-divergence. Estimates of this divergence with finite samples converges to the real $\mathcal{H} \Delta \mathcal{H}$-divergence. This divergence can be estimated by measuring the error when getting a classifier to discriminate between the unlabeled source and target examples; though, it is often intractable to find the theoretically-required divergence upper bound. Using the empirical source error $\hat{\epsilon}_S(h)$, the $\mathcal{H} \Delta \mathcal{H}$-divergence between source and target samples $d_{\mathcal{H} \Delta \mathcal{H}}(\mathcal{\hat{D}}_S, \mathcal{\hat{D}}_T)$, and ideal predictor error $\lambda^*$ using the optimal hypothesis for the source and target, the target error $\epsilon_T(h)$ can be bounded as shown in Equation \[theory1\] (using the form given by Zhao et al. [@zhao2019learning]), $\forall h \in \mathcal{H}$ with probability at least $1-\delta$ for $\delta \in (0,1)$. $$\epsilon_T(h) \leq \hat{\epsilon}_S(h) + \frac{1}{2} d_{\mathcal{H} \Delta \mathcal{H}}(\mathcal{\hat{D}}_S, \mathcal{\hat{D}}_T) + \lambda^* + O \left( \sqrt{\frac{d \log n + \log(\frac{1}{\delta})}{n}} \right)
\label{theory1}$$
Zhao et al. [@zhao2019learning] develop another upper bound that removes the reliance on $\lambda^*$. Let $\mathcal{H} \subseteq [0,1]^\mathcal{X}$, $\mathcal{\tilde{H}} \coloneqq \{\text{sgn} \left( |h(x) - h'(x)| - t \right) | h,h' \in \mathcal{H}, 0 \leq t \leq 1 \}$, $\langle \mathcal{D}_S, f_S \rangle$ and $\langle \mathcal{D}_T, f_T \rangle$ be the source and target domains (the true distributions, not empirical). The target error can then be bounded by the source error $\epsilon_S(h)$, the discrepancy between marginal distributions $d_\mathcal{\tilde{H}}(\mathcal{D}_S, \mathcal{D}_T)$, and the distance between the optimal source and target labeling functions $\forall h \in \mathcal{H}$, as shown in Equation \[theory2\]. $$\epsilon_T(h) \leq \epsilon_S(h) + d_\mathcal{\tilde{H}}(\mathcal{D}_S, \mathcal{D}_T) + \min\{ \mathbb{E}_{\mathcal{D}_S}[|f_S-f_T|], \mathbb{E}_{\mathcal{D}_T}[|f_S-f_T|] \}
\label{theory2}$$
Zhao et al. [@zhao2019learning] also develop an information-theoretic lower bound for target error. Let the labeling function $Y = f(X) \in \{0,1\}$, the prediction function $\hat{Y} = h(g(X)) \in \{0,1\}$, and $Z$ be the intermediate representation output by a shared feature extractor used on source and target domain data. If the Jensen-Shannon distance $d_{JS}(\mathcal{D}_S^Y, \mathcal{D}_T^Y) \geq d_{JS}(\mathcal{D}_S^Z, \mathcal{D}_T^Z)$ and the Markov chain $X \xrightarrow{g} Z \xrightarrow{h} \hat{Y}$ holds, then Equation \[theory3\] provides a lower bound on the source and target error. $$\epsilon_S(h \circ g) + \epsilon_T(h \circ g) \geq \frac{1}{2}\left( d_{JS}(\mathcal{D}_S^Y, \mathcal{D}_T^Y) - d_{JS}(\mathcal{D}_S^Z, \mathcal{D}_T^Z) \right)^2
\label{theory3}$$
### Different Hypothesis Spaces
Le et al. [@le2018theoretical] develop an upper bound that allows for different hypothesis spaces for source and target functions, possibly non-deterministic labeling, and any bounded or continuous loss. If $l$ is a bounded or continuous loss, $x \sim \mathbb{P}^s$ (source) and $x \sim \mathbb{P}^t$ (target), $T : \mathcal{X}^s \rightarrow \mathcal{X}^t$ and $K \coloneqq T^{-1}$ (bijective mapping), $R(\theta) = \mathbb{E}_{p(x,y)} [ l(y,h_\theta(x)) ]$ for $\theta$ parameterizing a hypothesis set $\mathcal{H} = \{ h_\theta | \theta \in \Theta \}$, $\Delta R(h^s, h^t) \coloneqq |R^t(h^t) - R^s(h^s)|$, $y \in \{-1,1\}$, $M$ is the number of labels, $\mathbb{P}^\# \coloneqq K_\# \mathbb{P}^t$ is the pushforward probability distribution transporting $\mathbb{P}^t$ via $K$, $\Delta p(y|x) \coloneqq p^t(y|T(x)) - p^s(y|x)$ for the true source and target labeling functions $p^s(y|x)$ and $p^t(y|x)$, where ${WS}_c(\mathbb{P}^s,\mathbb{P}^\#)$ denotes the Wasserstein-1 distance between the source and target distributions with a cost function $c(x,x') = 1_{x \neq x'}$ (1 if $x \neq x'$, otherwise 0), then Equation \[theory4\] provides an upper bound for the variance between a general loss on the source and target predictions. $$\Delta R(h^s, h^t) \leq M \left( {WS}_c(\mathbb{P}^s,\mathbb{P}^\#) + \min\{ \mathbb{E}_{\mathbb{P}^\#} [ \| \Delta p(y|x) \|_1 ], \mathbb{E}_{\mathbb{P}^s} [ \| \Delta p(y|x) \|_1 ]\} \right)
\label{theory4}$$
Semi-Supervised
---------------
In the semi-supervised case, a linear combination of the source and target errors is computed [@ben2010ml], called the $\alpha$-error. A bound can be calculated on the true $\alpha$-error based on the empirical $\alpha$-error. Finding the minimum $\alpha$-error depends on the empirical $\alpha$-error, the divergence between source and target, and the number of labeled source and target examples. Experimentation can be used to empirically determine the values of $\alpha$ that will perform well. Ben-David et al. [@ben2010ml] also demonstrate the process on sentiment classification, illustrating that the optimum uses non-trivial values.
The bound is given in Equation \[benLabelBound\]. If $S$ is a labeled sample of size $m$ with $(1-\beta) m$ points drawn from the source distribution and $\beta m$ from the target distribution, then with at least probability $1-\delta$ for $\delta \in (0,1)$: $$\begin{aligned}
\epsilon_T(\hat{h}) \leq
& \epsilon_T(h^*_T) + 4 \sqrt{\frac{\alpha^2}{\beta} + \frac{(1-\alpha)^2}{1-\beta}} \sqrt{\frac{2d \log(2(m+1)) + 2 \log(\frac{8}{\delta})}{m}} + \nonumber \\
& 2(1-\alpha) \left( \frac{1}{2} \hat{d}_{\mathcal{H} \Delta \mathcal{H}}(\mathcal{U}_S, \mathcal{U}_T) + 4 \sqrt{\frac{2d \log(2m') + \log(\frac{8}{\delta})}{m'}} + \lambda \right)
\label{benLabelBound}\end{aligned}$$
Here, $\hat{h} \in \mathcal{H}$ is the empirical minimizer of the $\alpha$-error on $S$ given by $\hat{\epsilon}_\alpha(h) = \alpha \hat{\epsilon}_T(h) + (1-\alpha) \hat{\epsilon}_S(h)$ and $h^*_T = \min_{h \in \mathcal{H}} \epsilon_T(h)$ is the target error minimizer.
The optimum $\alpha$ is then: $$\begin{aligned}
\alpha^*(m_T,m_S;D) =
\begin{cases}
1 & m_T \geq D^2 \\
\min\{ 1,\nu \} & m_T \leq D^2
\end{cases}\end{aligned}$$
Here, $m_S = (1-\beta)m$ is the number of source examples, $m_T = \beta m$ is the number of target examples, $D = \sqrt{d} / A$, and $$\nu = \frac{m_T}{m_T + m_S} \left( 1 + \frac{m_S}{\sqrt{D^2(m_S+m_T) - m_S m_T}} \right)$$ $$A = \frac{1}{2} \hat{d}_{\mathcal{H} \Delta \mathcal{H}}(\mathcal{U}_S, \mathcal{U}_T) + 4 \sqrt{\frac{2d \log(2m') + \log(\frac{4}{\delta})}{m'}} + \lambda$$ $$B = 4 \sqrt{\frac{2d \log(2(m+1)) + 2 \log(\frac{8}{\delta})}{m}}$$
Discussion
----------
### Unsupervised
Equation \[theory1\] indicates that if the optimal predictor error $\lambda^*$ on both source and target data is large, then there is no good hypothesis from training on the source domain that will work well on the target domain [@ben2010ml; @zhao2019learning]. However, as is more common in the application of domain adaptation, if $\lambda^*$ is small, then the bound depends on the source error and the $\mathcal{H} \Delta \mathcal{H}$-divergence [@ben2010ml]. The domain-invariant feature learning methods discussed in Section \[domainInvariance\] try minimizing these two terms [@zhao2019learning]: the source error via a task loss on labeled source data and divergence via a divergence measure such as MMD, with reconstruction, or adversarially. While Section \[results\] shows that on many datasets these methods work, there is no guarantee that such adaptation will increase performance (these are upper bounds), as shown by simple counterexamples [@zhao2019learning]. It may actually decrease performance if the marginal label distributions differ significantly between source and target [@zhao2019learning].
Equation \[theory2\] shows that the target error upper bound alternatively involves the marginal distributions and Equation \[theory3\] shows that the lower bound does too. These indicate the importance of aligning the label distributions. If the marginal label distributions are significantly different, then minimizing the source error and divergence between feature representations will actually increase the error [@zhao2019learning]. Thus over-training domain-invariant feature learning methods can increase target error, and Zhao et al. [@zhao2019learning] experimentally verified this. They found on MNIST, USPS, and SVHN adaptation that during training the target accuracy would initially rise rapidly but would eventually decrease again despite increasing source accuracy, an effect even more apparent with larger differences in the marginal label distributions. It is an open problem as to when the label distributions can be aligned without target labels [@zhao2019learning].
### Semi-Supervised
Equation \[benLabelBound\] indicates that when only source or target data is available, that data should be used (as we might expect). If the source and target are the same, then $\alpha^* = \beta$, which implies a uniform weighting of examples. Given enough target data, source data should not be used at all because it might increase the test-time error. Furthermore, without enough source data using it may also not be worthwhile, i.e., $\alpha^* \approx 0$ [@ben2010ml]. In this paper we focus on unsupervised domain adaptation, but these are important considerations if target labels can be obtained. For example, this shows that it may be better to perform semi-supervised adaptation if some labeled target examples are available rather than using the labeled target examples to hyperparameter tune an unsupervised adaptation method.
Applications
============
Domain adaptation has been applied in a variety of areas including computer vision, natural language processing, and for time-series data. Using domain adaptation in these various problems can save the human time that would be spent labeling the target data. In some cases such as image semantic segmentation, providing ground truth is very labor intensive. Each pixel-level annotated image in the Cityscapes dataset took on average 1.5 hours to complete [@cordts2016cvpr]. In addition, similar methods as described in this paper have been applied to the related problem of domain generalization and some other problems as well.
Computer Vision
---------------
Most of the methods surveyed in this paper are for computer vision tasks such as adapting a model trained on synthetic images to real photos (e.g., from synthetic numbers or signs, Table \[comparePerformance1\]), stock photos to real photos (e.g., Amazon to DSLR on the Office dataset, Table \[comparePerformance2\]), or simple to complex images (e.g., MNIST to SVHN, Table \[comparePerformance1\]). Others have been used in robotics for robot grasping [@bousmalis2018roboticgrasping], autonomous navigation [@yoo2017domain], and lifelong learning [@wulfmeier2018ieee], for semantic segmentation [@chen2018cvprroad; @luo2018taking; @lee2018spigan; @vu2018advent; @huang2018eccv; @zou2018eccv; @hong2018cvpr; @sankaranarayanan2018cvprsemantic; @tsai2018cvpr] including when additional information is available from a simulator [@lee2018spigan], in a medical context for chest X-ray segmentation [@chen2018seuda], 3D CT scans to X-ray segmentation [@zhang2018tdgan], MRI to CT scan segmentation [@chen2019synergistic], and MRI segmentation [@perone2018unsupervised], in low resource situations (where there are very few target data points) [@hosseiniasl2019augmented], in situations with different label sets for each domain [@sohn2018unsupervised], for object detection [@inoue2018cvpr; @chen2018cvpr; @hoffman2016fcns], and for person re-identification [@ganin2016jmlr; @deng2018cvpr; @bak2018eccv; @wei2018cvpr].
Natural Language Processing
---------------------------
Domain adaptation has been used in natural language processing such as for sentiment analysis (Table \[compareSentimentPerformance\], [@zhang2017aspect; @zhao2017principled]), other text classification [@liu2017adversarial; @zhang2017aspect] including weakly-supervised aspect-transfer from one aspect of a dataset to another [@zhang2017aspect], relation extraction [@fu2017domain], semi-supervised sequence labeling [@daume2007acl], semi-supervised question answering [@yang2017semi], sentence specificity [@ko2018domain], and neural machine translation [@chu2018survey; @britz2017effective; @chen2017cost].
Time Series
-----------
For time-series data, domain adaptation has been used for learning temporal latent relationships in health data across different population age groups [@purushotham2017variational] and to perform speech recognition [@zhao2017principled; @shinohara2016interspeech; @hosseiniasl2019augmented]. In a method addressing the related problem of domain generalization, time-series radio data was used for sleep-stage classification [@zhao2017icml].
Domain Generalization {#domainGeneralization}
---------------------
Domain-invariant feature learning approaches similar to those discussed in Section \[domainInvariance\] have been used for the related problem of domain generalization, where there are multiple source domains and an unseen target domain [@muandet2013domain]. Zhao et al. [@zhao2017icml] use an adversarial approach with a domain classifier to learn a model on a dataset collected from a number of people sleeping in various environments that will generalize well to new people and/or new environments (e.g., sleeping in a different room). Ghifary et al. [@ghifary2015iccv] use a reconstruction approach with a denoising autoencoder to improve object recognition generalizability, where the “noise” is different views (domains) of the data (e.g., rotation, change in size, or variation in lighting) and the autoencoder tries to reconstruct corresponding views of the object in other domains. Carlucci et al. [@carlucci2018agnostic] propose an adversarial approach combining domain adaptation and generalization while also doing domain mapping.
Other Problems
--------------
Adversarial losses like those used in adversarial domain adaptation methods have also been applied in multiple other settings. Wang et al. [@wang2017cvpr] created an adversarial spacial dropout network to add occlusions to images to improve the accuracy of object detection algorithms. They also created an adversarial spatial transformer network to add deformations such as rotations to objects to again increase object detection accuracy. Pinto et al. [@pinto2017icra] used adversarial agents to improve a robot’s ability to grasp an object via self-supervised learning by employing both shaking and snatching adversaries. Giu et al. [@guiteaching] used an adversarial loss to predict and demonstrate (i.e., robot will copy) human motion. Rippel et al. [@waveone2017; @rippel2018using] used a reconstruction and adversarial loss with an autoencoder for learning higher quality image compression at low bit rates. Sinclair [@sinclair2018sounderfeit] applied adversarial loss to clone a physical model for real-time sound synthesis. Adversarial techniques may also be applied to machine learning security, where the goal is to train a classifier robust to adversarial examples [@huang2011adversarial; @miyato2018virtual].
Research Directions {#researchDirections}
===================
As we have seen, the rapidly-growing body of research focused on unsupervised deep domain adaptation now encompasses many novel methods and components. Here we look at what could be explored in future research to further enhance this existing work.
Bi-Directional Adaptation
-------------------------
The more difficult domain adaptation problems are far from being solved. Tables \[comparePerformance1\] through \[comparePerformanceDatasets\] indicate that some domain adaptation problems are harder than others and point to the challenge that more work needs to be focused on these harder problems. While accuracy for SVHN$\rightarrow$MNIST ranges from 70.7% to 99.3%, for the reverse case of MNIST$\rightarrow$SVHN, the highest without highly-problem-specific hyperparameter tuning is 81.7% by Kumar et al. [@kumar2018nips] (though tuned on a small amount of labeled target data). This indicates how this reverse problem is much harder [@ganin2015icml; @french2018iclr]. As a result, few papers offer results for this direction. French et al. [@french2018iclr] were able to vastly improve performance up to 97.0%; however, this required developing a problem-specific unsupervised hyperparameter tuning method. Other methods may similarly benefit from such tuning. Continued work is needed to strengthen general-purpose bi-directional adaptation.
Hyperparameter Tuning {#hyperparameterTuningFuture}
---------------------
Some methods such as reverse validation and a problem-specific pixel intensity matching have been applied to hyperparameter tuning without requiring target labels (Section \[hyperparameterTuningExisting\]). While the reverse validation method appears promising, it was not used in most of the methods surveyed (only [@ganin2016jmlr; @pinheiro2018cvpr; @pei2018multi]). This may be because of the increase in computation cost [@perone2018unsupervised] or problems with the reverse validation accuracy not aligning with test accuracy [@bousmalis2016nips]. It is also possible researchers may just be unaware of the method since in the surveyed papers few mention the idea (only [@bousmalis2016nips; @perone2018unsupervised; @ganin2016jmlr; @pinheiro2018cvpr; @pei2018multi]). Problem-specific methods such as matching pixel intensity between domains as done by French et al. [@french2018iclr] are possible given some domain knowledge, but hyperparameter tuning methodologies should be developed that will work across a wider range of problems. This remains an open area of research.
Combining Promising Methods {#combinePromisingMethods}
---------------------------
French et al. [@french2018iclr], Co-DA [@kumar2018nips], CAN [@kang2019contrastive], AutoDIAL [@carlucci2017autodial], Generate to Adapt [@sankaranarayanan2018cvpr], and WDGRL [@shen2018wasserstein] are promising approaches based on Tables \[comparePerformance1\] through \[compareSentimentPerformance\]. French et al. uses a student and teacher network for self-ensembling, Co-DA trains multiple (e.g., two) adaptation networks while requiring diversity and agreement in addition to incorporating virtual adversarial training, CAN alternates between clustering and adaptation through minimizing intra-class discrepancy and maximizing inter-class margin, AutoDIAL adjusts batch normalization layer weights, Generate to Adapt uses an embedding-conditional GAN for adversarial domain adaptation, and WDGRL performs adversarial domain adaptation similar to DANN by using a domain classifier. These are largely independent ideas that if combined may result in additional performance gains.
For instance, the student network in French et al. that accepts either a source or target augmented image could be replaced by the AutoDIAL network to learn how much adaptation to perform at each level of the network. Or to combine with adversarial methods, the student and teacher networks’ outputs (or an intermediate layer’s outputs, as is being explored by Wang et al. [@wang2018domain]) could be fed to a gradient reversal layer followed by a domain classifier, in effect adding an adversarial loss term to the existing two terms used by French et al. Or since French et al. is based upon data augmentation, one might try replacing the existing stochastic data augmentation with a GAN since a GAN can be used for data augmentation (given enough unlabeled training data).
Balancing Classes
-----------------
In order to obtain high accuracy on the challenging problem of MNIST$\rightarrow$SVHN, French et al. [@french2018iclr] include an additional class-balance term in their loss function, which both improved training stability and helped the network avoid a degenerate local minimum. Though, this term was not required in their other experiments. Clearly, class balancing is an important concern; although, this depends on the dataset being used. Other methods may similarly benefit from balancing classes.
For instance, Hoffman et al. [@hoffman2018icml] note that the frequency-weighted intersection over union results in their paper were very close to the target-only model accuracy (an approximate upper bound). Thus, they conclude that domain mapping followed by domain-invariant feature learning is very effective for the common classes in the SYNTHIA dataset (season adaptation on a synthetic driving dataset). It is possible then that additional balancing of classes could help the not-as-common classes to perform better. In addition, data augmentation through occluding parts of the images may improve class balancing as would the adversarial spatial dropout network by Wang et al. [@wang2017cvpr] since the two best classes (road and sky) were likely in almost every image.
Incorporating Improved Image-to-Image Translation Methods
---------------------------------------------------------
Bousmalis et al. [@bousmalis2017cvpr] with PixelDA had difficulty applying their method with large domain differences. However, other image-to-image translation methods like XGAN [@royer2017xgan] have been developed that may support larger domain shifts. These methods could be extended to domain adaptation directly or also incorporating a semantic consistency loss (as explained in Section \[losses\]). This may allow for more substantial differences between domains. Similarly, image-to-image translation methods like StarGAN [@choi2018cvpr] have been developed for multiple domains, which could be extended for multi-domain adaptation.
Futher Experimental Comparison Between Methods
----------------------------------------------
As shown in Table \[comparePerformance1\], French et al. [@french2018iclr] outperforms all the other methods and Co-DA [@kumar2018nips] is quite close behind (with the advantage that it does not require highly-problem-specific tuning on MNIST$\rightarrow$SVHN). In Table \[comparePerformance2\], CAN [@kang2019contrastive] outperforms the others followed by Generate to Adapt [@sankaranarayanan2018cvpr]. Finally, in Table \[compareSentimentPerformance\], WDGRL [@shen2018wasserstein] generally performs the best. However, these methods are not all compared on the same dataset, making a direct comparison difficult. Additional experiments must be performed to see how these methods compare. Similarly, other promising approaches may outperform other methods on some datasets, which could be determined through additional experiments.
These comparisons can be made easier through developing a unified implementation of these various methods. Schneider et al. [@schneider2018salad] are developing such an open-source set of implementations of state-of-the-art domain adaptation (and domain generalization) methods. The results provided in individual papers have different hyperparameters, data augmentation, network architectures, etc. that can make direct comparisons challenging. Using a unified implementation of these methods can facilitate more clearly understanding what aspects of a method are responsible for performance gains and also support combining the novel elements from multiple methods.
Limitations of Datasets
-----------------------
Varying amounts of source and target data are available in different situations. The datasets used for comparisons (the image datasets listed in Table \[comparePerformanceDatasets\] and the Amazon review dataset) are relatively small when compared with the sizes of datasets commonly in use in deep learning, e.g., ImageNet [@5206848; @ILSVRC15] (though ImageNet is often used to pretrain adaptation networks). For example, Sankaranarayanan et al. [@sankaranarayanan2018cvpr] note how GANs require a lot of training data. This may limit GAN-based methods from being used on too small of source or target datasets. Modifications may need to be developed for such low resource situations, an area explored by Hosseini-Asl et al. [@hosseiniasl2019augmented]. Additionally, most domain adaptation datasets are for computer vision. To spur research in other application areas, other datasets could be created.
Other Applications {#otherApplications}
------------------
Other application areas may benefit from performing domain adaptation as have those discussed in Section \[applications\]. In particular, only a few methods were applied to time-series data. One time-series application that may benefit from adaptation is activity prediction, e.g., adapting from one type of sensor to another or from one person’s data to another’s. Some added challenges in this context may be the large differences in feature spaces due to the wide variety of sensors used (e.g., an event stream of fixed motion sensors turning on and off in a smart home vs. sampled motion and location data collected from smart phones or watches) or the difference in labels (e.g., one model may learn a “walk” activity while another learns “exercise” or may learn “read” while another model learns “school”). Applying domain adaptation in new areas may yield novel methods or components applicable in other areas as well.
Conclusions
===========
For supervised learning, deep neural networks are in prevalent use, but these networks require large labeled datasets for training. Unsupervised domain adaptation can be used to adapt deep networks to possibly-smaller datasets that may not even have target labels. Several categories of methods have been developed for this goal: domain-invariant feature learning, domain mapping, normalization statistics-based, and ensemble-based methods. These various methods have some unique and common elements as we have discussed. Additionally, theoretical results provide some insight into empirical observations. Some methods appear very promising, but further research is required for direct comparisons, novel method combinations, improved bi-directional adaptation, and use for novel datasets and applications.
This material is based upon work supported by the under Grant Nos. and .
[^1]: \[vatTraining\]also incorporate virtual adversarial training [@miyato2018virtual]
[^2]: also a self-labeled classification loss (learn label on source images, pseudo-label mapped target to source)
[^3]: maps to target domain so only have feature extractor for target (part of the classifier)
[^4]: unspecified; originally not applied to domain adaptation, but later used for this [@hoffman2018icml; @benaim2017nips; @fu2018geometry]
[^5]: semi-supervised for some classes, i.e., requires some labeled target data for some of the classes
[^6]: \[hyperparamTunedTarget\]hyperparameter tuned on some labeled target data
[^7]: \[renset50Net\]with ResNet-50 network
[^8]: \[inceptionNet\]with Inception-based network
[^9]: hyperparameter tuned on one $W$ labeled example per class on $A\rightarrow$W task (see [@long2016nips])
[^10]: \[resnet18Net\]with ResNet-18 network
[^11]: semi-supervised for some classes, but evaluated on 16 hold-out categories for which the labels were not seen during training
[^12]: using 30,000-dimensional feature vectors from marginalized stacked denoising autoencoders (mSDA) by Chen et al. [@chen2012marginalized], which is an unsupervised method of learning a feature representation from the training data
[^13]: \[features5000\]using 5000-dimensional unigram and bigram feature vectors
[^14]: using bag-of-words feature vectors including only the top 400 words, but suggest using deep text features in future work
[^15]: the best results on target data for various hyperparameters
[^16]: using bag-of-words feature vectors
[^17]: <http://yann.lecun.com/exdb/mnist/>
[^18]: \[ganinsite\]See Ganin’s website <http://yaroslav.ganin.net/> for links to download.
[^19]: This can be found on various sites and some Github repositories. One such place: <https://web.stanford.edu/~hastie/ElemStatLearn/data.html>
[^20]: <http://ufldl.stanford.edu/housenumbers>
[^21]: The synthetic dataset linked to on: <http://graphics.cs.msu.ru/en/research/projects/imagerecognition/trafficsign>
[^22]: <http://benchmark.ini.rub.de/?section=gtsrb&subsection=dataset>
[^23]: <http://ai.bu.edu/adaptation.html>
|
---
abstract: 'This paper takes a critical look at the usefulness of power law models of the Internet. The twin focuses of the paper are Internet traffic and topology generation. The aim of the paper is twofold. Firstly it summarises the state of the art in power law modelling particularly giving attention to existing open research questions. Secondly it provides insight into the failings of such models and where progress needs to be made for power law research to feed through to actual improvements in network performance.'
address:
- |
Department of Electronics and Electrical Engineering, University College London,\
email: [email protected]
- |
BT Research,\
email: [email protected]
- |
Department of Computer Science, University College London,\
email: [email protected]
author:
- 'Richard G. Clegg'
- 'Carla Di Cairano-Gilfedder'
- Shi Zhou
bibliography:
- 'networking.bib'
title: A critical look at power law modelling of the Internet
---
Internet, power laws, heavy-tails, long-range dependence, scale-free networks, network modelling
Introduction
============
Power laws describe a wide range of phenomena in nature and a large body of ongoing research investigates their applicability in fields such as computer science, physics, biology, social sciences and economics. Power law distributions are characterised by a slower than exponentially decaying probability tail, which loosely means that large values can occur with a non-negligible probability (see the next section for formal definitions). They can be used to characterise a variety of relations such as for example the distribution of income, city population, citations of scientific papers, word frequencies, computer file sizes and the number of daily hits to a given website. See [@mitzenmacher04] and [@newton05] and references therein for further examples.
The aim of this paper is not to be a general survey of power laws in networks but instead to be a critical look at open questions and the outcome of such research, in particular with regard to the question “How can power law modelling improve network performance?" The paper looks at two separate areas where power law research has been of interest in the Internet. The study of power laws in the analysis of Internet traffic characteristics has been ongoing since 1993 and in Internet topology generation since 1999.
In 1993, the seminal paper [@leland1993] (expanded in [@ltww94]) found evidence of the existence of power law relationships in network traffic by observing long-range correlation in Local Area Network (LAN) traffic. This brought the concept of self-similarity, and the related concept of Long-Range Dependence (LRD), into the field of network traffic and performance analysis. Before this finding, network traffic and performance studies had been mainly based on models, such as Poisson processes, which assume that traffic exhibits no long-term correlation. In networks with long-range correlated traffic, queuing performance can very different to that of traffic assumed independent or only having short-term correlations. Subsequently power law relationships have been observed in several other contexts on many different types of network.
In 1999 it was also discovered that the global Internet structure is characterised by a power law [@Faloutsos99]. That is, the probability distribution of a node’s connectivity (measured for example by the number of BGP peering relations that an autonomous system has) follows a power law. This discovery invalidated previous Internet models that were based on the classical random graphs. Since then a lot of efforts have been put into studying the Internet power law structure [@strogatz01; @krapivsky01; @albert02; @bornholdt02; @subramanian02; @Chen02; @dorogovtsev03a; @caldarelli03; @Pastor04; @chang06].
This paper reviews the measurements and models of the Internet topology, and comments upon whether the power law is in itself an adequate characterisation of the system. It questions whether models based on power laws provide a suitable platform for theoretical and simulation analysis of the Internet’s traffic and topological characteristics. Finally, it provides discussion of how such research could be of use in improving network performance (which, after all, should be the ultimate goal of networking research).
The structure of the paper is as follows. Section \[sec:defns\] provides the basic mathematical definitions used throughout the paper: heavy-tailed distributions, long-range dependence and statistical self-similarity. Section \[sec:traffic\] describes the use of power law relationships to model the statistical nature of Internet traffic. Section \[sec:topology\] discusses “scale-free networks" a power law relationship which describes the connectivity of networks.
Basic definitions {#sec:defns}
-----------------
In the sense meant in this paper, a power law relationship is a function, $f(x)$ with the form $f(x) = \alpha x^ \beta$ where $\alpha$ and $\beta$ are non zero constants. Several relationships of interest in the Internet have been shown to have this form asymptotically (usually as $x \rightarrow \infty$).
A random variable $X$ (which may be continuous or discrete) is said to have a [*heavy-tailed*]{} distribution if it satisfies $${{\mathbb{P}}\left[X > x\right]} e^{\varepsilon x} \rightarrow \infty, \text{ as }
x \rightarrow \infty.$$
Often a specific power law form is assumed for the distribution: $${{\mathbb{P}}\left[X > x\right]} \sim C x^{- \alpha},$$ for some $C > 0$ and some $\alpha \in (0,2)$. The symbol $\sim$ here and for the rest of this paper means [*asymptotically equal to*]{}, that is $f(x) \sim \phi(x) \leftrightarrow f(x)/\phi(x) \rightarrow 1$ as $x
\rightarrow
\infty$ (or occasionally, some other limit).
Let $\{X_1, X_2, \dots \}$ be a time series. The series is [*weakly-stationary*]{} if it has a constant and finite mean (${{\text{E}}\left[X_i\right]} = \mu$ for all $i$, where ${\text{E}}$ means expectation) and for all $i, j \in {\mathbb{N}}$ the covariance between $X_i$ and $X_j$ (i.e. ${{\text{E}}\left[(X_i - \mu)(X_j - \mu)\right]}$) depends only on $|j - i|$.
Weak stationarity is assumed for much that follows but in practice is not met by real network traffic over all timescales (for example, over a sufficiently long time the mean traffic level is not stationary, it varies with daily and weekly periodicity) and may not be met at all [@cleveland2000; @cao2001].
If the time series is weakly-stationary then the Auto Correlation Function (ACF) $\rho(k)$ is given by $$\rho(k) = \frac{{{\text{E}}\left[(X_t - \mu)(X_{t+k} - \mu)\right]}}{\sigma^2},$$ where $\mu$ is the mean and $\sigma^2$ is the variance.
The ACF allows the definition of [*long-range dependence*]{} which is sometimes called [*long memory*]{} or [*strong dependence*]{}. A standard reference on the topic is [@b94]. A commonly used definition is the following.
\[defn:lrd\] A weakly stationary time series is [*long-range dependent*]{} if the sum of the autocorrelation over all lags $\sum_{k=1}^\infty \rho(k)$ diverges.
\[defn:hurst\] The [*Hurst parameter*]{} is a commonly used measure of LRD. This makes the assumption that the ACF follows the specific functional form $$\rho(k) \sim C_\rho k^{-\alpha} = C_\rho k^{2-2H},
\label{eqn:lrdacf}$$ where $C_\rho > 0$ and $\alpha \in (0,1)$ and $H \in (1/2,1)$ is the [*Hurst parameter*]{}.
Note that sometimes it is this and not Definition \[defn:lrd\] which is taken as the definition of LRD. Other measures of LRD include Hurstiness [@ganesh2004 Chapter 8] and the ‘strength’ parameter used by [@cao2001].
LRD processes which meet Definition \[defn:lrd\] but not Definition \[defn:hurst\] will have no well-defined Hurst parameter. The value $H=1/2$ is usually taken to mean independent or short-range dependent data. Values of $H \in (0, 1/2)$ are sometimes termed anti-long-range dependence. Values of $H \leq 0$ or $H \geq 1$ do not give useful models [@b94 Section 2.3].
LRD can also be considered in the frequency domain. In this case, the characteristic of LRD is a pole in the spectral density (usually at zero).
Let $Y_t$ be a stochastic process in continuous time $t \geq 0$. The process is [*exactly self-similar*]{} with self-similarity parameter $H$ if for any choice of constant $c > 0$, the rescaled process $c^{-H}Y_{ct}$ is equal in distribution to the original process $Y_t$.
Note that a similar definition can be given for discrete time stochastic process.
\[defn:2nd\_order\_ss\] Let $Y_t$ be a stochastic process and $Y^{(m)}_t$ be the process derived from it by $Y^{(m)}_t = \frac{1}{m} \sum_{i=tm - (m-1)}^{tm}
Y_i$. A process $Y_t$ is [*exactly second-order self-similar*]{} if, for all $m$, the process $\{m^{1-H}Y^{(m)}_t\}$ has the same variance and autocorrelation function as $Y_t$. That is to say, for all $k \in {\mathbb{N}}$ and $m \in {\mathbb{N}}$, $${\text{var}\left(Y_k^{(m)}\right)} = \frac{{\text{var}\left(Y_k\right)}}{m^{2-2H}}$$ and $$\label{eqn:2nd_order_ss}
\rho^m(k) = \rho(k),$$ where $\rho(.)$ is the ACF of $Y_t$ and $\rho^m(.)$ is the ACF of $Y^{(m)}_t$.
Second-order self-similarity can also be defined in terms of the second central difference operator [@kriesten1999]. A process $Y_t$ is [*asymptotically second-order self-similar*]{} if holds as $k \rightarrow \infty$.
Finally it remains to define scale-free (or power law) networks.
\[defn:scale\_free\] Let $G$ be an undirected graph. Let $P_k$ be the probability that a randomly selected node in $G$ has degree $k$. The graph $G$ is [*scale-free*]{} if $P_k$ (the node-degree distribution) is heavy-tailed if: $$P_k \sim C k^{-\alpha},$$ where $C >0$ is a constant and $\alpha \in (0,2)$.
Similar definitions can be constructed for a directed graph. The in-node degree distribution and the out-node degree distributions are treated separately in this case.
A process which scales in a constant way is sometimes referred to as [*mono-fractal*]{}. A generalisation of this is a [*multi-fractal*]{} process which exhibits complex behaviour that changes over different timescales [@pw00]. When the multi-fractal behaviour can be approximated by a combination of two (or a small number of) monofractals then the process is sometimes described as having monofractal behaviour at different timescales rather than multifractal behaviour.
There are many connections to be made between these power laws, some more obvious than others. For example, a scale-free network is simply an example of a heavy-tailed distribution (as its node-degree distribution is heavy-tailed).
One connection which is sometimes less than clear from the literature is that exact second-order self-similarity as in Definition \[defn:2nd\_order\_ss\] implies LRD of the form given by . LRD of the form in implies asymptotic self-similarity. The details of this relationship can be found in [@kriesten1999] and [@gubner2005]. There is a more subtle connection between self-similarity and long-range dependence. If a self-similar process $Y_t$ has stationary increments and $H \in (0,1)$ then it can be shown (see [@b94 page 51]) that the increment process given by $X_i = Y_i - Y_{i - 1}$ for $i \in
{\mathbb{N}}$ has an ACF given by $\rho(k) \sim H(2H-1)k^{2H - 2}$, which implies that for $H \in (1/2,1)$ then the increment process is long-range dependent.
The connection between heavy-tails and long-range dependence is more subtle. One such connection is [@heath1998 Theorem 4.3] which states that in an on/off process with heavy-tailed on periods and off periods which fall off faster is a long-range dependent process. Other connections between power laws can be found in [@cb95; @wtsw97]. These papers show that multiplexing a high number of independent on/off sources with heavy-tailed strictly alternating on and/or off periods gives rise to self-similarity.
Power laws and Internet traffic {#sec:traffic}
================================
Nearly fifteen years ago, the seminal paper [@leland1993] found the existence of power law behaviour in Internet traffic. A time series describing LAN Ethernet packet traces at Bellcore showed evidence of second order self-similarity or long-range dependence. This paper for the first time questioned traditional modelling assumptions and showed that existing models (often based on Poisson processes) would not correctly estimate important characteristics of a network. Since this paper, many hundreds of papers have been published about the power law behaviour of Internet traffic. A recent edition of the journal [*Performance Evaluation*]{} was devoted to this topic and the editorial describes modelling of LRD and heavy-tails as “One of the most important research topics in performance modelling and evaluation in the last decade" [@liu2005].
Measuring long-range dependence
--------------------------------
The Hurst parameter is often used as an estimate of traffic’s LRD. This parameter however has to be used with prudence, as measuring traffic LRD and statistical self-similarity is a complex task which may be affected by many factors. Although, the estimation process can provide indication of the existence of long-range dependent characteristics, it does not unequivocally prove the existence of authentic LRD, as these characteristics may simply be due to traffic non-stationarity. In the time domain, the estimation of the Hurst parameter is characterised by the fall off of the ACF at high lag. However, the high lag measurements are those at which the fewest readings are available and the data is most unreliable. Similarly, in the frequency domain, the LRD is characterised by the behaviour of the spectrum at frequencies near zero, which are necessarily hard frequencies to measure. In terms of queuing performance, despite the common misconception, a high Hurst parameter does not always lead to worse performance or longer queues [@neidhardt98concept]. In fact, depending on the timescales of interest, traffic with a high Hurst parameter can lead to better performance than traffic with a low Hurst parameter. No single Hurst parameter estimator can be considered infallible, as this can hide LRD when it exists or create it when it does not [@kara04tenyears]. In addition, the Hurst parameter itself expresses the traffic scaling of the fluctuations around the mean and does not measure traffic burstiness.
It is certain that simply examining the ACF is not a robust way to estimate the Hurst parameter. In addition, a number of biases may be present in real-life data which could cause problems. These include periodicity (users and processes daily usage patterns) and trends (traffic volume changes throughout the measurement period) which violate the assumption of weak-stationarity. The topic of measuring LRD is beyond the scope of this paper, the reader is referred to [@taqqu1997; @bardet2003] for work which compares existing techniques.
Evidence for and against power law behaviour in Internet traffic
----------------------------------------------------------------
The original long-range dependence findings reported in 1993 [@leland1993] have subsequently been replicated in many different studies. In 1995, Floyd and Paxson [@pf95] found that WAN traffic is also consistent with self-similar scaling. These findings have been confirmed in the late nineties in [@cb97; @fgw98]. In particular, in [@cb97] the authors analyse WWW traffic and observe self-similarity in the patterns of recorded traffic and a heavy-tailed distribution in the sizes of the files transferred. They claim that heavy-tailed sizes of transferred files is the cause of the observed self-similarity. Also, in the late nineties, evidence was found that heavy-tailed distributions characterised a number of different measurements related to network traffic. In [@pkc96] the authors report on observations of heavy-tailed distributions of file sizes on web servers and also of CPU time taken by processes.
The paper [@hernandez04] analyses WWW flow duration distribution at a lightly utilised academic campus Internet access. It finds that the tail of the flow duration distribution does not stabilise. The suggestion is that the best fit to the data is with a power law which varies in time. In 2005, [@xia2005] also investigated the power law behaviour of WWW traffic and found evidence of self-similarity over a number of timescales.
The paper [@cao2001] is sometimes cited as evidence that LRD is not an important property of Internet traffic. The data they analyse was collected in 2000 on a 100 Mbps Bell Labs Ethernet link. Looking at inter-arrival times the authors find that when the traffic has more connections present the “strength" of the LRD is decreased. Note that this “strength" is not related to the Hurst parameter but could be considered analogous to the proportion of the traffic which exhibits LRD. Their conclusion is that as the number of connections in the network increases the traffic will remain long-range dependent, but that the strength of the LRD will be weaker, and the arriving traffic will look more like a Poisson process. In 2003, observations were recorded on university access links [@park05changetraf]. In the majority of the traces, the packet and byte count time series exhibit intermediate to heavy LRD, regardless of time of day or day of week. LRD is also found to be unaffected by traffic load and number of active connections. Therefore, in these access links, multiplexing of an increasing large number of TCP flows did not reduce correlation.
Behaviour at different timescales
---------------------------------
Some authors have claimed that different scaling behaviour occurs at different time scales. This matter still seems to be an open research question. LRD and self similarity are both “monofractal" models in the sense that they assume a constant scaling behaviour over all time-scales. Strictly speaking asymptotic self similarity and LRD only imply this behaviour in the limit (at high lags or low frequencies). Multi-fractal modelling allows this scaling behaviour to change at each time scale considered. The topic of multi-fractal modelling is beyond the scope of this paper. For a good introduction see [@riedi2003]. Some authors have claimed that a multi-fractal approach is necessary to replicate the behaviour of Internet traffic. However, others have argued that this is not the case and a mixture of different monofractal scalings at different timescales is necessary.
It has been argued (see [@rl97; @fgw98]) that protocol mechanisms (such as the TCP feedback mechanism) have the greatest impact at smaller timescales. At these timescales they claim that the traffic is consistent with multi-fractal scaling, but at larger timescales (larger than the typical RTT on the network being investigated) the traffic looks self-similar.
[@vehel97] compares the scaling behaviour of aggregated fractional Brownian motion processes with that of real traffic and concludes that it is not a good match and therefore suggests multifractal behaviour may be necessary to provide a good fit to real traffic traces.
In [@erramilli00performance] the relationship between wide-area traffic correlation and link utilisation is explored at different timescales. They find that at small timescales burstiness can impact on performance at low and intermediate utilisations, while correlations at larger timescales are more significant at intermediate and high utilisations.
The paper [@zhang03smalltime] analyses backbone traffic traces at multiple Tier 1 links and investigates its behaviour at small scales (less than one second). The presence of correlation at small timescales is attributed to the characteristics (and not the number) of the aggregated flows and affected by the presence of dense flows characterised by bursts of clustered packets. They conclude that, at small timescales, traffic has mainly a monofractal behaviour and even that the traffic is “almost independent" – this is a contrast to much of the other work discussed in this section.
The paper [@jiang04timeorigin] sheds more light as regards to the possible causes of traffic correlation at sub-RTT timescales on backbone links. This paper confirms the findings in [@zhang03smalltime] but in addition suggests that these clusters of bursts derive from TCP self-clocking mechanism and queuing delays. These cluster of bursts are produced by flows with large bandwidth-delay product relative to their window size.
Internet backbone traffic dated 2002–4 is also analysed in [@kara04tenyears] with the conclusion that the Poisson distribution can adequately model packet arrivals at smaller timescales (the threshold where behaviour changes from Poisson to LRD varies but is around 1000ms). It confirms the existence of LRD in packet and byte counts at timescales larger than a second.
In [@uhlig2004], the author considers the rate of TCP flow arrivals rather than the total traffic on a link. Several traces are investigated, collected between 1993 and 2002. The analysis finds different scaling behaviours over a range of timescales and concludes that the flow arrivals are uncorrelated at the smallest timescales, correlated at timescales between seconds and minutes and consistent with “LRD or self-similarity between minutes and hours" but non-stationary time-of-day behaviour prevails at longer time scales.
In [@hohn2005] the authors argue using analysis of several traces (taken between 1989 and 2002) that at longer timescales LRD is an appropriate model and at shorter timescales they refer to the behaviour as “pseudo-scaling", a process which gives the appearance of multifractality but “which does not have true multifractal scaling underlying it" – in other words that the multifractal scaling observed by earlier authors is unnecessary. The scale at which the behaviour changes differs according to the trace being examined.
In summary, consensus seems to have formed that LRD behaviour predominates when traffic is considered at a larger timescales (at least until the user related non-stationarity disturbs the observation). However, the shorter timescale behaviour is a matter for much debate with some authors suggesting something as simple as a Poisson model is adequate, others suggesting multi-fractal models are necessary and many taking positions in between these extremes.
Possible causes of long-range dependence in network traffic {#sec:causes}
-----------------------------------------------------------
The origins of power law behaviour in network traffic have not been unequivocally identified. Several possible causes for the presence of long-range dependence have been proposed in the literature. ***Heavy-tailed distributions***. A common suggestion is that the heavy-tailed nature of data transfers leads to LRD in the resultant traffic (see [@heath1998 Theorem 4.3]) Simulation studies have confirmed this experimentally: [@pkc96], for example, found it to hold over a range of link bandwidths and a range of buffer sizes. Also in [@fghw99] simulations show that heavy-tailed file sizes lead to self-similarity on large timescales but also that the delay behaviour interacts with the TCP feedback mechanism to greatly alter the structure of the traffic at shorter time scales. ***TCP protocol***. It has been argued [@vb00] that TCP congestion control alone can cause self-similarity regardless of the application layer traffic characteristics. This argument however is contested in [@figuerdo2005] which looks at the same data but over longer time scales and finds that it is not consistent with power law behaviour. Also [@hohn2002], by shuffling network samples, reordering traffic, and removing the effects of TCP mechanisms while leaving the effects of heavy-tailed traffic, is able to show that it is heavy-tailed traffic rather than TCP feedback mechanisms which leads to long-range dependence.
It has been suggested [@figueiredo2000auto] using evidence based upon Markov modelling that the TCP timeout mechanism can lead to “local long-range dependence" which they also refer to as “pseudo self-similarity", that is to say, self-similarity over a small number of timescales (note that this is not true self-similarity). The proposal in [@pe97] suggests that TCP retransmission mechanism can give rise to self-similarity. Also [@pkc96] concludes that TCP can preserve long-range dependence over time while [@veres03tcps] suggests that TCP can preserve correlation over space. ***Queuing/Routing effects***. Another possibility is that power law traffic arises as a result of the interaction of queues and routing on a network [@borella98measurement]. Simulation experiments shown that even when “packet inter-departure times are independent, arrival times at the destination exhibit LRD”, perhaps as a result of the routing algorithms . ***Multi-layers and timescales***. There is also the possibility that long-range dependence simply arises because of the combination of processes occurring at different timescales: user’s activity, session, and transmission processes. In fact, it can be seen that even under the assumption of Poisson distribution for all usage, session, and transmission processes, the mere presence of multiple layers may lead to correlated traffic [@rm99].
***Intrinsic traffic nature***. It has long been known that some types of traffic exhibit LRD at the source. For example, variable bit rate video traffic deriving from a single flow shows LRD in a time series of traffic [@gw94].
While no clear consensus has yet formed, many of the authors cited in this and the previous two sections agree that heavy-tails are the cause of the LRD observed in larger time scales. No consensus seems yet to have been reached on the behaviour of traffic at shorter time scales and this remains an important topic for traffic research. The lack of consensus in this is reflected in the number of possible causal models for the short timescale behaviour.
Effects on queuing {#sec:queuingeffects}
------------------
The effects of long-range correlated traffic on buffer dimensioning have been analysed by means of appropriate queuing models developed in [@azn95; @n94; @pm96; @tg98; @kherania2005] among others. These models apply to infinite buffers and only provide asymptotic results. Under the assumption of infinite length buffer and long-range dependent input traffic the main finding is that the distribution of queue length has slower than exponential decaying tail, as opposed to exponential observed for short-range dependent traffic. This decaying function has instead been described by other distributions such as for example a Weibull [@n94] and polynomial [@tg98]. In the case of finite buffer systems, it has been suggested that, in a network with long-range dependent traffic, the packet loss ratio is several orders of magnitude higher than with short-range dependent traffic [@chen95model]. The packet loss ratio could only be contained by choosing very large buffers which would have an impact on queuing delay [@chen95model]. However, in the mid-nineties other authors cast doubt on the usefulness of power law models of Internet traffic, by questioning the importance of capturing traffic long-range dependence in the case of finite buffers [@gb96; @re96]. They argue that correlation becomes irrelevant for small buffers and short timescales.
Traffic generation models {#sec:lrdgenmodels}
---------------------------
A variety of mathematical models have been suggested in the literature to capture the LRD in Internet traffic. For a comprehensive review of these models the reader is referred to [@pw00]. Only a short summary is provided here.
[*Fractional Brownian motion*]{} (fBm) is a non-stationary stochastic process which is a generalisation of the well-known Brownian motion, but with a dependence term between samples. It is a self-similar process and has a defined Hurst parameter H, with the Brownian motion obtained for $H =
1/2$. If $B_H(t)$ denotes the fBm then the difference process $Y_k(\cdot)$ defined as $Y_k(t) = B_H(t+k) - B_H(t)$ with $H \in (1/2, 1)$ is the [*fractional Gaussian noise*]{}(fGn) which is long-range dependent. Several methods exist for generating a fGn process, for example [@paxson1997].
Although fGn is mathematically attractive its simplicity means that it cannot capture a diversity of mathematical properties. The queue length distribution obtained with a fGn process decays according to the Weibull or “stretched exponential” distribution, which is heavy-tailed only in a weak sense [@roberts96].
[*Fractional Auto-Regressive Integrated Moving Average*]{} (FARIMA) [@b94 pages 59–66] models are an expansion of the classic time-series ARIMA models and allow modelling of long and short range dependence simultaneously and independently. Long-range dependence can also be generated by using [*chaotic maps*]{} as first proposed by Erramilli and Singh [@erramilli94chaotic]. However, modelling based on chaotic maps requires considerable experimentation, as these are very sensitive to initial conditions and their many parameters’ estimation is often a complex task [@rm99]. The queue length distribution obtained with a chaotic maps family has been found to decay according to the Weibull distribution [@pruthi95].
Another model is based on the superposition of [*heavy-tailed on/off*]{} sources [@wtsw97]. The process obtained by multiplexing many on/off sources with heavy-tailed distributions tends to a fGn process. Finally, another technique for modelling traffic is by means of [*Wavelet*]{} analysis [@burrus98]. This allows not only capturing the Hurst parameter but also synthesising a wide range of scaling behaviours and the replication of the multi-fractal spectrum [@riedi1999; @riedi2003].
An important criticism of these models is in their replication of queuing behaviour. While much work has been done to show that the models replicate certain representative traffic statistics, one of the primary motivations cited for using LRD models of queuing is estimating delays and buffer overflow probabilities. These models have not been shown to do this well, indeed while it has been shown that some mathematical models of LRD have very different queuing behaviour to non LRD versions of those models, it remains to be shown that LRD is necessary to replicate the queuing and delay performance of real traffic.
Criticisms and commentary
-------------------------
Although the majority of papers appear to replicate the finding that LRD is present in network traffic, some have questioned whether other models are more appropriate (for example multi-fractal models [@rl97; @fgw98]). Multifractals are in fact able to model varying scaling behaviour over different timescales, as they are characterised by a time dependent scaling coefficient. In addition, LRD appears at long timescales which are more relevant for network dimensioning and less for queuing behaviour. Others have also questioned whether LRD may be unimportant in practice, for example due to multiplexing gains [@cao2001].
Consensus seems to be forming on the origin of LRD behaviour (as discussed in section \[sec:causes\]) although some controversies remain. As regards to its effects, papers in the area often focus on the fact that LRD may impact on network performance by increasing delays or increasing the packet loss expected for a given buffer size. However this relationship is not a simple one and the presence of LRD does not always have a negative impact [@neidhardt98concept]. If a cause were unequivocally established the question would remain, “how might we go about eliminating LRD from the network given this cause?" If heavy-tailed file transfers are the cause then no clear method for resolving the problem is obvious. However, if TCP feedback mechanisms are a cause it would be difficult to change this without changing the protocol itself.
In order to understand the usefulness of power laws for practical studies, an important question to ask is whether LRD models generate traffic with the same queuing properties as real Internet traffic. If the models from Section \[sec:lrdgenmodels\] are to be useful then, when correctly tuned to the parameters of a genuine packet trace, they should have the same mean delay and buffer overflow probabilities as the genuine traffic. Huebner et al [@huebner1998] tested a Poisson model, a Weibull model, an autoregressive (AR(1)) model, a Pareto model, and a Fractional Brownian Motion model for generating traffic. None of the models tested produced a good match for queuing performance in all circumstances. The fBm model was useful only when the buffer size considered was large. Similarly, [@clegg2007] tests the queuing performance of fBm and three other LRD models based on Markov modulated processes as well as some non LRD models. The models are tuned so that their parameters (mean and Hurst parameter) match real network traffic and the queuing performance of each is tested in an infinite buffer simulation. In this case, none of the traffic models replicated the queuing performance of the real traffic and the LRD models often showed different performance from each other despite having the same mean and Hurst parameter. Of course, even if a model could be found which accurately reproduced a given queuing behaviour obtained with real traffic, this would not solve the entire problem since the statistical nature of Internet traffic arises at least in part from TCP feedback mechanisms, which in turn depends upon potentially changing traffic levels and congestion.
Theoretically, some interesting queuing theory results for systems with LRD input traffic have been achieved but these results are often asymptotic results for infinite buffer models. How applicable these would be in practical situations remains an open question although, of course, it may be hoped that future theoretical results will build on them.
Several questions therefore remain about LRD. Which LRD model, if any, is appropriate to generate traffic which has similar delay and buffer overflow probabilities to real Internet traffic when queued? Can future networks be designed to mitigate the potentially deleterious effects on performance which are said to result from LRD? Can analytical models be developed which give strong enough results to be practically applicable to real traffic on real networks?
Modelling Internet topology {#sec:topology}
============================
Topology is the connectivity graph of a network, upon which the network’s physical and engineering properties are based. The Internet contains millions of routers, which are grouped into tens of thousands of sub-networks, called Autonomous Systems (AS). The Internet topology can be studied at the router level and the AS level. Studies of the Internet topology very much depend on the availability and quality of measurement data. In the last decade a number of projects have provided more and more complete and accurate data on the Internet AS connectivity. By comparison it is more difficult to obtain router level data. So far there are more studies on the AS-level Internet topology than on the router-level.
In this paper the Internet topology is considered only at the AS level, in which a node is an AS network owned by an entity with a large Internet presence, such as an ISP or a large company; and a link represents a peering relationship between two AS nodes in the border gateway protocol (BGP) [@quoitin03]. Research on the structure and evolution of the Internet AS graph is relevant because the delivery of data traffic through the global Internet depends on the complex interactions between AS that exchange routing information using the BGP protocol.
Measuring Internet topology
---------------------------
Measurements of the Internet AS graph have been available since the late 1990s. There have been two types of measurements using different methodologies and data sources.
[*Passive measurements*]{} are constructed from BGP routing tables which contain information about links from an AS to its immediate neighbours. The Routing Information Service of RIPE [@ripe] is another important source of BGP data. The widely used BGP AS graphs are produced by the National Laboratory for Applied Network Research [@nlanr] and the RouteViews Project at the University of Oregon [@oregon]. They are connected to a number of operational routers on the Internet for the purpose of collecting BGP tables. The Topology Project at the University of Michigan [@michigan] has provided an extended version [@chang04] of the BGP AS graph by using additional data sources, such as the Internet Routing Registry (IRR) data and the Looking Glass (LG) data. BGP-based AS measurements may contain links that do not actually exist in the Internet, but a more serious problem is that the BGP measurements may miss a significant number of links [@he07].
[*Active measurements*]{} are based on the traceroute tool which sends probe packets to a given destination and captures the sequence of IP hops along the forward path from the source to the destination. The Internet research organisation CAIDA [@CAIDA] has developed a tool called [*skitter*]{} which probes around one million IPv4 addresses from 25 monitors around the world. Using the core BGP tables provided by RouteViews, CAIDA maps the IP addresses in the gathered traceroute data to AS numbers [@murray01] and constructs AS graphs on a daily basis. DIMES [@DIMES] is a more recent large-scale distributed measurement effort. It collects traceroute data by probing from more than $10,000$ software clients, installed by volunteers in over $90$ countries, to destinations assigned by a central server at random from a set of five million destination addresses. To further improve the completeness, DIMES merges the resulting AS graph with that of RouteViews. By using more monitors and a larger list of distinct addresses, DIMES produces larger AS graphs than skitter. The shortcoming of the traceroute measurements is that the translation from IP addresses to AS numbers is not trivial and could introduce many errors [@hyun03] and also, increasingly, firewalls block the probe packets. A recent study [@Oliveira07] suggested that traceroute measurements should probe destinations more frequently and avoid using a fixed list of destination addresses.
Power law degree distribution
-----------------------------
In graph theory, degree $k$ is defined as the number of links, or immediate neighbours, of a node. Degree is the principal parameter for characterising network connectivity. The first step in describing and discriminating between different networks is to measure the degree distribution $P(k)$, which is the probability of finding a node with degree $k$. In 1999 it was discovered that the Internet topology at the AS level (and the router level) exhibits a power law degree distribution $P(k)\sim C k^{-\gamma}$ [@Faloutsos99], where $C > 0$ is a constant and the exponent $\gamma\simeq 2.2\pm0.1$. This means on the Internet AS graph, a few nodes have very large numbers of links, whereas the vast majority of nodes have only a few links. Although different Internet AS graphs produced from different data sources vary in the numbers of nodes and links, all the Internet AS graphs are well characterised by a power law degree distribution [@mahadevan05b]. The power law distribution is an evidence that the Internet AS level topology has evolved into a complex, heterogeneous structure that is profoundly different from Internet models based on the random graph theory. This discovery profoundly changed the understanding of Internet topology. Since then there has been an international effort in characterising and modelling the Internet topology.
Power law or sampling bias?
---------------------------
A major problem of current measurements of the Internet AS graph is that these measurements, whether based on BGP, traceroute or other sources, miss a significant number of links [@chang04; @cohen06; @he07]. Some researchers suggested [@cohen06; @he07] that there could be as many as $35\%$ of the links in the AS level Internet that were still to be discovered. A series of papers [@Lakhina03; @Clauset05] reported that the traceroute type of measurement data collected from a small number of observers are not only incomplete but are possibly biased in such a way that graphs which in fact have Poisson degree distributions appear to exhibit a power law. There has been a debate on whether the power law degree distribution an integral property of the Internet AS graph or merely an artifact due to biased sampling methods.
There are two sides of the argument. Many researchers believe that the power law is an integral property of the Internet. Firstly all Internet AS graph measurements exhibit a power law degree distribution including the DIMES data which are collected from numerous observers distributed in thousands of AS networks around the world, as well as the BGP AS graph based on routing table data collected from many monitors and accumulated over many years. Secondly, a recent study [@DallAsta06] shows that if the larger real graph had a Poisson degree distribution and the observed power law were due to sampling bias, then the real graph’s average degree would be very large. In the case of the AS network the true average degree would have to be around one hundred. The observed average degree in the known sources is between five and seven so if this model were true it would require the unlikely proposition that less than one in ten edges have been observed. Surely this can not be true.
On the other hand, there are also many researchers who are sceptical about the power law degree distribution. Firstly, the visibility of the AS graph can be influenced to a great extent by which vantage points are used, not by how many. Secondly the analysis in [@DallAsta06] rejects the claim that the real AS graph may follow a Poisson degree distribution, but the real question is whether the Internet AS graph is characterised by a power law distribution or a different heavy-tail distribution which does not follow a power law.
Only better measurement data can settle this issue. The current situation is that all measurements are incomplete and bias in one way or another. There is an urgent need for improved methods to produce more complete and accurate data. A recent effort towards this direction is [@Oliveira07] which investigates both the completeness and the liveness problems in the measurement of Internet AS graph evolution.
Structures beyond the power law
-------------------------------
Degree distribution is a first-order topological property which is based on the connectivity information of individual nodes. When studying the Internet structure, it is important to look beyond the power law degree distribution because networks with exactly the same power law degree distribution can have completely different high order properties [@Tangmunarunkit02; @Li04; @Alderson05; @zhou07b].
High order properties are calculated on the connectivity information of a pair, a triad or a set of nodes. High order properties are able to explicitly determine lower order properties whereas the later only constrain the former. Researchers have introduced many high order topological properties, each of which has a distinct physical meaning, for example the degree-degree correlation [@Pastor01; @newman02; @newman03; @maslov04] which indicates whether high-degree nodes tend to connect with high-degree nodes (so-called ‘assortative mixing’) or low-degree nodes (‘disassortative mixing’); the rich-club coefficient [@Zhou04a; @zhou07b] which quantifies how tightly the best connected nodes connect with themselves; the clustering coefficient [@watts98] which measures the fraction of a node’s neighbours which are neighbours to each other; the average shortest path which is the average hop distance between any two nodes; the $k$-core decomposition [@Carmi07] which reveals a network’s underlying hierarchical structure; and the betweenness which measures how often a node or a link is on the shortest (fewest hop) path between two nodes.
The Internet topology can be describes a jellyfish [@tauro01], where a highly connected core is in the middle of the cap, and one-degree nodes form its legs. This intuitive model is simple yet very useful as it concisely illustrates a number of important properties of the Internet, including the dense core (rich-club) and the large number of low degree nodes (power law) which are directed connected with members of the core (disassortative mixing). The Internet has a small average distance between any two nodes because the rich-club functions as a ‘super’ traffic hub which provides a large selection of shortcuts for routing and the disassortative mixing ensures that the majority of network nodes, which are peripheral low-degree nodes, are always near the rich-club.
Our knowledge and understanding of the Internet topology have been improved significantly in recent years. However, it is still profoundly difficult to define the Internet topology and there are many unanswered questions: What are the key properties that fundamentally characterise the Internet topology? How do these properties relate to each other? What is the role each property plays on the network’s function and performance?
It is suggested [@Mahadevan06; @Mahadevan07] that for the Internet, the second order properties are sufficient for most practical purposes; while the third order properties essentially reconstruct the Internet AS and router level topologies exactly. A recent work [@zhou07b] pointed out that for the Internet the degree distribution and the rich-club coefficient restrict the degree-degree correlation to such a narrow range, that a reasonable model for the Internet can be produced by considering only the degree distribution and the rich-club coefficient. Note that although these studies provide new clues on how to choose topological properties for consideration in modelling the Internet topology, they do not constitute a ‘canonical’ set of metrics that are most relevant for the network’s function and performance.
Modelling Internet topology {#modelling-internet-topology}
---------------------------
Since the discovery of the power law degree distribution, a number of models have been proposed to generate Internet-like graphs [@krapivsky01; @albert02; @dorogovtsev03; @Pastor04; @Mitzenmacher05; @leskovec07]. Models from networking community, such as Tier, BRITE [@medina00], GT-ITM (Transit-Stub) and Inet [@winick02], often suffer from problems of no (or an incorrect) power law, inaccurate large-scale hierarchy, requiring parameter estimation or providing a mechanism for network evolution; and models from physicists [@Barabasi99; @dorogovtsev00; @bu02; @bianconi03; @caldarelli03; @krapivsky00] also have problems as they often are too general and do not incorporate any real network specifics.
In general there are two main approaches for generating topologies of complex networks [@rrfs05]. The equilibrium (top-down) approach is to construct an ensemble of static random graphs reproducing certain properties of observed networks and then to derive their other properties by the standard methods. The non-equilibrium approach (bottom-up) tries to mimic the actual dynamics of network growth: if this dynamics is accurately captured, then the modelling algorithm, when let to run to produce a network of the required size, will output the topology coinciding with the observations. It is clear that the more ambitious non-equilibrium approach has the potential to hold the ultimate truth. Classic examples of this approach include the Barabási-Albert (BA) model [@Barabasi99] and the HOT model [@Li04]. Many models owe their origins to the preferential attachment approach where new links attach to nodes with a probability proportional to the degree of that node.
The Positive-Feedback Preference (PFP) model proposed in 2004 [@Zhou04d] is an example of the non-equilibrium models for the Internet. The model is an extensive modification of the BA model. It is able to reproduce a large number of characteristics (including all topological properties mentioned above) of the Internet AS topology [@krapivsky08; @haddadi08; @zhou07a]. It uses two growth mechanisms inspired by observations on the Internet history data [@Pastor01; @vazquez02; @park04]. Firstly, the model starts from a small random graph and grows by two coupled actions called *interactive growth*, i.e. the attachment of new nodes to old nodes in the existing system and the addition of new links between these old nodes to other old nodes. This resembles the dynamics that when an Internet service provider (ISP) acquires new customers it reacts by increasing its number of connections to peering ISPs. Secondly, the preference probability that node $i$ acquires a new link (from a new node or a peer) is given as a function of the node’s degree $k_i$, $$\Pi(i) =
\frac{k_i^{1+\delta\log_{10}{k_i}}}
{\sum_j k_j^{1+\delta\log_{10}{k_j}}}, \delta=0.048.
\label{eq:PFP}$$ This is called the *positive-feedback preference*, which means a node’s ability of competing for a new link increases more and more rapidly with its growing number of links, like a positive-feedback loop. The consequence is that ‘the rich not just get richer, they get disproportionately richer’. This mechanism resembles the ‘winner-takes-all’ trend in the Internet development. More recently Chang *et al* [@chang06] proposed another bottom-up approach for generating Internet AS graph, where the Internet evolutionary process is modelled by identifying a set of criteria that an AS considers either in establishing a new peering relationship or in reassessing an existing relationship.
Practical responses to Internet power law modelling
---------------------------------------------------
It is suggested [@handley06] that the Internet power law structure is relevant to a number of issues, such as the severely biased distribution of traffic flow, the slow convergence of BGP routing tables [@labovits01] and the large-scale cascading failure caused by incidents or deliberate attacks [@park03]. As such, the power law property also provides novel insights into the solutions of these problems. For example it is shown that the power law property makes it possible to mitigate the distributed denial of service (DDoS) attacks by implementing route-based filtering on less than 20% of AS [@park01]; a compact routing scheme based on the power law property requires a significantly smaller routing table size [@krioukov04], and the power law property is relevant to the epidemic threshold for a network [@wang03].
Albert et al [@Albert2000] have reported that scale-free networks, i.e. networks having power law degree distributions, are robust to random failures but fragile to targeted attacks. This widely publicised work has generated a wave of studies on the robustness of various networks. This work, however, has generated some confusion in the Internet community. It should be noted that the Internet is much different from the generic BA model used in that study. Firstly the Internet AS topology does not follow a strict power law as in the BA model and the Internet’s high order topological properties are also significantly different from the BA model. Secondly it is unrealistic, if possible at all, to ‘attack’ an AS node, i.e. to wipe out an entire AS network and cut off all its connections with other networks. This is because an AS node can represent a network of thousands of routers which can spread across a number of continents. And finally it is important to realise that links on the Internet AS graph can represent different commercial relationships between AS networks, such that a ‘path’ of adjacent links between AS nodes on the Internet AS graph does not necessarily imply routing ‘reachability’ between the two AS. For example a customer AS does not transit traffic for its providers.
Criticisms and commentary
-------------------------
The discovery of a power law degree distribution in the Internet topology has attracted a huge amount of attention and there have been tremendous efforts to measure, characterise and model the Internet topology. Recent debate suggested that whether the power law degree distribution is an integral property of the Internet is still an open question. It is vital for researchers to look beyond the power law property and appreciate high order properties of the Internet topology.
There are generative models which well reproduce the Internet topology as a pure graph. However the reachability between two AS nodes on the Internet is not only affected by the underlying connectivity graph, but also constrained by many other factors, such as routing policies, capacities, demo-geographic distributions and local structures. Future Internet models should more closely reflect the Internet specifics in order to produce practically useful results.
As pointed out in [@Krioukov07], there is a need for more interdisciplinary communication among computer scientists, mathematicians, physicists and engineers. Such communication is much needed to facilitate the interdisciplinary flow of knowledge and enable the network research community to convert theoretical results into more practical solutions that matter for real networks, e.g. performance, revenue and engineering.
Conclusions {#sec:conclusions}
===========
An obvious question arising from this paper is whether there is a connection between the power law topology and the power laws observed in traffic levels. One likely mechanism for such an interaction would come from considering how traffic aggregates as a result of the topology. A starting point might be the work reported in [@arrow04] which combines power law topologies with simulations involving LRD sources. Further research in this area might well be fruitful.
Most authors agree that power law relationships are present in measurements of network traffic. Measurements of file size transfers appear consistent with a heavy-tailed distribution. Measurements of traffic levels per unit time and packet inter-arrival times fit the hypothesis of LRD. However, this only describes the long time-scale behaviour (at least below the time-scale where day-to-day non-stationarity affects measurement). The behaviour of network traffic at shorter time scales is still an open question with different authors proposing different models. On the origin of long-range dependence, consensus appears to have formed that heavy-tailed distribution of file sizes is the major cause but with alterations to the short term behaviour arising from TCP protocol interactions. However, some authors give other explanations for the short term behaviour and the matter cannot yet be said to be definitively settled.
While many models have been proposed which generate traffic with the appropriate power law behaviour, it remains to be shown which of these, if any, best fits real traffic traces. In particular, if LRD is of relevance for queuing and buffer behaviour, it is key that the model selected replicates the queuing performance of the real traffic and this is an important shortcoming. The models proposed to describe queuing behaviour with long-range dependent input traffic suggest that the tail of the queue occupancy distribution decays slower than exponentially. If the study of power laws is to result in a positive effect on network traffic engineering then: 1) it is important to find a power law based traffic generation model which replicates the queuing performance of the real traffic. 2) progress needs to be made in ways to either mitigate the effects of LRD or to plan a network by allowing for it.
Researchers have also made progress on measuring and modelling the Internet topology at the AS-level. More complete and accurate measurement data are needed to justify whether the power law degree distribution is indeed an integral property of the Internet. Much more research work is needed, for example, to identify the key topological properties that fundamentally characterise the Internet structure and to include the Internet specifics in topology models. It is encouraging that the power law modelling of Internet topology have begun to stimulate research which takes advantage of this network structure. There is an increasing recognition that effective engineering of the global Internet should be based on a detailed understanding of issues such as the large-scale structure of its underlying physical topology, the manner in which it evolves over time, and the way in which its constituent components contribute to its overall function [@floyd03].
In summary, for the research in power laws to truly have an engineering impact on the Internet, reliable and calibrated models are needed which match the characteristics of real data. It could be argued that there has been a certain level of success for topology generation but certainly not for traffic generation. The models should be capable of application as a design tool to allow engineers to improve real life network performance. As yet, this stage of research appears elusive in both fields.
|
---
abstract: 'The PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC) has measured electrons with $0.3 < p_{\rm T} < 9$ GeV/$c$ at midrapidity ($|y| < 0.35$) from heavy flavor (charm and bottom) decays in collisions at = 200 GeV. The nuclear modification factor $R_{\rm AA}$ relative to collisions shows a strong suppression in central collisions, indicating substantial energy loss of heavy quarks in the medium produced at RHIC energies. A large azimuthal anisotropy, $v_2$, with respect to the reaction plane is observed for $0.5 < p_{\rm T} < 5$ GeV/$c$ indicating substantial heavy flavor elliptic flow. Both and $v_2$ show a dependence different from those of neutral pions. A comparison to transport models which simultaneously describe $R_{\rm AA}(p_{\rm T})$ and $v_2(p_{\rm T})$ suggests that the viscosity to entropy density ratio is close to the conjectured quantum lower bound, [*i.e.*]{} near a perfect fluid.'
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title: Energy Loss and Flow of Heavy Quarks in Collisions at = 200 GeV
---
Experimental results from the Relativistic Heavy Ion Collider (RHIC) have established that dense partonic matter is formed in Au+Au collisions at RHIC [@wp_phenix; @wp_brahms; @wp_phobos; @wp_star]. Strong suppression observed for $\pi^0$ and other light hadrons at high transverse momentum ($p_{\rm T}$) [@ppg003; @ppg014; @sup_star; @ppg051] indicates partonic energy loss in the produced medium. The azimuthal anisotropy $v_2(p_{\rm T})$ [@ppg022; @star_v2_1] provides evidence that collective motion develops in a very early stage of the collision ($\tau\lesssim$ 5 fm/$c$), in accordance with hydrodynamical calculations [@hydro; @Hirano:2005wx]. The comparison of $v_2$ with several such models suggests [@Shuryak:2003xe; @Gyulassy:2004zy; @Kolb:2003dz] that the matter formed at RHIC is a near-perfect fluid with viscosity to entropy density ratio $\eta/s$ close to the conjectured quantum lower bound [@Kovtun:2004de]. Energy loss and flow are related to the transport properties of the medium at temperature $T$, in particular the diffusion coefficient $D \propto \eta/(sT)$.
Further insight into properties of the medium can be gained from the production and propagation of particles carrying heavy quarks (charm or bottom). A fixed-order-plus-next-to-leading-log (FONLL) perturbative QCD (pQCD) calculation [@fonll] describes the cross sections of heavy-flavor decay electrons in collisons at $\sqrt{s} = 200$ GeV within theoretical uncertainties [@ppg065]. In collisions the total yield of such electrons was found to scale with the number of nucleon-nucleon collisions as expected for point-like processes [@ppg035]. Energy loss via gluon radiation is expected to be reduced for heavy quarks due to suppression of forward radiation, thus increasing their expected thermalization time [@dk; @djord; @wied]. Consequently, a decrease of high suppression and of $v_2$ is expected from light to charm to bottom quarks, with the absolute values and their dependence sensitive to the properties of the medium. In contrast to these expectations a strong suppression of heavy-flavor decay electrons was discovered for $2 < p_{\rm T} < 5$ GeV/$c$ [@ppg056; @star_e], together with nonzero electron $v_2$ for $p_{\rm T} < 2$ GeV/$c$ [@ppg040].
This Letter presents spectra and the elliptic flow amplitude $v_2^{\rm HF}$ of electrons, $(e^++e^-)/2$, from heavy-flavor decays at midrapidity in collisions at = 200 GeV. An increase in statistics by more than a factor ten and reduced systematic uncertainties compared to earlier data [@ppg035; @ppg040; @ppg056] greatly extend the $p_{\rm T}$ range both for the determination of the centrality dependence of $R_{\rm AA}$ and for the measurement of $v_2^{\rm HF}$.
The data were collected by the PHENIX detector [@phenix] in the 2004 RHIC run. The minimum bias trigger and the collision centrality were obtained from the beam-beam counters (BBC) and zero degree calorimeters [@wp_phenix]. After selecting good runs, data samples of 8.1 and 7.0 $\times$ 10$^8$ minimum bias events in the vertex range $|z_{\rm vtx}| < 20$ cm are used for the spectra and $v_2$ analyses, respectively.
Charged particle tracks are reconstructed with the two PHENIX central arm spectrometers, each covering $\Delta\phi = \pi/2$ in azimuth and $|\eta| < 0.35$ in pseudo-rapidity [@phenix]. Tracks are confirmed by matching showers in the electromagnetic calorimeter (EMCal) within $2\sigma$ in position. Electron candidates have at least three associated hits in the ring imaging Čerenkov detectors (RICH) and fulfill a shower shape cut in the EMCal, where they deposit an energy, $E$, consistent with the momentum ($E/p - 1 > -2\sigma$). Below the Čerenkov threshold for pions ($p_{\rm T} < 5$ GeV/$c$) electron mis-identification is only due to random coincidences between hadron tracks and hits in the RICH. This small background ($<20$% at low in central collisions, less towards high and peripheral events) is subtracted statistically using an event mixing technique. Requiring at least five hits in the RICH and tightening the shower shape cut extends the electron measurement to 9 in $p_{\rm T}$, with negligible hadron background for $p_{\rm T} < 8$ GeV/$c$ and a hadron contamination of 20% for $8 < p_{\rm T} < 9$ GeV/$c$. The raw spectra are corrected for geometrical acceptance and reconstruction efficiency determined by a GEANT simulation. The centrality dependent efficiency loss $<2$% ($\approx23$%) for peripheral (central) events is evaluated by reconstructing simulated electrons embedded into real events.
The inclusive electron spectra consist of (1) “non-photonic” electrons from heavy-flavor decays, (2) “photonic" background from Dalitz decays and photon conversions (mainly in the beam pipe), and (3) “non-photonic” background from $K\rightarrow e\pi\nu$ ($K_{e3}$) and dielectron decays of vector mesons. Contribution (3) is small ($<$10% for $p_{\rm T}<$ 0.5 GeV/$c$, $<$2% for $p_{\rm T}>$ 2 GeV/$c$) compared to (2). The heavy-flavor signal and the ratio of non-photonic to photonic electrons, $R_{\rm NP}$, are determined via two independent and complementary methods described in detail in [@ppg065], where the identical detector configuration was used. At low $p_{\rm T}$ ($p_{\rm T} < 1.6$ GeV/$c$), where the heavy-flavor signal to background ratio is small (S/B $<$ 1), the “converter subtraction” method is used which employs a photon converter of 1.67% radiation length ($X_0$) installed around the beam pipe for part of the run. The converter multiplies the photonic background by a known, nearly independent factor $R_\gamma \sim 2.3$. The photonic background can then be determined by comparing the inclusive electron yield with and without the converter. For higher $p_{\rm T}$, where S/B is large, the “cocktail subtraction" method [@ppg056] is used. Here the background is calculated with a Monte Carlo hadron decay generator and subtracted from the data. At low the dominant background source is the $\pi^0$ Dalitz decay, which is calculated for each centrality using measured pion spectra [@ppg014; @ppg026] as input. In good agreement with measured data [@ppg051], the spectral shapes of other light hadrons $h$ ($\eta$, $\rho$, $\omega$, $\phi$, $\eta'$) are derived from the pion spectrum assuming a universal shape in $m_T = \sqrt{p_{\rm T}^2 +m_h^2}$ with a fixed constant ratio at high $p_{\rm T}$. Photon conversions in the beam pipe, air and helium bags (total: $0.4\% X_0$) are also included, along with background from $K_{e3}$ decays and both external and internal conversions of direct photons which are important for $p_{\rm T} > 4$ GeV/$c$. The agreement within the systematic uncertainties in the overlap region $0.3 < p_{\rm T} < 4$ GeV/$c$ of these two methods demonstrates that the absolute value of photonic backgrounds in the PHENIX aperture is well-understood.
The $v_2$ of inclusive electrons, $v_2^{inc}$, is measured as $v_2^{inc}=\langle \cos[2(\phi-\Phi_R)] \rangle / \sigma_R$ [@rpm], where $\Phi_R$ is the azimuthal orientation of the reaction plane measured with the resolution $\sigma_R$ using the BBC [@ppg022]. Since $\sigma_R$ is centrality dependent, $v_2$ is determined for narrow centrality bins (10%) and then averaged to calculate $v_2$ for minimum bias events. The $v_2$ of random hadronic background is subtracted statistically as described in [@ppg040].
The $v_2^{non-\gamma}$ of non-photonic electrons is obtained by subtracting the photonic electron $v_2^\gamma$ as: $v_2^{non-\gamma} = ((1+R_{\rm NP})v_2^{inc}-v_2^\gamma)/R_{NP}$. Here $v_{2}^\gamma$ is calculated via a Monte Carlo generator that includes $\pi^0$, $\eta$, and direct photons. The measured $v_2(p_{\rm T})$ of $\pi^\pm$,$\pi^0$ and $K^\pm$ [@ppg022; @ppg046] is used as input, assuming $v_2^{\pi^\pm}=v_2^{\pi^0}$, $v_2^\eta = v_2^{K^\pm}$, and $v_2^{{\rm direct}\gamma}=0$. A direct measurement of $v_2^\gamma$ using the converter subtraction method confirms the calculation within statistical uncertainties. The resulting $v_2^{non-\gamma}$ has a small contribution from $K_{e3}$ background which is simulated and subtracted to obtain $v_2^{\rm HF}$ of heavy-flavor decay electrons.
Three independent categories of systematic uncertainties are considered. (A) The inclusive electron spectra include uncertainties in the geometrical acceptance (5%), the reconstruction efficiency (3%), and the embedding correction ($\le$4%). (B) Uncertainties in the converter subtraction are mainly given by the uncertainty in $R_\gamma$ (2.7%) and in the relative acceptance of runs with and without the converter being installed (1%). (C) Uncertainties in the cocktail subtraction rise from 8% at $p_{\rm T} = 0.3$ GeV/$c$ to 13% at 9 GeV/$c$, dominated by systematic errors in the pion input and, at high $p_{\rm T}$, the direct photon spectrum. The $v_2$ measurement includes a systematic uncertainty of 5% due to the reaction plane uncertainty.
Figure \[fig1\] shows the invariant spectra of electrons from heavy-flavor decay for minimum bias events and in five centrality classes. The curves overlayed are the fit to the corresponding data from collisions [@ppg065] with the spectral shape taken from a FONLL calculation [@fonll] and scaled by the nuclear overlap integral $\langle T_{\rm AA} \rangle$ for each centrality class [@ppg014]. The insert in Fig. \[fig1\] shows the ratio of electrons from heavy-flavor decays to background. It increases rapidly with $p_{\rm T}$, exceeding unity for $p_{\rm T} >
1.8$ GeV/$c$, reflecting the small amount of material in the detector acceptance which makes the accurate measurement of heavy-flavor electron spectra and $v_2^{\rm HF}$ possible.
For all centralities, the spectra agree well with the reference at low but a suppression with respect to develops towards high $p_{\rm T}$. This is quantified by the nuclear modification factor $R_{\rm AA} = dN_{Au+Au}/(\langle T_{AA} \rangle d\sigma_{p+p})$, where $dN_{Au+Au}$ is the differential yield in and $d\sigma_{p+p}$ is the differential cross section in in a given bin. For $p_{\rm T} < 1.6$ GeV/$c$, $d\sigma_{p+p}$, is taken bin-by-bin from [@ppg065], whereas a fit to the same data (curves in Fig. \[fig1\]) is used at higher $p_{\rm T}$, taking systematic uncertainties in $d\sigma_{p+p}$ and $T_{\rm AA}$ into account.
![\[fig1\]Invariant yields of electrons from heavy-flavor decays for different centrality classes and for collisions, scaled by powers of ten for clarity. The solid lines are the result of a FONLL calculation normalized to the data [@ppg065] and scaled with $\langle T_{\rm AA} \rangle$ for each centrality class. The insert shows the ratio of heavy-flavor to background electrons for minimum bias collisions. Error bars (boxes) depict statistical (systematic) uncertainties.](fig1.eps){width="1.0\linewidth"}
\[hbt\] ![\[fig2\]$R_{\rm AA}$ of heavy-flavor electrons with above 0.3 and 3 GeV/$c$ and of $\pi^0$ with $p_{\rm T} > 4$ GeV/$c$ as function of centrality given by $N_{\rm part}$. Error bars (boxes) depict statistical (point-by-point systematic) uncertainties. The right (left) box at $R_{\rm AA} = 1$ shows the relative uncertainty from the reference common to all points for $p_{\rm T} > 0.3 (3)$ GeV/$c$.](fig2.eps "fig:"){width="1.0\linewidth"}
Figure \[fig2\] shows for electrons from heavy-flavor decays for two different ranges as a function of the number of participant nucleons, $N_{\rm part}$. For the integration interval $p_{\rm T} > 0.3$ GeV/$c$ containing more than half of the heavy-flavor decay electrons [@ppg065] is consistent with unity for all $N_{\rm part}$ in accordance with the binary scaling of the total heavy-flavor yield [@ppg035]. For $p_{\rm T} > 3$ GeV/$c$, the heavy flavor electron decreases systematically with centrality, while larger than of $\pi^0$ with $p_{\rm T} > 4$ GeV/$c$ [@ppg014]. Since above 3 GeV/$c$ electrons from charm decays originate mainly from $D$ mesons with above 4 GeV/$c$ this comparison indicates a smaller suppression of heavy-flavor mesons than observed for light mesons in this intermediate $p_{\rm T}$ range.
 $R_{\rm AA}$ of heavy-flavor electrons in 0-10% central collisions compared with $\pi^0$ data [@ppg014] and model calculations (curves I [@Armesto:2005mz], II [@vanHees], and III [@Moore:2004tg]). The box at $R_{\rm AA} = 1$ shows the uncertainty in $T_{AA}$. (b) $v_2^{\rm HF}$ of heavy-flavor electrons in minimum bias collisions compared with $\pi^0$ data [@ppg046] and the same models. Errors are shown as in Fig. \[fig2\].](fig3.eps){width="1.0\linewidth"}
Figure \[fig3\] shows the measured and $v_2^{\rm HF}$ of heavy-flavor electrons in 0-10% central and minimum bias collisions, and our corresponding $\pi^0$ data [@ppg014; @ppg046]. The data indicate strong coupling of heavy quarks to the medium. While at low $p_{\rm T}$ the suppression is smaller than that of $\pi^0$, $R_{AA}$ of heavy-flavor decay electrons approaches the $\pi^0$ value for $p_{\rm T} > 4$ GeV/$c$ although a significant contribution from bottom decays is expected at high $p_{\rm T}$. The large $v_2^{\rm HF}$ indicates that the charm relaxation time is comparable to the short time scale of flow development in the produced medium. It should be noted that much reduced uncertainties and the extended $p_{\rm T}$ range of the present data permit the comparisons of $R_{AA}$ and $v_2$ of the heavy and light flavors.
More quantitative statements require theoretical guidance. Figure \[fig3\] compares the $R_{\rm AA}$ and $v_2$ of heavy-flavor electrons with models calculating both quantities simultaneously. A perturbative QCD calculation with radiative energy loss (curves I) [@Armesto:2005mz] describes the measured $R_{\rm AA}$ reasonably well using a large transport coefficient $\hat{q} = 14$ GeV$^2$/fm, which also provides a consistent description of light hadron suppression. This value of $\hat{q}$ would imply a strongly coupled medium. In this model the azimuthal anisotropy is only due to the path length dependence of energy loss, and the data clearly favor larger $v_2^{\rm HF}$ than predicted from this effect alone.
Figure \[fig3\] also shows that the large $v_2^{\rm HF}$ is better reproduced in Langevin-based heavy quark transport calculations [@vanHees; @Moore:2004tg]. A calculation which includes elastic scattering mediated by resonance excitation (curves II) [@vanHees] is in good agreement with both the measured and $v_2$. This is achieved with a small heavy quark relaxation time $\tau$ which translates into a diffusion coefficient $D_{HQ} \times (2\pi T) = 4$-$6$ in this model [@vanHees]. Energy loss and flow are also calculated in [@Moore:2004tg] in terms of $D_{HQ}$ (curves III). While this model fails to simultaneously describe the measured and $v_2$ with one value for $D_{HQ}$, the range for $D_{HQ}$ leading to reasonable agreement with or $v_2$ is similar to that from [@vanHees], again implying that small $\tau$ and/or $D_{HQ} \times (2\pi T)$ are required to reproduce the data. Note that $D_{HQ}$ provides an upper bound for the bulk matter’s diffusion coefficient $D$. Using the observation [@Moore:2004tg] that $D \approx 6 \times \eta/(\epsilon+p)$ with $\epsilon+p = Ts$ at $\mu_B=0$ provides an estimate for the viscosity to entropy ratio $\eta/s \approx (\frac{4}{3}-2)/4\pi$, intriguingly close to the conjectured quantum lower bound $1/4\pi$ [@bound]. This result is consistent with estimates obtained in the light quark sector from elliptic flow [@roy] and fluctuation analyses [@gavin].
The conjecture of a bound on $\eta/s$ [@Kovtun:2004de] was obtained using the anti-de Sitter-space/conformal-field-theory correspondence [@Maldacena:1997re; @Witten:1998zw], which exploits a duality between strongly coupled gauge theories and semiclassical gravitational physics. Recently, such methods were applied to estimate $\hat{q}$[@Liu] and $D_{HQ}$ in a thermalized plasma [@Herzog:2006gh; @Gubser:2006bz; @Friess:2006aw]. These authors also find a small diffusion coefficient $D_{HQ} \times (2\pi T) \sim 1$.
In conclusion, we have observed large energy loss and flow of heavy quarks in collisions at = 200 GeV. The data provide strong evidence for the coupling of heavy quarks to the produced medium. A short relaxation time of heavy quarks and/or a small diffusion coefficient are required by the data. A model comparison suggests a viscosity to entropy ratio of the medium close to the quantum lower bound, [*i.e.*]{} near a perfect fluid.
We thank the staff of the Collider-Accelerator and Physics Departments at BNL for their vital contributions. We acknowledge support from the Department of Energy and NSF (U.S.A.), MEXT and JSPS (Japan), CNPq and FAPESP (Brazil), NSFC (China), MSMT (Czech Republic), IN2P3/CNRS, and CEA (France), BMBF, DAAD, and AvH (Germany), OTKA (Hungary), DAE (India), ISF (Israel), KRF and KOSEF (Korea), MES, RAS, and FAAE (Russia), VR and KAW (Sweden), U.S. CRDF for the FSU, US-Hungarian NSF-OTKA-MTA, and US-Israel BSF.
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---
abstract: 'We propose Federated Generative Adversarial Network (FedGAN) for training a GAN across distributed sources of non-independent-and-identically-distributed data sources subject to communication and privacy constraints. Our algorithm uses local generators and discriminators which are periodically synced via an intermediary that averages and broadcasts the generator and discriminator parameters. We theoretically prove the convergence of FedGAN with both equal and two time-scale updates of generator and discriminator, under standard assumptions, using stochastic approximations and communication efficient stochastic gradient descents. We experiment FedGAN on toy examples (2D system, mixed Gaussian, and Swiss role), image datasets (MNIST, CIFAR-10, and CelebA), and time series datasets (household electricity consumption and electric vehicle charging sessions). We show FedGAN converges and has similar performance to general distributed GAN, while reduces communication complexity. We also show its robustness to reduced communications.'
author:
- |
Mohammad Rasouli, Tao Sun, Ram Rajagopal,\
Stanford University, Stanford, CA, 94305\
`{rasoulim, luke18, ramr}@stanford.edu`\
bibliography:
- 'example\_paper.bib'
title: 'FedGAN: Federated Generative Adversarial Networks for Distributed Data'
---
Introduction {#intro}
============
Generative adversarial network (GAN) is proposed by [@goodfellow2014generative] for generating fake data similar to the original data and has found wide applications. In lots of cases data is distributed across multiple sources, with data in each source being too limited in size and diversity to locally train an accurate GAN for the entire population of distributed data. On the other hand, due to privacy constraints data can not be shared or pooled centrally. Therefore, a distributed GAN algorithm is required for training a GAN representing the entire population; such distributed GAN also allows generating publicly accessible data [@yonetani2019decentralized]. Current distributed GAN algorithms require large communication bandwidth among data sources or between data sources and an intermediary (to ensure convergence) due to architectures that separate generators from discriminators [@augenstein2019generative; @hardy2018md]. But in many applications communication bandwidth is limited, e.g. energy, mobile communications, finance, and sales [@yang2019federated]. Communication-efficient distributed GAN is an open problem. We propose an architecture which places local discriminators with local generators, synced occasionally through an intermediary.
Communication-efficient learning across multiple data sources, which are also subject to privacy constraints, is studied under federated learning [@konevcnyEtal-16arXiv; @McmahanEtal-16arXiv]. Therefore, we refer to distributed communication-efficient GAN as federated GAN (FedGAN). FedGAN can extend GAN applications to federated learning. For example, in lots of cases, even the pooled dataset is not large enough to learn an accurate model, and FedGAN can help producing more data similar to the original data for better training [@bowles2018gan].
The major challenge in GAN algorithms is their convergence since cost functions may not converge using gradient descent in the minimax game between the discriminator and the generator. Convergence is also the major challenge in federated learning since each source updates local model using multiple stochastic gradient descents (SGDs) before syncing with others through the intermediary [@dinh2019federated]; it becomes even more challenging when data at different sources are not independent and identically distributed (non-iid) [@wang2019adaptive]. Convergence of distributed GANs is an open problem. We theoretically prove our FedGAN algorithm converges even with non-iid sources under equal and two time step updates. We connect results from stochastic approximation for GAN convergence and communication-efficient SGD for federated learning to address FedGAN convergence.
We experiment our FedGAN on popular toy examples in the GAN literature including 2D system, mixed Gaussians, and Swiss role, on image datasets including MNIST, CIFAR-10, and CelebA, and on time series data form energy industry including household electricity consumption and electric vehicles charging, to show its convergence, efficiency, and robustness to reduced communications. We use energy industry since it is not currently studied in the federated learning but often involves distributed data, subject to privacy constraints, with limited communication infrastructure, and mostly in time series format [@stankovic2016measuring; @balachandran2014bandwidth]. To the best of our knowledge, there is no other algorithm for communication efficient distributed GAN to compare ours with. We compare the performance of FedGAN with a typical distributed GAN that has local discriminators and one generator in the intermediary which communicate frequently (similar to that in [@augenstein2019generative]).
The rest of the paper is as follows. We first review the relevant literature (Section \[sec:liter\_rev\]). Then we propose our FedGAN algorithm (Section \[sec:algo\]), discuss its communication and computation properties (Section \[sec:communi\_comput\]), and theoretically prove its convergence even when the data sources are non-iid (Section \[sec:conv\_analy\]). Next, we run experiments on toy examples (Section \[sec:toy\]), image dataset (Section \[sec:exp\_img\]), and time series dataset (Section \[sec:exp\_energy\]). Finally, we point out to some observations in our experiments and some open problems (Section \[sec:concl\]). Part of proofs and experiments are in appendices.
Literature Review {#sec:liter_rev}
=================
This work relates to three literature, GAN convergence, distributed GAN, and federated learning.
Convergence of GAN has been studied through convergence of dynamical systems using stochastic approximation by modeling GAN as a continuous two-player game solvable by gradient-based multi-agent learning algorithms [@chasnov2019convergence]. With equal step sizes of generator and discriminator updates the problem is single time-scale stochastic approximation [@konda2004convergence] for which certain conditions are developed (and tested [@mescheder2018training]) for convergence of the ODE representing the stochastic approximation of GAN, e.g. Hurwitz Jacobian at equilibrium [@khalil2002nonlinear], negative definite Hessians with small learning rate [@nowozin2016f; @ratliff2013characterization], consensus optimization regularization [@nagarajan2017gradient], and non-imaginary eigenvalues of the spectrum of the gradient vector field Jacobian [@mescheder2017numerics]. With different step sizes of generator and discriminator updates the problem is two time-scale stochastic approximation [@borkar1997stochastic] for which convergence is shown under global [@borkar1997stochastic] and local asymptotic stability assumptions [@karmakar2017two; @borkar2009stochastic]. [@heusel2017gans] proposes a two time-scale update rule (TTUR) for GAN with SGD and shows convergence under the those stability conditions. All of the above papers are for centrally trained GAN, while our FedGAN algorithm is distributed.
Distributed GANs are proposed recently. For iid data sources, [@hardy2018md] proposes a single generator at the intermediary and distributed discriminators which communicate generated data and the corresponding error. Also, discriminators exchange their parameters occasionally to avoid overfitting to local data. [@hardy2018gossiping] utilizes a gossip approach for distributed GAN which does not require an intermediary server. For non-iid data sources, [@yonetani2019decentralized] trains individual discriminators and updates the centralized generator to fool the weakest discriminator. All of the above algorithms require large communications, while our FedGAN is communication efficient. Also, to the best of our knowledge there is no theoretical result for convergence of distributed GAN, while we provide such results for FedGAN.
Federated learning, proposed for communication efficient distributed learning, runs parallel SGD on randomly selected subset of all the agents and updates the parameters with the averages of the trained models through an intermediary once in a while. Its convergence is proved for convex objective with iid data [@stich2018local], non-convex objective with iid data [@wang2018cooperative], strongly convex objective and non-iid data with all responsive agents [@wang2019adaptive] and some non-responsive agents [@li2019convergence], and non-convex objective and non-iid data [@yu2019parallel]. Our FedGAN study is with non-convex objective and non-iid data, and also involves GAN convergence challenges on top of distributed training issues.
A distributed GAN with communication constraints is proposed by [@augenstein2019generative] under FedAvg-GAN which has distributed discriminators but centralized generator, similar to distributed GAN in [@hardy2018md] with the difference of selecting a subset of agents for discriminator updating. This approach does not fully address the large communications required for distributed GAN as it needs communications in each generator update iteration. We overcome this issue by placing both the discriminators and the generators at the agents, and then communicating only every $K$ steps with intermediary to sync parameters across agents. An architecture similar to our FedGAN is envisioned in [@Rajagopal2019FederatedAL] and [@hardy2018md], but they do not provide theoretical studies for convergence, and their experiment results are very limited. We provide a complete study in this paper.
FedGAN Algorithm
================
In this section we propose FedGAN algorithm, discuss its communication and computation complexity, and theoretically prove its convergence.
Model and Algorithm {#sec:algo}
-------------------
We denote the training iteration horizon by $N$ and index time by $n$. Consider agents $\{1,2,..., B\}$ with local dataset of agent $i$ denoted by $\mathcal{R}_i$ and weight of agent $i$ denoted by $p_i:=\frac{|\mathcal{R}_i|}{\sum_{j=1,...,n} |\mathcal{R}_j|}$. $\mathcal{R}_i$ data comes from an individual distributions for agent $i$ (data in non-iid across agents). Assume each agent has local discriminator and generator with corresponding parameter vectors $\vw_n^i$ and $\vtheta_n^i$, loss functions $\mathcal{L}^i_D$ and $\mathcal{L}^i_G$, local true gradients $\vh^i(\vtheta^i_n, \vw^i_n)$ and $\vg^i(\vtheta^i_n, \vw^i_n)$, local stochastic gradients $\tilde{\vg}^i(\vtheta^i_n, \vw^i_n)$ and $\tilde{\vh}^i(\vtheta^i_n, \vw^i_n)$, and learning rates $a(n)$ and $b(n)$ at time $n$. We assume the learning rates are the same across agents. The gradients $\tilde{\vh}^i(\vtheta, \vw)$ and $\tilde{\vg}^i(\vtheta, \vw)$ are stochastic, since every agent uses a mini-batch of his local data for SGD.
There is an intermediary whose role is syncing the local generators and discriminators. The intermediary parameters at time $n$ are denoted by $\vw_n$ and $\vtheta_n$. Note that the intermediary does not train a generator or discriminator itself, and $\vw_n$ and $\vtheta_n$ are only obtained by averaging $\vtheta^i_n$ and $\vw^i_n$ across $i$.
The FedGAN algorithm is presented in Algorithm \[alg: fedgan\]. All agents run SGDs for training local generators and discriminators using local data. Every $K$ time steps of local gradient updates, the agents send their parameters to the intermediary which in turn sends back the average parameters to all agents to sync. We refer to $K$ by synchronization interval. $a(n)$, $b(n)$ and $K$ are tuning parameters of the FedGAN algorithm.
In our model privacy is the main reason agents do not share data and rather send model parameters. Adding privacy noise to the model parameters can further preserve privacy. We leave this as a future direction for this research. Also, we assume all agents participate in the communication process. There is a literature on federated learning which studies if only part of the agents send their parameters due to communication failures [@konevcnyEtal-16arXiv]. This could be an extension to this paper for FedGAN.
Communication and Computation Complexity {#sec:communi_comput}
----------------------------------------
FedGAN communications are limited to sending parameters to intermediary by all agents and receiving back the synchronized parameters every $K$ steps. For a parameter vector of size $M$ (we assume the size of generator and discriminator are the same order), the average communication per round per agent is $\frac{2\times 2M}{K}$. Increasing $K$ reduces the average communication, which may reduce the performance of trained FedGAN (we experiment FedGAN robustness to increasing $K$ in Section \[sec:exp\_img\] and leave its theoretical understanding for future research). For a general distributed GAN where the generator is trained at the intermediary, the communication involves sending discriminator parameters and generator parameters (or the fake generated data), and this communication should happen at every time step for convergence. The average communication per round per agent therefore is $2\times 2M$. This shows the communication efficiency of FedGAN.
Since each agent trains a local generator, FedGAN requires increased computations for agents compared to distributed GAN, but at the same order (roughly doubled). However, in FedGAN the intermediary has significantly lower computational burden since it only average the agents’ parameters.
Convergence Analysis {#sec:conv_analy}
--------------------
In this section, we show that FedGAN converges even with non-iid sources of data, under certain standard assumptions. We analyze the convergence of FedGAN for both equal time-scale updates and two time-scale updates (distinguished by whether $a(n)=b(n)$). While using equal time-scale update is considered standard, some recent progress in GANs such as Self-Attention GAN [@zhang2018self] advocate the use of two time-scale updates presented in [@heusel2017gans] for tuning hyper-parameters.
We extend the notations in this section. For a centralized GAN that pools all the distributed data together, we denote the generator’s and discriminator’s loss functions by $\mathcal{L}_G$ and $\mathcal{L}_D$, with true gradients $\vh(\vtheta, \vw):=\nabla _\vtheta \mathcal{L}_G$ and $\vg(\vtheta, \vw)=\nabla _\vw \mathcal{L}_D$. Also define ${\mM}^{(\vtheta)}:=\vh(\vtheta, \vw)-\sum_{i} p_i\tilde{\vh}^i(\vtheta, \vw)$ and ${\mM}^{(\vw)}:= \vg(\vtheta, \vw)-\sum_{i} p_i\tilde{\vg}^i(\vtheta, \vw)$. ${\mM}^{(\vtheta)}$ and ${\mM}^{(\vw)}$ are random variables due randomness in mini-batch stochastic gradient of $\tilde{\vh}^i,\tilde{\vg}^i$.
We make the following standard assumptions for rest of this section. The first four are with respect to the centralized GAN and are often used in stochastic approximation literature of GAN convergence. The last assumption is with respect to local GANs and is common in distributed learning.
- $\vh^i$ and $\vg^i$ are $L$-Lipschitz.
- $\sum_n a(n)=\infty$, $\sum_n a^2(n)<\infty$, $\sum_n b(n)=\infty$, $\sum_n b^2(n)<\infty$
- The stochastic gradient errors $\{{\mM}^{(\vtheta)}_{n}\} $ and $\{ {\mM}_{n}^{(\vw)}\}$ are martingale difference sequences w.r.t. the increasing $\sigma$-filed $\mathcal{F}_n=\sigma(\vtheta_l, \vw_l, {\mM}^{(\vtheta)}_{l}, {\mM}_{l}^{(\vw)}, l\leq n),n\geq 0$.
- $\sup_n||{\vtheta_n}|| < \infty$ and $\sup_n||{\vw_n}|| < \infty$.
- $\E||[\tilde{\vg}^i(\vtheta, \vw)]-\vg^i(\vtheta, \vw)||\leq \sigma_g$, $\E||[\tilde{\vh}^i(\vtheta, \vw)]-\vh^i(\vtheta, \vw)||\leq \sigma_h$ (bounded variance) and $||\vg^i(\vtheta, \vw)-\vg(\vtheta, \vw)||\leq \mu_{g}$ (bounded gradient divergence).
**(A1)**-**(A4)** are clear assumptions. In **(A5)**, the first bound ensure the closeness between the local stochastic gradients and local true gradients, while the second bound ensures closeness of local discriminator true gradient of non-iid sources and the discriminator true gradient of the pooled data. We next prove the convergence of FedGAN. To this end, we rely on the extensive literature that connects the convergence of GAN to the convergence of an ODE representation of the parameter updates [@mescheder2017numerics]. We prove the ODE representing the parameter updates of FedGAN asymptotically tracks the ODE representing the parameter update of the centralized GAN. We then use the existing results on convergence of centralized GAN ODE [@mescheder2018training; @nagarajan2017gradient]. Note that our results do not mean the FedGAN and centralized GAN converge to the same point in general.
**Equal time-scale update**. It has been shown in [@nagarajan2017gradient; @mescheder2017numerics] that under equal time-scale update, the centralized GAN tracks the following ODE asymptotically (we use $t$ to denote continuous time). $$\label{eq: ode_z}
\begin{pmatrix}\dot{{\vw}}(t)\\ \dot{{\vtheta}}(t)\end{pmatrix} =
\begin{pmatrix}{\vg}({\vtheta}(t),{\vw}(t))\\ {\vh}({\vtheta}(t),{\vw}(t))\end{pmatrix}.$$
We now show that when $a(t)=b(t)$, the updates in (\[eq:algo\_update\]) as specified by the proposed algorithm, also tracks the ODE in (\[eq: ode\_z\]) asymptotically. To this end, we further extend the notations. For centralized GAN, Let $\vz(t):=(\vw(t),\vtheta(t))^\top$ and define $\vq(\vz(t)):=\dot{\vz}(t)=(\vg(\vtheta(t)), \vh(\vw(t)))^\top$ from (\[eq: ode\_z\]). For FedGAN, define $\vz_n:=(\vw_n,\vtheta_n)^\top$ from (\[eq:algo\_2\]) and with a little abuse of notation, define time instants $t(0)=0$, $t(n)=\sum_{m=0}^{n-1} a(m)$. Define a continuous piece-wise linear function $\bar{\vz}(t)$ by $$\label{eq: z bar}
\bar{\vz}(t(n)):={\vz}_{n},$$ with linear interpolation on each interval $[t(n), t(n+1)]$. Correspondingly, define $\bar{\vw}(t(n))$ and $\bar{\vtheta}(t(n))$ to have $\bar{\vz}(t(n))=(\bar{\vw}(t(n)),\bar{\vtheta}(t(n)))^\top$.
Let $\vz^s(t)$ (correspondingly $\vz_s(t)$) denote the unique solution to (\[eq: ode\_z\]) starting (ending) at $s$ $$\label{eq:ode_z_s}
\dot{{\vz^s}}(t) = {\vq}({\vz^s}(t)), t\geq s \quad (t\leq s)$$ with $\vz^s(s)=\bar{\vz}(s)$ (with $\vz_s(s)=\bar{\vz}(s)$). Define $(\vw^s(t), \vtheta^s(t)$ (correspondingly $(\vw_s(t), \vtheta_s(t)$) to be the elements of $\vz^s(t)$ (elements of $\vz_s(t)$).
Now, in order to prove FedGAN follows (\[eq: ode\_z\]) asymptotically, it is sufficient to show that $\bar{\vz}(t)$ asymptotically tracks $\vz^s(t)$ and $\vz_s(t)$ as $s\to \infty$. The first step is to show the difference between the intermediary averaged parameters ${\vw}_{n},{\vtheta}_{n}$ and $\vv_n,\vphi_n$ defined below base on the centralized GAN updates in between synchronization intervals is bounded. If $n=\ell K$, then let $\vv_{n}=\vw_{n}$ and $\vphi_{n}=\vtheta_{n}$ otherwise, denote $n_1$ to be the largest multiplication of $K$ before $n$ and let $$\begin{aligned}
\vv_n = \vw_{n_1} + \sum_{k=n_1}^n a(k) \vg(\vphi_k, \vv_k), \quad
\vphi_n = \vtheta_{n_1} + \sum_{k=n_1}^n b(k) \vh(\vphi_k, \vv_k).
\end{aligned}$$ We prove the following and for this purpose. The distinction between proof here and in Theorem 1 in [@wang2019adaptive] is that we consider local SGD for federated learning.
We present the proofs of result in this Section in Appendix \[app: proof\].
\[lem: bound\_var\] $\E ||\vw^i_n-\vv_n||+\E||\vtheta^i_n-\vphi_n||\leq r_1(n):= \frac{\sigma_g+\mu_g+\sigma_h}{2L}[(1+2a(n-1)L)^{n\, \text{mod}\, K}-1]$.
\[lem: bound\_grad\_div\] $\E||{\vw}_{n}-\vv_{n}||+\E||{\vtheta}_{n}-\vphi_{n}||\leq r_2(n):=\frac{(\sigma_g+\sigma_h+\mu_g)}{2L}[(1+2a(n-1)L)^{K}-1]-a(n-1)\mu_gK $.
Next, using and , we prove which shows $\bar{\vz}(t)$ asymptotically tracks $\vz^s(t)$ and $\vz_s(t)$ as $s\to \infty$. This in turn proves (\[eq:algo\_update\]) asymptotically tracks the limiting ODE in (\[eq: ode\_z\]). The proof is modified from Lemma 1 in Chapter 2 of [@borkar2009stochastic] from centralized GAN to FedGAN.
\[thm: equal\_traj\] For any $T>0$ (a.s. stands for almost surly convergence) $$\lim_{s\rightarrow \infty} \sup_{t\in [s,s+T]} ||\bar{\vz}(t)-\vz^s(t)|| = 0, a.s.,\quad
\lim_{s\rightarrow \infty} \sup_{t\in [s-T,s]} ||\bar{\vz}(t)-\vz_s(t)|| = 0, a.s.$$
From Theorem \[thm: equal\_traj\] above and Theorem 2 of Section 2.1 in [@borkar2009stochastic], under equal time-scale update, FedGAN tracks the ODE in (\[eq: ode\_z\]) asymptotically.
We provide the convergence analysis of FedGAN with two time-scale updates in Appendix \[app: two\_time\].
Experiments
===========
In this section we experiment the proposed FedGAN algorithm using different datasets to show its convergence, performance in generating close to real data, and robustness to reducing communications (by increasing synchronization interval $K$). First in Section \[sec:toy\] we experiment with popular toy examples in the GAN literature, 2D system [@nagarajan2017gradient], mixed Gaussian [@Metz2016UnrolledGA] and Swiss Roll [@gulrajani2017improved]. Next, in Section \[sec:exp\_img\] we experiment with image datasets including MNIST, CFAR-10 and CelebA. Finally we consider time-series data in Section \[sec:exp\_energy\] by focusing on energy industry, including PG&E household electricity consumption and electric vehicles charging sessions from a charging station company. In all the experiments, data sources are partitioned into non-iid subsets each owned by one agent.
Toy Examples {#sec:toy}
------------
Three toy examples, 2D system mixed Gaussian and Swiss role, are presented in Appendix \[app: toy\]. These experiments show the convergence and performance of the FedGAN in generating data similar to real data. The first experiment, 2D system, also shows the robustness of FedGAN performance to increasing synchronization intervals $K$ for reducing communications.
Image Datasets {#sec:exp_img}
--------------
We test FedGAN on MNIST, CIFAR-10, and CelebA to show its performance on image datasets.
Both MNIST and CIFAR-10 consist of $10$ classes of data which we split across $B=5$ agents, each with two classes of images. We use the ACGAN neural network structure in [@odena2017conditional]. For a detailed list of architecture and hyperparameters and other generated images see Appendix \[app: net\_hyper\].
For the MNIST dataset, we set synchronization interval $K=20$. Figure \[fig:mnist\] presents the generated images of MNIST from FedGAN. It shows FedGAN can generate close to real images.
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For the CIFAR-10 dataset, we use the FID scores [@karmakar2017two] to compare the generated and real data and show the realness of the generated images from FedGAN. We check FedGAN performance robustness to reduced communications and increased synchronization intervals $K$ by setting $K=10, 20, 1000, 500, 3000, 6000$. We benchmark FedGAN performance against a typical distributed GAN similar to that in [@augenstein2019generative], where there are local discriminators that send the parameters to the intermediary after each update, and the intermediary sends back the average discriminator parameters plus the generated data of its updated centralized generator (to the best of our knowledge, there is no other algorithm for communication efficient distributed GAN to compare ours). Figure \[cifar10\_fid\] shows the results for FedGAN CIFAR-10. It can be observed that, even for large synchronization interval $K$, the FedGAN FID score is close to distributed GAN (except for the tail part). This indicates that FedGAN has high performance for image data, and furthermore its performance is robust to reducing the communications by increasing synchronization intervals $K$. The gap between the distributed GAN and FedGAN in the tail part required further investigation in the future.
Next, we experiment FedGAN algorithm on CelebA [@liu2015deep], a dataset of $202,599$ face images of celebrities. We split data across $B=5$ agents by first generating $16$ classes based on the combinations of four binary attributes of the images, Eyeglasses, Male, Smiling, and Young, and then allocating each of these $16$ classes to one of the $5$ agents (some classes divided between two agents to ensure equal size of data across agents). We check FedGAN robustness to reduced communications by setting synchronization intervals $K=10, 20, 50, 100, 200$, and also compare the performance with distributed GAN. We use the ACGAN neural network structure in [@odena2017conditional] (for details of data splitting, lists of structures and hyperparameters see Appendix \[app: net\_hyper\]). Figure \[fig:celebA\] shows the generated images with $K=50$ with $N=16000$ iterations (more generated images are presented in Appendix \[app: net\_hyper\]). Figure \[fig:celebA\_fid\] shows the performance of FedGAN is close to distributed GAN, and robust to reduced communications.
[0.45]{}
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Time Series Data in Energy Industry {#sec:exp_energy}
-----------------------------------
In this section, we experiment FedGAN for time series data. We particularly focus on energy industry where the communication and privacy constraints are important, and data often is in time series format. We experiment on household electricity consumption and electric vehicle (EV) charing sessions data. In order to measure the performance of FedGAN for time series data, absent of an accepted measure or score for measuring the realness of time series data, we cluster both the real data and generated data, and visually compare the top 9 cluster centroids for each.
For household electricity consumption, we use the hourly load profile data of $500k$ households in one year in California by Pacific Gas and Electric Company (PG&E) (dividing every single households data into separate daily profiles). The data includes both household characteristics and temporal features including regional average income, enrolled in low income tariff program/not, all electric/not, daily average temperature, weekday/weekend, month, tariff type, climate zone, house premise type.
For EV data, we use data from an electric vehicle (EV) charging stations company including $12.4$ million charging sessions where each session is defined by the plug-in and -off of an EV at a charging station. For each session, we observe start time, end time, 15-min charging power, charging time, and charges energy, and fully charged or not. We also observe characteristics of the charging station as well as the EV. For example an EV with battery capacity 24kW arriving at a high-tech workplace at 9:00am on Monday (see Appedix \[app: data\] for full detail).
For both experiments, we split the data in equal parts across $B=5$ agents (representing different utility companies or different EV charging companies), based on climate zones or category of charging stations (to ensure non-iid data across agents), and set synchronization interval $K=20$. We use a network structure similar to CGAN [@mirza2014conditional].
We separate $10\%$ of each agent’s data, train FedGAN on the rest of the data, and use the trained FedGAN to generate fake time series profiles for those $10\%$. We then apply k-means for both the real and generated data of those $10\%$. The k-means top 9 centroids are shown in Figure \[cluster\_real\] and \[cluster\_test\] for household electricity consumption, and in Figure \[cluster\_ev\] for EV charging sessions. Visually comparing them shows the performance of FedGAN on generating close to real profiles for time series data.
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![Top 9 k-means clusters for real PG&E daily household electricity consumption, and FedGAN generated profiles with $B=5$ and $K=20$. The consumption profiles are normalized.[]{data-label="cluster"}](img/test1_clst_ori.png)
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![Top 9 k-means clusters for real PG&E daily household electricity consumption, and FedGAN generated profiles with $B=5$ and $K=20$. The consumption profiles are normalized.[]{data-label="cluster"}](img/test1_clst_5.png)
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![Top 9 k-means clusters for real EV charging profiles, and FedGAN generated profiles with $B=5$ and $K=20$. The charging profiles are normalized.[]{data-label="cluster_ev"}](img/ev_cluster_ori.png)
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![Top 9 k-means clusters for real EV charging profiles, and FedGAN generated profiles with $B=5$ and $K=20$. The charging profiles are normalized.[]{data-label="cluster_ev"}](img/ev_cluster3.png)
Conclusions and Future Directions
=================================
We proposed an algorithm for communication-efficient distributed GAN subject to privacy constraints (FedGAN). We proved the convergence of our FedGAN for non-iid sources of data both with equal and two time-scale updates of generators and discriminators. We experimented FedGAN on toy examples (2D, mixed Gaussian, and Swiss roll), image data (MNIST, CIFAR-10, and CelebA), and time series data (household electricity consumption and EV charging sessions) to study its convergence and performance. We showed FedGAN has similar performance to general distributed GAN, while reduces communication complexity. We also showed its robustness to reduced communications.
There are some observations and open problems. First is experimenting FedGAN with other federated learning datasets such as mobile phone texts, and with other applications besides image classification and energy. Robustness to increasing agents number $N$ is required which requires large set of GPUs (beyond the engineering capacity of this research). Theoretically, an explanation for the FedGAN robustness to reduced communication, as well as identifying the rate of convergence are interesting. While privacy is the main reason agents do not share data in our FedGAN, adding privacy noise to the model parameters, for example by differential privacy, can further preserve privacy and should be studied. Finally, non-responsiveness of some agents in practice should be studied.
\[sec:concl\]
Broader Impact
==============
This work has the following potential positive impact in the society: it provides an algorithm for shared learning across agents with local data while preserving privacy and low communication cost, hence it helps democratizing data power. We have also emphasized energy domain as an application which is at the forefront of sustainability and reversing global warming; we particularly experimented on data for household demand prediction and electric vehicle charging station planning.
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to thank the Pacific Gas and Electric Company (PG&E) for providing the household energy consumption dataset and SLAC National Accelerator Laboratory for providing the EV dataset.
Supplementary Material for “FedGAN: Federated Generative Adversarial Networks for Distributed Data” {#supplementary-material-for-fedgan-federated-generative-adversarial-networks-for-distributed-data .unnumbered}
===================================================================================================
Convergence Analysis of FedGAN with Two Time-Scale Updates {#app: two_time}
==========================================================
To study the two time-scale FedGAN we add the following assumptions.
- $b(n)=\text{o}(a(n))$
- For each ${\vtheta}$, the ODE $\dot{{\vw}}(t)={\vg}({\vtheta}, {\vw}(t))$ has a local asymptotically stable attractor ${\lambda}({\vtheta})$ within a domain of attraction $H_\vtheta$ such that ${\vtheta}$ is Lipschitz. The ODE $\dot{{\vtheta}}(t)=\vh({\vtheta}(t),{\lambda}({\vtheta}(t)))$ has a local asymptotically stable equilibrium ${\vtheta^*}$ within a domain of attraction.
**(A6)** is the standard assumption to determine the relationship of the generator and discriminator learning rates. **(A7)** characterizes the local asymptotic behavior of the limiting ODE in (\[eq: ode\_z\]) and shows its local asymptotic stability. Both assumptions are regular conditions in the literature of two time-scale stochastic approximation [@borkar2009stochastic; @karmakar2017two]. In the literature of stochastic approximation, often global asymptotic stability assumptions are made but [@karmakar2017two] and Chapter 2 of [@borkar2009stochastic] relax them to local asymptotic stability which is a more practical assumption in GANs[^1]. The relaxed local stability assumption **(A7)** limits the convergence results to be conditioned on an unverifiable event i.e. $\{\vw_n\}$ and $\{\vtheta_n\}$ eventually belongs to some compact set of their region of attraction.
To prove the convergence of FedGAN for two time-scale updates, similar to the proof for equal time-scale update, we show the ODE representation of the FedGAN asymptotically tracks the ODE below representing the parameter update of the two time-scale centralized GAN $$\label{eq:ode_2_w}
\dot{{\vw}}(t) = \frac{1}{\epsilon}{\vg}(\vtheta(t), {\vw}(t)),\quad
\dot{{\vtheta}}(t) = {\vh}(\vtheta(t), {\vw}(t))$$ where $\epsilon \downarrow 0$ to ensure updating ${\vw}(t)$ is fast compared to updating $\vtheta(t)$ **(A6)**. Consequent to **(A6)**, $\vtheta(t)$ can be considered quasi-static while analyzing the updates of ${\vw}(t)$ and we can look at the following ODE in studying ${\vw}(t)$ (with small change of notation we drop the notational dependency on fixed $\vtheta$ for convenience) $$\dot{{\vw}}(t) = {\vg}({\vw}(t)).
\label{eq: sample_ode}$$ In order to show that the updates of ${\vw}_n$ asymptotically tracks (\[eq: sample\_ode\]), we follow the same idea as in equal time-scale update to construct the continuous interpolated trajectory $\bar{\vw}(t)$ defined immediately after (\[eq: z bar\]), and show that it asymptotically almost surely approaches the solution of (\[eq: sample\_ode\]). For this, we also use the construction of $\vw^s(t)$ and $w_s(t), t\leq s$ defined immediately after (\[eq:ode\_z\_s\]). We thus have the following lemma as a special case of .
\[lem: asym\_traj\] For any $T>0$, $$\lim_{s\rightarrow \infty} \sup_{t\in [s,s+T]} ||\bar{\vw}(t)-\vw^s(t)|| = 0, a.s.,\quad
\lim_{s\rightarrow \infty} \sup_{t\in [s-T,s]} ||\bar{\vw}(t)-\vw_s(t)|| = 0, a.s.$$
The proof can be directly obtained by consider the special case of where the dimension of $\vtheta$ is zero.
With , below shows the FedGAN tracks the ODE in (\[eq: sample\_ode\]) asymptotically when ${\vtheta}_{n}$ is fixed. We refer the reader for proof to [@borkar2009stochastic].
\[thm: single\_converge\] \[Theorem 2, Chapter 2, [@borkar2009stochastic]\]. Almost surely, the sequence $\{\vw_n\}$ generated by (\[eq:algo\_update\]) when ${\vtheta}_{n}$ is fixed to ${\vtheta}$ converges to a (possibly sample path dependent) compact connected internally chain transitive invariant set of (\[eq: sample\_ode\]).
The following Lemma \[lem: trans\_invar\] and Theorem \[thm: two\_time\] extend the results of Theorem \[thm: single\_converge\] to the case when both ${\vw}_{n}$ and ${\vtheta}_{n}$ could vary as in (\[eq:ode\_2\_w\]). Both proofs should be referred to the respective part of [@borkar2009stochastic]. Lemma \[lem: trans\_invar\] shows that $\vw_n$ asymptotically tracks $\lambda(\vtheta_n)$, and Theorem \[thm: two\_time\] shows the convergence of the proposed FedGAN asymptotically converges to $(\lambda(\vtheta^*), \vtheta^*)$, which is an equilibrium of the ODE in (\[eq:ode\_2\_w\]) representing centralized GAN with two time-scale updates. Lemma \[lem: trans\_invar\] is an adaption from Lemma 1 in Chapter 6 of [@borkar2009stochastic].
\[lem: trans\_invar\] For the two time-scale updates as specified in (\[eq:algo\_update\]), $(\vw_n,\vtheta_n)\rightarrow \{(\lambda(\vtheta), \vtheta)\}$ almost surely where $\{(\lambda(\vtheta), \vtheta)\}$ is the internally transitive invariant sets of the ODE $\dot{\vw}(t)=g(\vtheta(t),\vw(t))$, $\dot{\vtheta}(t)=0$.
Rewrite the generator update as
$${\vtheta}_{n+1}= {\vtheta}_{n}+a(n)[\epsilon_n+{\mM}_{n}^{(\vtheta')}]$$
where $\epsilon_n:=\frac{b(n)}{a(n)}{h}({\vtheta}_{n},{\vw}_{n})$ and ${\mM}_{n}^{(\vtheta')}:=\frac{b(n)}{a(n)}{\mM}_{n}^{(\vtheta)}$ for $n\geq 0$. From the third Extension in Section 2.2 of [@borkar2009stochastic], $\{\vv_n\}$ should converge to an internally chain transitive invariant set of $\dot{\vv(t)}=0$. Considering this, the proof follows directly from .
\[thm: two\_time\] \[Theorem 2, Chapter 6, [@borkar2009stochastic]\] $(\vw_n,\vtheta_n)\rightarrow (\lambda(\vtheta^*), \vtheta^*)$ almost surely.
Proof of Results in Section \[sec:conv\_analy\] {#app: proof}
===============================================
\[Proof of \] $$\label{eq: lem1_1}
\begin{split}
\E ||\vw^i_n-\vv_n||+\E||\vtheta^i_n-\vphi_n||&\leq \E||\vw^i_{n-1}-\vv_{n-1}-a(n-1)[\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg(\vv_{n-1}, \vphi_{n-1})]||\\
& + \E||\vtheta^i_{n-1}-\vphi_{n-1}-b(n-1)[\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh(\vv_{n-1}, \vphi_{n-1})]||\\
&\leq \E||\vw^i_{n-1}-\vv_{n-1}|| + \E||\vtheta^i_{n-1}-\vphi_{n-1}||\\
&+a(n-1)\E||\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg(\vv_{n-1},\vphi_{n-1})||\\
&+b(n-1)\E||\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh(\vv_{n-1},\vphi_{n-1})||
\end{split}$$ The latter part on the right hand side can be written as $$\label{eq: lem1_2}
\begin{split}
&a(n-1)\E||\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg(\vv_{n-1},\vphi_{n-1})||
+b(n-1)\E||\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh(\vv_{n-1},\vphi_{n-1})||\\
&=a(n-1)\E||[\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vw^i_{n-1}, \vtheta^i_{n-1})+\vg_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vv_{n-1},\vphi_{n-1})\\
&+\vg_i(\vv_{n-1},\vphi_{n-1})-\vg(\vv_{n-1},\vphi_{n-1})]||+b(n-1)\E||[\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vw^i_{n-1}, \vtheta^i_{n-1})\\
&+\vh_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vv_{n-1},\vphi_{n-1})+\vh_i(\vv_{n-1},\vphi_{n-1})-\vh(\vv_{n-1},\vphi_{n-1})]||\\
&\leq a(n-1)(\sigma_g+\mu_g)+b(n-1)\sigma_h + [a(n-1)+b(n-1)]L[\E||\vw^i_{n-1}-\vv_{n-1}||+\E||\vtheta^i_{n-1}-\vphi_{n-1}||]\\
&=a(n-1)[\sigma_g+\mu_g+\sigma_h+ 2L(\E||\vw^i_{n-1}-\vv_{n-1}|| + \E||\vtheta^i_{n-1}-\vphi_{n-1}||)]
\end{split}$$ Here $\sigma_g$ and $\sigma_h$ are bounds for the variances for discriminator and generator respectively, and $\mu_g$ is the bound for gradient divergence for discriminator (Assumption (**A5**)). The above inequality follows from Assumption (**A1**) and $\vh_i(\vv_{n-1},\vphi_{n-1})=\vh(\vv_{n-1},\vphi_{n-1})$ which holds because the fake data is generated based on parameters of the generator. Also, the last equality holds because $a(n-1)=b(n-1)$ in equal time-scale updates.
Considering (\[eq: lem1\_1\]) and (\[eq: lem1\_2\]), we have $$\begin{split}
\E ||\vw^i_n-\vv_n||&+\E||\vtheta^i_n-\vphi_n||\\
&\leq (1+2a(n-1)L)(\E||\vw^i_{n-1}-\vv_{n-1}|| + \E||\vtheta^i_{n-1}-\vphi_{n-1}||)+a(n-1)(\sigma_g+\mu_g+\sigma_h)\\
&\leq \frac{\sigma_g+\mu_g+\sigma_h}{2L}[(1+2a(n-1)L)^{n\, \text{mod}\, K}-1]
\end{split}$$ The last inequality can be holds by induction over $n$ and a mild assumption that the learning rate is unchanged within the same synchronization interval.
$$\begin{split}
&\E||{\vw}_{n}-\vv_{n}||+\E||{\vtheta}_{n}-\vphi_{n}||\\
&\leq \E||{\vw}_{n-1}-\vv_{n-1}+a(n-1)\sum_i p_i [\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vv_{n-1},\vphi_{n-1})]||\\
&+ \E||{\vtheta}_{n-1}-\vphi_{n-1}+b(n-1)\sum_i p_i [\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vv_{n-1},\vphi_{n-1})]||
\end{split}$$
We have $$\begin{split}
&\E||\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vv_{n-1},\vphi_{n-1})||+\E||\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vv_{n-1},\vphi_{n-1})||\\
&=\E||\tilde{\vg}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vw^i_{n-1}, \vtheta^i_{n-1})+\vg_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vg_i(\vv_{n-1},\vphi_{n-1})||\\
&+\E||\tilde{\vh}_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vw^i_{n-1}, \vtheta^i_{n-1})+\vh_i(\vw^i_{n-1}, \vtheta^i_{n-1})-\vh_i(\vv_{n-1},\vphi_{n-1})||\\
&\leq \sigma_g+\sigma_h+2L[\E||\vw^i_{n-1}-\vv_{n-1}||+\E||\vtheta^i_{n-1}-\vphi_{n-1}||]\\
&\leq \sigma_g+\sigma_h+2L r_1(n-1)
\end{split}$$ where the last inequality follows from .
Consequently, $$\begin{split}
&\E||{\vw}_{n}-\vv_{n}||+\E||{\vtheta}_{n}-\vphi_{n}|| \\
&\leq \E||{\vw}_{n-1}-\vv_{n-1}||+\E||{\vtheta}_{n-1}-\vphi_{n-1}||+a(n-1) [\sigma_g+\sigma_h+2L r_1(n-1)]\\
&=\E||{\vw}_{n-1}-\vv_{n-1}||+\E||{\vtheta}_{n-1}-\vphi_{n-1}||\\
&+a(n-1)((\sigma_g+\sigma_h+\mu_g)[(1+2a(n-1)L)^{n\, \text{mod}\, K}-1]+\sigma_g+\sigma_h)\\
&\leq a(n-1)(\sigma_g+\sigma_h+\mu_g)\sum_{j=1}^K (1+2a(n-1)L)^{j-1}-a(n-1)\mu_gK\\
&=a(n-1)(\sigma_g+\sigma_h+\mu_g) \frac{(1+2a(n-1)L)^{K}-1}{2a(n-1)L}-a(n-1)\mu_gK\\
&=\frac{(\sigma_g+\sigma_h+\mu_g)}{2L}[(1+2a(n-1)L)^{K}-1]-a(n-1)\mu_gK
\end{split}$$
The proof is modified from Lemma 1 in Chapter 2 of [@borkar2009stochastic]. We shall only prove the first claim, as the arguments for proving the second claim are completely analogous. Define $$\zeta^\vw_n=\sum_{m=0}^{n-1}a(m)\mM_{m}^{(\vw)}$$ $$\zeta^\vtheta_n=\sum_{m=0}^{n-1}b(m)\mM_{m}^{(\vtheta)}$$ Denote $\delta^{\vw}_{n,n+m}=\zeta^{\vw}_{n+m}-\zeta^{\vw}_n$, $\delta^{\vtheta}_{n,n+m}=\zeta^{\vtheta}_{n+m}-\zeta^{\vtheta}_n$. Let $\delta_{n,n+m} = (\delta^{\vw}_{n,n+m},\delta^{\vtheta}_{n,n+m})^\top$. Let $t'(n)=t(nK)$ and $[t]=\max\{t(k): t(k)\leq t\}$. (Note that $t$ is overloaded here both as a function and a variable). By construction, $$\begin{split}
&\vc^{\vw}_0 = \bar{\vw}(t'(n))\\
&\vc^{\vw}_k = \bar{\vw}(t'(n)) +\sum_{j=0}^{k-1} a(nK+j)\vg(\vc^{\vw}_{j}, \vc^{\vtheta}_{j}) + \delta^{\vw}_{nK,nK+k}
\end{split}$$
$$\begin{split}
&\vc^{\vtheta}_0 = \bar{\vtheta}(t'(n))\\
&\vc^{\vtheta}_k = \bar{\vtheta}(t'(n)) +\sum_{j=0}^{k-1} b(nK+j)\vh(\vc^{\vw}_{j}, \vc^{\vtheta}_{j}) + \delta^{\vtheta}_{nK,nK+j}
\end{split}
\label{eq: recons_seq1t}$$
Furthermore, denote $$\vc_k=(\vc^{\vw}_k, \vc^{\vtheta}_k)^\top=\bar{\vz}(t'(n))+\sum_{j=0}^{k-1}a(nK+j)\vf(\vc_j)+\delta_{nK,nK+j}
\label{eq: recons_seq1}$$
Also by construction,
$$\begin{split}
\vd^{\vw}_0 &= \bar{\vw}(t'(n))\\
\vd^{\vw}_k &=\bar{\vw}(t'(n))+\sum_{j=0}^{k-1} a(nK+j)\vg(\vd^{\vw}_{j}, \vd^{\vtheta}_{j})\\
&+\int_{t(nK)}^{t(nK+k)}(\vg( \vw^{t'(n)}(y),\vtheta^{t'(n)}(y))-\vg( \vw^{t'(n)}([y]), \vtheta^{t'(n)}([y])))dy
\end{split}$$
$$\begin{split}
\vd^{\vtheta}_0 &= \bar{\vtheta}(t'(n))\\
\vd^{\vtheta}_k &=\bar{\vtheta}(t'(n))+\sum_{j=0}^{k-1} b(nK+j)\vh(\vd^{\vw}_{j}, \vd^{\vtheta}_{j})\\
&+\int_{t(nK)}^{t(nK+k)}(\vh( \vw^{t(n)}(y),\vtheta^{t(n)}(y))-\vh( \vw^{t(n)}([[y]]), \vtheta^{t(n)}([y])))dy
\end{split}
\label{eq: recons_seq2t}$$
Further denote $$\vd_k=(\vd^{\vw}_k, \vd^{\vtheta}_k)^\top
\label{eq: recons_seq2}$$
Let $C_0=\sup_n||\vw_n||<\infty$, $C_1=\sup_n||\vtheta_n||<\infty$ almost surely (Assumption (**A4**)), let $L>0$ denote the Lipschitz constant of $\vg$ and $\vh$, and let $s\leq t\leq s+T$. Note that $||\vg(x)-\vg(0)||\leq L||x||$, and so $||\vg(x)||\leq ||\vg(0)||+L||x||$. Similar inequalities also hold for $\vh$. Since $\vw^s(t)=\bar{\vw}(s)+\int_s^t \vg(\vw^s(\tau), \vtheta^s(\tau))\vd\tau$ and $\vtheta^s(t)=\bar{\vtheta}(s)+\int_s^t \vh(\vw^s(\tau), \vtheta^s(\tau))\vd\tau$, for discriminator we have $$\begin{split}
||\vw^s(t)||&\leq ||\bar{\vw}(t)||+\int_s^t [||\vg(0,0)||+L||\vw^s(\tau)||+L||\vtheta^s(\tau)||]\vd\tau\\
&\leq (C_0+||\vg(0)||T)+L\int_s^t(||\vw^s(\tau)||+||\vtheta^s(\tau)||)\vd\tau
\end{split}$$ and for generator we have $$\begin{split}
||\vtheta^s(t)||&\leq ||\bar{\vtheta}(t)||+\int_s^t [||\vh(0,0)||+L||\vw^s(\tau)||+L||\vtheta^s(\tau)||]\vd\tau\\
&\leq (C_1+||\vh(0)||T)+L\int_s^t(||\vw^s(\tau)||+||\vtheta^s(\tau)||)\vd\tau
\end{split}$$
Adding the above two inequalities, we have $$||\vw^s(t)||+||\vtheta^s(t)||\leq (C_0+C_1+||\vg(0)||T+||\vh(0)||T)+2L\int_s^t(||\vw^s(\tau)||+||\vtheta^s(\tau)||)\vd\tau$$
By Gronwall’s inequality, it follows that $$||\vw^s(t)||+||\vtheta^s(t)||\leq (C_0+C_1+||\vg(0)||T+||\vh(0)||T)e^{2LT}, s\leq t\leq s+T$$
Thus, for all $s\leq t\leq s+T$ $$||\vg(\vw^s(t), \vtheta^s(t))||\leq C_T = ||\vg(0)||+L(C_0+C_1+||\vg(0)||T+||\vh(0)||T)e^{2LT}<\infty, a.s.$$ $$||\vh(\vw^s(t), \vtheta^s(t))||\leq C'_T = ||\vh(0)||+L(C_0+C_1+||\vg(0)||T+||\vh(0)||T)e^{2LT}<\infty, a.s.$$ Now, if $0 \leq j< mK$ and $t\in (t'(nK+j),t'(nK+j+1)]$, $$\begin{split}
|| \vw^{t(n)}(t)- \vw^{t(n)}(t'(nK+j))|| &\leq ||\int_{t'(nK+j)}^t \vg(\vw^{t(n)}(s), \vtheta^{t(n)}(s))ds||\\
&\leq C_T(t-t'(nK+j))\\
&\leq C_Ta(nK+j)
\end{split}$$ $$\begin{split}
|| \vtheta^{t(n)}(t)- \vtheta^{t(n)}(t'(nK+j)|| &\leq ||\int_{t'(nK+j)}^t \vh(\vw^{t(n)}(s), \vtheta^{t(n)}(s))ds||\\
&\leq C'_T(t-t'(nK+j))\\
&\leq C'_Ta(nK+j)
\end{split}$$ Thus, $$\begin{split}
&\int_{t(nK)}^{t(nK+k)}(\vg( \vw^{t'(n)}(t),\vtheta^{t’(n)}(t))-\vg( \vw^{t'(n)}([t]),\vtheta^{t’(n)}([t])))dt\\
&\leq\int_{t(nK)}^{t(nK+k)}L(||\vw^{t'(n)}(t)-\vw^{t'(n)}([t]))||+||\vtheta^{t'(n)}(t)-\vtheta^{t'(n)}([t]))||)dt\\
&=L \sum_{j=0}^{k-1} \int_{t(nK+j)}^{t(nK+j+1)}(||\vw^{t'(n)}(t)-\vw^{t’(n)}(t(nK+j)))|| +|| \vtheta^{t'(n)}(t)- \vtheta^{t’(n)}(t(nK+j)||)dt\\
&\leq (C_T+C'_T) L\sum_{j=0}^{k-1} a(nK+j) ^2 \\
&\leq (C_T+C'_T) L\sum_{j=0}^{\infty} a(nK+j) ^2 \xrightarrow{n \uparrow \infty }0, a.s.
\end{split}$$
Similarly, we have $$\begin{split}
&\int_{t(nK)}^{t(nK+k)}(\vh( \vw^{t'(n)}(t),\vtheta^{t’(n)}(t))-\vh( \vw^{t'(n)}([t]), \vtheta^{t’(n)}([t])))dt\\
&\leq (C_T+C'_T) L\sum_{j=0}^{\infty} a(nK+j) ^2 \xrightarrow{n \uparrow \infty }0, a.s.
\end{split}$$
Also by , , we have $$\sup_{k\geq 0}||\delta_{nK,nK+k}||\xrightarrow{n \uparrow \infty }0, a.s.$$
Subtracting (\[eq: recons\_seq2\]) from (\[eq: recons\_seq1\]) and taking norms, we have $$\begin{split}
||\vc_k-\vd_k|| &\leq L \sum_{j=0}^{k-1} a(nK+j)||\vc_j-\vd_{j}||\\
&+2(C_T+C'_T) L\sum_{j=0}^{\infty} a(nK+j) ^2+\sup_{k\geq 0}||\delta_{nK,nK+k}||, a.s.
\end{split}$$ Define $K_{T,n}=2(C_T+C'_T) L\sum_{j=0}^{\infty} a(nK+j) ^2+\sup_{k\geq 0}||\delta_{nK,nK+k}||$. Note that $K_{T,n}\rightarrow 0$ almost surely as $n\rightarrow \infty$ (by Assumption (**A2**)). Also, let $\vy_k=||\vc_k-\vd_k||$ and $q_j=a(nK+j)$. Thus, the above inequality becomes $$\vy_k\leq K_{T,n}+L\sum_{j=0}^{k-1}q_j\vy_j$$ Note that $y_0=0$ and $\sum_{j=0}^{k-1}q_j\leq KT$. The discrete Gronwall lemma tells that $$\label{eq: bound_lin_interpo}
\sup_k \vy_k\leq K_{T,n} e^{L KT}$$
Since $K_{T,n}\rightarrow 0$ almost surely as $n\rightarrow \infty$, the original statement of the Theorem follows directly from linear interpolation of (\[eq: bound\_lin\_interpo\]).
Toy Examples in Section \[sec:toy\]: 2D System, Mixed Gaussian and Swiss Roll {#app: toy}
=============================================================================
[@nagarajan2017gradient] experiments convergence of GANs using a 2D example. We extend this experiment for FedGAN setup. We consider a 2D system where both the true distribution for $x$ and the latent distribution for $z$ are uniform over $[-1, 1]$, the discriminator is $D(x) = \psi x^2$, and the generator is $G(z) = \theta z$. For the federated experiment, we assume there are $B=5$ agents by partitioning the data domain into $5$ equal segments with each agent’s data coming from one (e.g. Agent $1$’s true distribution will have be uniform over $[-1,-0.6)$). We set synchronization interval $K=1,5, 20, 50$.
The $(\theta, \phi)$ trajectory is plotted in Figure \[fig:2d\]. In all cases FedGAN converges to the same point of $(1, 0)$. At this point, the generator generated data with uniform distribution between $[-1,1]$, and the discriminator classifies all data in one class (fails to discriminate). This experiment shows robustness of the FedGAN to increasing parameter $K$ and reducing communications. In other words, reducing the communications has low impact on the algorithm result. Note that while $(1,0)$ is the convergence point of the centralized GAN, it may not be the case in general as we observe in later experiments.
[0.5]{}
![FedGAN parameters trajectory in a 2D experiment, generator $\theta$ and discriminator $\phi$ with $B=5$ agents and synchronization intervals $K=1,5,20, 50$. The red dot is the initial values.[]{data-label="fig:2d"}](img/gan_fed_iter1.png){width="\textwidth"}
[0.5]{}
![FedGAN parameters trajectory in a 2D experiment, generator $\theta$ and discriminator $\phi$ with $B=5$ agents and synchronization intervals $K=1,5,20, 50$. The red dot is the initial values.[]{data-label="fig:2d"}](img/gan_fed_iter5.png){width="\textwidth"}
[0.5]{}
![FedGAN parameters trajectory in a 2D experiment, generator $\theta$ and discriminator $\phi$ with $B=5$ agents and synchronization intervals $K=1,5,20, 50$. The red dot is the initial values.[]{data-label="fig:2d"}](img/gan_fed_iter20.png){width="\textwidth"}
[0.5]{}
![FedGAN parameters trajectory in a 2D experiment, generator $\theta$ and discriminator $\phi$ with $B=5$ agents and synchronization intervals $K=1,5,20, 50$. The red dot is the initial values.[]{data-label="fig:2d"}](img/gan_fed_iter50.png){width="\textwidth"}
[@Metz2016UnrolledGA] and [@gulrajani2017improved] train GAN on the popular 2D mixture of eight Gaussians arranged in a circle, and Swiss roll dataset. We extend both experiments to FedGAN by dividing the data into $B=4$ agents (each owning 2 Gaussians or different but equal-sized part of the roll). We set $K=5$. The neural network structure used is the same as that in [@kodali2017convergence].
In Figure \[fig:mix\_gau\] and \[fig:swiss\_roll\], for mixed Gaussian and Swiss role FedGAN experiments, the orange points represent the real data and the green points represent the generated data which are almost coinciding for $N=15000$ and $N=27000$ iterations correspondingly. These ensures high performance of FedGAN in generating fake samples representing the pooled data.
[0.5]{}
![Mixed Gaussians real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:mix_gau"}](img/8gau_f_0.jpg){width="\textwidth"}
[0.5]{}
![Mixed Gaussians real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:mix_gau"}](img/8gau_f_50.jpg){width="\textwidth"}
[0.5]{}
![Mixed Gaussians real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:mix_gau"}](img/8gau_f_100.jpg){width="\textwidth"}
[0.5]{}
![Mixed Gaussians real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:mix_gau"}](img/8gau_f_150.jpg){width="\textwidth"}
[0.5]{}
![Swiss role real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:swiss_roll"}](img/swiss_f_0.jpg){width="\textwidth"}
[0.5]{}
![Swiss role real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:swiss_roll"}](img/swiss_f_90.jpg){width="\textwidth"}
[0.5]{}
![Swiss role real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:swiss_roll"}](img/swiss_f_180.jpg){width="\textwidth"}
[0.5]{}
![Swiss role real data, and FedGAN generated data with $B=4$ agents and synchronization interval $K=5$.[]{data-label="fig:swiss_roll"}](img/swiss_f_270.jpg){width="\textwidth"}
Hyperparameters and Generated Images for Experiments in Section \[sec:exp\_img\] and \[sec:exp\_energy\] {#app: net_hyper}
=========================================================================================================
For our proposed FedGAN algorithm, the hyperparamters include those of local generators and discriminators. For the experiments in Section \[sec:exp\_img\], Table \[tab:cifar10\] lists the hyperparameters used for CIFAR-10, and Table \[tab:celebA\] shows those for CelebA. The hyperparameters used for MNIST are the same as CIFAR-10 except that the input dimension of discriminator and the output dimension of generator (number of channels of greyscale images) are both equal to $1$. More samples of generated images for CIFAR-10 and CelebA experiments are shown in Figure \[fig:cifar\_matrix\] and \[fig:celebA\_matrix\] respectively.
The hyperparameters for the PG&E and EV time series data in Section \[sec:exp\_energy\] are shown in Table \[tab:time\_series\].
Operation Kernel Strides Feature maps BN? Nonlinearity
------------------------------------------ ------------- ------------- -------------- ----- --------------
$G(z)$ - 62 $\times$ 1 $\times$ 1 input
Linear N/A N/A 1024 Y ReLU
Linear N/A N/A 128 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 64 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 3 N Tanh
$D(x)$ - 32 $\times$ 32 $\times$ 3 input
Convolution 4$\times$ 4 2$\times$ 2 64 N Leaky ReLU
Convolution 4$\times$ 4 2$\times$ 2 128 Y Leaky ReLU
Linear N/A N/A 1024 Y Leaky ReLU
Linear (binary) N/A N/A 1 Y Sigmoid
Linear (classify) N/A N/A 10 Y -
Generator Optimizer
Discriminator Optimizer
Batch size
Leaky ReLU slope
: CIFAR-10 hyperparameters. The learning rates for generator and discriminator are both equal to the same value, across all cases in Figure \[cifar10\_fid\]. BN stands for batch normalization.
\[tab:cifar10\]
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_10.png){width="\textwidth"}
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_20.png){width="\textwidth"}
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_100.png){width="\textwidth"}
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_500.png){width="\textwidth"}
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_3000.png){width="\textwidth"}
[0.5]{}
![Generated images for CIFAR-10 with $B=5$, $K=10,20,100,500,3000$ and distributed GAN, $N=30000$ iteration.[]{data-label="fig:cifar_matrix"}](img/cifar_distri.png){width="\textwidth"}
Operation Kernel Strides Feature maps BN? Nonlinearity
------------------------------------------ ------------- ------------- -------------- ----- --------------
$G(z)$ - 62 $\times$ 1 $\times$ 1 input
Linear N/A N/A 640 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 320 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 160 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 80 Y ReLU
Transposed Convolution 4$\times$ 4 2$\times$ 2 3 N Tanh
$D(x)$ - 48 $\times$ 48 $\times$ 3 input
Convolution 4$\times$ 4 2$\times$ 2 80 N Leaky ReLU
Convolution 4$\times$ 4 2$\times$ 2 160 Y Leaky ReLU
Convolution 4$\times$ 4 2$\times$ 2 320 Y Leaky ReLU
Convolution 4$\times$ 4 2$\times$ 2 640 Y Leaky ReLU
Linear (binary) N/A N/A 1 Y -
Linear (classify) N/A N/A 16 Y Sigmoid
Generator Optimizer
Discriminator Optimizer
Batch size
Leaky ReLU slope
: CelebA hyperparameters. The learning rates for generator and discriminator are for $K=10,20,50,100,200$ and distributed GAN respectively as in Figure \[fig:celebA\_fid\]. BN stands for batch normalization.
\[tab:celebA\]
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_10.jpg){width="\textwidth"}
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_20.jpg){width="\textwidth"}
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_50.jpg){width="\textwidth"}
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_100.jpg){width="\textwidth"}
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_200.jpg){width="\textwidth"}
[0.5]{}
![Generated images for CelebA with $B=5$, $K=10,20,50,100,200$ and distributed GAN, $N=47500$ iteration.[]{data-label="fig:celebA_matrix"}](img/celebA_distri.jpg){width="\textwidth"}
[lllll]{} Operation& Kernel &Strides &Feature maps & Nonlinearity\
$G(z)$ - (label dimension+1) $\times$ 24 input & & & &\
1d convolution & 5 & 1 & 64 & -\
1d convolution & 5 & 1 & 64 & ReLU\
\
1d convolution & 1 & 1 & 1 & -\
$D(x)$ - (label dimension+1) $\times$ 24 input & & & &\
1d convolution & 5 & 1 & 64 & -\
1d convolution & 5 & 1 & 64 & ReLU\
\
Linear & 1 & 1 & 1 & -\
Generator Optimizer &\
Discriminator Optimizer &\
Batch size &\
\[tab:time\_series\]
Supplementary EV Data Description in Section \[sec:exp\_energy\] {#app: data}
================================================================
For the EV dataset, we observe the following characteristics at the station level: station ID, connector type (e.g., J1772), POI category (e.g., workplace, retail, municipal), POI subcategory (e.g., commercial, high-tech), station zip code, and max power (e.g., 6.6kW, 24kW, 50kW). Also, at the user level, we observe: driver ID, home zip code, vehicle make, vehicle model, vehicle model year, battery capacity, and EV type (e.g., plugin, hybrid). For the charging profile generating problem considered in Section \[sec:exp\_energy\], we are using POI category/subcategory, max power, battery capacity, month, and day of week as labels for CGAN.
We split the charging dataset to $5$ agents based on the category of charging stations. Therefore, the data distribution across the agents is non-iid; each agent has a different distribution of charging profiles. Figure \[fig: ev\_distri\] shows this non-iid data distribution by comparing two charging stations (each belonging to a different agent), a high-tech workplace and a shopping center. The plots are all the charging sessions on Tuesdays for these two stations. Most charging sessions in the high-tech workplace happen during the day time which is consistent with people working close by. On the other hand, charging sessions in the shopping center last till midnight which is probably when it closes.
[0.5]{}
![Charging profiles on Tuesday for two stations from different categories.[]{data-label="fig: ev_distri"}](img/hightech_charge.png)
[0.5]{}
![Charging profiles on Tuesday for two stations from different categories.[]{data-label="fig: ev_distri"}](img/shop_charge.png)
[^1]: Note that another way of relaxation to local asymptotic stability is to assume that the initial parameter is in the region of attraction for a locally asymptotically stable attractor, which is difficult to ensure in practice.
|
---
abstract: 'Estimation, recognition, and near-future prediction of 3D trajectories based on their two dimensional projections available from one camera source is an exceptionally difficult problem due to uncertainty in the trajectories and environment, high dimensionality of the specific trajectory states, lack of enough labeled data and so on. In this article, we propose a solution to solve this problem based on a novel deep learning model dubbed *disjunctive factored four-way conditional restricted Boltzmann machine* (DFFW-CRBM). Our method improves state-of-the-art deep learning techniques for high dimensional time-series modeling by introducing a novel tensor factorization capable of driving forth order Boltzmann machines to considerably lower energy levels, at no computational costs. DFFW-CRBMs are capable of accurately estimating, recognizing, and performing near-future prediction of three-dimensional trajectories from their 2D projections while requiring limited amount of labeled data. We evaluate our method on both simulated and real-world data, showing its effectiveness in predicting and classifying complex ball trajectories and human activities.'
address:
- 'Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, the Netherlands.'
- 'Department of Computer Science, American University of Beirut, Beirut, Lebanon.'
- 'Department of Computer Science & Engineering, University of Washington, Seattle, USA.'
- 'Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA.'
author:
- Decebal Constantin Mocanu
- Haitham Bou Ammar
- Luis Puig
- Eric Eaton
- Antonio Liotta
bibliography:
- 'refs.bib'
title: 'Estimating 3D Trajectories from 2D Projections via Disjunctive Factored Four-Way Conditional Restricted Boltzmann Machines'
---
Deep learning, restricted Boltzmann machines, 3D trajectories estimation, activity recognition.
Introduction
============
Estimating and predicting trajectories in three-dimensional spaces based on two-dimensional projections available from *one* camera source is an open problem with wide-ranging applicability including entertainment [@3dtrajrobotics], medicine [@3dtrajmedicie], biology [@3dtrajbiology], physics [@3dtrajdarkmatter], etc. Unfortunately, solving this problem is exceptionally difficult due to a variety of challenges, such as the variability of states of the trajectories, partial occlusions due to self articulation and layering of objects in the scene, and the loss of 3D information resulting from observing trajectories through 2D planar image projections. A variety of techniques have considered variants of this problem by incorporating additional sensors, e.g., cameras [@3dtrajwith3cameras], radars [@3dtrajwithradar], which provide new data for geometric solvers allowing for accurate estimation and prediction. Though compelling, the success of these methods arrives at increased costs (e.g., incorporating new sensors) and computational complexities (e.g., handling more inputs geometrically).
The problem above, however, can be framed as a time-series estimation and prediction one, for which numerous machine learning algorithms can be applied. An emerging trend in machine learning for computer vision and pattern recognition is deep learning (DL) which has been successfully applied in a variety of fields, e.g., multi-class classification [@Larochelle+Bengio-2008], collaborative filtering [@Salakhutdinov07restrictedboltzmann], image quality assessment [@mocanu2014deep], reinforcement learning [@mnih15], transfer learning [@ecml2013dec], information retrieval [@Gehler06therate], depth estimation [@3ddepthnips2014], face recognition [@escalerafacerecognition], and activity recognition [@escaleraactivityrecognition]. Most related to this work are *temporal-based* deep learners, e.g., [@Shotton13; @temporalrbm], which we briefly review next. Extending on standard restricted Boltzmann machines (RBMs)[@originalrbm], temporal RBMs (TRBMs) consider a succession of RBMs, one for each time frame, allowing them to perform accurate prediction and estimation of time-series. Due to their complexity, such naive extensions require high computational effort before acquiring acceptable behavior. Conditional RBMs (CRBMs) remedy this problem by proposing an alternative extension of RBMs [@taylorcrbmicml]. Here, the architecture consists of two separate visible layers, representing history (i.e., values from previous time frames), and current values, and a hidden layer for latent correlation discovery. Though successful, CRBMs are only capable of modeling time series data with relatively “smooth” variations and similarly with other state-of-the-art neural network architectures for time series, e.g. recurrent neural networks, they can not learn within the same model different types of time-series. Thus, to model different types of non-linear time variations within the same model, the authors in [@taylorcrbmicml] extend CRBMs by allowing for a three-way weight tensor connection among the different layers. Computational complexity is then reduced by adapting a factored version (i.e., FCRBMs) of the weight tensor, which leads to a construction exhibiting accurate modeling and prediction results in a variety of experiments, including human motion styles [@gwtaylorhdts]. However, these methods fail to perform both classification and regression in one unified framework. Recently, Factored Four-Way Conditional Restricted Boltzmann Machines (FFW-CRBMs) have been proposed [@ffwcrbmprl]. These extend FCRBMs by incorporating a label layer and a four-way weight tensor connection among the layers to modulate the weights for capturing subtle temporal differences. This construction allowed FFW-CRBMs to perform both, i.e. classification and real-valued predictions, within the same model, and to outperform state-of-the-art specialized methods for classification or prediction [@ffwcrbmprl].
**Contributions:** In this paper we, first, propose the use of FFW-CRBMs to estimate 3D trajectories from their 2D projections, while at the same time being also capable to classify those trajectories. Though successful, we discovered that FFW-CRBMs require substantial amount of *labeled data* before achieving acceptable performance when predicting three-dimensional trajectories from two-dimensional projections. As FFW-CRBMs require *three-dimensional labeled* information for accurate predictions which is not typically available, secondly, in this paper, we remedy these problems by proposing an extension of FFW-CRBMs, dubbed Disjunctive FFW-CRBMs (DFFW-CRBMs). Our extension refines the factoring of the four-way weight tensor connecting the machine layers to settings where labeled data is scarce. Adopting such a factorization “specializes” FFW-CRBMs and ensures lower energy levels (approximately three times less energy on the overall dataset). This yields the sufficiency of a reduced training dataset for DFFW-CRBMs to reach similar classification performance to state-of-the-art methods and to at least double the performance on real-valued predictions. Importantly, such accuracy improvements come at the same computational cost of $\mathcal{O}\left(n^{2}\right)$ compared to FFW-CRBMs. Precisely, our machine requires limited labeled data (less than 10 $\%$ of the overall dataset) for: *i)* simultaneously classifying and predicting three-dimensional trajectories based on their two-dimensional projections, and *ii)* accurately estimating three-dimensional postures up to an arbitrary number of time-steps in the future.
We have extensively tested DFFW-CRBMs on both, simulated and real-world data, to show that they are capable of outperforming state-of-the-art methods in real-valued predictions and classifications. In the first set of experiments we evaluate its performance by predicting and classifying simulated three-dimensional ball trajectories (based on a real-world physics simulator) thrown from different initial spins. Given these successes, in the second set of experiments we predict and classify high-dimensional human poses and activities (up-to 32 human skeleton joints in 2D and 3D coordinates systems, corresponding to 160 dimensions) using real-world data showing that DFFW-CRBMs acquire double accuracy results at reduced labeled data sizes.
Background
==========
This section provides relevant background knowledge essential to the remainder of the paper. Firstly, restricted Boltzmann machines (RBMs), being at the basis of our proposed method, are surveyed. Secondly, Contrastive Divergence, a training algorithm for Deep Learning methods, is presented. The section concludes with a brief description of deep-learning based models for time series prediction and classification. \[sec:background\]
Restricted Boltzmann Machines
-----------------------------
Restricted Boltzmann machines (RBMs) [@originalrbm] are energy-based models for unsupervised learning. They use a generative model of the distribution of training data for prediction [@mocanugenerativereplay]. These models employ stochastic nodes and layers, making them less vulnerable to local minima [@gwtaylorhdts]. Further, due to their stochastic neural configurations, RBMs possess excellent generalization and density estimation capabilities [@bengiodl; @mocanumljxbm].
Formally, an RBM consists of visible and hidden binary layers connected by an undirected bipartite graph. More exactly, the visible layer $\textbf{v}=[v_{1},\dots,v_{n_{v}}]$ collects all visible units $v_{i}$ and represents the real-data, while the hidden layer $\textbf{h}=[h_{1},\dots,h_{n_{h}}]$ representing all the hidden units $h_{j}$ increases the learning capability by enlarging the class of distributions that can be represented to an arbitrary complexity. $n_{v}$ and $n_{h}$ are the number of neurons in the visible and hidden layers, respectively. $W_{ij}$ denotes the weight connection between the $i^{th}$ visible and $j^{th}$ hidden unit, and $v_{i}$ and $h_{j}$ denote the state of the $i^{th}$ visible and $j^{th}$ hidden unit, respectively. The matrix of all weights between the layers is given by $\textbf{W}\in \mathbb{R}^{n_{h}\times n_{v}}$. The energy function of RBMs is given by $$E\left(v,h\right)=-\sum_{i=1}^{n_{v}}\sum_{j=1}^{n_{h}}W_{ij}v_{i}h_{j}-\sum_{i=1}^{n_{v}}a_{i}v_{i} - \sum_{j=1}^{n_{h}}b_{j}h_{j}
\label{eq:rbmenergy}$$ where, $a_{i}$ and $b_{j}$ represent the biases of the visible and hidden layers, respectively. The joint probability of a visible and hidden configuration can be written as $P\left(v,h\right)=\frac{\exp(-E(v,h))}{Z}$ with $Z=\sum_{x,y} \exp\left(-E(x,y)\right)$. The marginal distribution, $p(v)=\sum_{h}p(v,h)$ , can be used to determine the probability of a data point represented by a state $v$.
Training an RBM via Contrastive Divergence
------------------------------------------
The RBMs parameters are trained by maximizing the likelihood function, typically by following the gradient of the energy function. Unfortunately, in RBMs, maximum likelihood estimation can not be applied directly due to intractability problems. These problems can be circumvented by using Contrastive Divergence (CD) [@hintoncd] to train the RBM. In CD, learning follows the gradient of: $$CD_{n}\propto D_{KL}(p_{0}(\textbf{x})||p_{\infty}(\textbf{x}))-D_{KL}(p_{n}(\textbf{x})||p_{\infty}(\textbf{x}))$$ where, $p_{n}(\cdot)$ is the distribution of a Markov chain running for $n$ steps and $D_{KL}$ symbolizes the Kullback-Leibler divergence [@Ponti2017470]. To find the update rules for the free parameters of the RBM (i.e weights and biases), the RBM’s energy function from Equation \[eq:rbmenergy\] has to be differentiated with respect to those parameters. Thus, in $CD_{n}$ the weight updates are done as follows: $
w^{\tau+1}_{ij}=w^{\tau}_{ij}+\alpha\left(\left\langle\langle h_{j}v_{i}\rangle_{p(\textbf{h}|\textbf{v};\textbf{W})}\right\rangle_{0}-\langle h_{j}v_{i}\rangle_{n}\right)
$ where $\tau$ is the iteration number, $\alpha$ is the learning rate, $
\left\langle\langle h_{j}v_{i}\rangle_{p(\textbf{h}|\textbf{v};\textbf{W})}\right\rangle_{0}=\frac{1}{N_I}\sum_{k=1}^{N_I}v^{(q)}_{i}P(h^{(q)}_{j}=1|\textbf{v}^{(q)};\textbf{W})
$ and $
\langle h_{j}v_{i}\rangle_{n}=\frac{1}{N_I}\sum_{k=1}^{N_I}v^{(q)(n)}_{i}
P(h^{(q)(n)}_{j}=1|\textbf{v}^{(q)(n)};\textbf{W})
$ where $N_I$ is the total number of input instances, and the superscript $^{(q)}$ shows the $q^{th}$ input instance. The superscript $^{(n)}$ indicates that the states are obtained after $n$ steps of Gibbs sampling on a Markov chain which starts at the original data distribution $p_{0}(\cdot)$. In practice, learning can be performed using just one step Gibbs sampling, which is carried in four sub-steps: (1) initialize visible units, (2) infer all the hidden units, (3) infer all the visible units, and (4) update the weights and the biases.
Factored Conditional Restricted Boltzmann Machine {#Sec:FCRBMs}
-------------------------------------------------
Conditional Restricted Boltzmann Machines (CRBM) [@gwtaylorhdts] are an extension of RBMs used to model time series data, for example, human activities. They use an undirected model with binary hidden variables connected to real-valued visible ones. At each time step $t$, the hidden and visible nodes receive a connection from the visible variables at the last $L$ time-steps. The history of the real-world values until time $t$ is collected in the real-valued history vector $\textbf{v}_{<t}$ with $n_{v_{<t}}=n_v(L-1)$ being the number of elements in $\textbf{v}_{<t}$. The total energy of the CRBM is given by: $$E=\sum_{i=1}^{n_v}\frac{(\hat{a}_{i,t}-v_{i,t})^{2}}{2\sigma^{2}_{i}}-\sum_{j=1}^{n_h}\hat{b}_{j,t}h_{j,t}-\sum_{i=1}^{n_v}\sum_{j=1}^{n_h}W_{ij}\frac{v_{i,t}}{\sigma_{i}}h_{j,t}$$ where $\hat{a}_{i,t}=a_{i}+\sum_{k=1}^{n_{v_{<t}}}A_{ki}v_{k,<t}$ and $\hat{b}_{j,t}=b_{j}+\sum_{k=1}^{n_{v_{<t}}}B_{kj}v_{k,<t}$ represent the “dynamic biases", with $k$ being the index of the elements from $\textbf{v}_{<t}$.
Taylor and Hinton introduced the Factored Condition Restricted Boltzmann Machine (FCRBM) [@gwtaylorhdts], which permits the modeling of different styles of time series within the same model, due to the introduction of multiplicative, three-way interactions and of a *preset* style label, $\mathbf{y}_{t}$. To reduce the computational complexity of this model, they factored the third order tensors between layer in products of matrices. Formally, FCRBM defines a joint probability distribution over the visible $\mathbf{v_{t}}$ and hidden $\mathbf{h_{t}}$ neurons. The joint distribution is conditioned on the past $L$ observations, $\mathbf{v_{<t}}$, model parameters, $\boldsymbol{\Theta}$, and the *preset* style label, $\mathbf{y}_{t}$. Interested readers are referred to [@gwtaylorhdts] for a more comprehensive discussion on CRBMs and FCRBMs.
Four-Way Conditional Restricted Boltzmann Machines
--------------------------------------------------
Due to the limitations exhibited by FCRBMs, e.g., the impossibility of performing classification without extensions, we proposed the four-way conditional restricted Boltzmann machines (FW-CRBMs) for performing prediction and classification in one unified framework [@ffwcrbmprl]. FW-CRBMs introduced an additional layer and a four-way multiplicative weight tensor interaction between neurons. Please note that, later on, other four-way models have been proposed but they can perform just classification an no prediction [@Elaiwat2016152].
![A high level depiction of the FFW-CRBM showing the four layer configuration and the factored weight tensor connection among them. Gaussian nodes shown on the history and visible layers represent real-valued inputs, while sigmoidal nodes on the hidden and label layers demonstrate binary values.[]{data-label="fig:ffwcrbm"}](ffwcrbm.png){width="0.6\linewidth"}
FW-CRBMs extended FCRBMs to include a label layer $\textbf{l}_{t}$ and a fourth order weight tensor connection $\mathbf{W}_{ijko} \in \mathbb{R}^{n_{\textbf{v}} \times n_{\textbf{h}} \times n_{\textbf{v}_{<t}} \times n_{\textbf{l}}}$, where $n_{\textbf{v}}$, $n_{\textbf{h}}$, $n_{\textbf{v}_{<t}}$, $n_{\textbf{l}}$ represent the number of neurons from the present, hidden, history and label layers, respectively. Though successful, FW-CRBMs exhibited high computational complexities (i.e., $\mathcal{O}\left(n^{4}\right)$) for tuning free parameters. Circumventing these problems, we factored the weight tensor into sums of products leading to more efficient machines (i.e., $\mathcal{O}\left(n^{2}\right)$) labeled as factored four-way conditional restricted Boltzmann machines (FFW-CRBMs). FFW-CRBMs, shown in Figure \[fig:ffwcrbm\], minimize the following energy functional $$\label{eq:eqffwcrbm123}
\begin{split}
\mathbf{E}&(\textbf{v}_{t},\textbf{h}_{t},\textbf{l}_{t}|\textbf{v}_{<t},\Theta) = \\
&\hspace{-0.5em}-\sum\limits_{i=1}^{n_v} \frac{{(v_{i,t}-a_i)}^2}{{\sigma_i}^2}-\sum\limits_{j=1}^{n_h} h_{j,t}b_j
-\sum\limits_{o=1}^{n_l} l_{o,t}c_o \\
&\hspace{-0.5em}-\sum\limits_{f=1}^{n_F}\sum\limits_{i=1}^{n_v}W_{if}^{v}\frac{v_{i,t}}{\sigma_i}\sum\limits_{j=1}^{n_h} W_{jf}^{h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{l}l_{o,t}
\end{split}$$ where ${n_F}$ is number of factors and $i$, $j$, $k$, and $o$ are the indices of the visible layer neurons $\mathbf{v}_{t}$, the hidden layer neurons $\mathbf{h}_{t}$, the history layer neurons $\mathbf{v}_{<t}$ and the labeled layer neurons $\mathbf{l}_{t}$ respectively. $\mathbf{W}^v$, $\mathbf{W}^h$, $\mathbf{W}^l$ symbolize the bidirectional and symmetric weights from the visible, hidden and label layers to the factors, respectively, while $\mathbf{W}^{v_{<t}}$ represents the directed weights from the history layer to the factors. As in the case of the three-way models [@modellingjointdensities], standard CD is unsuccessful in training also the four-way models, due to the need of predicting two output layers (i.e. label and present layers). Thus, in [@ffwcrbmprl] we proposed a sequential variant of CD, named sequential Markov chain contrastive divergence, more suitable for tuning the free parameters in FW-CRBMs.
FFW-CRBMs have shown good generalization and time series latent feature learning capabilities compared to state-of-the-art techniques including but not limited to, support vector machines, CRBMs, and FCRBMs [@ffwcrbmprl]. It is for these reasons that we believe that FFW-CRBMs can serve as a basis for predicting three-dimensional trajectories from two-dimensional projections. Unfortunately, FFW-CRBMs are not readily applicable to such a problem as they require substantial amount of labeled data for successful tuning. In this paper, we extend FFW-CRBMs to Disjunctive FFW-CRBMs (DFFW-CRBMs) by proposing a novel factoring process essential for predicting and classifying 3D trajectories from 2D projections. Our model, detailed next, reduces sample complexities of current methods and allows for lower energy levels compared to FFW-CRBMs leading to improved performance.
Disjunctive Factored Four Way Conditional restricted Boltzmann Machines {#sec:newmodel}
=======================================================================
![A high level depiction of DFFW-CRBMs showing the four layer configuration and the refined tensors factoring for increased accuracy and efficiency.[]{data-label="fig:dffwcrbm"}](dffwcrbm.png){width="0.6\linewidth"}
This section details disjunctive factored four way conditional restricted Boltzmann machines (DFFW-CRBMs), shown in Figure \[fig:dffwcrbm\]. Similarly to FFW-CRBMs, our model consists of four layers to represent visible, history, hidden, and label units. Contrary to the factoring adopted by FFW-CRBMs, however, our model incorporates two new factoring layers. The first, i.e., $F^{1}(f)$ in the figure, is responsible for specializing the machine to real-valued predictions through $\textbf{W}^{1l}$, $\textbf{W}^{1v}$, $\textbf{W}^{1h}$, and $\textbf{W}^{1v_{<t}}$, while the second, $F^{2}(f)$, specializes the machine to classification through the corresponding weight tensor collections. Such a specialization is responsible for reducing sample complexities needed by DFFW-CRBMs for successful parameter tuning as demonstrated in Section \[sec:experiments\], while the computational complexity of DFFW-CRBM remains the same as for FFW-CRBM (i.e., $\mathcal{O}\left(n^{2}\right)$) . Given our novel construction, DFFW-CRBMs require their own special mathematical treatment. Next, we detail each of the energy functional and learning rules needed by DFFW-CRBMs.
DFFW-CRBM’s Energy Function
---------------------------
The energy function of DFFW-CRBMs consists of three major terms. The first, i.e., $-\sum\limits_{i=1}^{n_v} \frac{{(v_{i,t}-a_i)}^2}{{\sigma_i}^2}-\sum\limits_{j=1}^{n_h} h_{j,t}b_j
-\sum\limits_{o=1}^{n_l} l_{o,t}c_o$ corresponds to the standard energy representing a specific submachine of DFFW-CRBMs (i.e. the energy given by the neurons of each layers and their biases) while the second two denote energies related to the first and second factoring layers, respectively: $$\begin{split}
\mathbf{E}&(\textbf{v}_{t},\textbf{h}_{t},\textbf{l}_{t}|\textbf{v}_{<t},\Theta) = \label{Eq1:EnergyDFFWCRBM}\\
&\hspace{-0.5em}-\underbrace{\sum\limits_{i=1}^{n_v} \frac{{(v_{i,t}-a_i)}^2}{{\sigma_i}^2}-\sum\limits_{j=1}^{n_h} h_{j,t}b_j
-\sum\limits_{o=1}^{n_l} l_{o,t}c_o}_{\text{standard three-layer energy}} \\
&\hspace{-0.5em}-\underbrace{\sum\limits_{f=1}^{n_{F^1}}\sum\limits_{i=1}^{n_v}W_{if}^{1v}\frac{v_{i,t}}{\sigma_i}\sum\limits_{j=1}^{n_h} W_{jf}^{1h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{1v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}}_{\text{first factoring layer}} \\
&\hspace{-0.5em}-\underbrace{\sum\limits_{f=1}^{n_{F^2}}\sum\limits_{i=1}^{n_v}W_{if}^{2v}\frac{v_{i,t}}{\sigma_i}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{2v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t}}_{\text{second factoring layer}}
\end{split}$$ Here, $n_{F^1}$ denotes the total number of factors for the weight tensor collection specializing DFFW-CRBMs to regression, while $n_{F^2}$ is total the number of factors responsible for classification. $i$, $j$, $k$, and $o$ represent the indices of the visible layer neurons $\mathbf{v}_{t}$, the hidden layer neurons $\mathbf{h}_{t}$, the history layer neurons $\mathbf{v}_{<t}$ and the labeled layer neurons $\mathbf{l}_{t}$, respectively. Furthermore, $\mathbf{W}^{1v}$ and $\mathbf{W}^{1h}$ represent the bidirectional and symmetric weight connections from the visible and hidden layers to the factors, while $\mathbf{W}^{1l}$ and $\mathbf{W}^{1v_{<t}}$ denote the *directed* weights from the label and history layers to the factors. Similarly, $\mathbf{W}^{2l}$ and $\mathbf{W}^{2h}$ represent the bidirectional and symmetric weights from the label and hidden layers to the factors, while $\mathbf{W}^{2v}$ and $\mathbf{W}^{2v_{<t}}$ denote the directed weights from the visible and history layers to the factors. Finally, the two groups of four weight matrices each noted with $\mathbf{W}^{1.}$ and $\mathbf{W}^{2.}$ belong to the factorized tensor specialization in regression and classification, respectively.
DFFW-CRBM’s Activation Probabilities
------------------------------------
Inference for DFFW-CRBM corresponds to determining values of the activation probabilities for each of the units. As shown in Figure \[fig:dffwcrbm\], units within the same layer do not share connections. This allows for parallel probability computation for all units within the same layer. The overall input of each hidden $s^{h}_{j,t}$, visible $s^{v}_{i,t}$, and labelled $s^{l}_{o,t}$ unit is given by: $$\begin{split}
{s}_{j,t}^{h}&=\sum\limits_{f=1}^{n_{F^1}}W_{jf}^{1h}\sum\limits_{i=1}^{n_v}W_{if}^{1v}\frac{v_{i,t}}{\sigma_i}\sum\limits_{k=1}^{n_{n_{v<t}}} W_{kf}^{1v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t} \\
&+\sum\limits_{f=1}^{n_{F^2}}W_{jf}^{1h}\sum\limits_{i=1}^{n_v}W_{if}^{2v}\frac{v_{i,t}}{\sigma_i}\sum\limits_{k=1}^{n_{n_{v<t}}} W_{kf}^{2v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t} \\
{s}_{i,t}^{v}&=\sum\limits_{f=1}^{n_{F^1}}W_{if}^{1v}\sum\limits_{j=1}^{n_h}W_{jf}^{1h}h_{j,t}\sum\limits_{k=1}^{n_{n_{v<t}}} W_{kf}^{1v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}
\\
{s}_{o,t}^{l}&=\sum\limits_{f=1}^{n_{F^2}}W_{of}^{2l}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{k=1}^{n_{n_{v<t}}} W_{kf}^{2v_{<t}}\frac{v_{k,<t}}{\sigma_k}\sum\limits_{i=1}^{n_v}W_{if}^{2v}\frac{v_{i,t}}{\sigma_i}
\label{Eq:inputDFFWCRBM}
\end{split}$$ Consequently, for each of the $j^{th}$ hidden, $i^{th}$ visible, and $o^{th}$ labelled units, the activation probabilities can be determined as $$\begin{split}
%\label{Eq:inferencef1}
%\label{Eq:inferencef2}
p(h_{j,t}=1|\mathbf{v}_t,\mathbf{v}_{<t},\mathbf{l}_t)&=\frac{1}{1+e^{-\left(b_j+{s}_{j,t}^{h}\right)}} \\
p(v_{i,t}=x|\mathbf{h}_t,\mathbf{v}_{<t},\mathbf{l}_t)&=\mathcal{N}\left(a_i+ {s}_{i,t}^{v},\sigma_i^2\right) \\
p(l_{o,t}=1|\mathbf{v}_t,\mathbf{v}_{<t},\mathbf{h}_t)&=\frac{1}{1+e^{-\left(c_o+{s}_{o,t}^{l}\right)}}
\label{Eq:Eq:inferenceDFFWCRBM}
\end{split}$$ where $\mathcal{N}(\cdot)$ represents the standard Gaussian distribution.
Parameter Tuning: Update Rules & Algorithm
------------------------------------------
### Update Rules
Generally, parameters, $\mathbf{\Theta}$, are updated according to: $$\mathbf{\Theta}_{\tau+1}=\mathbf{\Theta}_{\tau}+\underbrace{\rho\mathbf{\widetilde\Theta}_{\tau}+\alpha(\Delta\mathbf{\Theta}_{\tau+1}-\gamma\mathbf{\Theta}_{\tau})}_{\mathbf{\widetilde\Theta}_{\tau+1} \text{ update}}$$ where $\tau$ represents the update iteration, $\rho\in(0,1)$ is the momentum, $\alpha\in(0,1)$ denotes the learning rate, and $\gamma\in(0,1)$ is the weight decay. A more detailed discussion on the choice of these parameters is provided by Hinton in [@hintontrain]. Therein, the update rules are attained by deriving the energy functional with respect to free parameters (i.e., weights matrices, and the biases of each of the layers). In DFFW-CRBMs, a set of eight free parameters, corresponding to the connections between the factors and each of the layers, has to be inferred. These are presented below. Intuitively, each of these update equations, aims at minimizing the reconstruction error (i.e., the error between the original inputs and these reconstructed through the model). Moreover, each of the update equations include three main terms representing the connections between the factored weights and the corresponding layer of the machine, as per Figure \[fig:dffwcrbm\]. For instance, connections to only the hidden, history, and label layers suffice for updating $W^{1v}_{if}$. Thus, the update rules $\Delta\mathbf{\Theta}_{\tau}$ for each of the weights corresponding to the first factored layer, can be computed as: $$\begin{split}
{\Delta}W_{if}^{1v}&\propto{{\Bigg\langle}\displaystyle v_{i,t}\sum\limits_{j=1}^{n_h} W_{jf}^{1h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}{\Bigg\rangle}}_{0}\\
&\hspace{0em}-{{\Bigg\langle}\displaystyle v_{i,t}\sum\limits_{j=1}^{n_h} W_{jf}^{1h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{kf}^{1v_{<t}}&\propto{{\Bigg\langle}\displaystyle v_{k,<t}\sum\limits_{j=1}^{n_h} W_{jf}^{1h}h_{j,t}\sum\limits_{i=1}^{n_v} W_{if}^{1v}v_{i,t}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}\displaystyle v_{k,<t}\sum\limits_{j=1}^{n_h} W_{jf}^{1h}h_{j,t}\sum\limits_{i=1}^{n_v} W_{if}^{1v}v_{i,t}\sum\limits_{o=1}^{n_l} W_{of}^{1l}l_{o,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{of}^{1l}&\propto{{\Bigg\langle}l_{o,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{j=1}^{n_h}W_{jf}^{1h}h_{j,t}\sum\limits_{i=1}^{n_v}W_{if}^{1v}v_{i,t}
{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}l_{o,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{j=1}^{n_h}W_{jf}^{1h}h_{j,t}\sum\limits_{i=1}^{n_v}W_{if}^{1v}v_{i,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{jf}^{1h}&\propto{{\Bigg\langle}h_{j,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{i=1}^{n_v}W_{if}^{1v}v_{i,t}\sum\limits_{o=1}^{n_l}W_{of}^{1l}l_{o,t}
{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}h_{j,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{1v_{<t}}v_{k,<t}\sum\limits_{i=1}^{n_v}W_{if}^{1v}v_{i,t}\sum\limits_{o=1}^{n_l}W_{of}^{1l}l_{o,t}{\Bigg\rangle}}_{\lambda}
\end{split}$$ while for the second factoring we have: $$\begin{split}
{\Delta}W_{if}^{2v}&\propto{{\Bigg\langle}\displaystyle v_{i,t}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t}{\Bigg\rangle}}_{0}\\ &\hspace{0em}-{{\Bigg\langle}\displaystyle v_{i,t}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{k=1}^{n_{v<t}} W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{kf}^{2v_{<t}}&\propto{{\Bigg\langle}\displaystyle v_{k,<t}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{i=1}^{n_v} W_{if}^{2v}v_{i,t}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t}{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}\displaystyle v_{k,<t}\sum\limits_{j=1}^{n_h} W_{jf}^{2h}h_{j,t}\sum\limits_{i=1}^{n_v} W_{if}^{2v}v_{i,t}\sum\limits_{o=1}^{n_l} W_{of}^{2l}l_{o,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{of}^{2l}&\propto{{\Bigg\langle}l_{o,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{j=1}^{n_h}W_{jf}^{2h}h_{j,t}\sum\limits_{i=1}^{n_v}W_{if}^{2v}v_{i,t}
{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}l_{o,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{j=1}^{n_h}W_{jf}^{2h}h_{j,t}\sum\limits_{i=1}^{n_v}W_{if}^{2v}v_{i,t}{\Bigg\rangle}}_{\lambda} \\
{\Delta}W_{jf}^{2h}&\propto{{\Bigg\langle}h_{j,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{i=1}^{n_v}W_{if}^{2v}v_{i,t}\sum\limits_{o=1}^{n_l}W_{of}^{2l}l_{o,t}
{\Bigg\rangle}}_{0}\\&\hspace{0em}-{{\Bigg\langle}h_{j,t}\sum\limits_{k=1}^{n_{v<t}}W_{kf}^{2v_{<t}}v_{k,<t}\sum\limits_{i=1}^{n_v}W_{if}^{2v}v_{i,t}\sum\limits_{o=1}^{n_l}W_{of}^{2l}l_{o,t}{\Bigg\rangle}}_{\lambda}
\end{split}$$ and for the biases are: $$\begin{split}
{\Delta}a_{i}&\propto{{\langle}v_{i,t}{\rangle}}_{0}-{{\langle}v_{i,t}{\rangle}}_{\lambda}
\\
{\Delta}b_{j}&\propto{{\langle}h_{j,t}{\rangle}}_{0}-{{\langle}h_{j,t}{\rangle}}_{\lambda}
\\
{\Delta}c_{o}&\propto{{\langle}l_{o,t}{\rangle}}_{0}-{{\langle}l_{o,t}{\rangle}}_{\lambda}
\end{split}$$ where $\lambda$ represents a Markov chain step running for a total of $n$ steps and starting at the original data distribution, $\langle \cdot \rangle_{0}$ denotes the expectation under the input data, and $\langle \cdot \rangle_{\lambda}$ represents the model’s expectation.
### Sequential CD for DFFW-CRBMs
Algorithm \[Algo:AlgoOne\] presents a high-level description of the sequential Markov chain contrastive divergence [@ffwcrbmprl] adapted to train DFFW-CRBMs. It shows the two main steps needed for training such machines. Firstly, the visible layer is inferred by fixing the history and label layers. While in the second step the label layer is reconstructed by fixing the history and the present layers. Updating the weights involves the implementation of the rules derived in the previous section. These two procedures are then repeated for a pre-specified number of epochs, where at each epoch the reconstruction error is decreasing to reach the minimum of the energy function, guaranteeing a minimized divergence between the original data distribution and the one given by the model.
------------------------------------------------------------------------
------------------------------------------------------------------------
**Inputs:** TD - training data, $n$ - number of Markov Chain steps
------------------------------------------------------------------------
**Initialization:** $\mathbf{\Theta}$ $\leftarrow$ $\mathcal{N}(0,\sigma^2)$, Set $\alpha$, $\rho$, $\gamma$
------------------------------------------------------------------------
------------------------------------------------------------------------
\[Algo:AlgoOne\]
Experiments and Results {#sec:experiments}
=======================
This section extensively tests the performance of DFFW-CRBMs on both simulated as well as on real-world datasets. The major goal of these experiments was to assess the capability of DFFW-CRBM to predict three-dimensional trajectories from two-dimensional projection, given small amounts of labeled data (i.e., in the order of 9-10 % of the total dataset). As a secondary objective, the goal was to classify such trajectories to different spins (ball trajectories) or activities (human pose estimation). In the real-valued prediction setting, we compared our method to state-of-the-art FFW-CRBMs and FCRBMs, while for classification our method’s performance was tested against FFW-CRBMs and support vector machines with radial basis functions (SVM-RBFs) [@vapniksvm].
**Evaluation Metrics:** To assess the models’ performance, a variety of standard metrics were used. For classification, we used accuracy [@roc] in percentages, while for estimation tasks, we used the Normalized Root Mean Square Error (NRMSE) estimating distance between the prediction and ground truth, Pearson Correlation Coefficient (PCC) reflecting the correlations between predictions and ground truth, and the P-value to arrive at statistically significant predictions.
Ball Trajectory Experiments
---------------------------
We generated different ball trajectories thrown with different spins using the Bullet Physics Library[^1]. With this simulated dataset we targeted three objectives using small amounts (9 %) of labeled training data. First, we estimated 3D ball coordinates based on their 2D projections at each time-step $t$ (i.e., one-step prediction). Second, we aimed at predicting near-future (i.e., couple of time steps in the future) 3D ball coordinates recursively, while giving limited 2D sequence of coordinates as a starting point. Third, we classified various ball spins based on just 2D coordinates. We used four trajectory classes corresponding to four different ball spin types. For each class, a set of 11 trajectories each containing approximately 400 time-steps (amounting to a total of 17211 data instances) were sampled. To assess the performance of DFFW-CRBM, we performed 11-fold cross validation and reported mean and standard deviation results. Precisely, from each class of trajectories we used only *one labeled trajectory*[^2] to train the models and the other 10 were used for testing.
![Averaged energy levels of FFW-CRBM and DFFW-CRBM over all ball trajectories when the parameters (i.e. number of hidden neurons and factors) are varying. The training was done for 100 epochs.[]{data-label="fig:enleveltuneparameterballs"}](energylevels_parametertuneballs_v2-crop.pdf){width="0.6\linewidth"}
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------------ --------------- ---------------- ---------------- ----------------- -------------------- ---------------------
SVM-RBF FCRBM FFW-CRBM DFFW-CRBM
Accuracy\[%\] 39.26$\pm$4.63 N/A 37.49$\pm$3.66 **39.51$\pm$4.47**
NRMSE\[%\] N/A 18.38$\pm$8.07 19.53$\pm$33.24 **11.24$\pm$8.53**
PCC N/A -0.06$\pm$0.70 0.31$\pm$0.72 **0.62$\pm$0.61**
P-value N/A 0.51$\pm$0.29 0.40$\pm$0.29 **0.28$\pm$0.27**
After NRMSE\[%\] N/A 25.61$\pm$3.25 23.53$\pm$2.48 **9.52$\pm$6.12**
1 step PCC N/A 0.14$\pm$0.69 0.31$\pm$0.74 **0.95$\pm$0.14**
Multi-Step P-value N/A 0.50$\pm$0.29 0.38$\pm$0.25 **0.12$\pm$0.18**
3D After NRMSE\[%\] N/A 31.38$\pm$7.99 29.49$\pm$8.14 **19.93$\pm$10.27**
prediction 50 steps PCC N/A 0.05$\pm$0.69 -0.05$\pm$0.72 **0.20$\pm$0.66**
P-value N/A 0.51$\pm$0.26 0.47$\pm$0.26 0.51$\pm$0.26
------------ --------------- ---------------- ---------------- ----------------- -------------------- ---------------------
: Classification, present step 3D estimation, and multi-step 3D prediction for the balls trajectories experiment. Results, cross-validated and presented with mean and standard deviation, show that our method is capable of outperforming state-of-the-art techniques on all evaluation metrics.[]{data-label="tab:claspreddiffballs"}
**Deep Learner Setting:** The visible layers of both models (i.e. FFW-CRBM and DFFW-CRBM) were set to 5 neurons, three denoting 3D ball center coordinates (i.e. x, y, z), and two for its 2D projection at time $t$. The label layer consisted of 4 neurons (one for each of the different spins classes), while the history layers included 100 neurons corresponding to the last 50 history frames. One frame incorporates the 2D coordinates of the center of the ball projected in a two dimensional space. The number of hidden neurons was set to $10$, and the number of factors to $100$, as discussed in the next paragraph, and in Subsection \[subsec:har\]. A learning rate of $10^{-4}$ and momentum of $0.5$ were chosen. Weight decay factors were set to $0.0002$, and the number of the Markov chain steps for CD in the training phase, but also for the Gibbs sampling in the testing phase, was set to 3. All weights were initialized with $\mathcal{N}(0,0.3)$. Finally, data were normalized to have 0 mean and unit variance as explained in [@hintontrain], and the models were trained for 100 epochs.
**Importance of disjunctive Factoring:** To find the optimal number of hidden neurons and factors, we have performed exhaustive search by varying the number of hidden neurons from 10 to 100 and the number of factors from 10 to 160. To gain some insights on the behavioral differences between FFW-CRBMs and DFFW-CRBMs, even if the energy equation of DFFW-CRBM has an extra tensor, in Figure \[fig:enleveltuneparameterballs\] we illustrate on the same scale the heat-map of the averaged energy levels. They were computed using Equation \[eq:eqffwcrbm123\] for FFW-CRBM and Equation \[Eq1:EnergyDFFWCRBM\] for DFFW-CRBM, after both models were trained for 100 epochs. Though both models acquire the lowest energy levels in a configuration starting with 10-20 hidden neurons and a number of factors larger than 100, analyzing these results signifies the importance of the disjunctive factoring introduced in the paper. Namely, DFFW-CRBMs always acquire lower energy levels compared to FFW-CRBMs due to it’s “specialized” tensor factoring. Moreover, by averaging the energy levels from the aforementioned figure, we found that the average energy level of DFFW-CRBM is approximately three times smaller than the one of FFW-CRBM (i.e. $-6.63\pm2.09$ for DFFW-CRBM, and $-2.04\pm1.43$ for FFW-CRBM), thus anticipating the more accurate performance results, as showed next.
Figures \[fig:diffballstrajectories\] and \[fig:ballffwcrbm\] compare the capabilities of DFFW-CRBMs on estimating different 3D trajectories of balls picked at random to FFW-CRBMs, showing that our method is capable of achieving closely correlated transitions to the real trajectory. Interestingly, DFFW-CRBMs can handle discontinuities “less abruptly” compared to FFW-CRBMs. The cross-validation results showing the performance of all models of all ball trajectories are summarized in Table \[tab:claspreddiffballs\]. In terms of classification, SVM-RBF, FFW-CRBM, and DFFW-CRBM perform almost similarly, with a slightly advantage of DFFW-CRBM[^3]. In the case of 3D coordinates estimation from 2D projection at a time-step $t$, DFFW-CRBM clearly outperforms state-of-the-art methods with a NRMSE almost twice smaller than FCRBMs and FFW-CRBMs. Besides that, in this case, the mean value of the correlation coefficient for DFFW-CRBM is $0.62$, double than that for FFW-CRBM, while the one for FCRBM is powerless (i.e below zero). For the multi-step prediction of near-future 3D point coordinates, DFFW-CRBM has an even more significant improvement. It is worth highlighting that in this scenario, the average PCC value after one step prediction is almost perfectly $0.95$, while after 50 steps predicted into the future the mean PCC value is still positive and larger than those of the other methods. In a final set of experiments we tested the change in the accuracy of classification as a number of data points used. These are summarized in the bar-graph in Figure \[fig:increaseaccballs\], showing that our method slightly outperforms the state-of-the-art techniques in all cases.
{width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"} {width="0.19\linewidth"}
![Estimation of the 3D trajectory for the center of one ball from its 2D projection using FFW-CRBM (left) and DFFW-CRBM (right). The top figure presents the trajectory in the 3D space, while the bottom figure presents the Ox, Oy, Oz coordinates of the same trajectory in a 2D plot.[]{data-label="fig:ballffwcrbm"}](diff_data_estimation_ffwcrbm_3d_in_3d-crop.pdf){width="0.9\linewidth"}
![Estimation of the 3D trajectory for the center of one ball from its 2D projection using FFW-CRBM (left) and DFFW-CRBM (right). The top figure presents the trajectory in the 3D space, while the bottom figure presents the Ox, Oy, Oz coordinates of the same trajectory in a 2D plot.[]{data-label="fig:ballffwcrbm"}](diff_data_estimation_dffwcrbm_3d_in_3d-crop.pdf){width="0.93\linewidth"}
![Estimation of the 3D trajectory for the center of one ball from its 2D projection using FFW-CRBM (left) and DFFW-CRBM (right). The top figure presents the trajectory in the 3D space, while the bottom figure presents the Ox, Oy, Oz coordinates of the same trajectory in a 2D plot.[]{data-label="fig:ballffwcrbm"}](diff_data_estimation_ffwcrbm_3d_in_2d-crop.pdf){width="0.9\linewidth"}
![Estimation of the 3D trajectory for the center of one ball from its 2D projection using FFW-CRBM (left) and DFFW-CRBM (right). The top figure presents the trajectory in the 3D space, while the bottom figure presents the Ox, Oy, Oz coordinates of the same trajectory in a 2D plot.[]{data-label="fig:ballffwcrbm"}](diff_data_estimation_dffwcrbm_3d_in_2d-crop.pdf){width="0.9\linewidth"}
![Average classification accuracies with mean and standard deviation, over all balls trajectories, when the amount of training data is increased.[]{data-label="fig:increaseaccballs"}](plots_balls_class_more_datapoints-crop.pdf){width="0.6\linewidth"}
Human Activity Recognition {#subsec:har}
--------------------------
Given the above successes, next we evaluate the performance of our method on real-world data representing a variety of human activities. In each set of experiments, we targeted two main objectives and a third secondary one. The first two corresponded to estimating three-dimensional joint coordinates from two-dimensional projections as well as predicting such coordinates in near future, while the third involved classifying activities based on only two-dimensional joint coordinates. Please note that the third experiment is exceptionally hard due to the loss of three-dimensional information making different activities more similar.
**Human 2.6m dataset.** For all experiments, we used the real-world comprehensive benchmark database [@IonescuSminchisescu11; @h36m_pami], containing 17 activities performed by 11 professional actors (6 males and 5 females) with over 3.6 million 3D human poses and their corresponding images. Further, for 7 actors, the database accurately reports 32 human skeleton joint positions in 3D space, together with their 2D projections acquired at 50 frames per seconds (FPS).
![Averaged energy levels of FFW-CRBM and DFFW-CRBM for the human activities experiments when the parameters (i.e. number of hidden neurons and factors) are varying. The training was done for 100 epochs.[]{data-label="fig:enleveltuneparameter"}](energylevels_parametertune_cropped.pdf){width="0.6\linewidth"}
We used these seven actors being Subject 1 (S1), Subject 5 (S5), Subject 6 (S6), Subject 7 (S7), Subject 8 (S8), Subject 9 (S9), Subject 11 (S11) accompanied with their corresponding joint activities, such as Purchasing (A1), Smoking (A2), Phoning (A3), Sitting-Down (A4), Eating (A5), Walking-Together (A6), Greeting (A7), Sitting (A8), Posing (A9), Discussing (A10), Directing (A11), Walking (A12), and Waiting (A13). To avoid computational overhead, we have also reduced the temporal resolution of the data to 5 FPS leading to a total of 46446 training and testing instances. The instances were split between different subjects as: S1 (5514 instances), S5 (8748 instances), S6 (5402 instances),S7 (9081 instances), S8 (5657 instances), S9 (6975 instances), and S11 (5069 instances).
**Deep Learner Setting:** The visible layers of both the FFW-CRBM and DFFW-CRBM were set to 160 neurons corresponding to 96 neurons for the 3D coordinates of the joints, and 64 for their 2D projections at time $t$. The label layer consisted of 13 neurons (one for each of the activities), and the history layers included 320 neurons corresponding to 5 history frames each incorporating 2D joint coordinates. The size of the hidden layer was set to $10$ neurons, and the number of factors to $100$, as explained in the next paragraph. Furthermore, a learning rate of $10^{-5}$ was used to guarantee bounded reconstruction errors. The number of the Markov Chain steps in the training phase and of the Gibbs sampling in the testing phase were set to 3, and the weights were initialized with $\mathcal{N}(0,0.3)$. Further particularities, such as momentum and weight decay were set to $0.5$ and $0.0002$. Also, all data were normalized to have a 0 mean and unit standard deviation.
**Importance of disjunctive Factoring:** Similarly with the previous experiment on simulated balls trajectories, we searched for the optimal number of hidden neurons and factors, by performing exhaustive search and varying the number of hidden neurons and factors from 10 to 100 and from 10 to 160, respectively. Figure \[fig:enleveltuneparameter\] depicts on the same scale the averaged energy levels for both FFW-CRBM and DFFW-CRBM, after being trained for 100 epochs. As before, in the balls experiment, the energy levels of both models are more affected by the number of factors than the number of hidden neurons. Even if we are scrutinizing unnormalized energy levels, the fact that the energy levels of DFFW-CRBM are always much lower than the energy levels of FFW-CRBM reflects the importance of the disjunctive factoring. By quantifying and averaging all the energy levels for each model, we may observe that DFFW-CRBM has in average approximately three times less energy than FFW-CRBM (i.e. $-153.32\pm17.31$ for DFFW-CRBM, and $-48.57\pm33.92$ for FFW-CRBM).
### Training and Testing on The Same Person
Here, data from the same subject has been used for both training and testing. Emulating real-world 3D trajectory prediction settings where labeled data is scarce, we made use of only 10$\%$ of the available data for training and 90$\%$ for testing with the aim of performing accurate one and multi-step 3D trajectory predictions.
Results in Tables \[tab:accsameperson\], \[tab:onepredsameperson\], and \[tab:multipredsameperson\] show that DFFW-CRBMs are capable of achieving better performance than state-of-the-art techniques in both classification and prediction even when only using a small amount of training data. These results provide a proof-of-concept to the fact that DDFW-CRBMs are capable of accurately predicting (in both one-step and multi-step scenarios) 3D trajectories from their 2D projections by using only 10$\%$ of the data for training and 90$\%$ for testing.
Persons SVM-RBF FFW-CRBM DFFW-CRBM
--------- --------- ----------- -----------
S1 49.77 50.53 49.34
S5 36.92 38.82 40.21
S6 30.68 31.51 30.18
S7 38.50 37.03 37.94
S8 26.49 30.41 31.32
S9 24.69 28.12 22.63
S11 34.56 34.21 32.16
Average 34.51 **35.80** 34.83
: Classification accuracies in percentages for the human activities experiments, when training and testing data belong to the same person.[]{data-label="tab:accsameperson"}
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--------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ---------------------------- ---------------------------- ----------------------------
NRMSE \[%\] PCC P-value NRMSE \[%\] PCC P-value NRMSE \[%\] PCC P-value
S1 8.41$\pm$3.75 0.02$\pm$0.10 0.49$\pm$0.29 9.93$\pm$7.47 0.13$\pm$0.37 0.15$\pm$0.26 6.36$\pm$3.45 0.54$\pm$0.29 0.05$\pm$0.16
S5 6.70$\pm$2.44 -0.03$\pm$0.09 0.54$\pm$0.28 6.95$\pm$3.21 0.10$\pm$0.33 0.16$\pm$0.26 4.30$\pm$2.30 0.68$\pm$0.25 0.02$\pm$0.10
S6 4.41$\pm$1.93 0.03$\pm$0.09 0.53$\pm$0.28 4.50$\pm$2.37 0.01$\pm$0.28 0.21$\pm$0.29 3.19$\pm$1.64 0.50$\pm$0.32 0.05$\pm$0.16
S7 9.14$\pm$3.46 0.02$\pm$0.10 0.49$\pm$0.29 9.16$\pm$4.30 0.13$\pm$0.35 0.14$\pm$0.25 6.19$\pm$3.11 0.71$\pm$0.24 0.01$\pm$0.09
S8 8.31$\pm$3.37 -0.00$\pm$0.11 0.47$\pm$0.29 8.23$\pm$4.42 0.02$\pm$0.27 0.26$\pm$0.31 4.96$\pm$2.57 0.62$\pm$0.26 0.05$\pm$0.18
S9 7.25$\pm$2.74 0.00$\pm$0.09 0.55$\pm$0.28 8.40$\pm$5.05 0.01$\pm$0.27 0.22$\pm$0.29 4.63$\pm$2.34 0.54$\pm$0.32 0.05$\pm$0.17
S11 9.62$\pm$4.05 -0.00$\pm$0.10 0.54$\pm$0.28 9.89$\pm$5.94 0.06$\pm$0.32 0.15$\pm$0.25 6.82$\pm$3.82 0.53$\pm$0.35 0.04$\pm$0.14
Average $\approx$7.69$\pm$3.10 $\approx$0.01$\pm$0.09 $\approx$0.51$\pm$0.28 $\approx$8.15$\pm$4.68 $\approx$0.07$\pm$0.31 $\approx$0.18$\pm$0.27 **$\approx$5.21$\pm$2.75** **$\approx$0.59$\pm$0.29** **$\approx$0.04$\pm$0.14**
--------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ---------------------------- ---------------------------- ----------------------------
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----------- --------- ------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ---------------------------- ---------------------------- ----------------------------
Steps
Predicted NRMSE \[%\] PCC P-value NRMSE \[%\] PCC P-value NRMSE \[%\] PCC P-value
S1 7.68$\pm$3.71 0.02$\pm$0.09 0.51$\pm$0.28 7.78$\pm$4.95 0.17$\pm$0.32 0.17$\pm$0.27 5.91$\pm$3.45 0.55$\pm$0.26 0.06$\pm$0.16
S5 6.72$\pm$2.48 -0.05$\pm$0.09 0.50$\pm$0.28 6.41$\pm$2.70 0.19$\pm$0.37 0.18$\pm$0.29 3.88$\pm$1.68 0.68$\pm$0.25 0.01$\pm$0.08
After S6 4.40$\pm$2.17 0.04$\pm$0.09 0.52$\pm$0.28 4.27$\pm$2.31 0.11$\pm$0.31 0.21$\pm$0.30 3.17$\pm$1.83 0.48$\pm$0.32 0.05$\pm$0.16
1 step S7 9.07$\pm$3.20 0.02$\pm$0.11 0.46$\pm$0.29 8.78$\pm$3.52 0.27$\pm$0.35 0.06$\pm$0.17 6.52$\pm$3.05 0.73$\pm$0.17 0.00$\pm$0.02
S8 7.16$\pm$3.08 0.01$\pm$0.12 0.48$\pm$0.31 6.42$\pm$3.51 0.04$\pm$0.23 0.30$\pm$0.31 3.93$\pm$1.87 0.69$\pm$0.19 0.01$\pm$0.05
S9 6.98$\pm$2.64 -0.01$\pm$0.09 0.54$\pm$0.29 6.91$\pm$3.17 0.08$\pm$0.26 0.22$\pm$0.28 4.40$\pm$1.69 0.64$\pm$0.20 0.01$\pm$0.08
S11 9.55$\pm$4.05 -0.00$\pm$0.08 0.56$\pm$0.25 8.92$\pm$4.66 0.10$\pm$0.31 0.18$\pm$0.27 7.02$\pm$4.00 0.51$\pm$0.44 0.04$\pm$0.14
Average $\approx$7.37$\pm$3.05 $\approx$0.00$\pm$0.09 $\approx$0.51$\pm$0.28 $\approx$7.07$\pm$3.55 $\approx$0.14$\pm$0.31 $\approx$0.18$\pm$0.27 **$\approx$4.96$\pm$2.51** **$\approx$0.61$\pm$0.26** **$\approx$0.03$\pm$0.1**
S1 10.88$\pm$3.17 0.01$\pm$0.10 0.55$\pm$0.32 10.23$\pm$5.39 0.10$\pm$0.41 0.17$\pm$0.26 8.22$\pm$4.56 -0.03$\pm$0.29 0.14$\pm$0.24
S5 7.50$\pm$2.20 0.02$\pm$0.10 0.53$\pm$0.28 8.40$\pm$3.20 0.01$\pm$0.43 0.19$\pm$0.30 6.86$\pm$2.47 0.12$\pm$0.43 0.08$\pm$0.19
After S6 5.38$\pm$1.92 0.01$\pm$0.12 0.45$\pm$0.29 4.77$\pm$2.65 -0.02$\pm$0.24 0.22$\pm$0.30 4.44$\pm$2.17 0.12$\pm$0.32 0.12$\pm$0.23
50 steps S7 11.07$\pm$3.61 -0.03$\pm$0.09 0.52$\pm$0.30 10.68$\pm$4.04 0.11$\pm$0.44 0.15$\pm$0.26 9.31$\pm$3.60 0.17$\pm$0.33 0.12$\pm$0.25
S8 15.41$\pm$1.66 0.01$\pm$0.11 0.45$\pm$0.29 11.33$\pm$5.81 0.09$\pm$0.24 0.26$\pm$0.29 9.91$\pm$5.73 0.08$\pm$0.38 0.10$\pm$0.21
S9 9.25$\pm$2.10 0.01$\pm$0.11 0.48$\pm$0.28 8.56$\pm$3.23 0.03$\pm$0.25 0.25$\pm$0.28 7.48$\pm$3.60 -0.02$\pm$0.39 0.12$\pm$0.22
S11 14.39$\pm$2.71 -0.01$\pm$0.09 0.52$\pm$0.28 10.93$\pm$5.78 0.17$\pm$0.37 0.18$\pm$0.29 8.56$\pm$4.47 0.05$\pm$0.51 0.12$\pm$0.26
Average $\approx$10.55$\pm$2.48 $\approx$0.00$\pm$0.10 $\approx$0.5$\pm$0.29 $\approx$9.27$\pm$4.3 $\approx$0.07$\pm$0.34 $\approx$0.20$\pm$0.28 **$\approx$7.82$\pm$3.8** $\approx$0.07$\pm$0.37 **$\approx$0.11$\pm$0.23**
----------- --------- ------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ---------------------------- ---------------------------- ----------------------------
![Average classification accuracies with mean and standard deviation for the human activities experiments, over all subjects, when the data for training and testing the models come from the same person and the amount of training data is increased.[]{data-label="fig:increaseacc"}](plots_activities_class_more_datapoints-crop.pdf){width="0.6\linewidth"}
**Activity Recognition (Classification)** The goal in this set of experiments was to classify the 13 activities based on only their 2D projections. Please note that such a task is substantially difficult to solve due to the loss of information exhibited by the performed projection. Namely, activities different in 3D space might resemble high similarities in their 2D projections leading to low classification accuracies. Table \[tab:accsameperson\] reports the accuracy performance of DFFW-CRBMs, against state-of-the-art methods including SVMs and FFW-CRBMs. By averaging the results over all subjects, we can observe that all three models perform comparable. It is worth mentioning that the classification accuracy for random choice in this scenario would be $7.69\%$ and all models performs approximately 5 times better.
![Multi-step 3D prediction on the worst performer (S11) and best performer (S6) subjects, when the data for training and testing the models come from the same person.[]{data-label="fig:m3ds8"}](multistep_pred_subj_6and11-crop.pdf){width="0.6\linewidth"}
We also performed two more experiments to classify activities with more input data points to prove the correctness of the presented methods and show DFFW-CRBMs is capable of achieving state-of-the-art classification results. Here, we used 33$\%$ and 66$\%$ of the data to train the models, and the remaining for test. It is clear from Figure \[fig:increaseacc\] that all models increase in performance as the amount of training data increases, reaching around 55$\%$ accuracy when 66$\%$ of the data is used for training.
![Multi-step 3D prediction using cross-validation on all subjects, when the data for training and testing the models come from different persons.[]{data-label="fig:m3cv"}](multistep_pred_subj_differentpersons-crop.pdf){width="0.6\linewidth"}
**Estimating 3D Skeleton Coordinates from 2D Projections (Present Step Prediction)** In this task, we estimate the 3D joint coordinates from their 2D counterpart while using 10$\%$ training data. Results depicted in Table \[tab:onepredsameperson\] show that DFFW-CRBMs achieves better performance than FFW-CRBMs and FCRBM. Though FFW-CRBMs perform comparatively, it is worth noting that the PCC and P-values signify the fact that DFFW-CRBMs drastically outperform FFW-CRBMs in the sense that the predictions are correlated with ground truth, a property essential for accurate and reliable predictions.
**Prediction of 3D Skeleton Trajectories (Multi-Step Prediction)** Here, the goal was to perform multi-step predictions of the 3D skeleton joints based on only 2D projections. Starting from a 2D initial state, the model was executed autonomously by recursively feeding-back 2D outputs to perform next-step predictions. Definitely, the performance is expected to degrade since the prediction errors accumulate with time. Table \[tab:multipredsameperson\], showing the performance of the models after 1 and 50 step predictions, validate this phenomenon since all metrics show a decrease in both models’ performance over time. Table \[tab:multipredsameperson\], however, also signify that DFFW-CRBMs outperform FFW-CRBMs in both one and multi-step predictions achieving an average NRMSE of 7.82 compared to 9.27 NRMSE for FFW-CRBMs. Further results are summarized by Figure \[fig:m3ds8\] showing the minimum and maximum performance results of both models. In these experiments, clearly, DFFW-CRBM is the best performer in both, prediction errors and correlations.
### Testing Generalization Capabilities
**Motivation:** In the second set of human activities experiments, our goal was to determine to what extend can DFFW-CRBMs generalize across different human subjects and activities. The main motivation is that in reality subject-specific data is scarce, while data available from different users or domains is abundant. Results reported in Table \[tab:claspreddiffperson\] and Figure \[fig:m3cv\] show that DFFW-CRBMs are capable of generalizing beyond specific subjects due to their ability in learning latent features shared among a variety of tasks.
=0.03cm
------------ --------------- ---------------- ---------------- -------------------- ------------------- -------------------
SVM-RBF FCRBM FFW-CRBM DFFW-CRBM
Accuracy\[%\] 37.93$\pm$5.04 N/A **44.96$\pm$2.68** 44.49$\pm$6.60
NRMSE\[%\] N/A 7.58$\pm$3.62 7.52$\pm$3.63 **3.93$\pm$1.75**
PCC N/A -0.00$\pm$0.09 0.14$\pm$0.24 **0.79$\pm$0.16**
P-value N/A 0.52$\pm$0.28 0.21$\pm$0.28 **0.01$\pm$0.03**
After NRMSE\[%\] N/A 6.60$\pm$3.53 6.52$\pm$3.54 **3.95$\pm$1.99**
1 step PCC N/A -0.01$\pm$0.11 0.21$\pm$0.27 **0.81$\pm$0.14**
Multi-Step P-value N/A 0.49$\pm$0.29 0.14$\pm$0.24 **0.01$\pm$0.03**
3D After NRMSE\[%\] N/A 7.27$\pm$3.81 **7.24$\pm$3.84** 7.34$\pm$3.84
prediction 50 steps PCC N/A 0.01$\pm$0.11 0.13$\pm$0.46 **0.16$\pm$0.50**
P-value N/A 0.49$\pm$0.31 0.10$\pm$0.20 0.10$\pm$0.22
------------ --------------- ---------------- ---------------- -------------------- ------------------- -------------------
: Classification, present step 3D estimation, and multi-step 3D prediction, for the human activities experiments, when the training and the testing are done on different persons. The results are cross-validated and presented with mean and standard deviation.[]{data-label="tab:claspreddiffperson"}
**Experiments:** Here, data from 6 subjects was used to train the models, and predictions on an unseen subject were performed. The procedure was then repeated to cross-validate the results. Further, to emulate real-world settings only 10$\%$ of the data was used for training. During testing, however, all data from the all testing subjects was used increasing the tasks’ difficulty. The same three goals of the previous experiments were targeted.
**Activity Recognition (Classification):** Results reported in Table \[tab:claspreddiffperson\] show that DFFW-CRBMs achieve comparable results to FFW-CRBMs at an accuracy of 44.5 $\%$ both outperforming SVMs. Clearly, these classification results resemble higher accuracies when compared to these in Table \[tab:accsameperson\]. The reasons can be attributed back to the availability of similar domain data from other subjects signifying the latent feature similarities automatically learn by DFFW-CRBMs.
**Estimation of 3D Skeleton Coordinates from 2D Projections (Present Step Prediction):** Again, DFFW-CRBMs achieve better performance than FFW-CRBMs in present step estimation of the 3D skeleton joints from 2D projections, while both outperform FCRBM. It is worth highlighting that DFFW-CRBMs are capable of attaining a high average prediction correlation to ground-truth of almost 0.8.
**Prediction of 3D skeleton Trajectories (Multi-Step Prediction):** Finally, Figure \[fig:m3cv\] shows that DFFW-CRBMs are capable of surpassing FFW-CRBMs in multi-step predictions on unseen subjects achieving low prediction errors and high ground truth correlation.
Conclusion {#Sec:Conclusions}
==========
In this paper we proposed *disjunctive factored four-way conditional restricted Boltzmann machines* (DFFW-CRBMs). These novel machine learning techniques can be used for estimating 3D trajectories from their 2D projections using limited amounts of labeled data. Due to the new tensor factoring introduced by DFFW-CRBMs, these machines are capable of achieving substantially lower energy levels than state-of-the-art techniques leading to more accurate predictions and classification results. Furthermore, DFFW-CRBMs are capable of performing classification and accurate near-future predictions simultaneously in one unified framework.
Two sets of experiments, one on a simulated ball trajectories dataset and one on a real-world benchmark database, demonstrate the effectiveness of DFFW-CRBMs. The empirical evaluation showed that our methods are capable of outperforming state-of-the-art machine learning algorithms in both classification and regression. Precisely, DFFW-CRBM were capable of achieving substantially lower energy levels (approximately three times less energy on the overall datasets, independently on the number of factors or hidden neurons) than FFW-CRBM. This leads to at least double accuracies for real-valued predictions, while acquiring similar classification performance, at no increased computational complexity costs.
[^1]: http://bulletphysics.org, Last accessed on November $8^{th}$ 2016
[^2]: A labeled trajectory has complete information: the 3D ball coordinates, their 2D projections, and the spin (i.e. class).
[^3]: It is worth noting that in this scenario the random guess for classification would have an accuracy of $25\%$.
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---
author:
- 'P. Tozzi'
- 'R. Gilli'
- the CDFS Team
title: 'AGN and Galaxy evolution from Deep X-ray surveys'
---
Introduction
============
In the last years deep X–ray surveys with the [*Chandra*]{} and [*XMM–Newton*]{} satellites (Brandt et al. 2001; Rosati et al. 2002; Alexander et al. 2003; Hasinger et al. 2001), paralleled by multiwavelength campaigns (see, e.g., GOODS, Giavalisco et al. 2004), provided several crucial information on the evolution of the AGN and galaxy populations. The bold result from the two deepest X–ray fields, the Chandra Deep Field North (CDFN, observed for 2 Ms) and the Chandra Deep Field South (CDFS, observed for about 1Ms) is constituted by the resolution of the X–ray background (XRB) into single sources, mostly AGN, at a level between 80% and 90% (see Bauer et al. 2004 and the recently revised estimate by Hickox & Markevitch 2005), providing an almost complete census of the accretion history of matter onto supermassive black holes through the cosmic epochs. However, the most interesting outcomes go well beyond the demographic characterization of the extragalactic X–ray sky. Indeed, the physical and evolutionary properties of the AGN population are now revealing how they formed and how they are linked to their host galaxies. For the first time, the luminosity function of AGN has been measured up to high redshift. A striking feature is the [*downsizing*]{}, or [*anti–hierarchical*]{} behaviour, of the nuclear activity: the space density of the brightest Seyfert I and QSO is peaking at $z\geq 2$, while the less luminous Seyfert II and I peak at $z\leq 1$ (Ueda et al. 2003; Hasinger et al. 2005; La Franca et al. 2005). An analogous behaviour is presently observed in the cosmic star formation history: at low redshift star formation is mostly observed in small objects (see, e.g., Kauffmann et al. 2004), while at redshift 2 or higher, star formation activity is observed also in massive galaxies (with $M_* \sim 10^{11} M_\odot$, see, Daddi et al. 2004a).
The global picture, as outlined by the present data, requires a tight link between the formation of the massive spheroids and the central black holes, as witnessed by the relation between black hole and stellar masses or between black hole mass and the velocity dispersion of the bulge (Kormendy & Richstone 1995; Magorrian et al. 1998; Ferrarese & Merritt 2000). The anti–hierarchical behaviour in both star formation and AGN activity (which reflects in an anti–hierarchical supermassive black holes growth, see Merloni 2004; Marconi et al. 2004; for an alternative view see Hopkins et al. 2005), is envisaged by theoretical models where energy feedback is invoked to self–regulate both processes (see Fabian 1999; Granato et al. 2004).
In these Proceedings, we will describe a few observational results obtained from the latest analysis of the X–ray and optical data in the Chandra Deep Field South and North and which, in our view, are consistent with this picture. In detail, these results concern the following issues:
- the physical properties of AGN from the X–ray spectral analysis of faint X–ray sources;
- the missing fraction of the XRB;
- the distribution of obscured QSO and Compton–thick sources and their relation with the cosmic mass accretion history;
- star formation in high–z galaxies measured in the X–ray band thanks to stacking techniques;
- the effects of large scale structure onto nuclear activity.
Properties of faint AGN from X–ray spectral analysis
====================================================
The resolution of the XRB, the long–awaited result since its discovery in 1962, has been obtained simply by counting the point sources found in the deep X–ray images taken with the [*Chandra*]{} and [*XMM–Newton*]{} satellites. This result is clearly shown by the sharp images of the CDFS and the CDFN in Figure \[fig1\] and \[fig2\]. These images also show visually the solution of the so–called [*spectral paradox*]{}: fainter sources are more absorbed (and appear bluer in X–ray colors) than brighter ones, so that the total spectrum of the XRB, resulting from the summed contribution of the whole AGN population, has a slope $\Gamma \simeq 1.4$, flatter than that typical of the intrinsic nuclear emission $\Gamma = 1.8$ (as observed in unabsorbed AGN). The resolved fraction of the XRB has been recently revised slightly downwards to be about 80% in both bands (Hickox & Markevitch 2005). AGN makes the 83% and the 95% of the resolved fractions in the soft and in the hard band respectively. On the other hand, star forming galaxies contributes only 3% and 2% (Bauer et al. 2004).
However, quoting the resolved fraction of the XRB in the 2–8 keV band is somewhat misleading. Indeed, it has been pointed out that the resolved fraction is significantly decreasing with increasing energy (Worsley et al. 2005). In particular, above 5 keV, the resolved fraction can be as low as 50%. This finding opens again the issue of the resolution of the XRB, requiring the presence of a still undetected population of strongly absorbed AGN at moderate redshift, as can be inferred from the spectral shape of the missing XRB (see Worsley et al. 2005).
The issue of the missing XRB opens several questions on the physical properties of the X–ray sources and calls for a detailed X–ray spectral analysis of the faint AGN population. In a recent Paper (Tozzi et al. 2006) we went through a detailed X–ray spectral analysis of the large majority of the X–ray sources found in the CDFS (321 in the 1Ms catalog after excluding stars and low luminosity sources with $L_X < 10^{41}$ erg s$^{-1}$, see Giacconi et al. 2002). In particular, the knowledge of the spectroscopic or photometric redshift for almost all the X–ray sources (Szokoly et al. 2004; Zheng et al. 2004), allowed us to measure the value of the intrinsic absorption in terms of equivalent hydrogen column density $N_H$.
Summarizing the main results of our analysis, we found that the intrinsic spectral slope $\Gamma$ is always close to the average value $\Gamma=1.8$, without showing any significant dependence on redshift, intrisic luminosity, or intrinsic absorption. We find significant evidence of the 6.4 keV Fe line in 12% of the sources with spectroscopic redshift. We detect the presence of a soft component (possibly due to partial covering or to scattered emission) in only 8 sources. We measured the intrinsic column density $N_H$ for each source, after freezing the spectral slope $\Gamma=1.8$ for the faintest ones. The intrinsic $N_H$ distribution has been obtained after correcting for the detection probability as a function of the flux (the sky–coverage) and for sources below the limiting flux of the survey. We find that most of the AGN have high intrinsic absorption with $N_H > 10^{22}$ cm$^{-2}$ (see Figure \[nh\]). A fraction of the AGN (more than 10%) are Compton–thick sources, defined as sources with $N_H \geq 1.5 \times 10^{24}$ cm$^{-2}$. Only few of these elusive sources are actually detected (we identify only 14 Compton–thick candidates in the CDFS), buth their actual number density is expected to be high. Indeed, due to their spectral shape, well represented by a very hard reflection spectrum, only a small part of their population can be detected by [*Chandra*]{}, which is mostly sensitive in the soft band. We also find that about 80% of the sources (among the 139 AGN for which good optical spectra are available), follow a one–to–one correspondence between optical Type I (Type II) and X–ray unabsorbed (X–ray absorbed, defined as sources with $N_H \geq 10^{22}$ cm$^{-2}$) sources, as predicted by the original version of the unification model for AGN (Antonucci 1993).
The missing fraction of the XRB
===============================
The detailed knowledge of the spectral shape allows us to better evaluate the detection probability of each source class. This is an important aspect, since the probability of being included in a survey depends significantly on the spectral shape. Therefore, by a careful spectral analysis, we are able to estimate with high accuracy the contribution of the most elusive absorbed sources to the XRB. Indeed, the detection probability is lower for more absorbed sources, since the maximum detectability is achieved in the soft band; therefore, the contribution to the XRB of strongly absorbed sources found in the CDFS is higher than that of sources with the same energy flux but with no or little absorption. This aspect was not fully appreciated in previous works, where the detection probability was only a function of the flux and not of the spectral shape, with a consequent underestimate of the actual number density of the most absorbed sources.
We recompute the resolved XRB in the CDFS, and compare it to the total extragalactic XRB spectrum modeled as a power law with $\Gamma = 1.41$ and 1 keV normalization of 11.6 keV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ keV$^{-1}$ (De Luca & Molendi 2004). The contribution to the XRB from CDFS sources is obtained directly by summing the contributed flux from each source weighted by the inverse of the detection probability. The contributed flux in a given energy band is obtained by fitting the X–ray spectrum of each source with a power law plus an intrinsic absorption. We find that the decrease of the resolved fraction as a function of the energy range is not as pronounced as in Worsley et al. (2005), as shown in Figure \[xrb\] (Tozzi et al., in preparation). This implies that we are actually seeing a fraction of the population responsible for the missing XRB. Due to the strong absorption, most of these sources can not be detected even in the deepest [*Chandra*]{} or [*XMM–Newton*]{} surveys, and their discovery must wait for an higher energy, high–sensitivity X–ray mission, or make use of radio or submillimetric data (see Martìnez–Sansigre et al. 2005).
AGN and Galaxy formation: TypeII QSO and Compton–thick sources
==============================================================
Another interesting piece of information comes from the redshift distribution of the Compton–thick candidates in the CDFS. As shown in Figure \[cthick\], these sources are distributed in a wide range of redshift, a significant part of them around $z\sim 1$. Their redshift and their level of absorption match well with the values expected for the sources responsible of the missing XRB (see Worsley et al. 2005), reinforcing the reliability of our candidate sources.
Another interesting class of sources are the so called Type II QSO. Since we do not have an optical spectral classification for all the sources, we consider here absorbed QSO, simply defined on the basis of the X–ray properties as bright sources ($L>10^{44}$ erg s$^{-1}$) with $N_H>10^{22}$ cm$^{-2}$. Some of them have been shown to correspond to Type II QSO on the basis of optical and submm data (Norman et al. 2002, Mainieri et al. 2005a). Most of them are among the optically faint sources of the sample (Mainieri et al. 2005b). We select 54 sources with these properties distributed on a wide range of redshifts (see Figure \[qso2\]), corresponding to 80% of the sources with $L>10^{44}$ erg s$^{-1}$ in our sample. We remark that we explore here a limited luminosity range ($L < 10^{45}$ erg s$^{-1}$), given the small volume sampled. This confirms anyway that Type II QSO constitute a significant fraction of the AGN population.
These findings can be interpreted in the framework of the anti–hierarchical scenario, as described in the Introduction. Absorbed QSO may be sources associated to massive spheroids experiencing at the same time rapid growth of the central black hole and strong star formation activity. In this case, the absorption is not ascribed to circumnuclear matter, as in the simplest version of the unification model, but to the gas distributed on a much wider region strongly polluted by star formation processes. Indeed, the redshift distribution of the QSO population, and therefore of the absorbed ones which represent a significant fraction of it, peaks at a redshift $\geq 2$, an epoch when the number density of massive and powerful starburst galaxies is expected to be interestingly high, as found recently with near–IR observations (Daddi et al. 2004, see §5).
The strong energetic feedback from both nuclear activity and star formation in such large objects, is expected to inhibit further accretion and star formation events. On the other hand, smaller objects, where the feedback is less efficient, are able to retain gas that can be accreted subsequently, allowing for episodes of obscured accretion and star formation at lower redshifts.
Star formation seen in X-ray at high–z
======================================
X–ray emission witnessing star formation events is due to X–ray binaries and hot gas associated with superwinds and SNa remnants. The main advantage of X–ray studies of the cosmic star formation rate is to avoid problems of obscuration, which severely affects optical observations. The price to pay is that, since star forming galaxies have luminosities in the range $10^{39}$–$10^{42}$ erg s$^{-1}$, it is difficult to detect them at high redshifts even in the deepest X–ray surveys. The first normal star–forming galaxy X–ray luminosity function has been derived by Norman et al. (2004) in the combined Chandra Deep Field North and South. The results show an increasing cosmic star formation rate proportional to $(1+z)^{2.7}$, consistent with other star formation determination in different wavebands. However, the galaxy XLF is determined only at $z \leq 1$, still below the expected maximum of the cosmic star formation history.
On the other hand, other selection techniques, like the one using the B–z vs z–K color diagram (see Daddi et al. 2004b), are able to identify actively star forming galaxies at $z\geq 1.4$, with typical stellar masses larger than $10^{11} M_\odot$ and average star formation rate of $\simeq 200 M_\odot$ yr$^{-1}$. The estimated number density of these $z\sim 2$ star forming galaxies suggests that we are peering into the formation epoch of massive early–type galaxies (Daddi et al. 2004a).
The expected X–ray emission from each one of these high–z, starburst galaxies is below the flux limits of the deepest X–ray surveys. However, we can add together the images of the X–ray fields in the positions of the galaxies, to obtain a stacked image of all the optically selected star forming galaxies. We created the stacked images of 22 [*BzK*]{} selected galaxies in the CDFS in the soft and the hard bands. This can be considered an image of about 20 effective Ms of a typical $z\sim 2$ massive, star forming galaxy. The images are shown in Figure \[z2sb\]. In the soft band we detect a total $96 \pm 23$ net counts, which, for an average spectral slope of $\Gamma = 2.1$, corresponds to an average 2–10 keV rest frame luminosity of $9 \times 10^{41}$ erg s$^{-1}$, impliying a star formation rate of about $\sim 190 M_\odot$ yr$^{-1}$ (see Ranalli et al. 2003), in good agreement with the estimate from the reddening–corrected UV luminosities. On the other hand, we have no detection in the observed–frame hard band, confirming the steep spectral slope (hardness ratio $HR < -0.5$ at the 2$\sigma$ level) and therefore the non–AGN nature of these sources (Daddi et al. 2004b).
This findings confirm that stacking techniques on sources selected in other wavebands, are extremely useful in exploring the level of X–ray emission from star formation at high–z, an aspect which constitutes one of the most compelling scientific cases for the next generation X–ray facilities.
Large Scale Structure in deep X–ray surveys
===========================================
With the spectroscopic follow–up of X–ray sources detected in the CDFS, significant large scale structure has been discovered both in CDFS and CDFN, as shown in Figure \[gilli1\]. In particular, two prominent spikes have been found in the CDFS at $z=0.67$ and $z=0.73$ with 19 sources each (Gilli et al. 2003), corresponding to spikes found in the distribution of galaxies in the K20 survey (an ESO–VLT optical and near–infrared survey down to $K \leq 20$ covering part of the CDFS, see Cimatti et al. 2002). Comparing the X–ray and optical catalogs, we found that in the structure at $z=0.73$, the fraction of active galaxies is the same as in the field, while in the one at $z=0.67$ it is higher by a factor of 2. We also note that the structure at $z=0.73$ includes a cluster of galaxies, which may have biased downwards the estimate of the AGN fraction. This finding constitutes one of the first tantalizing hints (significant only at the 2$\sigma$ level) that large scale structure can trigger nuclear activity.
Investigation of the spatial clustering of X–ray sources in both fields (Gilli et al. 2005) points out a significant difference in the correlation lengths. We find $r_0 = 8.6 \pm 1.2 \, h^{-1}$ Mpc in the CDFS, and $r_0 = 4.2 \pm 0.4 \, h^{-1}$ Mpc in the CDFN, with similarly flat slope ($\gamma = 1.33 \pm 0.11$ and $1.42 \pm 0.07$ respectively), as shown in Figure \[gilli2\]. If we consider only AGN, we obtain higher correlation lenghts, in the range $5-10 \,
h^{-1}$ Mpc. Since at $z\sim 1$ late–type galaxies have a correlation lenght of $\sim 3.2 \, h^{-1}$ Mpc while early–type galaxies have $\sim ~6.6 \, h^{-1}$ Mpc (Coil et al. 2004), the high correlation lengths measured for AGN in the CDFS are consistent with the idea that at $z\sim 1$ AGN with Seyfert–like luminosities are hosted by massive galaxies. The difference in the correlation lengths measured between the two fields disappears when the two most prominent spikes in the CDFS are removed (see Figure \[gilli3\]). This shows that larger fields of view are needed to kill the cosmic variance and perform a proper investigation of the correlation properties of X–ray detected AGN. We mention two main projects, the COSMOS survey with the [*XMM–Newton*]{} satellite (Hasinger et al. in preparation), and the Extended CDFS with [*Chandra*]{}. The Extended CDFS complements the original 1Ms exposure with four [*Chandra*]{} ACIS-I fields with 250 ks each, bringing the linear size of the field from the former 16 arcmin to 32 arcmin. First results have been published by Lehmer et al. (2005), while spectroscopic follow up is currently under way. The main goal is to better understand the effect of large scale structure onto nuclear activity, and the evolution of the clustering of X–ray sources.
Conclusions
===========
We presented few selected topics which we find particularly relevant among the latest results from deep X–ray surveys. We can summarize our conclusions as follows:
- a population of strongly absorbed, possibly Compton–thick AGN at $z\sim 1$ is still missing to the census of the X–ray sky; the detailed X–ray spectral analysis of faint sources shows that we are detecting some of them, and help us in obtaining a complete reconstruction of the cosmic accretion history onto supermassive black holes;
- we find several absorbed sources (the so–called TypeII QSO) among the population of bright AGN, possibily witnessing the rapid growth of the super massive black holes associated to strong star formation events;
- thanks to stacking techniques, we detected the X–ray emission associated to massive star forming galaxies at redshift as high as $z\sim 2$, therefore peering with X–rays in the epoch of massive galaxy formation;
- investigation of large scale structure in the X–ray detected AGN distribution provides tantalizing hints of its effect on nuclear activity. Studies of spatial correlation of X–ray sources require larger fields of view to kill the cosmic variance and therefore evaluate properly the evolution of the AGN clustering properties.
Such observational findings are providing crucial information on the evolution of AGN and galaxies. Present–day data are consistent with a scenario where nuclear activity and star formation processes develop together in an anti–hierarchical fashion.
P.T. thanks the Organizers for providing a pleasant and stimulating scientific environment during the workshop.
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---
abstract: 'Few-shot image classification aims to classify unseen classes with limited labeled samples. Recent works benefit from the meta-learning process with episodic tasks and can fast adapt to class from training to testing. Due to the limited number of samples for each task, the initial embedding network for meta learning becomes an essential component and can largely affects the performance in practice. To this end, many pre-trained methods have been proposed, and most of them are trained in supervised way with limited transfer ability for unseen classes. In this paper, we proposed to train a more generalized embedding network with self-supervised learning (SSL) which can provide slow and robust representation for downstream tasks by learning from the data itself. We evaluate our work by extensive comparisons with previous baseline methods on two few-shot classification datasets ([*i.e.,*]{} MiniImageNet and CUB). Based on the evaluation results, the proposed method achieves significantly better performance, i.e., improve 1-shot and 5-shot tasks by nearly **3%** and **4%** on MiniImageNet, by nearly **9%** and **3%** on CUB. Moreover, the proposed method can gain the improvement of (**15%**, **13%**) on MiniImageNet and (**15%**, **8%**) on CUB by pretraining using more unlabeled data. Our code will be available at'
address: |
Alibaba Group, China\
{chen.cd, yuefeng.chenyf,daniel.lyh,maofeng.mf,heyuan.hy,[email protected]}
bibliography:
- 'ref.bib'
title: 'Self-supervised learning for few-shot image classification'
---
Self-supervised learning, Few-shot learning, Embedding network, Metric learning, Image classification
Introduction {#sec:intro}
============
Recent advances of deep learning techniques have made significant progresses in many areas. The main reason for such success is the ability to train a deep model that can retain profound knowledge from large scale labeled dataset [@deng2009imagenet; @ICASSP2017_Audioset]. This is somehow against the human learning behavior - one can easily classify objects from just a few examples with limited prior knowledge. How to computationally model such behavior motivates the recent researches in few-shot learning, where the focus is how to adapt the model to new data or tasks with restricted number of instances.
One popular solution for few-shot classification is to apply a meta-learning process, in which the dataset is divided into subsets for different meta tasks, in order to learn how to adapt the model according to the task change. The main challenge here is that the meta-learning could easily lead to over fitting, as only a few samples for each class are available. To solve this problem, an attention mechanism that achieves a good classification for the unlabeled samples by learning an embedding of the labeled ones has been proposed [@nips2016_vinyals2016matching_net]. By sub-sampling classes and the associated data therein to simulate few-shot tasks [@ravi2016optimization; @snell2017prototypical_net; @sung2018relation_net], the so called episodes can further benefit meta-learning by constructing a probability model to predict a decision boundary between classes. Recent meta-learning methods [@rusu2018LEO_arxiv; @qiao2018few_shot_activations] focus on retrieving transferable embedding from the dataset along with the relation between images and their class descriptions. This is done by decomposing the training into two stages, i.e., i). learning a robust and transferable embedding, and ii). fine-tuning the learned embedding for downstream classification task. All these works demonstrate that a robust pre-trained embedding network is essential to the performance of the few-shot image classification task. Current methods [@jiang2018learning_caml; @oreshkin2018tadam; @qiao2018few_shot_activations; @rusu2018LEO] with good performance mostly apply a ResNet12 [@he2016residual] or a wide ResNet [@zagoruyko2016wide_residul_WRN] as the embedding network and surpass the methods [@bauer2017discriminative_k_shot; @chen2019aug_few_shot] with deeper network. We argue that the abandon of large network are mainly because that all these methods are trained in a supervised way with limited labeled samples. In this paper, we propose to apply a much more larger embedding network with self-supervised learning (SSL) to incorporate with episodic task based meta-learning. According to the evaluation presented in Section \[sec:result\], the proposed method can significantly improve few-shot image classification performance over baseline methods in two common datasets. As a remark, under the same experiemnt setting, the proposed method improves 1-shot and 5-shot tasks by nearly **3%** and **4%** on MiniImageNet, by nearly **9%** and **3%** on CUB. Moreover, the proposed method can gain the improvement of **15%**, **13%** and **15%**, **8%** in two tasks on MiniImageNet and CUB dataset by pretraining using more unlabeled data. We also observe that self-supervised learning pre-train model can be robustly transferred to other dataset.
{width="0.95\linewidth"}
Related Work {#sec:rel}
============
**Few-shot learning** as an active research topic has been extensively studied. In this paper, we will primarily review recent deep-learning based approaches that are more relevant to our work. A number of works aim to improve the robustness of the training process. Garcia *et al.* [@garcia2017few_graph_gnn] propose a graph neural network according to the generic message passing inference method. Zhao *et al.* [@zhao2018msplit_few_shot] split the features to three orthogonal parts to improve the classification performance for few-short learning, allowing simultaneous feature selection and dense estimation. Chen *et al.* [@chen2019aug_few_shot] propose a Self-Jig algorithm to augment the input data in few-shot learning by synthesizing new images that are either labeled or unlabeled.
A popular strategy for few-shot learning is through meta-learning (also called learning-to-learn) with multi-auxiliary tasks [@nips2016_vinyals2016matching_net; @finn2017maml; @sung2018relation_net; @qiao2018few_shot_activations; @jiang2018learning_caml]. The key is how to robustly accelerate the learning progress of the network without suffering from over-fitting with limited training data. Finn *et al.* propose MAML [@finn2017maml] to search the best initial weights through gradient decent for network training, making the fine-tuning easier. REPTILE [@nichol2018reptile] simplifies the complex computation of MAML by incorporating an $L_2$ loss, but still performs in high dimension space. To reduce the complexity, Rusu *et al.* propose a network called LEO [@rusu2018LEO] to learn a low dimension latent embedding of the model. CAML [@jiang2018learning_caml] extends MAML by partitioning the model parameters into context parameters and shared parameters, enabling a bigger network without over-fitting. Another stream of meta-learning based approaches [@snell2017prototypical_net; @oreshkin2018tadam; @nips2016_vinyals2016matching_net; @sung2018relation_net] attempt to learn a deep embedding model that can effectively project the input samples to a specific feature space. Then the samples can be classified by the nearest neighbour (NN) criterion using a distance function such as Cosine distance or Euclidean distance, [*etc.*]{} Koch *et al.* [@koch2015siamese_few_shot] propose the Siamese network to extract embedding features from input images and converge images from the same class. Matching Network [@nips2016_vinyals2016matching_net] utilizes an augment neural network for feature embedding, forming the basis for metric learning.
**Self-supervised learning(SSL)** aims to learn robust representations from the data itself without class labels. The main challenge here is how to design the pretext tasks that are complex enough to exploit high-level compact semantic visual representations that are useful for solving downstream tasks. This is consistent with the mission of the pre-trained embedding network in few shot learning. The work of [@KolesnikovZB19] revisited some state-of-the art methods based on various classification based pretext tasks([*e.g.,*]{} Rotation, Exemplar, RelPatchLoc, Jigsaw). Recently, by maximizing mutual information between features extracted from multiple views of a shared context [@oord2018representation; @henaff2019data; @tian2019contrastive; @AMDIM], SSL archive comparable performance to supervised learning. Among them, Contrastive Predictive Coding [@oord2018representation; @henaff2019data] learns from two views (the past and future) and is applicable to sequential data. Contrastive Multiview Coding [@tian2019contrastive] extends the framework to learn representations from multiple views of a dataset. AMDIM [@AMDIM] learns features extracted from multiples views which produced by repeatedly applying data augmentation on the input image, achieve 68.1% accuracy on ImageNet.
Method {#sec:method}
======
Few-shot learning is a challenging problem as it has only limited data for training and need to verify the performance on the data for unseen classes. One popular solution for few-shot learning classification problem is to apply a meta learning on top of a pre-trained embedding network. Most of current methods are mainly focusing on the second stage [*i.e., meta learning stage*]{}. In this work, we follow this two stages paradigm but utilize self-supervised learning to train a large embedding network as our strong base.
Self-supervised learning stage {#subsec:self_supervised_learning}
------------------------------
Our goal is to learn representations that enhance the feature’s generalization. In our approach, we use Augmented Multiscale Deep InfoMax (AMDIM) [@AMDIM] as our self-supervised model. The pretext task is designed to maximize the mutual information between features extracted from multiple views of a shared context.
The mutual information (MI) measures the shared information between two random variables $X$ and $Y$ which is defined as the Kullback–Leibler (KL) divergence between the joint and the product of the marginals. $$\begin{split}
I( X,Y ) = D_{KL}\left( p(x,y) || p(x)p(y)\right) \\
= \sum \sum p(x,y) \log \frac{p(x|y)} {p(x)}
\end{split}$$ where $P(x,y)$ is the joint distribution and $P(x)$ and $P(y)$ are the marginal distributions of $X$ and $Y$. Estimating MI is challenging as we just have samples but not direct access to the underlying distribution. [@oord2018representation] proved that we can maximizes a lower bound on mutual information by minimizing the Noise Contrastive Estimation (NCE) loss based on negative sampling.
The core concept of AMDIM is to maximize mutual information between global features and local features from two views $(x_a, x_b)$ of the same image. Specifically, maximize mutual information between $\left<f_g(x_a), f_5(x_b)\right>$, $\left<f_g(x_a), f_7(x_b)\right>$ and $\left<f_5(x_a), f_5(x_b)\right>$. Where $f_g$ is the global feature, $f_{5}$ is encoder’s $5 \times 5$ local feature map as well as $f_7$ as the encoder’s $7 \times 7$ feature map. For example, the NCE loss between $f_g(x_a)$ and $f_5 ( x_b )$ is defined as below: $$\begin{split}
& \mathcal{L}_{amdim}\left(f_g(x_a) , f_5 ( x_b )\right) = \\
& -\log \frac{ \exp\{\phi(f_g(x_a), {{f_5}( x_b )}) \} }{\sum_{\widetilde{x_b} \in {\mathcal{N}_{x} \cup x_b}} \exp\{ \phi(f_g(x_a), {f_5}( \widetilde{x_b} )) \} }
\end{split}$$ $\mathcal{N}_x$ are the negative samples of image $x$, $\phi$ is the distance metric function. At last, the overall loss between $x_a$ and $x_b$ is as follows: $$\begin{split}
\mathcal{L}_{amdim}( x_a, x_b) = \mathcal{L}_{amdim}\left(f_g(x_a) , f_5 ( x_b )\right) \ + \\ \mathcal{L}_{amdim}\left(f_g(x_a) , f_7 ( x_b )\right) + \mathcal{L}_{amdim}\left(f_5(x_a) , f_5 ( x_b )\right)
\end{split}$$ The stage as shown in the Figure \[fig:archi\] gives a overview of the AMDIM self supervised learning method. The red and blue lines shows the local and global feature between two view $x_a$ and $x_b$. The detail of encoder network is defined in Table \[tab:Mode\_Architecture\].
Meta-learning stage {#subsec:meta_learing_general}
-------------------
Given an embedding network, meta-learning is applied to fine-tune it to fit the class changes requirement of few-shot classification. A typical meta learning can be considered as a $K$-way $C$-shot episodic classification problem with multi-tasks [@nips2016_vinyals2016matching_net]. For each classification task $T$, we have $K$ classes with $C$ samples from each class. The entire training dataset can be presented by $D=\{(x_1,y_1),\dots,(x_N,y_N)\}$ where $N$ is the total number of classes in $D$. For a specific task $T$, $V = \{y_i|i=1,\dots,K\}$ denotes the class labels associated therein. Here $K$ is the number of classes in support set for a single training task. The support set and query set can often be randomly selected from $D$: (**a**) the support set for task $T$ is denoted by $S=\{(x_i,y_i)|i=1,\dots,m\}$, where $m=C\times K$ ($K$-way $C$-shot); (**b**) the query set is $Q=\{(x_j,y_j)|j=1,\dots,n\}$ where $n$ is the number of samples selected for meta testing.
As mentioned in Section \[sec:rel\], the recent popular frameworks such as Snell *et al.* [@snell2017prototypical_net], are able to learn an embedding function to map all input samples to a mean vector $c$ in a description space to represent each class. For class $k$, it is represented by the centroid of embedding features of training samples and can be obtained as:
$$\label{equ:pnet_ck}
c_k = \frac{1}{\left | S \right |}\sum_{(x_i,y_i)\in S} f_g (x_i),$$
where $f_\phi(x_i)$ is the embedding function.
As a metric learning based method, we employ a distance function $d$ and produce a distribution over all classes given a query sample $q$ from query set $Q$:
$$\label{equ:loss1}
\begin{split}
p(y = k|q) = \frac{\exp(-d(f_{g}(q),c_k))}{\sum_{k'}\exp(-d(f_{g}(q),c_{k'}))}
\end{split}$$
In this paper, Euclidean distance is chosen as distance function $d$. As shown in Eq. \[equ:loss1\], the distribution is based on a softmax over distance between the embedding of the samples (in the query set) and the reconstructed features of the class. The loss in meta learning stage can then read:
$$\label{equ:loss2}
\mathcal{L}_{meta} = d(f_{g}(q),c_k) + \log\sum_{k'}d(f_{g}(q),c_{k'})$$
Experimental Results {#sec:result}
====================
In this section, we first introduce the dataset and training process used in our evaluation, then show quantitative comparisons against other baseline methods, finally we conduct a detailed study to validate the transfer ability of our approach.
Datasets {#subsec:dataset}
--------
MiniImageNet dataset, as proposed in [@nips2016_vinyals2016matching_net], is a benchmark to evaluate the performance of few-shot learning methods. This dataset is a subset randomly selected from ImageNet. MiniImageNet contains 60,000 images from only 100 classes, and each class has 600 images. We follow the data split strategy in [@ravi2016optimization] to sample images of 64 classes for training, 16 classes for validation, 20 classes for test.
Caltech-UCSD Birds-200-2011(CUB-200-2011) dataset, proposed in [@WahCUB_200_2011], is a dataset for fine-grained classification. The CUB-200-2011 dataset contains 200 classes of birds with 11788 images in total. For evaluation, we follow the split in [@hilliard2018few_conditional_embedding_maco]. 200 species of birds are randomly split to 100 classes for training, 50 classes for validation, and 50 classes for test.
Training Details {#subsec:trainingg_details}
----------------
Several recent works show that a typical training process can include a pre-trained network [@qiao2018few_shot_activations; @rusu2018LEO] or employ co-training [@oreshkin2018tadam] for feature embedding. This can significantly improve the classification accuracy. In this paper, we adopt the AMDIM [@AMDIM] SSL training framework to pre-train the feature embedding network. AmdimNet(ndf=192, ndepth=8, nrkhs=1536) is used for all datasets and the embedding dimension is 1536. Adam is chosen as the optimizer with a learning rate of $0.0002$. We use $128 \times 128$ as the input resolution among these datasets. For MiniImageNet dataset, 3 embedding models are trained. **Mini80-SSL** is self-supervised trained from 48,000 images (80 classes training and validation ) without labels. **Mini80-SL** is supervised training using same AmdimNet by cross entropy loss with labels. **Image900-SSL** is SSL trained from all images from ImageNet1K except MiniImageNet. For CUB dataset, **CUB150-SSL** is trained by SSL from 150 classes (training and validation). **CUB150-SL** is the supervised trained model. **Image1K-SSL** is SSL trained from all images from ImageNet1K without label.
Quantitative comparison {#subsec:Quantitative_comparison}
-----------------------
For MiniImageNet dataset, we evaluate our method in two common few-shot learning tasks [*i.e.,*]{} 1-shot 5-way task and 5-shot 5-way task against 18 baseline methods with different embedding networks including classical ones [@snell2017prototypical_net; @nips2016_vinyals2016matching_net; @finn2017maml] and recently proposed methods [@rusu2018LEO; @oreshkin2018tadam; @liu2018learning_TPN]. For CUB dataset, we follow the recent work [@chen2018a] to evaluate the robustness of the proposed framework with 7 other alternatives on this fine-grained dataset.
As detailed in Table \[tab:Mini\_result\], the proposed method outperforms all baselines in the tested tasks. In 1-shot 5-way test, our approach achieves $7.53\%$ and $2.27\%$ improvement over ProtoNet$^{+}$ [@snell2017prototypical_net] and LEO [@rusu2018LEO] respectively. The former is an amended variant of ProtoNet using pre-trained Resnet as embedding network and has same meta-learning stage with the proposed method, the later is the state-of-the-art method. In the experience for 5-Shot 5-Way, we observe a similar improvement in accuracy. Furthermore, we observe that the performance of our proposed method significantly increases when receiving more images/classes as input for pretrain. It gives $81.15\%$ improvement on 5-shot 5-ways test against $64.03\%$ for 1-Shot 5-Way. Table \[tab:CUB\_result\] illustrates our experiment on CUB dataset. Our proposed method yields highest accuracy from all trials. In the 1-shot 5-way test, we have $71.85\%$ gaining a margin of $20.54\%$ increment to the classic ProtoNet [@snell2017prototypical_net]. The improvement is more significant for 5-shot 5-way test. Our proposed method results is $84.29\%$ which introduces $2.39\%$ improvement to DN4-Da [@li_cvpr2019_revisiting]. Comparing to Baseline++ [@chen2018a], our method shows a significant improvement, i.e., $11.32\%$ and $4.95\%$ in both tests.
Ablation Study {#subsec:self_com}
--------------
As shown in the quantitative evaluation, the proposed method can significantly improve the performance in few-shot classification task by self-supervised pretrain using a large network. One concern may be raised is that if the gain of improvements of proposed network is simply due to the increment of network’s capacity. To prove the effectiveness of the proposed method, we train the embedding network with labeled data (Mini80-SL and CUB150-SL as detailed in Section \[subsec:trainingg\_details\]). As shown in Table \[tab:Mini\_result\] and Table \[tab:CUB\_result\], it performs even worse than the methods with simple 4 Conv blocks embedding networks as such big network under supervised learning with limited data can cause overfitting problem and cannot adjust to new unseen classes during testing. However, with SSL based pre-training a more generalized embedding network can be obtained and improve the results significantly. One may also concern about the effectiveness of the meta learning fine-tuning in the second stage. To test this, the pre-train embedding network is directly applied on the task with nearest neighbourhood(NN) classification. As shown in the test results on both dataset, meta-learning can effectively fine-tune the embedding network and achieve remarkable improvement.
We also include more data without labels during SSL pre-training and observe an more significant improvement of the result. As shown in Table \[tab:Mini\_result\], the proposed method can gain the improvement of 15% and 13% in two test tasks. As detailed analyzed in [@chen2018a], current few-shot learning methods can not efficiently transfer the domain of learning, [*i.e.,*]{} the training domain can not have huge gap with the testing set. In this paper, a transferability test is also conducted by pre-training the embedding network on ImageNet and applied on CUB dataset. As shown in Table \[tab:CUB\_result\], the proposed method with ImageNet pre-trained embedding network can be efficiently transferred to CUB dataset and gain an improvement of 15%,8% in both test tasks.
Conclusion {#sec:conclusion}
==========
In this paper, we propose to utilizes self-supervised learning to efficiently train a robust embedding network for few-shot image classification. The resulted embedding network is more generalized and more transferable comparing to other baselines. After fine-tuning by meta-learning process, the performance of the proposed method can significantly outperform all baselines based on the quantitative results using two common few-shot classification datasets. The current framework can be extended in several ways in the future. For instance, one direction is to combine these two stage together and develop an end-to-end method for this task. Another direction is to investigate the effectiveness of the proposed method on another few-shot tasks such as few-shot detection, [*etc.*]{}
|
---
abstract: |
Next generation of embedded Information and Communication Technology (ICT) systems are interconnected collaborative intelligent systems able to perform autonomous tasks. Training and deployment of such systems on Edge devices however require a fine-grained integration of data and tools to achieve high accuracy and overcome functional and non-functional requirements.
In this work, we present a modular AI pipeline as an integrating framework to bring data, algorithms and deployment tools together. By these means, we are able to interconnect the different entities or stages of
-1.25cm {width="9cm"}
particular systems and provide an end-to-end development of AI products. We demonstrate the effectiveness of the AI pipeline by solving an Automatic Speech Recognition challenge and we show that all the steps leading to an end-to-end development for Key-word Spotting tasks: importing, partitioning and pre-processing of speech data, training of different neural network architectures and their deployment on heterogeneous embedded platforms.
author:
- Miguel de Prado
- Jing Su
- Rozenn Dahyot
- Rabia Saeed
- Lorenzo Keller
- Noelia Vallez
subtitle: '**End-to-end integration of data, algorithms and deployment tools**'
title: 'AI Pipeline - bringing AI to you'
---
printcopyright
|
---
abstract: 'In the paper the Curie temperatures of selectively diluted planar Ising ferromagnet on the triangular lattice are calculated vs. concentration of magnetic atoms. Various analytical approaches are compared with the exact numerical calculations for finite clusters, as well as with the exact analytical solutions for the triangular and honeycomb lattices.'
author:
- 'T. Balcerzak'
- 'K. Sza[ł]{}owski'
- 'M. Žukovič'
- 'M. Borovský'
- 'A. Bobák'
- 'M. Jaščur'
title: Study of planar Ising ferromagnet on the triangular lattice with selective dilution
---
\[sec:level1\]Introduction
==========================
The studies of low-dimensional magnets have presented a topical item for many years. The investigations include both 1D and 2D magnets, as well as bilayers, thin films and multilayers. Many different theoretical methods have been employed; in some cases \[1–6\] the exact solutions are available.
As far as approximate methods are concerned, one should mention the Green Function (GF) \[7–9\] and spectral density \[10\] methods, Renormalization Group (RG) approach \[11, 12\], including Mean-Field Renormalization Group (MFRG) \[13, 14\] and Effective-Field Renormalization Group (EFRG) \[15\], Spin-Wave (SW) techniques \[16–18\], High Temperature Series Expansion (HTSE) \[19\] and Monte Carlo (MC) simulations \[20, 21\]. Many other approaches like Coherent Anomaly Method (CAM) together with transfer matrix technique \[22\] and Effective Field Methods (EFM) with correlations \[23\] should also be mentioned. Recently, Cluster Variational Method in the Pair Approximation (PA) has been adopted for studies of the bilayer \[24\] and bi-multilayer \[25\] systems.
The aim of the present paper is to study the ferromagnetic Ising model with spin $S=1/2$ on the Planar Triangular (PT) lattice with selective dilution. By the selective dilution we mean the dilution of only one sublattice, whereas the PT magnet can be, in general, decomposed into three interpenetrating sublattices. Regarding antiferromagnetism and the problem of frustration, such selectively diluted model has been considered by Kaya and Berker \[26\]. To the best of our knowledge, as far as ferromagnetism is concerned, the model has not been studied yet.
The model is interesting from the theoretical point of view, for by changing the selective dilution parameter we are able to pass continuously from the ideal PT lattice (without dilution) to the honeycomb lattice, where one sublattice is completely empty. On the other hand, for those two cases the exact solutions for the Curie temperatures do exist \[2\]. In this paper we will concentrate on the Curie temperature calculations for arbitrary concentration of magnetic atoms in the selectively diluted sublattice. Thus, we consider an intermediate situation between those two limiting cases, which were examined exactly by Wannier \[2\].
In the next Section, the outline of the theoretical methods will be given, and the respective formulas for the phase transition temperatures will be presented. With the help of numerical calculations we are able to compare the results of several theoretical approaches, namely the Molecular Field Approximation (MFA), Effective Field Theory (EFT), Pair Approximation (PA) method, as well as the Exact Calculation for Finite Clusters (ECFC). The results will be presented in the plots and discussed.
Theoretical methods
===================
Let us consider the ferromagnetic Ising model with spin $S=1/2$ on the PT lattice with selective dilution. The diluted lattice is illustrated in Fig.1. Following Kaya and Berker \[26\] we keep decomposition of the PT lattice into three interpenetrating sublattices $a$, $b$ and $c$, of which, for the ferromagnetic case the sublattices $a$ and $b$ will be equivalent. The sublattice $c$ is distinguished in the system by the random dilution of the spins. If we denote by $p$ the concentration of spins on $c$-sublattice, the case of $p=0$ corresponds to the honeycomb lattice, whereas $p=1$ stands for the non-diluted triangular lattice.
The Hamiltonian of the system is in the form of:
$$\label{eq0}
{\cal H} =- J\sum \limits_{\left( i_a,j_b\right) } S_{i_a}^{z}S_{j_b}^{z}
- J\sum \limits_{\left( i_a,j_c\right) } S_{i_a}^{z}S_{j_c}^{z}\xi_{j_c}
- J\sum \limits_{\left( i_b,j_c\right) } S_{i_b}^{z}S_{j_c}^{z}\xi_{j_c}$$
where $J>0$ is the exchange interaction coupling, and $\left(
i_\alpha,j_\beta \right)$ means the summation extending over nearest neighbour pairs of spins from sublattice $\alpha$ and $\beta$. $\xi_{j_c}$ are quenched, uncorrelated random variables chosen to be equal 1 with probability $p$ when the site $j_c$ is ocupied by a magnetic atom and 0 with probability $1-p$ otherwise.
![The triangular lattice divided into three interpenetrating sublattices: $a$, $b$ and $c$. The sublattice $c$ is randomly diluted. The thick grey line surrounds a cluster containing $3\times 4$ hexagons, used for exact calculations.[]{data-label="fig1"}](fig1.eps)
Molecular Field Approximation (MFA)
-----------------------------------
In this simplest approach the sublattice magnetizations $m_a=\left<
S_{i_a}^{z}\right>$, $m_b=\left< S_{i_b}^{z}\right>$ and $m_c=\left<
S_{i_c}^{z}\right>$ are described by three coupled MFA equations. In the vicinity of the Curie temperature the equations can be linearized and presented as follows: [ $$\begin{aligned}
\label{eq1}
m_a& = &\frac{3}{4}\beta_{\rm C}J\left(m_b+m_c\,p\right)\nonumber \\
m_b &= &\frac{3}{4}\beta_{\rm C}J\left(m_a+m_c\,p\right)\nonumber\\
m_c &= &\frac{3}{4}\beta_{\rm C}J\left(m_a+m_b\right)\end{aligned}$$ ]{}
where $\beta_{\rm C}=1/k_{\rm B} T_{\rm C}$, and $T_{\rm C}$ denotes the Curie temperature.
It is convenient to introduce the variable $t_{\rm C}=k_{\rm B} T_{\rm
C}/J$ which is a dimensionless Curie temperature. Then, assuming symmetry condition for the ferromagnet $m_a=m_b\ne m_c$, and demanding that the determinant of eqs.(2) must be zero, we obtain the equation for the Curie temperature in MFA: $$\label{eq2}
8 t_{\rm C}^2 - 6 t_{\rm C}- 9p=0.$$ The physical solution is then of the form: $$\label{eq3}
t_{\rm C}=\frac{3}{8}\left(1+\sqrt{1+8p}\right)$$ and is straightforward for numerical calculation.
Effective Field Theory (EFT)
----------------------------
By the EFT we mean the method proposed by Honmura and Kaneyoshi \[27\], which takes into account autocorrelations but neglects the spin-pair correlations. Among its many applications, the method has recently been applied for the triangular lattice with uniform dilution \[28\]. It is based on the exact Callen-Suzuki identity of the form: $$\label{eq4}
\left<S_{i_\alpha}^z\right>=\frac{1}{2}\left<\tanh \frac{1}{2k_{\rm B}T}
\left(J\sum_{j_{\beta} \in i_{\alpha}}S_{j_\beta}^z\xi_{j_\beta}\right)\right>.$$
where $\xi_{j_\beta}=1$, if $\beta=a,b$ and $j_{\beta}\in
i_{\alpha}$ denotes a lattice site being nearest neighbour of the site $i_{\alpha}$ .
Applying the differential operator method \[27\], together with the decoupling procedure for the mean value of multi-spin products, the local magnetizations can be calculated. The coupled equations for the sublattice magnetizations have polynomial form and can be linearized near the continuous phase transition points. For the system in question, with the general sublattice magnetizations $m_a$, $m_b$ and $m_c$, the Curie temperature can be found from the determinant equation: $$\label{eq5}
\det {\bf U} = 0,$$
where
$$\label{eq6}
{\bf U} = \left(
\begin{array}{ccc}
-1 & 3 a^2 a_{c}^3 b \tanh \left( x \right) |_{x=0} & 3 a^3 a_{c}^2 b \tanh \left( x \right) |_{x=0} \\
3 a^2 a_{c}^3 b \tanh \left( x \right) |_{x=0} & -1 & 3 a^3 a_{c}^2 b \tanh \left( x \right) |_{x=0} \\
3 p a^5 b \tanh \left( x \right) |_{x=0} & 3 p a^5 b \tanh \left( x \right) |_{x=0} & -1 \\
\end{array}
\right).$$
In eq.(7) the temperature-dependent coefficients are given in the following form: $$\begin{aligned}
\label{eq.7} a &=& \cosh \left(
\frac{1}{4t_{\rm C}} D \right) \nonumber\\ a_{c} &=& 1 - p + p \cosh \left(
\frac{1}{4t_{\rm C}} D \right)\nonumber\\ b &=& \sinh \left( \frac{1}{4t_{\rm
C}} D \right), \end{aligned}$$
where $D=\partial/\partial x$ is the differential operator. The equation(\[eq5\]) can be solved numerically only.
The Pair Approximation (PA) method
----------------------------------
The PA is one of the cluster variational methods and takes into account the nearest-neighbour correlations. Being more accurate than MFA and EFT it enables calculation of the Gibbs energy, and hence all thermodynamic properties. Contrary to MFA, in the PA method the local variational parameters (molecular fields acting on a pair) are no longer simply proportional to local magnetizations. A set of linearized equations for these parameters near the Curie temperature takes a form: $$\begin{aligned}
\label{eq.8}
\left(3C_1-2\right)\lambda + \left(3C_1-3\right)p\lambda_2&=&0\nonumber\\
\left(3C_1-3\right)\lambda + \left(6C_1-5\right)\lambda_1 + \left(3C_1p-3p+1\right)\lambda_2&=&0\nonumber\\
\left(3C_2-3\right)\lambda - \left(6C_2-5\right)\lambda_1 + \left(3C_2p-3p+1\right)\lambda_2&=&0.\nonumber\\\end{aligned}$$
The variational parameters have the following meaning:\
$\lambda$ is the field acting on a spin on the sublattice $a$ or $b$ and originating from spins on the sublattices $b$ or $a$, respectively;\
$\lambda_1$ is the field acting on the spin on the sublattice $c$ and originating from the sublattices $a$ or $b$;\
$\lambda_2$ is the field acting on the spin on the sublattice $a$ or $b$ and originating from the sublattice $c$.
The temperature-dependent coefficients have the following form: $$\begin{aligned}
\label{eq.9}
C_1&=&\frac{1}{2}\left(1+\frac{1}{x_{\rm C}}\right)\nonumber\\
C_2&=&\frac{1}{2}\left(1+x_{\rm C}\right)\end{aligned}$$
where $$\label{eq.10}
x_{\rm C}=\exp \left(\frac{1}{2t_{\rm C}}\right).$$
By setting the determinant of eqs.(9) equal to zero the Curie temperature can be found. The final result can be presented in the form of the algebraic equation: $$\label{eq.11}
3\left(1+5p\right)x_{\rm C}^3 -\left(13+30p\right)x_{\rm C}^2
+15\left(1+p\right)x_{\rm C} -9=0$$
where $x_{\rm C}$ is related to the Curie temperature, $t_{\rm C}=k_{\rm B} T_{\rm C}/J$, by the formula (11).
Exact Calculation for Finite Clusters (ECFC)
--------------------------------------------
Exact numerical diagonalization for finite systems is nowadays a powerful tool for the studies of magnetic properties \[29\]. The accuracy of this method improves with the increase of the cluster size, but simultaneously rapidly growing number of states, which should be taken into account, results in a corresponding huge increase of the calculation time. This requires increasingly powerful computers. However, in the case of spin systems with Ising interactions, all the system states and their energies can be listed explicitly, without resorting to diagonalization of the Hamiltonian. Hence, we call this approach Exact Calculation for Finite Clusters. The method bears some resemblance to Monte Carlo calculations, however, it uses all the system states to study its thermodynamics within canonical ensemble. The method is sensitive to the shape of a cluster and selection of the boundary conditions. It is known from the literature devoted to Monte Carlo simulations that the selection of periodic boundary conditions is evaluated as superior to other choices for planar lattices \[30,31\]. In particular, it guarantees the same number of nearest-neighbours for the atoms on the boundary and inside the cluster. In our calculations presented here, we based on a cluster consisting of $3 \times 4$ hexagons, and the boundary conditions were chosen as periodic. Such a cluster is illustrated in Fig.\[fig1\], where it is surrounded by the gray thick line. The maximum number of spins (for $p=1$, when all sites on the sublattice $c$ were occupied) amounted to 36, while for $p=0$ it was equal to 24.
The Curie temperatures were determined from the maxima of the specific heat curves for various concentrations $p$. The condition for the maximum can be found from the exact thermodynamic formula: $$\label{eq.12}
\left<E^3\right>-3\left<E^2\right>\left<E\right>+2\left<E\right>^3=
2\left[\left<E^2\right>-\left<E\right>^2\right]k_{\rm B}T_{\rm max},$$
where the mean values of the energy powers $E^n$ are calculated numerically with the Boltzmann distribution (taken at the temperature $T_{\rm max}$) over all possible states in the cluster. For our purpose, in eq.(13) $T_{\rm max}$ for a finite cluster is assumed to estimate $T_{\rm C}$, which is a common approach used in Monte Carlo studies, e.g. \[32,33\]. According to scaling relations, $T_{\rm max}$ is expected to converge to $T_{\rm C}$ in the limit of an infinite system size \[33,34\]. In the case of ECFC, the system size is severely limited by the computational resources and thus extrapolation to infinite system size would be not trustworthy, due to small-size corrections to scaling. Therefore, we present directly the obtained values of $T_{\rm max}$ for the largest system studied, i.e. the 3$\times$4 cluster and for comparison we provide also the numbers for smaller 3$\times$3 cluster.
The numerical results are presented in the next Section.
Numerical results and discussion
================================
The Curie temperatures for selectively diluted PT ferromagnet have been calculated based on the approximations presented in the previous Section. The results of $k_{\rm B} T_{\rm C}/J$ vs. concentration $p$ are illustrated in Fig.2. In the same figure two exact Wannier results \[2\] are shown, i.e., for $p=0$ (honeycomb lattice) and for $p=1$ (triangular lattice). For the pure honeycomb lattice we obtained the Curie temperature values $k_{\rm B} T_{\rm C}/J$ equal to: 3/4 (MFA); 0.5259 (EFT); $1/\left(2\ln3\right)\approx$0.4551 (PA); 0.4203 (ECFC for 3$\times$3 cluster) and 0.4128 (ECFC for 3$\times$4 cluster). These results can be compared with the exact Wannier solution $k_{\rm B} T_{\rm
C}/J=1/\left(2\ln(2+\sqrt{3})\right)\approx$0.3797. On the other hand, for the pure triangular lattice the Curie temperatures calculated in various approaches are: $k_{\rm B} T_{\rm C}/J$=3/2 (MFA); 1.2683 (EFT); $1/\left(2\ln(3/2)\right)\approx$1.2332 (PA); 0.9602 (ECFC for 3$\times$3 cluster) and 0.9520 (ECFC for 3$\times$4 cluster). The exact Wannier result in this case is $k_{\rm B} T_{\rm
C}/J=1/\ln3\approx$0.9102.
![The Curie temperatures vs. concentration for the triangular lattice with selective dilution, obtained by various theoretical approaches: MFA, EFT, PA and ECFC for 3$\times$4 cluster. Two exact Wannier results \[2\] for $p=0$ and $p=1$ are also indicated.[]{data-label="fig2"}](fig2.eps)
![The specific heat per lattice site vs. temperature for various concentrations of $c$-atoms. The presented results are obtained by the ECFC method for 3$\times$4 cluster.[]{data-label="fig3"}](fig3.eps)
The Curie temperature of the pure triangular lattice is higher than that of the honeycomb one, since the coordination number of the former doubles that of the latter. It is seen in Fig.2 that in each method the Curie temperature changes continuously with $p$; however, the change is not linear, not even in the MFA.
Fig.2 illustrates the accuracy of the methods described in the previous Section. As far as the analytical methods are concerned, we see that MFA is the least accurate; giving the highest Curie temperature. The EFT and PA are much more accurate methods. Noticeably, the numerical calculations performed on the finite clusters with periodic boundary conditions seem to be the most accurate. As pointed out in the previous Section, in this approach the Curie temperatures have been identified from the maxima of the specific heat, according to eq.(13).
In Fig.3 the specific heat curves per lattice site for finite clusters are illustrated vs. temperature for various concentrations $p$. The temperatures corresponding to the maxima of these curves are denoted by the circular markers in Fig.2. Although the location of the specific heat peaks can be determined numerically from the curves, we found that the application of the analytical formula (13) leads to more precise results. It should be noted that the specific heat in Fig.3 behaves correctly from the thermodynamic point of view, both in the low and high temperature limits.
Conclusions
===========
In the paper four approximate methods have been applied in order to study the low dimensional PT Ising ferromagnet with selective dilution. Within those methods the formulas for the Curie temperatures have been obtained. The phase diagram has been calculated for various values of the concentration parameter $p$ and the results of the different methods have been compared.
The most accurate method is the one based on the exact numerical calculations for finite clusters with the periodic boundary conditions. It is worth noticing that within this method all thermodynamic properties can be simultaneously calculated in the same computational time. However, regarding the determination of the Curie temperature, this method is applicable to the systems in which the maximum of the specific heat is unambiguously related to the phase transition temperature. This is not always the case; for instance, for the frustrated systems or even some paramagnets where the so-called Schottky maximum is observed. Therefore, the analytical methods, giving better physical insight and proper interpretation of the numerical results, are still important in such studies.
Among analytical methods the PA approach can be especially recommended, for it enables the self-consistent studies of all thermodynamic properties based on the Gibbs potential. It also gives satisfactory accuracy when compared with other approaches. As shown recently, this method can be applied to the Heisenberg systems as well \[24, 25\]. It should also be noted that for quantum systems the numerical diagonalization of the finite cluster Hamiltonian is much less efficient than ECFC for the classical Ising model.
For the model in question the antiferromagnetic interactions can also be considered. Then, for selective dilution $p > 0$ the frustrations will occur and the theoretical description becomes more complex. This problem should be a subject for separate paper.
The numerical calculations have been performed on the computer cluster HUGO at the P.J.Šafárik University in Košice.
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|
---
abstract: 'We discuss the conditions for a non-vanishing Dirac phase $\d$ and mixing angle $\theta_{13}$, sources of $C\!P$ violation in neutrino oscillations, to be uniquely responsible for the observed matter-antimatter asymmetry of the universe through leptogenesis. We show that this scenario, that we call $\d$-leptogenesis, is viable when the degenerate limit (DL) for the heavy right-handed (RH) neutrino spectrum is considered. We derive an interesting joint condition on $\sin\theta_{13}$ and the absolute neutrino mass scale that can be tested in future neutrino oscillation experiments. In the limit of hierarchical heavy RH neutrino spectrum (HL), we strengthen the previous result that $\d$-leptogenesis is only very marginally allowed, even when the production from the two heavier RH neutrinos is taken into account. An improved experimental upper bound on $\sin\theta_{13}$ and (or) an account of quantum kinetic effects could completely rule out this option in the future. Therefore, $\d$-leptogenesis can be also regarded as a motivation for models with degenerate heavy neutrino spectrum.'
author:
- |
[Alexey Anisimov$^a$, Steve Blanchet$^b$ and Pasquale Di Bari$^b$]{}\
$^a$[*Institut de Théorie des Phénomènes Physiques*]{}\
[*Ecole Polytechnique Fédérale de Lausanne*]{}\
[*CH-1015 Lausanne, Switzerland*]{}\
$^b$[*Max-Planck-Institut für Physik*]{}\
[*(Werner-Heisenberg-Institut)*]{}\
[*Föhringer Ring 6, 80805 München, Germany*]{}
title: ' **Viability of Dirac phase leptogenesis**'
---
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Introduction
============
Leptogenesis [@fy], a cosmological consequence of the see-saw mechanism [@seesaw], provides an attractive explanation for the baryon asymmetry of the Universe, one of the most long-standing cosmological puzzles. A lepton asymmetry is produced in the decays of the very heavy RH neutrinos predicted by the see-saw mechanism. In order for the ($B-L$ conserving) sphaleron processes to be able to convert part of the lepton asymmetry into a baryon asymmetry, very high temperatures, $T\gtrsim M_{\rm ew}\sim 100\,{\rm GeV}$, are required in the early Universe [@sphalerons].
In comparison with other models of baryogenesis, leptogenesis offers the unique advantage of relying on an ingredient of physics beyond the Standard Model, neutrino masses, already confirmed by the experiments. Furthermore and very interestingly, a quantitative analysis [@window] shows that the values of the atmospheric and of the solar neutrino mass scales, inferred from neutrino mixing experiments, favor leptogenesis to work in a mildly ‘strong wash-out regime’: inverse processes are strong enough to wash-out any contribution to the final asymmetry depending on the initial conditions but not too strong to prevent successful leptogenesis. In this way the observed matter-antimatter asymmetry can be unambiguously explained within a minimal extension of the SM where RH neutrinos with a Majorana mass term and Yukawa couplings are added to the SM Lagrangian and the see-saw limit is assumed. No particular assumptions on the initial conditions are required, in complete analogy with what happens in the calculation of the primordial nuclear abundances within standard Big Bang Nucleosynthesis.
In a typical $N_1$-dominated scenario, the asymmetry is dominantly produced from the decays of the lightest RH neutrino $N_1$. A necessary (but not sufficient) condition is the assumption of a mild hierarchy in the heavy neutrino mass spectrum, such that $M_2$, the mass of the next-to-lightest RH neutrino, is approximately three times larger than $M_1$, the mass of the lightest RH neutrino [@beyond].
In an unflavored analysis, a stringent lower bound on $M_1$ holds [@di]. In the decoupling limit, when the $N_1$-decay parameter $K_1\rightarrow 0$ and assuming an initial thermal abundance, one finds [^1] $M_1\gtrsim 5\times 10^8\,{\rm GeV}$ [@cmb]. However, there are different drawbacks for the saturation of this lower bound that is anyway strongly model dependent. A more significant and stringent lower bound, $M_1\gtrsim 3\times 10^9\,{\rm GeV}$, is obtained at the onset of the strong wash-out regime, $K_1\simeq K_{\star}\simeq 3.3$ [@annals; @flavorlep], where the final asymmetry does not depend on the initial conditions. This lower bound implies an associated lower bound [@annals; @giudice], $T_{\rm reh}\gtrsim 1.5\times
10^{9}\,{\rm GeV}$, on the value of the temperature at the beginning of the standard radiation dominated regime, the reheating temperature within inflation.
As we said, the assumption of a mild hierarchy in the heavy neutrino mass spectrum is not sufficient to guarantee that the $N_1$-dominated scenario holds. It is indeed possible that, for a proper choice of the see-saw parameters, a $N_2$-dominated scenario holds, where the final asymmetry is dominated by the contribution from the decays of the next-to-lightest RH neutrino $N_2$ [@geometry]. In this case the lower bound on $M_1$ does not hold any more and is replaced by a lower bound on $M_2$, however still implying a lower bound on $T_{\rm reh}$.
Even when flavor effects [@bcst; @endo; @pilaund; @nardi; @abada; @abada2] are taken into account [^2], these lower bounds do not get relaxed [@flavorlep; @abada3]. In particular, flavor effects do not help to alleviate the conflict with the upper bound on the reheating temperature coming from the avoidance of the gravitino problem when a supersymmetric framework is considered [@gravitino]. On the other hand, flavor effects relax the lower bound on $M_1$ for $K_1\gg K_{\star}$ [@abada] and, interestingly, it has been shown that the Dirac phase and, more efficiently, the Majorana phases can strongly enhance the amount of the relaxation [@flavorlep].
Besides this effect, there is an even more interesting role played by the Majorana and Dirac phases when flavor effects are taken into account. In an unflavored analysis, the final asymmetry does not depend on the low-energy phases and this represents a limit to the possibility of further tightening the link between leptogenesis predictions and low-energy neutrino experiments.
In [@nardi] it has been shown that, accounting for flavor effects, an asymmetry can be generated even when the total $C\!P$ asymmetry vanishes. This is possible because flavor effects introduce an additional source of $C\!P$ asymmetry stemming from low-energy phases. The flavor composition of the anti-lepton produced in the decay of the RH neutrino can be indeed different from the one of the $C\!P$ conjugated lepton. In this way a new intriguing scenario arises, where the Majorana and the Dirac phases, potentially observable in low-energy neutrino experiments, could act as the unique source of $C\!P$ violation responsible for the observed matter-antimatter asymmetry of the Universe. First calculations have been presented in [@abada2] for particular values of $M_1$ and within a two RH neutrino scenario, corresponding to a specific choice of the see-saw orthogonal matrix [@casas].
In [@flavorlep] it has been first shown that successful leptogenesis stemming only from low-energy phases is possible and the lower bound on $M_1$, with its dependence on $K_1$ and on the initial conditions, has been calculated for a specific choice of the see-saw orthogonal matrix in the HL. It has been found that for values of $K_1$ in the strong wash-out regime, the allowed region is very constrained when just Majorana phases are switched on, and it is even worse when only the Dirac phase is switched on, even for $\sin\theta_{13}$ close to its experimental upper bound. Compared to the usual cases where high-energy phases contribute to $C\!P$ violation as well, the lower bounds on $M_1$ and on the reheating temperature get much more stringent, especially in the strong wash-out regime and in particular for values of $K_1$ in the range favored by neutrino mixing experiments. Therefore, one can say that the asymmetry production from low-energy phases is somehow secondary compared to the usual case when leptogenesis proceeds from the high-energy phases contained in the see-saw orthogonal matrix. This conclusion has been recently confirmed also in [@antusch] in the context of MSSM. In [@pascoli; @pascoli2; @branco], the results have been generalized for an arbitrary choice of the orthogonal matrix and of the low-energy phases but without a study of the dependence on $K_1$ and on the initial conditions.
In this paper we focus on the particularly interesting case of $\d$-leptogenesis, where the Dirac phase, which has realistic chances to be observed in neutrino mixing experiments for not too small values of $\sin\theta_{13}$, is the only non-vanishing phase. We study the dependence of the final asymmetry on the initial conditions, finding the onset of the strong wash-out regime and showing the dependence of the $M_1$ lower bound on the important decay parameter $K_1$, whose value is related to the values of the neutrino masses and at the same time determines the efficiency of the asymmetry production (involving both the production of the heavy neutrinos and the wash-out). We first obtain that, in the HL, the possibility to explain the observed asymmetry is only marginally allowed and just limited to the less relevant weak wash-out regime, when the correct condition for the validity of the fully flavored regime is taken into account [@zeno]. Then we point out that this obstacle can be nicely circumvented going beyond the HL. Indeed, like in the unflavored case [@crv; @pilaftsis; @beyond], the flavored $C\!P$ asymmetries, and consequently the final $B-L$ asymmetry, get enhanced and the lower bounds on $M_1$ and on $T_{\rm reh}$ get relaxed. The possibility of $\d$-leptogenesis beyond the HL has been already studied in [@pascoli2] within resonant leptogenesis [@pilaftsis], where the heavy neutrino mass differences are equal to the resonance widths, for initial vanishing abundance and in [@selma] in the context of radiative leptogenesis [@radiative] with minimal flavor violation [@MFV].
We perform a general analysis in the degenerate limit (DL), where at least one of the degeneracies $\d_{ji}\equiv (M_j-M_i)/M_i\lesssim 0.01$. We show that in this case the strong wash-out regime always holds and the lower bound on $M_1$ can be expressed through $\d_{ji}$ and the quantity $\D\equiv\sin\theta_{13}\,\sin\d$. In the most extreme case of resonant leptogenesis this turns both into a lower bound on $\theta_{13}$ and into an upper bound on the absolute neutrino mass scale that depend on each other. In this way we find that $\d$-leptogenesis can indeed explain the observed matter-antimatter asymmetry of the Universe in the strong wash-out regime and therefore, like leptogenesis from high-energy phases, exhibits the same virtue of independence of the initial conditions.
In Section 2 we introduce the general framework and set the notation. In Section 3 we present the results in the HL. We confirm, in a more general way, the conclusions of [@flavorlep], showing that the allowed region for $\d$-leptogenesis is quite restricted, especially in the strong wash-out regime and considering that the asymmetry production has to switch off for $M_1\gtrsim 10^{12}\,{\rm GeV}$, when the unflavored case is recovered and $C\!P$ violation from low-energy phases turns off. We also verify that this conclusion holds even when the asymmetry production from the two heavier RH neutrinos is taken into account. On the other hand, we show that a $N_2$-dominated scenario can also be realized in $\d$-leptogenesis.
We conclude that one needs to go beyond the HL for successful $\d$-leptogenesis in the strong wash-out regime and in any case not to be just marginally allowed. Therefore, in Section 4 we study the DL showing that successful $\d$-leptogenesis is possible and we find a condition that relates $\d_{ji}$ to $M_i$ ($j=2,3$ and $i=1,2$) and to $\D$. We also find an upper bound on the absolute neutrino mass scale dependent on $\sin\theta_{13}$ that makes $\d$-leptogenesis falsifiable independently of the RH neutrino spectrum. In Section 5 we draw the conclusions.
General framework
=================
Adding to the Standard Model three RH neutrinos with a Majorana mass term $M$ and Yukawa couplings $h$, after spontaneous breaking a Dirac mass term, $m_D=v\,h$, is generated by the vev $v$ of the Higgs boson. In the see-saw limit, $M\gg m_D$, the spectrum of neutrino mass eigenstates splits in two sets, a very heavy one, $N_1,N_2$ and $N_3$ with masses respectively $M_1\leq M_2 \leq M_3$ and almost coinciding with the eigenvalues of $M$, and a light one, with masses $m_1\leq m_2\leq m_3$ corresponding to the eigenvalues of the light neutrino mass matrix given by the see-saw formula [@seesaw], m\_= - m\_D [1M]{} m\_D\^T . Neutrino mixing experiments measure two light neutrino mass squared differences. In a normal scheme one has $m^{\,2}_3-m_2^{\,2}=\Delta m^2_{\rm atm}$ and $m^{\,2}_2-m_1^{\,2}=\Delta m^2_{\rm sol}$, whereas in an inverted scheme one has $m^{\,2}_3-m_2^{\,2}=\Delta m^2_{\rm sol}$ and $m^{\,2}_2-m_1^{\,2}=\Delta m^2_{\rm atm}$. For $m_1\gg m_{\rm atm} \equiv
\sqrt{\Delta m^2_{\rm atm}+\Delta m^2_{\rm sol}}=
(0.052\pm 0.002)\,{\rm eV}$ [@concha] the spectrum is quasi-degenerate, while for $m_1\ll m_{\rm sol}\equiv \sqrt{\D m^2_{\rm sol}}
=(0.0089\pm 0.0002)\,{\rm eV}$ [@concha] is fully hierarchical.
In the early Universe, the decays of the heavy neutrinos into leptons and Higgs bosons produce, in general, a lepton number that is partly converted into a baryon number by sphaleron ($B-L$ conserving) processes if the temperature is higher than about $100\,{\rm GeV}$ [@sphalerons].
An important role is played by the decay parameters of the heavy neutrinos defined as $K_i\equiv \widetilde{\G}_i/H_{T=M_i}$, the ratios of the decay widths to the expansion rate when the RH neutrinos start to become non-relativistic at $T=M_i$. For $K_i \ll 1$ the bulk of the $N_i$ decays occurs when they are non-relativistic and the inverse decays are not effective anymore. In this case all decays occur out-of-equilibrium and the wash-out of the asymmetry is weak. On the other hand, for $K_i\gg 1$, the heavy neutrinos decays are balanced by inverse processes. In this case the heavy neutrino abundance tracks quite closely the equilibrium abundance and the wash-out of the asymmetry is potentially, but not necessarily, strong. The answer depends on a detailed description of flavor effects that are triggered by the charged lepton Yukawa interactions with a rate $\G_{\alpha}\simeq 5\times 10^{-3}\,T\,f^2_{\alpha}\,(\a=e,\m,\t)$ [@Campbell:1992jd], where the $f_{\alpha}$’s are the charged lepton Yukawa couplings in the diagonal basis.
If $\G_{\alpha}\ll \sum_{i}\,\G_{\rm ID}^i$ [^3], during all the relevant period of the asymmetry generation, then the lepton state coherence is preserved between decays and inverse decays and the unflavored regime, where flavor effects are negligible, holds. This requirement implies [@zeno] \[unflavored\] M\_1510\^[11]{}[GeV]{} . In the unflavored regime the condition $K_i\gg 1$ is also sufficient for the wash-out regime to be strong. It is important to stress that in this regime the only source of $C\!P$ violation is due to a different total decay rate into leptons and anti-leptons and, as it is well known, it stems uniquely from high-energy phases. Therefore, in the unflavored regime, $\d$-leptogenesis is not viable.
If the charged lepton Yukawa interactions are in equilibrium ($\G_{\a}> H$) and faster than inverse decays, \[condition\] \_\_[i]{}\_[ID]{}\^i , during the relevant period of the asymmetry generation, then the lepton quantum states lose coherence between the production at decay and the subsequent absorption in inverse processes. In this way the Higgs bosons interact incoherently with leptons of each flavor. In the limit case, when the quantum state becomes completely incoherent and is fully projected in one of the flavor eigenstates, each lepton flavor can be treated as a statistically independent particle species and a ‘fully flavored regime’ is obtained. One has to distinguish a two-flavor regime, for $M_1\gtrsim 10^{\rm 9}\,{\rm GeV}$, such that the condition Eq. (\[condition\]) is in any case satisfied only for $\a=\t$, and a three-flavor regime, where the condition Eq. (\[condition\]) applies also to $\a=\m$.
In the fully (two or three) flavored regime, classic Boltzmann equations can be used like in the unflavored regime, with the difference, in general, that now each single flavor asymmetry has to be tracked independently.
In the fully flavored regime there are two new effects compared to the unflavored regime [@nardi]. These can be understood introducing the projectors and writing them as the sum of two terms, P\_[i]{} & & |l\_[i]{}|l\_|\^2 = P\_[i]{}\^0 + [P\_[i]{}2]{}\
|[P]{}\_[i]{}& & ||[l]{}’\_[i]{}||[l]{}\_|\^2 = P\_[i]{}\^0 - [P\_[i]{}2]{} . The first effect is a reduction of the wash-out compared to the unflavored regime and is described by the tree level contribution $P_{i\a}^0=(P_{i\a}+\bar{P}_{i\a})/2$ setting the fraction of the total asymmetry, produced in $N_i$-decays, that goes into each single flavor $\a$. In the fully flavored regime, each single inverse decay involves an independent lepton flavor eigenstate and therefore does not wash out, in general, as much asymmetry as that one produced in each single decay but an amount reduced by $P_{i\a}^0$.
The second effect is an additional $C\!P$ violating contribution coming from a different flavor composition between $|l_i\rangle$ and $C\!P |\bar{l}_i'\rangle$. This can be described in terms of the projector differences $ \D\,P_{i\a}\equiv P_{i\alpha}-\bar{P}_{i\alpha}$, such that $\sum_{\a}\,\D\,P_{i\a}= 0$. Indeed, defining the flavored $C\!P$ asymmetries, \_[i]{} -[\_[i]{}-\_[i]{} \_[i]{}+\_[i]{}]{} , where $\G_{i\a}\equiv P^0_{i\a}\,\G_{\a}$ and $\bar{\G}_{i\a}\equiv P^0_{i\a}\,\bar{\G}_{\a}$, these can be now written as \_[i]{}=\_iP\^[0]{}\_[i]{}+ [P\_[i]{}2]{} , where $\ve_i\equiv \sum_\a\,{\ve_{i\a}}$ are the total $C\!P$ asymmetries. In the last expression one can see that the first term is the usual contribution due to a different decay rate into lepton and anti-leptons and the second is the additional contribution due to a possible different flavor composition between $|l_i\rangle$ and $C\!P |\bar{l}_i'\rangle$.
Taking into account only decays and inverse decays with proper subtraction of the resonant contribution from $\D L=2$ and $\D L=0$ processes [@giudice; @pilaund; @nardi], the set of effective classic Boltzmann equations valid in the fully three-flavored regime can be written as \[flke\] [dN\_[N\_i]{}dz]{} & = & -D\_i(N\_[N\_i]{}-N\_[N\_i]{}\^[eq]{}) (i=1,2,3)\
& = & \_i\_[i]{}D\_i(N\_[N\_i]{}-N\_[N\_i]{}\^[eq]{}) -\_iP\_[i]{}\^[0]{} W\_i\^[ID]{}N\_[\_]{} (=e,,) , where $z \equiv M_1/T$ and where we indicated with $N_X$ any particle number or asymmetry $X$ calculated in a portion of co-moving volume containing one heavy neutrino in ultra-relativistic thermal equilibrium, so that $N^{\rm eq}_{N_i}(T\gg M_i)=1$. Defining $x_i\equiv M_i^2/M_1^2$ and $z_i\equiv z\,\sqrt{x_i}$, the decay factors are given by D\_i =K\_ix\_iz , where $H$ is the expansion rate. The total decay rates, $\G_{D,i}\equiv \G_i+\bar{\G}_i$, are the product of the decay widths times the thermally averaged dilation factors $\langle 1/\gamma_i\rangle$, given by the ratio ${\cal K}_1(z_i)/ {\cal K}_2(z_i)$ of the modified Bessel functions. The equilibrium abundance and its rate are also expressed through the modified Bessel functions, N\_[N\_i]{}\^[eq]{}(z\_i)= [12]{}z\_i\^2[K]{}\_2 (z\_i) , = -[12]{}z\_i\^2[K]{}\_1 (z\_i) . Finally, the inverse decays wash-out terms are given by \[WID\] W\_i\^[ID]{}(z) = [14]{}K\_i[K]{}\_1(z\_i)z\_i\^3 . We are neglecting the non resonant contributions from $\D L=2$ and $\D L=0$ processes, a good approximation for $M_1\ll 10^{14}\,{\rm GeV}\,(m_{\rm atm}^2/\sum_i\,m_i^2)$, as we will always consider. We are also neglecting $\D L=1$ scatterings [@luty; @plum; @pilaund2; @abada2], giving a correction to a level less than $\sim 10\% $ [@flavorlep] and spectator processes [@buchplum; @nardi2] that, at least for a hierarchical heavy neutrino spectrum, produce a correction to a level less than $\sim 30\% $ [@nardi2; @abada3]. In the degenerate limit it cannot be excluded that the effect of spectator processes is more relevant and further studies are required. We are also neglecting thermal corrections [@giudice], that can give relevant (though with big theoretical uncertainties) corrections in the weak wash-out regime but negligible ones in the more important strong wash-out regime.
The evolution of the $N_{\D\a}$’s can be worked out in an integral form, N\_(z)=N\_\^[in]{} e\^[-\_iP\_[i]{}\^0\_[z\_[in]{}]{}\^zdz’W\_i\^[ID]{}(z’)]{} +\_i\_[i]{}\_[i]{}(z) , with the 9 efficiency factors given by \[ef\] \_[i]{}(z;K\_i,P\^[0]{}\_[i]{})=-\_[z\_[in]{}]{}\^zdz’[dN\_[N\_i]{}dz’]{} e\^[-\_iP\_[i]{}\^0\_[z’]{}\^zdz”W\_i\^[ID]{}(z”)]{} . The total final $B-L$ asymmetry is then given by $N_{B-L}^{\rm f}=\sum_{\a}\,N_{\D_\a}^{\rm f}$. Finally, assuming a standard thermal history and accounting for the sphaleron converting coefficient $a_{\rm sph}\sim 1/3$, the final baryon-to-photon number ratio can be calculated as \[etaB\] \_B=a\_[sph]{}[N\_[B-L]{}\^[f]{}N\_\^[rec]{}]{} 0.9610\^[-2]{}N\_[B-L]{}\^[f]{} , to be compared with the measured value [@WMAP3] \[etaBobs\] \_B\^[CMB]{} = (6.1 0.2)10\^[-10]{} . Notice that the efficiency factors depend only on the $P_{i\a}^0$ but not on the differences $\D P_{i\a}$. Notice also that, in the two-flavor regime, the individual electron and muon asymmetries are replaced by one kinetic equation for the sum, $N_{\D_{e\m}}\equiv N_{\D_{\m}}+N_{\D_e}$, where the individual flavored $C\!P$ asymmetries and projectors have also to be replaced by the their sum, namely $\ve_{1\,e+\m}\equiv \ve_{1\m}+\ve_{1e}$ and $P^0_{1\,e+\m}\equiv
P^0_{1\m}+P^0_{1 e}$ [@abada2]. The calculation is therefore somehow intermediate between the one-flavor approximation and the three-flavor regime, though the results are very similar to the three-flavor regime [@flavorlep].
The flavored $C\!P$ asymmetries are given by the following expression [@crv] \[veia\] \_[i]{}= \_[ji]{} { [Im]{}+ } , where \[xi\] (x)= [23]{}x . A parametrization of the Dirac mass matrix, particularly fruitful within leptogenesis, is obtained in terms of the see-saw orthogonal matrix $\O$ [@casas] \[Opar\] m\_D = UD\_m\^[1/2]{}ØD\_M\^[1/2]{} , where we defined $D_m\equiv {\rm diag}(m_1,m_2,m_3)$ and $D_M\equiv {\rm diag}(M_1,M_2,M_3)$. The matrix $U$ diagonalizes the light neutrino mass matrix $m_{\nu}$, such that $U^{\dagger}\,m_{\nu}\,U^{\star}=-D_m$, and it can be identified with the lepton mixing matrix in a basis where the charged lepton mass matrix is diagonal. Moreover, neglecting the effect of the running of neutrino parameters from high energy to low energy [@running], one can assume that the $U$ matrix can be identified with the PMNS matrix, partially measured in neutrino mixing experiments. For normal hierarchy we adopt the parametrization [@PDG] $$\label{Umatrix}
U=\left( \begin{array}{ccc}
c_{12}\,c_{13} & s_{12}\,c_{13} & s_{13}\,e^{-i\,\d} \\
-s_{12}\,c_{23}-c_{12}\,s_{23}\,s_{13}\,e^{i\,\d} &
c_{12}\,c_{23}-s_{12}\,s_{23}\,s_{13}\,e^{i\,\d} & s_{23}\,c_{13} \\
s_{12}\,s_{23}-c_{12}\,c_{23}\,s_{13}\,e^{i\,\d}
& -c_{12}\,s_{23}-s_{12}\,c_{23}\,s_{13}\,e^{i\,\d} &
c_{23}\,c_{13}
\end{array}\right)
\times {\rm diag(e^{i\,{\Phi_1\over 2}}, e^{i\,{\Phi_2\over 2}}, 1)}
\, ,$$ where $s_{ij}\equiv \sin\theta_{ij}$, $c_{ij}\equiv\cos\theta_{ij}$ and, neglecting the statistical errors, we will use $\theta_{12}=\pi/6$ and $\theta_{23}=\pi/4$, compatible with the results from neutrino mixing experiments. Moreover, we will adopt the $3\s$ range $s_{13}=0-0.20$, allowed from a global $3\n$ analysis for unitary $U$ [@concha], an approximation that holds with great precision in the see-saw limit with $M_i\gg 100\,{\rm GeV}$. Within the convention we are using, $m_1\leq m_2 \leq m_3$, the case of inverted hierarchy corresponds is obtained performing a cyclic permutation of columns in the PMNS matrix parametrization Eq. (\[Umatrix\]), such that the $i$-th column becomes the $(i+1)$-th. Since we are interested in understanding whether a non-vanishing Dirac phase can be the only source of $C\!P$ violation for successful leptogenesis, we will set the Majorana phases to zero. We will comment later on the effects of turning on the Majorana phases.
It will also prove useful to introduce the following parametrization for the see-saw orthogonal matrix, \[second\] Ø([ø]{}\_[21]{},[ø]{}\_[31]{},[ø]{}\_[32]{}) =R\_[12]{}(ø\_[21]{}) R\_[13]{}(ø\_[31]{}) R\_[23]{}(ø\_[32]{}) , where \[R\] Notice that, using the orthogonal parametrization, the decay parameters $K_i$ can be expressed as linear combinations of the neutrino masses [@fhy; @annals] \[Kimi\] K\_i = [m\_]{} = \_j[m\_jm\_]{}|Ø\_[ji]{}\^2| , where $\mti\equiv (m^{\dagger}_D\,m_D)_{ii}/M_i$ are the effective neutrino masses [@plum] and where $m_{\star}$ is the equilibrium neutrino mass [@annals] given by $$\label{d}
m_{\star} = {16\, \pi^{5/2}\,\sqrt{g_*} \over 3\,\sqrt{5}}\,
{v^2 \over M_{Pl}} \simeq 1.08\times 10^{-3}\,{\rm eV}\;.$$ Barring huge phase cancellations and special forms for $\O$, typically the $K_i$’s span within the range $[K_{\rm sol},K_{\rm atm}]$ where $K_{\rm sol}\equiv m_{\rm sol}/m_{\star}=8.2\pm 0.2$ and $K_{\rm atm}\equiv m_{\rm atm}/m_{\star}=48 \pm 2$.
Before entering into a detailed analysis focusing on $\d$-leptogenesis, we want to discuss some general features concerning the fully flavored regime and in particular the possibility to have important deviations from the unflavored case. For definiteness and simplicity, we refer to the two-flavor case within the $N_1$-dominated scenario, so that $N_{B-L}^{\rm f}\simeq \ve_{1\t}\,\k_{1\t}^{\rm f}+
\ve_{1,e+\m}\,\k_{1,e+\m}^{\rm f}$.
Consider first the ‘democratic case’, where $\D P_{1\a}=0$ and $P_{1\t}=P_{1\, e+\m}=1/2$. Summing the two equations for $\a=\t$ and $\a=e+\m$ one obtains a closed equation for the total asymmetry where the only effect compared to the unflavored case is that the wash-out is reduced by a factor two and the final asymmetry is obtained by replacing $K_1\rightarrow K_1/2$. Therefore, in the strong wash-out regime ($K_1\gg 1$), since $\k_{1\a}^{\rm f}\propto K_1^{-1.2}$ [@proc], one has approximately a factor two enhancement. Let us now consider $P_{1\m}^0< P_{1\t}^0$, still with $\D P_{1\a}=0$. Since approximately $\k_{1\a}^{\rm f}\propto (P_{1\a}^0)^{-1.2}$ and at the same time $\ve_{1\a}\propto P_{1\a}^0$, one has that the final asymmetry stays approximately constant compared to the democratic case with the two contributions from the $\m$ and $\t$ flavors comparable with each other. Therefore, for vanishing $\D P_{1\a}$, flavor effects produce just ${\cal O}(1)$ corrections compared to the unflavored approximation.
This conclusion changes when non-vanishing $\D P_{1\a}$ are considered. In this case there are two remarkable possibilities.
The [*first possibility*]{} is the so called one-flavor dominated scenario, relying on the fact that, for $P^0_{1\a}\rightarrow 0$, one has ${\rm max}(\D P_{1\a})\propto \sqrt{P_{1\a}^0}$ [@abada2]. Therefore, considering now for example $P_{1\t}\ll P_{1e+\m}\simeq 1$, one has that the asymmetry in the tauon flavor $\ve_{1\t}\,\k_{1\t}\propto (P_{1\t}^0)^{-0.7}$, showing that there can be a large enhancement compared to the unflavored case in the strong wash-out regime. This brings to a strong relaxation of the lower bounds on $M_1$ and $T_{\rm reh}$ at $K_1\gg 1$, though, as we already said, not to a relaxation of the usual lowest bounds at $K_1\rightarrow 0$ or at $K_1\simeq K_{\star}$. It should also be said that, as shown in [@zeno], the applicability of the one-flavor dominated scenario is strongly limited by the condition of validity of the fully flavored regime Eq. (\[condition\]).
The [*second possibility*]{} relies on the observation that, contrarily to $\ve_1$, the $\D P_{1\a}$’s depend on the low-energy phases as well and, even though $\ve_1=0$, they do not vanish if the Dirac or the Majorana phases do not vanish. Therefore, one can have a final asymmetry originating just from low-energy phases [@nardi]. This scenario represents, potentially, the most important novelty introduced by flavor effects compared to the unflavored picture and in the following Sections we will study it in detail, focusing on the case of $\d$-leptogenesis, when only the Dirac phase is switched on while $\O$ is real and the two Majorana phases vanish.
Before concluding this Section, we want to notice that one can have $\ve_i=0$ not only when the see-saw orthogonal matrix is real, but also when the absolute neutrino mass scale increases [@abada]. In this way, the low-energy phases can play an important role in circumventing the upper bound on the neutrino masses holding in the unflavored regime [@bound1; @window]. It is however still to be assessed whether the fully flavored regime can offer a sufficient description to solve this issue. In [@abada] the bound was found to be completely nullified by flavor effects. In [@zeno] it has been pointed out how this conclusion relies on a extension of the fully flavored regime beyond the regime of its validity given by the Eq. (\[condition\]). In [@riotto] the authors find that in the fully flavored regime, thanks to spectator processes, the bound holding in the unflavored regime, even though not nullified, is anyway relaxed to $m_1\lesssim 2\,{\rm eV}$.
The hierarchical limit
======================
Let us consider first $\d$-leptogenesis in the HL, such that $M_3 \gtrsim 3\,M_2 \gtrsim 3\,M_1$ [@beyond]. In the unflavored regime, this assumption typically implies a $N_1$-dominated scenario, where the final asymmetry is dominated by the contribution from the lightest RH neutrino decays, \[N1DS\] N\^[f]{}\_[B-L]{}. N\_[B-L]{}\^[f]{} |\_[N\_1]{} \_\_[1]{}\_[1]{} . Indeed, in general, in the HL one has two effects. The first effect is that the asymmetry production from the two heavier RH neutrinos, $N_2$ and $N_3$, is typically later on washed out by the $N_1$ inverse processes and $\k_3^{\rm f},\k_2^{\rm f}\ll \k_1^{\rm f}$. The second effect is a consequence of the fact that the total $C\!P$ asymmetries vanish in the limit when all particles running in the loops become massless and this yields typically $|\ve_3|\ll |\ve_2| \ll |\ve_1|$.
However, for a particular choice of the see-saw parameters, $\O\simeq R_{23}$ and $m_1\lesssim m_{\star}$, the contribution to the final asymmetry from the next-to-lightest RH neutrino $N_2$ is not only non-negligible but even dominant, giving rise to a $N_2$-dominated scenario [@geometry]. Indeed for $\O\simeq R_{23}$ different things happen simultaneously. First, $N_2$, even though decoupled from $N_1$, is still coupled to $N_3$ and in the HL the total $C\!P$ asymmetry $\ve_2$ does not vanish, since it receives a non suppressed contribution from graphs where $N_3$ runs in the loops. On the other hand, now one has $\ve_1=0$, since $N_1$ is essentially decoupled from the other two RH neutrinos. At the same time one also has $K_1\ll 1$, so that the wash-out from $N_1$ inverse processes is negligible. The final result is that $|\ve_2\,{\k_2}|\gg |\ve_{i\neq 2}\,\k_{i\neq 2}^{\rm f}|$ and the final asymmetry is dominantly produced from $N_2$-decays.
Therefore, in the unflavored approximation and in the HL, a condition $w_{32}\simeq 1$ in the $\O$-matrix parametrization (cf. Eq. (\[second\])) is sufficient to have a negligible asymmetry production from the two heavier RH neutrinos and to guarantee that the $N_1$-dominated scenario holds. This condition is even not necessary for $m_1\gg m_{\star}$, since in this case, due to the fact that $\mt\geq m_1$, one has necessarily $K_1\gg 1$ and a wash-out from $N_1$-inverse processes is anyway strong enough to suppress a possible contribution to the final asymmetry produced from $N_2$-decays.
When flavor effects are taken into account, the domain of applicability of the $N_1$-dominated scenario reduces somehow. There are two aspects to be considered.
The first aspect is that the wash-out from $N_1$ inverse processes becomes less efficient. Indeed the projectors $P_{1\a}$ can considerably reduce the wash-out of the asymmetry produced in the flavor $\a$ from $N_2$-decays [@vives]. This turns the condition $m_1\gg m_{\star}$ into a looser condition $m_1\gg m_{\star}/P_{1\a}$. Another effect is that $N_1$ inverse processes can make part of the asymmetry produced in $N_2$ decays somehow orthogonal to the the wash-out from $N_1$ inverse processes [@bcst; @nardi3]. Recently, it has been also pointed out that spectator processes can lead to a reduction of the wash-out from $N_1$-inverse processes as well [@shindou]. In this way the assumption $\k_{2\a}\ll \k_{1\a}$ is not valid in general.
The second aspect concerns the flavored $C\!P$ asymmetries. In the HL, from the general expression Eq. (\[veia\]), one has $$\begin{aligned}
\label{ve1a}
\ve_{1\alpha}&\simeq& \frac{3}{16 \p (h^{\dag}h)_{11}}
\sum_{j\neq 1} \frac{M_1}{M_j} {\rm Im}
\left[h_{\a 1}^{\star} h_{\a j}(h^{\dag}h)_{1 j}\right],\\ \label{ve2a}
\ve_{2\alpha}&\simeq&
\frac{3}{16 \p (h^{\dag}h)_{22}}
\left\{\frac{M_2}{M_3}{\rm Im}
\left[h_{\a 2}^{\star} h_{\a 3}(h^{\dag}h)_{2 3}\right]
-\frac{2}{3} {\rm Im}\left[h_{\a 2}^{\star} h_{\a 1}(h^{\dag}h)_{1 2}\right]
\right\} ,\\
\ve_{3\alpha}&\simeq& -\frac{1}{8\,\p (h^{\dag}h)_{33}}
\sum_{j\neq 3}
\left\{{\rm Im}\left[h_{\a 3}^{\star} h_{\a j}(h^{\dag}h)_{j 3}\right]\right\}.\end{aligned}$$ Different comments are in order. The $\ve_{1\a}$’s, like $\ve_1$, vanish for $\O=R_{23}$ while the $\ve_{2\a}$’s, like $\ve_2$, do not. On the other hand, in the HL, the $\ve_{2\a}$’s, contrarily to $\ve_2$, are not suppressed when $\o_{32}=0$ (a particular example is given by $\O=R_{12}$) but, like $\ve_2$, they vanish for $\O=R_{13}$.
This observation [@flavorlep] can also potentially contribute to enlarge the domain of applicability of the $N_2$-dominated scenario when flavor effects are taken into account. Another interesting observation is that the $\ve_{3\a}$’s, contrarily to $\ve_3$, do not vanish in the HL. This could open the door even to a $N_3$-dominated scenario, though this is possible only for $M_3\lesssim 10^{12}\,{\rm GeV}$, when flavor effects are effective in $N_3$ decays.
Therefore, when flavor effects are taken into account, the conditions of applicability of the $N_1$-dominated scenario become potentially more restrictive than in the unflavored case. There is a clear choice of the parameters, for $\O=R_{13}$ and $M_3\gtrsim 10^{12}\,{\rm GeV}$, where the $N_1$-dominated scenario holds. Indeed in this case, in the HL, one has that $\ve_{2\a}$ and $\ve_3$ are suppressed. This can be considered somehow opposite to the case $\O=R_{23}$, where the $N_2$-dominated scenario holds [@geometry].
In general, one can say that the asymmetry produced from the two heavier RH neutrinos is non-negligible if two conditions are satisfied. (i) The asymmetry generated from $N_{2,3}$-decays at $T\sim M_{2,3}$ has to be non-negligible compared to the asymmetry generated at $T\sim M_1$ from $N_1$-decays. This depends on an evaluation of the $C\!P$ asymmetries $\ve_{2,3}^{\a}$ and of the wash-out due to the same $N_{2,3}$-inverse processes. (ii) The asymmetry produced from $N_{2,3}$-decays has not to be afterwards washed-out by $N_1$-inverse processes. Notice that this second condition is subordinate to the first condition.
In the particular case of $\d$-leptogenesis, one has $\ve_2=\ve_3=0$. This means that the first condition can be satisfied only if $M_2, M_3\lesssim 10^{12}\,{\rm GeV}$ and this constitutes already an important limitation. In the following, we will consider different particular cases, verifying whether the production from the two heavier RH neutrinos can be neglected or not. We will find that the situation is actually similar to what happens in the unflavored case where, except for the case $\O\sim R_{23}$, a $N_1$-dominated scenario holds.
Let us therefore start showing in detail how to calculate the contribution to the final asymmetry from $N_1$-decays. The expression Eq. (\[ef\]) for the $\k_{1\a}$’s can be specialized as \[ef1\] \_[1]{}(z;K\_1,P\^[0]{}\_[1]{})= -\_[z\_[in]{}]{}\^zdz’[dN\_[N\_1]{}dz’]{} e\^[-P\_[1]{}\^0\_[z’]{}\^zdz”W\_1\^[ID]{}(z”)]{} . From the Eq. (\[ef\]), extending an analytic procedure derived within the one-flavor approximation [@annals], one can obtain simple analytic expressions for the $\k_{1\a}^{\rm f}$’s. In the case of an initial thermal abundance ($N_{N_1}^{\rm in}=1$), defining $K_{1\a}\equiv P^0_{1\a}\,K_1$, one has \[k1a\] \_[1]{}\^[f]{} (K\_[1]{}) (1-e\^[-[K\_[1]{}z\_B(K\_[1]{})2]{}]{}) , where z\_[B]{}(K\_[1]{}) 2+4K\_[1]{}\^[0.13]{}e\^[-[2.5K\_[1]{}]{}]{} . In the case of initial vanishing abundance ($N_{N_1}^{\rm in}=0$) one has to take into account two different contributions, a negative and a positive one, so that \_[1]{}\^[f]{} =\_[-]{}\^[f]{}(K\_1,P\_[1]{}\^[0]{})+ \_[+]{}\^[f]{}(K\_1,P\_[1]{}\^[0]{}) , whose analytic expressions, used to obtain all presented results, can be found in [@flavorlep].
The condition for the validity of the fully flavored regime Eq. (\[condition\]) can be specialized and re-cast like \[full\] M\_1. This condition neglects the effect of $\D L=1$ scatterings and of coherent scatterings, the first contributing with inverse decays to preserve the quantum state coherence, the second, conversely, in projecting it on the flavor basis [@zeno]. Both of them can be as large as the effect from inverse decays. Moreover, in a rigorous quantum kinetic description, it is likely that other subtle effects contribute to the determination of the exact value of $M_1$ below which the fully flavored regime can be assumed. Therefore, the condition (\[full\]) should be regarded as a very qualitative one. In the plots showing the $M_1$ lower bound, we will then distinguish four regions. All plots will be cut at $M_1 = 10^{12}\,{\rm GeV}$, since above this value, according to the condition (\[unflavored\]), the unflavored regime is recovered and the asymmetry production has to switch off. On the other hand, when the condition Eq. (\[full\]) is satisfied, one can expect the fully flavored regime to hold. There is an intermediate regime where a transition between the fully flavored regime and the unflavored regime takes place. This regime will be indicated in all plots with a squared region. This signals that, even though we still show the results obtained in the fully flavored regime, important corrections are expected, especially when $M_1$ gets close to $\sim 10^{12}\,{\rm GeV}$. Since this region describes a transition toward the unflavored regime, where the asymmetry production has to switch off, these corrections are expected to reduce the final asymmetry, making more stringent the lower bounds shown in the plots. Furthermore, since within current calculation, large corrections to the condition Eq. (\[full\]) cannot be excluded, we will also indicate, with a hatched region, that area where the condition Eq. (\[full\]) holds but a very conservative condition, \[conservative\] M\_1 does not. In this region some corrections to the presented results cannot be excluded but the fully flavored regime should represent a good approximation.
We anticipate that, in the $N_1$-dominated scenario, successful leptogenesis always requires $M_1\gtrsim 10^9\,{\rm GeV}$, where the two-flavor regime holds. Therefore, considering that we are assuming $\ve_1=\ve_{1\t}+\ve_{1,e+\m}=0$, the Eq. (\[N1DS\]) can be specialized into \[final\] . N\_[B-L]{}\^[f]{} |\_[N\_1]{} (\_[1]{}\^[f]{}-\_[1,e+]{}\^[f]{})\_[1]{} , showing that, in order to have a non-vanishing final asymmetry it has to be $P_{1\t}^0 \neq P_{1,e+\m}^0$. The tree-level projectors can be expressed, through the orthogonal parametrization Eq. (\[Opar\]), like \[P01a\] P\^0\_[1]{}=[|\_jU\_[j]{}Ø\_[j 1]{}|\^2 \_jm\_j|Ø\^2\_[j1]{}|]{} , that, from the Eq. (\[Kimi\]), also implies \[K1alpha\] K\_[1]{}= |\_jU\_[j]{}Ø\_[j 1]{}|\^2 . Let us now calculate the flavored $C\!P$ asymmetry $\ve_{1\t}$ from the general expression Eq. (\[ve1a\]). In terms of the orthogonal parametrization Eq. (\[Opar\]), this can be re-cast as [@abada2] \[e1alOm\] r\_[1]{}=-\_[h,l]{} [m\_lm\_[atm]{}]{} [Im]{}\[U\_[h]{}U\_[l]{}\^Ø\_[h1]{}Ø\_[l1]{}\] , where we defined $ r_{i\a}\equiv {\ve_{i\a}/ \overline{\ve}(M_i)}$, with |(M\_i)[M\_im\_[atm]{} v\^2]{} . For real $\O$, the Eq. (\[e1alOm\]) gets specialized into [@abada2] r\_[1]{}=-\_[h< l]{} [(m\_l-m\_h)m\_[atm]{}]{} Ø\_[h1]{}Ø\_[l1]{}[Im]{}\[U\_[h]{}U\_[l]{}\^\] . Taking $\a=\t$ and specifying the matrix elements $U_{\a j}$, from the Eq. (\[Umatrix\]), one has \[r1tau\] r\_[1]{}=-[m\_[atm]{}]{}\[A\_[12]{}+A\_[13]{}+A\_[23]{}\], where A\_[12]{} & = & - [(m\_2-m\_1) m\_[atm]{}\^2]{} Ø\_[11]{}Ø\_[21]{} [Im]{}\[(s\_[12]{}s\_[23]{}-c\_[12]{}c\_[23]{}s\_[13]{}e\^[i]{})\
& & (c\_[12]{}s\_[23]{}+s\_[12]{}c\_[23]{}s\_[13]{}e\^[-i]{}) e\^[-[i2]{}(\_2-\_1)]{}\] ,\
A\_[13]{} & = & [(m\_3-m\_1)m\_[atm]{}\^2]{} Ø\_[11]{}Ø\_[31]{}c\_[23]{}c\_[13]{} [Im]{}\[(s\_[12]{}s\_[23]{}-c\_[12]{}c\_[23]{}s\_[13]{}e\^[i]{}) e\^[[i2]{}\_1]{}\] ,\
A\_[23]{} & = & - [(m\_3-m\_2) m\_[atm]{}\^2]{} Ø\_[21]{}Ø\_[31]{}c\_[23]{}c\_[13]{} [Im]{}\[(c\_[12]{}s\_[23]{}+s\_[12]{}c\_[23]{}s\_[13]{}e\^[i]{}) e\^[[i2]{}\_2]{}\] . In the case of $\d$-leptogenesis ($\Phi_1=\Phi_2=0$) these expressions further specialize into A\_[12]{} & = & [(m\_2-m\_1)m\_[atm]{}\^2]{} Ø\_[11]{}Ø\_[21]{}s\_[23]{}c\_[23]{}\
A\_[13]{} & = & -[(m\_3-m\_1)m\_[atm]{}\^2]{} Ø\_[11]{}Ø\_[31]{}c\_[23]{}\^2c\_[12]{}c\_[13]{} ,\
A\_[23]{} & = & - [(m\_3-m\_2)m\_[atm]{}\^2]{} Ø\_[21]{}Ø\_[31]{}c\_[23]{}\^2s\_[12]{}c\_[13]{} , where remember that $\D\equiv \sin\theta_{13}\,\sin\d$.
It is now instructive to make some general considerations. Looking at the expression Eq. (\[final\]), one can see that, in order for the final $B-L$ asymmetry not to vanish, two conditions have to be simultaneously satisfied : $\ve_{1\t}\neq 0$ and $\k_{1\t}^{\rm f} \neq \k_{1,e+\m}^{\rm f}$. These two conditions are a specialization of the Sakharov necessary conditions to the case of $\d$-leptogenesis. Indeed, the first is the condition to have $C\!P$ violation and, as one could expect, from the expressions found for the terms $A_{ij}$, one can have $\ve_{1\t}\neq 0$ only if $\D\neq 0$. The second condition is a specialization of the condition of departure from thermal equilibrium in quite a non-trivial way. Indeed, in the case of $\d$ leptogenesis, in a full out-of-equilibrium situation where only decays are active, no final asymmetry is generated since $\ve_1=0$, implying that there is an equal number of decays into lepton and anti leptons. However, the presence of inverse processes can remove this balance, yielding a different wash-out rate for the $\t$ asymmetry and for the $e+\m$ asymmetry, such that, if $K_{1\t}\neq K_{1,e+\m}$, one has a net lepton number dynamical generation. From the expression (\[K1alpha\]), one can see that this is possible independently of the value of the Dirac phase that, therefore, is directly responsible only for $C\!P$ violation and not for lepton number violation, exactly as in neutrino mixing, where indeed lepton number is conserved. It should also be noticed that the $\ve_{1\a}$’s are expressed through quantities ${\rm Im}[U_{\a h}\,U^{\star}_{\a l}]$, that are invariant under change of the PMNS matrix parametrization [@nieves; @pascoli2]. Therefore, the final asymmetry depends correctly only on physical quantities.
Maximizing the asymmetry over all involved parameters for fixed $M_1$ and $K_1$ and imposing $\eta_B^{\rm max} \geq \eta_B^{\rm CMB}$ (cf. (\[etaB\]) and (\[etaBobs\])), a lower bound on $M_1$ is obtained [@flavorlep] M\_1 M\_1\^[min]{}(K\_1) , where we introduced the quantity [^4] \[barM1\] \_1 [N\_\^[rec]{}v\^2a\_[sph]{}]{} [\_B\^[CMB]{}m\_[atm]{}]{} =(6.250.4)10\^8[GeV]{} 510\^8[GeV]{} . The last inequality gives the $3\s$ value that we used to obtain all the results shown in the figures. We also defined [@flavorlep] \[xi1a\] \_1 \_[=,e+]{}\_[1]{} , \_[1]{} , that gives the deviation introduced by flavor effects compared to the unflavored approximation in the hierarchical light neutrino case ($m_1=0$). Notice that $r_{1\t}\propto \D$, implying $N_{B-L}^{\rm f}\propto \D$ as well. Therefore, the maximum asymmetry is obtained for $|\d|=\pi/2$ and $s_{13}=0.20$.
The calculation of the contribution to the asymmetry from $N_2$-decays proceeds in an analogous way. Again this can always be calculated in the two-flavor regime, since, in the HL, successful leptogenesis always implies $M_2\gtrsim 10^{9}\,{\rm GeV}$. Therefore, one can write an expression similar to the Eq. (\[final\]) for the contribution to the final asymmetry from $N_2$-decays, \[final2\] .N\_[B-L]{}\^[f]{}|\_[N\_2]{}(\_[2]{}\^[f]{}-\_[2,e+]{}\^[f]{})\_[2]{} . The difference is now in the calculation of the efficiency factors that are suppressed by the wash-out of the $N_1$ inverse processes. In the HL this additional wash-out factorizes and [@beyond; @flavorlep; @vives] \_[2]{}\^[f]{}(K\_[2]{}) e\^[-[38]{}K\_[1]{}]{} , where $K_{2\a}\equiv P_{2\a}^0\,K_2$. For the calculation of the tree-level projectors $P_{2\a}^0$ an expression analogous to the Eq. (\[P01a\]) holds.
The calculation of the contribution to the final asymmetry from $N_3$-decays proceeds in a similar way and analogous expressions hold. The only non trivial difference is that now, in the calculation of the efficiency factors, one has also to include the wash-out from the $N_2$ inverse processes, so that \[k3a\] \_[3]{}\^[f]{}(K\_[3]{}) e\^[-[38]{}(K\_[1]{}+K\_[2]{})]{} . Notice that in the calculation of $\k_{2\a}^{\rm f}$ ($\k_{3\a}^{\rm f}$) we are not including a possible effect where part of the asymmetry in the flavor $\a=e+\m$ produced in $N_2$ ($N_3$-decays) is orthogonal to $N_1$ inverse decays [@bcst; @nardi2] and is not washed out. This wash-out avoidance does not apply to the asymmetry in the $\t$ flavor. Therefore, as we have verified, in all cases we have considered the effect is negligible, since a $\t$-dominated scenario is always realized.
Let us now calculate the final asymmetry in some interesting cases.
$\O=R_{13}$
-----------
The first case we consider is $\O=R_{13}$, implying $A_{12}=A_{23}=0$ in the Eq. (\[r1tau\]). As we said already, it is easy to check from the Eq. (\[ve2a\]) that $\ve_{2\t}=0$ and therefore there is no asymmetry production from $N_2$-decays even if $M_2\lesssim 10^{12}\,{\rm GeV}$. On the other hand, one obtains r\_[3]{}= -[23]{}[(m\_3-m\_1) m\_[atm]{}]{} ø\_[31]{}c\_[12]{}c\_[23]{}\^2c\_[13]{} , essentially the same expression as for $r_{1\t}$ but with $\mt$ replaced by $\mttt$. Therefore, for $M_3\lesssim 10^{12}\,{\rm GeV}$, one has to worry about a potential non-negligible contribution from $N_3$ decays. However, when the wash-out from $N_1$ and $N_2$ inverse processes is taken into account, see Eq. (\[k3a\]), we always find that the contribution from $N_3$-decays is negligible and the $N_1$-dominated scenario holds.
The results are shown in Fig. 1 for $s_{13}=0.20$, $\d=-\pi/2$ and $m_1/m_{\rm atm}=0.1$, a choice of values that approximately maximizes the final asymmetry and yields the lower bound $M_1^{\rm min}(K_1)$.
In the left panel we show the tree level projectors $P^0_{1\a}$ and the $r_{1\a}$’s. It can be seen how for $K_1\gg 10$ one has $P^0_{1\t}\simeq P^0_{1,e+\m}\simeq 1/2$, while for $K_1\sim 10$ one has $P^0_{1\t}\ll P^0_{1,e+\m}$. In the central panel $\xi_1$ and the $\xi_{1\a}$’s are plotted and one can see how for $K_1\simeq 10$ a $\t$-dominance is realized. Finally, in the right panel, we show $M_1^{\rm min}(K_1)$ and we compare it with the lower bound in the unflavored approximation obtained for $\O=R_{13}$ (in this case $\O$ cannot be real) [@geometry]. One can see how, at $K_1\gg 10$, the asymmetry production rapidly dies, so that $\xi_1\ra 0$ and $M_1^{\rm min}(K_1)\ra \infty$. Notice that we plotted the lower bound both for initial thermal $N_1$-abundance and for initial vanishing $N_1$-abundance. We also indicated $K_{\star}$, defined as that value of $K_1$ such that for $K_1\gtrsim K_{\star}$ the dependence on the initial conditions can be neglected and the strong wash-out regime holds. One can notice that the intermediate regime between a fully flavored regime and the unflavored regime, the squared area, is quite extended. In this regime corrections to the results we are showing, obtained in the fully flavored regime, are expected, in a way that the unflavored regime should be recovered for $M_1\rightarrow 10^{12}\,{\rm GeV}$. In this limit the asymmetry production has to switch off and therefore one expects that the lower bound on $M_1$ has to become more restrictive and eventually, for $M_1\rightarrow 10^{12}\,{\rm GeV}$, the allowed region has to close up. Therefore, one can see that there is no allowed region in the strong wash-out regime. The hatched area, where corrections cannot be excluded within current theoretical uncertainties, cuts away almost completely any allowed region even in the weak wash-out regime. In conclusion, the allowed region where one can safely rely on the fully flavored regime according to current calculations, is very restricted and confined only to a small region in the weak wash-out regime.
$M_3\gg 10^{14}\,{\rm GeV}$
---------------------------
The second case we consider is the limit $M_3\gg 10^{14}\,{\rm GeV}$. In this limit one has necessarily $m_1\ll m_{\rm sol}$, implying $m_3\simeq m_{\rm atm}$, and also [@fgy; @ir; @ct] \[ss\] Ø= (
[ccc]{} 0 & 0 & 1\
& -Ø\_[31]{} & 0\
Ø\_[31]{} & & 0
) . Notice that this particular form of $\O$ corresponds to set $\o_{32}=1$ and $\o_{21}=1$ in the Eq. (\[R\]). Now in the expression for $r_{1\t}$ (cf. Eq. (\[r1tau\])) one has $A_{12}=A_{13}=0$ and therefore r\_[1]{} (1-[m\_2m\_[atm]{}]{}) Ø\_[31]{}c\_[23]{}\^2c\_[13]{}s\_[12]{} . If $M_2\gtrsim 10^{12}\,{\rm GeV}$, there is no contribution from the next-to-lightest RH neutrino decays anyway, since these occur in the unflavored regime where $\ve_2\simeq 0$. On the other hand, if $M_2\lesssim 10^{12}\,{\rm GeV}$, then one has to worry about a (flavored) asymmetry generation from $N_2$-decays. A calculation of $\ve_{2\a}$ shows that the first term in the Eq. (\[ve2a\]) vanishes while the second term gives r\_[2]{}=-[23]{}[m\_[atm]{}]{} (1-[m\_2m\_[atm]{}]{})Ø\_[31]{} c\_[23]{}\^2s\_[12]{}. This is an example of how the second term in the Eq. (\[ve2a\]) is not suppressed in the HL like the first term. However, like for the contribution from $N_3$-decays in the case $\O=R_{13}$, when the wash-out from $N_1$-inverse processes is taken into account one finds $\left. N_{B-L}^{\rm f}\right|_{N_2}\ll \left.N_{B-L}^{\rm f}\right|_{N_1}$ and a $N_1$-dominated scenario is realized anyway.
Notice that there is a strong dependence whether one assumes a normal or an inverted hierarchy. For normal hierarchy the results are shown in Fig. 2 for $\o_{31}>0$ and $\d=\pi/2$.
For inverted hierarchy the asymmetry is so suppressed that there is no allowed region. This means that for any choice of the parameters one always obtains $M_1^{\rm min}\gtrsim 10^{12}\,{\rm GeV}$.
Notice that results for $\d$-leptogenesis, in this particular case where $M_3\gg 10^{14}\,{\rm GeV}$, have been recently presented in [@pascoli2] for vanishing initial $N_1$ abundance. For example in [@pascoli2] the authors obtain a lower bound $\sin\theta_{13}\gtrsim 0.09$ imposing the existence of an allowed region for $M_1\lesssim 5\times 10^{11}\,{\rm GeV}$ while we would obtain $\sin\theta_{13}\gtrsim 0.05$. The difference is probably due to a ($\sim 30\%$) more conservative lower bound that we are using on $\overline{M}_1$ (see Eq. (\[barM1\])), a difference in the employed value of $m_{\star}$ (see Eq. (\[d\])), only partly understood in terms of the different convention for the Higgs v.e.v $v$. There is also a difference in the employed efficiency factor in the strong wash-out regime that, in our case, is about a factor 2 larger. Another likely minor source of difference is that we are not accounting for the effect of spectator processes encoded in the matrix $A$ that relates the $B/3-L_{\alpha}$ asymmetries to the $L_{\alpha}$ asymmetries [@bcst]. However, notice that here we do not want to emphasize too much a precise value of this lower bound on $\sin\theta_{13}$, since we believe this is anyway affected by much larger theoretical uncertainties on the validity of the fully flavored regime. It is however a good way to compare our results with those presented in [@pascoli2].
$\O=R_{12}$
-----------
The third case we consider is $\O=R_{12}$. This time one has $A_{13}=A_{23}=0$ in the Eq.(\[r1tau\]). In the case of normal hierarchy the $C\!P$ asymmetry, compared to the case $\O=R_{13}$, is suppressed by a factor $(m_{\rm sol}/m_{\rm atm})^{3/2}$, while it is essentially the same for inverted hierarchy. The projectors present very similar features to the case $\O=R_{13}$. One can also again calculate, for $M_2\lesssim 10^{12}\,{\rm GeV}$, the contribution from $N_2$-decays to the final asymmetry and again one finds that the first term in the Eq. (\[ve2a\]) vanishes, while the second produces a term $\propto M_1$, so that r\_[2]{}= [23]{}[(m\_2-m\_1) m\_[atm]{}]{} ø\_[21]{}s\_[23]{}c\_[23]{} . When the efficiency factors are taken into account, one finds that only in the case of normal hierarchy the contribution to the final asymmetry from $N_2$-decays can be comparable to that one from $N_1$-decays. However, in this case both productions are suppressed and there is no allowed region anyway in the end. In the case of inverted hierarchy, the contribution from $N_2$-decays is always negligible compared to that one from $N_1$-decays. Notice, moreover, that $\ve_{3\a}=0$ and therefore there is no contribution from $N_3$-decays. In conclusion, for $\O=R_{12}$, the lower bound on $M_1$ for normal hierarchy is much more restrictive than in the case $\O=R_{13}$, while it is very similar for inverted hierarchy. A production from the two heavier RH neutrinos can be neglected and the $N_1$-dominated scenario always holds when the asymmetry is maximized.
$\O=R_{23}$
-----------
The last interesting case is $\O=R_{23}$. From the Eq. (\[e1alOm\]) one can easily check that $\ve_{1\a}=0$. One can also easily check that, contrarily to the case $\O=R_{12}$, the second term in the Eq. (\[ve2a\]) vanishes while the first term does not and yields r’\_[2]{} = [(m\_3-m\_2)m\_[atm]{}]{} ø\_[32]{}s\_[12]{}c\_[23]{}\^2c\_[13]{} . Notice that this time $\ve_{2\t}\propto M_2$ and actually, more generally, one can see that this expression is obtained from the Eq. (\[r1tau\]) for $r_{1\t}$ in the case $\O=R_{13}$, just with the replacement $(M_1,\mt)\rightarrow (M_2,\mtt)$. At the same time one has $K_1=m_1/m_{\star}$ so that the wash-out from $N_1$-inverse processes vanishes for $m_1\rightarrow 0$. For $M_3\lesssim 10^{12}\,{\rm GeV}$ one has to worry about a possible contribution to the asymmetry also from $N_3$-decays. A straightforward calculation shows that $\ve_{3\a}=(2/3)\ve_{2\a}$ and therefore an asymmetry is produced at $T\sim M_3$. However, we verified, once more, that the wash-out from $N_2$-inverse processes is always strong enough that the contribution to the final asymmetry from $N_3$-decays is negligible.
In complete analogy with the unflavored case [@geometry], one has that the lower bound $M_1^{\rm min}(K_1)$ is replaced by a lower bound $M_2^{\rm min}(K_2)$ obtained for $\o_{32}>0$ and shown in the right panel of Fig. 3.
One can see that also in this case, within the validity of the condition Eq. (\[full\]), the allowed region is constrained to a small portion falling in the weak wash-out regime. Assuming the very conservative condition of validity for the fully flavored regime, outside the squared and hatched regions, there is no allowed region even in the weak wash-out regime.
One can wonder whether there is some choice of $\O$, beyond the special cases we analyzed, where the final asymmetry is much higher and the lower bound on $M_1$ much more relaxed, especially in the strong wash-out regime. We have checked different intermediate cases and we can exclude such a possibility. Therefore, the lower bound shown in Fig. 1 has to be considered, with good approximation, the lowest bound for any choice of real $\O$.
Another legitimate doubt is whether, going beyond the approximations we made, the lower bound in Fig. 1 can be considerably relaxed. However, the inclusion of non resonant $\D L=2$ or $\D L=1$ scattering does not produce large corrections. Recently the effect of the off-diagonal terms in the $A$ matrix has been considered, but it has been shown that it does not produce any relevant change in the final asymmetry [@abada3].
Relevant corrections, as already pointed out, can come only from a full quantum kinetic treatment, that should describe accurately the transition between the unflavored regime and the fully flavored regime.
The same kind of considerations holds for the $N_2$-dominated scenario, realized for $\O=R_{23}$. As soon as $\O$ deviates from $R_{23}$, the wash-out from $N_1$ inverse processes comes into play suppressing the final asymmetry and at the same time $\ve_{2\t}$ gets also suppressed. Therefore, the lower bound on $M_2$ is necessarily obtained for $\O=R_{23}$ in complete analogy with the unflavored approximation [@geometry].
In conclusion $\d$-leptogenesis in the HL is severely constrained, confirming the conclusions of [@flavorlep] and [@antusch]. In particular, imposing independence of the initial conditions, then not even a marginal allowed region seems to survive. Notice moreover that all plots have been obtained for $s_{13}=0.2$, the current $3\,\s$ upper bound value. Assuming that for values of $M_1$ above the condition Eq. (\[full\]) the unflavored regime is quickly recovered and therefore that the asymmetry production quickly switches off, then a one-order-of-magnitude improvement of the upper bound on $\sin\theta_{13}$ would essentially completely rule out $\d$-leptogenesis in the HL, even the marginally allowed regions falling in the weak wash-out regime.
Therefore, in the next section, we will consider the effect of close heavy neutrino masses in enhancing the $C\!P$ asymmetries and relaxing the lower bounds on $M_1, M_2$ and the related one on $T_{\rm reh}$. In the end of this section we want to mention that in the more general case of real $\O$ with non-vanishing Majorana phases, an upper bound $m_1\lesssim 0.1\,{\rm ev}$ has been obtained in the fully flavored regime [@branco]. This bound clearly applies also to $\d$-leptogenesis, but in this case, considering the results we have obtained and the expected quantum kinetic corrections to the fully flavored regime, the issue is actually whether an allowed region exists at all in the HL, even for $m_1=0$. Therefore, we do not even try to place an upper bound on $m_1$ in the HL. In the next section, we will show that actually for $\d$-leptogenesis an upper bound on $m_1$ holds even in the resonant limit, where the $C\!P$ asymmetries are maximally enhanced.
The degenerate limit
====================
In this section we show that going beyond the HL the lower bound on $M_1$ (or on $M_2$) can be considerably relaxed. Nevertheless, we will see that some interesting constraints on the involved parameters still apply. For simplicity, we can assume a full three-flavor regime holding for $M_1$ (or $M_2$) $\ll 10^{9}\,{\rm GeV}$, when also the muon-Yukawa interactions are faster than inverse decays. Therefore now, when we sum over the flavor index, it has to be meant $\a=e,\mu,\tau$. This assumption simplifies the calculation, since we do not have to describe a transition between the two and the three-flavor regime and because we can completely neglect the effect, envisaged in [@bcst; @nardi2], for which part of the asymmetry produced from $N_2$-decays is not touched by $N_1$-inverse decays. Indeed in a two-flavor regime, even though in the HL we have found that this effect is negligible in all cases we considered because a $\t$-dominance is always realized, in the DL it can become more relevant because the asymmetry is not necessarily produced dominantly in the $\t$-flavor.
In order to go beyond the HL, it is convenient to introduce the quantities \_[ji]{} = -1 . We are interested in the degenerate limit (DL), where at least one $\d_{ji}$ is small enough that both the asymmetry production from decays and the wash-out from inverse processes of the $N_i$’s and of the $N_j$’s can be approximately treated as if they occur at the same temperature, so that they can be simply added up. The DL is a good approximation for $|\delta_{ji}|\lesssim 0.01$ [@beyond]. If $i,j\neq 3$ and $M_3\gg M_2\simeq M_1$ then one has a partial DL and in this case the efficiency factors can be approximated, for thermal initial abundance, as [@beyond] \[k1\] \_[i]{}\^[f]{}\_[j]{}\^[f]{}(K\_[i]{}+K\_[j]{}) . In all considered cases, it will be always verified $K_{i\a}+K_{j\a}\gg 1$, so that the strong wash-out regime always applies and there is no need to consider the case of initial vanishing abundance. Another possibility is to have a partial DL with $i,j\neq 1$ so that $M_1\ll M_2\simeq M_3$. In this case one has to take into account the wash-out from the lightest RH neutrino and therefore \[k2\] \_[i]{}\^[f]{}\_[j]{}\^[f]{}(K\_[i]{}+K\_[j]{})e\^[-[38]{}K\_[1]{}]{} . Finally, in the full DL, one has $M_1\simeq M_2 \simeq M_3$ and \[k3\] \_[1]{}\^[f]{}\_[2]{}\^[f]{}\_[3]{}\^[f]{} (K\_[1]{}+K\_[2]{}+K\_[3]{}) . Let us now calculate the flavored $C\!P$ asymmetries. In the case of real $\O$, implying real $(h^{\dagger}\,h)_{ij}=(h^{\dagger}h)_{ji}$, the general expression Eq.’s (\[veia\]) can be conveniently specialized as \[veiarealO\] \_[i]{}= \_[ji]{}(h\^h)\_[i j]{}[Im]{} . In the DL one has approximately $\xi(x_j/x_i)\simeq 1/(3\,\d_{ji})$ and consequently \_[i]{} \_[ji]{}(h\^h)\_[i j]{} [Im]{}\_[ji]{}\^[-1]{}. We can again express the neutrino Yukawa coupling matrix through the orthogonal representation. This time the presence of the factor $\d_{ji}^{-1}$ does not allow to remove the sum on $j$, as it has been possible in the HL in order to derive the Eq. (\[e1alOm\]). However, considering the same special cases studied in the HL, only one term $j\neq i$ survives and we can write \_[i]{} \_[n,h<l]{}[m\_nm\_[atm]{}]{} Ø\_[ni]{}Ø\_[nj]{} \[Ø\_[hi]{}Ø\_[lj]{}-Ø\_[li]{}Ø\_[hj]{}\] [Im]{}\[U\^\_[h]{}U\_[l]{}\] . The same expression holds for $\ve_{j\alpha}$ simply exchanging the $i$ and $j$ indexes. We can always choose $j>i$, so that $M_j\geq M_i$. In all the particular cases we will consider it is realized $\ve_{k\a}=0$, for $k\neq i,j$, and moreover the following simplifications apply: \_nm\_nØ\_[ni]{}Ø\_[nj]{}=(m\_q-m\_p)Ø\_[ji]{}Ø\_[jj]{} \_[h<l]{}\[Ø\_[hi]{}Ø\_[lj]{}-Ø\_[li]{}Ø\_[hj]{}\] = , with $q>p$. Except for $M_3\gg 10^{14}\,{\rm GeV}$, in the other cases one has $q=j$ and $p=i$. The final asymmetry can then be expressed as N\_[B-L]{}\^[f]{} \_(\_[i]{}+\_[j]{}) \_\^[f]{}(K\_[i]{}+K\_[j]{},K\_[k]{})= [|(M\_i)3\_[ji]{}]{}g(m\_1,Ø\_[ji]{},\_[13]{},) , where g(m\_1,Ø\_[ji]{},\_[13]{},) & & [2K\_[atm]{}(K\_i+K\_j)K\_iK\_j]{} [(m\_q-m\_p)m\_[atm]{}\^2]{} Ø\_[ji]{}\
& & \[g\] \_\_\^[f]{}(K\_[i]{}+K\_[j]{},K\_[k]{}) [[Im]{}\[U\^\_[p]{}U\_[q]{}\]]{} and where $\k_{\a}^{\rm f}(K_{i\a}+K_{j\a},K_{k\a})=\k_{i\a}^{\rm f}=\k_{j\a}^{\rm f}$ is given by one of the three expressions Eq. (\[k1\]), Eq. (\[k2\]) or Eq. (\[k3\]), according to the particular case. It is interesting to notice that in the full DL the expression (\[k3\]) holds and as a consequence of the orthogonality of $\O$, one has K\_[1]{}+K\_[2]{}+K\_[3]{}=\_k[m\_km\_]{}|U\_[k]{}|\^2 . In the degenerate limit, since $U$ is unitary, this quantity tends to $m/m_{\star}$, independently of the flavor, and therefore the sum on the flavors in the Eq. (\[g\]) tends to vanish. This will contribute, as we will see, to place a stringent upper bound on the absolute neutrino mass scale in the full DL.
It is also worthwhile to notice that the sign of $\D$ cannot be predicted from the sign of the observed final asymmetry, since the sign of $g(m_1,\O_{ji},\theta_{13},\d)$ depends on the sign of $\O_{ji}$ that is undetermined. Notice also that ${{\rm Im}[U^{\star}_{\a h}\,U_{\a l}]/\D}$ does not depend on $\D$ but nevertheless there is a dependence of $g(m_1,\O_{ji},\theta_{13},\d)$ on $\d$ and on $\theta_{13}$ coming from the tree level projectors $P^0_{i\a}$ in the sum $K_{i\a}+K_{j\a}$. However, in any case, for $\D\rightarrow 0$ one has $g(m_1,\O_{ji},\theta_{13},\d)\,\D\rightarrow 0$, since the final asymmetry has to vanish when $\sin\theta_{13}$ or $\sin\d$ vanish.
The function $|g(m_1,\O_{ji},\theta_{13},\d)|$ can be maximized over $\O_{ji}$. Indeed for $m_1=0$, since $\k< 1$ and $K_i+K_j \leq K_{\rm atm}$, one has $g(m_1=0,K_i,\theta_{13},\d)<4$. Increasing $m_1$ there is a suppression due to the fact that $K_i\geq m_1/m_{\star}$ and $g_{\rm max}(m_1,\theta_{13},\d)$ decreases monotonically. Therefore, for any $m_1$, there is a lower bound on $M_1$ given by \[lbM1\] M\_1 M\_1\^[min]{}(m\_1,\_[13]{},) [\_[j1]{}||]{} . The $C\!P$ asymmetries, and consequently the final asymmetry, are maximally enhanced in the extreme case of resonant leptogenesis [@pilaftsis; @pilaund2] when the heavy neutrino mass degeneracy is comparable to the decay widths. This implies approximately to have $\d_{ji}^{\rm res}\simeq d\,\bar{\ve}(M_i)/3$ with $d=1\div 10$, that would correspond to have $\ve_1=1/d$ in the unflavored case with maximal phase. This can be taken as a conservative limit that implies, maximizing over $\d$, a lower bound \[lbsint\] \_[13]{} \_[13]{}\^[min]{} = [d\_B\^[CMB]{}N\_\^[rec]{} a\_[sph]{} [max]{}\_\[g\_[max]{}(m\_1,\_[13]{}\^[min]{},))\]]{} . Notice that, within the validity of perturbation theory, one cannot specify which is the exact value of $d$, that means the value of $\d_{ji}$ above which the expression for the $C\!P$ asymmetries given in the Eq. (\[ve1a\]) are valid [@abp] and therefore there is an uncertainty in the calculation of the maximum enhancement of the asymmetries in the resonant regime.
Let us now specialize the expressions for the four special cases we have already analyzed in the HL.
$M_3\gg 10^{14}\,{\rm GeV}$
---------------------------
Remember that in this case one has $(h^{\dagger}h)_{3j}=0$ implying $\ve_{3\a}=0$, a consequence of the fact that the heaviest RH neutrino decouples. Moreover $m_1\ll m_{\rm sol}$, such that terms $\propto m_1$ can be neglected, $m_3\simeq m_{\rm atm}$ and $m_2\simeq m_{\rm sol}$ for normal hierarchy or $m_2\simeq m_{\rm atm}\sqrt{1-m_{\rm sol}^2/m_{\rm atm}^2}$ for inverted hierarchy. Therefore, there is actually no dependence on $m_1$ in $g(m_1,\O_{ji},\theta_{13},\d)$ that we can indicate simply with $g(\O_{ji},\theta_{13},\d)$ and that is given by the expression (\[g\]) with $(i,j)=(1,2)$ and $(p,q)=(2,3)$ or explicitly g(Ø\_[21]{},\_[13]{},) & & [2(K\_1+K\_2)K\_[atm]{}K\_1K\_2]{} (1-[m\_2m\_[atm]{}]{}) Ø\_[21]{}\
\[gM3\] & & \_(K\_[1]{}+K\_[2]{}) [[Im]{}\[U\^\_[2]{}U\_[3]{}\]]{}.
In the case of normal hierarchy $|g(\O_{21},\theta_{13},\d)|$ slightly decreases when $\D$ increases and so the maximum is found for $\D=0$ and in this case the dependence on $\theta_{13}$ and on $\d$ disappears. Replacing $\O_{21}$ with $K_1$, in Fig. 4 we have plotted $|g(K_1,\D=0)|$ for central values of $m_{\rm sol}$ and $m_{\rm atm}$. Including the errors, one finds $g_{\rm max}\simeq 0.160 \pm 0.005$.
The ($3\s$) lower bounds on $M_1$ for normal hierarchy, from the general expression (\[lbM1\]), is then given by \[lbM1Minf\] M\_1 0.9 [10\^[10]{}]{}[GeV]{}[\_[21]{}||]{} . In the case of inverted hierarchy the situation is somehow opposite, since for $\theta_{13}=0$ the electron flavor contribution vanishes in the Eq. (\[gM3\]) and there is an exact cancellation between the $\tau$ and $\mu$ contributions. Consequently, the asymmetry increases for increasing values of $\theta_{13}$ and thus the maximum is found for $\sin\theta_{13}=0.2$ while $\d\simeq\pi/4$. In this case one has that ${\rm max}_{\theta_{13},\d}[g_{\rm max}(m_1=0,\theta_{13},\d)\,\D]
\simeq (9\pm 2)\times 10^{-8}$, that plugged in the Eq. (\[lbM1\]) gives at $3\s$ M\_1610\^[15]{}[GeV]{}[\_[21]{}]{}. It should be remembered that these conditions have been obtained in the three-flavor regime and in the DL and therefore are valid for $M_1\lesssim 10^9\,{\rm GeV}$. This implies $\delta_{21}\lesssim 10^{-1}\,|\D|$ for normal hierarchy and $\delta_{21}\lesssim 10^{-7}$ for inverted hierarchy.
Analogously the general expression (\[lbsint\]) gives, for normal and inverted hierarchy respectively, the following ($3\s$) lower bounds on $\sin\theta_{13}$: \[lbsintMinf\] \_[13]{} 3.310\^[-7]{}d \_[13]{}0.06d .
$\O=R_{13}$
-----------
In this particular case, the next-to-lightest RH neutrino is decoupled from the other two and this implies that $\ve_{2\a}=0$ for any $\a$ and that the $\ve_{1\a}$’s do not depend on $M_2$, in particular they do not get enhanced if $\d_{21}\rightarrow 0$. Therefore, one has necessarily to consider $\delta_{31}\lesssim 0.01$, implying a full DL with all three degenerate RH neutrino masses. The function $g(m_1,\O_{ji},\theta_{13},\d)$ is now obtained from the general expression (\[g\]) for $j=q=3$ and $i=p=1$, or explicitly g(m\_1,Ø\_[31]{},\_[13]{},) & & [2K\_[atm]{}(K\_1+K\_3)K\_1K\_3]{} [(m\_3-m\_1)m\_[atm]{}\^2]{} Ø\_[31]{}\
& & \[gR13\] \_(K\_[1]{}+K\_[2]{}+K\_[3]{}) [[Im]{}\[U\^\_[1]{}U\_[3]{}\]]{} . It is interesting to notice that in this case an $e$-dominance is realized. Moreover, one has that the dependence of $|g(m_1,\O_{31},\theta_{13},\d)|$ on $\theta_{13}$ and $\d$ is slight and the maximum is again for $\D=0$ and for $m_1=0$ and one finds $g_{\rm max}(0)=0.24\pm 0.01$ for normal hierarchy and $g_{\rm max}(0)=(3.1\pm 0.2)\times 10^{-3}$ for inverted hierarchy, so that the general expression (\[lbM1\]) for the lower bound on $M_1$ gives, at $3\s$ for normal and inverted hierarchy, M\_1 5.510\^[9]{}[GeV]{}[\_[31]{}||]{} M\_1 5 10\^[11]{}[GeV]{}[\_[31]{}||]{} , while the general expression (\[lbsint\]) in resonant leptogenesis gives \_[13]{}2.310\^[-7]{}d \_[13]{}1.510\^[-5]{}d . Increasing $m_1$, the value of $g_{\rm max}(m_1)$ decreases and the lower bound on $\sin\theta_{13}$ in resonant leptogenesis becomes more and more restrictive. This dependence is shown in Fig. 5 both for normal (left panel) and inverted (right panel) hierarchy and for $d=1$ (solid line) and $d=10$ (short-dashed line). Very interestingly, imposing the experimental ($3\s$) upper bound $\sin\theta_{13}\lesssim 0.20$, one obtain the upper bound $m_1 \lesssim \, (0.2-0.4)\,{\rm eV}$, depending on the value of $d$.
This upper bound will become more stringent if the experimental upper bound on $\sin\theta_{13}$ will improve, as expected in future experiments in the case of no discovery. The most stringent experimental upper bound that can be hopefully reached in future with neutrino factories is approximately $\sin\theta_{13}< 10^{-3}$ [@lindner]. This asymptotical upper bound is also shown in Fig. 5 and would imply an upper bound $m_1 \lesssim \, (0.05-0.1)\,{\rm eV}$ for normal hierarchy and $m_1 \lesssim \, (0.03-0.08)\,{\rm eV}$ for inverted hierarchy. Therefore, an interesting interplay between two measurable quantities is realized and this makes $\d$-leptogenesis falsifiable independently of the RH neutrino mass spectrum.
In the more conservative case of normal hierarchy, see left panel of Fig. 5, a good approximation is given by the fit m\_10.6 ([\_[13]{}- 2.310\^[-7]{}]{} )\^[0.25]{} [eV]{} . It is interesting that this upper bound holds in the extreme case of resonant leptogenesis and therefore holds for any RH neutrino spectrum. However, we have to verify whether it holds also for a different choice of $\O$.
$\O=R_{12}$
-----------
The situation for $\O=R_{12}$ is quite different compared to the previous cases. Now one has $i=p=1$ and $j=q=2$ and it is possible to have both a partial DL with $10^{14}\,{\rm GeV}\gtrsim M_3 \gg M_2\simeq M_1$ and a full DL. In the first case, the general expression Eq. (\[g\]) becomes g(m\_1,Ø\_[21]{},\_[13]{},) & & [2K\_[atm]{}(K\_1+K\_2)K\_1K\_2]{} [(m\_2-m\_1)m\_[atm]{}\^2]{} Ø\_[21]{}\
& & \[gR21\] \_(K\_[1]{}+K\_[2]{}) [[Im]{}\[U\^\_[1]{}U\_[2]{}\]]{} .
This time the contribution from the electron flavor vanishes. Furthermore, for normal hierarchy, there is an almost perfect cancellation between the $\m$ and the $\t$ contribution. In the left panel of Fig. 6 we show the lower bound on $\sin\theta_{13}$ versus $m_1$ and one can see how, compared to the previous case $\O=R_{13}$, this is much more restrictive. In particular, imposing $\sin\theta_{13}<0.2$, one obtains now a much more stringent upper bound $m_1\lesssim 0.06\,{\rm eV}$. On the other hand, for inverted hierarchy, the cancellation between the $\m$ and the $\tau$ flavor does not occur and one has a lower bound on $\sin\theta_{13}$, for $m_1\ll 0.01\,{\rm eV}$, shown in the right panel of Fig. 6, that is very similar to what has been obtained in the case $\O=R_{13}$. However, now there is no flavor cancellation for increasing values of $m_1$, because $K_{1\a}+K_{2\a}$ does not tend to a common value like $\sum_j\,K_{j\a}$. Therefore, one can see in Fig. 6 that this time the upper bound on $m_1$ is much looser, both compared to normal hierarchy and compared to $\O=R_{13}$.
In the full DL, the flavor cancellation at large $m_1$ occurs and the results are shown in Fig. 7. One can see how now for normal hierarchy the upper bound on $m_1$ is even much more restrictive and, for inverted hierarchy, one has a situation that is similar to the case $\O=R_{13}$.
$\O=R_{23}$
-----------
In this case the lightest RH neutrino decouples and $\ve_{1\a}=0$, independently of $M_1$. Therefore, there is no contribution to the final asymmetry from $N_1$ decays. On the other hand $\ve_{2\a}$ and $\ve_{3\a}$ do not vanish and therefore there is a contribution from the decays of the two heavier RH neutrinos. Still $N_1$ inverse processes have to be taken into account since they contribute to the wash-out. There are two different possibilities.
In a full DL the wash-out from $N_1$ inverse decays just cumulates with the wash-out from the two heavier. Therefore, this time, in the expression Eq. (\[g\]), one has $i=p=2$ and $j=q=3$ and $\k_{\a}^{\rm f}=\k(K_{1\a}+K_{2\a}+K_{3\a})$, explicitly g(m\_1,Ø\_[32]{},\_[13]{},) & & [2K\_[atm]{}(K\_2+K\_3)K\_2K\_3]{} [(m\_3-m\_2)m\_[atm]{}\^2]{} Ø\_[32]{}\
& & \[g32a\] \_(K\_[1]{}+K\_[2]{}+K\_[3]{}) [[Im]{}\[U\^\_[2]{}U\_[3]{}\]]{} . In Fig. 8 we show the dependence of the $\sin\theta_{13}$ lower bound on $m_1$. This time there is a bigger suppression than in the case $\O=R_{13}$, both for normal and for inverted hierarchy.
In the case $M_1\ll M_2\simeq M_3$ one has g(m\_1,Ø\_[32]{},\_[13]{},) & & [2K\_[atm]{}(K\_2+K\_3)K\_2K\_3]{} [(m\_3-m\_2)m\_[atm]{}\^2]{} Ø\_[32]{}\
& & \[g32b\] \_(K\_[2]{}+K\_[3]{}) e\^[-[38]{}K\_[1]{}]{} [[Im]{}\[U\^\_[2]{}U\_[3]{}\]]{} . The dependence of the lower bound on $\sin\theta_{13}$ on $m_1$ is shown in Fig. 9 for normal hierarchy. In this case the upper bound on $m_1$ is now slightly less stringent than in the previous cases.
For inverted hierarchy the final asymmetry production is so suppressed that there is no allowed region.
We can conclude this section noticing that these results show that $\d$-leptogenesis can be falsified. In the case of normal hierarchy, the current upper bound $\sin\theta_{13}\lesssim 0.2$ implies $m_1\lesssim 0.1\,{\rm eV}$, while, in future, a potential upper bound $\sin\theta_{13}\lesssim 10^{-3}$ would imply $m_1\lesssim {\cal O}(0.01\,{\rm eV})$, with a more precise determination depending on the possibility of improving the current estimation of the parameter $d$ in resonant leptogenesis.
Lights and shadows of $\d$-leptogenesis
=======================================
The most attractive feature of $\d$-leptogenesis is that a non-vanishing Dirac phase, the only see-saw phase that we can realistically hope to discover in future, acts as the only source of $C\!P$ violation responsible for the matter-antimatter asymmetry of the Universe. We think that this feature, despite of the objections that we are going to discuss, provides a strong motivation for $\d$-leptogenesis.
As we have seen, successful $\d$-leptogenesis implies stringent conditions on the RH neutrino masses, something quite interesting since they escape conventional experimental information. In particular we have seen that, except for a marginal allowed region in the weak wash-out regime, the HL is non-viable. We also observed that a definite conclusion on the existence of such a marginal allowed region, requires a full quantum kinetic treatment but in any case corrections are expected to shrink this already quite restricted allowed region.
Therefore, $\d$-leptogenesis motivates models with degenerate RH neutrino masses, with the most extreme limit represented by resonant leptogenesis. Even in this extreme limit however, imposing successful $\d$-leptogenesis, interesting conditions follow on quantities accessible in low-energy neutrino experiment: $\sin\theta_{13}$, the absolute neutrino mass scale, normal or inverted scheme, the Dirac phase itself. Therefore, an interesting aspect of $\d$-leptogenesis is that it is falsifiable independently of the heavy neutrino mass spectrum.
There are some objections to $\d$-leptogenesis. There is no clear theoretical motivation for $\d$-leptogenesis, more generally to choose a real orthogonal $\O$ matrix. Apparently, sequential dominated models [@king] could represent an interesting theoretical framework. Indeed in [@geometry] it was shown that these models correspond to have an $\O$ matrix that slightly deviates from the unit matrix or from all the other five that can be obtained from the unit matrix exchanging rows or columns. However, it has been noticed [@window; @geometry] that in the limit ${\rm Im}[\O]\rightarrow 0$ total $C\!P$ asymmetries $\ve_i$ do not necessarily vanish. Therefore, in this limit and taking vanishing Majorana phases, one does not necessarily obtain $\d$-leptogenesis. Writing $\O^2_{ij}=|\O^2_{ij}|\,{\rm exp}[i\,\varphi_{ij}]$, the correct condition to enforce $\ve_{i}\rightarrow 0$ is to take the limit $\varphi_{ij}\rightarrow 0$. This is a more demanding limit than ${\rm Im}[\O]\rightarrow 0$ and it is currently not motivated by generic sequential dominated models. This limit is not motivated either by radiative leptogenesis [@radiative] within the context of the minimal flavor violation principle [@MFV], as recently considered in [@burasbranco; @selma]. Therefore, there is no theoretical justification for $\d$-leptogenesis at the moment.
Another possible objection to $\d$-leptogenesis is that it cannot be distinguished from the general scenario, where all phases are present, even if a non-vanishing Dirac phase is discovered. Indeed a Dirac phase would give in this case a subdominant contribution. This objection is however related also to the first one. Indeed, since a theoretical model motivating $\d$-leptogenesis is required anyway, one can hope to find some specific prediction that makes the model testable and $\d$-leptogenesis together with it. Dirac phase leptogenesis would then become distinguishable from the general scenario, though in an indirect way.
This last objection can be also considered within a more particular case where $\O$ is still real but Majorana phases are present together with the Dirac phase. It has been noticed that the contribution to the final asymmetry from Majorana phases is in general dominant compared to that one coming from the Dirac phase [@flavorlep]. In the right panel of Fig. 1 we have compared the result on the $M_1$ lower bound for $\O=R_{13}$ obtained in $\d$-leptogenesis with the result when ${\rm Im}[\O_{ij}]=\d=0$ but $\Phi_1=-\pi/2$ (dotted lines). One can see that in the second case the lower bound is $\sim 2\div 3$ times more relaxed. This result can be easily understood analytically [@pascoli] and actually it can be also observed that there can be exact cancellations between the contribution to the final asymmetry from the Majorana phases and from the Dirac phase.
The presence of cancellations can be somehow regarded as a limit to $\d$-leptogenesis main motivation, since even though a Dirac phase will be discovered, it is not guaranteed that the observed asymmetry can be explained. This objection is however quite weak since it would be quite strange if Nature disposed a sufficient source of $C\!P$ violation but set up a second source that exactly cancels with the first one while the observed asymmetry is, in the end, explained still by a third one, for example the phases in $\O$. On the other hand, we can say that it would be certainly positive for $\d$-leptogenesis if in future experimental upper bounds on the Majorana phases are placed, for example from $\b\b0\n$ decay, thus constraining the contribution to the final asymmetry from Majorana phases [@pascoli]. This can be also regarded as a further prediction coming from $\d$-leptogenesis.
In conclusion, we have studied in detail a specific scenario of leptogenesis that is interesting especially in view of the many next planned experiments aiming at a discovery of $C\!P$ violation in neutrino mixing. Despite some important remarks and objections, we think that $\d$-leptogenesis realizes a very interesting link between a long-standing cosmological puzzle and $C\!P$ violation in neutrino oscillations, one of the most relevant experimental topics in high-energy physics during next years.
**Acknowledgments**\
It is a pleasure to thank S. Petcov for discussions during the Neutrino Oscillation Workshop 2006, held in Conca della Specchiulla (Italy). We also wish to thank G. Raffelt for interesting discussions and comments. This work was supported, in part, under the Marie Curie project “Leptogenesis, Seesaw and GUTs,” contract No. MEIF-CT-2006-022950.
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[^1]: More exactly, in [@cmb], it was found $4\times
10^8\,{\rm GeV}$. Here we are using a slightly higher value that is obtained, as we will see, when the reduced experimental error on the baryon asymmetry and on the atmospheric neutrino mass scale is taken into account.
[^2]: Flavor effects were first considered in [@bcst], and then in [@endo] in the particular case of 2 RH neutrinos. However, in these papers, it was found that flavor effects can only induce small corrections to the final asymmetry compared to the unflavored case. The possibility for a large enhancement was first found in [@pilaund] in the case of resonant leptogenesis and more generally in [@nardi; @abada; @abada2], where the typical factor 2–3 enhancement of the final asymmetry induced by flavor effects was also first understood. As for the potential role of low energy phases in providing an additional source of $C\!P$ violation relevant for leptogenesis, it was first discussed in [@nardi].
[^3]: Notice that more rigorously this condition should be written replacing the simple sum of the inverse decays rates with a sum weighted with projectors taking into account that the lepton produced by the decay of a RH neutrino $N_i$ is different by that lepton produced by the decay of a RH neutrino $N_{j\neq i}$ and therefore is not in general fully absorbed by the $N_{j\neq i}$ inverse decay [@bcst].
[^4]: Notice that $\overline{M_1}$ gives the lower bound on $M_1$ in the unflavored case for initial thermal abundance and in the limit $K_1\rightarrow 0$. Because of the improved determination of $\eta_B^{\rm CMB}/m_{\rm atm}$ from the 3 years WMAP data [@WMAP3] and from new data from neutrino oscillation experiments, in particular from the MINOS experiment, the error on $\overline{M}_1$ is halved compared to the previous estimation in [@annals].
|
---
abstract: 'We study a model of radiating gases that describes the interaction of an inviscid gas with photons. We show the existence of smooth traveling waves called ’shock profiles’, when the strength of the shock is small. Moreover, we prove that the regularity of the traveling wave increases when the strength of the shock tends to zero.'
author:
- |
Chunjin [Lin]{}$^\dag$, Jean-François [Coulombel]{}$^{\dag \ddag}$, Thierry [Goudon]{}$^{\dag \ddag}$\
\
[$\dag$ Team SIMPAF–INRIA Futurs & Université Lille 1, Laboratoire Paul Painlevé, UMR CNRS 8524]{}\
[Cité scientifique, 59655 VILLENEUVE D’ASCQ Cedex, France]{}\
[$\ddag$ CNRS]{}\
[E-mails: [[email protected]]{}, [[email protected]]{}]{},\
[[email protected]]{}
bibliography:
- 'lcg.bib'
title: Shock Profiles for Non Equilibrium Radiating Gases
---
Introduction and main results
=============================
We are interested in a system of PDEs describing astrophysical flows, where a gas interacts with radiation through energy exchanges. Similar questions arise in the modeling of reentry problems, or high temperature combustion phenomena. The gas is described by its density $\rho >0$, its bulk velocity $u \in {{\mathbb R}}$, and its specific total energy $E=e+u^2/2$, where $e$ stands for the specific internal energy. (Our analysis is restricted to a one-dimensional framework, but this is not a loss of generality, as shown below.) We consider a situation where the gas is not in thermodynamical equilibrium with the radiations, which are thus described by their own energy $n$. The evolution of the gas flow is governed by the system: $$\label{eulerevol}
\begin{cases}
\partial_t \rho +\partial_x (\rho \, u)=0 \, ,& \\
\partial_t (\rho \, u) +\partial_x (\rho \, u^2+P)=0\, ,& \\
\partial_t (\rho \, E) +\partial_x (\rho \, E \, u+P \, u)=n-\theta^4 \, ,&
\end{cases}$$ where the right-hand side in the last equation accounts for energy exchanges with the radiations, $P$ being the pressure of the gas, and $\theta$ its temperature. Throughout the paper, we always assume that the gas obeys the perfect gas pressure law: $$\label{pressure}
P=R \, \rho \, \theta =(\gamma-1) \, \rho \, e \, ,$$ where $R$ is the perfect gas constant, and $\gamma>1$ is the ratio of the specific heats at constant pressure, and volume. This assumption yields many algebraic simplifications, but we believe that our results still hold for a general pressure law satisfying the usual requirements of thermodynamics. System is completed by considering that radiations are described by a stationary diffusion regime that reads: $$\label{diffn}
-\partial_{xx} n =\theta^4 -n \, .$$ We detail in Appendix \[model\] how the system , can be formally derived by asymptotics arguments, starting from a more complete system involving a kinetic equation for the specific intensity of radiation.
As a matter of fact, the operator $(1-\partial_{xx})$ can be explicitly inverted, and can be recast as a convolution: $$\label{diffn2}
n(t,x) =\dfrac{1}{2} \, \displaystyle \int_{{{\mathbb R}}}
{\rm e}^{-|x-y|} \, \theta (t,y)^4 \, dy \, .$$ Let us introduce the quantity: $$\label{defq}
q(t,x) :=-\partial_x n \, (t,x)=\dfrac{1}{2} \displaystyle \int_{{{\mathbb R}}}
{\rm e}^{-|x-y|} \, {\rm sgn}(x-y) \, \theta (t,y)^4 \, dy \, ,$$ where sgn is the sign function: $${\rm sgn} (x)=\begin{cases}
1 &\text{if $x>0$,} \\
0 &\text{if $x=0$,} \\
-1 &\text{if $x<0$.}
\end{cases}$$ The quantity $q$ can be interpreted as the radiative heat flux. Then, we can rewrite , as follows: $$\label{Euler2}
\begin{cases}
\partial_t \rho +\partial_x (\rho \, u) =0\, ,& \\
\partial_t (\rho \, u) +\partial_x (\rho \, u^2+P)=0 \, ,& \\
\partial_t (\rho \, E) +\partial_x (\rho \, E \, u +P \, u +q) =0 \, ,&
\end{cases}$$ with $q$ given by . Recall that $E=e+u^2/2$, and $P$ is given by .
In this paper, we address the question of the influence of the energy exchanges on the structure of shock waves. More precisely, let us consider given states at infinity $(\rho_\pm, u_\pm,e_\pm)$, and let us asume that: $$\label{shock}
(\rho,u,e)(t,x) =\begin{cases}
(\rho_-,u_-,e_-) &\text{if $x<\sigma \, t$,} \\
(\rho_+,u_+,e_+) &\text{if $x>\sigma \, t$,}
\end{cases}$$ is a shock wave, with speed $\sigma$, solution to the standard Euler equations (that is, system with $q \equiv 0$). We refer to [@lax; @serrelivre; @smoller] for a detailed study of shock waves for the Euler equations. The question we ask is the following: does there exist a traveling wave $(\rho,u,e)(x-\sigma t)$ solution to , with $q$ given by , that satisfies the asymptotic conditions: $$\label{asym-cond}
\lim_{\xi \rightarrow \pm\infty} (\rho,u,e)(\xi) =(\rho_\pm,u_\pm,e_\pm) \, .$$ In other words, we are concerned with the existence of a shock profile, and a natural expectation (at least for shocks of small amplitude) is that the step shock is smoothed into a continuous profile, due to the dissipation introduced by . The analogous problem for the compressible Navier-Stokes system has been treated a long time ago, see [@gilbarg], without any smallness assumption on the shock wave. Concerning radiative transfer, a formal analysis of shock profiles has been performed in [@HB], together with rough numerical simulations. (We refer also to [@ZR; @MM] for the physical background.) The main purpose of this work is to make the analysis of [@HB] rigorous. Since we are only concerned in this paper with the existence of shock profiles, and not with their stability, the problem is purely one-dimensional (due to the Galilean invariance of the Euler equations). This is why we have directly restricted to the one-dimensional case. However, the formal derivation of Appendix \[model\] is made in several space dimensions.
Before stating our main results, let us mention that a simplified version of , has been introduced, and studied in [@SchoTad] and later in [@kawanishi; @kawanishi2]. This ’baby-model’ consists in a Burgers type equation: $$\partial_t u + \partial_x \Big( \dfrac{u^2}{2} \Big)=-\partial_x q \, ,$$ coupled to the diffusion equation: $$-\partial_{xx} q+q=-\partial_x u \, .$$ These two equations can be seen as a scalar version of , since they can be recast as: $$\label{Kburgers}
\partial_t u +\partial_x \Big( \dfrac{u^2}{2} \Big) =Ku-u \, ,$$ where $K$ is the integral operator already arising in : $$Ku (t,x)=\dfrac{1}{2} \displaystyle \int_{{{\mathbb R}}} {\rm e}^{-|x-y|} \, u(t,y) \, dy \, .$$ The thorough study of has motivated a lot of works; we mention in particular [@nishi; @marcati; @tadmor; @serre]. Clearly can be seen as a prototype for discussing , ; nevertheless, replacing , and by has two important consequences: the equation becomes scalar, and the ’diffusion’ $K-1$ applies to the unique unknown (while in , the ’diffusion’ appears through the radiative heat flux $q$ only in the third equation). Our work is a first attempt to extend the known results for to the more physical model , .
Let us now state our main results. The first result deals with the existence of smooth shock profiles when the strength of the shock is small:
\[exis\] Let $\gamma$ satisfy $$1< \gamma <\dfrac{\sqrt{7}+1}{\sqrt{7}-1} \simeq 2.215 \, ,$$ and let $(\rho_-,u_-,e_-)$ be fixed. Then there exists a positive constant $\delta$ (that depends on $(\rho_-,u_-,e_-)$, and $\gamma$) such that, for all state $(\rho_+,u_+,e_+)$ verifying:
- $\|(\rho_+,u_+,e_+)-(\rho_-,u_-,e_-)\| \le \delta$,
- the function is a shock wave, with speed $\sigma$, for the (standard) Euler equations,
then there exists a $C^2$ traveling wave $(\rho,u,e)(x-\sigma t)$ solution to , , .
As in the study of the ’baby-model’ , the existence of a smooth shock profile is linked to a smallness assumption on the shock strength, see [@kawanishi]. Here the smallness parameter $\delta$ may depend on the state $(\rho_-,u_-,e_-)$, while for , the smallness parameter is uniform (and even explicit!).
The restriction on the adiabatic constant $\gamma$ might be unnecessary, but it simplifies the proof, and it covers the main physical cases $1<\gamma \le 2$.
Our second result is also in the spirit of [@kawanishi], and deals with the smoothness of the shock profile constructed in the previous Theorem:
\[smooth\] Let $\gamma$ satisfy $$1< \gamma < \dfrac{\sqrt{7}+1}{\sqrt{7}-1} \simeq 2.215 \, ,$$ and let $(\rho_-,u_-,e_-)$ be fixed. Then there exists a decreasing sequence of positive numbers $(\delta_n)_{n \in {{\mathbb N}}}$ (the sequence depends on $(\rho_-,u_-,e_-)$, and $\gamma$) such that, for all $n\in {{\mathbb N}}$, and for all state $(\rho_+,u_+,e_+)$ verifying:
- $\|(\rho_+,u_+,e_+)-(\rho_-,u_-,e_-)\| \le \delta_n$,
- the function is a shock wave, with speed $\sigma$, for the (standard) Euler equations,
then there exists a $C^{n+2}$ traveling wave $(\rho,u,e)(x-\sigma t)$ solution to , , .
To a large extent, our analysis follows the arguments of [@HB], [@SchoTad] and [@kawanishi]. The proof of Theorem \[exis\] is presented in Section \[Proof\], while Section \[moresmooth\] is devoted to the proof of Theorem \[smooth\]. The investigation of strong shocks, as well as stability issues will be addressed in a forthcoming work.
Existence of smooth shock profiles {#Proof}
==================================
In this section, we prove Theorem \[exis\]. We first recall some basic facts on shock waves for the Euler equations. Then, we make some transformations on the traveling wave equation. Eventually, we prove Theorem \[exis\] by using an auxiliary system of Ordinary Differential Equations, that is introduced and studied in the last paragraph of this section.
Shock wave solutions to the Euler equations
-------------------------------------------
In this paragraph, we recall some basic facts about the (entropic) shock wave solutions to the Euler equations: $$\begin{cases}
\partial_t \rho +\partial_x (\rho \, u) =0 \, ,\\
\partial_t (\rho \, u) +\partial_x (\rho \, u^2 +P) =0 \, ,\\
\partial_t (\rho \, E) +\partial_x (\rho \, E \, u+ P \, u) = 0 \, ,
\end{cases}$$ where $P$, and $E$ are given as in the introduction. We refer to [@lax; @serrelivre; @smoller] for all the details, and omit the calculations. In all what follows, we only consider shock waves that satisfy Lax shock inequalities. We shall thus speak of $1$-shock waves, or $3$-shock waves.
We consider a fixed ’left’ state $(\rho_-,u_-,e_-)$. Then the ’right’ states $(\rho_+,u_+,e_+)$ such that $(\rho_\pm,u_\pm,e_\pm)$ define a $1$-shock wave, with speed $\sigma$, is a half-curve initiating at $(\rho_-,u_-,e_-)$. Introducing the notation $v_\pm=u_\pm -\sigma$, the Rankine-Hugoniot jump conditions can be rewritten as: $$\label{defjC1C2}
\begin{cases}
\rho_+ \, v_+ =\rho_- \, v_- =:j \, ,\\
\rho_+ \, v_+^2 +P_+ =\rho_- \, v_-^2 +P_- =:j \, C_1 \, ,\\
\rho_+ \, v_+ \big( e_+ +\dfrac{v_+^2}{2} \big) +P_+ \, v_+
=\rho_- \, v_- \big( e_- +\dfrac{v_-^2}{2} \big) +P_- \, v_- =:j \, C_2 \, .
\end{cases}$$ Observe that $v_-$ does not only depend on the ’left’ state $(\rho_-,u_-,e_-)$, but also on $(\rho_+,u_+,e_+)$, because $v_-$ is defined with the help of the shock speed $\sigma$. Consequently, the constants $j$, $C_1$, and $C_2$ depend on both $(\rho_-,u_-,e_-)$, and $(\rho_+,u_+,e_+)$.
For $1$-shocks, that is when the inequalities: $$u_+ -c_+ < \sigma <u_+ \, ,\quad \sigma < u_- -c_- \, ,$$ are satisfied ($c$ denotes the sound speed), all quantities $j$, $C_1$, and $C_2$ are positive. Moreover, when the strength of the shock tends to zero, that is when $(\rho_+,u_+,e_+)$ tends to $(\rho_-,u_-,e_-)$, one has $$\label{asymp}
\begin{pmatrix}
\sigma \\
j \\
C_1 \\
C_2 \end{pmatrix} \longrightarrow \begin{pmatrix}
u_- - c_- \\
\rho_- \, c_- \\
c_- +(\gamma-1) e_-/c_- \\
\gamma \, e_- +c_-^2/2 \end{pmatrix} \, .$$ Consequently, when the strength of the shock is small, all quantities $j$, $C_1$, $C_2$ are bounded away from zero. Recall also that $1$-shocks are compressive, in the sense that $\rho_+ >\rho_-$. This inequality immediately implies that $0<v_+ <v_-$. Eventually, the strength of the shock tends to zero if, and only if $u_+$ tends to $u_-$ (in that case, we also have $\rho_+ \rightarrow \rho_-$, and $e_+ \rightarrow e_-$ because of the Rankine-Hugoniot jump conditions).
In all what follows, we limit our discussion to the case of $1$-shocks for simplicity, but the extension to $3$-shocks is immediate.
Reduction of the traveling wave equation
----------------------------------------
In this paragraph, we derive, and transform the equation satisfied by traveling wave solutions to , . A traveling wave solution to , with speed $\sigma$ is a solution $(\rho,u,e)(x-\sigma t)$. For such solutions, the radiative heat flux $q$ also depends on the sole variable $x-\sigma t$: $$q(x-\sigma t)=\dfrac{1}{2} \displaystyle \int_{{{\mathbb R}}}
{\rm e}^{-|x-\sigma t-y|} \, {\rm sgn}(x-\sigma t-y) \, \theta (y)^4 \, dy \, ,$$ and reads: $$\begin{cases}
(\rho\, (u-\sigma))' =0 \, ,\\
(\rho \, u \, (u-\sigma)+(\gamma-1) \, \rho \, e)' =0 \, ,\\
(\rho \, (e+\frac{u^2}{2}) \, (u-\sigma)+(\gamma-1) \, \rho \, e \, u+q)' =0 \, ,
\end{cases}$$ where $'$ denotes differentiation with respect to the variable $\xi=x-\sigma t$. Introducing the new unknown $v=u-\sigma$, the above system is easily seen to be equivalent to: $$\label{Euler-xi}
\begin{cases}
(\rho \, v)'=0 \, ,\\
(\rho \, v^2+(\gamma-1)\, \rho \, e)' =0 \, ,\\
(\rho \, v \, (e+\frac{v^2}{2})+(\gamma-1) \, \rho \, v \, e+q)' =0 \, .
\end{cases}$$ Since we are looking for a shock profile, the traveling wave solution should also satisfy: $$\label{asym-cond2}
\lim_{\xi\to \pm\infty} (\rho,v,e) =(\rho_{\pm},v_{\pm},e_{\pm}) \, ,$$ where $v_{\pm}=u_{\pm}-\sigma$, and $(\rho_\pm,u_\pm,e_\pm)$ defines a $1$-shock wave with speed $\sigma$ for the Euler equations. Notice that the quantity $a=|u_+-u_-|/2$, that measures the strength of the shock, is invariant with respect to our change of velocity, that is $a=|u_+-u_-|/2=|v_+-v_-|/2$. Recall also that for $1$-shocks, there holds $v_->v_+>0$.
Observing that we have: $$q(\xi)=\dfrac{1}{2} \displaystyle \int_0^{+\infty} {\rm e}^{-y} \,
\big( \theta (\xi-y)^4 -\theta (\xi+y)^4 \big) \, dy \, ,$$ we conclude that $q$ tends to zero at $\pm \infty$ by Lebesgue’s Theorem (because $\theta$ is necessarily bounded since it has finite limits at $\pm \infty$). We can thus integrate the system - once, and reads equivalently: $$\label{system}
\begin{cases}
(\rho \, v)(\xi)=j \, ,\\
(\rho \, v^2+(\gamma-1) \, \rho \, e)(\xi)=j \, C_1 \, ,\\
(\rho \, v \, (e+\frac{v^2}{2})+(\gamma-1) \, \rho \, v \, e+q)(\xi)=j \, C_2 \, ,
\end{cases}$$ where the constants $j,$ $C_1$ and $C_2$ are given by the Rankine-Hugoniot conditions . For small shocks, the positive constants $j$, $C_1$, $C_2$ have the asymptotic behavior .
From the two first equations of , we derive the relations: $$\rho (\xi)=\dfrac{j}{v(\xi)} \, ,\quad
e(\xi)=\dfrac{(C_1-v(\xi)) \, v(\xi)}{\gamma-1} \, .$$ The third equation of thus reduces to: $$\label{v-q}
v(\xi)^2 -\dfrac{2 \, \gamma \, C_1}{\gamma+1} \, v(\xi)
+\dfrac{2 \, (\gamma-1) \, C_2}{\gamma+1}
=\dfrac{2 \, (\gamma-1)}{j \, (\gamma+1)} \, q(\xi) \, .$$ Using the equation of state , as well as the second equation of , we get: $$\theta (\xi)=\dfrac{(\gamma-1) \, e(\xi)}{R} =\dfrac{(C_1-v(\xi)) \, v(\xi)}{R} \, .$$ Consequently, can be recast as an integral equation with a single unknown function $v$: $$\label{eqv}
v(\xi)^2 -\dfrac{2 \, \gamma \, C_1}{\gamma+1} \, v(\xi)
+\dfrac{2 \, (\gamma-1) \, C_2}{\gamma+1}
=\dfrac{(\gamma-1)}{j \, (\gamma+1) \, R^4} \int_{{\mathbb R}}{\rm e}^{-|\xi-y|} \,
{\rm sgn} (\xi-y) \, v(y)^4 (C_1-v(y))^4 \, dy \, .$$ We are searching for a solution $v$ to , that satisfies the asymptotic conditions $v(\xi)\rightarrow v_\pm$, as $\xi \rightarrow \pm \infty$.
If we find a $C^2$ solution $v$ to that does not vanish, and that satisfies $v(\xi) \to v_{\pm}$ as $\xi \to \pm \infty$, then we obtain a $C^2$ shock profile $(\rho,u,e)$ by simply setting: $$\rho(\xi)=\dfrac{j}{v(\xi)} \, ,\quad u(\xi)=v(\xi)+\sigma \, ,\quad
e(\xi)=\dfrac{(C_1-v(\xi))\, v(\xi)}{\gamma-1} \, .$$ In particular, if $v(\xi) \in [v_+,v_-]$ for all $\xi$, then $v$ does not vanish.
Since the heat flux $q$ vanishes at $\pm \infty$, can be also rewritten as: $$(v(\xi)-v_-)(v(\xi)-v_+) =
\dfrac{(\gamma-1)}{j \, (\gamma+1) \, R^4} \int_{{\mathbb R}}{\rm e}^{-|\xi-y|} \,
{\rm sgn} (\xi-y) \, v(y)^4 \, (C_1-v(y))^4 \, dy \, .$$
We are going to rewrite as a second order differential equation, that will be easier to study than the integral equation . Indeed, assuming that $v$ is a $C^2$ function of $\xi$, and differentiating twice with respect to $\xi$, we get (see [@HB] for the details of the computations): $$\label{eqv1}
(v-\dfrac{\gamma \, C_1}{\gamma+1}) \, v''+(v')^2
-\dfrac{4 \, (\gamma-1)}{j \, (\gamma+1) \, R^4} \, (C_1-v)^3 \, v^3
\, (C_1-2v) \, v' -\dfrac{1}{2} \, (v-v_-) \, (v-v_+) =0 \, .$$ Conversely, if $v$ is a $C^2$ solution to that satisfies $v(\xi) \to v_{\pm}$ as $\xi \to \pm \infty$, then $v$ is also a solution to . If in addition $v$ takes its values in the interval $[v_+,v_-]$, then we can construct a $C^2$ shock profile, and thus prove Theorem \[exis\].
The differential equation can be simplified by introducing the new unknown function $\hat{v}=v-(v_- +v_+)/2$, and by rewriting the second order differential equation as a first order system: $$\label{ode}
\left\{
\begin{array}{lll}
\hat{v}' &=& w \, ,\\
\hat{v} \, w' &=& -w^2-f(\hat{v}) \, w+\dfrac{\hat{v}^2-a^2}{2} \, ,
\end{array}
\right.$$ where $f$ is the following polynomial function: $$\label{fun f}
f(\hat{v})=\dfrac{4 \, (\gamma-1)}{j \, R^4 \, (\gamma+1)} \,
\left( \dfrac{C_1}{\gamma+1}-\hat{v} \right)^3 \,
\left( \hat{v}+\dfrac{\gamma\, C_1}{\gamma+1} \right)^3 \,
\left( 2\hat{v}+\dfrac{(\gamma-1) \, C_1}{\gamma+1} \right) \, .$$ We recall that $a=|v_- -v_+|/2$, and that $a$ measures the strength of the shock.
\[f0\] The asymptotic behavior of $j$, $C_1$, and $C_2$ shows that when the strength of the shock tends to zero ($a \to 0^+$), the limit of $f(0)$ is given by: $$f(0) \rightarrow \dfrac{4 \, \gamma^3 \, (\gamma -1)^2}{R^4 \, (\gamma +1)^8}
\, \dfrac{(c_- +(\gamma-1) \, \frac{e_-}{c_-})^7}{\rho_- \, c_-} >0 \, .$$
Since $v_+ <v_-$ for a $1$-shock, we are searching for a solution to that is defined on all ${{\mathbb R}}$, and that satisfies: $$\label{asy-ode}
\lim_{\xi \to -\infty} (\hat{v},w)(\xi) = (a,0) \, ,\quad
\lim_{\xi \to +\infty} (\hat{v},w)(\xi) = (-a,0) \, .$$ To prove Theorem \[exis\], we are thus reduced to showing the existence of a heteroclinic orbit for that connects the stationary solutions $(\pm a,0)$. Due to the previous transformation $\hat{v}=v-(v_-+v_+)/2$, if $\hat{v}$ takes its values in $[-a,a]$, then $v=\hat{v}+(v_-+v_+)/2$ will take its values in the interval $[v_+,v_-]$, and therefore will not vanish.
The system is ’singular’ at $\hat{v}=0$. Nevertheless, we are searching for a smooth solution connecting $(\pm a,0)$, so that $\hat{v}$ vanishes in at least one point. Because $w'=\hat{v}''$ should also have a limit at this point, a $C^2$ shock profile can exist only if the equation: $$w^2 +f(0) \, w +\dfrac{a^2}{2} =0 \, ,$$ has real roots. The corresponding discriminant condition turns out to be much less simple than the one found in [@kawanishi] for the ’baby model’ . (In particular, $f(0)$ depends on the shock through the constants $j$, and $C_1$). This is a first ’nonexplicit’ restriction on the shock strength to derive the existence of a smooth shock profile.
Due to the singular nature of the system at $\hat{v}=0$, it is more convenient to work on an auxiliary system of ODEs, where the singularity has been eliminated (at least formally) thanks to a change of variables. This procedure was already used in [@kawanishi]. In the next paragraph, we shall introduce this auxiliary system, and complete the proof of Theorem \[exis\].
Existence of a heteroclinic orbit
---------------------------------
We begin with a result on an auxiliary system of ODEs, where the singularity at $\hat{v}=0$ has been eliminated:
\[lem\] Assume that $\gamma$ satisfies $1<\gamma<(\sqrt{7}+1)/(\sqrt{7}-1)$, and consider the following system of ODEs: $$\label{ode-ref}
\left\{
\begin{array}{rcl}
V' & = & V\, W \, ,\\
W' & = & -W^2-f(V) \, W+\dfrac{(V^2-a^2)}{2} \, .
\end{array}
\right.$$ There exists a positive constant $a_0$, that depends only on $(\rho_-,u_-,e_-)$, and $\gamma$ such that if the shock strength $a$ satisfies $a\in (0,a_0]$, the following properties hold:
- $f(0)^2-2a^2>0$, and we define $w_0 := \big(
-f(0)+\sqrt{f(0)^2-2a^2} \big) /2 <0$.
- There exists a solution $(V_\flat,W_\flat)$ to that is defined on all ${{\mathbb R}}$, and that satisfies $$\lim_{\eta \rightarrow -\infty} (V_\flat,W_\flat)(\eta) =(a,0) \, ,\quad
\lim_{\eta \rightarrow +\infty} (V_\flat,W_\flat)(\eta) =(0,w_0) \, .$$ Furthermore, $V_\flat$ is decreasing, and the convergence of $V_\flat$ to $0$ as $\eta \rightarrow +\infty$ is exponential.
- There exists a solution $(V_\sharp,W_\sharp)$ to that is defined on all ${{\mathbb R}}$, and that satisfies $$\lim_{\eta \rightarrow -\infty} (V_\sharp,W_\sharp)(\eta) =(-a,0) \, ,\quad
\lim_{\eta \rightarrow +\infty} (V_\sharp,W_\sharp)(\eta) =(0,w_0) \, .$$ Furthermore, $V_\sharp$ is increasing, and the convergence of $V_\sharp$ to $0$ as $\eta \rightarrow +\infty$ is exponential.
Assuming that the result of Proposition \[lem\] holds, the existence of a heteroclinic orbit for connecting $(\pm a,0)$ can be derived by following the analysis of [@SchoTad; @kawanishi]. We briefly recall the method. Using the solution $(V_\flat,W_\flat)$, we introduce the change of variable: $$\Xi_\flat (\eta) = -\int_{\eta}^{+\infty} V_\flat (\zeta) \, d\zeta \, .$$ Since $V_\flat$ tends to $0$ exponentially as $\eta$ tends to $+\infty$, $\Xi_\flat$ is well-defined, and it is an increasing $C^\infty$ diffeomorphism from ${{\mathbb R}}$ to $(-\infty,0)$. Then $(\hat{v}_\flat,w_\flat) :=(V_\flat,W_\flat)
\circ \Xi_\flat^{-1}$ is a $C^\infty$ solution to on the interval $(-\infty,0)$, that satisfies: $$\lim_{\xi \rightarrow -\infty} (\hat{v}_\flat,w_\flat)(\xi) =(a,0) \, ,\quad
\lim_{\xi \rightarrow 0^-} (\hat{v}_\flat,w_\flat)(\xi) =(0,w_0) \, .$$ Similarly, with the help of the solution $(V_\sharp,W_\sharp)$ we can construct a $C^\infty$, decreasing diffeomorphism $\Xi_\sharp$ from ${{\mathbb R}}$ to $(0,+\infty)$, and a $C^\infty$ solution $(\hat{v}_\sharp,w_\sharp)$ to on the interval $(0,+\infty)$. This solution $(\hat{v}_\sharp,w_\sharp)$ connects $(0,w_0)$ and $(-a,0)$, as $\xi$ varies from $0^+$ to $+\infty$. Let us now ’glue’ the solutions $(\hat{v}_\flat,w_\flat)$, and $(\hat{v}_\sharp,w_\sharp)$, by defining: $$\label{defsolution}
(\hat{v},w) (\xi) :=\begin{cases}
(\hat{v}_\flat,w_\flat) (\xi) &\text{if $\xi<0$,}\\
(\hat{v}_\sharp,w_\sharp) (\xi) &\text{if $\xi>0$,}
\end{cases}$$ and extend the functions $\hat{v}$, and $w$ at $0$ by setting $(\hat{v},w)(0)=(0,w_0)$. In this way, $\hat{v}$, and $w$ are continuous on ${{\mathbb R}}$, and $C^\infty$ on ${{\mathbb R}}\setminus \{ 0\}$. It remains to show that $\hat{v} \in C^2({{\mathbb R}})$, that $(\hat{v},w)$ solves on ${{\mathbb R}}$, and that $\hat{v}$ takes its values in $(-a,a)$.
Observe first of all that $\hat{v}$ is a decreasing function, because of the monotonicity properties of $V_\flat,V_\sharp,\Xi_\flat,\Xi_\sharp$. Using the asymptotic behavior of $V_\flat$, $V_\sharp$ at $-\infty$, we get that $\hat{v}(\xi) \in (-a,a)$ for all $\xi \in {{\mathbb R}}$.
Let us now note that the above construction of $(\hat{v},w)$ shows that $(\hat{v},w)$ is a solution to on ${{\mathbb R}}\setminus \{ 0\}$. In particular, $\hat{v}' (\xi)=w(\xi)$ if $\xi \neq 0$. Moreover, $w$ is continuous on ${{\mathbb R}}$, so $\hat{v} \in C^1({{\mathbb R}})$, and $\hat{v}'(0)=w(0)=w_0$. To prove that $\hat{v}\in C^2({{\mathbb R}})$, it is sufficient to show that $w\in
C^1({{\mathbb R}})$, which is equivalent to showing that $w'$ has a limit at $0$ (because we already know that $w$ is $C^\infty$ on ${{\mathbb R}}\setminus \{ 0\}$). To prove that $w'$ has a limit at $0$, we are going to study the asymptotic behavior of $(V_\flat,W_\flat)$, and $(V_\sharp,W_\sharp)$ at $+\infty$. More precisely, let us denote $U(V,W)$ the vector field associated to the ODE : $$\label{defU}
U(V,W)=\begin{pmatrix}
V \, W \\
-W^2- f(V) \, W+\dfrac{(V^2-a^2)}{2} \end{pmatrix} \, ,$$ where $f$ is given by . The Jacobian matrix of $U$ at $(0,w_0)$ is: $$\begin{pmatrix}
w_0 & 0 \\
-f'(0) \, w_0 & -2\, w_0-f(0) \end{pmatrix} = \begin{pmatrix}
\lambda_1^{(0)} & 0 \\
b_0 & \lambda_2^{(0)} \end{pmatrix} \, .$$ For $a$ sufficiently small, one checks that $\lambda_2^{(0)}<\lambda_1^{(0)}<0$ (see Proposition \[lem\] for the definition of $w_0$). The eigenvectors corresponding to the eigenvalues $\lambda_1^{(0)}$ and $\lambda_2^{(0)}$ are: $$e_1^{(0)}=\begin{pmatrix}
f(0)+3 \, w_0\\
b_0 \end{pmatrix} \, ,\quad e_2^{(0)}=\begin{pmatrix}
0\\
1 \end{pmatrix} \, .$$ The standard theory of autonomous ODEs, see e.g. [@pontriaguine], shows that there are exactly two solutions to that tend to $(0,w_0)$ as $\eta$ tends to $+\infty$, and that are tangent to the straight line $(0,w_0)
+{{\mathbb R}}\, e_2^{(0)}$. Moreover, all the other solutions to that tend to $(0,w_0)$ as $\eta$ tends to $+\infty$ are tangent to the straight line $(0,w_0) +{{\mathbb R}}\, e_1^{(0)}$. Now, it is rather simple to see that the two solutions to that tend to $(0,w_0)$ as $\eta$ tends to $+\infty$, and that are tangent to the straight line $(0,w_0) +{{\mathbb R}}\, e_2^{(0)}$, satisfy $V \equiv 0$, and: $$W'=-W^2-f(0) \, W -\dfrac{a^2}{2} \, .$$ Because the solutions $(V_\flat,W_\flat)$, and $(V_\sharp,W_\sharp)$ given by Proposition \[lem\] cannot satisfy $V_\flat \equiv 0$, and $V_\sharp \equiv 0$, we can conclude that the solutions $(V_\flat,W_\flat)$, and $(V_\sharp,W_\sharp)$ are tangent to $(0,w_0) +{{\mathbb R}}\, e_1^{(0)}$ as $\eta$ tends to $+\infty$. In particular, this yields: $$\label{asympderiv}
\lim_{\eta \rightarrow +\infty} \dfrac{W_\flat' (\eta)}{V_\flat' (\eta)}
=\lim_{\eta \rightarrow +\infty} \dfrac{W_\sharp' (\eta)}{V_\sharp' (\eta)}
=\dfrac{-f'(0) \, w_0}{f(0)+3 \, w_0} \, .$$ (A quick verification shows that $f(0)+3 \, w_0>0$ for small enough $a$.) From the construction of the solutions $(\hat{v}_\flat,w_\flat)$, and $(\hat{v}_\sharp,w_\sharp)$, we get: $$\lim_{\xi \rightarrow 0^-} w_\flat' (\xi)
=\lim_{\xi \rightarrow 0^+} w_\sharp' (\xi)
=\dfrac{-f'(0) \, w_0^2}{f(0)+3 \, w_0} \, .$$ As a consequence, when $a$ is small enough, $w \in C^1({{\mathbb R}})$, and therefore $\hat{v} \in C^2({{\mathbb R}})$. Moreover, $(\hat{v},w)$ solves on ${{\mathbb R}}\setminus \{ 0\}$, so by continuity, it solves on ${{\mathbb R}}$. This completes the proof of Theorem \[exis\], provided that the result of Proposition \[lem\] holds.
Proof of Proposition \[lem\]
----------------------------
In this paragraph, we prove Proposition \[lem\], which will complete the proof of Theorem \[exis\]. At first, we define the set: $$P = \Big\{ (V,W) | V \in [-a,a], W^2+f(V) \, W-\dfrac{V^2-a^2}{2}=0 \Big\} \, ,$$ so that the points $(\pm a,0)$ belong to $P$. The following Lemma gives a description of $P$ for $a>0$ small enough. We refer to figure \[K\] for a schematic picture.
![The set $P =P_1 \cup P_2$.[]{data-label="K"}](EnsembleP.eps)
\[geo\] Assume that $1 < \gamma < (\sqrt{7}+1)/(\sqrt{7}-1)$. Then there exists a constant $a_0>0$, that only depends on $(\rho_-,u_-,e_-)$ and $\gamma$ such that if the shock strength $a$ satisfies $a\in (0,a_0]$, we have the following results:
- For all $V \in [-a,a]$, $f(V)^2+2\, (V^2-a^2)>0$. We can thus define $$\begin{aligned}
\forall \, V \in [-a,a] \, ,\quad
{{\mathbb W}}_1(V) &= \dfrac{-f(V)+\sqrt{f(V)^2 +2\, (V^2-a^2)}}{2} \, ,\\
{{\mathbb W}}_2(V) &= \dfrac{-f(V)-\sqrt{f(V)^2 +2\, (V^2-a^2)}}{2} \, .\end{aligned}$$
- $P = P_1\cup P_2$, where $P_1$ and $P_2$ are two curves defined by $$P_1 =\big\{ (V,{{\mathbb W}}_1(V)) | V \in [-a,a] \big\} \, , \quad
P_2 =\big\{ (V,{{\mathbb W}}_2(V)) | V \in [-a,a] \big\} \, ,$$ so that the points $(\pm a,0)$, and $(0,w_0)$ belong to $P_1$.
- There exists a unique point $\overline{V} \in \, (-a,0)$ such that ${{\mathbb W}}_1$ is increasing on the interval $[\overline{V},a]$, and ${{\mathbb W}}_1$ is decreasing on the interval $[-a,\overline{V}]$.
- For all $V \in [\overline{V},0]$, one has ${{\mathbb W}}_2(V) < {{\mathbb W}}_1(\overline{V})$.
Let us first define a function $\Delta$ by setting: $$\Delta (V) := f(V)^2 +2 \, (V^2-a^2) \, .$$ Using , for $a$ small enough, we have: $$\dfrac{C_1}{\gamma +1} -a \ge \kappa >0 \, ,\quad
\dfrac{\gamma \, C_1}{\gamma +1} -a \ge \kappa >0 \, ,\quad
\dfrac{(\gamma -1)\, C_1}{\gamma +1} -2a \ge \kappa >0 \, ,$$ where $\kappa$ is a positive constant that only depends on $(\rho_-,u_-,e_-)$ and $\gamma$. Moreover, also shows that $j \ge \kappa$ for $a \in (0,a_0]$, up to restricting $\kappa$. Consequently, there exist $a_0>0$, and $\kappa>0$ such that for $a \in (0,a_0]$, we have $f(V) \ge \kappa$, and $\Delta (V) \ge \kappa$ for all $V\in [-a,a]$. This directly shows that the set $P$ is the union of the two curves $P_1$, and $P_2$. It is rather clear from the definition of $P_1$ that $(\pm a,0)$, and $(0,w_0)$ belong to $P_1$ (recall that $w_0$ is defined in Proposition \[lem\]). Observe also that ${{\mathbb W}}_2(V) <{{\mathbb W}}_1 (V) \le 0$ for $V \in [-a,a]$, and ${{\mathbb W}}_1(V)<0$ if $V \in (-a,a)$.
The functions ${{\mathbb W}}_1$, and ${{\mathbb W}}_2$ are $C^\infty$ on $[-a,a]$. Moreover, we compute the relation: $$\label{derivW1}
\forall \, V \in [-a,a] \, ,\quad
\sqrt{\Delta (V)} \, {{\mathbb W}}_1'(V) =V-{{\mathbb W}}_1(V) \, f'(V) \, ,$$ and from , we also compute $$\begin{gathered}
\label{derivf}
f'(V)=\dfrac{14 \, (\gamma-1)}{j \, R^4 \, (\gamma+1)} \\
\left( \dfrac{C_1}{\gamma+1}-\hat{v} \right)^2 \,
\left( \hat{v}+\dfrac{\gamma\, C_1}{\gamma+1} \right)^2 \,
\left( 2\hat{v}+\dfrac{(\gamma-1) \, C_1}{\gamma+1} +\dfrac{C_1}{\sqrt{7}} \right)
\left( -2\hat{v}-\dfrac{(\gamma-1) \, C_1}{\gamma+1} +\dfrac{C_1}{\sqrt{7}} \right) \, .\end{gathered}$$ As we have done for $f$, and $\Delta$, a careful analysis shows that for $1<\gamma<(\sqrt{7}+1)/(\sqrt{7}-1)$, and for $a$ small enough, one has $f'(V) \ge \kappa>0$ for all $V \in [-a,a]$, because each term in the product is positive. Using this information in , we can already conclude that ${{\mathbb W}}_1$ is increasing on the interval $[0,a]$ (see figure \[K\]). Moreover, the relation also shows that ${{\mathbb W}}'_1(0)>0$, and ${{\mathbb W}}'_1(-a)<0$. Consequently, there exists some $\overline{V} \in (-a,0)$ such that ${{\mathbb W}}_1'(\overline{V})=0$. Let us prove that $\overline{V}$ is the only zero of ${{\mathbb W}}_1'$. We claim that it is sufficient to show the following property: $$\label{property}
{{\mathbb W}}_1' (V) =0 \Longrightarrow {{\mathbb W}}_1'' (V) >0 \, .$$ Indeed, if the property holds true, then any point where ${{\mathbb W}}_1'$ vanishes is a local strict minimum. If there existed two such local strict minima $-a<\overline{V}_1<\overline{V}_2<a$, then ${{\mathbb W}}_1$ would admit a local maximum $\overline{V}_3 \in (\overline{V}_1,\overline{V}_2)$, which is obviously impossible. Therefore let us prove that the property holds true.
Differentiating with respect to $V$, we obtain that if ${{\mathbb W}}_1'(\overline{V})=0$, then $$\sqrt{\Delta (\overline{V})} \, {{\mathbb W}}_1''(\overline{V}) = 1 -f''(\overline{V})
\, {{\mathbb W}}_1(\overline{V}) \, .$$ Observing that $$|f''(\overline{V}) \, {{\mathbb W}}_1(\overline{V})| \le C \, |{{\mathbb W}}_1(\overline{V})|
\le C \, \dfrac{a^2 -\overline{V}^2}{f(\overline{V})} \le
\dfrac{C \, a^2}{\kappa} \, ,$$ for suitable positive constants $C$, and $\kappa$ (that are independent of $a \in (0,a_0]$), we can conclude that ${{\mathbb W}}_1''(\overline{V})>0$, provided that $a$ is small enough. This completes the proof that ${{\mathbb W}}_1'$ has a unique zero $\overline{V} \in (-a,0)$, and therefore ${{\mathbb W}}_1$ is decreasing on $[-a,\overline{V}]$, and is increasing on $[\overline{V},a]$.
For the last point of the lemma, we use the relation: $${{\mathbb W}}_1'(V) +{{\mathbb W}}_2'(V)=-f'(V)<0 \, .$$ Because ${{\mathbb W}}_1'(V)\ge 0$ for $V \in [\overline{V},0]$, we get ${{\mathbb W}}_2'(V)<0$ for $V \in [\overline{V},0]$. Thus for $V \in [\overline{V},0]$, we have ${{\mathbb W}}_2(V) \le {{\mathbb W}}_2 (\overline{V}) <{{\mathbb W}}_1 (\overline{V})$, and the proof of the Lemma is complete.
Using Lemma \[geo\], we are going to prove Proposition \[lem\]. The analysis follows [@gilbarg].
As we have already seen in the preceeding paragraph, the point $(w_0,0)$ is a stable node of . We now study the nature of the equilibrium points $(\pm a,0)$. Recall that the vector field associated to is denoted $U$, see . The Jacobian matrix of $U$ at $(a,0)$ is: $$\begin{pmatrix}
0 & a \\
a & f(a) \end{pmatrix} \, ,$$ so it has exactly one negative eigenvalue $\mu_1$, and one positive eigenvalue $\mu_2$ (the equilibrium point $(a,0)$ is a saddle point): $$\mu_1 =\dfrac{-f(a)-\sqrt{f(a)^2+4 \, a^2}}{2} \, ,\quad
\mu_2 =\dfrac{-f(a)+\sqrt{f(a)^2+4 \, a^2}}{2} \, .$$ An eigenvector associated to $\mu_2$, and is $r_2=(a,\mu_2)$. Moreover, using the relation , we can check that for $a$ small enough, the following inequality holds: $$\label{ineg1}
0 <\dfrac{\mu_2}{a} < \dfrac{a}{f(a)} ={{\mathbb W}}_1'(a) \, ,$$ where the function ${{\mathbb W}}_1$ is defined in Lemma \[geo\]. Let us now define a compact set $K_1$ by: $$K_1 := \Big\{ (V,W) \in [0,a] \times {{\mathbb R}}\, | \, {{\mathbb W}}_1 (V) \le W \le 0 \Big\} \, ,$$ Then the inequalities show that for $s<0$ small enough, the point $(a,0) + s \, r_2$ belongs to the interior of $K_1$. We refer to figure \[compactK1\] for a detailed picture of the situation.
![The compact set $K_1$.[]{data-label="compactK1"}](compactK1.eps)
From the standard theory of autonomous ODEs, see e.g. [@pontriaguine], we know that there exists a maximal solution $(V_\flat,W_\flat)$ to that tends to the saddle point $(a,0)$ as $\eta$ tends to $-\infty$, and that is tangent to the half-straight line $(a,0) +{{\mathbb R}}^- \, r_2$. This solution is defined on an open interval $(-\infty,\eta_*)$ (with possibly $\eta_*=+\infty$). For large negative $\eta$, the preceeding analysis shows that $(V_\flat,W_\flat) (\eta)$ belongs to the interior of $K_1$. Moreover, $(V_\flat,W_\flat)$ cannot reach the boundary of $K_1$. Indeed $V_\flat$ cannot identically vanish so $(V_\flat,W_\flat)(\eta) \not \in \partial K_1 \cap
\{ V=0\}$. Similarly, we have $(V_\flat,W_\flat)(\eta) \neq (a,0)$. Eventually, on the set: $$\Big\{ (V,0) \, | \, V \in (0,a) \Big\} \cup
\Big\{ (V,{{\mathbb W}}_1(V)) \, | \, V \in \, (0,a) \Big\} \, ,$$ the vector field $U$ is not zero, and is directed towards the interior of $K_1$. Therefore the solution $(V_\flat,W_\flat)$ cannot reach $\partial K_1$, so it takes its values in the compact set $K_1$. The maximal solution $(V_\flat,W_\flat)$ is thus defined on ${{\mathbb R}}$. It cannot reach the boundary of $K_1$, so $W_\flat$ takes negative values, which means that $V_\flat$ is decreasing (because $V_\flat'=V_\flat \, W_\flat$). Because $(V_\flat,W_\flat)$ takes values in the interior of $K_1$, the function $W_\flat$ is also decreasing. This shows that $(V_\flat,W_\flat)(\eta)$ has a limit as $\eta$ tends to $+\infty$, and this limit is necessarily be a stationary solution of . The only possibility is that $(V_\flat,W_\flat)(\eta)$ tends to $(0,w_0)$ as $\eta$ tends to $+\infty$. The convergence is necessarily exponential, because the Jacobian matrix of $U$ at $(0,w_0)$ has two negative eigenvalues, see e.g. [@pontriaguine].
To construct the other solution $(V_\sharp,W_\sharp)$, we argue similarly by defining a compact set $K_2$: $$K_2 := \Big\{ (V,W) \in [-a,\overline{V}] \times {{\mathbb R}}| {{\mathbb W}}_1 (V) \le W \le 0 \Big\}
\cup \Big\{
(V,W) \in [\overline{V},0] \times {{\mathbb R}}| {{\mathbb W}}_1 (\overline{V}) \le W \le 0 \Big\} \, ,$$ see figure \[compactK2\]. The Jacobian matrix of $U$ at $(-a,0)$ has one negative eigenvalue $\nu_1$, and one positive eigenvalue $\nu_2$, with: $$\nu_2 = \dfrac{-f(-a)+\sqrt{f(-a)^2+4 \, a^2}}{2} \, .$$ An eigenvector associated to the eigenvalue $\nu_2$ is $R_2=(-a,\nu_2)$. As was done earlier, we check that the inequalities: $${{\mathbb W}}_1'(-a)=\dfrac{-a}{f(-a)} < \dfrac{\nu_2}{-a} <0 \, ,$$ hold true. Therefore, one can reproduce the above analysis, and show that there exists a solution $(V_\sharp,W_\sharp)$ to that takes its values in $K_2$ (and is thus defined on ${{\mathbb R}}$), and that tends to $(-a,0)$ at $-\infty$. Moreover, $(V_\sharp,W_\sharp)$ can not reach the boundary of $K_2$, so $V_\sharp$ is increasing. It only remains to study the monotonicity of $W_\sharp$. This is slightly more complicated than for $W_\flat$. Observe that $K_2$ is the union of the sets: $$\begin{aligned}
K_2^1 &:=\Big\{ (V,W) \in [-a,0] \times {{\mathbb R}}\, | \, {{\mathbb W}}_1 (V) \le W \le 0 \Big\} \, ,\\
K_2^2 &:=\Big\{ (V,W) \in [\overline{V},0] \times {{\mathbb R}}\, | \,
{{\mathbb W}}_1 (\overline{V}) \le W \le {{\mathbb W}}_1(V) \Big\} \, .\end{aligned}$$ When $(V_\sharp,W_\sharp)$ takes its values in the interior of $K_2^1$, the function $W_\sharp$ is decreasing (this is the case for large negative $\eta$). At the opposite, when $(V_\sharp,W_\sharp)$ takes its values in the interior of $K_2^2$, the function $W_\sharp$ is increasing, because thanks to Lemma \[geo\], we have: $$\begin{aligned}
W_\sharp'(\eta)
&=-W_\sharp(\eta)^2 -f(V_\sharp(\eta)) \, W_\sharp(\eta)
+\dfrac{V_\sharp(\eta)^2 -a^2}{2} \\
&=\big(
{{\mathbb W}}_1 (V_\sharp(\eta)) -W_\sharp (\eta) \big) \,
\big( W_\sharp(\eta) -{{\mathbb W}}_2 (V_\sharp(\eta)) \big) \\
&\ge \big( {{\mathbb W}}_1 (V_\sharp(\eta)) -W_\sharp (\eta) \big) \,
\big( {{\mathbb W}}_1(\overline{V}) -{{\mathbb W}}_2 (V_\sharp(\eta)) \big) >0 \, .\end{aligned}$$ Moreover, if $(V_\sharp,W_\sharp) (\eta_0)$ belongs to the interior of $K_2^2$ for some $\eta_0 \in {{\mathbb R}}$, then $(V_\sharp,W_\sharp) (\eta)$ belongs to the interior of $K_2^2$ for all $\eta \ge \eta_0$ (because it cannot reach the boundary of $K_2^2$ for $\eta \ge \eta_0$). Summing up, either $(V_\sharp,W_\sharp)(\eta)$ belongs to $K_2^1$ for all $\eta$, and $W_\sharp$ is monotonic on ${{\mathbb R}}$, either $(V_\sharp,W_\sharp)(\eta)$ belongs to $K_2^2$ for all $\eta$ greater than some $\eta_0$, and $W_\sharp$ is monotonic on $[\eta_0,+\infty)$. In any case, the function $W_\sharp$ is monotonic on a neighborhood of $+\infty$, and thus has a limit at $+\infty$. This shows that $(V_\sharp,W_\sharp)(\eta)$ tends to $(0,w_0)$ as $\eta$ tends to $+\infty$, and the convergence is exponential. As a matter of fact, we have seen in the preceeding paragraph that $(V_\sharp,W_\sharp)$ is tangent to the straight line $(0,w_0)+{{\mathbb R}}\, e_1^{(0)}$ as $\eta$ tends to $+\infty$, so one can check that $(V_\sharp,W_\sharp) (\eta)$ belongs to the interior of $K_2^2$ for large positive $\eta$. This means that $W_\sharp$ is decreasing on some interval $(-\infty,\eta_0)$, and increasing on $[\eta_0,+\infty)$. The proof of Proposition \[lem\] is now complete.
![The compact set $K_2$.[]{data-label="compactK2"}](compactK2.eps)
Additional regularity of shock profiles {#moresmooth}
=======================================
As should be clear from the preceeding section, the key point in the construction of a shock profile is Proposition \[lem\] that gives the existence of two heteroclinic orbits for the system . To prove Theorem \[smooth\], we are going to study the behavior of the derivatives of $(V_\flat,W_\flat)$, and $(V_\sharp,W_\sharp)$ near $+\infty$. The proof of Theorem \[smooth\] follows from an induction argument. To make the arguments clear, we deal with the first case separately. In all what follows, $(V_\flat,W_\flat)$, and $(V_\sharp,W_\sharp)$ are the solutions to that are defined in Proposition \[lem\], and $(\hat{v},w)$ denotes the solution to that is defined by . We have the following:
\[n=3\] Under the assumptions of Proposition \[lem\], there exists a positive constant $a_1 \le a_0$ (that depends on $(\rho_-,u_-,e_-)$, and $\gamma$), such that for all $a \in (0,a_1]$, one has $w \in C^2 ({{\mathbb R}})$, $\hat{v} \in C^3 ({{\mathbb R}})$, and: $$w(\xi)=w_0 +w_1 \, \hat{v} (\xi) +w_2 \, \hat{v} (\xi)^2 +o(\hat{v} (\xi)^2) \, ,\quad
\text{\rm as $\xi \rightarrow 0$,}$$ for some suitable constants $w_1,w_2$ ($w_0$ has already been defined in Proposition \[lem\]).
Recall that $V_b$, and $V_\sharp$ do not vanish on ${{\mathbb R}}$, so we can introduce some $C^\infty$ functions $W_{\flat,1}$, and $W_{\sharp,1}$ that are defined by: $$W_\flat = w_0 + V_\flat \, W_{\flat,1} \, ,\quad
W_\sharp = w_0 + V_\sharp \, W_{\sharp,1} \, .$$ Substituting in shows that $(V_\flat,W_{\flat,1})$, and $(V_\sharp,W_{\sharp,1})$ are solutions to the system: $$\label{w_1}
\begin{cases}
V' = V (w_0+V \, W_1) \, , &\\
W_1' = -w_0 \, \dfrac{f(V)-f(0)}{V} -2 \, V \, W_1^2 -3 \, w_0 \, W_1
-f(V) \, W_1 +\dfrac{V}{2} \, . &
\end{cases}$$ Moreover, we already know from Proposition \[lem\], and that: $$\lim_{\eta \rightarrow +\infty} (V_\flat,W_{\flat,1}) (\eta)
=\lim_{\eta \rightarrow +\infty} (V_\sharp,W_{\sharp,1}) (\eta)
=\Big( 0,\dfrac{-f'(0) \, w_0}{f(0)+3 \, w_0} \Big) \, .$$
We denote $U_1 (V,W_1)$ the vector field associated with : $$U_1 (V,W_1) := \begin{pmatrix}
V (w_0+V \, W_1) \\
-w_0 \, \dfrac{f(V)-f(0)}{V} -2 \, V \, W_1^2 -3 \, w_0 \, W_1
-f(V) \, W_1 +\dfrac{V}{2} \end{pmatrix} \, .$$ Recall that $f$ is a polynomial function of degree $7$, see , thus $F(V) :=(f(V)-f(0))/V$ is a polynomial function of degree $6$, and we have $F(0)=f'(0)$, $F'(0)=f''(0)/2$. Obviously the system of ODEs admits the equilibrium point $(0,w_1)$, where: $$w_1 :=\dfrac{-F(0) \, w_0}{f(0)+3\, w_0} =\dfrac{-f'(0) \, w_0}{f(0)+3\, w_0} \, .$$ We are now going to study the nature of the equilibrium point $(0,w_1)$, and show that for $a$ small enough, this equilibrium point is a stable node for . Then we shall show that $w \in C^2 ({{\mathbb R}})$, and $\hat{v} \in C^3 ({{\mathbb R}})$. In the end, we shall derive the asymptotic expansion near $\xi=0$.
: the Jacobian matrix of $U_1$ at $(0,w_1)$ is: $$\begin{pmatrix}
w_0 & 0\\
\dfrac{1}{2} -2\, w_1^{2} -f'(0)\, w_1 -\dfrac{f''(0)}{2} \, w_0 & -f(0)-3\, w_0
\end{pmatrix} =\begin{pmatrix}
\lambda_1^{(1)} & 0 \\
b_1 & \lambda_2^{(1)} \end{pmatrix} \, .$$ Using Remak \[f0\], we can conclude that for sufficiently small $a$, that is $a \in (0,a_1]$ for some positive number $a_1$ less than $a_0$, one has $\lambda_2^{(1)} <\lambda_1^{(1)}<0$, that is, $f(0)+4\, w_0>0$. Moreover, the eigenvectors corresponding to the eigenvalues $\lambda_1^{(1)}$, and $\lambda_2^{(1)}$ are: $$e_1^{(1)} =\begin{pmatrix}
f(0) +4\, w_0\\
b_1 \end{pmatrix} \, ,\quad
e_2^{(1)} =\begin{pmatrix}
0\\
1 \end{pmatrix} \, .$$ Consequently, $(0,w_1)$ is a stable node of , and there are exactly two solutions to that tend to $(0,w_1)$ as $\eta$ tends to $+\infty$, and that are tangent to the straight line $(0,w_1) +{{\mathbb R}}\, e_2^{(1)}$. All the other solutions to that tend to $(0,w_1)$ as $\eta$ tends to $+\infty$ are tangent to the straight line $(0,w_1) +{{\mathbb R}}\, e_1^{(1)}$. As in the preceeding section, we can thus conclude that: $$\label{eq:0}
\lim_{\eta \rightarrow +\infty} \dfrac{W_{\flat,1}'(\eta)}{V_\flat'(\eta)}
=\lim_{\eta \rightarrow +\infty} \dfrac{W_{\sharp,1}'(\eta)}{V_\sharp'(\eta)}
=\dfrac{b_1}{f(0)+4\, w_0} \, .$$
: if we let $g_1$ denote the second coordinate of the vector field $U_1$, we have $W_{\flat,1}'=g_1(V_\flat,W_{\flat,1})$, and $W_{\sharp,1}'=g_1(V_\sharp,W_{\sharp,1})$. Differentiating once with respect to $\eta$, and using , we end up with: $$\label{eq:1}
\lim_{\eta \rightarrow +\infty} \dfrac{W_{\flat,1}''(\eta)}{V_\flat'(\eta)}
=\lim_{\eta \rightarrow +\infty} \dfrac{W_{\sharp,1}''(\eta)}{V_\sharp'(\eta)}
= \ell_1 \, ,$$ where the real number $\ell_1$ can be explicitely computed (but its exact expression is of no use). Following the analysis of the preceeding section, we define some functions $w_{\flat,1} := W_{\flat,1} \circ \Xi_\flat^{-1}$, and $w_{\sharp,1}
:= W_{\sharp,1} \circ \Xi_\sharp^{-1}$. First of all, yields: $$\label{eq:2}
\lim_{\xi \rightarrow 0^-} w_{\flat,1}' (\xi)
=\lim_{\xi \rightarrow 0^+} w_{\sharp,1}' (\xi)
=\dfrac{b_1 \, w_0}{f(0)+4\, w_0} \, .$$ Observe now that we have the relations: $$\hat{v}_\flat \, w_{\flat,1}' =W_{\flat,1}' \circ \Xi_\flat^{-1} \, ,\quad
\hat{v}_\sharp \, w_{\sharp,1}' =W_{\sharp,1}' \circ \Xi_\sharp^{-1} \, ,$$ and combining with , we get: $$\label{eq:3}
\begin{split}
\lim_{\xi \rightarrow 0^-} (\hat{v}_\flat \, w_{\flat,1}')' (\xi)
=\lim_{\eta \rightarrow +\infty} \dfrac{W_{\flat,1}''(\eta)}{V_\flat (\eta)}
=\lim_{\eta \rightarrow +\infty}
\dfrac{W_{\flat,1}''(\eta) \, W_\flat (\eta)}{V_\flat' (\eta)} =\ell_1 \, w_0 \, ,\\
\lim_{\xi \rightarrow 0^+} (\hat{v}_\sharp \, w_{\sharp,1}')' (\xi)
=\lim_{\eta \rightarrow +\infty} \dfrac{W_{\sharp,1}''(\eta)}{V_\sharp (\eta)}
=\lim_{\eta \rightarrow +\infty}
\dfrac{W_{\sharp,1}''(\eta) \, W_\sharp (\eta)}{V_\sharp' (\eta)} =\ell_1 \, w_0 \, .
\end{split}$$ Differentiating twice the relations $w_\flat =w_0 +\hat{v}_\flat \, w_{\flat,1}$, and $w_\sharp =w_0 +\hat{v}_\sharp \, w_{\sharp,1}$, we obtain: $$\begin{gathered}
w_\flat'' =\hat{v}_\flat'' \, w_{\flat,1} +\hat{v}_\flat' \, w_{\flat,1}'
+(\hat{v}_\flat \, w_{\flat,1}')' =w_\flat' \, w_{\flat,1}
+w_\flat \, w_{\flat,1}' +(\hat{v}_\flat \, w_{\flat,1}')' \, ,\\
w_\sharp'' =w_\sharp' \, w_{\sharp,1}
+w_\sharp \, w_{\sharp,1}' +(\hat{v}_\sharp \, w_{\sharp,1}')' \, .\end{gathered}$$ Using , and , we get $w_\flat'' (0^-)=w_\sharp'' (0^+)$. Using the definition , this shows that $w \in C^2 ({{\mathbb R}})$, and using $\hat{v}'=w$, we obtain $\hat{v} \in C^3 ({{\mathbb R}})$.
: note that we have the following expansions near $\xi=0$: $$\begin{aligned}
&w(\xi) = w(0) +w'(0) \, \xi +\dfrac{w''(0)}{2} \, \xi^2 +o(\xi^2) \, ,\\
&\hat{v}(\xi) = w(0) \, \xi +\dfrac{w'(0)}{2} \, \xi^2 +o(\xi^2) \, ,\end{aligned}$$ with $w(0)=w_0 <0$. We can thus combine these expansions, and derive: $$w(\xi) = w_0 +\alpha \, \hat{v}(\xi) +\beta \, \hat{v}(\xi)^2
+o(\hat{v}(\xi)^2) \, ,$$ for some appropriate real numbers $\alpha$, and $\beta$, that we are going to determine. From the relation $w_\flat (\xi)=w_0+\hat{v}_\flat(\xi) \,
w_{\flat,1} (\xi)$, and using that $w_{\flat,1} (\xi)$ tends to $w_1$ as $\xi$ tends to $0^-$, we first get $\alpha=w_1$. Then from , and from the relation $\hat{v}_\flat'(0^-)=w_0$, we can obtain: $$w_{\flat,1} (\xi)=w_1 +\dfrac{b_1}{f(0)+4\, w_0} \, \hat{v}_\flat (\xi)
+o(\hat{v}_\flat (\xi)) \, ,\quad \text{as $\xi \rightarrow 0^-$.}$$ We thus obtain $\beta=b_1/(f(0)+4\, w_0)$, which yields: $$w(\xi) = w_0 +w_1 \, \hat{v}(\xi) +w_2 \, \hat{v}(\xi)^2 +o(v(\xi)^2) \, ,$$ where $w_2 :=b_1/(f(0)+4\, w_0)$. This latter expansion will be generalized to any order in what follows.
We now turn to the proof of Theorem \[smooth\]. More precisely, we are going to prove the following result, that is a refined version of Theorem \[smooth\]:
\[smooth2\] Let the assumptions of Proposition \[lem\] be satisfied. Then there exists a nonincreasing sequence of positive numbers $(a_n)_{n \in {{\mathbb N}}}$ such that, for all integer $n$, if $a \in (0,a_n]$, then $w \in C^{n+1} ({{\mathbb R}})$, and $\hat{v}
\in C^{n+2} ({{\mathbb R}})$. Moreover, $w$ admits the following asymptotic expansion near $\xi=0$: $$\label{expansion}
w(\xi) =w_0+w_1 \, \hat{v}(\xi) +\cdots +w_{n+1} \, \hat{v} (\xi)^{n+1}
+o(\hat{v} (\xi)^{n+1}) \, ,$$ where the real numbers $w_0,\dots,w_{n+1}$ are defined by: $$\begin{cases}
w_0 =\dfrac{-f(0)+\sqrt{f(0)^2 -2\, a^2}}{2} \, ,& \\
w_k =\dfrac{b_{k-1}}{f(0) +(k+2)\, w_0} \, ,&\text{\rm for $k=1,\dots,n+1$,}
\end{cases}$$ and the real numbers $b_0,\dots,b_n$ are given by: $$\begin{cases}
b_0 = -f'(0) \, w_0 \, ,& \\
b_1 =\dfrac{1}{2}-2 \, w_1^2 -f'(0) \, w_1 -\dfrac{f''(0)}{2} \, w_0 \, ,& \\
b_k =-\displaystyle \sum_{i=1}^{k+1} \dfrac{f^{(i)}(0)}{i!} \, w_{k+1-i}
-\displaystyle \sum_{i=1}^k (i+1) \, w_i \, w_{k+1-i} \, ,&
\text{\rm for $k=2,\dots,n$.}
\end{cases}$$
The case $n=0$ has been proved in the preceeding section, while the case $n=1$ is proved in Proposition \[n=3\]. (The reader can check that the definition of $w_0$, $w_1$, $w_2$, $b_0$, and $b_1$ coincide with our previous notations.) We prove the general case by using an induction with respect to $n$, and we thus assume that the result of Theorem \[smooth2\] holds up to the order $n \ge 1$. We are going to construct $a_{n+1}$ so that the conclusion of Theorem \[smooth2\] holds for $a \in (0,a_{n+1}]$. In particular, the real numbers $w_0,\dots,w_{n+1}$, and $b_0,\dots,b_n$ are given as in Theorem \[smooth2\], and we can already define the real number $b_{n+1}$ by the formula: $$b_{n+1} :=-\displaystyle \sum_{i=1}^{n+2} \dfrac{f^{(i)}(0)}{i!} \, w_{n+2-i}
-\displaystyle \sum_{i=1}^{n+1} (i+1) \, w_i \, w_{n+2-i} \, .$$ (Observe indeed that this definition only involves $w_0,\dots,w_{n+1}$, and not $w_{n+2}$.)
: because $V_\flat$, and $V_\sharp$ do not vanish, we can introduce some functions $W_{\flat,n+1}$, and $W_{\sharp,n+1}$ by the relations: $$W_\flat =w_0 +w_1 \, V_\flat +\cdots +w_n \, V_\flat^n
+W_{\flat,n+1} \, V_\flat^{n+1} \, ,\quad
W_\sharp =w_0 +w_1 \, V_\sharp +\cdots +w_n \, V_\sharp^n
+W_{\sharp,n+1} \, V_\sharp^{n+1} \, .$$ Thanks to Taylor’s formula, we can write the polynomial function $f$ as: $$f(V)=f(0)+f'(0) \, V+\dfrac{f''(0)}{2} \, V^2 +\cdots
+\dfrac{f^{(n)}(0)}{n!} \, V^n +V^{n+1} \, F_{n+1} (V) \, ,$$ where $F_{n+1}$ is a polynomial function such that: $$F_{n+1} (0)=\dfrac{f^{(n+1)}(0)}{(n+1)!} \, ,\quad
F_{n+1}' (0)=\dfrac{f^{(n+2)}(0)}{(n+2)!} \, .$$ Substituting the expression of $W_\flat$, and $W_\sharp$ in shows (after a tedious computation!) that $(V_\flat,W_{\flat,n+1})$, and $(V_\sharp,W_{\sharp,n+1})$ are solutions to the following system of ODEs: $$\label{eq:k}
\begin{cases}
V' = V \, (w_0+w_1 \, V +\cdots +w_n \, V^n +W_{n+1} \, V^{n+1}) \, ,& \\
W_{n+1}' =g_{n+1} (V,W_{n+1}) \, ,&
\end{cases}$$ where the function $g_{n+1}$ is given by: $$\begin{aligned}
g_{n+1} (V,W_{n+1}) := &-(n+2) \, W_{n+1} \left( \sum_{k=0}^n w_k \, V^k
+V^{n+1} \, W_{n+1} \right) -W_{n+1} \, \sum_{k=0}^n (k+1) \, w_k \, V^k \notag\\
&-W_{n+1} \, f(V) -F_{n+1} (V) \, \sum_{k=0}^n w_k \, V^k +b_n
+\dfrac{f^{(n+1)}(0)}{(n+1)!} +V \, Q_{n+1}(V) \, ,\label{defgn+1}\end{aligned}$$ and $Q_{n+1}$ is a polynomial function that satisfies: $$Q_{n+1}(0)=b_{n+1} +(n+4) \, w_1 \, w_{n+1} +f'(0) \, w_{n+1}
+\dfrac{f^{(n+1)}(0)}{(n+1)!} \, w_1 +\dfrac{f^{(n+2)}(0)}{(n+2)!} \, w_0 \, .$$ When $n=1$, one has $Q_2 \equiv Q_2(0)=0$ (see the above definition for $b_2$). Using the expansion , which is part of the induction assumption, we also know that: $$\lim_{\eta \rightarrow +\infty} (V_\flat,W_{\flat,n+1}) (\eta)
=\lim_{\eta \rightarrow +\infty} (V_\sharp,W_{\sharp,n+1}) (\eta)
=(0,w_{n+1}) =\Big( 0,\dfrac{b_n}{f(0)+(n+2)\, w_0} \Big) \, .$$
With the above definitions for $g_{n+1}$, and $Q_{n+1}$, we can check that $(0,w_{n+1})$ is a stationary solution to . (Recall that $w_{n+1}$ is defined as in Theorem \[smooth2\] by the induction assumption.) We can also evaluate the Jacobian matrix of the vector field associated with the system of ODEs : $$\begin{pmatrix}
w_0 & 0 \\
b_{n+1} & -f(0)-(n+3) \, w_0 \end{pmatrix} =
\begin{pmatrix}
\lambda_1^{(n+1)} & 0 \\
b_{n+1} & \lambda_2^{(n+2)} \end{pmatrix} \, .$$ There exists a positive number $a_{n+1} \le a_n$ such that for all $a \in
(0,a_{n+1}]$, one has $\lambda_2^{(n+2)}<\lambda_1^{(n+2)}<0$, or equivalently $f(0)+(n+4) \, w_0>0$. In that case, the eigenvectors corresponding to the eigenvalues $\lambda_1^{(n+1)}$ and $\lambda_2^{(n+1)}$ are: $$e_1^{(n+1)} =\begin{pmatrix}
f(0)+(n+4) \, w_0 \\
b_{n+1} \end{pmatrix} \, ,\quad e_2^{(n+1)}=\begin{pmatrix}
0 \\
1 \end{pmatrix} \, .$$ Using the same argument as in the proof of Proposition \[n=3\], we can conclude that the solutions $(V_\flat,W_{\flat,n+1})$, and $(V_\sharp,W_{\sharp,n+1})$ of are tangent to the straight line $(0,w_{n+1}) +{{\mathbb R}}\, e_1^{(n+1)}$ as $\eta$ tends to $+\infty$. In particular, this yields: $$\label{lim0}
\lim_{\eta \rightarrow +\infty} \dfrac{W_{\flat,n+1}'(\eta)}{V_\flat'(\eta)}
=\lim_{\eta \rightarrow +\infty} \dfrac{W_{\sharp,n+1}'(\eta)}{V_\sharp'(\eta)}
=\dfrac{b_{n+1}}{f(0)+(n+4)\, w_0} =:w_{n+2} \, .$$
: let us define the function $\widetilde{w}_{n+1}$ by the formula: $$\widetilde{w}_{n+1} (\xi) :=\begin{cases}
W_{\flat,n+1} \circ \Xi_\flat^{-1} (\xi) &\text{if $\xi<0$,}\\
w_{n+1} &\text{if $\xi=0$,}\\
W_{\sharp,n+1} \circ \Xi_\sharp^{-1} (\xi) &\text{if $\xi>0$.}
\end{cases}$$ With this definition, $\widetilde{w}_{n+1}$ is continuous, and we have the relation: $$\label{recurrence0}
w=w_0 +w_1 \, \hat{v} +\dots+w_n \, \hat{v}^n
+\widetilde{w}_{n+1} \, \hat{v}^{n+1} \, .$$ Moreover, using , we obtain: $$\label{recurrence4}
\lim_{\xi \rightarrow 0^-} \dfrac{\widetilde{w}_{n+1}' (\xi)}{\hat{v}'(\xi)}
=\lim_{\xi \rightarrow 0^+} \dfrac{\widetilde{w}_{n+1}' (\xi)}{\hat{v}'(\xi)}
=w_{n+2} \, ,$$ which yields $\widetilde{w}_{n+1}'(0^+)=\widetilde{w}_{n+1}'(0^-)$. Therefore, we have $\widetilde{w}_{n+1} \in C^1 ({{\mathbb R}})$. Moreover, using , we can compute: $$\label{recurrence1}
\widetilde{w}_{n+1}' \, \hat{v} =g_{n+1} (\hat{v},\widetilde{w}_{n+1}) \, ,$$ so we get $\widetilde{w}_{n+1}' \, \hat{v} \in C^1({{\mathbb R}})$.
: we use an induction argument to show that $w \in C^{n+2} ({{\mathbb R}})$ (which will imply immediately $\hat{v} \in C^{n+3} ({{\mathbb R}})$). More precisely, we assume that for some $k \in \{ 0,\dots,n \}$, we have: $$\label{recurrence2}
\widetilde{w}_{n+1} \, \hat{v}^k \in C^{k+1}({{\mathbb R}}) \, ,\quad
w \in C^{k+1} ({{\mathbb R}}) \, ,\quad
\widetilde{w}_{n+1}' \, \hat{v}^{k+1} \in C^{k+1} ({{\mathbb R}}) \, .$$ We are going to show that this property implies the same property with $k$ replaced by $k+1$. (Observe that step 2 above shows that the property holds for $k=0$.)
We note that $\hat{v} \in C^{k+2} ({{\mathbb R}})$, because $\hat{v}'=w \in C^{k+1} ({{\mathbb R}})$. Moreover, we have $\widetilde{w}_{n+1} \, \hat{v}^{k+1} =(\widetilde{w}_{n+1} \, \hat{v}^k) \,
\hat{v} \in C^{k+1} ({{\mathbb R}})$, and we also have: $$(\widetilde{w}_{n+1} \, \hat{v}^{k+1})'=\widetilde{w}_{n+1}' \, \hat{v}^{k+1}
+(k+1) \, (\widetilde{w}_{n+1} \, \hat{v}^k) \, w \in C^{k+1} ({{\mathbb R}}) \, .$$ Therefore, we get $\widetilde{w}_{n+1} \, \hat{v}^{k+1} \in C^{k+2} ({{\mathbb R}})$.
Using the relation , we immediately obtain $w \in C^{k+2}({{\mathbb R}})$.
We have $\widetilde{w}_{n+1}' \, \hat{v}^{k+2}=(\widetilde{w}_{n+1}' \, \hat{v}^{k+1})
\, \hat{v} \in C^{k+1} ({{\mathbb R}})$, and using , we derive: $$\begin{gathered}
\label{recurrence3}
\big( \widetilde{w}_{n+1}' \, \hat{v}^{k+2} \big)' =\big(
g_{n+1} (\hat{v},\widetilde{w}_{n+1}) \, \hat{v}^{k+1} \big)' \\
= (\partial_1 g_{n+1}) (\hat{v},\widetilde{w}_{n+1}) \, \hat{v}^{k+1} \, w
+(\partial_2 g_{n+1}) (\hat{v},\widetilde{w}_{n+1}) \, \widetilde{w}_{n+1}' \, \hat{v}^{k+1}
+(k+1) \, g_{n+1} (\hat{v},\widetilde{w}_{n+1}) \, \hat{v}^k \, w \, ,\end{gathered}$$ where $\partial_1 g_{n+1}$ (resp. $\partial_2 g_{n+1}$) denotes the partial derivative of $g_{n+1}$ with respect to its first (resp. second) variable. From the definition , we see that $g_{n+1} (\hat{v},\widetilde{w}_{n+1})$ can be decomposed as follows: $$g_{n+1} (\hat{v},\widetilde{w}_{n+1})=-(n+2) \, \widetilde{w}_{n+1}^2 \, \hat{v}^{n+1}
+\widetilde{w}_{n+1} \, P_1(\hat{v}) +P_0(\hat{v}) \, ,$$ where $P_0$, and $P_1$ are polynomial functions. Using this decomposition, and the induction assumption , we can show that each term of the sum in the right-hand side of belongs to $C^{k+1} ({{\mathbb R}})$. Consequently $\widetilde{w}_{n+1}' \, \hat{v}^{k+2}$ belongs to $C^{k+2} ({{\mathbb R}})$, and holds with $k$ replaced by $k+1$. Because holds for $k=0$, we get that holds for $k=n+1$, so we have proved $w \in C^{n+2} ({{\mathbb R}})$, and $\hat{v} \in C^{n+3}({{\mathbb R}})$.
: it remains to show that $w$ satisfies the asymptotic expansion at the order $n+1$. Using , and $\widetilde{w}_{n+1} \in C^1 ({{\mathbb R}})$, we obtain: $$\widetilde{w}_{n+1} (\xi)-w_{n+1}=w_{n+2} \, \hat{v} (\xi) +o(\hat{v} (\xi))
\, ,\quad \text{as $\xi \rightarrow 0$.}$$ Plugging this expansion in , we obtain at the order $n+1$, so the proof of the induction is complete.
Once we know that the function $\hat{v}$ belongs to $C^{n+2} ({{\mathbb R}})$, for $a \in (0,a_n]$, then $v=\hat{v}+(v_-+v_+)/2$ also belongs to $C^{n+2} ({{\mathbb R}})$, and we have already seen in the previous section that $v$ does not vanish because $v(\xi)>v_+>0$ for all $\xi$. Moreover, the components $(\rho,u,e)$ of the shock profile are given by: $$\rho(\xi)=\dfrac{j}{v(\xi)} \, ,\quad u(\xi)=v(\xi)+\sigma \, ,\quad
e(\xi)=\dfrac{(C_1-v(\xi))\, v(\xi)}{\gamma-1} \, ,$$ so one has $(\rho,u,e) \in C^{n+2} ({{\mathbb R}})$, and the proof of Theorem \[smooth\] is complete. (Recall that the strength of the shock tends to zero if, and only if $a=|u_+-u_-|/2$ tends to zero.)
Formal derivation of the model {#model}
==============================
It is worth describing how the model , can be obtained from a more complete physical system. The derivation we propose below remains formal – a rigorous proof being certainly delicate and beyond the scope of this work – and we refer to [@BD; @GL; @LHM; @MM] for further details. Let us introduce the specific intensity of radiation $f(t,x,v)$, that depends on a time variable $t\ge 0$, a space variable $x\in {{\mathbb R}}^N$, and a direction $v\in
{{\mathbb S}}^{N-1}$. We make the ’grey assumption’, which means that the frequency dependence is ignored (all photons have the same frequency). Photons are subject to two main interaction phenomena:
- scattering produces changes in the direction of the photons,
- absorption/emission where photons are lost/produced through a transfer mecanism with the surrounding gas.
The scattering phenomenon is described by the operator: $$Q_s(f)(t,x,v)=\sigma_s \, \Big(
\displaystyle \int_{{{\mathbb S}}^{N-1}} f(t,x,v') \, dv'-f(t,x,v) \Big) \, ,$$ (with $dv$ the normalized Lebesgue measure on ${{\mathbb S}}^{N-1}$), and the absorption/emission phenomenon is described by the operator: $$Q_a(f)(t,x,v)=\sigma_a \, \Big(
\dfrac{\sigma}{\pi} \, \theta (t,x)^4 -f(t,x,v) \Big) \, ,$$ where we used the Stefan-Boltzmann emission law, $\theta$ being the temperature of the gas, and $\sigma$ the Stefan-Boltzmann constant. In these definitions, the coefficients $\sigma_{s,a}$ are given positive quantities. These phenomena are both characterized by a typical mean free path, denoted $\ell_s,\ell_a$ respectively. Therefore, the evolution of the specific intensity is driven by: $$\label{kindim}
\dfrac{1}{c} \, \partial_t f +v \cdot \nabla_x f =\dfrac{1}{\ell_s} \, Q_s(f)
+\dfrac{1}{\ell_a} \, Q_a(f)=Q(f) \, ,$$ where $c$ stands for the speed of light. The equation is coupled to the Euler system describing the evolution of the fluid: $$\label{eulerevolbis}
\begin{cases}
\partial_t \rho +\nabla_x \cdot (\rho \, u) =0 \, ,& \\
\partial_t (\rho \, u) +\nabla_x \cdot (\rho \, u\otimes u) +\nabla_x P
=-\dfrac{1}{c} \displaystyle \int_{{{\mathbb S}}^{N-1}} v \, Q(f) \, dv \, ,& \\
\partial_t (\rho \, E) +\nabla_x \cdot (\rho \, E \, u+P\, u)
=-\displaystyle \int_{{{\mathbb S}}^{N-1}} Q(f) \, dv \, .&
\end{cases}$$ The equations , are thus coupled by the exchanges of both momentum and energy, and by the Stefan-Boltzmann emission law. Observe that only the emission/absorption operator enters into the energy equation since the scattering operator is conservative (this would be different if Doppler corrections were taken into account). Note also that the total energy: $$\dfrac{1}{c} \displaystyle \int_{{{\mathbb R}}^N} \int_{{{\mathbb S}}^{N-1}} f \, dv \, dx
+\displaystyle \int_{{{\mathbb R}}^N} \rho \, E \, dx \, ,$$ is (formally) conserved. Writing the system , and the kinetic equation in the dimensionless form, we can make four dimensionless parameters appear:
- $\mathcal C$, the ratio of the speed of light over the typical sound speed of the gas,
- $\mathcal L_s$, the Knudsen number associated to the scattering,
- $\mathcal L_a$, the Knudsen number associated to the absorption/emission,
- $\mathcal P$, which compares the typical energy of radiation and the typical energy of the gas.
We thus obtain the rescaled equations: $$\label{adim}
\begin{cases}
\dfrac{1}{\mathcal C} \, \partial_t f + v \cdot \nabla_x f
=\dfrac{1}{\mathcal L_s} \, Q_s(f)+\dfrac{1}{\mathcal L_a} \, Q_a(f) \, ,& \\
\partial_t \rho +\nabla_x \cdot (\rho \, u)=0 \, ,& \\
\partial_t (\rho \, u) +\nabla_x \cdot (\rho \, u \otimes u) +\nabla_x P
=\dfrac{\mathcal P}{\mathcal L_s} \, \sigma_s \,
\displaystyle \int_{{{\mathbb S}}^{N-1}} v \, f(v) \, dv \, ,& \\
\partial_t (\rho \, E) +\nabla_x \cdot (\rho \, E \, u+P\, u)
=-\dfrac{\mathcal P}{\mathcal L_a} \, \sigma_a \, \Big(
\theta^4 -\displaystyle \int_{{{\mathbb S}}^{N-1}} f (v) \, dv \Big) \, .&
\end{cases}$$ System , is then obtained in two steps. First of all, we assume $\mathcal C \gg 1$. Next, we keep $\mathcal P$ of order 1, and we are concerned here with a regime where scattering is the leading phenomenon: the mean free paths are rescaled according to: $$\mathcal L_s \simeq \dfrac{1}{\mathcal C} \, ,\qquad
\mathcal L_a \simeq \mathcal C \, .$$ The asymptotics can be readily understood by means of the Hilbert expansion: $$f=f^{(0)} +\dfrac{1}{\mathcal C} \, f^{(1)} +\dfrac{1}{\mathcal C^2} \, f^{(2)}+\dots$$ Identifying the terms arising with the same power of $1/\mathcal C$, we get:
- at the leading order, $f^{(0)}$ belongs to the kernel of the scattering operator, so that is does not depend on the microscopic variable $v$: $f^{(0)}(t,x,v)=n(t,x)$,
- the relation $Q_s(f^{(1)})=v \cdot \nabla_x f^{(0)}$ then leads to: $f^{(1)}(t,x,v)=-\frac{1}{\sigma_s}\, v \cdot \nabla_x n(t,x)$,
- integrating the equation for $f^{(2)}$ over the sphere yields: $$\partial_t n -\dfrac{1}{N \, \sigma_s} \, \Delta_x n
=\sigma_a \, ( \theta^4 -n) \, .$$
Note also that in the momentum equation, we have: $$\dfrac{\sigma_s}{\mathcal L_s} \displaystyle \int_{{{\mathbb S}}^{N-1}} v \, f(v) \, dv
\simeq \sigma_s \, \displaystyle \int_{{{\mathbb S}}^{N-1}} v \, f^{(1)}(v) \, dv
=-\dfrac{1}{N} \, \nabla_x n \, .$$ Finally, we obtain the limit system: $$\label{lim1}
\begin{cases}
\partial_t \rho +\nabla_x \cdot (\rho \, u)=0\, ,& \\
\partial_t (\rho \, u) +\nabla_x \cdot (\rho \, u \otimes u) +\nabla_x P
=-\dfrac{\mathcal P}{N} \, \nabla_x n \, ,& \\
\partial_t (\rho \, E) +\nabla_x \cdot (\rho \, E \, u +P\, u)
=-\mathcal P \, \sigma_a \, ( \theta^4-n ) \, ,& \\
\partial_t n -\dfrac{1}{N \, \sigma_s} \, \Delta_x n =\sigma_a \,
( \theta^4-n ) \, .
\end{cases}$$ The system describes a nonequilibrium regime, where the material and the radiations have different temperatures ($\theta\neq n^{1/4}$); the equilibrium regime would correspond to assuming that the emission/absorption is the leading contribution.
After this first asymptotics, we perform a second asymptotics where we set: $$\mathcal P \ll 1 \, ,\qquad \mathcal P \, \sigma_a=1 \, ,\qquad
N\, \sigma_s=1/\sigma_a \, .$$ This leads to , . Of course, one might wonder how this second approximation modifies the shock profiles compared to , in particular when we get rid of the radiative pressure in the momentum equation. We refer to [@MM page 579] for some aspects of this problem.
|
---
abstract: 'We construct the first analytic self-gravitating skyrmions with higher baryon charge in four dimensions for the $SU(3)$-Skyrme-Einstein-$\Lambda $ theory by combining the generalized hedgehog ansatz with the approach developed by Balachandran et al. to describe the first (numerical) example of a non-embedded solution. These are genuine $SU(3)$ analytic solutions instead of trivial embeddings of $SU(2)$ into $SU(3)$. The geometry is that of a Bianchi IX universe. The Skyrme ansatz is chosen in such a way that the *Skyrme field equations are identically satisfied in the sector with baryon charge 4* (we call these configurations diBaryons anyway to emphasize the importance of the non-embedded ansatz). The field equations reduce to a dynamical system for the three Bianchi IX scale factors. Particular solutions are explicitly analyzed. Traversable wormholes with NUT-AdS asymptotics supported by a topologically non-trivial $SU(3)$-sigma soliton are also constructed. The self-gravitating solutions admit also a suitable flat limit giving rise to Skyrmions of charge 4 confined in a box of finite volume maintaining the integrability of the $SU(3)$ Skyrme field equations. This formalism discloses a novel transition at finite baryon density arising from the competition between embedded and non-embedded solutions in which the non-embedded solutions prevail at high density while are suppressed at low densities.'
author:
- 'Eloy Ayón-Beato'
- Fabrizio Canfora
- Marcela Lagos
- Julio Oliva
- Aldo Vera
title: 'Gravitating diBaryons in the $SU(3)$-Skyrme-Einstein theory, their flat limit and a novel finite-density transition'
---
Introduction
============
One of the most important results in low-energy quantum chromodynamics (QCD) is that its effective action becomes the Skyrme model [@witten0]; a Bosonic action for a $SU(N)$-valued scalar field [@skyrme], the physical case being $SU(3)$. Its solitons, dubbed *Skyrmions*, represent Fermionic states whose topological charge is the baryon number [@witten0; @bala0], see also [@All; @ANW; @manton]. These results have been also generalized to curved space-times [@curved1f]. The Skyrme model has been deeply analyzed not only by its emergence in low energy QCD but also due to its relevance in General Relativity. For instance, black holes with a non-trivial Skyrme hair were found using numerical tools in [@lucock; @droz] providing the first genuine counterexample to the well-known no-hair conjecture; unlike other supposedly hairy black holes that were unstable at the end [@Bizon:1994dh], see also [@numerical1; @numerical2]. Cosmological applications of the Skyrme model have also been considered [@cosmo; @cosmo2].
Many of the important results in both the Skyrme model and the Einstein-Skyrme system have been derived numerically due to the highly non-linear character of the field equations and, until very recently, there were no analytic solutions with baryon charge in these models. Analytic results are not just of academic interest. For instance, it was known that large isospin chemical potentials lead to Skyrmion instabilities on flat space. However, it is just lately that this critical behavior in the chemical potential has been fully understood thanks to an analytic formula [@canfora9; @canfora8; @Canfora:2018clt] [@crist1] [@crist2]. Although the most relevant case corresponds to the $SU(3)$ group, many of the theoretical and numerical works have been performed for the $SU(2)$ case, which is already quite difficult in itself but not as much as the former one that not only exhibits five more generators infinitesimally, but also a non-constant curvature group manifold. However, since genuine features of the $SU(3)$ Skyrme model could have very important physical consequences it is important to pursue any hint contributing to its understanding. In the seminal works [@bala0; @Bala1] the first numerical example of a *non-embedded* solution was constructed. This is a genuine feature of the $SU(3)$ Skyrme model since the solution has spherical symmetry and at the same time baryon charge equal to 2; in contrast to the $SU(2)$ Skyrme model where the only stable solution with spherical symmetry, in the hedgehog sense, has unit baryon charge. These pioneering ideas have been generalized in [@kopeilovich; @ioannidou1; @ioannidou2; @ioannidou3]. Remarkably enough, in the present paper we provide the first analytic examples of these types. Very recently, following the techniques developed in [@canfora2; @yang1; @ferreira; @canfora10; @Giacomini:2017xno; @Astorino:2018dtr], the first analytic self-gravitating $SU(2)$ Skyrmions have been constructed in [@canfora6]. Here it will be shown that the configurations found in [@canfora6] can be generalized to genuine self-gravitating and topologically non-trivial $SU(3)$ configurations. The resulting ansatz is based on the $SO(3)$ subgroup of $SU(3)$ that has been introduced in [@bala0; @Bala1]. Moreover, we can define a suitable flat limit of these self-gravitating configurations in which they remain at the same time topologically non-trivial and integrable. Such a limit corresponds to confining the dibaryon in a finite volume. This construction discloses a novel finite density transition: the non-embedded solutions prevail at high Baryon density while the trivially-embedded solutions prevail at low Baryon density.
The paper is organized as follows: in the second section, the Einstein-Skyrme action is introduced. In the third section, the gravitating diBaryon is constructed. In the fourth section, the NUT-AdS wormhole is described. In the fifth section, diBaryons living within a finite flat spatial volume are introduced. In the sixth section, the possibility of a phase transition arising from the competition between embedded and non-embedded solutions is discussed. Eventually, some conclusions and perspectives are presented.
The Einstein-Skyrme theory
==========================
The Einstein-Skyrme system in presence of a cosmological constant is described by the action $$I[g,U]=\int d^{4}x\sqrt{-g}\left( \frac{R-2\Lambda }{2\kappa }
+\frac{K}{4}\mathrm{Tr}[A^{\mu }A_{\mu }
+\frac{\lambda }{8}F_{\mu \nu }F^{\mu \nu
}]\right) , \label{skyrmeaction}$$ where $R$ is the Ricci scalar, $A_{\mu }=U^{-1}\nabla _{\mu }U$, with $U\in
SU(N)$ and $\nabla _{\mu }$ the covariant derivative. Moreover $F_{\mu \nu
}=[A_{\mu },A_{\nu }]$, $\Lambda $ is the cosmological constant, $\kappa $ the gravitational constant, and the positive couplings $K$ and $\lambda $ are fixed by experimental data. In our convention $c=\hbar =1$ and Greek indices run over the four dimensional space-time with mostly plus signature.
The complete Einstein-Skyrme field equations read $$\nabla ^{\mu }A_{\mu }+\frac{\lambda }{4}\nabla ^{\mu }[A^{\nu },F_{\mu \nu
}]=0\ ,\quad G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\ ,
\label{skyrme-einstein}$$ where $G_{\mu \nu }$ is the Einstein tensor, and the Skyrme energy-momentum tensor is defined by $$\begin{aligned}
T_{\mu \nu }& =-\frac{K}{2}\mathrm{Tr}\left[ A_{\mu }A_{\nu }
-\frac{1}{2}g_{\mu \nu }A^{\alpha }A_{\alpha }
+\frac{\lambda }{4}\left( g^{\alpha \beta
}F_{\mu \alpha }F_{\nu \beta }\right. \right. \notag \\
& \left. \left. -\frac{1}{4}g_{\mu \nu }F_{\alpha \beta }F^{\alpha \beta
}\right) \right] . \label{tmunu}\end{aligned}$$ The winding number for a given solution is given by $$B=\frac{1}{24\pi ^{2}}\int \rho _{B}\ ,\quad \rho _{B}=\text{Tr}[\epsilon
^{ijk}A_{i}A_{j}A_{k}]\ . \label{winding}$$ When the topological density $\rho _{B}$ is integrated on a space-like surface, $B$ represents the baryon number of the configuration.
$SU(3)$ self-gravitating dibaryon
=================================
Before presenting the new results, first we will shortly describe the trivial embedding into $SU(3)$ of the $SU(2)$ self-gravitating solution found in [@canfora6].
The $SU(2)$ embedded self-gravitating Skyrmion
----------------------------------------------
The $SU(2)$ generalized hedgehog ansatz reads $$U(x^{\mu })=Y^{0}(x^{\mu })I\pm Y^{i}(x^{\mu })\Lambda _{i},\ \left(
Y^{0}\right) ^{2}+Y^{i}Y_{i}=1,$$ $$\Lambda _{i}=(\lambda _{1},\lambda _{2},\lambda _{3}),\
Y^{0}=\cos{\alpha},\ Y^{i}=n^{i}\sin {\alpha }, \label{sessea1}$$ where $\ n^{1}=\sin{\Theta }\cos{\Phi},\ n^{2}=\sin{\Theta }\sin {\Phi },\
n^{3}=\cos {\Theta }\,$ and $\left\{ \lambda _{j}\right\} _{j=1,..,8}$ are the Gell-Mann matrices. Notice that $\{\lambda _{1},\lambda _{2},\lambda
_{3}\}$ generate the $SU(2)$ subgroup of $SU(3)$. The above scalar functions are chosen in [@canfora6] as $$\Phi =\frac{\gamma +\varphi }{2}\ ,\ \tan \Theta =\frac{\cot \left( \frac{
\theta }{2}\right) }{\cos \left( \frac{\gamma -\varphi }{2}\right) }\ ,\
\tan \alpha =\frac{\sqrt{1+\tan ^{2}\Theta }}{\tan \left( \frac{\gamma
-\varphi }{2}\right) }\ , \label{sessea2}$$ while the metric is $$ds^{2}=-dt^{2}+\rho \left( t\right) ^{2}\left[ (d\gamma +\cos \theta
d\varphi )^{2}+d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}\right] \ ,
\label{simplem}$$ with the range of coordinates $$0\leq \gamma <4\pi \ ,\;\;\;\ 0\leq \theta <\pi \ ,\;\;\;\ 0\leq \varphi
<2\pi \ , \label{range1}$$ uniquely fixed by requiring the regularity of the metric. With this ansatz one can verify that the Skyrme field equations are identically satisfied, while the Einstein equations reduce to [@canfora6] $$\dot{\rho}^{2}=\frac{\Lambda }{3}\rho ^{2}+\frac{\lambda \kappa K}{32\rho
^{2}}+\frac{\kappa K-2}{8}\ ,\quad \ddot{\rho}=\frac{\Lambda }{3}\rho
-\frac{\lambda \kappa K}{32\rho ^{3}}\ . \label{isoskSU(2)}$$ Using Eqs. (\[winding\]) and (\[range1\]) the baryon number of this configuration turns out to be $B=1$.
The new $SU(3)$ non-embedded self-gravitating dibaryon
------------------------------------------------------
To construct a self-gravitating dibaryon, we use the remarkable ansatz introduced in [@bala0; @Bala1] for a dibaryon in flat space-time. This ansatz is constructed with the subalgebra of the Gell-Mann matrices generating the subgroup $SO(3)\subseteq SU(3)$, namely $\{\lambda
_{7},-\lambda _{5},\lambda _{2}\}$. The $U$ field reads $$\begin{aligned}
U_{B}& =\exp \left( i\psi \right) \mathbf{1}_{3\times 3}+i\sin \left( \chi
\right) \exp \left( -\frac{i\psi }{2}\right) \mathbf{T} \notag \\
& +\left( \cos \left( \chi \right) \exp \left( -\frac{i\psi }{2}\right)
-\exp \left( i\psi \right) \right) \mathbf{T}^{2}, \label{bala}\end{aligned}$$ $$\mathbf{T}=\vec{\Lambda}\cdot \hat{n},\ \hat{n}=(\sin \Theta \cos \Phi ,\sin
\Theta \sin \Phi ,\cos \Theta ),\ \vec{\Lambda}=(\lambda _{7},-\lambda
_{5},\lambda _{2}). \label{bala2}$$ In Refs. [@bala0; @Bala1] $\psi $ and $\chi $ are the radial profiles of the spherically symmetric $SU(3)$ Skyrmion, while $\Theta $ and $\Phi $ are chosen as the spherical angles. Here, we promote the four functions to new profiles depending on the coordinates of a generally curved space-time, which must be determined by solving the Skyrme field equations.
In the flat case, the baryon charge can be different from zero even if the profile $\psi $ vanishes [@bala0; @Bala1] but the field equations require $\psi $ to be non-trivial. On the other hand, in the Einstein-Skyrme case one can take $\psi =0$ for the curved space-times we consider as it will be now discussed.
At this point, we need an educated ansatz for the four functions appearing in the above dibaryon-like ansatz. A slight modification of the generalized hedgehog ansatz of [@canfora6] defined by $\psi =0$ and $$\Phi =\frac{\gamma +\varphi }{2}\ ,\ \tan \Theta =\frac{\cot \left( \frac{
\theta }{2}\right) }{\cos \left( \frac{\gamma -\varphi }{2}\right) }\ ,\
\tan \left( \frac{\chi }{2}\right) =\frac{\sqrt{1+\tan ^{2}\Theta }}{\tan
\left( \frac{\gamma -\varphi }{2}\right) }\ , \label{skbala}$$ does the job. Indeed, the $SU(3)$ Skyrme configuration (\[bala\])-(\[skbala\]) *identically satisfies the Skyrme field equations on any metric of the form (\[simplem\])-(\[range1\]).*
Thus, Einstein equations with the energy-momentum tensor in Eq. (\[tmunu\]) corresponding to the Skyrme configurations defined in Eqs. (\[bala\])-(\[skbala\]) reduce to $$\dot{\rho}^{2}=\frac{\Lambda }{3}\rho ^{2}+\frac{\lambda \kappa K}{8\rho ^{2}
}+\frac{2\kappa K-1}{4}\ ,\quad \ddot{\rho}=\frac{\Lambda }{3}\rho -\frac{
\lambda \kappa K}{8\rho ^{3}}\ . \label{isosk}$$ One can write down the most general solution of Eqs. (\[isosk\]) following the analysis of [@canfora10]. The above Eq. (\[isosk\]) in the $SU(3)$ case are very similar to the $SU(2)$ one in Eq. (\[isoskSU(2)\]), but one can notice that the coefficients appearing are different; in fact, the $SU(3)$ energy-momentum contribution is four times that of $SU(2)$. This difference also appears when computing the topological charge density $\rho_{B}^{\text{SU(3)}}$ of the $SU(3)$ ansatz, defined in Eqs. (\[bala\])-(\[skbala\]), and the corresponding $\rho_{B}^{\text{SU(2)}}$ of the $SU(2) $ ansatz, defined in Eqs. (\[sessea1\]) and (\[sessea2\]), $$\rho _{B}^{\text{SU(3)}}=6\sin {\theta }\ ,\qquad \rho _{B}^{\text{SU(2)}}=
\frac{3}{2}\sin {\theta }\ .$$ Taking into account Eqs. (\[winding\]) and (\[range1\]), the topological charge of the $SU(3)$ configuration is $B=4$. These are genuine $SU(3)$ configurations: in $SU(2)$ one cannot obtain configurations with topological charge 4 compatible with the metric in Eq. (\[simplem\]).
Another interesting quantity is the ratio $$\Delta =\frac{\text{Vol}\left( SU(3)\right) }{\text{Vol}\left( SU(2)\right)}
=\frac{1}{2\sqrt{2}}\left(\frac{2K\kappa-1}{K\kappa-2}\right)^{\frac{3}{2}}\,$$ between the three-dimensional volume of the spatial section of the static gravitating dibaryons (corresponding to the static solutions of Eq. (\[isosk\])) and the static gravitating Skyrmions of [@canfora6](corresponding to the static solutions of Eq. (\[isoskSU(2)\])). It reveals the non-trivial strong and gravitational interactions of the system (as, in the case of four non-interacting solitons, one should expect $\Delta =4$).
The above regular solutions of the Einstein-$\Lambda $-$SU(3)$-Skyrme system defined in Eqs. (\[simplem\]), (\[range1\]), (\[bala\]), (\[bala2\]) and (\[skbala\]), are the first analytic self-gravitating Skyrmions of higher baryonic charge in $(3+1)$-dimensions. We decided to keep the original name to emphasize the importance of the non-embedded ansatz.
The fact that the baryon charge is 4 instead of 2 as in [@bala0; @Bala1], is related to the compactness of the $t=\text{const.}$ hypersurfaces of the metric (\[simplem\]). Here, rather than requiring boundary conditions at spatial infinity as in [@bala0; @Bala1], one has to require periodic boundary conditions for $U_{B}$ compatible with the compact spatial metric (\[simplem\])-(\[range1\]). This charge 4 arises due to the fact that, unlike what happens in [@bala0; @Bala1], the present gravitating diBaryons must wrap around three compact spatial directions instead of two.
Bianchi IX
----------
The above construction can be further extended to a Bianchi IX cosmology. Indeed, the $SU(3)$ *Skyrme field equations on the metric* $$\begin{aligned}
ds^{2}& =-dt^{2}+I_{1}^{2}(\cos \theta d\gamma +d\phi )^{2} \notag \\
& +I_{2}^{2}(\cos \phi d\theta +\sin \theta \sin \phi d\gamma
)^{2}+I_{3}^{2}(\sin \phi d\theta -\sin \theta \cos \phi d\gamma )^{2}\ ,
\label{BianchiIX}\end{aligned}$$ *where $I_{j}=I_{j}(t)$, are still identically satisfied with the same ansatz defined in Eqs. (\[bala\]), (\[bala2\]) and (\[skbala\])!* The reason behind this quite remarkable fact is that the left-invariant forms that appear in the construction of the most general Bianchi IX metric are proportional to the left-invariant forms that appear when computing the skyrmion derivatives $$\left( U_{B}\right) ^{-1}\partial _{\mu }U_{B}=\Omega _{\mu }^{a}\lambda
_{a}\ ,$$ with the $U_{B}$ defined in Eqs. (\[bala\]), (\[bala2\]) and (\[skbala\]). Since the $\Omega _{\mu }^{a}$ characterizing the Skyrmionic configuration play, at the same time, also the role of drei-bein of the spatial metric, huge simplifications appear in the field equations. Hence, the complete set of coupled Einstein-$\Lambda$-$SU(3)$-Skyrme field equations (\[skyrme-einstein\]) and (\[tmunu\]) for the metric (\[BianchiIX\]) with the ansatz in Eqs. (\[bala\]), (\[bala2\]) and (\[skbala\]), reduce to a consistent dynamical system for the three Bianchi IX scale factors
$$\begin{aligned}
I^{(4)}-2\mathcal{I}+4\Lambda \mathcal{I}_{(3)}^{2}-4\mathcal{I}_{(3)}\left(
I_{1}^{\prime }I_{2}^{\prime }I_{3}+I_{1}^{\prime }I_{2}I_{3}^{\prime
}+I_{1}I_{2}^{\prime }I_{3}^{\prime }\right) +\frac{K\kappa }{2}(\lambda
I^{(2)}+4\mathcal{I})& =0\ , \\
\left( 4I_{1}^{4}-I^{(4)}-2\mathcal{I}+4I_{2}^{2}I_{3}^{2}\right)
I_{2}I_{1}^{\prime }+\left( 4I_{1}^{4}-2I^{(4)}+4(1-\Lambda
I_{1}^{2})I_{2}^{2}I_{3}^{2}\right) I_{1}I_{2}^{\prime }& \\
{}+4\mathcal{I}_{(3)}I_{3}\left( I_{1}I_{1}^{\prime }I_{2}^{\prime
2}+I_{2}(I_{1}I_{2})^{\prime }I_{1}^{\prime \prime }\right)
+\frac{K\kappa }{2}\left[ \lambda I_{1}\left( 2I_{1}(I_{1}I_{2})^{\prime
}-I^{(2)}I_{2}^{\prime }\right) +4I_{2}\left( \mathcal{I}I_{1}^{\prime
}-2I_{2}I_{3}^{2}(I_{1}I_{2})^{\prime }\right) \right] & =0\ , \\\end{aligned}$$
and $$\begin{aligned}
\left( 4I_{2}^{4}-I^{(4)}-2\mathcal{I}+4I_{1}^{2}I_{3}^{2}\right)
I_{1}I_{2}^{\prime }+\left( 4I_{2}^{4}-2I^{(4)}+4(1-\Lambda
I_{2}^{2})I_{1}^{2}I_{3}^{2}\right) I_{2}I_{1}^{\prime }& \\
{}+4\mathcal{I}_{(3)}I_{3}\left( I_{2}I_{2}^{\prime }I_{1}^{\prime
2}+I_{1}(I_{1}I_{2})^{\prime }I_{2}^{\prime \prime }\right)
+\frac{K\kappa }{2}\left[ \lambda I_{2}\left( 2I_{2}(I_{1}I_{2})^{\prime
}-I^{(2)}I_{1}^{\prime }\right) +4I_{1}\left( \mathcal{I}I_{2}^{\prime
}-2I_{1}I_{3}^{2}(I_{1}I_{2})^{\prime }\right) \right] & =0\ ,\end{aligned}$$
where we have defined $I^{(2)}=I_{1}^{2}+I_{2}^{2}+I_{3}^{2}$, $I^{(4)}=I_{1}^{4}+I_{2}^{4}+I_{3}^{4}$, $\mathcal{I}
=I_{1}^{2}I_{2}^{2}+I_{1}^{2}I_{3}^{2}+I_{2}^{2}I_{3}^{2}$, $\mathcal{I}_{(3)}=I_{1}I_{2}I_{3}$.
As the SU(3) Skyrme field equations are automatically satisfied, the above dynamical system for the scale factors $I_{j}\left( t\right) $ describes the dynamical evolution of the four self-gravitating Skyrmions. The analysis of such system can reveal many interesting features about the interplay between the gravitational and the strong interactions of these four baryons. A comparison with the numerical gravitating Skyrmions constructed in [@satosawado] is useful. Their numerical solutions with Baryon charge 2 (which are based on the $SU(2)$ subgroup of $SU(3)$) are asymptotically flat while the present analytic solutions with Baryon charge 4 (which are based on a non-embedded ansatz) either have compact spatial sections (in the Bianchi IX sector) or possess two asymptotically NUT-AdS regions (see the section below).
NUT-AdS Wormhole
================
In this section we will consider the limit of vanishing Skyrme coupling $\lambda =0$. Following [@canfora6], a double-Wick rotation of the ansatz in Eq. (\[skbala\]) leads to $\psi =0$ and $$\Phi =\frac{t+\varphi }{2},\
\tan \Theta =\frac{\cot\left(\frac{\theta}{2}\right)}
{\cos \left(\frac{t-\varphi}{2}\right) },\
\tan \left( \frac{\chi}{2}\right) =
\frac{\sqrt{1+\tan ^{2}\Theta }}{\tan \left( \frac{t-\varphi }{2}\right) },
\label{balawor}$$ for the $SU(3)$ non-linear sigma model. Interestingly enough, the field equations of the $SU(3)$ non-linear sigma model with the ansatz defined in Eqs. (\[bala\]), (\[bala2\]) and (\[balawor\]) on the metric $$ds^{2}=\rho (\gamma )^{2}\left[ -Q^{2}(dt+\cos {\theta }d\varphi
)^{2}+d\theta ^{2}+\sin ^{2}{\theta }d\varphi ^{2}\right] +d\gamma ^{2}\ ,
\label{balawormetric}$$ are identically satisfied. A direct computation shows that the coupled Einstein-$\Lambda $-$SU(3)$-non-linear sigma model field equations (\[skyrme-einstein\]) and (\[tmunu\]) with $\lambda =0$ are satisfied for $$\rho (\gamma )=\sqrt{\frac{3(\kappa K-2)}{4|\Lambda |}}
\cosh \left( \sqrt{{\frac{|\Lambda |}{3}}}{\gamma }\right) \ ,\quad \
Q^{2}=\kappa K\ ,$$ provided $\Lambda <0$. Notice that the square of the NUT parameter $Q^{2}$ is four times the result for $SU(2)$ [@canfora6]. The above metric represents a traversable Lorentzian wormhole with NUT-AdS asymptotic supported by a reasonable physical source (see [@canfora6] [@cawor]); the non-linear sigma models do not violate energy conditions. The well-known no-go results on the existence of wormholes are avoided due to the presence of a NUT parameter. The $SU(3)$ wormhole throat is larger than the one of the $SU(2)$ configuration in [@canfora6; @cawor] since the present matter field possesses a higher topological charge.
Dibaryons at finite density
===========================
The first analytic solutions with non-vanishing baryon charge in the $SU(2)$ Skyrme model on flat space have been found by adapting the self-gravitating solution found in [@canfora6] to a flat space with finite volume in [@canfora9]. A similar construction also works in the present case. The sector explored below describes four low-energy baryons confined within a finite volume in flat space.
Consider the $SU(3)$ Skyrme configuration defined by $\psi =0$ and $$\Phi =\frac{\gamma +\varphi }{2},\
\tan \Theta = \frac{\tan H\left(
t,z\right) }{\cos \left( \frac{\gamma -\varphi }{2} \right) },\
\tan \left(\frac{\chi }{2}\right) =\frac{\sqrt{1+\tan ^{2}\Theta }}
{\tan \left( \frac{\gamma -\varphi }{2}\right) }, \label{flatBox}$$ together with Eqs. (\[bala\]) and (\[bala2\]) in a flat metric of the form $$ds^{2}=-dt^{2}+L_0^2\left[ dz^{2}+d\gamma ^{2}+d\varphi ^{2}\right] \ ,
\label{flatmetric}$$ where $L_{0}$ has dimension of length and represents the size of the box in which the baryons are confined, and the dimensionless coordinates have the following ranges $$0\leq z\leq 2\pi \ ,\quad 0\leq \gamma \leq 4\pi \ ,\quad 0\leq \varphi \leq
2\pi \ .$$
The topological density is again different from the $SU(2)$ case since $$\rho _{B}^{\text{SU(3)}}=-12\sin (2H)\partial _{z}H\ ,\qquad
\rho _{B}^{\text{SU(2)}}=-3\sin (2H)\partial _{z}H\ .$$ Considering the boundary conditions as $H(t,0)=0$ and $H(t,2\pi)=\pm\frac{\pi}{2}$, the $SU(3)$ topological charge is $\pm 4$ (which is four times the charge found in [@canfora9; @canfora8; @Canfora:2018clt]). Concretely, these Skyrmions confined to a finite volume can only have charges $-4$, $0$, or $4$ (while in the $SU(2)$ case defined in [@canfora9; @canfora8; @Canfora:2018clt], the corresponding Skyrmions can only have charges $-1$, $0$, or $1$). This confirms the genuine $SU(3)$ nature of the ansatz defined in Eqs. (\[bala\]), (\[bala2\]) and (\[flatBox\]).
The $SU(3)$ Skyrme field equations on the flat metric (\[flatmetric\]) for the baryon-like ansatz (\[bala\]), (\[bala2\]) and (\[flatBox\]) become a single partial differential equation for a scalar a profile $$\Box H-\frac{\lambda }{8L_{0}^{2}(2L_{0}^{2}+\lambda )}\sin (4H)=0\ .$$ Hence, the eight coupled $SU(3)$ Skyrme field equations collapse to a single integrable PDE. This opens the intriguing possibility to analyze many interesting non-trivial properties of these multi-Skyrmions confined to a finite volume using Sine-Gordon theory such as possible phase transitions between the trivial embedding of the $SU(2)$ solutions [@canfora9] and the present non-embedded $SU(3)$ solutions living in the same box.
A novel transition at finite Baryon density
===========================================
In the present section we will show that, within the same box of finite volume, there is a competition between embedded and non-embedded configurations with the same Baryonic charge, the control parameter of the transition being the Baryon density. We show here below that at high Baryon density non-embedded solutions are favoured over embedded solutions (and vice-versa at low Baryon density).
A suitable ansatz to describe $SU(3)$ configurations living in the flat box defined above that are trivial embedding of $SU(2)$ into $SU(3)$ is $$\begin{aligned}
\vec{\Lambda} &=&(\lambda _{1},\lambda _{2},\lambda _{3})\ ,\quad
\Phi =\frac{p\gamma +q\varphi }{2}\ , \label{flatBox2} \\
\quad \tan \Theta &=&\frac{\tan H\left( t,z\right) }
{\cos \left( \frac{p\gamma -q\varphi }{2}\right) }\ ,\quad
\tan \left( \chi \right) =\frac{\sqrt{1+\tan ^{2}\Theta }}
{\tan \left( \frac{p\gamma -q\varphi }{2}\right) }\ , \notag\end{aligned}$$ where $p$ and $q$ are non-vanishing integer numbers (see [@last]) and $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ the first three Gell-Mann matrices.
As we explained in Section V, as well as in [@last], both in the diBaryon case (Eqs. (\[bala\]) and (\[flatBox\])) and in the case of trivially embedded solutions (Eqs. (\[bala\]) and (\[flatBox2\])) the consistent boundary conditions for the profile $H(t,r)$ are $$H(t,0)=0\ ,\ \ H(t,2\pi )=\pm \frac{\pi }{2}\ ,$$
The topological densities read $$\rho _{B}^{\text{SU(3)}}=-12\sin (2H)\partial _{z}H\ ,\qquad
\rho _{B}^{\text{SU(2)}}=-3pq\sin (2H)\partial _{z}H\ .$$ We know that the topological charge in the $SU(3)$ case is $B^{SU(3)}=4$ , while the topological charge in the trivially embedded solutions with the above slightly generalized ansatz is $$B=pq\ ,$$ where the integers $p$ and $q$ are the ones appearing in the ansatz in Eqs. (\[bala\]) and (\[flatBox2\]) (see also [@last]).
Thus, if we choose $$pq=4\ ,$$ then we will have two inequivalent configurations[^1] (the one in Eqs. (\[bala\]) and (\[flatBox\]) as well as the one in Eqs. (\[bala\]) and (\[flatBox2\])) with the same charge living in the same box. The natural question is:
*Which type of configuration is energetically favoured?*
Thanks to the remarkable properties of the ansatz described above, one can answer this question explicitly. The first step is to notice that in both cases (the one in Eqs. (\[bala\]) and (\[flatBox\]) as well as the one in Eqs. (\[bala\]) and (\[flatBox2\])) the $SU(3)$ Skyrme field equations in the flat metric (\[flatmetric\]) reduce to a single partial differential equation of the form $$\Box H-\beta \sin (4H)=0\ . \label{SineGordon}$$ with $$\beta_{\text{SU(3)}}=\frac{\lambda }{8L_{0}^{2}(2L_{0}^{2}+\lambda )}\,\quad
\beta_{\text{SU(2)}}=\frac{p^{2}q^{2}\lambda }{4L_{0}^{2}(4L_{0}^{2}
+\lambda (p^{2}+q^{2}))}\ ,$$ for (\[flatBox\]) and (\[flatBox2\]), respectively.
For static configurations $H(t,r)=H(r)$, Eq. (\[SineGordon\]) admits a first integral $$\left( H^{\prime }\right) ^{2}+\frac{1}{2}\beta \cos (4H)=I_{0}\ ,$$ with the integration constant defined using the boundary conditions as $$\int_{0}^{2\pi }dr=\int_{0}^{\frac{\pi }{2}}\frac{dH}{\sqrt{I_{0}
-\frac{1}{2}\beta \cos (4H)}}=2\pi \ .$$ The energy density for these configurations are given by $$\begin{aligned}
T_{00}^{\text{SU(3)}}& =-\frac{K}{16L_{0}^{4}}\left( 16L_{0}^{2}
+\lambda (1-\cos(4H))+\frac{16}{L^{2}}(2L_{0}^{2}+\lambda )H^{\prime 2}
\right) \ , \\
T_{00}^{\text{SU(2)}}& =-\frac{K}{16L_{0}^{4}}\biggl(8L_{0}^{2}(p^{2}+q^{2})
+p^{2}q^{2}\lambda (1-\cos
(4H))+8(4L_{0}^{2}+(p^{2}+q^{2})\lambda )H^{\prime 2}\biggl)\ .\end{aligned}$$
---------------- ------------------- ------------------- ------------------- -------------
$L_0$ $E^{SU(2)}_{2,2}$ $E^{SU(2)}_{1,4}$ $E^{SU(2)}_{4,1}$ $E^{SU(3)}$
\[0.5ex\] 0.01 283.131 416.22 416.22 147.81
0.1 29.627 44.317 44.317 16.198
0.5 12.288 21.925 21.925 10.082
1 16.040 31.359 31.359 15.675
1.4 20.455 41.033 41.033 20.899
2 27.637 56.414 56.414 29.054
3 40.093 82.814 82.814 42.933
5 65.557 136.421 136.421 70.999
10 129.980 271.440 271.440 141.528
---------------- ------------------- ------------------- ------------------- -------------
: Phase transition $SU(3)$ vs. $SU(2)\subset SU(3)$ at finite density.[]{data-label="T1"}
TABLE \[T1\][^2] shows the energy comparison between the $SU(2)$ and the $SU(3)$ configurations with baryon charge $B=4$, while it varies the size of the box (measured by $L_{0}$). We have fixed $K=2$, $\lambda =1$ (which means that we are measuring length in $fm$). One can see that a phase transition appears at[^3] $L_{0}^{\ast }\approx 1.4$. When $L_{0}<L_{0}^{\ast }$ then the diBaryon are favoured over the trivially embedded solutions (and vice-versa when $L_{0}>L_{0}^{\ast }$). The appearance of this transition is related to the fact that diBaryons deal more efficiently with the well known repulsive interactions between Skyrmions. From the plots of the energy densities below for the two types of configurations one can see that, at high Baryon density, it pays off that the diBaryon is “less peaked” around its maximum while at low Baryon densities the energy densities of both types of solutions become relatively flat and, in such cases, the trivially embedded solutions prevail.
Fig. 1 shows the energy density of the three relevant configurations, for three different values of the size of the box.
At last, we make a small observation about the quantization of the solutions. Since, with the above ansatz, the hedgehog property holds (as the 8 coupled $SU(3)$ field equations reduce to a single PDE for the profile $H$). The small fluctuations of the profile $H$ around these solutions are described by the effective action obtained replacing the ansatz itself into the original $SU(3)$ Skyrme action ([@coleman], [@shifman]). Thus, the quantization of the diBaryon living in the finite box defined above can be analyzed using known results on the sine-Gordon theory plus the semiclassical quantization of the Isospin degrees of freedom described in [@Bala1]. This observation also shows that the flat configurations constructed in the manuscript are stable, at least, under the perturbations which keep the hedgehog properties. Namely, the flat solutions considered here are stable under the following type of perturbations (see the analysis in [@coleman] and [@shifman]): $$H(t,r)\rightarrow H(t,r)+\varepsilon u\left( t,r\right) \ ,\ \ 0<\varepsilon
\ll 1\ .$$ We hope to come back on the interesting but rather difficult problem of the full stability analysis (which must be analyzed numerically) in a future publication.
Conclusions and perspectives
============================
The first regular analytic self-gravitating skyrmions of higher baryon charge have been constructed. The space-time corresponds to a general Bianchi IX cosmology whose three scales factors evolve accordingly, while the Skyrme field equations are identically satisfied in the sector with baryon charge 4. All these solutions disclose genuine features of the $SU(3)$ Skyrme model. Traversable wormholes with NUT-AdS asymptotic supported by an $SU(3)$ regular solitonic solution of the resulting non-linear sigma model are also constructed. Also, a suitable flat limit in a finite volume of the self-gravitating Skyrme configurations can also be defined. In this case, the eight $SU(3)$ Skyrme field equations become the Sine-Gordon field theory without sacrificing the higher baryon charge. This flat limit explicitly describes a charge 4 baryons confined in a box. The present formalism discloses the existence of a novel transition between embedded and non-embedded configurations with the same Baryonic charge. The control parameter of this transition is the Baryon density: at high Baryon densities, it is energetically convenient to have non-embedded solutions while, at low Baryon densities, trivially embedded solutions prevail. To the best of authors knowledge, this phenomenon related to the competition between embedded and non-embedded solutions at finite Baryon density is new. The reason is that the present techniques are especially suitable to deal with Skyrmions and diBaryons at finite density as these techniques allow to determine how relevant physical properties of these solitons depend on the Baryon density. Without these informations it would have been impossible to disclose such possibility.
M.L. appreciates the support of FONDECYT postdoctoral grant 3190873. This work has been partially funded by the Fondecyt grants 1160137, 1181047, and the Conacyt grant A1-S-11548. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.
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[^1]: Using the elegant arguments in [@bala0], one can easily see that these two configurations cannot be deformed into each other using global isospin transformations.
[^2]: The notation $E_{X,Y}^{SU(2)}$ means “the energy of the $SU(2)$ configuration in Eqs. (\[bala\]) and (\[flatBox2\]) with $p=X$ and $q=Y$.” In the table $XY=4$ as we need to compare configurations with the same topological charge.
[^3]: It is worth to mention that at this scale, the Skyrme model is still well within its range of validity.
|
---
abstract: 'Learning interpretable disentangled representations is a crucial yet challenging task. In this paper, we propose a weakly semi-supervised method, termed as *Dual Swap Disentangling (DSD)*, for disentangling using both labeled and unlabeled data. Unlike conventional weakly supervised methods that rely on full annotations on the group of samples, we require only limited annotations on paired samples that indicate their shared attribute like the color. Our model takes the form of a dual autoencoder structure. To achieve disentangling using the labeled pairs, we follow a “encoding-swap-decoding” process, where we first swap the parts of their encodings corresponding to the shared attribute, and then decode the obtained hybrid codes to reconstruct the original input pairs. For unlabeled pairs, we follow the “encoding-swap-decoding” process twice on designated encoding parts and enforce the final outputs to approximate the input pairs. By isolating parts of the encoding and swapping them back and forth, we impose the dimension-wise modularity and portability of the encodings of the unlabeled samples, which implicitly encourages disentangling under the guidance of labeled pairs. This dual swap mechanism, tailored for semi-supervised setting, turns out to be very effective. Experiments on image datasets from a wide domain show that our model yields state-of-the-art disentangling performances.'
author:
- |
Zunlei Feng\
Zhejiang University\
`[email protected]`\
Xinchao Wang\
Stevens Institute of Technology\
`[email protected]`\
Chenglong Ke\
Zhejiang University\
`[email protected]`\
Anxiang Zeng\
Alibaba Group\
`[email protected]`\
Dacheng Tao\
University of Sydney\
`[email protected]`\
Mingli Song\
Zhejiang University\
`[email protected]`\
bibliography:
- 'nips\_2018.bib'
title: Dual Swap Disentangling
---
Introduction
============
Disentangling aims at learning dimension-wise interpretable representations from data. For example, given an image dataset of human faces, disentangling should produce representations or encodings for which part corresponds to interpretable attributes like facial expression, hairstyle, and color of the eye. It is therefore a vital step for many machine learning tasks including transfer learning ( [@lake2017building]), reinforcement learning ( [@higgins2017darla]) and visual concepts learning ( [@Higgins2017SCAN]).
Existing disentangling methods can be broadly classified into two categories, supervised approaches and unsupervised ones. Methods in the former category focus on utilizing annotated data to explicitly supervise the input-to-attribute mapping. Such supervision may take the form of partitioning the data into subsets which vary only along some particular dimension ( [@Kulkarni2015Deep; @Bouchacourt2017Multi]), or labeling explicitly specific sources of variation of the data ( [@Kingma2014Semi; @Siddharth2017Learning; @Perarnau2016Invertible; @Wang2017Tag]). Despite their promising results, supervised methods, especially for deep-learning ones, usually require a large number of training samples which are often expensive to obtain.
Unsupervised methods, on the other hand, do not require annotations but yield disentangled representations that are usually uninterpretable and dimension-wise uncontrollable. In other words, the user has no control over the semantic encoded in each dimension of the obtained codes. Taking a photo of a human face for example, the unsupervised approach fails to make sure that one of the disentangled codes will contains the feature of hair color. In addition, existing methods produce for each attribute with a single-dimension code, which sometimes has difficulty in expressing intricate semantics.
In this paper, we propose a *weakly semi-supervised* learning approach, dubbed as *Dual Swap Disentangling (DSD)*, for disentangling that combines the best of the two worlds. The proposed DSD takes advantage of limited annotated sample pairs together with many unannotated ones to derive dimension-wise and semantic-controllable disentangling. We implement the DSD model using an autoencoder, training on both labeled and unlabeled input data pairs and by swapping designated parts of the encodings. Specifically, DSD differs from the prior disentangling models in the following aspects.
- Limited Weakly-labeled Input Pairs. Unlike existing supervised and semi-supervised models that either require strong labels on each attribute of each training sample ( [@Kingma2014Semi; @Perarnau2016Invertible; @Siddharth2017Learning; @Wang2017Tag; @Banijamali2017JADE]), or require fully weak labels on a group of samples sharing the same attribute ( [@Bouchacourt2017Multi]), our model only requires *limited pairs of samples*, which are much cheaper to obtain.
- Dual-stage Architecture. To our best knowledge, we propose the first dual-stage network architecture to utilize unlabeled sample pairs for semi-supervised disentangling, to facilitate and improve over the supervised learning using a small number of labeled pairs.
- Multi-dimension Attribute Encoding. We allow multi-dimensional encoding for each attribute to improve the expressiveness capability. Moreover, unlike prior methods ( [@Kulkarni2015Deep; @Chen2016InfoGAN; @Higgins2016beta; @Burgess2017Understanding; @Bouchacourt2017Multi; @Chen2018Isolating; @Gao2018Auto; @Kim2018Disentangling]), we do not impose any over-constrained assumption, such as each dimension being independent, into our encodings.
We show the architecture of DSD in Fig. \[fig:DSD\_framework\]. It comprises two stages, primary-stage and dual-stage, both are utilizing the same autoencoder. During training, the annotated pairs go through the primary-stage only, while the unannotated ones go through both. For annotated pairs, again, we only require weak labels to indicate which attribute of the two input samples is sharing. We feed such annotated pairs to the encoder and obtained a pair of codes. We then designate which dimensions correspond to the specific shared attribute, and swap these parts of the two codes to obtain a pair of hybrid codes. Next we feed the hybrid codes to the decoder to reconstruct the final output of the labeled pairs. We enforce the reconstruction to approximate the input since we swap only the shared attribute, in which way we encourage the disentangling of the specific attribute in the designated dimensions and thus make our encodings dimension-wise controllable.
The unlabeled pairs during training go through both the primary-stage and the dual-stage. In the primary-stage, unlabeled pairs undergo the exact same procedure as the labeled ones, i.e., the encoding-swap-decoding steps. In the dual-stage, the decoded unlabeled pairs are again fed into the same autoencoder and parsed through the encoding-swap-decoding process for the second time. In other words, the code parts that are swapped during the primary-stage are swapped back in the second stage. With the guidance and constraint of labeled pairs, the dual swap strategy can generate informative feedback signals to train the DSD for the dimension-wise and semantic-controllable disentangling. The dual swap strategy, tailored for unlabeled pairs, turns out to be very effective in facilitating supervised learning with a limited number of samples.
Our contribution is therefore the first dual-stage strategy for semi-supervised disentangling. Also, require limited weaker annotations as compared to previous methods, and extend the single-dimension attribute encoding to multi-dimension ones. We evaluate the proposed DSD on a wide domain of image datasets, in term of both qualitative visualization and quantitative measures. Our method achieves results superior to the current state-of-the-art.
Related Work
============
Recent works in learning disentangled representations have broadly followed two approaches, (semi-)supervised and unsupervised. Most of existing unsupervised methods ([@Burgess2017Understanding; @Chen2018Isolating; @Gao2018Auto; @Kim2018Disentangling; @Dupont2018Joint]) are based on two most prominent methods InfoGAN ([@Chen2016InfoGAN]) and $\beta$-VAE ([@Higgins2016beta]). They however impose the independent assumption of the different dimensions of the latent code to achieve disentangling. Some semi-supervised methods ([@Bouchacourt2017Multi; @Siddharth2017Learning]) import annotation information into $\beta$-VAE to achieve controllable disentangling. Supervised or semi-supervised methods like ([@Kingma2014Semi; @Perarnau2016Invertible; @Wang2017Tag; @Banijamali2017JADE]), they focus on utilizing annotated data to explicitly supervise the input-to-attribute mapping. Different with above methods, our method does not impose any over-constrained assumption and only require limited weak annotations.
We also give a brief review here about *swapping scheme*, *group labels*, and *dual mechanism*, which relate to our dual-stage model and weakly-labeled input. For *swapping*, [@Xiao2017DNA] propose a supervised algorithm called DNA-GAN which can learn disentangled representations from multiple semantic images with swapping policy. The significant difference between our DSD and DNA-GAN is that the swapped codes correspond to different semantics in DNA-GAN. DNA-GAN requires lots of annotated multi-labeled images and the annihilating operation adopted by DNA-GAN is destructive. Besides, DNA-GAN is based on GAN, which also suffers from the unstable training of GAN. For *group information*, [@Bouchacourt2017Multi] propose the Multi-Level VAE (ML-VAE) model for learning a meaningful disentanglement from a set of grouped observations. The group used in the ML-VAE requires that observations in the same group have the same semantics. However, it also has the limitation on increased reconstruction error. For *dual mechanism*, [@Zhu2017Unpaired] use cycle-consistent adversarial networks to realize unpair image-to-image translation. [@Xia2016Dual] adopt the dual-learning framework for machine translation. However, they all require two domain entities, such as image domains (sketch and photo) and language domains (English and French). Different with above two works, our dual framework only needs one domain entities.
Method
======
In this section, we give more details of our proposed DSD model. We start by introducing the architecture and basic elements of our model, then show our training strategy for labeled and unlabeled pairs, and finally summarize the complete algorithm.
Dual-stage Autoencoder
----------------------
The goal of our proposed DSD model is to take both weakly labeled and unlabeled sample pairs as input, and train an autoencoder that accomplishes dimension-wise controllable disentangling. We show a visual illustration of our model in Fig. \[fig:DSD\_framework\], where the dual-stage architecture is tailored for the self-supervision on the unlabeled samples. In what follows, we describe DSD’s basic elements: input, autoencoder, swap strategy and the dual-stage design in detail.
#### Input
DSD takes a pair of samples as input denoted as $ (\mathcal{I}_A,\mathcal{I}_B)$, where the pair can be either weakly labeled or unlabeled. Unlike conventional weakly supervised methods like [@Bouchacourt2017Multi] that rely on full annotations on the group of samples, our model only requires limited and weak annotations as we only require the labels to indicate which attribute, if any, is sharing by a pair of samples.
#### Autoencoder
DSD conducts disentangling using an autoencoder trained in both stages. Given a pair of input $ (\mathcal{I}_A,\mathcal{I}_B)$, weakly labeled or not, the encoder $f_{\phi}$ first encodes them to two vector representations $\mathcal{R}_A=f_{\phi}(\mathcal{I}_A)=[a_1,a_2,...,a_n]$ and $\mathcal{R}_B=f_{\phi}(\mathcal{I}_B)=[b_1,b_2,...,b_n]$, and then the decoder $f_{\varphi}$ decodes the obtained codes or encodings to reconstruct the original input pairs, i.e., $ \overline{\mathcal{I}_A}=f_{\varphi}(\mathcal{R}_A) $ and $\overline{\mathcal{I}_B}=f_{\varphi}(\mathcal{R}_B)$. We would expect the obtained codes $\mathcal{R}_A$ and $\mathcal{R}_B$ to possess the following two properties: i) they include as much as possible information of the original input $ \mathcal{I}_A$ and $\mathcal{I}_B$, and ii) they are disentangled and element-wise interpretable. The first property, as any autoencoder, is achieved through minimizing the following original autoencoder loss: $$\label{eq3}
\mathbf{\mathcal{L}_o}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)=||\mathcal{I}_A-\overline{\mathcal{I}_A}||^2_2+||\mathcal{I}_B-\overline{\mathcal{I}_B}||^2_2.$$ The second property is further achieved via the swapping strategy and dual-stage design, described in what follows.
#### Swap Strategy
If given the knowledge that the pair of input $\mathcal{I}_A$ and $\mathcal{I}_B$ are sharing an attribute, such as the color, we can designate a specific part of their encodings, like $a_k$ of $\mathcal{R}_A$ and $b_k$ of $\mathcal{R}_B$, to associate the attribute semantic with the designated part. Assume that $\mathcal{R}_A$ and $\mathcal{R}_B$ are disentangled, swapping their code parts corresponding to the shared attribute, $a_k$ and $b_k$, should not change their encoding or their hybrid reconstruction $\ddot{\mathcal{I}_A}$ and $\ddot{\mathcal{I}_B}$. Conversely, enforcing the reconstruction after swapping to approximate the original input should facilitate and encourage disentangling for the specific shared attribute. Notably, here we allow each part of the encodings to be multi-dimensions, i.e., $a_k,b_k \in R^m, m \geq 1$, so as to improve the expressiveness of the encodings.
#### Dual-stage
For labeled pairs, we know what their shared attribute is and can thus swap the corresponding parts of the code. For unlabeled ones, however, we do not have such knowledge. To take advantage of the large volume of unlabeled pairs, we implement a dual-stage architecture that allows the unlabeled pairs to swap random designated parts of their codes to produce the reconstruction during the primary-stage and then swap back during the second stage. Through this process, we explicitly impose the element-wise modularity and portability of the encodings of the unlabeled samples, and implicitly encourages disentangling under the guidance of labeled pairs.
Labeled Pairs {#sec:labeledPairs}
-------------
For a pair of labeled input $ (\mathcal{I}_A,\mathcal{I}_B)$ in group $\mathcal{G}_k$, meaning that they share the attribute corresponding to the $k$-th part of their encodings $\mathcal{R}_A$ and $\mathcal{R}_B$, we swap their $k$-th part and get a pair of hybrid codes $ \ddot{\mathcal{R}_A}=[a_1,a_2,...,b_k,...,a_n]$ and $ \ddot{\mathcal{R}_B}=[b_1,b_2,...,a_k,...,b_n]$. We then feed the hybrid code pair $ \ddot{\mathcal{R}_A}$ and $ \ddot{\mathcal{R}_B}$ to the decoder $f_{\varphi}$ to obtain the final representation $ \ddot{\mathcal{I}_A} $ and $ \ddot{\mathcal{I}_B}$. We enforce the reconstructions $\ddot{\mathcal{I}_A} $ and $ \ddot{\mathcal{I}_B}$ to approximate $ (\mathcal{I}_A,\mathcal{I}_B)$, and encourage disentangling of the $k$-th attribute. This is achieved by minimizing the swap loss $$\label{eq4}
\mathbf{\mathcal{L}_s}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)=||\mathcal{I}_A-\ddot{\mathcal{I}_A}||^2_2+||\mathcal{I}_B-\ddot{\mathcal{I}_B}||^2_2,$$ so that the $k$-th part of $\mathcal{R}_A$ and $\mathcal{R}_B$ will only contain the shared semantic. The theoretical proof for the disentanglement of labeled pairs is provided in the supplementary material.
We take the total loss $\mathbf{\mathcal{L}_p}$ for the labeled pairs to be the sum of the original autoencoder loss $\mathbf{\mathcal{L}_o}$ and swap loss $\mathbf{\mathcal{L}_s}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)$: $$\label{eq6}
\mathbf{\mathcal{L}_p}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)=\mathbf{\mathcal{L}_o}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)+\alpha \mathbf{\mathcal{L}_s}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi),$$ where $ \alpha $ is a balance parameter.
\[alg:alg1\]
Initialize $ \phi^1 $ and $ \varphi^1 $. Random sample $k \in \{1,2,...,n \}$. Sample paired observation $ (\mathcal{I}_A,\mathcal{I}_B)$ from group $\mathcal{G}_k$. Encode $\mathcal{I}_A$ and $\mathcal{I}_B$ into $\mathcal{R}_A$ and $\mathcal{R}_B$ with encoder $f_{\phi^t}$. Swap the $k$-th part of $\mathcal{R}_A$ and $\mathcal{R}_B$ and get two hybrid representations $ \ddot{\mathcal{R}_A}$ and $ \ddot{\mathcal{R}_B}$. Construct $\mathcal{R}_A$ and $\mathcal{R}_B$ into $ \Bar{\mathcal{I}_A}=f_{\varphi^t}(\mathcal{R}_A) $ and $ \Bar{\mathcal{I}_B}=f_{\varphi^t}(\mathcal{R}_B)$. Construct $\ddot{\mathcal{R}_A}$ and $\ddot{\mathcal{R}_B}$ into $ \ddot{\mathcal{I}_A}=f_{\varphi^t}(\ddot{\mathcal{R}_A}) $ and $ \ddot{\mathcal{I}_B}=f_{\varphi^t}(\ddot{\mathcal{R}_B})$. Update $ \phi^{t+1}, \varphi^{t+1} \leftarrow \phi^{t}, \varphi^{t}$ by ascending the gradient estimate of $\mathbf{\mathcal{L}_p}(\mathcal{I}_A,\mathcal{I}_B;\phi^t,\varphi^t)$. Sample unpaired observation $ (\mathcal{I}_A,\mathcal{I}_B)$ from unannotated observation set $\mathbb{G}$. Encode $\mathcal{I}_A$ and $\mathcal{I}_B$ into $\mathcal{R}_A$ and $\mathcal{R}_B$ with encoder $f_{\phi^{t+1}}$. swap the $k$-th part of $\mathcal{R}_A$ and $\mathcal{R}_B$ and get two hybrid representations $ \ddot{\mathcal{R}_A}$ and $ \ddot{\mathcal{R}_B}$. Construct $\mathcal{R}_A$ and $\mathcal{R}_B$ into $ \Bar{\mathcal{I}_A}=f_{\varphi^{t+1}}(\mathcal{R}_A) $ and $ \Bar{\mathcal{I}_B}=f_{\varphi^{t+1}}(\mathcal{R}_B)$. Construct $\ddot{\mathcal{R}_A}$ and $\ddot{\mathcal{R}_B}$ into $ \ddot{\mathcal{I}_A}=f_{\varphi^{t+1}}(\ddot{\mathcal{R}_A}) $ and $ \ddot{\mathcal{I}_B}=f_{\varphi^{t+1}}(\ddot{\mathcal{R}_B})$. Encode $ (\ddot{\mathcal{I}_A},\ddot{\mathcal{I}_B})$ into $\ddot{\mathcal{R}'_A}$ and $\ddot{\mathcal{R}'_B}$ with encoder $f_{\phi^{t+1}}$. Swap the $k$-th parts of $\ddot{\mathcal{R}'_A}$ and $\ddot{\mathcal{R}'_B}$ backward and get $ \mathcal{R}'_A$ and $ \mathcal{R}'_B$. Construct $\mathcal{R}'_A$ and $\mathcal{R}'_B$ into $ \Bar{\Bar{\mathcal{I}_A}}=f_{\varphi^{t+1}}(\mathcal{R}'_A) $ and $ \Bar{\Bar{\mathcal{I}_B}}=f_{\varphi^{t+1}}(\mathcal{R}'_B)$. Update $ \phi^{t+2}, \varphi^{t+2} \leftarrow \phi^{t+1}, \varphi^{t+1}$ by ascending the gradient estimate of $\mathbf{\mathcal{L}_u}(\mathcal{I}_A,\mathcal{I}_B;\phi^{t+1},\varphi^{t+1})$.
Unlabeled Pairs {#sec:unlabeledPairs}
---------------
Unlike the labeled pairs that go through only the primary-stage, unlabeled pairs go through both the primary-stage and the dual-stage, in other words, the “encoding-swap-decoding” process is conducted twice for disentangling. Like the labeled pairs, in the primary-stage the unlabeled pairs $ (\mathcal{I}_A,\mathcal{I}_B)$ also produce a pair of hybrid outputs $ \ddot{\mathcal{I}_A}$ and $ \ddot{\mathcal{I}_B}$ through swapping a random $k$-th part of $\mathcal{R}_A$ and $\mathcal{R}_B$. In the dual-stage, the two hybrids $\ddot{\mathcal{I}_A}$ and $ \ddot{\mathcal{I}_B}$ are again fed to the same encoder $f_{\phi}$ and encoded as new representations $\ddot{\mathcal{R}'_A}=[a'_1,a'_2,...,b'_k,...,a'_n]$ and $ \ddot{\mathcal{R}'_B}=[b'_1,b'_2,...,a'_k,...,b'_n]$. We then swap back the $k$-th part of $\ddot{\mathcal{R}'_A}$ and $\ddot{\mathcal{R}'_B}$ and denote the new codes as $\mathcal{R}'_A=[a'_1,a'_2,...,a'_k,...,a'_n]$ and $\mathcal{R}'_B=[b'_1,b'_2,...,b'_k,...,b'_n]$. These codes are fed to the decoder $f_{\varphi}$ to produce the final output $ \Bar{\Bar{\mathcal{I}_A}}=f_{\varphi}(\mathcal{R}'_A)$ and $ \Bar{\Bar{\mathcal{I}_B}}=f_{\varphi}(\mathcal{R}'_B)$.
We minimize the reconstruction error of dual swap output with respect to the original input, and write the dual swap loss $\mathbf{\mathcal{L}_d}$ as follows: $$\label{eq5}
\mathbf{\mathcal{L}_d}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)=||\mathcal{I}_A-\Bar{\Bar{\mathcal{I}_A}}||^2_2+||\mathcal{I}_B-\Bar{\Bar{\mathcal{I}_B}}||^2_2.$$ The dual swap reconstruction minimization here provides a unique form of self-supervision. That is, by swapping random parts back and forth, we encourage the element-wise separability and modularity of the obtained encodings, which further helps the encoder to learn disentangled representations under the guidance of limited weak labels.
The total loss for the unlabeled pairs is consists of the original autoencoder loss $\mathbf{\mathcal{L}_o}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)$ and dual autoencoder loss $\mathbf{\mathcal{L}_d}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)$: $$\label{eq7}
\mathbf{\mathcal{L}_u}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)=\mathbf{\mathcal{L}_o}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)+\beta \mathbf{\mathcal{L}_d}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi),$$ where $ \beta$ is the balance parameter. As we will show in our experiment, adopting the dual swap on unlabeled samples and solving the objective function of Eq. \[eq7\], yield a significantly better result as compared to only using unlabeled samples during the primary-stage without swapping, which corresponds to optimizing over the autoencoder loss alone.
Complete Algorithm
------------------
Within each epoch during training, we alternatively optimize the autoencoder using randomly-sampled labeled and unlabeled pairs. The complete algorithm is summarized in Algorithm \[alg:alg1\]. Once trained, the encoder is able to conduct disentangled encodings that can be applied in many applications.
Experiments
===========
To validate the effectiveness of our methods, we conduct experiments on five image datasets of different domains: a synthesized Square dataset, MNIST ( [@Haykin2009GradientBased]), Teapot ( [@Moreno2016Overcoming; @Eastwood2018A]), CAS-PEAL-R1 ( [@Gao2008The]), and Mugshot ( [@Shen2016Automatic]). We firstly qualitatively assess the visualization of DSD’s generative capacity by performing swapping operation on the parts of latent codes, which verifies the *disentanglement* and *completeness* of our method. To evaluate the *informativeness* of the disentangled codes, we compute the classification accuracies based on DSD encodings. We are not able to use the framework of [@Eastwood2018A] as it is only applicable to methods that encode each semantic into a single dimension code.
Qualitative Evaluation
----------------------
We show in Fig. \[fig:visualizeResult\] some visualization results on the five datasets. For each dataset, we show input pairs, the swapped attribute, and results after swapping. We provide more results and implementation details (supervision rates, network architecture, code length and the number of semantic) in our supplementary material.
**Square** We create a synthetic image dataset of $60,000$ image samples ( $30,000$ pair images), where each image features a randomly-colored square at a random position with a randomly-colored background. Visual results of DSD on Square dataset are shown in Fig. \[fig:visualizeResult\](a), where DSD leads to visually plausible results.
**Teapot** The Teapot dataset used in [@Eastwood2018A] contains $200,000$ $64 \times 64$ color images of an teapot with varying poses and colors. Each generative factor is independently sampled from its respective uniform distribution: azimuth $(z_{0})\thicksim U \lceil 0,2\pi \rceil$, elevation $ (z_1)\thicksim U \lceil 0,2\pi \rceil$, red $ (z_2)~\thicksim U \lceil 0; 1\rceil$, green $(z4) \thicksim U \lceil 0; 1\rceil$. Fig. \[fig:visualizeResult\](b) shows the visual results on Teapot, where we can see that the five factors are again evidently disentangled.
**MNIST** In the visual experiment, we adopt InfoGAN to generate $5,000$ paired samples, for which we vary the following factors: digital identity ($0-9$), angle and stroke thickness. The whole training dataset contains $50,000$ samples: $5,000$ generated paired samples and $45,000$ real unpaired samples collected from the original dataset. Semantics swapping for MNIST are shown in Fig. \[fig:visualizeResult\](c), where the digits swap one attribute but preserve the other two. For example, when swapping the angle, the digital identity and thickness are kept unchanged. The generated images again look very realistic.
**CAS-PEAL-R1** CAS-PEAL-R1 contains $30,900$ images of $1,040$ subjects, of which $438$ subjects wear $6$ different types of accessories ($3$ types of glasses, and $3$ types of hat). There are images of $233$ subjects that involve at least $10$ lighting changes and at most $31$ lighting changes. Fig. \[fig:visualizeResult\](d) shows the visual results with swapped light, hat and glasses. Notably, the covered hairs by the hats can also be reconstructed when the hats are swapped, despite the qualities of hybrid images are not exceptional. This can be in part explained by the existence of disturbed paired samples, as depicted in the last column. This pair of images is in fact labeled as sharing the same hat, although the appearances of the hats such as the wearing angles are significantly different, making the supervision very noisy.
**Mugshot** We also use the Mugshot dataset which contains selfie images of different subjects with different backgrounds. This dataset is generated by artificially combining human face images in [@Shen2016Automatic] with $1,000$ scene photos collected from internet. Fig. \[fig:visualizeResult\](e) shows the results of the same mugshot through swapping different backgrounds, which are visually impressive. Note that, in this case we only consider two semantics, the foreground being the human selfie and the background being the collected scene. The good visual results can be partially explained by the fact that the background with different subjects has been observed by DSD during training.
Quantitative Evaluation
-----------------------
To quantitatively evaluate the *informativeness* of disentangled codes, we compare our methods with $4$ methods: InfoGAN ( [@Chen2016InfoGAN]), $\beta$-VAE ( [@Higgins2016beta]), Smi-VAE ( [@Siddharth2017Learning]) and basic Autoencoder. We first use InfoGAN to generate $5,0000$ pair digital samples, and then train all methods on this generated dataset. For InfoGAN and $\beta$-VAE , the lengths of their codes are set as $5$. To fairly compare with the above two methods, the codes’ length of Smi-VAE, Autoencoder and our DSD are taken to be $5\times 3$. In this condition, we can compare part of codes ($length=5$) that correspond to digit identity with whole codes ($length=5$) of InfoGAN and $\beta$-VAE and variable ($length=1$) that correspond to digit identity. After training all the models, real MNIST data are encoded as codes. Then, $55,000$ training samples are used to train a simple knn classifier and remaining $10,000$ are used as test samples. Table \[score\_table\] gives the classification accuracy of different methods, where the InfoGAN achieves the worst accuracy score. The DSD($0.5$) achieves best accuracy score, which further validates the informativeness of our DSD.
**Model** $\beta$-VAE(1) $\beta$-VAE(6) InfoGAN Semi-VAE Autoencoder DSD(0.5) DSD(1)
----------- ---------------- ---------------- ----------- ----------- ------------- --------------- ------------
**Acc** 0.22/0.72 0.25/0.71 0.19/0.51 0.22/0.57 0.66/0.93 **0.76**/0.91 0.742/0.90
: The accuracy score comparison among different models. DSD(n) denotes the DSD with $n$ supervision rate paired samples. Accuracy(**Acc**) values are shown as “$q/p$”, where $q$ is the accuracy obtained using the digital identity part of the codes for classification, and $p$ is the accuracy obtained using the whole codes.[]{data-label="score_table"}
In addition, we summarize different methods’ requirements in terms of label annotations into the Table \[data\_table\]. DSD is the only one that requires limited and weak labels, meaning that it requires the least amount of human annotation.
DC-IGN DNA-GAN TD-GAN Semi-DGM Semi-VAE JADE ML-VAE DSD
----------- -------- --------- -------- ----------- ----------- ----------- -------- -----------
**Label** strong strong strong strong strong strong *weak* *weak*
**Rate** 100 % 100 % 100 % *limited* *limited* *limited* 100 % *limited*
: Comparison of the required annotated data. **Label** indicates whether the method require strong label or weak label. **Rate** indicates the proportion of annotated data required for training. Name abbreviation with corresponding methods is given as following: DC-IGN ( [@Kulkarni2015Deep]), DNA-GAN ( [@Xiao2017DNA]), TD-GAN ( [@Wang2017Tag]), Semi-DGM ( [@Kingma2014Semi]), Semi-VAE ( [@Siddharth2017Learning]), ML-VAE ( [@Bouchacourt2017Multi]), JADE( [@Banijamali2017JADE]) and our DSD. []{data-label="data_table"}
Supervision Rate
----------------
We also conduct experiments to demonstrate the impact of the supervision rate for DSD’s disentangling capabilities, where we set the rates to be $0.0,0.1,0.2,...,1.0$. From Fig. \[fig:Ten\_score\](a), we can see that different supervision rates do not affect the convergence of DSD. Lower supervision rate will however lead to the overfitting if the epoch number greater than the optimal one. Fig. \[fig:Ten\_score\](d) shows the classification accuracy of DSD with different supervision rates. With only $20\%$ paired samples, DSD achieves comparable accuracy as the one obtained using $100\%$ paired data, which shows that the dual-learning mechanism is able to take good advantage of unpaired samples. Fig. \[fig:Ten\_score\](c) shows some hybrid images that are swapped the digital identity code parts. Note that, images obtained by DSD with supervision rates equal to $0.2,0.3,0.4,0.5$ and $0.7$ keep the angles of the digits correct while others not. These image pairs are highlighted in yellow.
Primary vs Dual
---------------
To verify the effectiveness of dual-learning mechanism, we compare our DSD (dual framework) with a basic primary framework that also requires paired and unpaired samples. The difference between the primary framework and DSD is that there is no swapping operation for unpaired samples in the primary framework. Fig. \[fig:Ten\_score\](b) gives the training and validation loss curves of the dual framework and primary framework with different supervision rates, where we can find that different supervision rates have no visible impacts on the convergence of dual framework and primary framework. From Fig. \[fig:Ten\_score\](d), we can see that accuracy scores of the dual framework are always higher than accuracies of the primary framework in different supervision rate, which proves that codes disentangled by the dual framework are informativeness than those disentangled by the primary framework. Fig. \[fig:Ten\_score\](c) gives the visual comparison between the hybrid images in different supervision rate. It is obvious that hybrid images of the primary framework are almost the same with original images, which indicates that the swapped codes contain redundant angle information. In other words, the disentanglement of the primary framework is defective. On the contrary, most of the hybrid images of dual framework keep the angle effectively, indicating that swapped coded only contains the digital identity information. These results show that dual framework (DSD) is indeed superior to the primary framework.
Discussion and Conclusion
=========================
In this paper, we propose the Dual Swap Disentangling (DSD) model that learns disentangled representations using limited and weakly-labeled training samples. Our model requires the shared attribute as the only annotation of a pair of input samples, and is able to take advantage of the vast amount of unlabeled samples to facilitate the model training. This is achieved by the dual-stage architecture, where the labeled samples go through the “encoding-swap-decoding” process once while the unlabeled ones go through the process twice. Such self-supervision mechanism for unlabeled samples turns out to be very effective: DSD yields results superior to the state-of-the-art on several datasets of different domains. In the future work, we will take semantic hierarchy into consideration and potentially learn disentangled with even fewer labeled pairs.
Supplemental Material {#supplemental-material .unnumbered}
=====================
Theoretical Proof {#section:proof}
-----------------
**Proposition 1.** Let $\mathcal{I}$ denotes object which is consist of $n$ independent semantics, $D=\{ G_j=\{(\mathcal{I}^1_A,\mathcal{I}^1_B),...,(\mathcal{I}^t_A,\mathcal{I}^t_B)\},j=1,2,...,n\}$ denotes the whole pair group image dataset, where $G_k$ is consist of paired observations by sharing a semantic similarity and the paired observations share a common semantics. For all paired observations $(\mathcal{I}_A,\mathcal{I}_B) \in G_j, j=1,2,...,n$, minimizing the interchanging autoencoder loss $\mathbf{\mathcal{L}_p}(\mathcal{I}_A,\mathcal{I}_B;\phi,\varphi)$ will disentangle $\mathcal{I}$ into semantic parts $[r_1,r_2,...,r_n]$, where part $r_j$ will only contain $j$th semantics.
*Proof of Proposition 1.* Define the independent semantic information in $\mathcal{I}$ as $\mathcal{S}=\{s_1,s_2,...,s_n\}$. For the paired observations $(\mathcal{I}_A,\mathcal{I}_B) \in G_j, j=1,2,...,n$ which have common semantics $s_j$, semantic information in $i$th part of $\mathcal{R}_A=[a_1,a_2,...,a_n]$ can be written as $\mathbb{S}(a_i)=\{\lambda^i_1 s^a_1,\lambda^i_2 s^a_2,...,\lambda^i_j s_j,...,\lambda^i_n s^a_n\}$, where $\lambda^i_1,...,\lambda^i_n$ is the semantic rate. Through minimizing the original autoencoder loss : $ ||\mathcal{I}_A-\overline{\mathcal{I}_A}||^2_2 \rightarrow 0 $, $\mathcal{R}_A$ will contain all the semantic information $\{s^a_1, s^a_2,...,s^a_n\}$ of $\mathcal{I}_A$. So, $ \lambda^1_j \oplus \lambda^2_j \oplus ,...,\oplus \lambda^n_j \rightarrow 1$, where $ \oplus $ means non-coupled addition. Trough minimizing the interchanging autoencoder loss $ ||\mathcal{I}_A-\ddot{\mathcal{I}_A}||^2_2 \rightarrow 0 $, the $j$th part of $R_A$ will only contain the information of $s_j$. So, for $\lambda^j_{\tau} \in \{\lambda^j_{\tau},\tau \neq j , \tau=1,2,...,n \} $, $\lambda^j_{\tau}\rightarrow 0$ and semantic rate $\lambda^j_j \in [0,1]$. Then, semantic information in $a_i$ can be written as $\mathbb{S}(a_j)=\{0\times s^a_1,0\times s^a_2,...,\lambda^j_j s_j ,...,0 \times s^a_n\}, j=1,2,...,n$. With $ ||\mathcal{I}_A-\overline{\mathcal{I}_A}||^2_2 \rightarrow 0 $, $R_A$ should contain all the semantic information $\{ s^a_1, s^a_2,...,s^a_n\}$. So, for all $j \in \{1,2,...,n\}$, $\lambda^j_j \rightarrow 1$, which means that part $a_j$ will only contain $j$th semantics $s^a_j$. In the same way, for all $j \in \{1,2,...,n\}$, part $b_j$ will only contain $j$th semantics $s^b_j$. In summary, for all $j \in \{1,2,...,n\}$, part $r_j$ will only contain $j$th semantics $s_j$.
Experiment Setup
----------------
For all generative models, we use the ResNet architectures shown in Table \[tab:net64\] and Table \[tab:net32\] for the encoder / discriminatior (D) / auxilary network (Q) and the decoder / generator (G). Adam optimizer ( [@Kingma2014Adam]) is adopted with learning rates of $1e^{-4}$ ($64\times 64$ network) and $0.5e^{-4}$ ($32\times 32$ network). The batch size is $64$. For the stable training of InfoGAN, we fix the latent codes’ standard deviations to $1$ and use the objective of the improved Wasserstein GAN ( [@Gulrajani2017Improved]), simply appending InfoGAN’s approximate mutual information penalty. We use layer normalization instead of batch normalization. In our experiment, the visual results are generated with the $64\times 64$ network architecture and other quantitative results are generated with the $32\times 32$ network architecture. For the above two network architecture, $\alpha$ and $\beta$ are all set as $5$ and $0.2$, respectively.
------------------------------------------------------- ---------------------------------------------
(r)[1-1]{} (r)[2-2]{} $3 \times 3~64$ conv. FC $4\cdot4\cdot8\cdot64$
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~64$ conv BN, ReLU, $3 \times 3~512$ conv, $\uparrow$
BN, ReLU, $3 \times 3~128$ conv, $\downarrow$ BN, ReLU, $3 \times 3~512$ conv
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~128$ conv BN, ReLU, $3 \times 3~256$ conv, $\uparrow$
BN, ReLU, $3 \times 3~256$ conv, $\downarrow$ BN, ReLU, $3 \times 3~256$ conv
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~256$ conv BN, ReLU, $ \times 3~128$ conv, $\uparrow$
BN, ReLU, $3 \times 3~512$ conv, $\downarrow$ BN, ReLU, $3 \times 3~128$ conv
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~512$ conv BN, ReLU, $3 \times 3~64$ conv, $\uparrow$
BN, ReLU, $3 \times 3~512$ conv, $\downarrow$ BN, ReLU, $3 \times 3~64$ conv
(r)[1-1]{} (r)[2-2]{} FC Output BN, ReLU, $3 \times 3~3$ conv, tanh
------------------------------------------------------- ---------------------------------------------
: Network architecture for image size $64 \times 64$. Each network has 4 residual blocks (all but the first and last rows). The input to each residual block is added to its output (with appropriate downsampling/upsampling to ensure that the dimensions match). Downsampling $\downarrow$ is performed with mean pooling and $\uparrow$ indicates nearest-neighbour upsampling.[]{data-label="tab:net64"}
------------------------------------------------------- ---------------------------------------------
(r)[1-1]{} (r)[2-2]{} $3 \times 3~32$ conv. FC $4\cdot4\cdot8\cdot32$
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~32$ conv BN, ReLU, $3 \times 3~256$ conv, $\uparrow$
BN, ReLU, $3 \times 3~64$ conv, $\downarrow$ BN, ReLU, $3 \times 3~128$ conv
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~64$ conv BN, ReLU, $3 \times 3~128$ conv, $\uparrow$
BN, ReLU, $3 \times 3~128$ conv, $\downarrow$ BN, ReLU, $3 \times 3~64$ conv
(r)[1-1]{} (r)[2-2]{} BN, ReLU, $3 \times 3~128$ conv BN, ReLU, $ \times 3~64$ conv, $\uparrow$
BN, ReLU, $3 \times 3~256$ conv, $\downarrow$ BN, ReLU, $3 \times 3~32$ conv
(r)[1-1]{} (r)[2-2]{} FC Output BN, ReLU, $3 \times 3~3$ conv, tanh
------------------------------------------------------- ---------------------------------------------
: Network architecture for image size $32 \times 32$. []{data-label="tab:net32"}
Dataset and Experiment Result
-----------------------------
In the experiment, the latent codes’ length and semantic number for the $5$ datasets is set as follows: Square $(15,3)$, MNIST$(15,3)$, Teapot $(50,5)$,CAS-PEAL-R1 $(40,4)$ and Mugshot $(100,2)$.
**Square** The Square dataset contains $60,000$ image samples ( $30,000$ pair images). The training, validation and testing dataset are set as $ \{(20,000),(9,000)$ and $(1,000)\}$, respectively. More visual results of DSD on Square dataset are shown in Fig. \[fig:geometryResult\].
**MNIST** In the quantitative evaluation, we adopt InfoGAN to generate $5,0000$ labeled pair digital samples. Through setting different supervision rates, we can get different supervision ratio dataset. More semantics swapping for MNIST are shown in Fig. \[fig:mnist\]. Usually, the digits keep other two semantic unchanged and only the swapped semantic is changed. However, the hybrid images that are swapped digital identity code usually contain some thickness semantic. The reason for such issue is that digital identity often has a close tie with thickness. For example, the dataset usually contains more thin digit “1” than thin digit “8”.
**Teapot** In our experiment, the Teapot dataset contains $50,000$ traning, $10,000$ validation and $10,000$ testing samples. Fig. \[fig:teapotResult\] shows the visual results on Teapot.
**CAS-PEAL-R1** We sample $50,000$ pair samples from original CAS-PEAL-R1. They are divided into $ \{(40,000),(9,000),(1,000)\}$ for training, validation and testing. Due to the existence of disturbed paired samples, the quality of generated hybrids is not as good as other datasets. However, when the hats are swapped, the covered hairs by the hats can also be reconstructed. More visual results are shown in Fig. \[fig:faceResult\].
**Mugshot** For Mugshot dataset, we divided it into $ \{(20,000),(9,000),(1,000)\}$ for training, validation and testing. Fig. \[fig:photoResult\] shows the results of the same mugshot through swapping different backgrounds. As that Mugshot dataset perfectly conforms to the pairing requirement of DSD, the quality of hybrid is imposing.
|
---
abstract: 'New self-consistent parameter sets are presented and discussed for muon collider rings at center-of-mass energies of 10, 30 and 100 TeV. All three parameter sets attain luminosities of ${\cal{L} {\rm }}= {\rm 3 \times 10^{35}}\;{\,{\rm cm^{-2}.s^{-1}}}$. The parameter sets benefit from new insights gained at the HEMC’99 workshop [@hemc99] that considered the feasibility of many-TeV muon colliders.'
author:
- 'B.J. King, BNL, Upton, NY 11973, USA'
title: ' FURTHER STUDIES ON THE PROSPECTS FOR MANY-TEV MUON COLLIDERS [^1] '
---
INTRODUCTION
============
Table 1 of this paper presents self-consistent parameter sets for muon collider storage rings at center-of-mass energies of ${{\rm E_{CoM}}}= 10$, 30 and 100 TeV. The parameter sets have benefitted and evolved from previous attempts at defining plausible parameter sets for many-TeV muon colliders. It is helpful to begin by reviewing these previous studies and their motivation in order to provide a context for the discussion of the current parameters.
Parameter sets for muon collider rings at energies up to ${{\rm E_{CoM}}}=100$ TeV were presented in 1998 [@epac98] and 1999 [@pac99]. Following this, a much improved level of understanding was then obtained from the first substantial dedicated study of such many-TeV muon colliders, which took place at the week-long HEMC’99 workshop [@hemc99]. The majority of the studies at HEMC’99 either assumed or critiqued straw-man parameter sets [@hemc99specs], one at ${{\rm E_{CoM}}}=10$ and two at 100 TeV, that were provided expressly for this purpose.
Besides presenting an overview of the HEMC’99 parameter sets, reference [@hemc99specs] also reviewed the feed-back on the parameters that was provided by the workshop. This paper should be referred to for many discussions that remain relevant for the current parameter sets of table 1.
The 48 participants at HEMC’99 considered side-by-side the accelerator challenges and the high energy physics (HEP) potential of many-TeV muon colliders. The HEP motivation for the workshop was very strong because experimental discoveries in HEP normally come from advances in energy reach, as has been emphasized and discussed in, for example, references [@Willis] and [@hemc99intro]. HEP discussions specific to many-TeV muon colliders can be found in [@pr98] and, mainly, in the HEMC’99 Proceedings [@hemc99].
Of the three many-TeV parameter sets in table 1, those at 10 TeV and 100 TeV evolved directly from the corresponding 10 TeV and (the first of the) 100 TeV parameter sets for HEMC’99, taking into account the constructive criticisms that emerged from the workshop. A mid-point energy was considered valuable for examining parameter trends with increasing energy, and the 30 TeV parameter set provides such an interpolation between the lower and higher energy sets.
Invaluable benchmarks for all of these many-TeV studies were provided by lower energy parameters that have been studied and evaluated [@Snowmass; @status] by the Muon Collider Collaboration (MCC). The first column of table 1 shows, for comparison, the range of parameters for the muon colliders in the range ${{\rm E_{CoM}}}=0.1$, 3 TeV from the MCC’s status report [@status].
DISCUSSION ON PARAMETER SETS
============================
The energy scale and some other parameter choices in table 1 were strongly influenced by considerations of synchrotron radiation. This imposes a natural cut-off scale for circular muon storage rings in the range ${{\rm E_{CoM}}}\sim 100$ TeV since the synchrotron radiation loss at such energies has risen rapidly to become comparable to the beam power. At HEMC’99, Telnov made the additional observation [@Telnov] that the quantum nature of the sychrotron radiation could lead to beam heating, rather than cooling, for sufficiently high beam energies and small emittances. This observation effectively invalidated the more aggressive of the two HEMC’99 parameter sets at 100 TeV – which therefore won’t be discussed further in this paper – and also cast some doubt on the 100 TeV parameter set with the larger emittance.
The synchrotron radiation concerns were addressed in the 100 TeV parameter set in table 1 by:
1. raising the emittance in each of the transverse coordinates by the large factor of 90. This should comfortably address Telnov’s concern and result in net synchrotron cooling by raising the horizontal emittance to well above the quantum break-even value
2. increasing the collider ring circumference by a factor of two and, correspondingly, reducing the average bending magnetic field by a factor of two, to 5.3 Tesla
3. reducing the average beam current by nearly a factor of 2, to 4 mA.
The combined effect of the second and third changes was to reduce the synchrotron radiation to 50 MW, down from the previous, somewhat problematic level of 195 MW in the HEMC’99 parameter set. Although still a factor of 2.5 larger than the synchrotron power at LEP II, this reduced level was considered very appropriate for a far future collider at the energy frontier.
These changes should also help to address reservations expressed by Harrison [@Harrison] at HEMC’99 about the feasibility of 10 Tesla cosine theta dipoles in the presence of large amounts of synchrotron radiation. Besides lowering the average required magnetic field by a factor of two, it is noted that the synchrotron radiation power deposited per unit length around the collider ring has fallen by almost a factor of 8 from the HEMC’99 parameter set at 100 TeV.
In addition to the adjustments just mentioned that were specific to the 100 TeV parameter set, all three many-TeV parameter sets in table 1 were made more conservative than the HEMC’99 parameter sets in several areas:
- in recognition of the difficulty and novelty of ionization cooling, the phase space densities in table 1 were all scaled back to coincide with the upper end of the parameter choices from reference [@status] for lower energy muon colliders, i.e. $2.4 \times 10^{22}$ ${\rm m}^{-3}$.
- the final focus parameters are perhaps the most difficult of all for a non-specialist to evaluate. As has been discussed in references [@epac98; @pac99; @hemc99specs], the final focus difficulty can be usefully benchmarked to other muon collider and ${{\rm e^+e^-}}$ collider parameter sets according to the value of 3 parameters in particular: the $\beta^*$ in the x and y coordinates and of two other defined parameters, the so-called “demagnification factor” and “chromaticity quality factor”. All three benchmark parameters have been somewhat relaxed in response to feed-back [@hemc99specs] from the studies by final focus lattice experts at HEMC’99. Further explicit magnet lattice designs, now for each of the three parameter sets in table 1, would be invaluable for assessing whether the new, more relaxed parameters have reached an acceptable level of plausibility
- the average beam currents and resulting beam powers were reduced so that the worst case, at 100 TeV, had a summed beam plus synchrotron power of 180 MW, i.e. comparable to the 170 MW beam power that has been under consideration for the Accelerator Production of Tritium project [@APT]
- the beam-beam tune disruption parameter was lowered slightly for all three sets to a value, in the worst case, of $\Delta \nu = 0.091$. This is not far above the impressive new LEP II record of $\Delta \nu = 0.083$ that was reported in this conference [@LEPIItuneshift].
The unavoidable cost of these relaxed machine parameters was to lower the luminosity to ${\cal{L} {\rm }}= {\rm 3 \times 10^{35}}$ ${\,{\rm cm^{-2}.s^{-1}}}$ for each of the 10, 30 and 100 TeV parameters. This is a reduction to 30% of the luminosities, ${\cal{L} {\rm }}= {\rm 1 \times 10^{36}}$ ${\,{\rm cm^{-2}.s^{-1}}}$, of the corresponding HEMC’99 parameter sets for 10 TeV and 100 TeV. To put this in perspective, the new luminosities are still orders of magnitude higher than at any existing colliders and are also higher than any speculated parameters the author is aware of for plausible future machines other than muon colliders.
SUMMARY
=======
The extremely high constituent particle energies and luminosities of the parameter sets presented in table 1 continue to emphasize the impressive potential of muon colliders for exploring the energy frontier of elementary particle physics. Therefore, further paper studies and simulations for many-TeV muon colliders should continue to play a valuable role in our field. More specifically, the parameter sets presented in this paper would certainly benefit from feed-back and constructive criticism by experts in areas such as the design of final focus lattices.
-------------------------------------------------------------- ------------------------------------------------------------------------------ ---------------------- ---------------------- ----------------------
A B C
0.1 to 3 TeV 10 TeV 30 TeV 100 TeV
luminosity, ${\cal L}$ \[${\rm 10^{35}\: cm^{-2}.s^{-1}}$\] $8 \times 10^{-5}$$\rightarrow$0.5 3.0 3.0 3.0
$\int {\cal L}$dt \[${\rm fb^{-1}/year}$\] 0.08$\rightarrow$540 3000 3000 3000
No. of $\mu\mu \rightarrow {\rm ee}$ events/det/year 650$\rightarrow$10 000 2600 290 26
No. of 100 GeV SM Higgs/year 4000$\rightarrow$600 000 $4 \times 10^6$ $5 \times 10^6$ $6 \times 10^6$
CoM energy spread, ${\rm \sigma_E/E}$ \[$10^{-3}$\] 0.02$\rightarrow$1.1 0.42 0.080 0.071
circumference, C \[km\] 0.35$\rightarrow$6.0 15 39 200
ave. bending B field \[T\] 3.0$\rightarrow$5.2 7.0 8.1 5.2
($\mu^-$ or) $\mu^+$/bunch, ${\rm N_0[10^{12}}]$ 2.0$\rightarrow$4.0 2.9 2.0 1.6
($\mu^-$ or) $\mu^+$ bunch rep. rate, ${\rm f_b}$ \[Hz\] 15$\rightarrow$30 15 7.5 5
6-dim. norm. emit., $\epsilon_{6N} 170$\rightarrow$170 125 85 70
[10^{-12}{\rm m}^3$\]
$\epsilon_{6N} 2.0$\rightarrow$2.0 1.5 1.0 0.83
[10^{-4}{\rm m}^3.{\rm MeV/c}^3$\]
P.S. density, ${\rm N_0}/\epsilon_{6N} 1.2$\rightarrow$2.4 2.3 2.4 2.3
[10^{22}{\rm m}^{-3}$\]
x,y emit. (unnorm.) \[${\rm \pi.\mu m.mrad}$\] 3.5$\rightarrow$620 0.84 0.19 0.040
x,y normalized emit. \[${\rm \pi.mm.mrad}$\] 50$\rightarrow$290 40 27 19
long. emittance \[${\rm 10^{-3}eV.s}$\] $0.81\rightarrow24$ 28 40 68
fract. mom. spread, $\delta$ \[$10^{-3}$\] 0.030$\rightarrow$1.6 0.50 0.20 0.075
relativistic $\gamma$ factor, ${\rm E_\mu/m_\mu}$ 473$\rightarrow$14 200 47 300 142 000 473 000
time to beam dump, ${\rm t_D} [\gamma \tau_\mu]$ no dump no dump no dump no dump
effective turns/bunch 450$\rightarrow$780 1040 1200 780
ave. current \[mA\] 17$\rightarrow$30 29 12 4.0
beam power \[MW\] 1.0$\rightarrow$29 70 72 128
synch. rad. critical E \[MeV\] $5 \times 10^{-7} \rightarrow 0.012 0.12 1.75
8 \times 10^{-4}$
synch. rad. E loss/turn \[GeV\] $7 \times 10^{-9} \rightarrow 0.017 0.52 25
3 \times 10^{-4}$
synch. rad. power \[MW\] $1\times10^{-7}\rightarrow$0.010 0.48 6.0 50
beam + synch. power \[MW\] 1.0$\rightarrow$29 70 78 180
power density into magnet liner \[kW/m\] 1.0$\rightarrow$1.7 2.0 0.84 0.48
spot size, $\sigma_{x,y}$ $[\mu {\rm m}]$ 3.3$\rightarrow$290 1.7 0.88 0.47
bunch length, $\sigma_z$ \[mm\] 3.0$\rightarrow$140 3.4 4.0 5.4
$\beta^*_{x,y}$ \[mm\] 3.0$\rightarrow$140 3.4 4.0 5.4
ang. divergence, $\sigma_\theta$ \[mrad\] 1.1$\rightarrow$2.1 0.50 0.22 0.086
beam-beam tune disruption, $\Delta \nu$ 0.015$\rightarrow$0.051 0.079 0.079 0.091
pinch enhancement factor, ${\rm H_B}$ 1.00$\rightarrow$1.01 1.06 1.06 1.09
beamstrahlung frac. E loss/collision negligible $2.3 \times 10^{-8}$ $1.0 \times 10^{-7}$ $5.5 \times 10^{-7}$
max. poletip field of quads., ${\rm B_{5\sigma}}$ \[T\] 6$\rightarrow$12 12 12 12
max. full aper. of quad., ${\rm A_{\pm5\sigma}}$\[cm\] 14$\rightarrow$24 21 25 31
quad. gradient, $2{\rm B_{5\sigma} / A_{\pm5\sigma}}$\[T/m\] 50$\rightarrow$90 120 97 77
${\rm \beta_{max} [km]}$ 1.5$\rightarrow$150 520 3200 24 000
ff demag., $M \equiv \sqrt{\beta_{\rm max}/\beta^*}$ 220$\rightarrow$7100 12 000 28 000 67 000
chrom. quality factor, $Q \equiv M \cdot \delta$ 0.007$\rightarrow$11 6.2 5.7 5.0
collider reference depth, D\[m\] 10$\rightarrow$300 100 100 100
ave. rad. dose in plane \[mSv/yr\] $2 \times 10^{-5}$$\rightarrow$0.02 1.2 4.8 20
str. sec. len. for 10x ave. rad. \[m\] 1.3$\rightarrow$2.2 0.95 1.6 8.4
$\nu$ beam distance to surface \[km\] 11$\rightarrow$62 36 36 36
$\nu$ beam radius at surface \[m\] 4.4$\rightarrow$24 0.75 0.25 0.075
-------------------------------------------------------------- ------------------------------------------------------------------------------ ---------------------- ---------------------- ----------------------
\[colliderpara\]
[9]{}
B.J. King, [*Discussion on Muon Collider Parameters at Center of Mass Energies from 0.1 TeV to 100 TeV*]{}, Proc. EPAC’98, BNL–65716. available from LANL preprint archive as [*physics/9908016*]{}. B.J. King, [*Muon Colliders from 10 TeV to 100 TeV*]{}, Proc. PAC’99, New York, 1999, pp. 3038-40, available from LANL preprint archive as [*physics/9908018*]{}. HEMC’99 workshop, “Studies on Colliders and Collider Physics at the Highest Energies: Muon Colliders at 10 TeV to 100 TeV”, Montauk, NY, U.S.A., 27 Sept-1 Oct, 1999, Proceedings published by American Institute of Physics, web page: http://pubweb.bnl.gov/people/bking/heshop/ . B.J. King, [*Parameter Sets for 10 TeV and 100 TeV Muon Colliders, and their Study at the HEMC’99 Workshop*]{}, Proc. HEMC’99 \[3\], also available from the LANL preprint archive as [*physics/0005008*]{}. Bill Willis, [*Muon Collider Workshop Summary*]{}, Proc. HEMC’99 \[3\]. B.J. King, [*Prospects for Colliders and Collider Physics to the 1 PeV Energy Scale*]{}, Proc. HEMC’99 \[3\], also available from the LANL preprint archive as [*hep-ex/0005008*]{}. B.J. King, [Muon Colliders: New Prospects for Precision Physics and the High Energy Frontier]{}, Proc. Second Latin American Symposium on High Energy Physics, San Juan, Puerto Rico, 8-11 April, 1998, Also available from the LANL preprint archive as [*hep-ex/9908041*]{}. The Muon Collider Collaboration, [*${ \mu^+\mu^-}$ Collider: A Feasibility Study*]{}, BNL-52503, Fermilab-Conf-96/092, LBNL-38946, July 1996. The Muon Collider Collaboration, [*Status of Muon Collider Research and Development and Future Plans*]{}, Phys. Rev. ST Accel. Beams, 3 August, 1999. Valery Telnov, [*Limit on Horizontal Emittance in High Energy Muon Colliders due to Synchrotron Radiation*]{}, Proc. HEMC’99 \[3\]. Mike Harrison, [*Magnet Challenges: Technology and Affordability*]{}, oral presentation at HEMC’99 \[3\]. The Accelerator Production of Tritium Project, `http://apt.lanl.gov`. R. Assmann [*et al.*]{}, [*LEP Operation and Performance with 100 GeV Colliding Beams*]{}, these proceedings.
[^1]: Submitted to Proc. EPAC 2000. This work was performed under the auspices of the U.S. Department of Energy under contract no. DE-AC02-98CH10886.
|
---
author:
- Pat Scott
- Martin Asplund
- Nicolas Grevesse
- Maria Bergemann
- 'A. Jacques Sauval'
bibliography:
- 'CObiblio.bib'
- 'AbuGen.bib'
- 'FePeakGeneral.bib'
- 'Sc.bib'
- 'Ti.bib'
- 'V.bib'
- 'Cr.bib'
- 'Mn.bib'
- 'Co.bib'
- 'Ni.bib'
- 'Others.bib'
- 'MA.bib'
date: 'Received 1 May 2014 / Accepted 1 Sep 2014'
subtitle: 'II. The iron group elements Sc to Ni'
title: The elemental composition of the Sun
---
Introduction {#intro}
============
Cosmic abundances of the transition metals Sc – Ni ($21\le Z\le 28$) tend to form a ‘peak’ around iron. This behaviour approximately tracks the variation in average binding energy per nucleon with $Z$, and reflects the predominantly common origin of iron peak nuclei in core-collapse and thermonuclear supernovae [e.g. @Pagel97]. Variations of abundances within the group provide information on nuclear physics and the physical environments in which the elements were processed. To compare such analyses with theories of stellar structure and evolution, galactic chemical evolution, supernova nucleosynthesis and the formation history of the solar system, accurate solar abundances of the iron group elements are required. In this paper, we present a reanalysis of the solar composition of the iron peak elements Sc, Ti, V, Cr, Mn, Fe, Co and Ni, using a realistic 3D hydrodynamic solar model atmosphere.
This paper is part of a series detailing, and updating, the chemical composition of the Sun presented in @AGSS [hereafter ]. This paper covers the iron group nuclei Sc – Ni. @AGSS_NaCa [hereafter ] deals with the intermediate-mass elements Na – Ca, whereas @AGSS_heavy [hereafter ] is devoted to the heavy elements Cu – Th. Later studies will describe the analysis of the light elements C, N and O, as well as summarise and compare the solar photospheric abundances with the meteoritic evidence, indications from helioseismology and the solar neighbourhood. In an earlier series of papers (@AspI; @AspII; @AspIII; @2001ApJ...556L..63A, @2002ApJ...573L.137A, @AspIV; @AspV; @AspVI; @ScottVII; @AGS05, hereafter ; , @Scott09Ni), we examined the abundances of all elements up to Ca, as well as Fe and Ni, using a predecessor of the current 3D solar model atmosphere. The only 3D solar analyses of any iron group elements to date have been of nickel [@Scott09Ni] and iron itself (; also the 1D calculations of @MB_fe based on an averaged 3D model).
In Sect. \[previouswork\] we summarise the current state of knowledge about the solar abundances of the iron peak elements. We describe the observational data we employ in Sect. \[observations\], then give brief recapitulations of our solar model atmosphere, line synthesis code (Sect. \[model\]) and abundance calculations (Sect. \[calculations\]). In Sect. \[atomicdata\] we justify our selection of atomic data, spectral lines and non-LTE (NLTE) corrections. Our results are presented in Sect. \[results\], discussed in Sect. \[discussion\], compared to previous compilations in Sect. \[compilations\] and summarised in Sect. \[conclusions\].
Previous solar analyses of the iron group {#previouswork}
=========================================
*Scandium:* Earlier reference compilations of the solar composition (@GS98, hereafter ; ) included the scandium abundance $\log \epsilon_\mathrm{Sc}=3.05\pm0.08$ from @Youssef89, derived using spectrum synthesis of Sc<span style="font-variant:small-caps;">ii</span> lines and the @HM ([-@HM]: hereafter ) model atmosphere. A more recent study, adopted as the standard in @Grevesse07, was that of @Neuforge93, who used the model with both Sc<span style="font-variant:small-caps;">i</span> and Sc<span style="font-variant:small-caps;">ii</span> lines. Despite using accurate oscillator strengths (the same values as we use in this paper in fact), abundance scatter was high ($\log \epsilon_\mathrm{Sc}=3.14\pm0.12$ for Sc<span style="font-variant:small-caps;">i</span>, $3.20\pm0.07$ for Sc<span style="font-variant:small-caps;">ii</span>). The dominant ionisation stage of scandium is Sc<span style="font-variant:small-caps;">ii</span>, so one expects Sc<span style="font-variant:small-caps;">ii</span> lines to return the most reliable abundances. However, the Sc<span style="font-variant:small-caps;">ii</span> results could not be reconciled with the meteoritic value, nor could close agreement between ionisation stages be claimed. @Zhang08 performed a detailed analysis of NLTE effects on Sc<span style="font-variant:small-caps;">i</span> and Sc<span style="font-variant:small-caps;">ii</span> lines in the Sun, finding large NLTE corrections to abundances from Sc<span style="font-variant:small-caps;">i</span> lines. This finally reconciled abundances from the neutral and once-ionised species: with the MAFAGS-ODF 1D model atmosphere (based on opacity distribution functions: ODF; @Fuhrmann97), @Zhang08 found $\log \epsilon_\mathrm{Sc}=3.07$–$3.13$, depending upon the oscillator strengths adopted.
*Titanium:* @Blackwell87 presented a thorough study of the solar abundance of Ti using a large number of Ti<span style="font-variant:small-caps;">i</span> lines with both the and <span style="font-variant:small-caps;">marcs</span> [@MARCS75] solar photospheric models (see Sect. \[model\]). These results were corrected by @Grevesse89 for a systematic shift in Ti<span style="font-variant:small-caps;">i</span> oscillator strengths of $0.056$dex (see Sect. \[Tigfs\]), resulting in $\log \epsilon_\mathrm{Ti}=4.99\pm0.04$ with the model. With the <span style="font-variant:small-caps;">marcs</span> model, the result would have been $0.10$dex smaller. Using Ti<span style="font-variant:small-caps;">ii</span> (the dominant species) instead, along with the model, @Bizzarri93 found $\log \epsilon_\mathrm{Ti}=5.04\pm0.04$, in very good agreement with the Ti<span style="font-variant:small-caps;">i</span> result. The first NLTE analysis of the solar Ti abundance was performed by @Bergemann11, who found strong NLTE effects in [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}line formation, and a severe dependence upon the adopted solar model atmosphere and rates of inelastic collisions with H<span style="font-variant:small-caps;">i</span> and $e^{-}$. From [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, @Bergemann11 found a mean abundance of $\log \epsilon_\mathrm{Ti}=4.93$–$4.98$, depending mostly on the adopted oscillator strengths. The most recent results are from @Lawler13 and @Wood13, who found $\log \epsilon_\mathrm{Ti}=4.97\pm0.04$ and $\log \epsilon_\mathrm{Ti}=4.98\pm0.03$ from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines respectively, using spectrum synthesis with the model and new laboratory oscillator strengths.
*Vanadium:* The most recent derivations of the solar abundance of vanadium are due to @Whaling85 [ with V<span style="font-variant:small-caps;">i</span> and the model: $\log \epsilon_\mathrm{V}=3.99\pm0.01$] and @Biemont89 [ with V<span style="font-variant:small-caps;">i</span>, V<span style="font-variant:small-caps;">ii</span> and the model: $\log
\epsilon_\mathrm{V}=4.00\pm0.02$]. The latter is the previously-adopted reference abundance (; ; @Grevesse07), giving more weight to the V<span style="font-variant:small-caps;">i</span> data, which is derived from a much larger number of lines than the V<span style="font-variant:small-caps;">ii</span> result. The error estimate given is probably unrealistically low however, as we describe in Sect. \[Vgfs\]. Both these studies assumed that V lines form in LTE. During the refereeing phase of our paper, we became aware of a recent determination of the solar V abundance using newly determined experimental transition probabilities for V<span style="font-variant:small-caps;">ii</span> [@Wood14V2]. Using spectrum synthesis with the HM model for a set of 15 often heavily blended V<span style="font-variant:small-caps;">ii</span> lines, they estimated $\log \epsilon_\mathrm{V}=3.95\pm0.01$ ($\sigma=0.05$dex).
*Chromium:* Recent compilations (; @Grevesse07) recommended a Cr abundance of $\log
\epsilon_\mathrm{Cr}=5.64\pm0.10$, derived from two papers. Using various $gf$-values available at the time, @Biemont78c found $\log \epsilon_\mathrm{Cr}=5.67\pm0.03$ with Cr<span style="font-variant:small-caps;">i</span> lines and the model, and $\log
\epsilon_\mathrm{Cr}=5.64\pm0.03$ using the VAL [@VAL76] model. @Blackwell87, using the accurate $gf$-values measured at Oxford and different solar spectra, found $\log
\epsilon_\mathrm{Cr}=5.68\pm0.06$ with the model. As for Ti<span style="font-variant:small-caps;">i</span>, with the <span style="font-variant:small-caps;">marcs</span> model this would have been $0.10$dex smaller. @Sobeck07 measured new $gf$-values for Cr<span style="font-variant:small-caps;">i</span> lines (see Sect. \[Crgfs\]) and used them to revise the solar abundance assuming LTE. When two highly discrepant outlying lines are removed, the results are $\log \epsilon_\mathrm{Cr}=5.64\pm0.05$ with the model and $\log
\epsilon_\mathrm{Cr}=5.53\pm0.05$ with <span style="font-variant:small-caps;">marcs</span>. @Sobeck07 also used a small number of Cr<span style="font-variant:small-caps;">ii</span> lines with $gf$-values from @Nilsson06. These lines lead to higher abundances and much larger dispersions: $\log
\epsilon_\mathrm{Cr}=5.77\pm0.13$ () and $\log
\epsilon_\mathrm{Cr}=5.67\pm0.13$ (<span style="font-variant:small-caps;">marcs</span>). @Bergemann10 investigated NLTE effects in solar Cr line formation for the first time, finding corrections of order $+$0.05–0.10dex to abundances from [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}with the 1D MAFAGS-ODF model. Using [[Cr<span style="font-variant:small-caps;">ii</span>]{} ]{}lines and $gf$-values from @Nilsson06, @Bergemann10 confirmed the high abundance and large scatter seen by @Sobeck07. With the complete exclusion of inelastic collisions with hydrogen, chosen so as to satisfy Cr ionisation balance for the Sun and a number of late-type stars, they also found a high abundance from [Cr<span style="font-variant:small-caps;">i</span>]{}: $\log \epsilon_\mathrm{Cr}=5.74\pm0.05$.
*Manganese:* Previous reference solar manganese abundances (; ; @Grevesse07) came from @Booth84b, who found $\log \epsilon_\mathrm{Mn}=5.39\pm0.03$ using the model and Mn<span style="font-variant:small-caps;">i</span> lines. The derived abundance is almost 3$\sigma$ below the meteoritic value [@Lodders09], quite a striking discrepancy when one considers that agreement between photospheric and meteoritic values is typically quite good (cf. @AG89, hereafter ; ; @Lodders09). The errors on the photospheric value have probably been underestimated however, as revealed by a detailed investigation of Mn<span style="font-variant:small-caps;">i</span> oscillator strengths and line selection (Sect. \[Mngfs\]). @Bergemann07 made a detailed NLTE analysis of a large number of Mn<span style="font-variant:small-caps;">i</span> lines in the solar flux spectrum, showing that NLTE abundance corrections are of order $+0.08$dex for solar lines. Their analysis with the MAFAGS-ODF model produced an abundance of $\log \epsilon_\mathrm{Mn}=5.36\pm0.10$. This work was subsequently revised with improved oscillator strengths by @BW07, giving $\log \epsilon_\mathrm{Mn}=5.37\pm0.05$ with the same model. Using the model, the result was $\log \epsilon_\mathrm{Mn}=5.46\pm0.08$, in reasonable agreement with the meteoritic value but exhibiting an uncomfortably high scatter.
*Iron:* @GS99 and @AspII summarised the long and well-known debate as to whether the solar abundance of Fe is equal to or higher than seen in meteorites. Discrepant results in older studies of the solar Fe abundance using 1D solar models appeared to be due to differences in the adopted $gf$-values, equivalent widths, microturbulent velocities and collisional damping parameters, as well as differences in computer codes. @GS99 succeeded in reconciling LTE abundances from [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines by modifying the temperature structure of the model, so as to remove the observed trend with excitation potential in abundances from [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines.
The first pioneering work aimed at determining the solar Fe abundance using a 3D solar model that we are aware of was by , who used two different 3D models (with what nowadays is obviously very modest numerical resolution and simplified radiative transfer). Perhaps not surprisingly, their derived Fe abundance showed a large difference between [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}($\log \epsilon_\mathrm{Fe} \approx 7.0$) and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}($\log \epsilon_\mathrm{Fe} \approx 7.6$) lines when using either equivalent widths or line depths. @AspII analysed [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines with a more realistic 3D model, albeit still in LTE. They found abundances from weak [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines to be independent of the excitation energy, and in very good agreement with both [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}results and the meteoritic abundance: $\log
\epsilon_\mathrm{Fe}=7.45\pm0.05$. Both @GS99 and @AspII found that abundances derived from [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, using $gf$-values available at the time, showed a very large scatter, 0.10dex. @Caffau11 analysed a set of [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines using a 3D solar model computed with the CO$^5$BOLD code @COBOLD and improved $gf$-values from [@Melendez09], finding $\log \epsilon_\mathrm{Fe}=7.52\pm0.06$.
and @MB_fe carried out NLTE calculations of Fe line formation, using the most up-to-date theoretical and experimental atomic data to construct their model atoms. , using the MAFAGS-OS[^1] models, obtained $\log \epsilon_\mathrm{Fe}=7.56\pm0.09$ from [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines, and rather discrepant results from [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines ($7.41$–$7.56$dex depending on the adopted $gf$-values). @MB_fe also investigated NLTE Fe line formation with the [<span style="font-variant:small-caps;">marcs</span>]{} and [$\langle\mathrm{3D}\rangle$]{} (Sect. \[model\]) model atmospheres, finding values fully consistent with the meteoritic abundance, and, in view of the small NLTE effects for the Sun, with the result of @AspII. With the [$\langle\mathrm{3D}\rangle$]{} model, they found a mean abundance of $\log \epsilon_\mathrm{Fe}=7.46\pm0.02$dex.
*Cobalt:* The Co content of the Sun was derived by @Cardon82 under the assumption of LTE using Co<span style="font-variant:small-caps;">i</span> lines, giving $\log \epsilon_\mathrm{Co}=4.92\pm0.08$ with the model atmosphere. This was the reference value adopted by , , , @Grevesse07 and @Lodders09, although it only overlaps the meteoritic value because of the rather large errors. Recently, @Bergemann10Co re-analysed a series of Co lines in flux, taking into account departures from NLTE. They found large NLTE corrections, of order $+$0.15dex. Using a MAFAGS-ODF solar photospheric model, they derived an NLTE Co abundance of $\log \epsilon_\mathrm{Co}=4.95\pm0.04$.
*Nickel:* We recently provided a revised solar nickel abundance in the context of the Ni-blended forbidden oxygen line at 630nm [@Scott09Ni]. Using the 3D model of @AspI, that analysis gave $\log \epsilon_\mathrm{Ni}=6.17\pm0.05$. Here we update those results using an improved 3D solar model atmosphere . @Wood14 found $\log \epsilon_\mathrm{Ni}=6.28\pm0.06$ by employing spectrum synthesis of [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}lines, the model and new laboratory oscillator strengths. The previous reference solar Ni abundance [@GS98; @AGS05; @Grevesse07] was $\log \epsilon_\mathrm{Ni}=6.25\pm0.09$, from an HM-based analysis of [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}by @Biemont80.
Observations
============
We compared theoretical line profiles to the Fourier Transform Spectrograph (FTS) spectral intensity atlas of @Brault87 [see also @Neckel99] at solar disk-centre ($\mu=1$). We removed the solar gravitational redshift of 633 ms$^{-1}$ from the observed spectrum, and convolved simulated profiles with an instrumental sinc function of width $\Delta\sigma = \frac{c}{R}= 0.857$ kms$^{-1}$, reflecting the FTS resolving power $R=350\,000$ [@Neckel99].
Our adopted equivalent widths are the integrated values we previously obtained in full $\chi^2$-based profile fits, using the earlier version of the 3D model [@AspI] and the observed FTS spectrum of @Brault87. We masked sections of profiles perturbed by nearby lines from the fitting procedure. We fitted local continua independently using nearby clear sections of the spectrum. We were sure to use the same spectral regions to integrate both the observed and theoretical profiles. As a cross-check, we also directly measured the equivalent widths of all lines on two different disc-centre solar atlases: the FTS atlas mentioned above [@Brault87], and the atlas of @LiegeAtlas recorded with a classical double-pass spectrometer at the Jungfraujoch high-altitude station. We noted excellent agreement between these two sets of measurements, and with the equivalent widths derived from the fitted 3D profiles (i.e. to within 1–2%). In order to ensure that our 1D and 3D abundances were derived consistently, for the 1D analyses we used the same equivalent widths as in the 3D analysis (i.e. those arising from the earlier 3D line profile fits).
Solar model atmospheres and spectral line formation {#model}
===================================================
We use the improved 3D model atmosphere introduced in and described in more detail in . We carried out comparative calculations with four 1D models: , <span style="font-variant:small-caps;">marcs</span> [@MARCS75; @MARCS97; @OSMARCS], [miss]{} [@MISS] and [$\langle\mathrm{3D}\rangle$]{}. The [$\langle\mathrm{3D}\rangle$]{} model is a temporal average of the 3D model, contracted into the vertical dimension with horizontal averages taken over surfaces of common optical depth. The reader is directed to for further details of these model atmospheres.
We obtain NLTE abundances by applying NLTE corrections to the values we obtained in LTE.[^2] This is not strictly correct unless full 3D NLTE calculations are carried out; for computational reasons, this is not the case for any of the elements we investigate here. 3D NLTE line formation is still very challenging, and only very few such studies have been undertaken to date . Instead, we apply NLTE abundance corrections computed using 1D model atmospheres. For most elements (Ti, Cr, Mn, Fe, Co), we computed NLTE corrections for solar disk-centre intensity profiles of the selected lines, using the , <span style="font-variant:small-caps;">marcs</span>, and [$\langle\mathrm{3D}\rangle$]{} models. For calculating 3D+NLTE abundances we adopt offsets computed with the [$\langle\mathrm{3D}\rangle$]{} model, which we expect to be a close approximation to the full 3D NLTE problem, given that radiative transfer proceeds primarily vertically. For [miss]{} we adopt the offsets computed with the model. We performed statistical equilibrium calculations with the DETAIL code [@Giddings81; @Butler85]. For Sc, we rely on NLTE corrections from the literature, while V and Ni have not been exposed to a NLTE study. Our NLTE calculations are described in detail in Sect. \[atomicdata\].
We do not discuss the NLTE line formation in detail in this work, as this aspect has been extensively discussed previously [e.g. @Bruls93; @Bergemann07; @Zhang08; @Bergemann10Co; @Bergemann10; @Bergemann11; @MB_fe], along with descriptions of the adopted atomic models. In short, the Fe-group elements are predominantly singly-ionised in the solar atmosphere, and departures from LTE are significant only for the neutral species, which are overionised. NLTE effects on the lines of singly-ionised atoms are typically negligible. One remaining uncertainty in NLTE calculations is the unknown cross-sections for inelastic collisions between hydrogen and the element in question. In the absence of quantum mechanical calculations that still only exist for lighter elements, most NLTE studies rely on the classical and therefore uncertain formula of @Drawin69, which at best should be considered an order-of-magnitude estimate. Therefore, a scaling factor $S_{\rm H}$ for the Drawin cross-sections is used. At least with iron lines and averaged 3D models, the unscaled @Drawin69 formula ($S_\mathrm{H}=1$) leads to ionisation balance and consistent inferred effective temperatures and surface gravities across a substantial sample of metal-poor stars [@MB_fe]. Wherever possible, we therefore prefer to use $S_\mathrm{H}=1$ for iron-group elements; we do this for all elements where we calculate our own NLTE abundance corrections (Ti, Cr, Mn, Fe and Co). For Sc, not having a model atom of our own to draw on, we must rely on results from the literature assuming $S_\mathrm{H}=0.1$ [@Zhang08]. Indeed, this parameter remains quite uncertain, and will likely differ across lines, elements and stars. In the absence of detailed quantum mechanical calculations to rely on, or better, solar observations to guide our choices, our selection is by necessity somewhat arbitrary. However, we argue that it is a reasonable approach to adopt the same scaling factor $S_\mathrm{H}$ for all Fe-peak elements as empirically estimated for Fe. Reliable atomic physics computations are urgently needed for inelastic H collisions, not only for these but also other elements.
Abundance calculations {#calculations}
======================
We derived abundances as per : by matching equivalent widths of simulated and observed line profiles, and including isotopic and hyperfine components in our calculations as blends.
As in , the final uncertainties of our 3D+NLTE abundance results are the sum in quadrature of a systematic term and a statistical one. We take the statistical term to be the standard error of the mean abundance. We calculate the systematic term as the sum in quadrature of uncertainties due to the mean temperature structure (half the mean difference between the [$\langle\mathrm{3D}\rangle$]{} and HM results), atmospheric inhomogeneities (half the mean difference between the 3D and [$\langle\mathrm{3D}\rangle$]{} results), and departures from LTE (the greater of 0.03dex and half the mean NLTE correction).
Atomic data and line selection {#atomicdata}
==============================
For each element and ionisation stage, we performed an extensive search of the atomic literature for the most reliable oscillator strengths, hyperfine splitting constants, isotopic separations, wavelengths, excitation potentials, transition designations and partition functions. We preferred to make our own independent critical selection rather than relying on any existing compilation, though the NIST Atomic Transition Probability Bibliographical Database [@NISTbib] proved invaluable for this task. We used the compilations of @Martin88, @Fuhr88, @Doidge95 and especially @Morton03 as guides and secondary comparators.
We extracted radiative broadening parameters from the Vienna Atomic Line Database [VALD, @VALD]. We treated collisional broadening of neutral lines via the Anstee-Barklem-O’Mara technique [@Anstee95; @Barklem97; @Barklem98]. The broadening parameters $\sigma$ and $\alpha$ we used were previously calculated for many individual lines [@Barklem00]. For others we interpolated within the tables of [@Anstee95] or [@Barklem97]. No such data exist for ionised iron-peak elements except iron itself , so we employed the classical @Unsold broadening recipe for such lines, with an enhancement factor of $2.0$. The same is true for the small number of neutral lines that lie outside the Anstee-Barklem-O’Mara tables. This scaling factor reflects the approximate proportionality typically seen between accurate modern broadening calculations and the @Unsold treatment, as observed over a large range of lines for which modern data are available. We note that most of the lines for which we have to resort to using scaled @Unsold broadening are weak and thus insensitive to the adopted damping.
We typically only used a line if it had a $gf$-value available from the source that we deemed most reliable. Each candidate line was checked for blends, by inspection of the solar spectra [@Brault87; @LiegeAtlas] and the tables of @Moore66. Line strengths were also checked in @Moore66, and only lines weaker than were generally allowed; in some circumstances, these requirements were relaxed slightly.[^3] The selected lines were assigned a relative ranking from 1 to 3 based upon their appearance in the observed spectrum, with rankings sometimes also adjusted to reflect differences in uncertainties in atomic data. These rankings were used to weight the contribution of each line to mean abundances. Note that the rankings are only indications of relative merit within a line list, so the same rank for lines of different species does not necessarily imply the same line quality.
Our adopted lines, oscillator strengths, NLTE corrections, equivalent widths, excitation potentials and derived abundances for all elements are given in Table \[table:lines\]. We provide isotopic and hyperfine splitting data separately in Table \[table:hfs\]. The isotopic ratios given for individual elements are taken from , but the original data are the terrestrial ratios recommended by @IUPAC98. Our chosen partition functions are from Barklem & Collet (in preparation), and our ionisation energies from NIST data tables. These data are given in Table \[table:partition\].
Scandium {#Scatomicdata}
--------
Wavelengths, excitation potentials and transition identifications come from @Kaufman88 for Sc<span style="font-variant:small-caps;">i</span>, and from @Johansson80 for Sc<span style="font-variant:small-caps;">ii</span>. Scandium exhibits hyperfine but not isotopic structure, as it has just one stable isotope [@IUPAC98]: $^{45}$Sc, with spin $I=\frac{7}{2}$.
### Oscillator strengths {#Scgfs}
For both Sc<span style="font-variant:small-caps;">i</span> and Sc<span style="font-variant:small-caps;">ii</span>, we prefer the $gf$-values of @Lawler89. These authors obtained emission FTS branching fractions (BFs), which they set to an absolute scale using the time-resolved laser-induced fluorescence (TRLIF) lifetimes of @Marsden88. These techniques are currently the most accurate means available for determining relative spectral intensities and radiative lifetimes respectively, and their combination is the most reliable way of determining absolute atomic $gf$-values. For Sc<span style="font-variant:small-caps;">ii</span>, accurate lifetimes are also available from @Vogel85, where results are in excellent agreement with those of @Marsden88; using the former or the latter data would result in $gf$-values differing by less than 0.01 dex.
Three very good solar lines (624.56, 630.07 and 632.08nm) were not measured by @Lawler89. We derived $gf$-values for these lines from existing experimental (@CB62, ) and theoretical [@Kuruczweb] data, using the lifetimes of @Vogel85 for normalisation. However, the resulting scatter in the abundances from these lines (with all models) left us ultimately unconvinced as to the accuracy of the oscillator strengths, so we chose to discard these lines.
### Hyperfine structure {#Schfs}
The HFS of Sc<span style="font-variant:small-caps;">i</span> has been studied extensively. The atomic-beam magnetic-resonance technique (ABMR; also known as laser-rf double resonance or ABMR-LIRF when detected using laser-induced resonance fluorescence) was employed by @Childs71 to give highly accurate data for the 3d4s$^2$ $^2$D$_{3/2,\,5/2}$ levels. For the 3d4s4p levels, the data with lowest uncertainties are the FTS results of @Aboussaid96. In some cases @Aboussaid96 provide more than one measurement for a given level; we take the average of these measurements, weighted according to their uncertainties. For the ($^3$F)4s $^2$F$_{5/2,\,7/2}$ levels, theoretical results presented by @Basar04 are the only data available. @Ertmer76 also presented ABMR data for the ($^3$F)4s $^4$F$_{3/2,\,9/2}$ levels. We note that optogalvanic spectroscopy (OGS) data presented by @Singh91 for the ($^1$D)4s $^2$D$_{3/2,\,5/2}$ levels are not reliable, due to errors in their relative intensity formula pointed out by @Aboussaid96, and confirmed by @Bieron02 and @Basar04. @Singh91 also measured the ($^3$F)4p $^4$G levels, though these were not affected by this error; for these levels we thus adopt either the data of @Singh91 ( for $^4$G$_{5/2,\,11/2}$) or @Ertmer76 [for $^4$G$_{7/2,\,9/2}$], based upon the size of the quoted uncertainties in each case.
Work on the HFS of Sc<span style="font-variant:small-caps;">ii</span> is rather less common. The most recent and accurate data that we could find come from @Villemoes92 and @Mansour89, the latter of whom employed the ultra-high-resolution ABMR technique. We use the results of both these studies where available, adopting an average weighted according to the stated uncertainties; in practice this means that the results of @Mansour89 dominate due to their smaller error bars. Where data are not available from both @Villemoes92 and @Mansour89, we turn to each of these studies individually, followed by the experiments of @Young88 and then @Arnesen82. Apart from the recent work by @Zhang08, previous determinations of the solar Sc abundance have not considered the effects of HFS in [Sc<span style="font-variant:small-caps;">i</span>]{}, and only incompletely considered the effects in [Sc<span style="font-variant:small-caps;">ii</span>]{}.
### NLTE corrections {#ScNLTE}
NLTE formation of solar Sc<span style="font-variant:small-caps;">i</span> and Sc<span style="font-variant:small-caps;">ii</span> lines has been thoroughly investigated by @Zhang08, using the MAFAGS-ODF model. As might be expected from the minority status of neutral Sc in the Sun and its quite low ionisation potential (6.56eV), @Zhang08 found very large NLTE corrections to Sc<span style="font-variant:small-caps;">i</span> abundances: about $+0.15$dex for flux profiles of the lines of interest in our analysis, when employing the standard @Drawin69 recipe for treating collisions with hydrogen rescaled by a factor $S_\mathrm{H}=0.1$. Corrections to Sc<span style="font-variant:small-caps;">ii</span> abundances were less severe (about $-0.01$dex for lines of interest to us). In the absence of any calculations for intensity profiles and/or in 3D, we simply adopt these results for disk centre in Table \[table:lines\], noting that this way the NLTE corrections may be slightly overestimated. For lines not studied by @Zhang08, given the size of corrections and the error likely induced by neglecting NLTE, we use the typical correction observed with similar lines. Although we have NLTE corrections available in both flux and disk-centre intensity for most other iron-group elements, we choose not to rescale the NLTE flux corrections for Sc by the mean ratio of those corrections in order to estimate intensity corrections. This is because the ratio of intensity to flux corrections, although sometimes substantially less than 1, is quite line specific; the line-to-line scatter in this ratio, across other elements, is actually comparable to the offset of the mean ratio from 1. Dedicated calculations of Sc NLTE intensity abundance offsets, for the lines and model atmospheres that we employ here, would be most welcome.
### Line selection {#Sclineselection}
We applied our line selection criteria (see the beginning of this Section) to all Sc<span style="font-variant:small-caps;">i</span> and Sc<span style="font-variant:small-caps;">ii</span> lines in the solar spectrum measured by @Lawler89. We also compared with the previous work of @Biemont74a, @Neuforge93, @Youssef89 and @Reddy03, retaining the five Sc<span style="font-variant:small-caps;">i</span> and nine Sc<span style="font-variant:small-caps;">ii</span> lines given in Table \[table:lines\]. We note that the $gf$-value of the very good Sc<span style="font-variant:small-caps;">ii</span> line at 660.5nm has a large uncertainty [$>$$40$%; @Lawler89]. Rather than exclude this line, we reduced its weight (as indicated by the asterisk beside its weight in Table \[table:lines\]).
Titanium {#Tiatomicdata}
--------
Our adopted wavelengths, transition designations and excitation potentials for Ti<span style="font-variant:small-caps;">i</span> come from @Forsberg91. For Ti<span style="font-variant:small-caps;">ii</span>, we took wavelengths from @Pickering01 [with erratum: @Pickering02] where possible, based upon unpublished work of Zapadlik et al. in Lund. Otherwise, wavelengths came from @Huldt82, as did all excitation potentials and transition identifications. Ti has five stable isotopes [@IUPAC98]: $^{46}$Ti (8.2% by number on Earth), $^{47}$Ti (7.4%), $^{48}$Ti (73.7%), $^{49}$Ti (5.4%) and $^{50}$Ti (5.2%). $^{47}$Ti has a nuclear spin of $I=\frac{5}{2}$ and $^{49}$Ti has $I=\frac{7}{2}$.
### Oscillator strengths {#Tigfs}
@Nitz98 and @BW06 produced Ti<span style="font-variant:small-caps;">i</span> oscillator strengths by combining their own FTS BFs with accurate TRLIF lifetimes from @Salih90 and @Lawler91, respectively.
@Lawler13 have recently expanded and improved the work of @Nitz98, providing accurate oscillator strengths for nearly a thousand lines by combining their FTS and eschelle BFs with the lifetimes of @Salih90 and @Lawler91.
@Grevesse89 earlier produced accurate $gf$ values by renormalising the relative oscillator strengths of @Blackwell2 [@Blackwell1; @Blackwell3; @Blackwell4], which had been obtained by absorption spectroscopy in the Oxford furnace. As opposed to the original Oxford works, in which relative oscillator strengths were set to an absolute scale using less accurate beam-foil lifetimes from @Roberts73b and the absolute data of @Bell75, @Grevesse89 set their new values to an absolute scale using the accurate TRLIF lifetimes of @Rudolph82.
We prefer the data of @Lawler13 where possible, but we also performed some preliminary calculations of abundances arising from lines in common between @Nitz98, @BW06 and the revised Oxford $gf$-values, in order to establish which set of data was the next most reliable. Based upon the internal scatter and relative agreement between different lists, we concluded that the @Nitz98 values are to be preferred very slightly over the revised Oxford values, while the @BW06 $gf$-values are surprisingly discrepant.
High-quality Ti<span style="font-variant:small-caps;">ii</span> oscillator strengths are available from the FTS BFs and TRLIF lifetimes of @Bizzarri93, and the extensive FTS and eschelle work by @Wood13. The FTS study of @Pickering01 also produced $gf$-values for many lines, where fractions for some branches were completed using theoretical oscillator strengths of weak lines from @Kuruczweb. @Pickering01 set different BFs to an absolute scale using either the @Bizzarri93 lifetimes or lifetimes derived from the theoretical @Kuruczweb $gf$-values. Our preliminary investigations with lines common to the lists of @Bizzarri93 and @Pickering01 revealed a much larger abundance scatter with the @Pickering01 data; we thus prefer the @Wood13 and @Bizzarri93 oscillator strengths to those from @Pickering01.
### Isotopic and hyperfine structure {#Tihfs}
Much complimentary data exist on the isotopic splitting of Ti<span style="font-variant:small-caps;">i</span> lines, though unfortunately only for two of the lines we use here. The data we use come from laser fluorescence spectroscopy [LFS, also known simply as laser-induced fluorescence, LIF; @Gangrsky95]. We prefer the data of @Gangrsky95 over the less accurate work of @Cruz94 and previous results from the same group [@Anastassov94]. Isotopic separations can be estimated for many of our our chosen Ti<span style="font-variant:small-caps;">ii</span> lines using the LFS measurements of @Nouri10.
It has been consistently found that the hyperfine $A$ constants for $^{47}$Ti and $^{49}$Ti are essentially equal, and $B(47) / B(49)\approx1.22$, for all levels [@Channappa65; @Aydin90; @Stachowska94; @Gangrsky95]. We therefore use the experimental values for the relevant isotope where available, but use rescaled experimental data from the other where it does not exist for both isotopes. Data on hyperfine structure for the Ti<span style="font-variant:small-caps;">i</span> lines for which we have isotopic information are best obtained from @Gangrsky95 and @Aydin90. In cases of overlap, the ABMR data of @Aydin90 have smaller uncertainties than those of @Johann81, whereas the LFS data of @Gangrsky95 is preferable to @Aydin90’s LFS. LFS data from @Jin09 is of similar quality to, and agrees well with, that of @Gangrsky95. The only HFS data on Ti<span style="font-variant:small-caps;">ii</span> are the experimental ABMR and corresponding theoretical values produced by @Berrah92, and the LFS data of @Nouri10.
### NLTE corrections {#TiNLTE}
The NLTE line formation of Ti lines has been extensively discussed by @Bergemann11. Our NLTE calculations rely on the same model atom, although we adopt a different scaling factor to the @Drawin69 formula for inelastic collision cross-sections ($S_\mathrm{H}=1$ rather than $S_\mathrm{H}=3$; cf Sect. \[model\]). We computed [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}NLTE abundance corrections for disk-centre intensity with the [$\langle\mathrm{3D}\rangle$]{}, [<span style="font-variant:small-caps;">marcs</span>]{} and 1D model atmospheres; we adopt the [$\langle\mathrm{3D}\rangle$]{} results as an approximation to the real 3D NLTE corrections. It is interesting that even with the relatively large value $S_\mathrm{H}=1$, the resulting NLTE corrections for the [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}lines are significant. For the [$\langle\mathrm{3D}\rangle$]{} model, they range from $0.04$ to $0.09$dex, whereas the use of the model reduces them by a factor of two, mainly because its reduced temperature gradient makes over-ionisation less pronounced. @Bergemann11 found minimal NLTE effects on the relatively weak [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines we consider, so we simply adopt the LTE results for this species.
### Line selection {#Tilineselection}
We applied our selection criteria to numerous solar lines, including those used in previous works by @Blackwell87, @Reddy03 and @Bizzarri93. We ultimately retained 34 lines of Ti<span style="font-variant:small-caps;">i</span> and 14 of Ti<span style="font-variant:small-caps;">ii</span> (Table \[table:lines\]). Twenty-four of our Ti<span style="font-variant:small-caps;">i</span> lines we included by @Lawler13.
Vanadium {#Vatomicdata}
--------
For V<span style="font-variant:small-caps;">i</span> we sourced wavelengths and excitation potentials from @Davis78b, calculating wavelengths of missing lines from the stated energy levels. We adopted transition identifications from @Whaling85, with corrections to and following consultation with @Davis78b and @Martin88. V<span style="font-variant:small-caps;">ii</span> wavelengths and transition identifications came from @Biemont89, with excitation potentials from @Sugar85.
Vanadium has two stable isotopes: $^{51}$V ($I=7/2$) and $^{50}$V ($I=6$). The isotopes are present in the ratio $^{51}$V / $^{50}$V $\approx 400$ on Earth [@IUPAC98]; because of this large ratio, isotopic structure is of no importance for vanadium lines. V<span style="font-variant:small-caps;">i</span> and V<span style="font-variant:small-caps;">ii</span> lines are given in Table \[table:lines\] with corresponding atomic data.
### Oscillator strengths {#Vgfs}
The best V<span style="font-variant:small-caps;">i</span> oscillator strengths available are those of @Whaling85, who measured both TRLIF lifetimes and FTS BFs. In some cases we correct this data for arithmetic errors in converting from BFs to transition probabilities, as per @Martin88. There are also a few accurate $gf$-values from @Doerr85b, who combined TRLIF lifetimes with BFs from hook absorption and hollow cathode emission. We prefer the data of @Whaling85, as their lifetime uncertainties are lower than @Doerr85b’s, and obtaining BFs by FTS is generally considered the most reliable method available.
For V<span style="font-variant:small-caps;">ii</span>, until very recently the most accurate $gf$-values come from the FTS BFs and TRLIF lifetimes of @Biemont89. In addition to their own, these authors drew on a large number of accurate TRLIF lifetimes measured by @Karamatskos86 to arrive at their final oscillator strengths. @Karamatskos86 had also obtained FTS BFs, and also produced mostly accurate $gf$ values, but their results disagree with those of @Biemont89 below around . @Biemont89 suggest that this is likely due to an FTS calibration error by @Karamatskos86, so we prefer @Biemont89’s results in general. However, we do choose the $gf$ value of @Karamatskos86 over that of @Biemont89 for the line, as in this case the uncertainty of @Biemont89’s measurement is 50%, whereas that of @Karamatskos86’s is 8%. @Schade87 also produced TRILF lifetimes, which agree nearly perfectly with those of @Karamatskos86, and exhibit similar errors. @Biemont89 preferred the lifetimes of @Karamatskos86 because they were more extensive, but also because in the one case of disagreement, the errors of @Karamatskos86 are smaller.
During the final stages of refereeing of our article, we became aware of new experimental FTS+LIF measurements of V<span style="font-variant:small-caps;">ii</span> transition probabilities for a large number of UV/optical lines by the Wisconsin group [@Wood14V2]. Without a doubt, these should be the most accurate V<span style="font-variant:small-caps;">ii</span> data available now. Although it was too late to adopt these new $gf$-values, below we discuss how our results would have changed had we done so.
By comparing the claimed uncertainties of the vanadium abundances stated by @Biemont89 with the internal uncertainties of the sets of $gf$-values used to derive the abundances, we note that the uncertainty of their vanadium abundance is almost certainly underestimated.
### Hyperfine structure {#Vhfs}
Quite a lot of good data exists on the HFS of V<span style="font-variant:small-caps;">i</span>, with little overlap between the levels investigated by different authors. Based on the uncertainties assigned to levels common to different studies, we placed the data into a preferential tier system. In this system, no tier contained more than one value for any given level. In the top tier were the ABMR and LFS data of @Childs79, the ABMR results of @ElKashef92, @Unkel89, @Johann81 and @Childs67b, and the FTS data of @Palmeri97. The second tier consisted of an earlier FTS study by @Palmeri95 and the crossed-beam results of @Cochrane98. On the third tier were additional results from @Unkel89 using LFS, FTS data of @Lefebvre02 and Doppler-free LFS results from @Gough85. @Whaling85 and @Biemont89 included the effects of HFS in their analysis of V<span style="font-variant:small-caps;">i</span>, though we are now able to draw upon better HFS data.
Until the recent fast-ion-beam LFS work of @Armstrong11, no HFS data existed in the literature for ionised vanadium. @Biemont89 estimated hyperfine broadening of V<span style="font-variant:small-caps;">ii</span> lines empirically, adding multiple line components by eye to approximately reproduce line shapes and sufficiently desaturate modelled solar lines. We have done something similar for the one line (399.7nm) where HFS data are not available from @Armstrong11, iteratively altering the hyperfine $A$ constants of the two levels involved until we achieved a synthetic spectral line that looked qualitatively similar to the observed line. To account for the effects of convective velocities upon line shapes, it was necessary to use a 3D model for this exercise. Due to the computational demands of recalculating the radiative transfer every time however, we performed these calculations on a single snapshot of the earlier 3D model [@AspI] only.[^4] The results of this estimation procedure, along with all other data pertaining to our chosen [[V<span style="font-variant:small-caps;">i</span>]{} ]{}and [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, are given in Table \[table:lines\].
### NLTE corrections {#VNLTE}
NLTE formation of solar vanadium lines has not yet been investigated. Like Sc, Ti and Cr, the rather low ionisation energy of V means that it is predominantly singly-ionised in the solar atmosphere. As the minority species, [[V<span style="font-variant:small-caps;">i</span>]{} ]{}is expected to exhibit significant NLTE effects. In the absence of any better guidance, we adopt a blanket NLTE correction of $+0.1$dex for all [[V<span style="font-variant:small-caps;">i</span>]{} ]{}lines; this is of a similar order as the mean NLTE offsets observed in [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}($+0.15$dex), [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}($+0.06$dex) and [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}($+0.03$dex). A dedicated NLTE study of V is sorely needed.
### Line selection {#Vlineselection}
We retained 32 V<span style="font-variant:small-caps;">i</span> lines (Table \[table:lines\]) from previous analyses by @Biemont78b, @Whaling85, @Reddy03 and @McWilliam94. The solar V<span style="font-variant:small-caps;">ii</span> lines are very poor quality, because of severe blending, and are ultimately only really useful as weak supporting indicators of the solar vanadium abundance. After careful analysis of various lines used by [@Youssef89], @Biemont89 and @McWilliam95, we choose to keep only the five lines in Table \[table:lines\].
Chromium {#Cratomicdata}
--------
We sourced Cr<span style="font-variant:small-caps;">i</span> and Cr<span style="font-variant:small-caps;">ii</span> excitation potentials from @Sugar85, and used them to calculate wavelengths. Where possible, we took transition identifications from @Sobeck07 for Cr<span style="font-variant:small-caps;">i</span> and @Nilsson06 for Cr<span style="font-variant:small-caps;">ii</span>. Otherwise, we sourced transitions from VALD and checked them against the NIST database [@NIST]. Chromium has four stable isotopes [@IUPAC98]: $^{50}$Cr (4.3%), $^{52}$Cr (83.8%), $^{53}$Cr (9.5%) and $^{54}$Cr (2.4%). Only $^{53}$Cr has non-zero nuclear spin ($I=\frac{3}{2}$).
### Oscillator strengths {#Crgfs}
Highly accurate Cr<span style="font-variant:small-caps;">i</span> oscillator strengths have been produced by @Sobeck07, who measured FTS BFs and normalised them with the extensive, very accurate TRLIF lifetimes of @Cooper97. Other accurate lifetimes have been measured by TRLIF [@Hannaford81; @Kwiatkowski81; @Kwong80; @Measures77] and level-crossing [@Becker77]; these data all agree well with @Cooper97’s, and have comparable uncertainties. Other accurate $gf$-values were produced by @Tozzi85, also based upon FTS BFs but normalised to @Kwiatkowski81’s lifetimes, and @Blackwell84 [@Blackwell86], who measured relative $gf$-values using absorption spectroscopy and set them to an absolute scale with the lifetimes of @Hannaford81, @Kwiatkowski81 and @Becker77. To complete their systems of lines, @Blackwell84 also drew upon some of the relative oscillator strengths carefully measured by @Huber77 using the hook method. Because they are all of high quality, we use $gf$-values from @Sobeck07, @Blackwell84 [@Blackwell86] and @Tozzi85 without any preference for data from one source or another; where data overlap, we take the mean of the $\log gf$-values available from each of these sources.
Oscillator strengths for Cr<span style="font-variant:small-caps;">ii</span> were recently produced by @Gurell10 and @Nilsson06, who each combined their own FTS BFs with accurate TRLIF lifetimes; @Gurell10 used their own lifetimes, whereas @Nilsson06 utilised a mixture of TRLIF lifetimes from @Schade90, @Bergeson93b and their own work. Unfortunately, @Gurell10 measured no useful solar lines, and @Nilsson06 only very few of them. The small number of $gf$-values available from @Nilsson06 for good solar lines also return abundances that are highly inconsistent with each other. In the absence of any good $gf$-values for the unblended Cr<span style="font-variant:small-caps;">ii</span> lines in the solar spectrum, we default to using the theoretical @Kuruczweb oscillator strengths. Given that the @Kuruczweb $gf$-values are known to often be inaccurate, especially for weak transitions, this is not a satisfactory situation; high-quality atomic data is urgently needed for Cr<span style="font-variant:small-caps;">ii</span>.
### Isotopic and hyperfine structure {#Crhfs}
The only data on the isotopic splitting of Cr<span style="font-variant:small-caps;">i</span> lines come from the recent LFS work of @Furmann05 and the much older Fabry-Perot spectroscopy of @Heilig67; we use the former. HFS of $^{53}$Cr<span style="font-variant:small-caps;">i</span> has been measured very accurately with ABMR by @Jarosz07. No data exist on the isotopic splitting of Cr<span style="font-variant:small-caps;">ii</span> lines, nor HFS of $^{53}$Cr<span style="font-variant:small-caps;">ii</span>.
### NLTE corrections {#CrNLTE}
We computed [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}NLTE abundance corrections in intensity at disk-centre, for the [$\langle\mathrm{3D}\rangle$]{}, [<span style="font-variant:small-caps;">marcs</span>]{} and 1D model atmospheres, using the Cr model atom of @Bergemann10. For the majority of our [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}lines, the corrections are in the range $+0.02$ to $+0.04$dex for the [$\langle\mathrm{3D}\rangle$]{} model, and typically a factor of two lower for the semi-empirical model. As for the other iron-peak elements except Sc (cf. Sect. \[model\]), we used a scaling factor of $S_\mathrm{H}=1$ to the @Drawin69 recipe for inelastic collisions with H. @Bergemann10 excluded inelastic collisions from their model atom ($S_\mathrm{H}=0$), in order to obtain ionisation balance with MAFAGS-ODF model atmospheres in a larger sample of late-type stars. This, together with the fact that they considered flux spectra, explains the rather large differences (of order $\sim$0.1dex) between our NLTE abundance corrections and theirs for solar [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}lines.
For [Cr<span style="font-variant:small-caps;">ii</span>]{}, @Bergemann10 found that LTE is an excellent approximation even without inelastic hydrogen collisions; we therefore do not apply any NLTE corrections for [Cr<span style="font-variant:small-caps;">ii</span>]{}.
### Line selection {#Crlineselection}
Based on the solar analyses by @Sobeck07 and @Biemont78c, we selected the 29 best Cr<span style="font-variant:small-caps;">i</span> and 10 best Cr<span style="font-variant:small-caps;">ii</span> lines in the solar spectrum. These are given in Table \[table:lines\].
Manganese {#Mnatomicdata}
---------
We took Mn<span style="font-variant:small-caps;">i</span> wavelengths and transition designations from @Adelman89, and excitation potentials from @Corliss77. Mn has just a single stable isotope [@IUPAC98]: $^{55}$Mn, with $I=\frac{5}{2}$. Mn<span style="font-variant:small-caps;">i</span> lines and atomic data used in the current study are given in Table \[table:lines\]. None of the Mn<span style="font-variant:small-caps;">ii</span> lines that we investigated were ultimately of sufficient quality for abundance determination.
### Oscillator strengths {#Mngfs}
FTS Mn<span style="font-variant:small-caps;">i</span> BFs have been most recently measured by @DenHartog11, @BW07 and @BW05b. @BW07 used highly accurate TRLIF lifetimes from @Schnabel95, along with one other lifetime from the laser-excited delayed coincidence of @Marek75 to produce accurate oscillator strengths. @BW05b used their own TRLIF lifetimes to also convert their BFs into accurate oscillator strengths, though for some levels slightly more accurate TRLIF lifetimes are also available from @Schnabel95. @DenHartog11 also measured TRLIF lifetimes with which to convert their BFs into very accurate $gf$-values, and averaged their data with previous accurate measurements in order to produce a set of recommended values.
Until these three recent studies, the most commonly used Mn oscillator strengths were those of @Booth84c. These data were measured as a set of relative $gf$-values in the Oxford furnace and set to an absolute scale using the laser-excited delayed coincidence lifetimes of @Becker80 and @Marek75, as well as the phase-shift results of @Marek73. @Booth84c measured three different systems of lines. The first system consisted of eight lines with excitation potentials of around , and was set to an absolute scale using a single averaged lifetime from @Marek75 and @Marek73. The second system (24 lines with excitation potential ) was normalised using an average of the absolute scales implied by lifetimes of six different levels, taken from @Becker80. The 27 lines of the third system (with excitations ) were normalised using a pyrometry link to the second system, setting the two systems to the same absolute scale.
One concern with the $gf$-values of @Booth84c were some odd discrepancies with the BFs derived earlier by @Greenlee79 ([-@Greenlee79], ). There is no immediate reason for the BFs by to be unreliable. However, if one compares $gf$-values given for the lines by @Booth84c with $gf$-values for the same lines derived using BFs and either @Becker80 or @Schnabel95 lifetimes, an odd dichotomy appears. Whilst we expect both sets to be reliable, the $gf$-values of @Booth84c are consistently higher than the -@Becker80 or -@Schnabel95 values. This is confirmed when the $gf$-values of @BW07 are compared with the data of @Booth84c: the values of @Booth84c are systematically larger, by $0.13$dex ($\pm0.02$). This is however not the case for the system, where the two sets agree very well. These discrepancies have often been ignored in the literature.
The obvious question is whether the pyrometry link utilised by @Booth84c was indeed accurate, seeing as the discrepancy only exists for the lines. It seems that poor pyrometry is an unlikely explanation for a $\sim$40% difference. Clearly something is amiss, but we cannot explain the discrepancy with any confidence. The confusion in the oscillator strengths is our main reason for concluding that the stated uncertainty in the solar manganese abundance of @Booth84b [$\log \epsilon_\mathrm{Mn}=5.39\pm0.03$] probably substantially underestimated the true error. In the end, two of our adopted [Mn<span style="font-variant:small-caps;">i</span>]{} lines are affected by the uncertainties in the @Booth84c $gf$-values, as we explain below.
Wherever possible, we use the oscillator strengths of @BW07 or those recommended by @DenHartog11. In cases of overlap, we use the recommended @DenHartog11 values wherever the uncertainty of @DenHartog11’s own measured value is smaller than the error given by [@BW07]; the differences are however tiny ($\sim$0.01dex or less). For the 408.3nm line, where the value recommended by @DenHartog11 is the average of their own very accurate value and the slightly less accurate result of @BW05b, we adopt @DenHartog11’s own raw result rather than the recommended value.
For lines without $gf$-values available from either [@DenHartog11] or @BW07, we derive new oscillator strengths from the BFs of and the lifetimes of @Schnabel95. For the two good lines (426.59 and 445.70nm) measured only by @Booth84c, we use the $gf$-values of @Booth84c but renormalise them to the absolute scale of @BW07, i.e. decrease them by 0.13dex. Consummate with this rather approximate $gf$ derivation, we only give these lines a weighting of 1 in the final mean abundance. These lines are marked with asterisks in Table \[table:lines\]. For the remaining line in the system (542.0nm), we continue to use the original oscillator strength of @Booth84c.
### Hyperfine structure {#Mnhfs}
A wealth of data exists on HFS in Mn<span style="font-variant:small-caps;">i</span>, which we have classified into a similar tier system as for other elements. The best original data come from the extremely accurate spin-exchange results of @Davis71, the ABMR of @Johann81, ABMR by @D79, interference spectroscopy by @B87 and laser-atomic-beam spectroscopy by @Kronfeldt85. The second and third tiers consist of FTS and OGS data obtained by @BW05c and @Basar03 respectively. We do not use the $B$ values of @Basar03 for the odd levels however, because in our opinion their accuracy is insufficient to clearly distinguish them from zero. The next most accurate data come from @Lefebvre03, followed by @Luc72, @Handrich69 and @Walther62. The solar abundance determinations of @BW07 and @Bergemann07 include extensive HFS data from many of the sources listed above.
### NLTE corrections {#MnNLTE}
The NLTE formation of solar [[Mn<span style="font-variant:small-caps;">i</span>]{} ]{}lines was considered by @Bergemann07, using the 1D theoretical MAFAGS-ODF model atmosphere. Differences between the LTE and NLTE abundances determined using the solar flux spectrum were typically found to be around $+0.07$dex for the lines of interest to us. We performed NLTE calculations with the same model atom, but adopted a scaling factor $S_\mathrm{H}=1$ to the @Drawin69 formula (cf. Sect. \[model\]), instead of @Bergemann07’s default of $S_\mathrm{H}=0.05$. We calculated corrections in disk-centre intensity with the [$\langle\mathrm{3D}\rangle$]{}, [<span style="font-variant:small-caps;">marcs</span>]{} and 1D model atmospheres; as for other elements we adopt the [$\langle\mathrm{3D}\rangle$]{} results as proxies for the 3D case. The NLTE abundance corrections depend on the line properties, i.e. upper and lower excitation potentials, equivalent width and HFS. For example, the saturated $408.2$nm line ($E_\mathrm{exc}=2.2$eV) has an NLTE correction of only $+0.016$ dex. In contrast, the $542.0$nm line, with roughly the same equivalent width but different upper level, has an NLTE correction of $+0.07$ dex. NLTE effects in the solar [[Mn<span style="font-variant:small-caps;">i</span>]{} ]{}lines are not very sensitive to the adopted efficiency of inelastic hydrogen collisions. Reducing $S_\mathrm{H}$ to 0.05 increases the NLTE corrections for all investigated lines by a maximum of $\sim$0.02dex. Our adopted NLTE corrections are given in Table \[table:lines\].
### Line selection {#Mnlineselection}
The system yields better lines for solar abundance determination than the or systems, as the lines are formed lower in the photosphere, and are therefore less prone to uncertainties associated with the temperature structure of the model atmosphere. Even amongst the lines however, most usable Mn<span style="font-variant:small-caps;">i</span> lines are not particularly weak, so we were forced to consider mostly lines of intermediate strength. The large HFS of many of these lines should at least mitigate the effects of line strength, by desaturating profiles and lowering formation heights. Unfortunately, apart from [[Mn<span style="font-variant:small-caps;">i</span>]{} ]{}408.3nm, all the lines with BFs available from @BW05b are too weak or blended to be useful in the Sun, so most of our chosen lines came from @BW07. After considering previous solar abundance analyses [e.g. @Blackwell72a; @Biemont75a; @Booth84b; @BW07; @Bergemann07], we retained the 14 lines given in Table \[table:lines\].
Iron {#Featomicdata}
----
We used wavelengths from @Nave94 for Fe<span style="font-variant:small-caps;">i</span>, and from @2013ApJS..204....1N for Fe<span style="font-variant:small-caps;">ii</span>. Excitation potentials and transition designations for both species were taken from VALD. Iron has four stable isotopes [@IUPAC98]: $^{54}$Fe (5.8%), $^{56}$Fe (91.8%), $^{57}$Fe (2.1%) and $^{58}$Fe (0.3%). The only one of these with non-zero nuclear spin is $^{57}$Fe, with $I=\frac{1}{2}$. Iron lines therefore exhibit virtually no isotopic or hyperfine structure.
### Oscillator strengths {#Fegfs}
The best $gf$-values for Fe<span style="font-variant:small-caps;">i</span> have been obtained by quite different techniques. The Oxford dataset [see @Oxford6 and references therein] is based on absorption spectroscopy: very precise relative $gf$-values were measured in the Oxford furnace, and then normalised to an absolute scale using one line for which the absolute $gf$-value is known with high precision ($\pm$0.02 dex). Two other groups at Hannover [@Bard91; @Bard94] and at Madison [@OBrian91] used emission spectroscopy, measuring lifetimes and BFs. These three sources provide our adopted $gf$-values. When $gf$-values were available from more than one of these sets for any given line, we adopted an unweighted mean of the values from the different sets. The exception to this rule was a group of three lines where we gave less weight to the @OBrian91 data, because of their larger uncertainties for these specific lines. For one line ([[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}829.4nm) where the error on the $gf$-value remains large, we degrade the weight of the line in our analysis by one unit, as indicated by the asterisk in Table \[table:lines\]; the uncertainty of the other [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}$gf$-values given in Table \[table:lines\] is probably of order 5–10%. Newer oscillator strengths are also available from @Ruffoni13, but for the only line in our list to have been remeasured ([[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}578.4nm), the newer oscillator strength results in a clearly discrepant abundance (by $\approx$$0.1$dex).
Fe<span style="font-variant:small-caps;">ii</span> oscillator strengths increased in accuracy over the past 20 years as progressively more accurate TRLIF lifetimes were measured by @Hannaford92, @Schnabel99 and @Schnabel04, and used to normalise earlier FTS and grating spectrometer emission BFs from @Heise90 and @Kroll87. Probably the most accurate $gf$-values now come from the compilation of @Melendez09, who used these and other experimental lifetimes to recalibrate and average a raft of theoretical and experimental BFs; we adopt these data for all our Fe<span style="font-variant:small-caps;">ii</span> lines. All of our [Fe<span style="font-variant:small-caps;">ii</span>]{} lines have laboratory-based rather than astrophysical $gf$-values from @Melendez09.
### NLTE corrections {#FeNLTE}
We computed NLTE corrections for [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines using the Fe model atom of @MB_fe, which was constructed from the most up-to-date theoretical and experimental atomic data available for [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [Fe<span style="font-variant:small-caps;">ii</span>]{}. We computed the disk-centre intensity spectrum using a scaling factor $S_\mathrm{H}=1$ to the @Drawin69 recipe for inelastic H collisions, as preferred by the analysis of @MB_fe (cf. Sect. \[model\]). We did the calculations with the [$\langle\mathrm{3D}\rangle$]{}, [<span style="font-variant:small-caps;">marcs</span>]{} and 1D model atmospheres, resulting in mean NLTE corrections of $+0.01$dex. Larger [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}NLTE effects were advocated by , who also used an extended Fe model atom, but a lower efficiency for hydrogen collisions ($S_\mathrm{H}=0.1$). This choice, together with the fact that considered flux spectra, mostly explains the difference with our results. ’s NLTE corrections were $+0.04$dex for lines with excitation energies up to 1eV, and about $+0.03$dex for higher-excitation lines. NTLE corrections are negligible for [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}[@MB_fe], so we adopt LTE results for the ionised lines.
### Line selection {#Felineselection}
We selected the 22 best [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and 9 best [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, i.e. those lines for which equivalent widths are easily measured, that do not show any intractable trace of blending, and are not too strong. Our selected lines and atomic data are given in Table \[table:lines\]. For Fe<span style="font-variant:small-caps;">i</span>, we made sure to have a sample of lines that covers a large range of excitation potentials (0–4.6eV) to probe the performance of the different atmospheric models over a range of heights.
Cobalt {#Coatomicdata}
------
We took excitation potentials of Co<span style="font-variant:small-caps;">i</span> from @Pickering96b, as well as wavelengths and transition designations where available; otherwise, we sourced wavelengths and transitions from @Cardon82. The only stable isotope of cobalt is $^{59}$Co [@IUPAC98], which has nuclear spin $I=\frac{7}{2}$. Our chosen Co<span style="font-variant:small-caps;">i</span> lines and atomic data are given in Table \[table:lines\]. None of the Co<span style="font-variant:small-caps;">ii</span> lines in the solar spectrum are suitable for abundance analyses.
### Oscillator strengths {#Cogfs}
The most reliable Co<span style="font-variant:small-caps;">i</span> oscillator strengths currently available come from @Nitz99, who measured FTS BFs and set them to an absolute scale using their own TRLIF lifetimes [@Nitz95]. The next most accurate data are those of @Cardon82, who measured BFs that they set to an absolute scale using the TRLIF lifetimes of @Marek77 and @Figger75. For some lines, the $gf$-values of @Cardon82 are accurate to better than 10%, which is comparable to the accuracy obtained by @Nitz99; for other lines the uncertainties are much larger, of order 20–30%. BFs contemporary with those of @Cardon82 are also available from @Guern82, but we prefer the data of @Cardon82 as they are based upon FTS recordings and include a more complete set of branches.
### Hyperfine structure {#Cohfs}
We used stated uncertainties to classify the wealth of data available on Co<span style="font-variant:small-caps;">i</span> HFS into a similar tier system as for other elements. Our first choice of HFS data were the ABMR results of @Childs68, and the combined Doppler-free and Doppler-limited LFS / OGS results of @Guthlorien90. The next most accurate data come from the FTS of @Pickering96a. Also available are unpublished data obtained by J. Ibrahim-Rüd and R. Wenzel, reproduced in the paper of @Guthlorien90. We fit these data into the hierarchy on a level-by-level basis around @Guthlorien90, @Pickering96a and @Childs68. @Bergemann10Co included extensive HFS data in their calculation of the solar Co abundance, showing that neglect or inaccurate treatment of HFS can lead to severe errors in derived abundances.
### NLTE corrections {#CoNLTE}
Non-LTE formation of solar Co<span style="font-variant:small-caps;">i</span> and Co<span style="font-variant:small-caps;">ii</span> lines has been investigated by @Bergemann10Co using the MAFAGS-ODF models. The results indicate large departures from LTE in Co<span style="font-variant:small-caps;">i</span>, leading to NLTE abundance corrections of $+0.1$–$0.2$dex at $S_\mathrm{H}=0.05$ [@Bergemann10Co cf. their Table 4], resembling the situation with [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}lines [@Zhang08]. We use the same Co model atom, but adopt $S_\mathrm{H}=1$ (cf. Sect. \[model\]) and disk-centre intensity spectra for the [$\langle\mathrm{3D}\rangle$]{}, [<span style="font-variant:small-caps;">marcs</span>]{} and 1D model atmospheres. This leads to somewhat smaller NLTE abundance corrections, of order $+0.09$dex for [$\langle\mathrm{3D}\rangle$]{} and $+0.07$ dex for the model.
### Line selection {#Colineselection}
Unfortunately, there are rather few good lines in the solar spectrum with oscillator strengths available from @Nitz99, so the bulk of our lines have $gf$ values drawn from @Cardon82. For some of the cleanest weak lines, the @Cardon82 $gf$-values have rather large uncertainties (over 20% in some cases). We include such lines because of their excellent profiles, but downgrade their weightings; affected lines are marked with an asterisk in Table \[table:lines\]. From the lines considered in the abundance analyses of @Cardon82, @Biemont78b, @Kerola76 and @Holweger71, we retained the 13 transitions given in Table \[table:lines\].
Nickel {#Niatomicdata}
------
We obtained [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}wavelengths and excitation potentials from @Litzen93. Transition identities came from @Wickliffe97, except for , where the transition designation is from VALD [@VALD]. Our selected [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}lines are given in Table \[table:lines\]. We also considered [Ni<span style="font-variant:small-caps;">ii</span>]{}, but it ultimately played a very minimal role in our analysis; we omit it from Table \[table:lines\], give a truncated discussion of its atomic data and line selection in this section, and discuss only briefly the mean implied Ni abundance in Sect. \[Niresults\].
Nickel has five stable isotopes, so exhibits significant isotopic structure [as seen by e.g. @Brault81; @Melendez99a]. These are $^{58}$Ni, $^{60}$Ni, $^{61}$Ni, $^{62}$Ni and $^{64}$Ni, present in the approximate ratios 68:26:1:4:1 [@IUPAC98]. In practice, the isotopic structure of nickel lines is dominated by $^{58}$Ni and $^{60}$Ni due to their much greater natural abundances. As an even-$Z$ element, all the even-$A$ nuclei of nickel have $I=0$, so nickel lines do not exhibit any HFS apart from $^{61}$Ni. The contribution of $^{61}$Ni to the HFS of nickel is minimal, given its very low abundance relative to $^{58}$Ni and $^{60}$Ni. For spectroscopic purposes, one can thus effectively regard nickel as consisting of four isotopes, and devoid of HFS.
### Oscillator strengths {#Nigfs}
High-quality [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}oscillator strengths are available from the FTS BFs of @Wickliffe97, which the authors placed on an absolute scale using the TRLIF lifetimes of @Bergeson93a. These have recently been updated and greatly extended by @Wood14. A small number of high-quality $gf$-values are also available from @Johansson03, based upon FTS BFs and a single TRLIF lifetime. @Bergeson93a used their new lifetimes to produce other accurate $gf$-values from the BFs of @Blackwell89, but these lines are all in the UV, so of little use to us because the ultraviolet solar spectrum is so crowded. Accurate oscillator strengths for optical lines of [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}do not exist, so we turned to the extensive theoretical transition probabilities of @Fritzsche00.
### Isotopic structure {#Nihfs}
Wherever available, we employ isotopic separations from @Wood14, who fitted the isotopic shifts of a large number of [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}energy levels to earlier spectroscopic data. Much of the power of that analysis can be attributed to the accurate FTS wavelengths of $^{58}$Ni and $^{60}$Ni line components recorded by @Litzen93. As in @Wood14’s analysis, we model $^{58}$Ni and $^{60}$Ni components explicitly, and estimate the contribution of the remaining isotopes by placing them in a single line component, which we offset from $^{60}$Ni by the same amount as $^{60}$Ni is offset from $^{58}$Ni. These data are included in Table \[table:hfs\].
### NLTE corrections {#NiNLTE}
The only explicit investigation of non-LTE effects on solar nickel line formation so far has been that of @Bruls93, who looked at the [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}676.8nm line often used for helioseismology. Although @Bruls93 did not give any explicit NLTE abundance correction for this line, his Fig. 4 would imply a correction of about $+$0.06dex. This line corresponds to a transition between low-lying atomic levels and is thus formed higher than those we employ here. It may therefore be expected to show stronger NLTE effects than our weaker high-excitation lines. Because @Bruls93 completely neglected inelastic H collisions, his results can probably be taken as an upper limit for possible NLTE effects. We therefore do not expect significant departures from NLTE for our own weak, high-excitation lines, and simply adopt LTE results for [Ni<span style="font-variant:small-caps;">i</span>]{}. Further investigation of NLTE Ni line formation [e.g. @Vieytes13] would be welcome however, as this expectation bears additional verification.
### Line selection {#Nilineselection}
From the most accurate $gf$-values available for [Ni<span style="font-variant:small-caps;">i</span>]{}, we have retained the 16 weak, unblended lines of Table \[table:lines\]. We also included the slightly stronger line, because of its pristine appearance in the solar spectrum and the quality of its atomic data.
Although the situation with [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}is better than for [[Mn<span style="font-variant:small-caps;">ii</span>]{} ]{}or [Co<span style="font-variant:small-caps;">ii</span>]{}, most of the [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}lines in the IR are too weak to be useful for abundance purposes, and those in the optical are generally at very short wavelengths and severely blended. We attempted to use the lines at 340.2, 342.1, 345.4 and ; all are perturbed to some degree, so the scatter in resultant abundances probably reflects both large intrinsic errors in the theoretical $gf$-values and the crowding in this spectral region.
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Derived solar elemental abundances {#results}
==================================
We have derived the solar abundances of Sc, Ti, V, Cr, Mn, Fe, Co and Ni from each of the lines given in Table \[table:lines\]. The interpolated theoretical 3D line profiles show good agreement with the observed solar spectrum, as can be seen in the sample of lines shown in Figs. \[fig:profiles\] and \[fig:profiles2\]. This agreement is a result of the inhomogeneous, three-dimensional temperature and velocity structure of the 3D model atmosphere, and the inclusion of HFS and isotopic structure wherever necessary. A small systematic deviation of the theoretical profiles from the observed spectrum can be seen in the cores and wings of some lines: the lines are slightly too deep in the core and too shallow in the wings, which may signal NLTE effects (note that full 3D line formation calculations have not been attempted). These small discrepancies may also indicate that the photospheric velocity structure of the 3D model, whilst certainly highly realistic, is not quite perfect.
The 3D abundance results are given for each line in Table \[table:lines\], including NLTE corrections where possible. In Table \[table:lines\] we also give the LTE abundances derived from each line with the [$\langle\mathrm{3D}\rangle$]{}, HM, <span style="font-variant:small-caps;">marcs</span> and [miss]{} 1D models. In Figs. \[fig:sc\]–\[fig:ni\] we plot the 3D abundances as a function of line strength and excitation potential. We also show in these figures the difference between the abundances derived using the 3D and HM models, and between those derived from the 3D and [$\langle\mathrm{3D}\rangle$]{} models, as a function of line strength and excitation potential.
In the following sections we discuss the results for each element in detail, comparing with previous determinations of the solar abundance and with the meteoritic values. The latter we take from the recent careful compilation and analysis of @Lodders09, renormalised to the photospheric abundance of silicon determined in ($\log\epsilon_\mathrm{Si}=7.51$, as already done in ). We also describe the updates we have made for specific elements since . In addition to those updates, before reiterating on all abundance calculations a final time, we updated all partition functions and ionisation potentials (Table \[table:partition\]), and updated our equation-of-state tables and base atmospheric composition to the published mixture. We summarise our full results in Table \[table:abuns\], including our final recommended abundances. We compare these results to previous solar abundance compilations in Table \[table:compilations\]. We remind the reader that the error treatment used in the following sections is summarised in Sect. \[calculations\].
Scandium {#Scresults}
--------
For [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}lines, the 3D+NLTE result is $\log \epsilon_\mathrm{Sc}=3.14\pm0.09$ ($\pm$0.01 stat, $\pm$0.09 sys). The corresponding result from [[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}lines is $\log \epsilon_\mathrm{Sc}=3.17\pm0.04$ ($\pm$0.02 stat, $\pm$0.04 sys), in good agreement with the [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}result. Taking all [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}lines together, our final recommended Sc abundance becomes $$\log \epsilon_\mathrm{Sc}=3.16\pm0.04\ (\pm0.01\ \mathrm{stat},\ \pm0.04\ \mathrm{sys}).$$ Intriguingly, this value is 0.11dex, or more than two standard deviations, larger than the meteoritic value [$3.05\pm0.02$; @Lodders09]. We leave speculation as to the importance of this and other photospheric-meteoritic differences for future work.
We see from Table \[table:abuns\] and the right-hand panels of Fig. \[fig:sc\] that the model atmosphere plays only a minor role for [[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, as is generally true for the dominant ionisation stage. On the other hand, abundances derived from [[Sc<span style="font-variant:small-caps;">i</span>]{} ]{}are extremely sensitive to the choice of model, as expected for a low ionisation neutral species. We also see in Fig. \[fig:sc\] that 3D Sc abundances do not show any significant trend with excitation potential or line strength. Abundance scatter is generally higher with Sc<span style="font-variant:small-caps;">ii</span> than Sc<span style="font-variant:small-caps;">i</span> lines, reflecting the greater uncertainty in the [[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}$gf$-values.
Our results are in good agreement with the Sc abundance of @Zhang08. Their value of $\log \epsilon_\mathrm{Sc}=3.13\pm0.05$ was essentially based on [[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, and derived using the same experimental $gf$-values as we do here, along with a theoretical 1D photospheric model. Compared to the result we presented in ($\log \epsilon_\mathrm{Sc}=3.16\pm0.04$), in this analysis we have discarded the three lines discussed in Sect. \[Scgfs\] as having excellent profiles but unsatisfactory oscillator strengths.
Titanium {#Tiresults}
--------
Titanium is in principle an ideal case: a large number of good solar lines with accurate transition probabilities, very minor HFS and isotopic broadening, and extensive NLTE calculations available for the minority species ([Ti<span style="font-variant:small-caps;">i</span>]{}).
Nonetheless, the derived 3D+NLTE Ti abundances from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}($\log \epsilon_\mathrm{Ti}=4.88\pm0.05$; $\pm$0.01 stat, $\pm$0.05 sys), and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}($\log \epsilon_\mathrm{Ti}=4.97\pm0.04$; $\pm$0.01 stat, $\pm$0.03 sys) show a 0.09dex discrepancy. Even more puzzlingly, this discrepancy is not present in either the mean or [$\langle\mathrm{3D}\rangle$]{} results. Referring to Fig. \[fig:ti\], 3D+NLTE abundances from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}show no perceptible trend with line strength or excitation potential, whereas the HM and [$\langle\mathrm{3D}\rangle$]{} results show a clear trend with line strength for both ionisation stages. Given the rather large 3D corrections for [Ti<span style="font-variant:small-caps;">i</span>]{}, and the fact that we always *assume* the NLTE corrections for the 3D model to be equal to those calculated for the [$\langle\mathrm{3D}\rangle$]{} model, we suspect that the NLTE corrections are somewhat underestimated in [$\langle\mathrm{3D}\rangle$]{} compared to full 3D. Alternatively, our chosen efficiency of inelastic collisions with hydrogen ($S_\mathrm{H}=1$) may be somewhat too high in this case; had we instead adopted $S_\mathrm{H}=0.05$, the 3D+NLTE abundance from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}would have been $\log \epsilon_\mathrm{Ti}=4.93$.
Considering the sensitivity of the [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}NTLE corrections to $S_{\rm H}$, for our final recommended Ti abundance we take a weighted mean of the [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}results, with the weightings determined by the respective uncertainties of the two results. This favours [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}due to its smaller systematic uncertainty, resulting in $$\log \epsilon_\mathrm{Ti}=4.93\pm0.04\ (\pm0.01\ \mathrm{stat},\ \pm0.04\ \mathrm{sys}).$$ Here we have estimated the errors by considering all [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines equally in a single list, rather than by using the same statitistical weighting procedure as for the mean. Had we instead used the latter procedure, the final uncertainty would be just $0.03$dex; owing to the tension between [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}, and for consistency with other elements where we include both ionisation stages in a single list, we think it more appropriate to adopt the larger estimate. The final Ti abundance is in excellent agreement with the meteoritic value [$4.91\pm0.03$; @Lodders09], but the difference between the results returned by the two ionisation stages remains troubling.
@Bergemann11 found that Ti ionisation balance would be best satisfied with the MAFAGS-OS model if one were to adopt the @BW06 and @Pickering01 $gf$-values, giving $\log \epsilon_\mathrm{Ti}=4.94\pm0.05$ from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}lines and $\log \epsilon_\mathrm{Ti}=4.95\pm0.06$ for [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines. Using more accurate $gf$-values (@Bizzarri93) would lead to a larger discrepancy between [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [Ti<span style="font-variant:small-caps;">ii</span>]{}: $\log \epsilon_\mathrm{Ti}=4.93\pm0.04$ from [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}lines and $\log\epsilon_\mathrm{Ti}=4.98\pm0.04$ for [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}lines. As in this paper, @Bergemann11 observed a strong dependence of [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}abundances upon the chosen model atmosphere and collisional efficiency parameters.
Compared to the result we adopted in ($\log \epsilon_\mathrm{Ti}=4.95\pm0.05$), we now have dedicated NLTE calculations available in intensity for [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}based on the work of @Bergemann11. We have also now dropped the [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}line at 522.4nm from our line list due to blending, adopted isotopic splitting and HFS data for [Ti<span style="font-variant:small-caps;">ii</span>]{}, and employed accurate new oscillator strengths for both ionisation stages [@Lawler13; @Wood13].
Vanadium {#Vresults}
--------
Quality atomic data exists for both ionisation stages of V, but the only clear solar lines are of [V<span style="font-variant:small-caps;">i</span>]{}. Unfortunately, only [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines are expected to form in LTE; the magnitude of NLTE corrections for [[V<span style="font-variant:small-caps;">i</span>]{} ]{}is unknown, so we apply an [*ad hoc*]{} correction of $+0.1$dex, as discussed in Sect. \[Vgfs\].
The minority species, [V<span style="font-variant:small-caps;">i</span>]{}, is more strongly affected by the temperature structure of the model atmosphere than [V<span style="font-variant:small-caps;">ii</span>]{}. The 3D LTE abundance from [[V<span style="font-variant:small-caps;">i</span>]{} ]{}lines is $\log \epsilon_\mathrm{V}=3.79\pm0.04$ ($1\sigma$ dispersion), whereas with the model we obtain $\log \epsilon_\mathrm{V}=3.97\pm0.03$ ($1\sigma$). For [V<span style="font-variant:small-caps;">ii</span>]{}, the 3D and LTE results agree well: $\log \epsilon_\mathrm{V}=4.00\pm0.05$ ($1\sigma$) in 3D, $\log \epsilon_\mathrm{V}=4.01\pm0.05$ ($1\sigma$) with .
These numbers are not a complete surprise: we know that ‘includes’ NLTE effects to some degree by way of its empirical temperature construction, as effects of departures from LTE can be partially mimicked by adjusting the spatially-averaged temperature structure of the model atmosphere, a phenomenon dubbed ‘NLTE masking’ by . Similarly, the 3D [[V<span style="font-variant:small-caps;">i</span>]{} ]{}results exhibit a strong dependence upon excitation potential, whereas the results from do not (Fig. \[fig:v\]). Because we applied the same NLTE correction to all [V<span style="font-variant:small-caps;">i</span>]{} lines, the trend also remains in NLTE. In reality we expect more pronounced NLTE effects for lower-excitation lines, as these are sensitive to higher atmospheric layers, where lower densities and temperatures make LTE an increasingly poor approximation.
Although low-excitation lines are most sensitive to the temperature structure, and therefore less reliable as abundance indicators, we have no way to know whether our universal NLTE correction of $+0.1$dex is more accurate at high or low excitation potential. To avoid introducing any further systematic bias into our result, we therefore retain both the high- and low-excitation lines in our sample of [[V<span style="font-variant:small-caps;">i</span>]{} ]{}lines. A dedicated NLTE study of V line formation in the Sun would clarify matters substantially.
Given the abysmal nature of the [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines in the solar spectrum, abundances from these lines are dominated by systematic and statistical errors in the determination of equivalent widths. We therefore trust the absolute values of the [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}results even less than our *ad hoc* [[V<span style="font-variant:small-caps;">i</span>]{} ]{}NLTE correction, and adopt the NLTE-corrected [[V<span style="font-variant:small-caps;">i</span>]{} ]{}3D result as our recommended value: $$\log \epsilon_\mathrm{V}=3.89\pm0.08\ (\pm0.01\ \mathrm{stat},\ \pm0.08\ \mathrm{sys}).$$
Our result is significantly lower than the stated -based abundances of @Whaling85 [$3.99\pm0.01$] and @Biemont89 [$4.02\pm0.02$]; their quoted uncertainties only consider statistical errors, not systematic errors stemming from, for example, the $gf$-values, model atmospheres or LTE line formation. Our 3D result is also below the meteoritic value [$3.96\pm0.02$; @Lodders09], but the mutual uncertainties overlap. Compared to ($\log \epsilon_\mathrm{V}=3.93\pm0.08$), here we have added laboratory HFS data for [V<span style="font-variant:small-caps;">ii</span>]{}. We also discarded two [[V<span style="font-variant:small-caps;">i</span>]{} ]{}lines (619.9nm and 624.3nm), because we are suspicious as to the accuracy of the experimental branching fractions measured from their shared upper level.
As noted in Sect. \[Vgfs\], @Wood14V2 have recently measured new experimental transition probabilities for a large number of [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines. Had we adopted their values for our five lines, the 3D-based [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}abundance would be 0.02dex lower, and thus in slightly better agreement with the [[V<span style="font-variant:small-caps;">i</span>]{} ]{}results. @Wood14V2 also performed spectrum synthesis (using the HM model) rather than fitting equivalent widths as we do here, which further reduces the inferred [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}abundance. Comparing the results obtained with the HM model by both @Wood14V2 and us, and taking into account the differences in the adopted $gf$-values, we estimate that employing spectrum synthesis with our 3D models would have reduced the abundance by a further 0.02dex. Our final 3D [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}abundance would then have become $\log \epsilon_\mathrm{V}=3.96$, in perfect agreement with the meteoritic value. We note that @Wood14V2 employed a larger set of 15 [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, resulting in a mean abundance of $\log \epsilon_\mathrm{V}=3.95\pm0.05$ ($1\sigma$), which should be contrasted with our HM-based value of $\log \epsilon_\mathrm{V}=4.01\pm0.05$ ($1\sigma$). Adopting the oscillator strengths of @Wood14V2 and taking into account the $-0.02$dex impact of spectrum synthesis on our lines, our HM abundance would become $\log \epsilon_\mathrm{V}=3.97\pm0.04$, in perfect agreement with @Wood14V2 for the five lines in common. In other words, there is a real possibility that our 3D [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}result should be decreased by about 0.04dex, bringing it into better agreement with [V<span style="font-variant:small-caps;">i</span>]{}. However, we still argue that [[V<span style="font-variant:small-caps;">i</span>]{} ]{}is a better indicator of the solar V abundance, in spite of the uncertainty in the NLTE effects.
Chromium {#Crresults}
--------
For Cr<span style="font-variant:small-caps;">i</span>, we have a large number of very good solar lines (Sect. \[CrNLTE\]) and very accurate $gf$-values (Sect. \[Crgfs\]). Our derived NLTE abundance from [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{} lines is $\log \epsilon_\mathrm{Cr}=5.60\pm0.04$ ($\pm$0.01 stat, $\pm$0.04 sys). Using the very few recent experimental $gf$-values for Cr<span style="font-variant:small-caps;">ii</span> lines [@Nilsson06], we found a very large scatter in abundances. We therefore recommended (Sect. \[Crgfs\]) the theoretical $gf$-values of @Kuruczweb as the best currently available. With these data we find $\log \epsilon_\mathrm{Cr}=5.65\pm0.04$ ($\pm$0.02 stat, $\pm$0.04 sys) from [[Cr<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, in good agreement with both the [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}result and the meteoritic abundance [$5.64\pm0.01$; @Lodders09]. We therefore adopt the mean result from all Cr lines $$\log \epsilon_\mathrm{Cr}=5.62\pm0.04\ (\pm0.01\ \mathrm{stat},\ \pm0.03\ \mathrm{sys})$$ as our final recommended solar Cr abundance.
No significant trends are visible with line strength or excitation potential in the results with any model (Fig. \[fig:cr\], left panels). Our NLTE results with the model agree very well with the LTE abundances derived from the Cr<span style="font-variant:small-caps;">i</span> lines by @Biemont78c [$\log \epsilon_\mathrm{Cr}=5.67$], @Blackwell87 [$\log \epsilon_\mathrm{Cr}=5.68$] and @Sobeck07 [$\log \epsilon_\mathrm{Cr}=5.64$], which however is somewhat of a coincidence given the significant NLTE corrections. The NLTE result of @Bergemann11, obtained by choosing $S_\mathrm{H}=0$ so as to impose ionisation balance with the MAFAGS-ODF model, is $0.12$ dex higher: $\log \epsilon_\mathrm{Cr}=5.74$.
For [Cr<span style="font-variant:small-caps;">ii</span>]{}, our results are much smaller than those of @Sobeck07 [$\log \epsilon_\mathrm{Cr}=5.67\pm0.13$ with <span style="font-variant:small-caps;">marcs</span>, $\log \epsilon_\mathrm{Cr}=5.77\pm0.13$ with ] or @Bergemann10 [$\log \epsilon_\mathrm{Cr}=5.79\pm0.12$ with MAFAGS-ODF]. Even using the theoretical $gf$-values of Kurucz, our dispersion is also much smaller ($\sigma=0.06$) than either of these results. The substantially larger scatter seen with @Nilsson06 $gf$-values [as used by @Sobeck07; @Bergemann10] than with semi-empirical Kurucz values is worrisome; it remains to be seen if the experimental measurements were affected by a systematic error of some kind.
The result we give here is slightly updated with respect to that in ($\log \epsilon_\mathrm{Cr}=5.64\pm0.04$), as we now have dedicated NLTE intensity calculations for [[Cr<span style="font-variant:small-caps;">i</span>]{} ]{}(Sect. \[CrNLTE\]) for our specific 1D models with $S_\mathrm{H}=1$, based on the work of @Bergemann10.
Manganese {#Mnresults}
---------
Following our consideration of the most reliable lines and oscillator strengths for [[Mn<span style="font-variant:small-caps;">i</span>]{} ]{}(Sect. \[Mngfs\]), we find a final NLTE Mn abundance of $$\log \epsilon_\mathrm{Mn}=5.42\pm0.04\ (\pm0.01\ \mathrm{stat},\ \pm0.04\ \mathrm{sys}),$$ slightly smaller than the meteoritic value [$5.48\pm0.01$; @Lodders09].
Our result is somewhat larger than the earlier LTE result of @Booth84b [$\log \epsilon_\mathrm{Mn}=5.39$]. This shift can mainly be attributed to the positive NLTE abundance corrections and our more accurate oscillator strengths. Our and <span style="font-variant:small-caps;">marcs</span> abundances ($\log \epsilon_\mathrm{Mn}=5.47$ and 5.37, respectively) are in good agreement with the corresponding results of @BW07 [$\log \epsilon_\mathrm{Mn}=5.46$ and 5.37, respectively]. No significant trends with equivalent width or excitation potential are visible in Fig. \[fig:mn\].
Compared to the Mn abundance adopted in ($\log \epsilon_\mathrm{Mn}=5.43\pm0.04$), we now have dedicated NLTE intensity calculations using the model atom of @Bergemann07 for the individual Mn lines and 1D models we employ (instead of relying on the MAFAGS-ODF model), and adopted $S_\mathrm{H}=1$ (instead of $S_\mathrm{H}=0.05$). We have also updated four oscillator strengths (Table \[table:lines\]) with the new data of @DenHartog11.
Iron {#Feresults}
----
Both [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}should be good indicators of the Fe abundance, as we have several clean solar lines, small NLTE corrections and accurate oscillator strengths. Our derived Fe abundance from [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines ($\log \epsilon_\mathrm{Fe}=7.45\pm0.04$; $\pm$0.01 stat, $\pm$0.04 sys) overlaps the [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}result ($\log \epsilon_\mathrm{Fe}=7.51\pm0.04$; $\pm$0.01 stat, $\pm$0.04 sys) to within the mutual uncertainties, but the agreement is not perfect. This may indicate a small error in the [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}oscillator strengths (as they are at least partially based on theoretical results, which are not always accurate), or perhaps slightly too high an adopted value of $S_\mathrm{H}$ (resulting in slightly too low NLTE corrections). A similar size discrepancy exists in the results (Table \[table:abuns\]), but reversed in sign: the neutral species returns an abundance $0.06$dex higher. The only significant trend visible in Fig. \[fig:fe\] is in the difference between the 3D and or [$\langle\mathrm{3D}\rangle$]{} results from [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}as a function of line strength: stronger [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines appear to show larger positive corrections due to 3D effects.
Considering all our adopted [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines, our final 3D+NLTE Fe abundance is $$\log \epsilon_\mathrm{Fe}=7.47\pm0.04\ (\pm0.01\ \mathrm{stat},\ \pm0.04\ \mathrm{sys}),$$ in very good agreement with the meteoritic value [$7.45\pm0.01$; @Lodders09].
Our derived Fe abundance from [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines is in perfect agreement with the 3D result of @Caffau11 [$\log \epsilon_\mathrm{Fe}=7.51$], although the standard deviation of our result is smaller (0.04 vs. 0.06dex). For lines in common, the equivalent widths employed in the two studies agree to within a few percent, so the difference in scatter presumably reflects a difference in the quality of the line selection. Both our [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}abundances are consistent with those found in @AspII [$\log \epsilon_\mathrm{Fe}=7.44\pm0.05$ and $7.45\pm0.10$ respectively]. The difference in the scatter of the [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}result is in this case due to our use of the improved @Melendez09 oscillator strengths. The difference in the central value, whilst not statistically significant, probably reflects a slight difference in the temperature gradient between the two versions of the 3D model. Our results are also in full agreement with @MB_fe [$\log \epsilon_\mathrm{Fe}=7.46\pm0.02$], who investigated NLTE line formation of Fe with the same [$\langle\mathrm{3D}\rangle$]{} solar model atmosphere as we employ here.
Recently, revisited the issue of the solar Fe abundance in light of 3D magneto-hydrodynamic simulations of the solar atmosphere for different magnetic field strengths ($B_{\rm z} = 0-200$G). Their 3D models were calculated with the same [stagger]{} code as we employ, but with less up-to-date opacities and equation-of-state. They found quite substantial effects on the derived Fe abundance due to the presence of magnetic fields: in some cases up to $+0.15$dex for the strongest magnetic fields. For typical [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines employed here and elsewhere, with small or negligible Landé factors, the effects are much more sedate: $\approx +0.04$dex for $B_{\rm z} = 200$G. Most of this is an indirect effect: it is not Zeeman broadening (which in any case would strengthen the line and thus lead to lower inferred Fe abundance), but the impact of magnetic fields on the atmospheric temperature structure that matters most. With magnetic dissipation included, the higher atmospheric layers are heated relative to the non-magnetic case, with the difference amounting to $\approx$$130$K at $\log \tau_{\rm 500} = -2$ for the $B_{\rm z} = 200$G case . As a consequence, the number density of [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}is decreased and a higher Fe abundance is required to reproduce the observed [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines; although did not consider typical [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines used for abundance purposes, the expectation is that those lines should be rather insensitive to the different temperature structures, as they are formed in significantly deeper layers. At face value, the agreement between the [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{} and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{} results would be improved, especially since observations of the quiet Sun suggest the presence of a ubiquitous mixed-polarity magnetic field with an average strength of $\approx 100$G [@2004Natur.430..326T].
We intend to return to this important issue in the future, but in the meantime we note that the case for a significant upward revision of the solar Fe abundance (and by consequence many other elements) due to the presence of magnetic fields is not as unequivocal as argued by . Firstly, at magnetic fields of 100G, the effect is in fact rather minor: $\approx 0.02$dex for lines similar to those we use. Secondly, our recommended Fe abundance is based on both [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}lines. Thirdly, found that 3D MHD models of the solar atmosphere perform worse than simulations without magnetic fields against a number of key observational diagnostics, including the continuum centre-to-limb variation; they thus conclude that current MHD solar models are in fact less realistic than the one employed by us. In view of these findings, we recommend our 3D+NLTE value based on a 3D hydrodynamic solar model, but caution that further studies into the importance of magnetic fields are needed.
In , we adopted the result from [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}($\log \epsilon_\mathrm{Fe}=7.50\pm0.04$) as our reference abundance. Here we also utilise [Fe<span style="font-variant:small-caps;">i</span>]{}, because we now have dedicated NLTE calculations available for our lines with the [$\langle\mathrm{3D}\rangle$]{} model. Relative to the analysis, we have dropped two [[Fe<span style="font-variant:small-caps;">i</span>]{} ]{}lines: 657.4nm, because it sits in the wing of H$\alpha$, and 660.9nm, because of its relatively large line strength.
Cobalt {#Coresults}
------
From our selection of weak [[Co<span style="font-variant:small-caps;">i</span>]{} ]{}lines, we find a mean NLTE Co abundance of $$\log \epsilon_\mathrm{Co}=4.93\pm0.05\ (\pm0.01\ \mathrm{stat},\ \pm0.05\ \mathrm{sys}).$$ This is somewhat higher than the meteoritic value [$4.87\pm0.01$; @Lodders09], but still marginally consistent to within the mutual errors. Our result agrees well with that of @Bergemann11 [$4.95\pm0.04$], although in that paper a different model atmosphere, flux spectra and $S_\mathrm{H}=0.05$ were used, resulting in larger NLTE corrections than we see here with $S_\mathrm{H}=1$ ($+0.14$ vs. $+0.08$dex). Our mean LTE result ($\log \epsilon_\mathrm{Co}=4.94$) is also consistent with the -based abundance derived by @Cardon82 [$\log \epsilon_\mathrm{Co}=4.92$]. Our result exhibits a smaller dispersion however, reflecting the care we took in our line selection: $\sigma=0.06$ in our results, $\sigma=0.08$ in @Cardon82’s. The dispersions of our 3D LTE and NLTE results were $\sigma=0.05$dex, similar to those of @Bergemann10Co, which is indicative of the intrinsic uncertainty of the oscillator strengths.
No substantial trend in abundances with line strength can be seen in Fig. \[fig:co\]. A weak trend with excitation potential is visible in the 3D results: lines with $\chi_\mathrm{exc}>3$eV lead to an abundance of $\log \epsilon_\mathrm{Co}=4.90\pm0.02$($1\sigma$), whereas lower-excitation lines return an abundance of $\log \epsilon_\mathrm{Co}=4.97\pm0.05$($1\sigma$). This may be an effect of imperfect $gf$-values, NLTE corrections or the temperature structure of the model atmosphere. Inspection of the lower right panel of Fig. \[fig:co\] reveals that the trend is more severe with the [$\langle\mathrm{3D}\rangle$]{} model than the full 3D model, and yet more severe again with the model. Using the model, the high-excitation lines give $\log \epsilon_\mathrm{Co}=4.92\pm0.02$($1\sigma$), whereas the low-excitation lines return $\log \epsilon_\mathrm{Co}=5.04\pm0.05$($1\sigma$); the switch to 3D atmospheric modelling is a clear improvement for solar analysis of Co.
Relative to ($\log \epsilon_\mathrm{Co}=4.99\pm0.07$), the main update to the Co abundance here is that we calculate NLTE intensity corrections [based on @Bergemann10Co] specifically for our different 1D models rather than the MAFAGS-ODF model, and use $S_\mathrm{H}=1$ instead of $S_\mathrm{H}=0.05$. This accounts for $0.03$dex of the reduction; the remaining $0.03$dex comes from the updated opacities, equation of state, ionisation potentials and partition functions.
Nickel {#Niresults}
------
The mean 3D nickel abundance from [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}lines $$\log \epsilon_\mathrm{Ni}=6.20\pm0.04\ (\pm\!<\!0.01\ \mathrm{stat},\ \pm0.04\ \mathrm{sys})$$ is in excellent agreement with the meteoritic value [$6.20\pm0.01$; @Lodders09]. [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}lines indicated widely varying abundances, though the mean values they return with each model are broadly consistent with [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}results. Using the theoretical $gf$-values of @Fritzsche00 results in a far lower abundance scatter than any other $gf$-values, leading us to believe that these are currently the most accurate oscillator strengths available for optical [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}lines. Given the uncertainty in the mean [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}value, we adopt the 3D [[Ni<span style="font-variant:small-caps;">i</span>]{} ]{}result as the most reliable estimate of the solar abundance. [[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{}is the most model-sensitive of our ionised species.
No trends with line strength or excitation potential can be seen in the 3D results (Fig. \[fig:ni\]). The abundance corrections due to 3D effects have a clear dependence upon line strength, and a smaller correlation with excitation potential. In contrast, the effect of the mean temperature structure is 0.04dex regardless of line strength.
Our result is consistent with that presented in @Scott09Ni [$\log \epsilon_\mathrm{Ni}=6.17\pm0.05$], but slightly higher due to the improved temperature structure of the improved 3D model we use here. Compared to the result we reported in ($\log \epsilon_\mathrm{Ni}=6.22\pm0.04$), our result here is slightly lower because we employed new $gf$-values and an expanded set of isotopic separations from @Wood14 (in and @Scott09Ni we used $gf$-values from @Wickliffe97 and isotopic separations from @Litzen93). Improvements in the overall opacity, equation of state, ionisation potential and partition function also play a small role in the difference from .
Species 3D [$\langle\mathrm{3D}\rangle$]{} HM <span style="font-variant:small-caps;">marcs</span> [miss]{} 3D$-$HM 3D$-$[$\langle\mathrm{3D}\rangle$]{} Recommended Meteoritic
----------------------------- ------------------------------------------------------ --------------- --------------------------------- ------ ----------------------------------------------------- ---------- --------- -------------------------------------- --------------- ---------------
$\log \epsilon_\mathrm{Sc}$ Sc<span style="font-variant:small-caps;">i</span> 3.14$\pm$0.09 3.21 3.28 3.18 3.23 $-$0.14 $-$0.07 $3.16\pm0.04$ $3.05\pm0.02$
Sc<span style="font-variant:small-caps;">ii</span> 3.17$\pm$0.04 3.16 3.19 3.14 3.19 $-$0.02 0.01
Sc all 3.16$\pm$0.04 3.18 3.22 3.15 3.21 $-$0.07 $-$0.03
$\log \epsilon_\mathrm{Ti}$ Ti<span style="font-variant:small-caps;">i</span> 4.88$\pm$0.05 4.94 4.99 4.90 4.93 $-$0.11 $-$0.06 $4.93\pm0.04$ $4.91\pm0.03$
Ti<span style="font-variant:small-caps;">ii</span> 4.97$\pm$0.04 4.94 4.97 4.91 4.97 0.00 0.02
Ti all 4.90$\pm$0.04 4.94 4.99 4.90 4.94 $-$0.08 $-$0.04
$\log \epsilon_\mathrm{V}$ V<span style="font-variant:small-caps;">i</span> 3.89$\pm$0.08 3.99 4.07 3.96 4.00 $-$0.18 $-$0.10 $3.89\pm0.08$ $3.96\pm0.02$
(V<span style="font-variant:small-caps;">ii</span>) 4.00$\pm$0.04 3.98 4.01 3.95 4.01 $-$0.01 0.02
$\log \epsilon_\mathrm{Cr}$ Cr<span style="font-variant:small-caps;">i</span> 5.60$\pm$0.04 5.62 5.66 5.57 5.63 $-$0.06 $-$0.02 $5.62\pm0.04$ $5.64\pm0.01$
Cr<span style="font-variant:small-caps;">ii</span> 5.65$\pm$0.04 5.62 5.63 5.56 5.65 $+$0.03 0.04
Cr all 5.62$\pm$0.04 5.62 5.65 5.57 5.64 $-$0.04 $-$0.01
$\log \epsilon_\mathrm{Mn}$ Mn<span style="font-variant:small-caps;">i</span> 5.42$\pm$0.04 5.43 5.47 5.37 5.42 $-$0.04 $-$0.00 $5.42\pm0.04$ $5.48\pm0.01$
$\log \epsilon_\mathrm{Fe}$ Fe<span style="font-variant:small-caps;">i</span> 7.45$\pm$0.04 7.46 7.52 7.41 7.46 $-$0.07 $-$0.00 $7.47\pm0.04$ $7.45\pm0.01$
Fe<span style="font-variant:small-caps;">ii</span> 7.51$\pm$0.04 7.46 7.46 7.42 7.49 $+$0.05 0.05
Fe all 7.47$\pm$0.04 7.46 7.50 7.41 7.47 $-$0.03 0.01
$\log \epsilon_\mathrm{Co}$ Co<span style="font-variant:small-caps;">i</span> 4.93$\pm$0.05 4.96 4.99 4.92 4.96 $-$0.06 $-$0.03 $4.93\pm0.05$ $4.87\pm0.01$
$\log \epsilon_\mathrm{Ni}$ Ni<span style="font-variant:small-caps;">i</span> 6.20$\pm$0.04 6.20 6.24 6.15 6.23 $-$0.04 0.00 $6.20\pm0.04$ $6.20\pm0.01$
(Ni<span style="font-variant:small-caps;">ii</span>) 6.30$\pm$0.10 6.23 6.24 6.19 6.26 $+$0.06 0.08
Comments and discussion {#discussion}
=======================
Sensitivity to temperature: 3D vs. [$\langle\mathrm{3D}\rangle$]{} vs. HM {#sensitivity}
-------------------------------------------------------------------------
Table \[table:abuns\] shows that the results for the once-ionised species are typically less model-dependent than those of the neutral species; this is to be expected for these dominant species. We notice that the model-dependence of the abundances of neutral species increases with decreasing ionisation potential, whereas the model-dependence of abundances from ionised lines increases with ionisation potential. This reflects the general rule that the more in majority a species is, the less sensitive its lines will be to the ionisation balance, and therefore less affected by the temperature structure of the model atmosphere.
From Table \[table:abuns\], we see that the differences 3D$-$[$\langle\mathrm{3D}\rangle$]{} vary widely between different neutral species. From values of 0.06–0.10dex for [Sc<span style="font-variant:small-caps;">i</span>]{}, [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [V<span style="font-variant:small-caps;">i</span>]{}, they decrease to 0.00–0.03dex for the rest of the neutrals, reflecting the lower ionisation energies and therefore more severe minority status of [Sc<span style="font-variant:small-caps;">i</span>]{}, [[Ti<span style="font-variant:small-caps;">i</span>]{} ]{}and [[V<span style="font-variant:small-caps;">i</span>]{} ]{}compared to the other neutrals.
When looking at the plots in Figs. \[fig:sc\]–\[fig:ni\], we clearly see that the 3D$-$[$\langle\mathrm{3D}\rangle$]{} abundance difference is also related to the excitation potentials of individual lines, or more precisely, the difference between the ionisation and excitation energies ($E_{\rm ion}-E_{\rm exc}$). This difference is the most important parameter for the temperature sensitivity of lines of minor species like the neutral iron group elements. Lines with lower excitation energies are typically more sensitive than higher-excitation lines to higher atmospheric layers and the presence of atmospheric inhomogeneities, as seen in our 3D$-$ and 3D$-$[$\langle\mathrm{3D}\rangle$]{} results, respectively. A similar argument holds also for line strengths: stronger lines are typically formed higher, so show larger sensitivity to both the mean structure and horizontal inhomogeneities.
Sensitivity to collisional broadening and HFS {#controls}
---------------------------------------------
Collisional broadening is now well determined for neutral species and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}. Even extreme collisional sensitivity should therefore not be a major source of error when using neutral lines and [[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{}in the current analysis. For other ionic lines, the enhancement factor used with the classical @Unsold broadening recipe is a potential source of error. As [Sc<span style="font-variant:small-caps;">ii</span>]{}, [[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{}and [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}lines were mostly insensitive to its variation though, even with ionic lines the broadening treatment should contribute very little to our uncertainties.
The sensitivity of derived abundances to hyperfine (and by implication, isotopic) structure varies greatly with different lines, species and abundance-determination techniques. Clearly, lines with large HFS (i.e. transitions between levels with large $A$, $B$, and/or $J$ values, or in nuclei with large $I$) will be most affected. If one finds abundances using equivalent widths, the strongest lines are those most sensitive to the HFS treatment. This is because the spreading of a strong line into multiple components causes it to become either partially or wholly desaturated, whereas a single component would be more saturated. This means that completely neglecting HFS or isotopic structure often leads to overestimated abundances, a common concern in past 1D analyses [e.g. @vonderHeide68; @Holweger71; @Kurucz93; @Prochaska00]. Broadening by HFS or isotopic structure modifies the depth of line formation in general for all lines (pushing them deeper into the photosphere), so it can play a role even for fainter lines, even when equivalent widths are used for fitting rather than profile fits. Furthermore, it is very important in combination with NLTE line formation, especially when NLTE abundance corrections are computed from differences between LTE and NLTE equivalent widths.
To ascertain the overall impact of HFS on our abundances, we also computed all [$\langle\mathrm{3D}\rangle$]{} abundances with HFS neglected, and calculated the mean HFS correction $\Delta_\mathrm{HFS}\equiv\log\epsilon_\mathrm{no\,HFS}-\log\epsilon_\mathrm{HFS}$ for our sample of lines. We found that [[Mn<span style="font-variant:small-caps;">i</span>]{} ]{}was by far the species most affected, with $\Delta_\mathrm{HFS}=0.16$dex. [[Co<span style="font-variant:small-caps;">i</span>]{} ]{}was the next most strongly affected ($\Delta_\mathrm{HFS}=0.05$dex), followed by [[V<span style="font-variant:small-caps;">ii</span>]{} ]{}($\Delta_\mathrm{HFS}=0.04$dex), [[V<span style="font-variant:small-caps;">i</span>]{} ]{}($\Delta_\mathrm{HFS}=0.03$dex) and [Sc<span style="font-variant:small-caps;">i</span>]{}/[[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{}(both $\Delta_\mathrm{HFS}=0.01$dex). HFS in Ti and Cr had virtually no effect.
Previous Solar Abundance Compilations {#compilations}
=====================================
Table \[table:compilations\] compares the values we recommend here with those adopted in some of the most commonly-used compilations: , , , and @Lodders09. We note however that with the exception of , all of the others are in fact compilations of results from the literature, all with their own methodologies, spectrum synthesis codes, model atmospheres, and error estimation procedures, which makes the recommended solar values a rather inhomogeneous mixture. In particular, none of the previous studies have attempted to account for systematic errors in the quoted abundance uncertainties.
Not surprisingly, the solar abundances that we present here are quite similar to those of . As outlined in detail in Sect. \[results\] however, we have updated them following a complete re-assessment of all analysis ingredients, including continuous opacities, equation-of-state, line selection, atomic data and NLTE abundance corrections. In most cases this has resulted in very minor changes. Cobalt ($-0.06$dex, see Sect. \[Coresults\]) is the notable exception, explained mostly by improved NLTE calculations.
For the Fe-peak elements, only included 1D-based analyses with the exception of Fe [@AspII], although it still updated the recommended values for a few elements relative to . was in turn primarily based on . The main difference between the latter two is the adopted Fe value, where still preferred a high value (0.2dex larger than derived here); see @GS99 for a detailed description of the reasons for the long-standing debate on the solar Fe abundance. Since then the preferred Fe value has not changed drastically, in spite of the advent of 3D hydrodynamic model atmospheres, more complete NLTE calculations and improved $gf$-values – which is reassuring.
Compared with @Lodders09, our solar abundances for the Fe-peak elements are similar overall, but there are some rather large isolated differences. These include Sc ($+0.06$dex), V ($-0.11$dex) and Mn ($+0.05$dex). As outlined in Sect. \[results\], we are confident that our analysis is the most reliable and accurate possible today.
Z el. This work AG89 GS98 AGS05 AGSS09 LPG09
---- ----- --------------------- ------ ------ ------- -------- -------
21 Sc $ 3.16\pm 0.04 $ 3.10 3.17 3.05 3.15 3.10
22 Ti $ 4.93\pm 0.04 $ 4.99 5.02 4.90 4.95 4.90
23 V $ 3.89\pm 0.08 $ 4.00 4.00 4.00 3.93 4.00
24 Cr $ 5.62\pm 0.04 $ 5.67 5.67 5.64 5.64 5.64
25 Mn $ 5.42\pm 0.04 $ 5.39 5.39 5.39 5.43 5.37
26 Fe $ 7.47\pm 0.04 $ 7.67 7.50 7.45 7.50 7.45
27 Co $ 4.93\pm 0.05 $ 4.92 4.92 4.92 4.99 4.92
28 Ni $ 6.20\pm 0.04 $ 6.25 6.25 6.23 6.22 6.23
: The present-day solar photospheric abundances for the Fe-peak elements Sc to Ni that we recommend here, compared with oft-used solar abundance compilations: , , , and @Lodders09 (LPG09)[]{data-label="table:compilations"}
Conclusions
===========
We have determined the abundances of all the iron group elements in the Sun. For our analysis, we have carefully assessed all relevant atomic data, made very stringent line selections, employed a highly realistic 3D model for the solar atmosphere and accounted for departures from LTE. We have attempted to quantify the remaining systematic uncertainties stemming from possible errors in atmospheric and line-formation modelling, and to properly account for statistical errors.
Our final recommended abundances of Sc, Ti, V, Cr, Mn, Fe, Co and Ni are given in Table \[table:abuns\]. The derived abundances generally show good agreement with the meteoritic values, and between different ionisation stages, but some discrepancies remain. Trends in abundances with excitation potential or line strength are largely absent in the 3D results, but are visible in a number of results from 1D models. The level of agreement between theoretical and observed line profiles with the 3D model is clearly satisfactory. Nonetheless, theoretical profiles computed in 3D systematically underestimate the line width by a small amount, suggesting that some additional work on improving the atmospheric velocity field or NLTE effects is still required before perfect agreement can be claimed. Nevertheless, we are confident that the solar photospheric abundances that we present here are the most accurate possible by today’s standards.
We thank Dan Bayliss, Mike Bessell, Remo Collet, Peter Hannaford, Wolfgang Hayek, Lyudmila Mashonkina, Tiago Pereira, Chris Sneden and Regner Trampedach for helpful discussions, and the referee for constructive feedback. PS, NG and MA variously thank the Max Planck Institut für Astrophysik, Garching, the Centre Spatial de Liège, the Department of Astrophysics, Geophysics and Oceanography, University of Liège and Mount Stromlo Observatory for support and hospitality during the production of this paper. We acknowledge further support from IAU Commission 46, the Lorne Trottier Chair in Astrophysics, the (Canadian) Institute for Particle Physics, the Banting Fellowship scheme as administered by the Natural Science and Engineering Research Council of Canada, the UK Science & Technology Facilities Council (PS), the Australian Research Council (MA) and the Royal Belgian Observatory (NG).
[r@c@[ ]{}r@[ ]{}l@[ ]{}r@[ ]{}l@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}r@[ ]{}c]{}\
474.3821 & & ($^3$F)4s &$^4$F$_\frac{9}{2}$ &($^3$F)4p &$^4$D$_\frac{7}{2}$ & 1.448 & 0.422 & 1 & 0.82 & 1& 2.974 & 3.039 & 3.104 & 3.006 & 3.065 & $+$0.15& 3.124\
508.1561 & & ($^3$F)4s &$^4$F$_\frac{9}{2}$ &($^3$F)4p &$^4$F$_\frac{9}{2}$ & 1.448 & 0.469 & 1 & 1.00 & 2& 2.989 & 3.056 & 3.121 & 3.022 & 3.080 & $+$0.16& 3.149\
535.6097 & & ($^3$F)4s &$^2$F$_\frac{7}{2}$ &($^3$F)4p &$^2$D$_\frac{5}{2}$ & 1.865 & 0.168 & 1 & 0.20 & 3& 2.974 & 3.030 & 3.089 & 2.994 & 3.056 & $+$0.15& 3.124\
567.1828 & & ($^3$F)4s &$^4$F$_\frac{9}{2}$ &($^3$F)4p &$^4$G$_\frac{11}{2}$ & 1.448 & 0.495 & 1 & 1.24 & 2& 3.024 & 3.092 & 3.159 & 3.058 & 3.114 & $+$0.15& 3.174\
623.9800 & & 4s$^2$ &$^2$D$_\frac{3}{2}$ &($^3$D)4sp&$^4$D$_\frac{3}{2}$ & 0.000 & $-$1.780 & 1 & 0.22 & 1& 2.985 & 3.133 & 3.224 & 3.108 & 3.146 & $+$0.15& 3.135\
& & & & & & & & & & & & & & &\
\
442.0661 & & 3d$^2$ &$^3$F$_4$ &4p &$^3$F$_3$ & 0.618 & $-$2.273 & 1 & 1.51 & 2& 3.109 & 3.109 & 3.140 & 3.093 & 3.143 & $-$0.01& 3.099\
443.1362 & & 3d$^2$ &$^3$F$_3$ &4p &$^3$F$_2$ & 0.605 & $-$1.969 & 1 & 2.92 & 1& 3.165 & 3.162 & 3.193 & 3.142 & 3.195 & $-$0.01& 3.155\
535.7202 & & 3d$^2$ &$^3$P$_2$ &4p &$^1$P$_1$ & 1.507 & $-$2.111 & 1 & 0.43 & 2& 3.131 & 3.130 & 3.153 & 3.110 & 3.164 & 0.00& 3.131\
564.1000 & & 3d$^2$ &$^3$P$_1$ &4p &$^3$P$_2$ & 1.500 & $-$1.131 & 1 & 3.65 & 1& 3.246 & 3.236 & 3.255 & 3.205 & 3.264 & $-$0.02& 3.226\
565.8362 & & 3d$^2$ &$^3$P$_0$ &4p &$^3$P$_1$ & 1.497 & $-$1.208 & 1 & 3.14 & 1& 3.221 & 3.212 & 3.232 & 3.183 & 3.241 & $-$0.01& 3.211\
566.7164 & & 3d$^2$ &$^3$P$_1$ &4p &$^3$P$_1$ & 1.500 & $-$1.309 & 1 & 2.81 & 1& 3.245 & 3.238 & 3.258 & 3.212 & 3.268 & $-$0.01& 3.235\
566.9055 & & 3d$^2$ &$^3$P$_1$ &4p &$^3$P$_0$ & 1.500 & $-$1.200 & 1 & 3.26 & 1& 3.256 & 3.243 & 3.263 & 3.214 & 3.272 & $-$0.01& 3.246\
568.4214 & & 3d$^2$ &$^3$P$_2$ &4p &$^3$P$_1$ & 1.507 & $-$1.074 & 1 & 3.56 & 2& 3.174 & 3.164 & 3.183 & 3.133 & 3.193 & $-$0.02& 3.154\
660.4578 & & 3d$^2$ &$^1$D$_2$ &4p &$^1$D$_2$ & 1.357 & $-$1.309 & 1 & 3.54 & 1\* & 3.214 & 3.202 & 3.219 & 3.173 & 3.227 & $-$0.01& 3.204\
& & & & & & & & & & & & & & &\
\
428.1363 & & ($^4$F)4s &$^5$F$_1$ &($^4$F)4p &$^5$D$_2$ & 0.813 & $-$1.260 & 2 & 2.40 & 1& 4.787 & 4.856 & 4.938 & 4.825 & 4.878 & $+$0.056 & 4.843\
446.5805 & & ($^4$P)4s &$^5$P$_2$ &($^4$P)4p &$^5$P$_3$ & 1.739 & $-$0.130 & 2 & 3.56 & 2& 4.829 & 4.862 & 4.933 & 4.822 & 4.886 & $+$0.050 & 4.879\
475.8118 & & ($^2$H)4s &$^3$H$_5$ &($^2$H)4p &$^3$H$_5$ & 2.249 & 0.510 & 2 & 4.18 & 3& 4.792 & 4.809 & 4.876 & 4.763 & 4.832 & $+$0.053 & 4.845\
475.9269 & & ($^2$H)4s &$^3$H$_6$ &($^2$H)4p &$^3$H$_6$ & 2.256 & 0.590 & 2 & 4.60 & 2& 4.810 & 4.821 & 4.889 & 4.772 & 4.842 & $+$0.053 & 4.863\
496.4715 & & ($^3$F)4sp&$^5$G$_2$ &4s($^4$F)5s&$^5$F$_2$ & 1.969 & $-$0.820 & 3 & 0.77 & 2& 4.826 & 4.880 & 4.942 & 4.845 & 4.905 & $+$0.060 & 4.886\
502.2866 & & ($^4$F)4s &$^5$F$_3$ &($^4$F)4p &$^5$G$_3$ & 0.826 & $-$0.330 & 2 & 6.99 & 1& 4.817 & 4.804 & 4.896 & 4.753 & 4.808 & $+$0.065 & 4.882\
511.3439 & & ($^4$F)4s &$^3$F$_3$ &($^3$P)4sp &$^3$D$_2$ & 1.443 & $-$0.700 & 2 & 2.45 & 2& 4.782 & 4.838 & 4.912 & 4.802 & 4.860 & $+$0.049 & 4.831\
514.5459 & & ($^4$F)4s &$^3$F$_4$ &($^3$P)4sp &$^3$D$_3$ & 1.460 & $-$0.540 & 2 & 3.31 & 1& 4.838 & 4.884 & 4.959 & 4.845 & 4.904 & $+$0.048 & 4.886\
514.7477 & & 4s$^2$ &$^3$F$_2$ &($^3$F)4sp &$^3$F$_3$ & 0.000 & $-$1.940 & 2 & 3.46 & 1& 4.781 & 4.871 & 4.968 & 4.840 & 4.882 & $+$0.086 & 4.867\
515.2184 & & 4s$^2$ &$^3$F$_3$ &($^3$F)4sp &$^3$F$_4$ & 0.021 & $-$1.950 & 2 & 3.32 & 2& 4.782 & 4.873 & 4.970 & 4.843 & 4.885 & $+$0.084 & 4.866\
521.9699 & & 4s$^2$ &$^3$F$_3$ &($^3$F)4sp &$^3$F$_2$ & 0.021 & $-$2.220 & 2 & 2.24 & 2& 4.783 & 4.896 & 4.991 & 4.868 & 4.909 & $+$0.081 & 4.864\
522.3620 & & ($^3$F)4sp&$^5$F$_2$ &4s($^4$F)5s&$^5$F$_2$ & 2.092 & $-$0.490 & 2 & 1.22 & 1& 4.824 & 4.874 & 4.935 & 4.836 & 4.899 & $+$0.054 & 4.878\
524.7288 & & ($^3$F)4sp&$^5$F$_3$ &4s($^4$F)5s&$^5$F$_2$ & 2.103 & $-$0.640 & 3 & 0.85 & 1& 4.802 & 4.853 & 4.914 & 4.817 & 4.879 & $+$0.057 & 4.859\
525.2098 & & 4s$^2$ &$^3$F$_4$ &($^3$F)4sp &$^3$F$_3$ & 0.048 & $-$2.360 & 4 & 1.64 & 1& 4.768 & 4.890 & 4.984 & 4.863 & 4.905 & $+$0.079 & 4.847\
529.5774 & & 4s$^2$ &$^3$P$_2$ &($^1$D)4sp &$^3$D$_3$ & 1.067 & $-$1.590 & 2 & 1.06 & 2& 4.824 & 4.907 & 4.984 & 4.875 & 4.927 & $+$0.065 & 4.889\
549.0147 & & ($^4$F)4s &$^3$F$_4$ &($^4$F)4p &$^5$D$_3$ & 1.460 & $-$0.840 & 2 & 2.03 & 1& 4.795 & 4.857 & 4.929 & 4.821 & 4.878 & $+$0.051 & 4.846\
566.2147 & & ($^3$F)4sp&$^5$D$_4$ &4s($^4$F)5s&$^5$F$_5$ & 2.318 & 0.010 & 3 & 2.12 & 1& 4.802 & 4.843 & 4.903 & 4.802 & 4.866 & $+$0.057 & 4.859\
568.9459 & & ($^3$F)4sp&$^5$D$_2$ &4s($^4$F)5s&$^5$F$_3$ & 2.297 & $-$0.360 & 3 & 1.13 & 1& 4.824 & 4.871 & 4.930 & 4.833 & 4.896 & $+$0.058 & 4.882\
570.2658 & & ($^3$F)4sp&$^5$D$_1$ &4s($^4$F)5s&$^5$F$_2$ & 2.292 & $-$0.590 & 3 & 0.70 & 1& 4.815 & 4.864 & 4.922 & 4.827 & 4.889 & $+$0.059 & 4.874\
571.6441 & & ($^3$F)4sp&$^5$D$_2$ &4s($^4$F)5s&$^5$F$_2$ & 2.297 & $-$0.720 & 3 & 0.54 & 1& 4.828 & 4.878 & 4.935 & 4.841 & 4.903 & $+$0.058 & 4.886\
586.6429 & & 4s$^2$ &$^3$P$_2$ &($^4$F)4p &$^3$D$_3$ & 1.067 & $-$0.790 & 2 & 4.46 & 2& 4.833 & 4.887 & 4.971 & 4.849 & 4.899 & $+$0.042 & 4.875\
592.2088 & & 4s$^2$ &$^3$P$_0$ &($^4$F)4p &$^3$D$_1$ & 1.046 & $-$1.380 & 2 & 1.79 & 2& 4.802 & 4.886 & 4.964 & 4.853 & 4.903 & $+$0.048 & 4.850\
609.2789 & & ($^2$G)4s &$^3$G$_5$ &($^4$F)4p &$^3$G$_5$ & 1.887 & $-$1.380 & 2 & 0.36 & 2& 4.881 & 4.942 & 5.008 & 4.909 & 4.964 & $+$0.057 & 4.938\
625.8099 & & ($^4$F)4s &$^3$F$_3$ &($^3$F)4sp &$^3$G$_4$ & 1.443 & $-$0.390 & 2 & 5.05 & 3& 4.912 & 4.943 & 5.023 & 4.901 & 4.952 & $+$0.045 & 4.957\
630.3753 & & ($^4$F)4s &$^3$F$_3$ &($^3$F)4sp &$^3$G$_3$ & 1.443 & $-$1.580 & 2 & 0.68 & 1& 4.912 & 4.986 & 5.061 & 4.955 & 5.004 & $+$0.056 & 4.968\
631.2234 & & ($^4$F)4s &$^3$F$_4$ &($^3$F)4sp &$^3$G$_4$ & 1.460 & $-$1.550 & 2 & 0.68 & 2& 4.892 & 4.966 & 5.040 & 4.935 & 4.985 & $+$0.055 & 4.947\
659.9104 & & 4s$^2$ &$^1$D$_2$ &($^3$F)4sp &$^1$F$_3$ & 0.900 & $-$2.029 & 5 & 0.80 & 2& 4.857 & 4.955 & 5.038 & 4.927 & 4.970 & $+$0.083 & 4.940\
735.7726 & & ($^4$F)4s &$^3$F$_3$ &($^3$F)4sp &$^3$F$_3$ & 1.443 & $-$1.020 & 2 & 1.99 & 2& 4.814 & 4.886 & 4.960 & 4.854 & 4.901 & $+$0.054 & 4.868\
842.6504 & & ($^4$F)4s &$^5$F$_3$ &($^3$F)4sp &$^5$D$_2$ & 0.826 & $-$1.197 & 6 & 4.66 & 2& 4.832 & 4.903 & 4.992 & 4.871 & 4.904 & $+$0.071 & 4.903\
843.5648 & & ($^4$F)4s &$^5$F$_4$ &($^3$F)4sp &$^5$D$_3$ & 0.836 & $-$0.967 & 6 & 5.76 & 1& 4.805 & 4.859 & 4.951 & 4.825 & 4.855 & $+$0.073 & 4.878\
867.5371 & & 4s$^2$ &$^3$P$_2$ &($^3$F)4sp &$^3$D$_3$ & 1.067 & $-$1.500 & 2 & 1.85 & 2& 4.774 & 4.869 & 4.948 & 4.838 & 4.880 & $+$0.075 & 4.849\
868.2979 & & 4s$^2$ &$^3$P$_1$ &($^3$F)4sp &$^3$D$_2$ & 1.053 & $-$1.790 & 2 & 1.07 & 2& 4.773 & 4.876 & 4.954 & 4.846 & 4.888 & $+$0.075 & 4.848\
869.2328 & & 4s$^2$ &$^3$P$_0$ &($^3$F)4sp &$^3$D$_1$ & 1.046 & $-$2.130 & 2 & 0.52 & 2& 4.765 & 4.874 & 4.951 & 4.845 & 4.887 & $+$0.074 & 4.839\
873.4711 & & 4s$^2$ &$^3$P$_1$ &($^3$F)4sp &$^3$D$_1$ & 1.053 & $-$2.240 & 2 & 0.41 & 1& 4.771 & 4.880 & 4.957 & 4.851 & 4.894 & $+$0.075 & 4.846\
& & & & & & & & & & & & & & &\
\
440.9520 & & ($^3$P)4s &$^4$P$_\frac{3}{2}$ &($^3$F)4p &$^4$D$_\frac{3}{2}$ & 1.231 & $-$2.530 & 7 & 3.81 & 2& 4.938 & 4.923 & 4.951 & 4.894 & 4.956 & &\
444.4524 & & 3d$^3$ &$^2$G$_\frac{7}{2}$ &($^3$F)4p &$^2$F$_\frac{7}{2}$ & 1.116 & $-$2.200 & 7 & 5.99 & 1& 4.960 & 4.923 & 4.952 & 4.881 & 4.951 & &\
449.3525 & & ($^1$D)4s &$^2$D$_\frac{3}{2}$ &($^3$F)4p &$^4$F$_\frac{5}{2}$ & 1.080 & $-$2.780 & 7 & 3.18 & 1& 4.897 & 4.887 & 4.916 & 4.862 & 4.920 & &\
458.3396 & & 3d$^3$ &$^4$P$_\frac{3}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.165 & $-$2.840 & 7 & 3.02 & 2& 4.985 & 4.977 & 5.005 & 4.953 & 5.010 & &\
460.9253 & & 3d$^3$ &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.180 & $-$3.320 & 7 & 1.16 & 1& 4.933 & 4.932 & 4.959 & 4.914 & 4.966 & &\
465.7212 & & ($^3$P)4s &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^2$F$_\frac{7}{2}$ & 1.243 & $-$2.290 & 7 & 5.18 & 1& 4.988 & 4.956 & 4.983 & 4.917 & 4.986 & &\
470.8656 & & 3d$^3$ &$^2$P$_\frac{3}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.237 & $-$2.350 & 7 & 5.06 & 1& 4.951 & 4.923 & 4.949 & 4.885 & 4.953 & &\
471.9533 & & ($^3$P)4s &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.243 & $-$3.320 & 7 & 1.22 & 1& 5.013 & 5.011 & 5.038 & 4.993 & 5.043 & &\
476.4518 & & 3d$^3$ &$^2$P$_\frac{3}{2}$ &($^3$F)4p &$^4$F$_\frac{5}{2}$ & 1.237 & $-$2.690 & 7 & 3.35 & 1& 4.968 & 4.956 & 4.982 & 4.928 & 4.989 & &\
479.8535 & & ($^1$D)4s &$^2$D$_\frac{3}{2}$ &($^3$F)4p &$^4$G$_\frac{5}{2}$ & 1.080 & $-$2.660 & 7 & 4.29 & 1& 4.990 & 4.970 & 4.997 & 4.937 & 5.001 & &\
486.5597 & & 3d$^3$ &$^2$G$_\frac{7}{2}$ &($^3$F)4p &$^4$G$_\frac{5}{2}$ & 1.116 & $-$2.700 & 7 & 3.50 & 1& 4.877 & 4.866 & 4.893 & 4.838 & 4.898 & &\
533.6770 & & 3d$^3$ &$^2$D2$_\frac{5}{2}$ &($^3$F)4p &$^2$F$_\frac{7}{2}$ & 1.582 & $-$1.600 & 7 & 7.20 & 2& 4.991 & 4.922 & 4.944 & 4.870 & 4.942 & &\
538.1013 & & 3d$^3$ &$^2$D2$_\frac{3}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.566 & $-$1.970 & 7 & 5.66 & 1& 5.004 & 4.963 & 4.983 & 4.918 & 4.988 & &\
541.8760 & & 3d$^3$ &$^2$D2$_\frac{5}{2}$ &($^3$F)4p &$^2$F$_\frac{5}{2}$ & 1.582 & $-$2.130 & 7 & 4.81 & 3& 4.999 & 4.970 & 4.990 & 4.930 & 4.997 & &\
& & & & & & & & & & & & & & &\
\
458.6370 & & 4s$^2$ &$^4$F$_\frac{7}{2}$ &($^4$F)4sp&$^4$G$_\frac{9}{2}$ & 0.040 & $-$0.793 & 8 & 4.14 & 2& 3.750 & 3.849 & 3.943 & 3.820 & 3.865 & $+$0.1& 3.850\
459.4119 & & 4s$^2$ &$^4$F$_\frac{9}{2}$ &($^4$F)4sp&$^4$G$_\frac{11}{2}$ & 0.069 & $-$0.672 & 8 & 5.26 & 2& 3.762 & 3.862 & 3.956 & 3.833 & 3.878 & $+$0.1& 3.862\
463.5172 & & 4s$^2$ &$^4$F$_\frac{9}{2}$ &($^4$F)4sp&$^4$G$_\frac{9}{2}$ & 0.069 & $-$1.924 & 8 & 0.45 & 1& 3.753 & 3.886 & 3.975 & 3.861 & 3.905 & $+$0.1& 3.853\
482.7452 & & 4s$^2$ &$^4$F$_\frac{7}{2}$ &($^4$F)4sp&$^4$D$_\frac{7}{2}$ & 0.040 & $-$1.478 & 8 & 1.30 & 1& 3.742 & 3.875 & 3.965 & 3.849 & 3.892 & $+$0.1& 3.842\
487.5486 & & 4s$^2$ &$^4$F$_\frac{7}{2}$ &($^4$D)4sp&$^4$G$_\frac{5}{2}$ & 0.040 & $-$0.806 & 8 & 4.22 & 1& 3.754 & 3.853 & 3.948 & 3.822 & 3.866 & $+$0.1& 3.854\
488.1555 & & 4s$^2$ &$^4$F$_\frac{9}{2}$ &($^4$F)4sp&$^4$D$_\frac{7}{2}$ & 0.069 & $-$0.657 & 8 & 5.39 & 1& 3.749 & 3.848 & 3.943 & 3.818 & 3.861 & $+$0.1& 3.849\
562.6019 & & ($^5$D)4s&$^4$D$_\frac{1}{2}$ &($^5$D)4p &$^4$D$_\frac{1}{2}$ & 1.043 & $-$1.252 & 8 & 0.31 & 1& 3.834 & 3.923 & 3.999 & 3.894 & 3.943 & $+$0.1& 3.934\
564.6108 & & ($^5$D)4s&$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$D$_\frac{1}{2}$ & 1.051 & $-$1.187 & 8 & 0.37 & 2& 3.855 & 3.941 & 4.019 & 3.912 & 3.961 & $+$0.1& 3.955\
565.7438 & & ($^5$D)4s&$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$D$_\frac{3}{2}$ & 1.064 & $-$1.018 & 8 & 0.50 & 3& 3.842 & 3.928 & 4.004 & 3.898 & 3.947 & $+$0.1& 3.942\
566.8361 & & ($^5$D)4s&$^4$D$_\frac{7}{2}$ &($^5$D)4p &$^4$D$_\frac{5}{2}$ & 1.081 & $-$1.021 & 8 & 0.49 & 1& 3.849 & 3.934 & 4.011 & 3.905 & 3.954 & $+$0.1& 3.949\
567.0847 & & ($^5$D)4s&$^4$D$_\frac{7}{2}$ &($^4$F)4sp&$^2$G$_\frac{9}{2}$ & 1.081 & $-$0.425 & 8 & 1.69 & 3& 3.823 & 3.907 & 3.983 & 3.876 & 3.926 & $+$0.1& 3.923\
570.3586 & & ($^5$D)4s&$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$F$_\frac{5}{2}$ & 1.051 & $-$0.212 & 8 & 2.57 & 2& 3.823 & 3.900 & 3.978 & 3.867 & 3.917 & $+$0.1& 3.923\
572.7046 & & ($^5$D)4s&$^4$D$_\frac{7}{2}$ &($^5$D)4p &$^4$F$_\frac{9}{2}$ & 1.081 & $-$0.012 & 8 & 3.62 & 3& 3.810 & 3.888 & 3.966 & 3.855 & 3.905 & $+$0.1& 3.910\
572.7655 & & ($^5$D)4s&$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$F$_\frac{3}{2}$ & 1.051 & $-$0.875 & 8 & 0.78 & 2& 3.876 & 3.961 & 4.041 & 3.933 & 3.980 & $+$0.1& 3.976\
573.1249 & & ($^5$D)4s&$^4$D$_\frac{5}{2}$ &($^4$F)4sp&$^2$G$_\frac{7}{2}$ & 1.064 & $-$0.732 & 8 & 0.97 & 2& 3.836 & 3.924 & 4.000 & 3.894 & 3.943 & $+$0.1& 3.936\
573.7065 & & ($^5$D)4s&$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$F$_\frac{5}{2}$ & 1.064 & $-$0.736 & 8 & 0.92 & 2& 3.828 & 3.915 & 3.991 & 3.885 & 3.934 & $+$0.1& 3.928\
600.2294 & & 4s$^2$ &$^4$P$_\frac{5}{2}$ &($^5$D)4p &$^4$D$_\frac{7}{2}$ & 1.218 & $-$1.773 & 8 & 0.07 & 1& 3.843 & 3.926 & 4.000 & 3.897 & 3.946 & $+$0.1& 3.943\
603.9728 & & ($^5$D)4s&$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$P$_\frac{5}{2}$ & 1.064 & $-$0.652 & 8 & 1.12 & 3& 3.834 & 3.917 & 3.994 & 3.886 & 3.934 & $+$0.1& 3.934\
608.1441 & & ($^5$D)4s&$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$P$_\frac{3}{2}$ & 1.051 & $-$0.579 & 8 & 1.26 & 3& 3.788 & 3.875 & 3.951 & 3.844 & 3.893 & $+$0.1& 3.888\
609.0208 & & ($^5$D)4s&$^4$D$_\frac{7}{2}$ &($^5$D)4p &$^4$P$_\frac{5}{2}$ & 1.081 & $-$0.062 & 8 & 3.07 & 3& 3.799 & 3.869 & 3.947 & 3.835 & 3.884 & $+$0.1& 3.899\
611.1650 & & ($^5$D)4s&$^4$D$_\frac{1}{2}$ &($^5$D)4p &$^4$P$_\frac{1}{2}$ & 1.043 & $-$0.714 & 8 & 0.99 & 3& 3.784 & 3.876 & 3.951 & 3.846 & 3.895 & $+$0.1& 3.884\
611.9528 & & ($^5$D)4s&$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$P$_\frac{3}{2}$ & 1.064 & $-$0.320 & 8 & 1.95 & 2& 3.779 & 3.857 & 3.933 & 3.824 & 3.874 & $+$0.1& 3.879\
613.5363 & & ($^5$D)4s&$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$P$_\frac{1}{2}$ & 1.051 & $-$0.746 & 8 & 0.94 & 1& 3.807 & 3.896 & 3.972 & 3.866 & 3.914 & $+$0.1& 3.907\
624.2828 & & ($^5$D)4s&$^6$D$_\frac{1}{2}$ &($^4$F)4sp&$^6$D$_\frac{3}{2}$ & 0.262 & $-$1.552 & 8 & 0.83 & 3& 3.716 & 3.850 & 3.936 & 3.823 & 3.863 & $+$0.1& 3.816\
625.1823 & & ($^5$D)4s&$^6$D$_\frac{7}{2}$ &($^4$F)4sp&$^6$D$_\frac{7}{2}$ & 0.287 & $-$1.342 & 8 & 1.29 & 3& 3.745 & 3.874 & 3.962 & 3.847 & 3.887 & $+$0.1& 3.845\
625.6903 & & ($^5$D)4s&$^6$D$_\frac{5}{2}$ &($^4$F)4sp&$^6$D$_\frac{5}{2}$ & 0.275 & $-$2.006 & 8 & 0.31 & 2& 3.740 & 3.878 & 3.964 & 3.851 & 3.892 & $+$0.1& 3.840\
627.4653 & & ($^5$D)4s&$^6$D$_\frac{3}{2}$ &($^4$F)4sp&$^6$D$_\frac{1}{2}$ & 0.267 & $-$1.673 & 8 & 0.71 & 1& 3.774 & 3.906 & 3.995 & 3.880 & 3.919 & $+$0.1& 3.874\
628.5160 & & ($^5$D)4s&$^6$D$_\frac{5}{2}$ &($^4$F)4sp&$^6$D$_\frac{3}{2}$ & 0.275 & $-$1.512 & 8 & 0.88 & 3& 3.724 & 3.854 & 3.940 & 3.826 & 3.867 & $+$0.1& 3.824\
629.2824 & & ($^5$D)4s&$^6$D$_\frac{7}{2}$ &($^4$F)4sp&$^6$D$_\frac{5}{2}$ & 0.287 & $-$1.471 & 8 & 1.02 & 1& 3.768 & 3.895 & 3.983 & 3.869 & 3.908 & $+$0.1& 3.868\
653.1401 & & 4s$^2$ &$^4$P$_\frac{5}{2}$ &($^5$D)4p &$^4$P$_\frac{5}{2}$ & 1.218 & $-$0.836 & 8 & 0.57 & 2& 3.809 & 3.895 & 3.967 & 3.864 & 3.913 & $+$0.1& 3.909\
& & & & & & & & & & & & & & &\
\
376.0222 & & 3d$^4$ &$^3$F$_4$ &($^4$F)4p &$^3$F$_3$ & 1.687 & $-$1.153 & 9 & 3.64 & 1& 3.958 & 3.942 & 3.970 & 3.917 & 3.975 & &\
386.6740 & & 3d$^4$ &$^3$P$_1$ &($^4$F)4p &$^5$D$_2$ & 1.428 & $-$1.550 & 9 & 3.29 & 1& 4.015 & 4.000 & 4.030 & 3.978 & 4.034 & &\
395.1960 & & 3d$^4$ &$^3$P$_2$ &($^4$F)4p &$^3$D$_3$ & 1.476 & $-$0.740 &10 & 6.47 & 1& 3.950 & 3.904 & 3.933 & 3.861 & 3.932 & &\
399.7117 & & 3d$^4$ &$^3$P$_2$ &($^4$F)4p &$^5$F$_3$ & 1.476 & $-$1.230 & 9 & 5.01 & 1& 4.062 & 4.041 & 4.069 & 4.008 & 4.073 & &\
403.6777 & & 3d$^4$ &$^3$P$_2$ &($^4$F)4p &$^5$F$_2$ & 1.476 & $-$1.594 & 9 & 3.17 & 1& 4.015 & 4.006 & 4.036 & 3.986 & 4.041 & &\
& & & & & & & & & & & & & & &\
\
437.3259 & & 4s$^2$ &$^5$D$_2$ &($^5$D)4sp &$^5$F$_1$ & 0.983 & $-$2.323 &11 & 3.76 & 1& 5.584 & 5.615 & 5.698 & 5.578 & 5.636 & $+$0.031 & 5.615\
452.9838 & & ($^4$G)4s &$^5$G$_6$ &($^4$G)4p &$^5$G$_5$ & 2.544 & $-$1.380 &12 & 1.77 & 1& 5.598 & 5.626 & 5.685 & 5.589 & 5.653 & $+$0.023 & 5.621\
453.5127 & & ($^4$G)4s &$^5$G$_3$ &($^4$G)4p &$^5$G$_4$ & 2.544 & $-$0.993 &13 & 3.15 & 2& 5.576 & 5.593 & 5.653 & 5.550 & 5.619 & $+$0.023 & 5.599\
454.1060 & & ($^4$G)4s &$^5$G$_4$ &($^4$G)4p &$^5$G$_3$ & 2.545 & $-$1.143 &13 & 2.50 & 1& 5.578 & 5.600 & 5.660 & 5.560 & 5.627 & $+$0.023 & 5.601\
463.3259 & & ($^5$D)4sp&$^7$F$_3$ &4s5s &$^7$D$_4$ & 3.125 & $-$1.110 &14 & 0.93 & 2& 5.534 & 5.560 & 5.610 & 5.522 & 5.589 & $+$0.046 & 5.580\
470.0599 & & ($^4$P)4s &$^5$P$_1$ &($^3$P2)4sp&$^5$S$_2$ & 2.710 & $-$1.255 &15 & 1.51 & 2& 5.581 & 5.608 & 5.665 & 5.570 & 5.636 & $+$0.027 & 5.608\
470.8017 & & ($^5$D)4sp&$^7$F$_5$ &4s5s &$^7$D$_4$ & 3.168 & 0.090 &16 & 5.67 & 1& 5.595 & 5.578 & 5.633 & 5.516 & 5.599 & $+$0.066 & 5.661\
474.5270 & & ($^4$P)4s &$^5$P$_3$ &($^3$P2)4sp&$^5$D$_4$ & 2.708 & $-$1.380 &14 & 1.22 & 2& 5.536 & 5.566 & 5.622 & 5.528 & 5.595 & $+$0.026 & 5.562\
478.9340 & & ($^4$G)4s &$^5$G$_6$ &($^5$D)4sp &$^5$F$_5$ & 2.544 & $-$0.348 &16 & 5.99 & 2& 5.528 & 5.501 & 5.567 & 5.443 & 5.519 & $+$0.019 & 5.547\
480.1048 & & 4s$^2$ &$^3$F$_4$ &($^4$G)4p &$^3$F$_3$ & 3.122 & $-$0.131 &15 & 4.79 & 2& 5.613 & 5.600 & 5.657 & 5.544 & 5.623 & $+$0.045 & 5.658\
488.5733 & & ($^4$G)4s &$^5$G$_3$ &($^5$D)4sp &$^5$P$_2$ & 2.544 & $-$1.055 &15 & 2.82 & 2& 5.558 & 5.579 & 5.639 & 5.536 & 5.604 & $+$0.025 & 5.583\
493.6336 & & 4s$^2$ &$^3$F$_4$ &($^4$G)4p &$^3$H$_4$ & 3.113 & $-$0.237 &16 & 4.27 & 1& 5.591 & 5.586 & 5.642 & 5.533 & 5.610 & $+$0.037 & 5.628\
495.3714 & & 4s$^2$ &$^3$F$_4$ &($^4$G)4p &$^3$H$_4$ & 3.122 & $-$1.480 &14 & 0.47 & 1& 5.562 & 5.590 & 5.639 & 5.551 & 5.619 & $+$0.035 & 5.597\
522.0913 & & ($^5$D)4sp&$^7$D$_1$ &4s5s &$^7$D$_1$ & 3.385 & $-$0.890 &14 & 1.09 & 2& 5.599 & 5.622 & 5.669 & 5.581 & 5.650 & $+$0.025 & 5.624\
524.1454 & & ($^4$P)4s &$^5$P$_1$ &($^5$D)4sp &$^5$P$_1$ & 2.710 & $-$1.920 &14 & 0.35 & 3& 5.450 & 5.484 & 5.540 & 5.449 & 5.512 & $+$0.025 & 5.475\
527.2008 & & ($^5$D)4sp&$^7$P$_3$ &4s5s &$^7$D$_4$ & 3.449 & $-$0.421 &16 & 2.29 & 1& 5.603 & 5.620 & 5.667 & 5.573 & 5.646 & $+$0.026 & 5.629\
528.7201 & & ($^5$D)4sp&$^7$P$_2$ &4s5s &$^7$D$_3$ & 3.438 & $-$0.888 &16 & 1.00 & 2& 5.608 & 5.630 & 5.677 & 5.589 & 5.657 & $+$0.025 & 5.633\
530.0743 & & 4s$^2$ &$^5$D$_2$ &($^6$S)4p &$^5$P$_3$ & 0.983 & $-$2.083 &17 & 5.48 & 2& 5.558 & 5.566 & 5.652 & 5.522 & 5.575 & $+$0.035 & 5.593\
530.4184 & & ($^5$D)4sp&$^7$P$_4$ &4s5s &$^7$D$_4$ & 3.464 & $-$0.681 &16 & 1.45 & 2& 5.612 & 5.633 & 5.679 & 5.590 & 5.659 & $+$0.025 & 5.637\
531.2871 & & ($^5$D)4sp&$^7$P$_3$ &4s5s &$^7$D$_3$ & 3.449 & $-$0.556 &16 & 1.85 & 1& 5.599 & 5.618 & 5.665 & 5.573 & 5.644 & $+$0.026 & 5.625\
531.8810 & & ($^5$D)4sp&$^7$P$_2$ &4s5s &$^7$D$_2$ & 3.438 & $-$0.679 &16 & 1.49 & 3& 5.596 & 5.617 & 5.663 & 5.573 & 5.643 & $+$0.026 & 5.622\
534.0474 & & ($^5$D)4sp&$^7$P$_2$ &4s5s &$^7$D$_1$ & 3.438 & $-$0.730 &16 & 1.48 & 1& 5.642 & 5.662 & 5.710 & 5.620 & 5.688 & $+$0.026 & 5.668\
562.8621 & & 4s$^2$ &$^3$G$_3$ &($^4$G)4p &$^3$H$_4$ & 3.422 & $-$0.756 &16 & 1.36 & 2& 5.597 & 5.618 & 5.665 & 5.575 & 5.644 & $+$0.033 & 5.630\
571.9809 & & ($^4$D)4s &$^5$D$_3$ &($^5$D)4sp &$^5$D$_4$ & 3.013 & $-$1.620 &16 & 0.43 & 1& 5.503 & 5.535 & 5.586 & 5.496 & 5.562 & $+$0.026 & 5.529\
578.1163 & & ($^4$D)4s &$^5$D$_4$ &($^5$D)4sp &$^5$D$_3$ & 3.011 & $-$1.000 &14 & 1.56 & 1& 5.498 & 5.525 & 5.577 & 5.483 & 5.551 & $+$0.002 & 5.500\
578.5024 & & ($^6$S)4p &$^5$P$_3$ &($^6$S)4d &$^5$D$_3$ & 3.321 & $-$0.380 &15 & 3.13 & 1& 5.595 & 5.610 & 5.659 & 5.560 & 5.633 & $+$0.029 & 5.624\
584.4592 & & ($^4$D)4s &$^5$D$_3$ &($^5$D)4sp &$^5$D$_2$ & 3.013 & $-$1.770 &14 & 0.40 & 2& 5.616 & 5.647 & 5.698 & 5.609 & 5.673 & $+$0.026 & 5.642\
688.2477 & & ($^5$D)4sp&$^7$P$_2$ &($^6$S)4d &$^7$D$_2$ & 3.438 & $-$0.375 &15 & 3.13 & 1& 5.626 & 5.640 & 5.687 & 5.592 & 5.659 & $+$0.025 & 5.651\
688.2997 & & ($^5$D)4sp&$^7$P$_2$ &($^6$S)4d &$^7$D$_1$ & 3.438 & $-$0.420 &15 & 2.96 & 3& 5.636 & 5.651 & 5.698 & 5.604 & 5.671 & $+$0.025 & 5.661\
& & & & & & & & & & & & & & &\
\
455.4990 & & d$^5$ &$^4$F$_\frac{7}{2}$ &($^5$D)4p &$^4$D$_\frac{7}{2}$ & 4.071 & $-$1.249 &18 & 4.66 & 1& 5.632 & 5.596 & 5.606 & 5.538 & 5.627 & &\
458.8200 & & d$^5$ &$^4$F$_\frac{7}{2}$ &($^5$D)4p &$^4$D$_\frac{5}{2}$ & 4.071 & $-$0.594 &18 & 7.47 & 2& 5.648 & 5.565 & 5.576 & 5.492 & 5.588 & &\
484.8237 & & ($^3$F)4s &$^4$F$_\frac{7}{2}$ &($^5$D)4p &$^4$F$_\frac{7}{2}$ & 3.864 & $-$1.160 &18 & 6.11 & 3& 5.689 & 5.621 & 5.629 & 5.555 & 5.648 & &\
523.7328 & & d$^5$ &$^4$F$_\frac{9}{2}$ &($^5$D)4p &$^4$F$_\frac{9}{2}$ & 4.073 & $-$1.087 &18 & 5.36 & 2& 5.610 & 5.554 & 5.557 & 5.490 & 5.581 & &\
524.6768 & & ($^3$F)4s &$^4$P$_\frac{1}{2}$ &($^5$D)4p &$^4$P$_\frac{3}{2}$ & 3.714 & $-$2.436 &18 & 1.62 & 1& 5.654 & 5.652 & 5.665 & 5.620 & 5.684 & &\
527.9877 & & d$^5$ &$^4$F$_\frac{9}{2}$ &($^5$D)4p &$^4$F$_\frac{7}{2}$ & 4.073 & $-$1.909 &18 & 2.02 & 1& 5.580 & 5.576 & 5.587 & 5.539 & 5.608 & &\
531.0686 & & d$^5$ &$^4$F$_\frac{3}{2}$ &($^5$D)4p &$^4$F$_\frac{5}{2}$ & 4.072 & $-$2.144 &18 & 1.32 & 1& 5.564 & 5.566 & 5.578 & 5.534 & 5.597 & &\
531.3561 & & d$^5$ &$^4$F$_\frac{5}{2}$ &($^5$D)4p &$^4$F$_\frac{5}{2}$ & 4.073 & $-$1.473 &18 & 3.49 & 1& 5.544 & 5.523 & 5.529 & 5.473 & 5.553 & &\
550.2068 & & ($^3$G)4s &$^4$G$_\frac{9}{2}$ &($^5$D)4p &$^4$F$_\frac{7}{2}$ & 4.168 & $-$2.049 &18 & 1.84 & 2& 5.743 & 5.741 & 5.750 & 5.704 & 5.770 & &\
612.9226 & & ($^3$G)4s &$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$D$_\frac{5}{2}$ & 4.750 & $-$2.478 &18 & 0.29 & 1& 5.749 & 5.759 & 5.768 & 5.731 & 5.785 & &\
& & & & & & & & & & & & & & &\
\
408.2945 & & ($^5$D)4s &$^6$D$_\frac{3}{2}$ &($^5$D)4p &$^6$D$_\frac{5}{2}$ & 2.178 & $-$0.365 &19 & 8.97 & 2& 5.380 & 5.316 & 5.392 & 5.256 & 5.329 & $+$0.016 & 5.396\
426.5928 & & ($^5$D)4s &$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$P$_\frac{3}{2}$ & 2.941 & $-$0.400 &20 & 5.85 & 1\* & 5.376 & 5.354 & 5.420 & 5.301 & 5.376 & $+$0.076 & 5.452\
445.3013 & & ($^5$D)4s &$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$D$_\frac{1}{2}$ & 2.941 & $-$0.620 &21 & 5.19 & 2& 5.368 & 5.371 & 5.436 & 5.322 & 5.395 & $+$0.070 & 5.438\
445.7041 & & ($^6$S)4sp&$^6$P$_\frac{5}{2}$ &($^7$S)4sd &$^6$D$_\frac{3}{2}$ & 3.073 & $-$0.685 &20 & 4.33 & 1\* & 5.392 & 5.400 & 5.460 & 5.353 & 5.426 & $+$0.065 & 5.457\
447.0142 & & ($^5$D)4s &$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$D$_\frac{3}{2}$ & 2.941 & $-$0.560 &21 & 5.22 & 2& 5.411 & 5.389 & 5.454 & 5.336 & 5.411 & $+$0.061 & 5.472\
449.8897 & & ($^5$D)4s &$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$D$_\frac{5}{2}$ & 2.941 & $-$0.460 &21 & 5.54 & 1& 5.398 & 5.364 & 5.430 & 5.309 & 5.385 & $+$0.054 & 5.452\
450.2223 & & ($^5$D)4s &$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^4$D$_\frac{7}{2}$ & 2.920 & $-$0.430 &21 & 5.81 & 2& 5.349 & 5.328 & 5.393 & 5.273 & 5.348 & $+$0.055 & 5.404\
467.1688 & & ($^5$D)4s &$^4$D$_\frac{7}{2}$ &($^5$D)4p &$^4$F$_\frac{5}{2}$ & 2.888 & $-$1.660 &21 & 1.23 & 1& 5.379 & 5.412 & 5.472 & 5.377 & 5.440 & $+$0.061 & 5.440\
470.9710 & & ($^5$D)4s &$^4$D$_\frac{7}{2}$ &($^5$D)4p &$^4$F$_\frac{7}{2}$ & 2.888 & $-$0.487 &19 & 6.88 & 2& 5.353 & 5.360 & 5.427 & 5.306 & 5.379 & $+$0.065 & 5.418\
473.9110 & & ($^5$D)4s &$^4$D$_\frac{3}{2}$ &($^5$D)4p &$^4$F$_\frac{3}{2}$ & 2.941 & $-$0.604 &19 & 5.82 & 3& 5.346 & 5.359 & 5.424 & 5.310 & 5.382 & $+$0.065 & 5.411\
500.4891 & & ($^5$D)4s &$^4$D$_\frac{5}{2}$ &($^5$D)4p &$^6$F$_\frac{7}{2}$ & 2.920 & $-$1.636 &22 & 1.31 & 2& 5.404 & 5.434 & 5.493 & 5.398 & 5.461 & $+$0.064 & 5.468\
525.5330 & & 4s$^2$ &$^4$G$_\frac{11}{2}$ &($^5$D)4p &$^4$F$_\frac{9}{2}$ & 3.133 & $-$0.858 &19 & 3.69 & 2& 5.326 & 5.355 & 5.413 & 5.313 & 5.381 & $+$0.069 & 5.395\
538.8538 & & 4s$^2$ &$^4$P$_\frac{5}{2}$ &($^5$D)4p &$^4$D$_\frac{7}{2}$ & 3.373 & $-$1.620 &21 & 0.49 & 1& 5.334 & 5.361 & 5.416 & 5.322 & 5.389 & $+$0.064 & 5.398\
542.0368 & & ($^5$D)4s &$^6$D$_\frac{7}{2}$ &($^6$S)4sp &$^6$P$_\frac{5}{2}$ & 2.143 & $-$1.462 &23 & 7.86 & 3& 5.315 & 5.361 & 5.434 & 5.323 & 5.382 & $+$0.072 & 5.387\
& & & & & & & & & & & & & & &\
\
444.5472 & & 4s$^2$ &$^5$D$_2$ &4s4p($^3$P)&$^7$F$_2$ & 0.087 & $-$5.412 & 24 & 3.80 & 2& 7.419 & 7.463 & 7.568 & 7.436 & 7.474 & $+$0.016 & 7.435\
524.7050 & & 4s$^2$ &$^5$D$_2$ &4s4p($^3$P)&$^7$D$_3$ & 0.087 & $-$4.961 & 24 & 6.40 & 3& 7.472 & 7.449 & 7.559 & 7.412 & 7.440 & $+$0.022 & 7.494\
549.1832 & & 3d$^8$ &$^3$F$_2$ &($^2$P)4p &$^3$D$_3$ & 4.186 & $-$2.188 & 25 & 1.23 & 1& 7.441 & 7.452 & 7.500 & 7.411 & 7.481 & $+$0.006 & 7.447\
560.0224 & & 4s4p($^3$P)&$^3$P$_1$ &4s($^4$D)5s&$^5$D$_1$ & 4.260 & $-$1.420 & 25 & 3.65 & 1& 7.369 & 7.367 & 7.412 & 7.316 & 7.390 & $+$0.007 & 7.376\
566.1346 & & 4s4p($^3$P)&$^3$P$_0$ &4s($^4$D)5s&$^5$D$_1$ & 4.284 & $-$1.756 & 25 & 2.22 & 2& 7.414 & 7.419 & 7.465 & 7.374 & 7.445 & $+$0.006 & 7.420\
570.5465 & & ($^4$F)4p &$^5$F$_1$ &4s($^4$D)5s&$^5$D$_1$ & 4.301 & $-$1.355 & 25 & 3.96 & 2& 7.418 & 7.409 & 7.455 & 7.356 & 7.431 & $+$0.005 & 7.423\
577.8453 & & 4s$^2$ &$^3$F2$_3$ &($^4$F)4p &$^3$D$_3$ & 2.588 & $-$3.440 & 25 & 2.04 & 2& 7.403 & 7.427 & 7.495 & 7.391 & 7.450 & $+$0.005 & 7.408\
578.4658 & & 4s4p($^3$P)&$^5$F$_3$ &4s($^6$D)5s&$^5$D$_4$ & 3.396 & $-$2.532 & 25 & 2.58 & 2& 7.418 & 7.429 & 7.486 & 7.387 & 7.453 & $+$0.007 & 7.425\
585.5077 & & ($^4$F)4p &$^3$F$_3$ &($^4$F)4d &$^5$H$_4$ & 4.608 & $-$1.478 & 25 & 2.20 & 1& 7.422 & 7.426 & 7.468 & 7.379 & 7.452 & $+$0.006 & 7.428\
595.6694 & & ($^4$F)4s &$^5$F$_5$ &4s4p($^3$P)&$^7$P$_4$ & 0.859 & $-$4.552 & 24 & 5.02 & 3& 7.430 & 7.441 & 7.538 & 7.408 & 7.443 & $+$0.017 & 7.447\
615.1618 & & ($^4$P)4s &$^5$P$_3$ &($^4$F)4p &$^5$D$_2$ & 2.176 & $-$3.282 & 26 & 4.88 & 3& 7.445 & 7.437 & 7.514 & 7.397 & 7.447 & $+$0.012 & 7.457\
624.0646 & & ($^4$P)4s &$^5$P$_1$ &4s4p($^3$P)&$^3$P$_2$ & 2.223 & $-$3.287 & 27 & 4.76 & 3& 7.469 & 7.461 & 7.538 & 7.421 & 7.472 & $+$0.012 & 7.481\
631.1500 & & ($^4$P)4s &$^3$P$_2$ &($^4$F)4p &$^3$D$_2$ & 2.831 & $-$3.141 & 25 & 2.66 & 1& 7.470 & 7.485 & 7.549 & 7.447 & 7.505 & $+$0.008 & 7.478\
649.8939 & & ($^4$F)4s &$^5$F$_3$ &4s4p($^3$P)&$^7$F$_3$ & 0.958 & $-$4.695 & 24 & 4.39 & 3& 7.488 & 7.516 & 7.610 & 7.486 & 7.519 & $+$0.015 & 7.503\
651.8367 & & ($^4$P)4s &$^3$P$_2$ &($^4$F)4p &$^3$D$_3$ & 2.831 & $-$2.448 & 27 & 5.72 & 2& 7.429 & 7.389 & 7.459 & 7.343 & 7.396 & $+$0.012 & 7.441\
669.9142 & & ($^2$F)4s &$^3$F$_4$ &($^2$P)4p &$^3$D$_3$ & 4.593 & $-$2.101 & 25 & 0.81 & 2& 7.515 & 7.469 & 7.543 & 7.422 & 7.471 & $+$0.006 & 7.489\
679.3259 & & 3d$^8$ &$^3$F$_4$ &4s4p($^3$P)&$^5$G$_4$ & 4.076 & $-$2.326 & 25 & 1.25 & 1& 7.420 & 7.431 & 7.479 & 7.390 & 7.455 & $+$0.006 & 7.426\
683.7006 & & ($^2$F)4s &$^3$F$_4$ &($^2$H)4p &$^3$G$_4$ & 4.593 & $-$1.687 & 25 & 1.77 & 1& 7.466 & 7.468 & 7.509 & 7.422 & 7.492 & $+$0.006 & 7.472\
685.4823 & & ($^2$F)4s &$^3$F$_4$ &4s4p($^3$P)&$^1$H$_5$ & 4.593 & $-$1.926 & 25 & 1.22 & 1& 7.506 & 7.512 & 7.553 & 7.469 & 7.536 & $+$0.006 & 7.512\
740.1685 & & 3d$^8$ &$^3$F$_2$ &($^4$P)4p &$^3$D$_1$ & 4.186 & $-$1.500 & 25 & 4.16 & 3& 7.381 & 7.371 & 7.417 & 7.323 & 7.387 & $+$0.008 & 7.389\
791.2867 & & ($^4$F)4s &$^5$F$_5$ &4s4p($^3$P)&$^7$D$_4$ & 0.859 & $-$4.848 & 29 & 4.57 & 2& 7.451 & 7.489 & 7.586 & 7.462 & 7.486 & $+$0.017 & 7.468\
829.3515 & & ($^2$D)4s &$^3$D$_2$ &($^4$F)4p &$^3$D$_2$ & 3.301 & $-$2.203 & 30 & 5.85 & 1\* & 7.471 & 7.448 & 7.509 & 7.401 & 7.453 & $+$0.011 & 7.482\
& & & & & & & & & & & & & & &\
\
462.0513 & & 4s &$^4$F$_\frac{7}{2}$ &4p &$^4$D$_\frac{7}{2}$ & 2.828 & $-$3.210 & 31 & 5.40 & 1& 7.474 & 7.405 & 7.416 & 7.350 & 7.436 & &\
526.4804 & & 4s &$^4$G$_\frac{5}{2}$ &4p &$^4$D$_\frac{3}{2}$ & 3.230 & $-$3.130 & 31 & 4.74 & 3& 7.556 & 7.500 & 7.503 & 7.445 & 7.530 & &\
541.4072 & & 4s &$^4$G$_\frac{7}{2}$ &4p &$^4$D$_\frac{7}{2}$ & 3.221 & $-$3.580 & 31 & 2.73 & 2& 7.483 & 7.464 & 7.471 & 7.424 & 7.496 & &\
643.2676 & & 4s$^2$ &$^6$S$_\frac{5}{2}$ &4p &$^6$D$_\frac{5}{2}$ & 2.891 & $-$3.570 & 31 & 4.30 & 3& 7.515 & 7.463 & 7.462 & 7.416 & 7.488 & &\
651.6077 & & 4s$^2$ &$^6$S$_\frac{5}{2}$ &4p &$^6$D$_\frac{7}{2}$ & 2.891 & $-$3.310 & 31 & 5.69 & 3& 7.569 & 7.485 & 7.482 & 7.432 & 7.504 & &\
722.2392 & & 4s &$^4$D$_\frac{3}{2}$ &4p &$^4$D$_\frac{1}{2}$ & 3.889 & $-$3.260 & 31 & 2.03 & 1& 7.519 & 7.504 & 7.501 & 7.466 & 7.530 & &\
722.4479 & & 4s &$^4$D$_\frac{1}{2}$ &4p &$^4$D$_\frac{1}{2}$ & 3.889 & $-$3.200 & 31 & 2.10 & 1& 7.480 & 7.464 & 7.461 & 7.425 & 7.490 & &\
751.5831 & & 4s &$^4$D$_\frac{7}{2}$ &4p &$^4$D$_\frac{5}{2}$ & 3.903 & $-$3.390 & 31 & 1.47 & 2& 7.455 & 7.445 & 7.444 & 7.411 & 7.472 & &\
771.1721 & & 4s &$^4$D$_\frac{7}{2}$ &4p &$^4$D$_\frac{7}{2}$ & 3.903 & $-$2.500 & 31 & 5.04 & 3& 7.500 & 7.431 & 7.417 & 7.378 & 7.448 & &\
& & & & & & & & & & & & & & &\
\
521.2688 & &($^4$F)4sp &$^4$F$_\frac{9}{2}$ &4s($^5$F)5s&$^4$F$_\frac{9}{2}$ & 3.514 & $-$0.110 & 32 & 1.91 & 3& 4.807 & 4.822 & 4.873 & 4.785 & 4.851 & $+$0.072 & 4.879\
528.0627 & &($^4$F)4sp &$^4$G$_\frac{9}{2}$ &4s($^5$F)5s&$^4$F$_\frac{7}{2}$ & 3.629 & $-$0.030 & 32 & 1.78 & 2& 4.820 & 4.833 & 4.883 & 4.795 & 4.862 & $+$0.077 & 4.897\
530.1044 & &4s$^2$ &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^4$D$_\frac{5}{2}$ & 1.710 & $-$1.940 & 33 & 1.79 & 1& 4.859 & 4.899 & 4.973 & 4.869 & 4.922 & $+$0.100 & 4.959\
535.2041 & &($^4$F)4sp &$^4$G$_\frac{11}{2}$ &4s($^5$F)5s&$^4$F$_\frac{9}{2}$ & 3.576 & 0.060 & 32 & 2.38 & 2& 4.823 & 4.836 & 4.886 & 4.796 & 4.864 & $+$0.082 & 4.905\
548.3353 & &4s$^2$ &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^4$D$_\frac{7}{2}$ & 1.710 & $-$1.410 & 33 & 4.64 & 3& 4.800 & 4.837 & 4.913 & 4.804 & 4.857 & $+$0.099 & 4.899\
564.7233 & &($^3$P)4s &$^2$P$_\frac{3}{2}$ &($^3$F)4p &$^2$D$_\frac{5}{2}$ & 2.280 & $-$1.560 & 32 & 1.24 & 2\* & 4.837 & 4.869 & 4.936 & 4.836 & 4.894 & $+$0.084 & 4.921\
593.5390 & &($^3$P)4s &$^4$P$_\frac{5}{2}$ &($^3$F)4p &$^4$D$_\frac{7}{2}$ & 1.883 & $-$2.610 & 33 & 0.32 & 1& 4.849 & 4.891 & 4.963 & 4.861 & 4.914 & $+$0.087 & 4.936\
608.2423 & &($^4$F)4sp &$^4$F$_\frac{9}{2}$ &($^3$F)5s &$^4$F$_\frac{9}{2}$ & 3.514 & $-$0.520 & 32 & 1.00 & 3& 4.857 & 4.873 & 4.924 & 4.836 & 4.901 & $+$0.072 & 4.929\
609.3141 & &4s$^2$ &$^4$P$_\frac{3}{2}$ &($^4$F)4sp &$^4$D$_\frac{3}{2}$ & 1.740 & $-$2.440 & 32 & 0.79 & 2\* & 4.935 & 4.978 & 5.050 & 4.950 & 4.998 & $+$0.085 & 5.020\
618.9005 & &4s$^2$ &$^4$P$_\frac{5}{2}$ &($^4$F)4sp &$^4$D$_\frac{5}{2}$ & 1.710 & $-$2.450 & 32 & 0.89 & 2\* & 4.956 & 4.998 & 5.071 & 4.971 & 5.018 & $+$0.085 & 5.041\
642.9913 & &4s$^2$ &$^2$G$_\frac{7}{2}$ &($^4$F)4sp &$^2$F$_\frac{5}{2}$ & 2.137 & $-$2.410 & 32 & 0.31 & 2\* & 4.854 & 4.892 & 4.960 & 4.861 & 4.914 & $+$0.087 & 4.941\
645.4995 & &($^4$F)4sp &$^4$D$_\frac{7}{2}$ &($^3$F)5s &$^4$F$_\frac{9}{2}$ & 3.632 & $-$0.250 & 32 & 1.34 & 2& 4.826 & 4.840 & 4.889 & 4.803 & 4.867 & $+$0.084 & 4.910\
741.7386 & &($^1$D)4s &$^2$D$_\frac{3}{2}$ &($^4$F)4sp &$^4$D$_\frac{5}{2}$ & 2.042 & $-$2.070 & 32 & 1.00 & 2\* & 4.880 & 4.920 & 4.989 & 4.892 & 4.939 & $+$0.090 & 4.970\
& & & & & & & & & & & & & & &\
\
474.0166 & &($^3$F)4sp &$^5$G$_4$ &($^2$D)4d &$^3$G$_5$ & 3.480 & $-$1.720 & 34 & 1.60 & 1& 6.192 & 6.197 & 6.246 & 6.160 & 6.228 & &\
481.1977 & &($^2$D)4p &$^3$P$_1$ &($^2$D)4d &$^3$P$_0$ & 3.658 & $-$1.450 & 35 & 2.13 & 1& 6.238 & 6.241 & 6.287 & 6.201 & 6.271 & &\
481.4598 & &($^3$F)4sp &$^5$G$_2$ &4s($^4$F)5s&$^5$F$_3$ & 3.597 & $-$1.630 & 34 & 1.59 & 1& 6.216 & 6.218 & 6.265 & 6.179 & 6.248 & &\
487.4793 & &($^3$F)4sp &$^5$G$_3$ &4s($^4$F)5s&$^5$F$_4$ & 3.543 & $-$1.440 & 34 & 2.35 & 1& 6.177 & 6.178 & 6.225 & 6.137 & 6.207 & &\
488.6711 & &($^2$D)4p &$^3$D$_2$ &4s($^4$F)5s&$^5$F$_2$ & 3.706 & $-$1.810 & 34 & 0.90 & 1& 6.201 & 6.207 & 6.253 & 6.170 & 6.238 & &\
490.0971 & &($^3$F)4sp &$^5$G$_4$ &4s($^4$F)5s&$^5$F$_5$ & 3.480 & $-$1.660 & 34 & 1.79 & 1& 6.195 & 6.196 & 6.244 & 6.156 & 6.226 & &\
497.6135 & &($^3$F)4sp &$^5$F$_4$ &($^2$D)4d &$^3$G$_4$ & 3.606 & $-$1.260 & 34 & 2.86 & 2& 6.179 & 6.176 & 6.222 & 6.132 & 6.204 & &\
515.7981 & &($^3$F)4sp &$^5$F$_4$ &4s($^4$F)5s&$^5$F$_5$ & 3.606 & $-$1.510 & 34 & 1.86 & 3& 6.169 & 6.169 & 6.215 & 6.128 & 6.198 & &\
550.4095 & &($^3$F)4sp &$^3$G$_5$ &4s($^4$F)5s&$^5$F$_4$ & 3.834 & $-$1.690 & 34 & 0.97 & 1& 6.207 & 6.211 & 6.254 & 6.170 & 6.240 & &\
551.0009 & &($^2$D)4p &$^1$F$_3$ &($^2$D)4d &$^3$G$_4$ & 3.847 & $-$0.880 & 34 & 3.75 & 2& 6.189 & 6.176 & 6.219 & 6.125 & 6.201 & &\
553.7105 & &($^2$D)4p &$^1$F$_3$ &4s($^4$F)5s&$^5$F$_4$ & 3.847 & $-$2.220 & 34 & 0.31 & 3& 6.213 & 6.220 & 6.263 & 6.182 & 6.250 & &\
574.9280 & &($^3$F)4sp &$^3$G$_3$ &($^2$D)4d &$^3$G$_4$ & 3.941 & $-$1.920 & 34 & 0.44 & 2& 6.145 & 6.152 & 6.193 & 6.113 & 6.181 & &\
617.6820 & &($^3$F)4sp &$^3$F$_4$ &($^2$D)4d &$^3$G$_5$ & 4.088 & $-$0.260 & 34 & 6.64 & 2& 6.225 & 6.193 & 6.234 & 6.130 & 6.208 & &\
620.4605 & &($^3$F)4sp &$^3$F$_4$ &4s($^4$F)5s&$^5$F$_4$ & 4.088 & $-$1.080 & 34 & 2.11 & 3& 6.211 & 6.206 & 6.245 & 6.161 & 6.232 & &\
622.3991 & &($^3$F)4sp &$^3$F$_3$ &($^2$D)4d &$^3$G$_4$ & 4.105 & $-$0.910 & 34 & 2.79 & 3& 6.197 & 6.194 & 6.233 & 6.148 & 6.220 & &\
637.8258 & &($^3$F)4sp &$^3$D$_3$ &($^2$D)4d &$^3$G$_4$ & 4.154 & $-$0.820 & 34 & 3.20 & 3& 6.225 & 6.221 & 6.259 & 6.173 & 6.245 & &\
641.4588 & &($^3$F)4sp &$^3$D$_3$ &4s($^4$F)5s&$^5$F$_4$ & 4.154 & $-$1.160 & 34 & 1.68 & 2& 6.215 & 6.213 & 6.251 & 6.169 & 6.239 & &\
**References:**\
1. @Lawler89
2. @Lawler13
3. @Nitz98
4. @Blackwell1, as corrected by @Grevesse89
5. @Blackwell3, as corrected by @Grevesse89
6. @Blackwell2, as corrected by @Grevesse89
7. @Wood13
8. @Whaling85, with corrected for arithmetic error in converting from BFs to $A$ values as per @Martin88
9. @Biemont89
10. @Karamatskos86
11. mean of @Sobeck07 and @Blackwell84
12. mean of @Sobeck07 and @Tozzi85
13. mean of @Sobeck07, @Tozzi85 and @Blackwell86
14. @Sobeck07
15. @Blackwell86
16. mean of @Sobeck07 and @Blackwell86
17. mean of @Sobeck07, @Tozzi85 and @Blackwell84
<!-- -->
18. @Kuruczweb
19. @DenHartog11
20. @Booth84c, renormalised to the absolute scale of @BW07 (lines with excitation $\mathrm{potential}\approx3$eV; see Sect. \[Mngfs\])
21. @BW07
22. derived from BFs of @Greenlee79 and lifetimes of @Schnabel95
23. @Booth84c
24. mean of Oxford data [@Oxford1; @Oxford2; @Oxford3; @Oxford4; @Oxford5; @Oxford6] and @OBrian91
25. Hannover data [@Bard91; @Bard94]
26. mean of Oxford (see Ref. 24) and Hannover data (see Ref. 25)
27. mean of Hannover data (see Ref. 25) and @OBrian91, with double weight to Hannover
28. mean of Oxford data (see Ref. 24) and @OBrian91, with double weight to Oxford
29. Oxford data (see Ref. 24)
30. mean of Hannover data (see Ref. 25) and @OBrian91
31. @Melendez09
32. @Cardon82
33. @Nitz99
34. @Wood14
35. @Johansson03
[r c@c r r ccc r r c]{}\
474.3821 & $^{45}$Sc & $9/2$ & 285.967 & $-$15.460 & 1 && $7/2$ & & &\
508.1561 & $^{45}$Sc & $9/2$ & 285.967 & $-$15.460 & 1 && $9/2$ & & &\
535.6097 & $^{45}$Sc & $7/2$ & $-$25.000 & & 2 && $5/2$ & & &\
567.1828 & $^{45}$Sc & $9/2$ & 285.967 & $-$15.460 & 1 && $11/2$ & 55.000 & 25.000 & 3\
623.9800 & $^{45}$Sc & $3/2$ & 269.556 & $-$26.346 & 4 && $3/2$ & 348.320 & & 5\
& & & & & & & & & &\
\
442.0661 & $^{45}$Sc & 4 & 38.357 & $-$16.456 & 6 && 3 & 205.400 & $-$70.000 & 7\
443.1362 & $^{45}$Sc & 3 & 113.674 & $-$12.615 & 6 && 2 & 366.800 & $-$40.000 & 7\
535.7202 & $^{45}$Sc & 2 & $-$27.732 & 22.127 & 8 && 1 & & &\
564.1000 & $^{45}$Sc & 1 & $-$107.501 & $-$12.300 & 8 && 2 & 106.117 & $-$20.200 & 8\
565.8362 & $^{45}$Sc & 0 & 0.000 & 0.000 & N && 1 & 255.155 & 11.753 & 8\
566.7164 & $^{45}$Sc & 1 & $-$107.501 & $-$12.300 & 8 && 1 & 255.155 & 11.753 & 8\
566.9055 & $^{45}$Sc & 1 & $-$107.501 & $-$12.300 & 8 && 0 & 0.000 & 0.000 & N\
568.4214 & $^{45}$Sc & 2 & $-$27.732 & 22.127 & 8 && 1 & 255.155 & 11.753 & 8\
624.5641 & $^{45}$Sc & 2 & $-$27.732 & 22.127 & 8 && 3 & 99.730 & 21.495 & 8\
630.0746 & $^{45}$Sc & 2 & $-$27.732 & 22.127 & 8 && 2 & 125.423 & 8.769 & 8\
632.0843 & $^{45}$Sc & 1 & $-$107.501 & $-$12.300 & 8 && 1 & 304.788 & 3.824 & 8\
660.4578 & $^{45}$Sc & 2 & 149.361 & 7.818 & 6 && 2 & 215.700 & 18.000 & 9\
& & & & & & & & & &\
\
\
586.6429 & $^{50}$Ti & 2 & 0.000 & 0.000 & N && 3 & 0.000& 0.000 & N\
586.6439 & $^{49}$Ti & 2 & $-$25.216 & $-$39.202 & 10 && 3 & & &\
586.6448 & $^{48}$Ti & 2 & 0.000 & 0.000 & N && 3 & 0.000& 0.000 & N\
586.6458 & $^{47}$Ti & 2 & $-$25.216 & $-$47.826 & 10 && 3 & & &\
586.6468 & $^{46}$Ti & 2 & 0.000 & 0.000 & N && 3 & 0.000& 0.000 & N\
592.2088 & $^{50}$Ti & 0 & 0.000 & 0.000 & N && 1 & 0.000& 0.000 & N\
592.2097 & $^{49}$Ti & 0 & 0.000 & 0.000 & N && 1 &$-$140.600& 0.000 & 11\
592.2107 & $^{48}$Ti & 0 & 0.000 & 0.000 & N && 1 & 0.000& 0.000 & N\
592.2117 & $^{47}$Ti & 0 & 0.000 & 0.000 & N && 1 &$-$140.700& 0.000 & 11\
592.2128 & $^{46}$Ti & 0 & 0.000 & 0.000 & N && 1 & 0.000& 0.000 & N\
& & & & & & & & & &\
\
4444.524 & $^{50}$Ti & $7/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
4444.530 & $^{49}$Ti & $7/2$ & $-$54.374 & 26.422 & 12 && $7/2$ & $-$31.500& $-$14.000 & 12\
4444.536 & $^{48}$Ti & $7/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
4444.542 & $^{47}$Ti & $7/2$ & $-$54.374 & 32.235 & 12 && $7/2$ & $-$31.500& $-$17.080 & 12\
4444.547 & $^{46}$Ti & $7/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
4493.520 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4493.521 & $^{47}$Ti & $3/2$ & 97.013 & $-$19.453 & 12 && $5/2$ & & &\
4493.523 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4493.524 & $^{49}$Ti & $3/2$ & 97.013 & $-$23.733 & 12 && $5/2$ & & &\
4493.525 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4583.396 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4583.403 & $^{49}$Ti & $3/2$ & $-$6.630 & $-$24.100 & 13 && $5/2$ & $-$84.210& $-$44.000 & 12\
4583.409 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4583.415 & $^{47}$Ti & $3/2$ & $-$6.630 & $-$29.402 & 13 && $5/2$ & $-$84.210& $-$53.680 & 12\
4583.421 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4609.253 & $^{50}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4609.259 & $^{49}$Ti & $5/2$ & 11.520 & 38.400 & 13 && $5/2$ & $-$84.210& $-$44.000 & 12\
4609.265 & $^{48}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4609.271 & $^{47}$Ti & $5/2$ & 11.520 & 46.848 & 13 && $5/2$ & $-$84.210& $-$53.680 & 12\
4609.277 & $^{46}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4708.656 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4708.659 & $^{49}$Ti & $3/2$ & 53.334 & $-$23.471 & 12 && $5/2$ & $-$84.210& $-$44.000 & 12\
4708.662 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4708.665 & $^{47}$Ti & $3/2$ & 53.334 & $-$28.635 & 12 && $5/2$ & $-$84.210& $-$53.680 & 12\
4708.668 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4764.518 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4764.521 & $^{49}$Ti & $3/2$ & 53.334 & $-$23.471 & 12 && $5/2$ & & &\
4764.524 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4764.527 & $^{47}$Ti & $3/2$ & 53.334 & $-$28.635 & 12 && $5/2$ & & &\
4764.530 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4798.529 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4798.530 & $^{47}$Ti & $3/2$ & 97.013 & $-$19.453 & 12 && $5/2$ & & &\
4798.532 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4798.533 & $^{49}$Ti & $3/2$ & 97.013 & $-$23.733 & 12 && $5/2$ & & &\
4798.535 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4865.597 & $^{50}$Ti & $7/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4865.605 & $^{49}$Ti & $7/2$ & $-$54.374 & 26.422 & 12 && $5/2$ & & &\
4865.611 & $^{48}$Ti & $7/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
4865.618 & $^{47}$Ti & $7/2$ & $-$54.374 & 32.235 & 12 && $5/2$ & & &\
4865.625 & $^{46}$Ti & $7/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5336.770 & $^{50}$Ti & $5/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
5336.774 & $^{49}$Ti & $5/2$ & & & && $7/2$ & $-$31.500& $-$14.000 & 12\
5336.778 & $^{48}$Ti & $5/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
5336.782 & $^{47}$Ti & $5/2$ & & & && $7/2$ & $-$31.500& $-$17.080 & 12\
5336.786 & $^{46}$Ti & $5/2$ & 0.000 & 0.000 & N && $7/2$ & 0.000& 0.000 & N\
5381.013 & $^{50}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5381.017 & $^{49}$Ti & $3/2$ & & & && $5/2$ & $-$84.210& $-$44.000 & 12\
5381.021 & $^{48}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5381.025 & $^{47}$Ti & $3/2$ & & & && $5/2$ & $-$84.210& $-$53.680 & 12\
5381.029 & $^{46}$Ti & $3/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5418.760 & $^{50}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5418.764 & $^{49}$Ti & $5/2$ & & & && $5/2$ & $-$84.210& $-$44.000 & 12\
5418.768 & $^{48}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
5418.771 & $^{47}$Ti & $5/2$ & & & && $5/2$ & $-$84.210& $-$53.680 & 12\
5418.775 & $^{46}$Ti & $5/2$ & 0.000 & 0.000 & N && $5/2$ & 0.000& 0.000 & N\
& & & & & & & & & &\
\
458.6370 & $^{51}$V & $7/2$ & 249.739 & 5.081 & 14 && $9/2$ & 408.197& & 15\
459.4119 & $^{51}$V & $9/2$ & 227.132 & 7.822 & 14 && $11/2$ & 448.669& & 15\
463.5172 & $^{51}$V & $9/2$ & 227.132 & 7.822 & 14 && $9/2$ & 408.197& & 15\
482.7452 & $^{51}$V & $7/2$ & 249.739 & 5.081 & 14 && $7/2$ & 606.150& & 15\
487.5486 & $^{51}$V & $7/2$ & 249.739 & 5.081 & 14 && $5/2$ & 611.067& & 15\
488.1555 & $^{51}$V & $9/2$ & 227.132 & 7.822 & 14 && $7/2$ & 606.150& & 15\
562.6019 & $^{51}$V & $1/2$ & 1276.000 & & 16 && $1/2$ & 1100.238& & 15\
564.6108 & $^{51}$V & $3/2$ & 6.966 & $-$10.854 & 16 && $1/2$ & 1100.238& & 15\
565.7438 & $^{51}$V & $5/2$ & $-$143.432 & $-$1.196 & 16 && $3/2$ & 141.202& & 15\
566.8361 & $^{51}$V & $7/2$ & $-$160.219 & 10.229 & 16 && $5/2$ & 15.289& & 15\
567.0847 & $^{51}$V & $7/2$ & $-$160.219 & 10.229 & 16 && $9/2$ & 94.644& & 15\
570.3586 & $^{51}$V & $3/2$ & 6.966 & $-$10.854 & 16 && $5/2$ & 215.851& & 17\
572.7046 & $^{51}$V & $7/2$ & $-$160.219 & 10.229 & 16 && $9/2$ & 89.038& & 15\
572.7655 & $^{51}$V & $3/2$ & 6.966 & $-$10.854 & 16 && $3/2$ & 634.361& & 15\
573.1249 & $^{51}$V & $5/2$ & $-$143.432 & $-$1.196 & 16 && $7/2$ & 431.551& & 15\
573.7065 & $^{51}$V & $5/2$ & $-$143.432 & $-$1.196 & 16 && $5/2$ & 215.851& & 17\
600.2294 & $^{51}$V & $5/2$ & 112.835 & & 18 && $7/2$ & $-$17.088& & 15\
603.9728 & $^{51}$V & $5/2$ & $-$143.432 & $-$1.196 & 16 && $5/2$ & $-$89.800& 8.000 & 16\
608.1441 & $^{51}$V & $3/2$ & 6.966 & $-$10.854 & 16 && $3/2$ &$-$286.400& $-$6.000 & 16\
609.0208 & $^{51}$V & $7/2$ & $-$160.219 & 10.229 & 16 && $5/2$ & $-$89.800& 8.000 & 16\
611.1650 & $^{51}$V & $1/2$ & 1276.000 & & 16 && $1/2$ &$-$795.200& & 16\
611.9528 & $^{51}$V & $5/2$ & $-$143.432 & $-$1.196 & 16 && $3/2$ &$-$286.400& $-$6.000 & 16\
613.5363 & $^{51}$V & $3/2$ & 6.966 & $-$10.854 & 16 && $1/2$ &$-$795.200& & 16\
619.9191 & $^{51}$V & $7/2$ & 382.367 & 2.268 & 14 && $9/2$ & 503.460& 3.300 & 19\
624.2828 & $^{51}$V & $1/2$ & 751.478 & 3.337 & 14 && $3/2$ & 594.690& $-$4.400 & 19\
624.3110 & $^{51}$V & $9/2$ & 406.851 & 14.324 & 14 && $9/2$ & 503.460& 3.300 & 19\
625.1823 & $^{51}$V & $7/2$ & 382.367 & 2.268 & 14 && $7/2$ & 514.350& $-$1.200 & 19\
625.6903 & $^{51}$V & $5/2$ & 373.518 & $-$5.459 & 14 && $5/2$ & 537.440& $-$4.000 & 19\
627.4653 & $^{51}$V & $3/2$ & 405.604 & $-$8.107 & 14 && $1/2$ & 939.940& 0.000 & 19\
628.5160 & $^{51}$V & $5/2$ & 373.518 & $-$5.459 & 14 && $3/2$ & 594.690& $-$4.400 & 19\
629.2824 & $^{51}$V & $7/2$ & 382.367 & 2.268 & 14 && $5/2$ & 537.440& $-$4.000 & 19\
653.1401 & $^{51}$V & $5/2$ & 112.835 & & 18 && $5/2$ & $-$89.800& 8.000 & 17\
& & & & & & & & & &\
\
371.8152 & $^{51}$V & 3 & 250.910 & & 20 && 4 & 178.223& & 20\
376.0222 & $^{51}$V & 4 & 171.400 & & 20 && 3 & 301.130& & 20\
386.6740 & $^{51}$V & 1 & $-$73.330 & & 20 && 2 & & &\
395.1960 & $^{51}$V & 2 & 0.000 & & 20 && 3 & 160.220& & 20\
399.7117 & $^{51}$V & 2 & 50.000 & & 21 && 3 & 200.000& & 21\
403.6777 & $^{51}$V & 2 & 0.000 & & 20 && 2 & 239.500& & 20\
& & & & & & & & & &\
\
\
452.98384& $^{50}$Cr & 6 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
452.98396& $^{52}$Cr & 6 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
452.98404& $^{53}$Cr & 6 & $−$112.000 & 8.300 & 22 && 5 & 0.000& 0.000 & N\
452.98412& $^{54}$Cr & 6 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
& & & & & & & & & &\
\
408.2945 & $^{55}$Mn & $3/2$ & 469.391 & $-$65.091 & 23 && $5/2$ & $-$26.981& & 24\
426.5928 & $^{55}$Mn & $3/2$ & 50.965 & & 24 && $3/2$ &$-$293.797& & 24\
445.3013 & $^{55}$Mn & $3/2$ & 50.965 & & 24 && $1/2$ & 1067.261& & 24\
445.7041 & $^{55}$Mn & $5/2$ & 467.410 & $-$73.460 & 25 && $3/2$ & 683.527& 224.844 & 26\
447.0142 & $^{55}$Mn & $3/2$ & 50.965 & & 24 && $3/2$ & 191.867& & 24\
449.8897 & $^{55}$Mn & $3/2$ & 50.965 & & 24 && $5/2$ & 92.936& & 24\
450.2223 & $^{55}$Mn & $5/2$ & $-$137.905 & & 24 && $7/2$ & 44.969& & 24\
467.1688 & $^{55}$Mn & $7/2$ & $-$161.888 & & 24 && $5/2$ & 284.803& & 24\
470.9710 & $^{55}$Mn & $7/2$ & $-$161.888 & & 24 && $7/2$ & 170.882& & 24\
473.9110 & $^{55}$Mn & $3/2$ & 50.965 & & 24 && $3/2$ & 668.537& & 24\
500.4891 & $^{55}$Mn & $5/2$ & $-$137.905 & & 24 && $7/2$ & 137.905& & 27\
525.5330 & $^{55}$Mn & $11/2$& 405.265 & & 28 && $9/2$ & 131.909& & 24\
538.8538 & $^{55}$Mn & $5/2$ & 89.938 & & 24 && $7/2$ & 44.969& & 24\
542.0368 & $^{55}$Mn & $7/2$ & 458.930 & 21.701 & 23 && $5/2$ &$-$549.000& & 23\
& & & & & & & & & &\
\
521.2688 & $^{59}$Co & $9/2$ & 810.039 & $-$59.958 & 29 && $9/2$ & 1076.855& 149.896 & 29\
528.0627 & $^{59}$Co & $9/2$ & 517.142 & 179.875 & 29 && $7/2$ & 846.914& 89.938 & 29\
530.1044 & $^{59}$Co & $5/2$ & 178.900 & $-$170.000 & 30 && $5/2$ & 464.678& & 29\
535.2041 & $^{59}$Co & $11/2$& 771.966 & 209.855 & 29 && $9/2$ & 1076.855& 149.896 & 29\
548.3353 & $^{59}$Co & $5/2$ & 178.900 & $-$170.000 & 30 && $7/2$ & 478.169& 149.896 & 29\
564.7233 & $^{59}$Co & $3/2$ & 332.000 & 101.000 & 30 && $5/2$ & 491.660& 0.000 & 29\
593.5390 & $^{59}$Co & $5/2$ & 1124.800 & 144.000 & 30 && $7/2$ & 478.169& 149.896 & 29\
608.2423 & $^{59}$Co & $9/2$ & 810.039 & $-$59.958 & 29 && $9/2$ & 401.722& & 29\
609.3141 & $^{59}$Co & $3/2$ & 317.780 & 119.917 & 29 && $3/2$ & 702.400&$-$15.000 & 31\
618.9005 & $^{59}$Co & $5/2$ & 178.900 & $-$170.000 & 30 && $5/2$ & 696.118& 29.979 & 29\
642.9913 & $^{59}$Co & $7/2$ & 839.400 & $-$97.000 & 30 && $5/2$ & 1046.276& & 29\
645.4995 & $^{59}$Co & $7/2$ & 751.500 & 31.000 & 30 && $9/2$ & 401.722& & 29\
741.7386 & $^{59}$Co & $3/2$ & 389.730 & & 29 && $5/2$ & 696.118& 29.979 & 29\
& & & & & & & & & &\
\
\
474.0134 & $^{61,62,64}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
474.0150 & $^{\phantom{61,62,}60}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
474.0166 & $^{\phantom{61,62,}58}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
481.1977 & $^{\phantom{61,62,}58}$Ni & 1 & 0.000 & 0.000 & N && 0 & 0.000& 0.000 & N\
481.1993 & $^{\phantom{61,62,}60}$Ni & 1 & 0.000 & 0.000 & N && 0 & 0.000& 0.000 & N\
481.1993 & $^{61,62,64}$Ni & 1 & 0.000 & 0.000 & N && 0 & 0.000& 0.000 & N\
497.6114 & $^{61,62,64}$Ni & 4 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
497.6125 & $^{\phantom{61,62,}60}$Ni & 4 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
497.6135 & $^{\phantom{61,62,}58}$Ni & 4 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
550.9994 & $^{61,62,64}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
551.0002 & $^{\phantom{61,62,}60}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
551.0009 & $^{\phantom{61,62,}58}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
574.9257 & $^{61,62,64}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
574.9280 & $^{\phantom{61,62,}60}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
574.9304 & $^{\phantom{61,62,}58}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
617.6777 & $^{61,62,64}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
617.6798 & $^{\phantom{61,62,}60}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
617.6820 & $^{\phantom{61,62,}58}$Ni & 4 & 0.000 & 0.000 & N && 5 & 0.000& 0.000 & N\
622.3949 & $^{61,62,64}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
622.3971 & $^{\phantom{61,62,}60}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
622.3991 & $^{\phantom{61,62,}58}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
637.8206 & $^{61,62,60}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
637.8233 & $^{\phantom{61,62,}60}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
637.8258 & $^{\phantom{61,62,}58}$Ni & 3 & 0.000 & 0.000 & N && 4 & 0.000& 0.000 & N\
& & & & & & & & & &\
**References:**\
1. @Ertmer76 [ABMR]
2. @Basar04 [theoretical calculations]
3. @Singh91
4. @Childs71
5. @Aboussaid96
6. @Mansour89
7. @Young88
8. statistically-weighted average of @Mansour89 and @Villemoes92
9. @Arnesen82
10. @Aydin90
11. @Gangrsky95
12. @Berrah92
13. @Nouri10
14. @Childs67b
15. @Palmeri95
16. @Childs79
17. @Lefebvre02
<!-- -->
18. @Johann81
19. @Cochrane98
20. @Armstrong11
21. estimated from solar profiles by trial and error with a single snapshot of an earlier version of the 3D model [@AspI]
22. @Jarosz07
23. @D79
24. @BW05c
25. @Handrich69
26. @Luc72
27. @Lefebvre03
28. @Johann81
29. @Pickering96a
30. @Guthlorien90
31. unpublished work of R. Wenzel, reproduced in @Guthlorien90
32. no HFS because $J=0$ or $I=0$
------------------------------------------------------------- -------------------- ------- ------- ------- --------
Species $E_{\rm ion}$ (eV)
3000K 5000K 8000K 12000K
[[Sc<span style="font-variant:small-caps;">i</span>]{} ]{} 6.562 9.65 11.95 21.54 49.08
[[Sc<span style="font-variant:small-caps;">ii</span>]{} ]{} 12.800 17.80 22.74 29.45 37.65
[[Ti<span style="font-variant:small-caps;">i</span>]{} ]{} 6.828 20.88 29.61 54.83 114.24
[[Ti<span style="font-variant:small-caps;">ii</span>]{} ]{} 13.580 44.03 55.45 72.24 95.29
[[V<span style="font-variant:small-caps;">i</span>]{} ]{} 6.746 34.67 47.50 79.09 152.62
[[V<span style="font-variant:small-caps;">ii</span>]{} ]{} 14.660 31.77 43.36 64.24 98.87
[[Cr<span style="font-variant:small-caps;">i</span>]{} ]{} 6.766 7.61 10.26 20.20 52.33
[[Cr<span style="font-variant:small-caps;">ii</span>]{} ]{} 16.500 6.03 7.09 12.15 27.16
[[Mn<span style="font-variant:small-caps;">i</span>]{} ]{} 7.434 5.99 6.34 9.95 24.10
[[Fe<span style="font-variant:small-caps;">i</span>]{} ]{} 7.902 22.01 27.78 43.05 81.05
[[Fe<span style="font-variant:small-caps;">ii</span>]{} ]{} 16.190 34.20 43.21 56.33 78.50
[[Co<span style="font-variant:small-caps;">i</span>]{} ]{} 7.881 24.44 33.43 48.16 76.50
[[Ni<span style="font-variant:small-caps;">i</span>]{} ]{} 7.640 26.34 30.76 36.35 48.84
[[Ni<span style="font-variant:small-caps;">ii</span>]{} ]{} 18.170 8.30 10.83 15.73 23.21
------------------------------------------------------------- -------------------- ------- ------- ------- --------
: Our adopted ionisation energies $E_{\rm ion}$ and partition functions $U(T)$ for relevant ionisation stages of the iron group elements.[]{data-label="table:partition"}
[^1]: MAFAGS-OS models are successors to MAFAGS-ODF models by @Fuhrmann97, relying on opacity sampling instead of ODFs.
[^2]: An NLTE abundance correction is defined as a difference in abundance required to equalise NLTE and LTE equivalent widths.
[^3]: Even with the 3D model, weaker lines are to be preferred because they lie on the linear section of the curve of growth and are less sensitive to errors in the treatment of broadening or the atmospheric temperature structure, which is less certain in the higher parts of the atmosphere where stronger lines are formed.
[^4]: Interestingly, the resulting hyperfine constants for this line are in reasonable agreement with the experimental measurements of @Wood14V2, which appeared only at the end of the refereeing stage of this paper.
|
---
abstract: 'The motion of a pair of counter-rotating point vortices placed in a uniform flow around a circular cylinder forms a rich nonlinear system that is often used to model vortex shedding. The phase portrait of the Hamiltonian governing the dynamics of a vortex pair that moves symmetrically with respect to the centerline—a case that can be realized experimentally by placing a splitter plate in the center plane—is presented. The analysis provides new insights and reveals novel dynamical features of the system, such as a nilpotent saddle point at infinity whose homoclinic orbits define the region of nonlinear stability of the so-called Föppl equilibrium. It is pointed out that a vortex pair properly placed downstream can overcome the cylinder and move off to infinity upstream. In addition, the nonlinear dynamics resulting from antisymmetric perturbations of the Föppl equilibrium is studied and its relevance to vortex shedding discussed.'
author:
- 'G. L. Vasconcelos'
- 'M. N. Moura'
- 'A. M. J. Schakel'
date: 'November 18, 2011'
title: Vortex motion around a circular cylinder
---
Introduction {#intro}
============
Flow around a circular cylinder is a classical topic in hydrodynamics that is of fundamental importance to many scientific fields with numerous applications [@mmz; @sf]. Of particular interest is the formation, at moderate Reynolds numbers, of vortex eddies behind a circular cylinder, which then go unstable at higher Reynolds numbers and evolve into a Karman vortex street [@saffman; @vvd]. Since an analytic treatment of the problem in terms of the Navier-Stokes equation is difficult and the computational cost of direct numerical simulation very high, a particularly useful approach to study the basic features of vortex shedding from bluff bodies is to consider the dynamics of point vortices in an inviscid fluid.
A point-vortex model for the formation of two recirculating, symmetric eddies in the wake of a circular cylinder was first introduced by Föppl [@foeppl]. He obtained stationary solutions for a pair of vortices behind the cylinder in a uniform stream and found that the centers of the eddies observed in the experiments lie on the locus of such equilibria—now called the Föppl curve. In addition, Föppl found that these equlibria, although stable against perturbations that are symmetric with respect to the centerline, were unstable against nonsymmetric perturbations. This instability is believed to constitute the origin of the vortex shedding process that leads to the formation of the Karman vortex street [@tang]. It was later found out independently by several authors [@smith; @soibelman1; @cai] that Föppl’s stability analysis for symmetric perturbations was in error in that the stationary solution behind the cylinder is not exponentially but only marginally stable. Physically, marginal stability implies, for instance, that if a splitter plate is placed behind the cylinder in the center plane of the wake to suppress vortex shedding [@roshko; @roshko2; @cai2003], oscillating forces on the cylinder may still arise owing to the cyclic motion of the vortices around their equilibrium position [@laat].
Despite many contributions to the problem, it is fair to say that the nonlinear dynamics of the Föppl system is not yet fully understood. In particular, a more complete picture of vortex-pair dynamics in the presence of symmetric perturbations is lacking, and several aspects of the nonlinear dynamics for nonsymmetric perturbations remain unclear. To address these two issues is the main motivation of the present paper. It should be emphasized at the outset that a better understanding of the dynamical structure underlying the Föppl model is of interest not only because of its practical relevance for vortex shedding, but also in its own theoretical right from the viewpoint of nonlinear dynamics.
The Föppl model has inspired a number of studies on several related problems, such as the modeling of vortex wake behind slender bodies in terms of multiple pairs of point vortices [@seath; @weihs; @miller; @protas3], the Hamiltonian structure of a circular cylinder interacting dynamically with point vortices [@marsden; @shashi2006; @borisov2007], the control of vortex shedding [@tang2000; @protas1; @protas2], and the stability of symmetric and asymmetric vortex pairs over three-dimensional slender conical bodies [@cai2003; @cai2005; @bridges]. The related problem of desingularization of the Föppl pair in terms of vortex patches of finite area was also studied [@elcrat1; @elcrat2]. A recent review on vortex motion past solid bodies with additional references to the Föppl model and related problems can be found in Ref. \[28\].
After formulating the problem of a pair of counter-rotating point vortices placed in a uniform stream around a circular cylinder in Sec. \[sec:2\], we begin our analysis of the Föppl system in Sec. \[sec:3\] by studying its Hamiltonian dynamics restricted to the invariant subspace where the vortices move symmetrically with respect to the centerline. A phase portrait of the system is presented that fully characterizes the dynamics within this symmetric subspace. In particular, we point out that in addition to the two previously known sets of equilibira, namely, the Föppl equilibrium and the equilibrium on the axis bisecting the cylinder perpendicularly to the uniform flow, the system possesses a hitherto unnoticed nilpotent saddle at infinity. We show furthermore that the homoclinic orbits associated with this nilpotent saddle delimit the region of closed orbits around the Föppl equilibrium. We proceed in Sec. \[sec:4\] to study the linear and nonlinear dynamics resulting from antisymmetric perturbations of the Föppl equilibrium. In the linear regime, a mistake that went undetected in Föppl’s expressions [@foeppl] for the corresponding eigenvalues is now corrected. As for the nonlinear dynamics, the unstable manifold associated with the Föppl equilibrium is computed numerically and its close relation to the vortex shedding instability is pointed out. The linear stability analysis of the equilibria on the normal line with respect to symmetric and antisymmetric perturbations is also presented—for the first time, it seems—and the respective nonlinear dynamics is investigated numerically. A discussion of the physical relevance of our findings and our main conclusions are presented in Sec. \[sec:discuss\].
Problem Formulation {#sec:2}
===================
![A pair of vortices behind a circular cylinder in a uniform stream.[]{data-label="fig:1"}](fig1.eps){width="60.00000%"}
We consider the motion of a pair of point vortices of same strength and opposite polarities around a circular cylinder of radius $a$ and in the presence of a uniform stream of velocity $U$, as illustrated in Fig. \[fig:1\]. It is convenient to work in the complex $z$-plane, where $z=x+iy$, and place the center of the cylinder at the origin. The upper and lower vortices are located at positions $z_1=x_1 + iy_1$ and $z_2=x_2 + iy_2$, respectively. The complex potential $w(z)=\phi(x,y)+i\psi(x,y)$, with $\phi$ being the velocity potential and $\psi$ the stream function, is given by [@milne] $$w(z)=U \left(z+\frac{a^2}{z}\right)+\frac{\Gamma }{2\pi i}\ln
\frac{z-z_1}{z- {a^2}/{\bar{z}_1}}-\frac{\Gamma }{2\pi i}\ln \frac{z- z_2}{z-{a^2}/{\bar{z}_2}} ,
\label{eq:1}$$ where $\Gamma$ is the circulation of the vortex at $z_{1}$ and bar denotes complex conjugation. In Eq. (\[eq:1\]), the first two terms represent the incoming flow and its image (a doublet at the origin) with respect to the cylinder, the third term gives the contributions to the complex potential from the upper vortex and its image, and similarly the last term contains the contributions from the lower vortex and its image. As can be inferred from Fig. \[fig:1\], a necessary condition for a steady configuration to exist is that the upper (lower) vortex be of negative (positive) circulation, hence only the case $\Gamma<0$ is of interest to us here.
In dimensionless variables $$z'=\frac{z}{a}, \quad t'=\frac{U}{a}t, \quad w'=\frac{w}{Ua}, \quad \kappa=-\frac{\Gamma}{2\pi Ua}>0,
\label{eq:non}$$ the complex potential (\[eq:1\]) becomes $$w(z)= z+\frac{1}{z}+ i\kappa \ln
\frac{(z-{z_1})\left(1-\bar{z}_{2} z\right)}{(z-
z_2)\left(1-\bar{z}_1z\right) } ,
\label{eq:12}$$ where the prime notation has been dropped. According to standard theory of point vortices in an inviscid fluid, any given vortex moves with the velocity of the flow computed at the position of that vortex, excluding its own contribution to the flow. It then follows from Eq. (\[eq:12\]) that the velocity ${\bf u}_1=(u_1, v_1)$ of the vortex located at $z_1$ is given by $$u_{1}-iv_{1}=1-\frac{1}{z_{1}^2}- i\kappa \left(\frac{1}{z_{1}-z_2} - \frac{\bar{z}_1}{1-z_1\bar{z}_1}+ \frac{\bar{z}_2}{1-z_{1}\bar{z}_2}\right) ,
\label{eq:22}$$ or more explicitly
\[eq:24\] $$\begin{aligned}
u_1=1 -\frac{x_1^2-y_1^2}{r_1^4}- \kappa & \left(\frac{y_1-y_2}{r_1^2+r_2^2-2(x_1x_2+y_1y_2)} + \frac{y_1}{r_1^2-1}- \frac{y_1r_2^2-y_2}{1+r_1^2r_2^2-2(x_1x_2+y_1y_2)} \right),
\label{eq:24a}
\\
v_1= -2\frac{x_1y_1}{r_1^4} + \kappa & \left(\frac{x_1-x_2}{r_1^2+r_2^2-2(x_1x_2+y_1y_2)} + \frac{x_1}{r_1^2-1} -\frac{x_1r_2^2-x_2}{1+r_1^2r_2^2-2(x_1x_2+y_1y_2)}\right) ,
\label{eq:24b}\end{aligned}$$
where $r_i^2=x_i^2+y_i^2$, $i=1,2$. The velocity ${\bf u}_2=(u_2, v_2)$ of the second vortex is obtained by simply interchanging the indexes $1\leftrightarrow 2$ in Eq. (\[eq:24\]) and letting $\kappa\to-\kappa$.
Dynamics on the Symmetric Subspace {#sec:3}
==================================
It is not difficult to see from Eq. (\[eq:24\]) that if the vortices are initially placed at positions symmetrically located with respect to the centerline, i.e., $z_2(0)= \overline{z}_{1}(0)$, then this symmetry is preserved for all later times, i.e., $z_2(t) =\overline{z}_1(t)$ for $t>0$. In this section, we study the dynamics within this invariant symmetric subspace, where the motion of the lower vortex is simply the mirror image of that of the upper vortex with respect to the centerline. Symmetry can be enforced experimentally by placing a splitter plate behind the cylinder in the center plane of the wake [@foeppl; @roshko].
With $x_2=x_1$ and $y_2=-y_1$, Eq. (\[eq:24\]) reduces to
\[eq:4\] $$u=1-\frac{x^2-y^2}{r^4}+ \kappa y \left[\frac{r^2+1}{(r^2-1)^2+4y^2} -\frac{1}{r^2-1}
- \frac{1}{2y^2}\right],
\label{eq:4a}$$ $$v=-2\frac{xy}{r^4} - \kappa x \left[\frac{r^2-1}{(r^2-1)^2+4y^2}-\frac{1}{r^2-1}\right].
\label{eq:4b}$$
Here, the subscripts have been dropped with the understanding that in the remainder of the section we restrict our attention to the upper vortex.
Hamiltonian dynamics and phase portrait
---------------------------------------
As is well known, the equations of motion for point vortices in a two-dimensional inviscid flow, first derived by Kirchhoff, can be formulated as a Hamiltonian system [@saffman; @vvd]. The dynamics of point vortices in the presence of closed, rigid boundaries was shown by Lin [@lin1941] to be also Hamiltonian with the same canonical sympletic structure as in the absence of boundaries. For a vortex pair placed in a uniform stream around a circular cylinder, the phase space is four-dimensional and has a two-dimensional (2D) invariant subspace corresponding to symmetric orbits. The Hamiltonian restricted to the 2D symmetric subspace is given by [@zannetti] $$H(x,y)=y\left(1-\frac{1}{r^2}\right)-\frac{\kappa}{2}\ln\frac{y(r^2-1)}{\sqrt{(r^2-1)^2+4y^2}}.
\label{eq:H}$$ The corresponding dynamical equations $$\label{eq:HJ}
\dot{x}=\frac{\partial H}{\partial y}, \quad \dot{y}=-\frac{\partial H}{\partial x},$$ where dot denotes time derivative, yield Eq. (\[eq:4\]) upon identifying $(u, v)$ with $(\dot{x},
\dot{y})$.
A phase portrait of this Hamiltonian system for $\kappa=45/32$ is presented in Fig. \[fig:phase\], where the curves shown are (unevenly spaced) level sets of the Hamiltonian (\[eq:H\]). \[For convenience, these curves were obtained from a direct numerical integration of Eq. (\[eq:4\]).\] A detailed description of the main features of this phase portrait will be given below, starting with an analysis of the various equilibrium points and their stability. The related problem of the symmetric “moving Föppl system,” where the cylinder advances through the fluid followed by the vortex pair, was recently considered by Shashikanth [*et al.*]{} [@marsden], but there the phase portrait [@shashi2006] is quite different from the one shown in Fig. \[fig:phase\], because of the additional degrees of freedom related to the velocity of the moving cylinder.
![Phase portrait for the symmetric Föppl system with $\kappa=45/32$. The isolated black dots are the Föppl equilibria. The dashed curves are the stable and unstable branches of the separatrix associated with the equilibrium point on the normal line, and the thick solid lines are the homoclinic loops of the equilibrium point at infinity; see text.[]{data-label="fig:phase"}](fig2.eps){width="60.00000%"}
Equlibrium points
-----------------
The equilibrium positions for the vortex are obtained by setting $u=v=0$ in Eq. (\[eq:4\]). Three types of equilibrium points can be identified.
### Föppl equilibria
The locus of possible equilibrium positions $(x_0,y_0)$ for the upper vortex found by Föppl [@foeppl] is the curve $$r_0^2-1= 2 r_0 y_0,
\label{eq:pair}$$ with corresponding strength $$\kappa =\frac{(r_0^2+1)(r_0^2-1)^2}{r_0^5}.
\label{eq:7}$$ Along the Föppl curve (\[eq:pair\]), the vortex strength increases with distance from the center of the cylinder and diverges linearly for $r_0 \to \infty$. For the equilibrium point on the edge of the cylinder ($r_0 \to 1$), the strength vanishes. Notice that Eq. (\[eq:pair\]) yields two branches of solution: one in which the vortex pair is behind the cylinder ($x_0>0$) and the other where the vortex pair is in front of the cylinder ($x_0<0$). The former case models the formation of vortex eddies behind a cylinder in a uniform stream and was the primary motivation of Föppl’s original study [@foeppl]. The latter case has attracted far less attention because it is not usually observed in experiments. We note, however, that recirculating eddies are observed in front of a circular cylinder near a plane boundary when the gap between the cylinder and the plane is sufficiently small [@lin]. In this context, the Föppl equlibrium upstream of the cylinder may eventually be relevant for flows around a half-cylinder placed on a plane wall (or for the closely related situation where a splitter plate is attached to the front of the cylinder), although we are unaware of specific experiments in this setting.
### Equilibria on the normal line
This corresponds to the upper vortex being located on the line bisecting the cylinder perpendicularly to the incoming flow [@weihs], that is, $$x=0, \qquad y= b, \qquad b>1,
\label{eq:b}$$ with strength $$\kappa =\frac{2(b^2-1)(b^2+1)^2}{b(b^4+4b^2-1)}.
\label{eq:kappa}$$ As in the Föppl solution, the strength tends to zero when the edge of the cylinder is reached ($b \to 1$) and diverges linearly with distance from the center of the cylinder. At large distances, the vortex strength for this equilibrium is about twice that of a Föppl pair located at the same distance from the origin.
### Equilibrium at infinity
Equation (\[eq:4\]) also yields equilibrium points at the positions $$x=\pm\infty, \qquad y_\infty=\frac{\kappa}{2} .
\label{eq:yc}$$ To the best of our knowledge, the existence of this additional equilibrium point at infinity was not noted before. Its physical origin, however, can be easily understood, as it corresponds to the equilibrium configuration for a vortex pair placed in a uniform stream (without the cylinder). At points infinitely far from the cylinder, the flow induced by the image system (inside the cylinder) becomes negligible and hence a stationary configuration is possible if the vortices with given circulation $\pm \kappa$ are placed at the appropriate distance ($=\kappa$) from each other.
Stability analysis {#sec:sasym}
------------------
The linear stability analysis of the equilibria described above is presented next, together with a discussion of the *nonlinear* stability of the Föppl equilibrium.
### Föppl equilibria
Consider a perturbation of the Föppl equilibrium (\[eq:pair\]) parameterized as: $z=z_0+{\Delta z}$, where ${\Delta z} =
\xi+i\eta$, with $\xi$ and $\eta$ being infinitesimal (real) quantities. Linearization of Eq. (\[eq:4\]) then yields the following dynamical system $$\left(\begin{array}{c} \dot{\xi}\cr \dot{\eta}\end{array}\right)=
A \left(\begin{array}{c}{\xi}\cr{\eta}\end{array}\right),$$ where the matrix $A$ reads $$A_{11} =-A_{22}=-\frac{x_0(r_0^4 - 3 r_0^2+2)}{r_0^8},$$ $$A_{12} =\frac{
4r_0^8+5r_0^6+2r_0^4-5r_0^2+2}{2r_0^9},$$ $$A_{21} = -\frac{2x_0^2(r_0^4+r_0^2+2)}{r_0^7(r_0^2+1)}.$$ Its eigenvalues $\lambda$ are given by $$\lambda^2 =- \frac{3 r_0^6+5 r_0^4+13 r_0^2-5}{r_0^{10}}<0,
\label{eq:ls}$$ for $r_0>1$. The eigenvalues are thus purely imaginary, and not a complex pair with negative real part as found by Föppl [@foeppl]. In other words, the Föppl equilibrium is a center and not a stable focus. Our equation (\[eq:ls\]) agrees with the expression for the eigenvalues of the symmetric modes obtained in Ref. \[7\] from the linearization of the full 4D dynamical system. As can be seen from Fig. \[fig:phase\], the Föppl solution is in fact a nonlinearly stable center, meaning that when the vortex is displaced from its equilibrium position by a small (but finite) amount, it executes a periodic motion around that point, corresponding to the closed orbits in the figure. This periodic motion around the Föppl equilibrium has been observed in numerical simulations of the model carried out by de Laat and Coene [@laat]. Note that since the eigenvalues given in Eq. (\[eq:ls\]) do not depend explicitly on the coordinate $x_0$, it follows that the two Föppl equilibria, downstream and upstream of the cylinder, have identical stability properties, as is evident from Fig. \[fig:phase\]. This means, in particular, that if vorticity can be generated upstream of the cylinder then stationary recirculating eddies could form in front of the cylinder—a situation observed, for instance, in flows around a cylinder placed above a plane wall [@lin].
### Equilibria on the normal line {#sec:C2}
Linearization of Eq. (\[eq:4\]) around the equilibrium point $z=ib$ yields for the matrix $A$: $$A_{11} =A_{22}=0,$$ $$A_{12} =\frac{b^8+10 b^6-8 b^4+14 b^2-1}{b^3(b^2-1)(b^4+ 4 b^2-1 )},$$ $$A_{21} = \frac{2(b^2-1)(3b^2-1)}{b^3(b^4+ 4 b^2-1 )}.$$ The eigenvalues $\lambda$ of this matrix are determined by $$\lambda^2=\frac{2 \left(3 b^2-1\right) \left(b^8+10 b^6-8 b^4+14 b^2-1\right)}{b^6 (b^4+ 4 b^2-1 )^2}>0,$$ which yields a pair of real eigenvalues, $\lambda_\pm=\pm\sqrt{\lambda^2}$. The equilibrium point on the normal line is therefore a saddle, having a stable and unstable direction, as is also evident from the phase portrait shown in Fig. \[fig:phase\]. The eigenvectors ${\bf w}_\pm$ associated with the eigenvalues $\lambda_\pm$, respectively, read $${\bf w}_\pm= \left(\begin{array}{c} \pm \sqrt{ A_{12}/A_{21}} \cr 1\end{array}\right).
\label{eq:theta}$$ Although it was known from numerical simulations [@laat] that the equilibrium point on the normal line is unstable (against generic symmetric perturbations), it seems that an explicit linear stability analysis for this case was not carried out before, perhaps because these equilibria were not considered physically relevant since they are not observed in experiment [@foeppl]. However, when the full nonlinear dynamics is considered, the stable and unstable eigendirections ${\bf w}_\pm$ give origin to the respective stable and unstable separatrices, indicated by the dashed curves in Fig. \[fig:phase\]. In this sense, the existence of an equilibrium point on the normal line is dynamically felt by a vortex even if it is placed far from this “unphysical” equilibrium.
### Equilibrium at infinity
![Trajectories near a nilpotent saddle.[]{data-label="fig:saddle"}](fig3.eps){width="60.00000%"}
The matrix $A$ of the linearized system around the equilibrium point at infinity is given by $$A= -\frac{2}{\kappa} \left(\begin{array}{cc} 0 & 1 \cr 0 & 0\end{array}\right),$$ which is nilpotent and has two zero eigenvalues. To study the stability of this equilibrium point, one needs to examine the nonlinear contributions. To this end, we note that for $|x| \to \infty$ and $y \approx y_\infty$, Eq. (\[eq:4b\]) assumes the form $$\label{eq:doty}
\dot{y} = -\frac{\kappa}{x^3} .$$ It then follows from a theorem in ordinary differential equations [@perko] that, in view of the cubic term in Eq. (\[eq:doty\]), the equilibrium point is a *degenerate* or *nilpotent* saddle [@abraham], for which the two eigenvectors are the same. The behavior of trajectories in the neighborhood of a generic nilpotent saddle is illustrated in Fig. \[fig:saddle\]. The behavior near the nilpotent saddle at $x=\pm\infty$ and $y=y_\infty$ can be described as follows. A vortex placed very far downstream and below (above) the line $y=y_\infty$ will move away from (towards) the equilibrium point at $x=\infty$. Similarly, a vortex placed very far upstream will move away from (towards) the equilibrium point at $x=-\infty$ if $y>y_\infty$ ($y<y_\infty$).
The stable and unstable separatrices associated with the nilpotent saddle at infinity form two homoclinic loops [@abraham], called nilpotent saddle loops, which are indicated in Fig. \[fig:phase\] by thick solid lines and correspond to the level curves passing through this equilibrium point: $$H(x,y)=H(\pm\infty,y_\infty)=\frac{\kappa}{2} \left(1- \ln \frac{\kappa}{2} \right).$$ The nilpotent saddle loops encircle the Föppl equilibria and define their region of nonlinear stability, in the sense that vortex trajectories are closed for initial positions inside the loops and unbounded otherwise. In this way, the nilpotent saddle at infinity, which went unnoticed until now, allows us to fully characterize the nonlinear stability of the Föppl equilibrium.
For unbounded orbits, the long-time asymptotic behavior depends on the location of the vortex initial position with respect to the separatrices associated with the equilibrium point on the normal line. A vortex placed downstream of the cylinder between the nilpotent saddle loop and the separatrices of the equilibrium point on normal line will eventually be convected away by the free stream; see Fig. \[fig:phase\]. In particular, if the vortex starts very far behind the cylinder at a position that is below the nilpotent saddle loop and above the stable separatrix, it first moves towards the cylinder, turns around the Föppl equilibrium, and is then “reflected” back to infinity. Even more surprising trajectories arise if the vortex is placed downstream below the stable separatrix, for it will be close enough to its image below the centerline to be able to overcome the cylinder and move off to infinity upstream. (A related phenomenon occurs in the inviscid coupled motion of a cylinder initially at rest and a vortex pair starting at infinity with no imposed background flow [@eames]. When the cylinder is less dense than the fluid, it is found that if the vortices are released sufficiently above the centerline they reverse relative to the moving cylinder; otherwise, they move over and past the cylinder.) Unbounded trajectories for the Föppl system also result for initial positions upstream of the cylinder: i) if placed above the stable separatrix, the vortex moves downstream to infinity; and ii) if placed between the stable separatrix and the nilpotent saddle loop, the vortex goes around the Föppl equlibrium in front of the cylinder and returns to infinity upstream; see Fig. \[fig:phase\]. It is again the hitherto unnoticed nilpotent saddle at infinity, together with the precise nature of the equilibrium point on the normal line, that allows us to go beyond linear stability analysis and capture the full phase portrait in the symmetric subspace.
We stress that closed orbits exist only when the flow is symmetric. Nonsymmetric perturbations inevitably cause the vortex pair to move off to infinity, as we demonstrate next.
Nonsymmetric Dynamics {#sec:4}
=====================
In this section, the effect of antisymmetric perturbations on the equilibria of the Föppl system is studied. We begin by observing that the dynamics of two counter-rotating point vortices possesses a *conjugation symmetry*. To describe this symmetry, let $z_1(t;z_{1,0},z_{2,0})$ and $z_2(t;z_{1,0},z_{2,0})$ denote the trajectories of the upper and lower vortices, respectively, with initial positions $z_{1,0}$ and $z_{2,0}$. For the dynamical system defined by Eq. (\[eq:24\]) and the corresponding equation for the second vortex, one can verify that the following relations hold
\[eq:cc\] $$z_1(t;\overline{z}_{2,0},\overline{z}_{1,0})=\overline{z_2(t;z_{1,0},z_{2,0})} ,$$ $$z_2(t;\overline{z}_{2,0},\overline{z}_{1,0})=\overline{z_1(t;z_{1,0},z_{2,0})} .$$
In other words, for any given pair of initial positions, $z_{1,0}$ and $z_{2,0}$, there exists a “conjugate pair” of initial positions, $\overline{z}_{2,0}$ and $\overline{z}_{1,0}$, such that the vortex trajectories of the first pair are the complex conjugate of those of the second pair.
Any perturbation of a vortex-pair equilibrium can be written as the superposition of a symmetric perturbation and an antisymmetric one. To be precise, antisymmetric perturbations are of the form $$z_1=z_0+\Delta z, \qquad z_2=\overline{z}_0-\overline{\Delta z},
\label{eq:daz}$$ where $z_0$ denotes a generic equilibrium point and ${\Delta z} = \xi+i\eta$. Since the antisymmetric subspace of the full 4D phase space is invariant under linear dynamics, we can focus on the upper vortex in carrying out our linear stability analysis.
Föppl equilibria {#sec:unst}
----------------
Linearization of Eq. (\[eq:24\]) around the Föppl equilibrium (\[eq:pair\]) with respect to antisymmetric perturbations (\[eq:daz\]) yields $$\left(\begin{array}{c} \dot{\xi}\cr \dot{\eta}\end{array}\right)=
B \left(\begin{array}{c}{\xi}\cr{\eta}\end{array}\right) ,$$ where the matrix $B$ is given by $$B_{11}=-B_{22}=\frac{x_0\left(r_0^4+3 r_0^2-2\right)}{r_0^8},
\label{eq:B11}$$ $$B_{12} = \frac{3 r_0^6-5 r_0^2+2}{2 r_0^9},$$ $$B_{21} = \frac{4 r_0^8+3 r_0^6 -4 r_0^4-5 r_0^2+2}{2 r_0^9}.
\label{eq:B21}$$ This matrix has a pair of real eigenvalues, $\lambda_{\pm}=\pm\sqrt{\lambda^2}$, where $$\lambda^2 =\frac{3 r_0^6+3 r_0^4-3 r_0^2 + 1}{r_0^{10}}.
\label{eq:las}$$ The Föppl equilibrium is therefore a saddle with respect to antisymmetric perturbations, while it is a center with respect to symmetric perturbations, as seen earlier. That is, the Föppl equilibrium is a [*saddle-center*]{} of the full 4D dynamical system [@hs]. We note in passing that, although Föppl obtained a pair of real eigenvalues for the case of antisymmetric perturbations, his original formulae for the eigenvalues are in error [@footnote]. Our expression (\[eq:las\]) is in agreement with the eigenvalues of the skew-symmetric modes obtained by Smith [@smith] from the linearization of the full dynamical system. The eigenvectors ${\bf
w}_{\pm}$ associated with the eigenvalues $\lambda_\pm$ are readily computed, with the result $${\bf w}_\pm = \left(\begin{array}{c} (\lambda_\pm+B_{11})/B_{21} \cr 1\end{array}\right) .
\label{eq:waf}$$
![Vortex trajectories for antisymmetric perturbations of the Föppl equilibrium for $\kappa=45/32$, in which case $x_0=\sqrt{55}/4$ and $y_0=\pm3/4$ (black dots). The solid and dashed curves are the trajectories starting along the unstable directions ${\bf w}_+$ and $-{\bf w}_+$, respectively, while the short straight lines indicate the axes defined by the stable direction ${\bf w}_-$. The dotted lines represent the loci of the Föppl equilibria.[]{data-label="fig:anti"}](fig4.eps){width="60.00000%"}
In Fig. \[fig:anti\], we show in solid curves the pair of vortex trajectories obtained by slightly displacing the vortices from their equilibrium positions in the directions defined by the unstable eigenvector ${\bf w_+}$, while the trajectories obtained by slightly displacing the vortices in the opposite directions are shown in dashed curves. The latter pair of trajectories is the complex conjugate of the former by conjugation symmetry. Note that for the first pair of trajectories, the lower vortex initially moves towards the centerline and upstream, while the upper vortex moves away from the centerline and downstream. At later times, the vortex pair moves off to infinity with the lower vortex trailing behind the upper vortex. For the second pair of trajectories, the upper and lower vortices switch role; see Fig. \[fig:anti\]. In the flow of a real fluid past a cylinder, the two basic instabilities associated with displacements along the unstable directions $\pm{\bf w_+}$ happen alternately and constitute the origin of vortex shedding that leads to the formation of the Karman vortex street [@tang]. In like manner, the suppression of vortex shedding by placing a splitter plate behind the cylinder [@roshko; @roshko2] is consistent with the fact that the Föppl equlibrium is nonlinearly stable with respect to symmetric perturbations; see Sec. \[sec:discuss\] for further discussions on vortex shedding and its suppression by a splitter plate.
For small, generic antisymmetric perturbations, the vortices move along trajectories that follow closely the ones depicted in Fig. \[fig:anti\]. Whether a vortex pair eventually moves up or down is determined by the initial position of the upper vortex relative to the stable direction ${\bf w}_-$, which is indicated in Fig. \[fig:anti\] by the short straight line passing through the Föppl equilibrium. If the initial position of the upper vortex is to the right (left) of the stable direction, then the vortex pair asymptotically moves upwards (downwards). This explains the behavior seen in the numerical simulations reported in Ref. \[25\], where nearby initial positions around the Föppl equilibrium were found to lead to close-by trajectories.
Since any degree of antisymmetry in the initial perturbation causes the vortex pair to move off to infinity, the Föppl equilibrium is unstable under generic perturbations. As an example, Fig. \[fig:loop\] shows vortex trajectories obtained by displacing the Föppl pair (at $r_0=2$) by the amounts $\Delta z_1=\Delta z_2=-0.25+i0.005$. During the linear stage, the trajectories are a superposition of a symmetric orbit and a growing mode associated with the antisymmetric component of the perturbation, which ultimately leads to asymptotic trajectories with the vortices moving parallel to each other.
![Trajectories resulting from a generic perturbation ${\Delta z}_1=\Delta z_2=-0.25 + i0.005$ of the Föppl pair at $r_0=2$ (black dots).[]{data-label="fig:loop"}](fig5.eps){width="60.00000%"}
Equilibria on the normal line {#sec:4b}
-----------------------------
For antisymmetric perturbations of the equilibrium (\[eq:b\]) on the normal line, the matrix $B$ assumes the form $$B_{11}=B_{22}=0,$$ $$B_{12}=\frac{2 \left(3 b^6+b^4+5 b^2-1\right)}{b^3(b^2-1)\left(b^4+4b^2-1\right)},$$ $$B_{21} = \frac{b^2-1}{b^3},$$ with eigenvalues $\lambda$ given by $$\lambda^2 =\frac{2 \left(3 b^6+b^4+5 b^2-1\right)}{b^6 \left(b^4+ 4 b^2-1 \right)}>0.$$ This yields a pair of real eigenvalues, $\lambda_{\pm}=\pm\sqrt{\lambda^2}$, with respective eigenvectors: $${\bf w}_\pm = \left(\begin{array}{c} \pm \sqrt{B_{12}/B_{21}} \cr 1\end{array}\right) .
\label{eq:theta}$$
![Vortex trajectories (solid curves) associated with the unstable direction ${\bf w}_+$ of the equilibrium at $z=\pm2i$ (black dots). The dashed curves are trajectories resulting from the antisymmetric perturbation $\Delta z= 0.16$.[]{data-label="fig:normal"}](fig6.eps){width="60.00000%"}
In Fig. \[fig:normal\], we show the vortex trajectories (solid curves) obtained by slightly displacing the vortices from their equilibrium position along the unstable direction ${\bf w}_+$ for $b=2$. The initial motion here is somewhat similar to what is seen for a Föppl pair, in the sense that one vortex moves upstream towards the centerline and the other moves downstream away from the centerline. The main difference is that for later times, the vortices now end up moving upstream. The long-time dynamics in this case is also more sensitive on the initial conditions: for somewhat larger perturbations, the vortices are eventually carried away by the free stream. An example where this happens is indicated by the dashed curves in Fig. \[fig:normal\], which represent the vortex trajectories for the antisymmetric perturbation $\Delta z=0.16$.
As already argued in Sec. \[sec:C2\], although the equilibrium point on the normal line is not directly observed in experiments, it is important to know its instability properties under both symmetric and antisymmetric perturbations. This knowledge contributes to a better understanding not only of the full nonlinear dynamics of the Föppl system but also of more general flows, such as the case of stationary vortex patches above and below the cylinder in a uniform stream, where similar unstable modes are observed [@elcrat2].
Discussion and Conclusions {#sec:discuss}
==========================
In this paper, we have investigated a two-dimensional vortex model for the formation of recirculating eddies behind a fixed cylinder placed on a uniform stream. The model, which was first introduced by Föppl [@foeppl] almost a century ago, has two main simplifying assumptions: i) the fluid is treated as inviscid and hence the flow is potential, and ii) the size of the vortex core is neglected and so the vortices are considered to be point-like. In spite of these simplifications, the model is known to be in qualitative agreement with real flows past a cylinder, as was already pointed out by Föppl in his original paper. Several novel features of the Föppl model have been obtained in the present work, which help one to better understand the basic dynamics of vortex shedding behind a cylinder.
In real flows, governed by the Navier-Stokes equations, stationary vortices behind a cylinder are formed at moderate Reynolds number ($Re < 50$). As the Reynolds number increases past $Re\approx50$, the configuration loses its symmetry and becomes unstable. New vortices then start to form alternately on both sides of the cylinder, while the vortices further downstream break away and develop into a Karman vortex street, as described by Föppl [@foeppl]. It has been argued by Roshko [@roshko] that “possibly the breaking away should be regarded as primary, resulting in asymmetry.” The analysis presented in Sec. \[sec:unst\] makes it clear that the reverse scenario is more plausible: the asymmetrical disturbances induce the instability of the vortex pair which then breaks away from the cylinder. As vorticity is continuously generated from the separated boundary layer on both sides of the cylinder, new vortices are formed and alternately shed into the far wake of the cylinder according to the unstable modes shown in Fig. \[fig:anti\]. Direct numerical simulations (DNS) of two-dimensional flows past a cylinder performed by Tang and Aubry [@tang] have confirmed that the mechanism for the instability of the symmetric eddies in real flows is qualitatively described by the instability of the point-vortex model.
It is experimentally observed [@roshko; @roshko2] that vortex shedding is suppressed if a splitter plate is installed behind the cylinder in the center plane of the wake. The presence of the splitter plate tends to enforce symmetry of the flow with respect to the centerline, thus effectively reducing the appearance of antisymmetric disturbances behind the cylinder. The suppression of vortex shedding in this case is thus entirely consistent with the fact that the Föppl equilibria of the vortex-point model is nonlinearly stable against symmetric perturbations and that vortex shedding is induced by unstable antisymmetric modes, as discussed above. This scenario has been confirmed by DNS of flows past a cylinder with symmetry imposed along the centerline recently performed by Kumar [*et al.*]{} [@kumar]. The problem of stationary configurations for vortex flows past a cylinder with patches of constant vorticity has also been studied numerically by Elcrat [*et al.*]{} [@elcrat1; @elcrat2]. These authors found two families of solutions, representing desingularized versions of the Föppl and the normal equilibria, respectively, which have the same stability properties as the corresponding point-vortex equilibria.
In conclusion, we have seen that the Föppl model, where a pair of counter-rotating point vortices move around a circular cylinder in the presence of a uniform stream, is a rich nonlinear dynamical system whose features—notably its stability properties—bear a direct relevance to our understanding of the vortex shedding mechanism in real flows. The results obtained here should, in principle, carry over to more general geometries, such as vortex motion around a plate or around a cylinder with noncircular cross section.
This work was supported in part by the Brazilian agencies CNPq and FACEPE. One of the authors (AMJS) acknowledges financial support from CAPES, Brazil through a visiting professor scholarship.
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---
abstract: 'In this paper, we study radiative decays of $X_b$, the counterpart of the famous $X(3872)$ in the bottomonium-sector as a candidate for meson-meson molecule, into the $\gamma \Upsilon(nS)$ ($n=1$, $2$, $3$). Since it is likely that the $X_b$ is below the $B\bar B^*$ threshold and the mass difference between the neutral and charged bottom meson is small compared to the binding energy of the $X_b$, the isospin violating decay mode $X_b\to \Upsilon (nS)\pi^+\pi^-$ would be greatly suppressed. This will promote the importance of the radiative decays. We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate the partial widths. Our results show that the partial widths into $\gamma \Upsilon(nS)$ are about $1$ keV, and thus the branching fractions may be sizeable, considering the fact the total width may also be smaller than a few MeV like the $X(3872)$. These radiative decay modes are of great importance in the experimental search for the $X_b$ particularly at hadron collider. An observation of the $X_b$ will provide a deeper insight into the exotic hadron spectroscopy and is helpful to unravel the nature of the states connected by the heavy quark symmetry.'
author:
- 'Gang Li$^{1}$ and Wei Wang$^{2}$'
title: 'Hunting for the $X_b$ via Radiative Decays'
---
Introduction {#sec:introduction}
============
In the past decades, there has been great progress in hadron spectroscopy thanks to the unprecedented data sample accumulated by the B factories and hadron-hadron colliders. A number of charmonium-like and bottomonium-like states have been discovered on these experimental facilities so far but not all of them can be placed in the ordinary $\bar qq$ (for reviews, see Refs. [@Brambilla:2010cs; @Godfrey:2008nc; @Drenska:2010kg; @Bodwin:2013nua]).
The $X(3872)$ is the first and perhaps the most renowned exotic candidate. It was first discovered in 2003 by Belle in the $B^+\to K^++ J/\psi \pi^+\pi^-$ final state [@Choi:2003ue] and subsequently confirmed by the BaBar Collaboration [@Aubert:2004ns]. Complementary observation is also found in proton-proton/antiproton collisions at the Tevatron [@Abazov:2004kp; @Aaltonen:2009vj] and LHC [@Chatrchyan:2013cld; @Aaij:2013zoa]. Though the existence is well established, the nature of the $X(3872)$ is still ambiguous due to a few peculiar properties. First, compared to typical hadronic widths the total width is tiny. Only an upper bound has been measured experimentally: $\Gamma<1.2$ MeV [@Beringer:1900zz]. The mass lies closely to the $D^0\overline D^{*0}$ threshold, $M_{X(3872)}-M_{D^0}-M_{D^{*0} }=(-0.12\pm0.24)$ MeV [@TheBABAR:2013dja], which leads to speculations that the $X(3872)$ is presumably a meson-meson molecular state [@Tornqvist:2004qy; @Hanhart:2007yq].
These peculiar features have stimulated considerable research interest in investigating the production and decays of the $X(3872)$ towards understanding its nature. A very important aspect involves the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration. In this viewpoint, it would be also valuable to look for the analogue in the bottom sector, referred to as $X_b$ following the notation suggested in Ref. [@Hou:2006it], as states related by heavy quark symmetry may have universal behaviours. Since the $X_b$ is expected to be very heavy and its $J^{PC}$ of is $1^{++}$, it is less likely for a direct discovery at the current electron-positron collision facilities, though the Super KEKB may provide an opportunity in $\Upsilon(5S,6S)$ radiative decays [@Aushev:2010bq].
In Ref. [@GMW], the production of the $X_b$ at the LHC and the Tevatron has been investigated, along the same line with the studies on the search for exotic states at hadron colliders [@Bignamini:2009sk; @Artoisenet:2009wk; @Artoisenet:2010uu; @Esposito:2013ada; @Ali:2011qi; @Ali:2013xba; @Guo:2013ufa]. It is shown that the production rates at the LHC and the Tevatron are sizeable [@GMW]. On the other hand, the search for the $X_b$ also depends on reconstructing the $X_b$, which motivates us to study the $X_b$ decays. Since this meson is expected to be far below threshold, the isospin violating decay mode for instance $X_b\to \Upsilon\pi^+\pi^-$ is highly suppressed, and this may explain the escape of $X_b$ in the recent CMS search [@Chatrchyan:2013mea]. As a consequence, radiative decays of the $X_b$ will be of high priority, on which we will focus in this paper. As we will show in the following, these modes have sizeable decay widths.
To calculate the radiative decays, we study the intermediate meson loop contributions, which have been one of the important nonperturbative transition mechanisms in various transitions, and their impact on the heavy quarkonium transitions, also referred to as coupled-channel effects, has been noticed for a long time [@Lipkin:1986bi; @Lipkin:1988tg; @Moxhay:1988ri]. The intermediate meson loops mechanism has been applied to study the production and decays of ordinary and exotic states [@Guo:2009wr; @Guo:2010ak; @Wang:2013cya; @Liu:2013vfa; @Guo:2013zbw; @Wang:2013hga; @Cleven:2013sq; @Chen:2011pv; @Li:2012as; @Li:2013yla; @Voloshin:2013ez; @Voloshin:2011qa; @Bondar:2011ev; @oai:arXiv.org:1002.2712; @Chen:2011pu; @Chen:2012yr; @Chen:2013coa; @Chen:2013bha] and B decays [@Du:1998ss; @Chen:2000ih; @Liu:2007qs; @Lu:2005mx; @Colangelo:2003sa; @Liu:2008tv; @Cheng:2004ru; @Colangelo:2002mj], and a global agreement with experimental data is found. Thus this approach may be an effective approach to handle the $X_b$ radiative decays.
The paper is organized as follows. In Sec. \[sec:formula\], we will introduce the formalism used in this work. Based on this framework, numerical results are presented in Sec. \[sec:results\] and the summary will be given in Sec. \[sec:summary\].
Radiative decays {#sec:formula}
================
![Feynman diagrams for the radiative decays $X_b \to \gamma \Upsilon(nS)$ with the $B{\bar B}^*$ as the intermediate states.[]{data-label="fig:loops"}](fig_diagram.eps){width="80.00000%"}
The calculation of contributions from the meson loops requests the leading order effective Lagrangian. Based on the heavy quark symmetry, we employ the relevant effective Lagrangian for the $\Upsilon(nS)$ [@Colangelo:2003sa; @Casalbuoni:1996pg] $$\begin{aligned}
\mathcal{L}_{\Upsilon(nS) B^{(*)} B^{(*)}} &=&
ig_{\Upsilon BB} \Upsilon_{\mu} (\partial^\mu B \bar{B}- B
\partial^\mu \bar{B})-g_{\Upsilon B^* B} \varepsilon^{\mu \nu
\alpha \beta}
\partial_{\mu} \Upsilon_{\nu} (\partial_{\alpha} B^*_{\beta} \bar{B}
+ B \partial_{\alpha}
\bar{B}^*_{\beta})\nonumber\\
&&-ig_{\Upsilon B^* B^*} \big\{
\Upsilon^\mu (\partial_{\mu} B^{* \nu} \bar{B}^*_{\nu}
-B^{* \nu} \partial_{\mu}
\bar{B}^*_{\nu})+ (\partial_{\mu} \Upsilon_{\nu} B^{* \nu} -\Upsilon_{\nu}
\partial_{\mu} B^{* \nu}) \bar{B}^{* \mu} \nonumber\\
&& +
B^{* \mu}(\Upsilon^\nu \partial_{\mu} \bar{B}^*_{\nu} -
\partial_{\mu} \Upsilon^\nu \bar{B}^*_{\nu})\big\}, \label{eq:h1}\end{aligned}$$ where ${{B}^{(*)}}=\left(B^{(*)+},B^{(*)0}\right)$ and ${\bar B^{(*)T}}=\left(B^{(*)-},\bar{B}^{(*)0}\right)$ correspond to the bottom meson isodoublets. [ $\epsilon^{\mu\nu\alpha\beta}$ is the anti-symmetric Levi-Civita tensor and $\epsilon^{0123}= -1$. Due to the heavy quark symmetry, the following relationships of the couplings are valid [@Casalbuoni:1996pg; @Colangelo:2003sa] $$\begin{aligned}
g_{\Upsilon(nS) BB} = 2g_n \sqrt{m_{\Upsilon(nS)}} m_B \ ,
\quad g_{\Upsilon(nS) B^* B} = \frac {g_{\Upsilon(nS) BB}} {\sqrt{m_B m_{B^*}}} \ ,
\quad g_{\Upsilon(nS) B^* B^*} = g_{\Upsilon(nS) B^* B} \sqrt{\frac {m_{B^*}} {m_B}} m_{B^*},\end{aligned}$$ where $g_n = \sqrt{m_{\Upsilon(nS)}}/(2m_B f_{\Upsilon(nS)})$; $m_{\Upsilon(nS)}$ and $f_{\Upsilon(nS)}$ denote the mass and decay constant of $\Upsilon(nS)$, respectively. The decay constant $f_{\Upsilon(nS)}$ can be extracted from the $\Upsilon(nS)\to e^+e^-$: $$\begin{aligned}
\Gamma(\Upsilon(nS) \to e^+e^-) = \frac {4\pi\alpha^2} {27} \frac {f_{\Upsilon(nS)}^2} {m_{\Upsilon(nS)}},\end{aligned}$$ where $\alpha = 1/137$ is the electromagnetic fine-structure constant. Using the masses and leptonic decay widths of the $\Upsilon(nS)$ states: $\Gamma(\Upsilon(1S)
\to e^+e^-) =1.340 \pm 0.018$ keV, $\Gamma(\Upsilon(2S) \to e^+e^-)
=0.612 \pm 0.011$ keV, $\Gamma(\Upsilon(3S) \to e^+e^-) =0.443 \pm
0.008$ keV [@Beringer:1900zz], one can obtain $f_{\Upsilon(1S)} =
715.2 $ [MeV]{}, $f_{\Upsilon(2S)} = 497.5 $ [MeV]{}, and $f_{\Upsilon(3S)} = 430.2 $ [MeV]{}.]{}
We consider the iso-scalar $X_b$ as a $S$-wave molecular state with the positive charge parity given by the superposition of $B^0 {\bar B}^{*0}+c.c$ and $B^- {\bar B}^{*+}+c.c$ hadronic configurations as $$\begin{aligned}
|X_b\rangle= \frac {1} {2} [ (|B^0{\bar B}^{*0}\rangle - |B^{*0} {\bar B}^0\rangle) + (| B^+ B^{*-}\rangle - | B^- B^{*+}\rangle ) ].\end{aligned}$$ The coupling of $X_b$ to the bottomed meson is based on the effective Lagrangian $$\begin{aligned}
{\cal L} = \frac {1} {2} X_{b\mu}^{\dagger} [x_1(B^{*0\mu} {\bar B}^0 - B^{0} {\bar B}^{*0\mu})+x_2(B^{*+\mu} B^- - B^+ B^{*-\mu})] + h.c.,\end{aligned}$$ [ where $x_i$ denotes the coupling constant. ]{}
For a bound state below an $S$-wave two-hadron threshold, the effective coupling of this state to the two-body channel is related to the probability of finding the two-hadron component in the physical wave function of the bound states and the binding energy, $E_{X_b}=m_B+m_{B^*}-m_{X_b}$ [@Weinberg:1965zz; @Baru:2003qq; @Guo:2013zbw] $$\begin{aligned}
\label{eq:coupling-Xb}
x_i^2 \equiv 16\pi (m_B+ m_{B^*})^2 c_i^2 \sqrt{\frac {2E_{X_b}}{\mu}} ,\end{aligned}$$ where $c_i=1/{\sqrt 2}$, $\mu=m_Bm_{B^*}/(m_B+m_{B^*})$ is the reduced mass.
The magnetic coupling of the photon to heavy bottom meson is described by the Lagrangian [@Hu:2005gf; @Amundson:1992yp] $$\begin{aligned}
{\cal L}_\gamma = \frac {e\beta Q_{ab}} {2} F^{\mu\nu} {\rm Tr}[H_b^\dagger \sigma_{\mu\nu} H_a ] + \frac {e Q^\prime} {2m_{Q}} F^{\mu\nu} {\rm Tr}[H_a^\dagger H_a \sigma_{\mu\nu}],\end{aligned}$$ [ with $$\begin{aligned}
H&=&\left( \frac{1+ \rlap{/}{v} }{2} \right)
[\mathcal{B}^{*\mu}
\gamma_\mu -\mathcal{B}\gamma_5],\end{aligned}$$]{} where $Q= {\rm diag}\{2/3, -1/3, -1/3\}$ is the light quark charge matrix, $\beta$ is an unknown parameter and $Q^\prime$ is the heavy quark electric charge (in units of $e$). [ In the nonrelativistic constituent quark model $\beta\simeq 3.0$ GeV$^{-1}$, which has been adopted in the study of radiative $D^*$ decays [@Amundson:1992yp]. Note heavy quark symmetry ensures that $\beta$ is the same in the $b$ and $c$ systems, so we take the same value as Ref. [@Amundson:1992yp].]{} The first term is the magnetic moment coupling of the light quarks, while the second one is the magnetic moment coupling of the heavy quark and hence is suppressed by $1/m_Q$.
The decay amplitudes for the transitions in Fig. \[fig:loops\] can be expressed in a generic form in the effective Lagrangian approach as follows, $$\begin{aligned}
M_{fi}=\int \frac {d^4 q_2} {(2\pi)^4} \sum_{B^* \ \mbox{pol.}}
\frac {V_1V_2V_3} {a_1 a_2 a_3}{\cal F}(m_2,q_2^2)\end{aligned}$$ where $V_i$ and $a_i = q_i^2-m_i^2 \ (i=1,2,3)$ are the vertex functions and the denominators of the intermediate meson propagators. For example, in Fig. \[fig:loops\] (a), $V_i \
(i=1,2,3)$ are the vertex functions for the initial $X_b$, final bottominum and photon, respectively. $a_i \
(i=1,2,3)$ are the denominators for the intermediate $B^+$, $B^{*-}$ and $B^+$ propagators, respectively. In addition, we introduce a dipole form factor, $$\begin{aligned}
\label{ELA-form-factor}
{\cal F}(m_{2}, q_2^2) \equiv \left(\frac
{\Lambda^2-m_{2}^2} {\Lambda^2-q_2^2}\right)^2,\end{aligned}$$ where $\Lambda\equiv m_2+\alpha\Lambda_{\rm QCD}$ and the QCD energy scale $\Lambda_{\rm QCD} = 220$ MeV. This form factor is supposed to compensate the off-shell effects arising from the intermediate exchanged particle and the non-local effects of the vertex functions [@Li:1996yn; @Locher:1993cc; @Li:1996cj], and phenomenological studies have suggested $\alpha\sim 2$. The explicit expression of the transition amplitudes can be found in Appendix (A.6) in Ref. [@Zhao:2013jza], where radiative decays of charmonium are studied extensively based on the effective Lagrangian approach.
Numerical Results {#sec:results}
=================
The existence of the $X_b$ was predicted in both the tetraquark model [@Ali:2009pi] and hadronic molecular calculations [@Tornqvist:1993ng; @Guo:2013sya; @Karliner:2013dqa]. The mass of the lowest-lying $1^{++}$ $\bar b \bar q bq$ tetraquark was predicted to be 10504 MeV in Ref. [@Ali:2009pi], while the mass of the $B\bar B^*$ molecule based on the mass of the $X(3872)$ is a few tens of MeV higher [@Guo:2013sya; @Karliner:2013dqa]. In Ref. [@Guo:2013sya], the mass was predicted to be $(10580^{+9}_{-8})$ MeV, corresponding to a binding energy of $(24^{+8}_{-9})$ MeV. These studies have provided a range for the binding energy, for which in the following we will choose a few illustrative values: $E_{X_b} =(1, 2, 5, 20)$ MeV.
![ The dependence of partial widths of $X_b \to \gamma\Upsilon(1S)$ on the $E_{X_b}$ with $\alpha=2.0$ (solid lines) and $\alpha=3.0$ (dashed lines), respectively. Panels (b) and (c) corresponds to the ones in the $X_b \to \gamma\Upsilon(2S)$ and $3S$, respectively. []{data-label="fig:WidthOnEXb"}](fig_mass_1S.eps "fig:"){width="45.00000%"} ![ The dependence of partial widths of $X_b \to \gamma\Upsilon(1S)$ on the $E_{X_b}$ with $\alpha=2.0$ (solid lines) and $\alpha=3.0$ (dashed lines), respectively. Panels (b) and (c) corresponds to the ones in the $X_b \to \gamma\Upsilon(2S)$ and $3S$, respectively. []{data-label="fig:WidthOnEXb"}](fig_mass_2S.eps "fig:"){width="45.00000%"}\
![ The dependence of partial widths of $X_b \to \gamma\Upsilon(1S)$ on the $E_{X_b}$ with $\alpha=2.0$ (solid lines) and $\alpha=3.0$ (dashed lines), respectively. Panels (b) and (c) corresponds to the ones in the $X_b \to \gamma\Upsilon(2S)$ and $3S$, respectively. []{data-label="fig:WidthOnEXb"}](fig_mass_3S.eps "fig:"){width="45.00000%"}
![ (a) The $\alpha$-dependence of the ratios of $R_1$ (solid line), and $R_2$ (dashed line) defined in Eq. (\[eq:ratio\]) with $E_{X_b}=1$ MeV. (b), (c), and (d) corresponds to $E_{X_b}=2$ MeV, $5$ MeV, and $20$ MeV, respectively.[]{data-label="fig:3"}](fig_ratio_deltaE_1.eps "fig:"){width="45.00000%"} ![ (a) The $\alpha$-dependence of the ratios of $R_1$ (solid line), and $R_2$ (dashed line) defined in Eq. (\[eq:ratio\]) with $E_{X_b}=1$ MeV. (b), (c), and (d) corresponds to $E_{X_b}=2$ MeV, $5$ MeV, and $20$ MeV, respectively.[]{data-label="fig:3"}](fig_ratio_deltaE_2.eps "fig:"){width="45.00000%"}\
![ (a) The $\alpha$-dependence of the ratios of $R_1$ (solid line), and $R_2$ (dashed line) defined in Eq. (\[eq:ratio\]) with $E_{X_b}=1$ MeV. (b), (c), and (d) corresponds to $E_{X_b}=2$ MeV, $5$ MeV, and $20$ MeV, respectively.[]{data-label="fig:3"}](fig_ratio_deltaE_5.eps "fig:"){width="45.00000%"} ![ (a) The $\alpha$-dependence of the ratios of $R_1$ (solid line), and $R_2$ (dashed line) defined in Eq. (\[eq:ratio\]) with $E_{X_b}=1$ MeV. (b), (c), and (d) corresponds to $E_{X_b}=2$ MeV, $5$ MeV, and $20$ MeV, respectively.[]{data-label="fig:3"}](fig_ratio_deltaE_20.eps "fig:"){width="45.00000%"}\
![ (a) The ratio $R_1$ defined in Eq. (\[eq:ratio\]) in terms of the $E_{X_b}$ with $\alpha=2.0$ (solid line) and $\alpha=3.0$ (dashed line). (b) The same notation with (a) except for $R_2$ defined in Eq. (\[eq:ratio\]).[]{data-label="fig:ratio_mass_R"}](fig_ratio_mass_R1.eps "fig:"){width="45.00000%"} ![ (a) The ratio $R_1$ defined in Eq. (\[eq:ratio\]) in terms of the $E_{X_b}$ with $\alpha=2.0$ (solid line) and $\alpha=3.0$ (dashed line). (b) The same notation with (a) except for $R_2$ defined in Eq. (\[eq:ratio\]).[]{data-label="fig:ratio_mass_R"}](fig_ratio_mass_R2.eps "fig:"){width="45.00000%"}\
-------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------
$\gamma\Upsilon(1S)$ $\gamma\Upsilon(2S)$ $\gamma\Upsilon(3S)$ $\gamma\Upsilon(1S)$ $\gamma\Upsilon(2S)$ $\gamma\Upsilon(3S)$
$E_{X_b} = 1$ MeV 0.12 0.34 0.22 0.41 0.96 0.46
$E_{X_b} = 2$ MeV 0.19 0.42 0.28 0.62 1.18 0.57
$E_{X_b} = 5$ MeV 0.28 0.53 0.33 0.92 1.53 0.70
$E_{X_b} = 20$ MeV 0.36 0.66 0.30 1.20 1.96 0.66
-------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------
: Predicted partial widths (in unit of keV) of the $X_b$ decays. The parameter in the form factor is chosen as $\alpha =2.0$ and $\alpha =3.0$. []{data-label="tab:results"}
Choosing two values for the cutoff parameter $\alpha$, we have predicted the partial decay widths and the numerical results are collected in Table \[tab:results\]. From this table, we can see that the widths for the $X_{b}$ radiative decays are about $1$ keV. It is noteworthy to recall that the upper bound for the $\Gamma(X(3872))$ is $1.2$ MeV [@Beringer:1900zz]. If the $X_{b}$ were similarly narrow, our results would indicate a sizeable branching fractions, at least $10^{-3}$, for these radiative decay modes.
In Fig. \[fig:WidthOnEXb\], we present the partial widths for the $X_b\to \gamma \Upsilon(1S)$ (panel a), $\gamma\Upsilon(2S)$ (panel b), and $\gamma\Upsilon(3S)$ (panel c) in terms of the $E_{X_b}$ with $\alpha=2.0$ (solid lines) and $3.0$ (dashed lines), respectively. The uncertainties caused by the cutoff parameter indicate our limited knowledge on the applicability of the effective Lagrangian. However fortunately the $\alpha$ dependence of the partial widths are not drastically sensitive, which indicates a reasonable cutoff of the ultraviolet contributions by the empirical form factors. In this figure, there exists an evident enhancement structure around $E_{X_b}=20$ MeV resulting from the cusp effect. As can be seen from this figure, this enhancement structure is independent of the cutoff parameter $\alpha$.
It would be interesting to further clarify the uncertainties arising from the introduction of the form factors by studying the ratios between different partial decay widths. We define the following ratios $$\begin{aligned}
R_1= \frac {\Gamma(X_b\to \gamma\Upsilon(2S))} {\Gamma(X_b\to \gamma\Upsilon(1S))}, \quad R_2= \frac {\Gamma(X_b\to \gamma\Upsilon(3S))} {\Gamma(X_b\to \gamma\Upsilon(1S))}, \label{eq:ratio}\end{aligned}$$ which are plotted in Fig. \[fig:3\] for the dependence on the cutoff parameter and Fig. \[fig:ratio\_mass\_R\] for the dependence on binding energy. Since the first coupling vertices are the same for those decay channels when taking the ratio, so the ratio only reflects the open threshold effects through the intermediate bottomed meson loops. The ratios are less sensitive to the cutoff parameter, which is a consequence of the fact that the involved loops are the same. As can be seen from this figure, when the cutoff parameter $\alpha$ increases, the ratios decrease. These predictions can be tested by the experimental measurements in future.
Summary {#sec:summary}
=======
Our understanding of hadron spectroscopy will be greatly improved by studies of exotic states that may defy the conventional models of $q\bar q$ meson spectroscopy, and accordingly great progress has been made in the past decades. One of the most important aspects in the study of exotics is the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule. Such task requests a large amount of efforts on both experimental and theoretical sides in future.
In this work, we have investigated the radiative decays of the $X_b$, the counterpart of the famous $X(3872)$ in the bottomonium-sector as a candidate for meson-meson molecule, into the $\gamma \Upsilon(nS)$. Since this state may be far below the $B\bar B^*$ threshold, the isospin violating decay mode $X_b\to \Upsilon\pi^+\pi^-$ would be highly suppressed, and stimulate the importance of the radiative decays. We have made used of the effective Lagrangian based on the heavy quark symmetry, and explore the rescattering mechanism. Our results have shown that the partial widths for the $X_b\to \gamma \Upsilon(nS)$ are about $1$ keV, and thus the branching fractions may be sizeable, taking into account the fact the total width may also be smaller than a few MeV like $X(3872)$. This study of radiative decays and the previous work on production rates in hadron-hadron collisions have indicated a promising prospect to find the $X_b$ at hadron collider in particular the LHC, and we suggest our experimental colleagues to perform an analysis. Such attempt will likely lead to the discovery of the $X_b$ and thus enrich the exotics garden in the heavy quarknoium sector.
Acknowledgements {#sec:acknowledgements .unnumbered}
================
The authors are very grateful to Feng-Kun Guo, Xiao-Hai Liu, Qian Wang, and Qiang Zhao for useful discussions. W.W. thanks Ulf-G. Mei[ß]{}ner and Feng-Kun Guo for the collaboration of Ref. [@GMW]. This work is supported in part by the National Natural Science Foundation of China (Grant No. 11275113), the China Postdoctoral Science Foundation (Grant No. 2013M530461), and the DFG and the NSFC through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”.
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|
---
author:
- Olaf Merkert
bibliography:
- 'phd-thesis.bib'
date: 'Anno accademico 2015-2016'
title: Reduction and specialization of hyperelliptic continued fractions
---
Introduction {#sec:org6b1216b}
============
This thesis investigates how prime factors arise in denominators of polynomial continued fractions, with a focus on continued fractions of the square root of a polynomial. This is strongly related to the problem of reducing polynomial continued fractions modulo a prime.
Continued fractions have a very long history – those of rational numbers express the Euclidean Algorithm which was already known in ancient Greece. In modern times, mathematicians such as Lagrange and Galois studied continued fractions of irrational numbers, in particular quadratics (for example square roots). Even today, continued fractions of real numbers remain an important research topic in number theory and other branches of mathematics.
We write a continued fraction as $$\alpha = [a_0, a_1, a_2, \dots] = a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \ddots}}.$$ For the classical continued fractions with $\alpha \in \R$, the *partial quotients* $a_n$ are integers, positive for $n \geq 1$. Instead, one may also take the $a_n \in \Q[X]$ to be polynomials, non-constant for $n \geq 1$, to build the continued fraction of a Laurent series in $\inv X$, i.e. $\alpha \in \laurinx \Q$. The role of the nearest integer is then played by the polynomial part of the Laurent series.
We are interested for which $n$ a given prime number $\pp$ divides the denominator of the coefficients of the $a_n$ (for brevity, we say the “prime $\pp$ appears in the denominator of $a_n$”). We are especially interested when it first appears and whether it can disappear again.
Of particular interest is the continued fraction of $\sqrt{D}$, where $D \in \Q[X]$ is a monic non-square polynomial of even degree $2d$. It was first considered by Abel in 1826 [@abel-1826-ueber-integ-differ], who used it to study the integration in elementary terms of certain algebraic functions. Abel showed that periodicity of this continued fraction is equivalent to the existence of a non-trivial solution $p, q \in \Q[X]$, $q \neq 0$ of the polynomial Pell equation $p^2 - D \, q^2 = 1$ (see Chapter \[sec:org953fbfb\] and Theorem \[thm-pellian-iff-cf-periodic\]). We say that $D$ is *Pellian* if such a solution exists. Later, Chebyshev expanded upon these results [@chebyshev-1857-sur-integration].
We call continued fractions of this type *hyperelliptic* because they encode information about the (hyper)elliptic curve $Y^2 = D(X)$, given that $d \geq 1$ and $D$ is also square-free. For example, if $O_\pm$ are the two points at infinity in a smooth model, the class of $\pd{O_+} - \pd{O_-}$ is torsion in the Jacobian of the curve $D$ is Pellian, i.e. the continued fraction is periodic (see Theorem \[thm-pellian-iff-torsion\]).
Note that the polynomials of degree $2d$, after some normalisation, form an affine variety of dimension $2d-2$. The Pellian polynomials are then contained in a denumerable union of subvarieties of dimension at most $d-1$ (see [@zannier-2014-pell-survey], a survey focusing on the geometric aspects of the polynomial Pell equation). This implies that unlike positive square-free integers which are always “Pellian”, most polynomials $D$ are not Pellian, and usually we do not expect a periodic continued fraction. But other results for the classical continued fractions have direct analogues for polynomial continued fractions, see for example [@schmidt-2000-continued-fractions-diophantine].
Let us also introduce the *canonical convergents* which are defined via the recurrence relations $$\begin{aligned}
p_n &= a_n \, p_{n-1} + p_{n-2}, &
q_n &= a_n \, q_{n-1} + q_{n-2}\end{aligned}$$ and $p_0 = a_0, \; p_{-1} = q_0 = 1, \; q_{-1} = 0$. These imply that $p_n, q_n \in \Q[X]$ are coprime for any integer $n \geq 0$, via the identity $p_n \, q_{n-1} - q_n \, p_{n-1} = (-1)^{n+1}$. The canonical convergents arise by calculating the numerator and denominator of the finite continued fraction $$\frac{p_n}{q_n} = [a_0, a_1, \dots, a_n] = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{ \ddots + \dfrac{1}{a_{n}}}}.$$ Note that they are usually not monic nor have content $1$. This is related to prime numbers suddenly appearing in the denominators of the coefficients of the $a_n$, something van der Poorten was already aware of (see [@poorten-2001-non-periodic-continued]).
This follows from the fact that $D_\pp$, the reduction of $D$ modulo $\pp$, is Pellian unless it is a square (the Jacobian over $\F_\pp$ is finite, all points on it are torsion), so the continued fraction of $\sqrt{D_\pp}$ is automatically periodic. This leads to one of the main results of this thesis:
\[thm-intro-infinite-poles-rationals\] Let $\sqrt{D} = [a_0, a_1, a_2, \dots]$. If $D \in \Q[X]$ is not Pellian, then for all prime numbers $\pp$ except finitely many, $\pp$ appears in infinitely many polynomials $a_n$ in a denominator (of the coefficients).
We prove this in Theorem \[thm-infinite-poles-number-field\] more generally for arbitrary number fields. Note that the formula for multiplying polynomial continued fractions with a constant, $$\label{intro-eq-cf-mult-const}
\pp^e \, [b_0, b_1, b_2, \dots] = [\pp^e \, b_0, \pp^{-e} \, b_1, \pp^{e} \, b_2, \dots], \quad (e \in \Z),$$ raises the question if – at least for a fixed prime $\pp$ – the infinite occurrences in the denominators of the $a_n$ arise in this rather trivial way. Indeed, this is not the case; we can show that for any $e \in \Z$, the continued fraction of $\sqrt{\pp^{-2e} D} = [b_0, b_1, b_2, \dots]$ enjoys the property that $\pp$ appears in infinitely many $b_n$ as a denominator.
The primes which are excluded in Theorem \[thm-intro-infinite-poles-rationals\] are the prime $2$, any primes appearing already in a denominator of $D$ and those with $D_\pp$ square. For technical reasons, we may also need to exclude further primes, depending on the first occurrence of an $a_n$ with minimal degree. These primes can be determined effectively, too (see Remark \[minimal-an-degree-effective\]). The prime $2$ is of course excluded because we are taking square roots.
\[intro-counterexample-infinite-good-reduction\] This result is true only for $\sqrt{D}$, and does not apply to other elements of the hyperelliptic function field $\Q(X, \sqrt{D})$. With an analogue of the fact that there are infinitely many primes $\pp$ such that $2^n \not\equiv 5 \mod \pp$ for all $n$, we construct an example of type $\alpha = \ifracBb{r + \sqrt{D}}{X}$ where there are infinitely many primes $\pp$ that never appear in the denominators of the $a_n$ (see Theorem \[thm-good-reduction-infinite-primes\] in Section \[sec:org53fcc2b\]). The proof relies on the Čebotarev density theorem, and represents a variant of the results of Schinzel [@schinzel-1960-the-congruence-a] and Corrales-Rodrigáñez-Schoof [@corrales-schoof-1997-support-problem-its].
For $\deg D = 4$, another more explicit approach avoids the issue of excluding additional primes. This is described in the rather technical Theorem \[thm-genus1-zero-patterns\] and Corollary \[cor-infinite-poles-deg4\]. The former has another consequence for the Gauss norm of the convergents.
Recall that, given some valuation on a field $K$, we may extend the valuation to polynomials. Define the valuation of a polynomial in $K[X]$ as the minimum of the valuation on the coefficients (see Section \[sec:orge83be0b\] for details). The corresponding absolute value is usually called a *Gauss norm*. For $D \in \Q[X]$, we naturally use the $\pp$-adic valuation $\nu_\pp$. A negative $\nu_\pp(f)$ then indicates that $\pp$ appears in at least one denominator of the coefficients of the polynomial $f$.
As a special case of Corollary \[cor-genus1-unbounded-gauss-norm\], we obtain:
\[thm-intro-genus1-unbounded-gauss-norm\] Let $D$ be a non-Pellian polynomial of degree $4$, and let $\pp$ an odd prime with $D_\pp$ square-free and the class of $\pd{O_+} - \pd{O_-}$ of *even* torsion order $m$ in the (finite) Jacobian of the elliptic curve $Y^2 = D_\pp(X)$. Then $$\begin{aligned}
(-1)^n \nu_\pp(a_{n}) &\geq 2 \floor{\ifracBb{n-1}{m}}_\Z + 2 \floor{\ifracBb{n+1}{m}}_\Z,\\
(-1)^n \nu_\pp(q_{n}) &\geq 2 \floor{\ifracBb{n+1}{m}}_\Z,\end{aligned}$$ where $\floor{\cdot}_\Z$ denotes the floor function. In particular, the Gauss norms of the partial quotients and the convergents are unbounded both from above and below.
In the case of *odd* torsion order $m$, the negative valuations are possibly cancelled out by positive valuations coming from phenomena as in ; this currently prevents any similar prediction (see Example \[ex-cfp1-zero-pattern-deg4\], in particular table \[cfr-mod19-valuations-table\]). Moreover, the precise growth of these Gauss norms is not understood at all right now. This is an even bigger issue for $\deg D > 4$, where we have to keep track of further unknowns. This makes an exact estimation of the valuations for higher degrees much more difficult.
The Gauss norms are also related to the height of polynomials. However, we have no information on the archimedean place and the $2$-adic valuations, so we have to be careful if we want to compare with known results about the height of the convergents (see Section \[sec:orgba963e8\]).
Indeed, the convergents $(p_n, q_n)$ are also Padé approximations of $\sqrt{D}$, i.e. they satisfy $$\label{intro-convergent-order-inequality}
\ord_\infty (p_n - \sqrt{D} \, q_n) > \deg q_n, % \quad \text{ where } p, q \in \mino{\Q[X]}$$ where $\ord_\infty$ is the non-archimedean valuation with $\ord_\infty X = -1$ and which makes $\laurinx \Q$ the completion of $\Q(X)$. In other words $p_n - \sqrt{D} \, q_n$ has a zero of high order at infinity.
Then by a general result of Bombieri and Paula Cohen [@bombieri-cohen-1997-siegels-lemma-pade] on the height of Padé approximations, it follows in the non-periodic case that the logarithmic projective height of the convergents grows quadratically in $n$. In this thesis, we have worked out the details of a simpler proof for the hyperelliptic case suggested by Zannier, see Theorem \[convergent-height-lower-bound\] and Theorem \[convergents-upper-proj-height-bound\] for lower respectively upper bounds. This leads to upper bounds for the projective height of the partial quotients as well (see Corollary \[partial-quotients-upper-proj-height-bound\]). The corresponding lower bounds for the height of the partial quotients require different arguments, see [@zannier-2016-hyper-contin-fract].
The main approach to prove results like Theorem \[thm-intro-infinite-poles-rationals\] and \[thm-intro-genus1-unbounded-gauss-norm\] is to study reduction of continued fractions modulo primes. This is interesting in itself, as it gives an example of a map between two “spaces” of continued fractions. Chapter \[sec:orgd5f1900\] contains a general exposition of reduction of continued fractions, using the theory of discrete valuation rings.
The idea is to compare the continued fractions of $\sqrt{D}$ and $\sqrt{D_\pp}$. Their partial quotients are contained in $\Q[X]$ respectively in $\F_p[X]$. A naive approach would be to try to reduce the partial quotients, but this does not capture the structure of the continued fraction sufficiently. Instead we have to try to reduce the complete quotients $\alpha_n = [a_n, a_{n+1}, \dots]$ of $\sqrt{D}$ which are Laurent series in $\inv X$ over $\Q$.
We say that a continued fraction has *good reduction in $\pp$* if we can reduce the complete quotients of $\sqrt{D}$ and obtain exactly the complete quotients of $\sqrt{D_\pp}$. If this fails, we speak of *bad reduction of the continued fraction*. The latter is the usual situation for non-Pellian $D$ over $\Q$ – and this is a key ingredient for the proof of Theorem \[thm-intro-infinite-poles-rationals\]. Other equivalent characterisations for good reduction of the continued fraction are given in Theorem \[cf-good-red-partial-quotients\]. Note that this notion of good or bad reduction for the continued fraction of $\sqrt{D}$ is very different from the good or bad reduction of the corresponding (hyper)elliptic curve.
If the continued fraction of $\sqrt{D}$ is periodic, it trivially has good reduction at almost all primes $\pp$. This implies that the period length of the continued fraction of $\sqrt{D_\pp}$ is essentially independent of $\pp$. This can also be stated and deduced directly in terms of reducing minimal solutions of the polynomial Pell equation, and has recently been used by Platonov [@platonov-2014-number-theoretic-properties], also together with Benyash-Krivets [@benyash-platonov-2007-groups-s-units] and Petrunin [@platonov-petrunin-2012-the-torsion-problem], to construct hyperelliptic curves over $\Q$ of genus $2$, where the Jacobian contains a torsion point of a specific order. These examples are relevant for the uniform boundedness conjecture for torsion points of abelian varieties.
Van der Poorten’s approach to reduction of continued fractions deals primarily with reduction of the convergents: the inequality essentially characterises the convergents up to a common factor of small degree, constant if $p$ and $q$ are coprime (see Corollary \[cf-convergent-classification\]). If we normalise $p_n$ and $q_n$ correctly, their reduction modulo $\pp$ remains a convergent of $\sqrt{D_\pp}$. Moreover, the following theorem holds (both for Pellian and non-Pellian $D$):
\[thm-vdp-intro\] If the prime $\pp$ does not appear in a denominator in $D$, then the reductions modulo $\pp$ of the normalised convergents $(\normal{p_n}, \normal{q_n})$ of $\sqrt{D}$ yield *all* the convergents of $\sqrt{D_\pp}$.
Unfortunately, the proofs given by van der Poorten (there are slightly different versions in [@poorten-1998-formal-power-series], [@poorten-1999-reduction-continued-fractions] and [@poorten-2001-non-periodic-continued]) do not appear to be complete. So one of the main goals of Chapter \[sec:orgd5f1900\] is to give a more precise statement and a rigorous proof of van der Poorten’s result (as in Theorem \[convergent-reduction-surjective\]).
As might be expected, the reduction of the convergents is strongly related to the reduction of the continued fraction. For example, the bad reduction of the continued fraction is caused by two (or more) convergents of $\sqrt{D}$ reducing to the same convergent modulo $\pp$ – see Proposition \[cf-good-reduction-lambda-bijective\] and example \[ex-cfp2-zero-pattern-deg6\], in particular table \[cf2-mod3-degrees-table\].
Finally, we remark that periodicity of the continued fraction of $\sqrt{D}$ is equivalent to $\deg a_n = d$ for at least one $n \geq 1$, where $2d = \deg D$ (see Corollary \[cor-pq-degree-periodicity\]). Bad reduction of the continued fraction is also determined by how these degrees increase under reduction (see the discussion in Section \[sec:org84e8497\]) which connects periodicity of the continued fraction of $\sqrt{D_\pp}$ and occurrences of $\pp$ in the denominators. The interplay with the normalisation factors of the canonical convergents then allows us to exclude issues related to , and leads to a proof of Theorem \[thm-intro-infinite-poles-rationals\].
**On specialization**
The reduction theory for continued fractions of Chapter \[sec:orgd5f1900\] applies also to specialization. Instead of reducing $D \in \Q[X]$ modulo a prime, we take for example $D \in \C(t)[X]$, and try to specialize $t$ to some $t_0 \in \C$. Searching for the values $t_0$ of $t$ that specialize to a periodic continued fraction of $\sqrt{D_{t=t_0}}$ corresponds to a special case of the relative Manin-Mumford conjecture, which in turn is a consequence of Pink’s conjecture. Recall that periodicity is equivalent to the class of $\pd{O_+} - \pd{O_-}$ being torsion in the Jacobian of the curve $Y^2 = D(X)$.
The periodicity of the reduction of the continued fraction was a crucial ingredient for the proof of Theorem \[thm-intro-infinite-poles-rationals\]. It is therefore natural to ask for specialization analogues of this theorem. The answer depends on the geometry:
For example Masser and Zannier showed that for $D = X^6 + X + t$, the continued fraction of $\sqrt{D}$ is non-periodic, the Jacobian of the curve $Y^2 = D(X)$ is simple and there are only finitely many $t_0 \in \C$ such that $\sqrt{D_{t=t_0}}$ has a periodic continued fraction (see [@masser-zannier-2015-torsion-points-on], here we have reformulated the results in the language of continued fractions). For these $t_0$, all of them algebraic numbers, we can reuse the arguments from Theorem \[thm-intro-infinite-poles-rationals\] and show that $t-t_0$ appears in infinitely many $a_n$ of the generic continued fraction as a denominator of a coefficient.
However, from the results of Masser and Zannier follows also that there are infinitely many $t_1 \in \closure{\Q}$ for which $t-t_1$ appears at least once as a denominator of a coefficient of some $a_n$. They might appear infinitely often, but we will show that this can happen only for the trivial reason that we excluded in Theorem \[thm-intro-infinite-poles-rationals\]. More precisely we can find $e \in \Z$ (perhaps not effectively), such that in $$(t-t_1)^e \, \sqrt{D} = [b_0, b_1, b_2, \dots], \qquad b_n \in \C(t)[X]$$ the “prime” $(t-t_1)$ appears only in finitely many $b_n$ as a denominator. We will discuss this in more detail in Section \[sec:orgf9f9759\].
Acknowledgements {#sec:org9942b95}
----------------
First and foremost, I would like to thank my supervisor Prof. Umberto Zannier, for pointing me to interesting mathematical problems and sharing his mathematical insight. You have helped me to see number theory in a new light, and improved my understanding of various problems. This thesis would not exist without his input and support. Thank you for answering my many questions and teaching me not to give up and to be independent. I am indebted to you and Scuola Normale Superiore for offering me the chance to pursue a Perfezionamento.
I also would like to dearly thank Prof. David Masser for introducing me to the polynomial Pell equation, and sending me towards Pisa in the first place. I thank Prof. Vistoli for teaching me some algebraic geometry.
A very big “thank you” goes to Lars, for many discussions about mathematics and other more trivial topics, putting up with me as a flatmate, and actually reading a draft of this thesis.
Big thanks also to Laura, Fabrizio, Michele and Soli, for countless lunches, game nights and for working together. Thanks for all your help, and for listening to me, even if I made rather less sense. Special thanks to Laura for helping me from my first day in Italy, and to Michele for participating in many sometimes crazy activities.
I would like to thank Francesca for working together, and being a very diligent mathematician.
My referees I am indebted to for their suggestions and careful perusal of my thesis.
I wish to thank all the wonderful and interesting people I met at Scuola Normale Superiore, for silly and serious conversations and reminding me that there are people in this world. Many of you I consider now my friends.
Let me thank in particular Josefine for showing me Florence and the beach, Alex for early morning runs and literally talking to everybody, Sara for teaching me about real friendship, Alexey for extraordinary observations and highly entertaining discussions, Clélie for not being afraid to talk of anything, Błażej for making me a better table tennis player, and Giacomo for his delicious chinese cooking and strange questions.
Thanks to Mario and Simone for explaining Italy, and Michele (the other one) for explaining biology with a passion. Thanks to Ilir, Marcello and Renata for being loyal hikers, to Umesh for playing table-tennis, to Adam for trying to take silly things seriously,to Elisa and Henry for chatting about fotography and to François and Max for reminding me that I am german.
Thanks also to all the people I spent time with at conferences, for interesting discussions and experiences from other places. Harry and Jung-Kyu, thanks for inviting me to visit the math department of Basel every once in a while.
To Aki, even if we have never met in real life, thank you for the countless hours in the skys of Georgia, Nevada and elsewhere, and in the woods of Chernarus, and for sharing your knowledge of aviation.
Finally, I want to thank my parents, my brother Sven and my sister Heike, for your support (logistical and otherwise) and for *always* believing that I could complete my PhD. It looks like you were right in the end.
Notation reference {#sec:org69c4145}
------------------
Symbol Description
-------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------
$\N$ natural numbers: $\{ 1, 2, 3, \dots \}$
$\N_0$ natural numbers with $0$: $\{ 0, 1, 2, 3, \dots \}$
$\K$ field of characteristic $\neq 2$
$\Fr(R)$ fraction field of integral domain $R$
$\laurinx \K$ Laurent series in $\inv X$ with coefficients in $\K$
$\ord(f)$ zero-order at $X = \infty$, sometimes denoted $\ord_\infty$
$\LC(f)$ leading coefficient of polynomial or Laurent series
$\Batest{\K}$ $\{ (p, q) \in \K[X]^2 \mid q \neq 0 \}$
$\Coset{\alpha}{\K}$ set of convergents of $\alpha$
$\Baset{\alpha}{\K}$ set of best-approximations of $\alpha$
$D$, $d$ polynomial, with $\deg D = 2d$ and $\LC(D)$ a square
$\sol{D}$ solutions of polynomial Pell equation
$\solu{D}$ solutions of unit-norm equation
$\sigma$ involution $\sqrt{D} \to - \sqrt{D}$
$\O$, $\mm = \spann{\pi}$ discrete valuation ring and maximal ideal with uniformiser
$K$, $k$ fraction field and residue field of $\O$, of characteristic $\neq 2$
$\nu$ valuation, usually of $\O$
$\laurinx K_\nu$ Laurent series with coefficient valuations bounded from below
$\normal{x}$ normalisation of $x \in \laurinx K_\nu$ to valuation $\nu(\normal{x}) = 0$
$\Red{x} = \RedM{x}$ reduction/specialization of $x \in \laurinx \O$
$\Redn{x} = \RedM{\normal{x}}$ reduction of normalisation
$\pp$ prime *number* $\pp$ (positive integer)
$\PP$ prime *ideal* $\PP$ (usually over $\pp$)
$\CF(\alpha)$ continued fraction of $\alpha$
$a_n$ partial quotient of $\alpha$
$\alpha_n$ complete quotient of $\alpha$
$(p_n, q_n)$ canonical convergent of $\alpha$
$g_n$ normalisation factor of canonical convergent, $\nu(g_n) = \nu(q_n)$.
$\vartheta_n$ $p_n - \alpha \, q_n$ normalised to $\nu(\vartheta_n) = 0$
$c_n$ partial quotient of $\gamma = \Red{\alpha}$
$\gamma_n$ complete quotient of $\gamma$
$(u_n, v_n)$ canonical convergent of $\gamma$
$h_n$ correction factor (in $k[X]$) for reduced convergents
$\lambda : \N_0 \to \N_0$ $(\Redn{p_n}, \Redn{q_n}) = h_n \cdot (u_{\lambda(n)}, v_{\lambda(n)})$, see Corollary \[definition-convergent-reduction-map-lambda\]
$\pd{P}$ point as divisor
$\j{P}$ divisor class of point
$\di{D}$ divisor (bold)
$\jdi{D}$ divisor class
$\CCa, \CC$ smooth affine and projective models of $Y^2 = D(X)$
$O_\pm$, $\OO$ the two points of $\CC$ at infinity; $\OO = \pd{O_+} - \pd{O_-}$
$\sigma$ conjugation of points, $Y \to -Y$
Pell equation {#sec:org953fbfb}
=============
We begin by exploring some well-known basic properties of the Pell equation over polynomials, usually called the *polynomial Pell equation*. We also explain how to write square roots of polynomials in $X$ as Laurent series in $\inv X$, and use this to show that the group of solutions of the polynomial Pell equation has rank at most $1$.
Given a base field $\K$ with $\Char \K \neq 2$, let $D \in \K[X]$ a *non-constant* polynomial and consider the *polynomial Pell equation* $$\label{pell}
% \tag{P_1}
p^2 - D \, q^2 = 1.$$ Clearly, there *always* exist the trivial solutions $(p,q) = (\pm 1,0)$, so naturally we ask if there exist other solutions $(p,q) \in \K[X]^2$ with $q \neq 0$, which we call the *non-trivial solutions*. If this is the case, we say $D$ is *Pellian*. If $\K$ is finite, one may show as for the classical Pell equation over $\Z$ that $D$ is always Pellian. If $\K$ is infinite, it is unlikely that $D$ is Pellian – because $D$ Pellian is equivalent to a torsion condition on a point in the Jacobian of a (hyper)elliptic curve, see Chapter \[sec:org99a8e17\] for details.
\[pell-necessary-conditions\] Suppose $D$ is Pellian. Then $\deg D$ must be even, and the leading coefficient $\LC(D)$ is a square in $\K$. However $D$ cannot be a square in $\K[X]$.
By the hypotheses $D$ non-constant and $q \neq 0$, we have $\deg (D \, q^2) > 0$. Then $p^2$ must cancel out the non-constant terms, hence $\deg p^2 = \deg (D \, q^2)$ which implies $\deg D = 2 (\deg p - \deg q)$ and that $\LC(D) = \ifrac{\LC(p)^2}{\LC(q)^2}$ is a square.
Finally, we show that $D$ is not a square in $\K[X]$: It is obvious that for $D = 1$, i.e. $p^2 - q^2 = (p-q)(p+q) = 1$ there are only constant solutions because $\units{\K[X]} = \units{\K}$. So if $D = E^2$ with $E \in \K[X] \setminus \K$, then for any solution $(p, q)$ we must have $p, E \, q$ constant which implies $q = 0$.
So these three conditions are necessary (but not sufficient) for the existence of non-trivial solutions.[^1]
Multiplication law and unit-norm equation {#sec:org6aa78e0}
-----------------------------------------
We assume from now on that $D$ has even degree $2d$, is not a square, but $\LC(D)$ is a square in $K$ (for example $1$ if $D$ is monic).
The set of solutions $\sol{D}$ (including trivial solutions) of carries an abelian group structure[^2] via the multiplication $$(p, q) * (p', q') = (p \, p' + D \, q \, q', p \, q' + p' \, q)$$ which comes from the map $$\sol{D} \longto \units{\HER}, \qquad (p,q) \mapsto p + q \, \sqrt{D}$$ which is an (injective) group homomorphism (see Section \[sec:orgc9b1f16\] below).
Note that $(p, q) * (p, -q) = (p^2 - D \, q^2, 0) = (1,0)$ for any Pell solution, so $(1,0)$ is the neutral element, and $(p, -q)$ is the inverse of $(p, q)$.
Actually, we will not really work with . To study the structure of the solution set, it is far more convenient to relax to the unit-norm equation (see [@hellegouarch-mcquillan-1987-unites-de-certains] for a general treatment) $$\label{pellu}
% \tag{P}
p^2 - D \, q^2 = \omega \in \units{\K}$$ where $\omega$ is an arbitrary unit of $\K$. Clearly, any Pell solution satisfies also this equation. The converse does of course not hold, but from a non-trivial solution of we can recover a non-trivial solutions of :
Suppose has a non-trivial solution $(p,q) \in \K[X]^2$ (with $q \neq 0$). Then $D$ is Pellian.
The multiplication law from above generalises to , with $(p, q) * (p, -q) = (\omega, 0)$, hence $$(p, q) * (p, q) * (p, -q) * (p, -q) = (\omega^2, 0).$$ Set $$(p', q') = (\inv\omega, 0) * (p, q) * (p, q) = \inv\omega \cdot (p^2 + D \, q^2, 2 \, p \, q),$$ so that $(p', q')$ remains in $\K[X]$ and is clearly a solution of . As observed in the proof of Proposition \[pell-necessary-conditions\], $q \neq 0$ implies $p \neq 0$, hence $\inv\omega \, 2 \, p \, q \neq 0$, so $(p', q')$ is a non-trivial Pell solution.
From now on, we refer also to as the *Pell equation*, and mostly forget about . We denote by $\solu{D}$ the set of all solutions of . We will see that for the purposes of this thesis, it is more natural to work with the unit-norm equation.
Units of hyperelliptic function fields {#sec:orgc9b1f16}
--------------------------------------
The quadratic field extension $\HEF$ of $\K(X)$ is called a hyperelliptic function field – specifically it is the function field of the hyperelliptic curve $\CCa : Y^2 = D(X)$ which we will study in more detail in Chapter \[sec:org99a8e17\]. The subring $\HER$ of $\HEF$ is the integral closure of $\K[X]$, describing the regular functions on the affine curve. For now, we show that the units of $\HER$ correspond to solutions of the Pell equation . See also [@hellegouarch-mcquillan-1987-unites-de-certains] for generalisations to other algebraic functions.
The map $$\pi: \solu{D} \longto \units{\HER}, \quad (p,q) \mapsto p + q\, \sqrt{D}$$ is bijective, and via the multiplication $$ on $\solu{D}$ gives an isomorphism of abelian groups.
Observe that there is a single non-trivial $\K(X)$-automorphism $\sigma$ of $\HEF$, defined by $\sigma(\sqrt{D}) = -\sqrt{D}$.
Actually, we defined $$ as the pullback under $\pi$ of the multiplication on $\HER$, so clearly $$\pi(\phi * \psi) = \pi(\phi) * \pi(\psi) \text{ for all } \phi, \psi \in \solu{D}.$$ And by the identity $$(p + q \, \sqrt{D} )(p - q \, \sqrt{D}) = p^2 - D \, q^2 = \omega \in \units \K$$ it follows that $\im \pi \subset \units \HER$, so $\pi$ is well defined.
Recall that we assume that $D$ is not a square, so the ring $\HER$ is a free rank 2 module over $\K[X]$ with basis $(1, \sqrt{D})$: this implies that $\pi$ is injective.
It remains to check that $\pi$ is also surjective: Let $\phi= p + q \, \sqrt{D} \in \units{\HER}$ with $p, q \in \K[X]$. Then we have $$\phi \cdot \sigma(\phi) = (p + q \, \sqrt{D})(p - q \, \sqrt{D}) = p^2 - D \, q^2 \in \K[X]$$ Applying the same argument to the inverse $1/\phi$, we find $p^2 - D \, q^2 \in \units{\K[X]} = \units \K$, so $(p,q)$ is a solution of . This proves that $\pi$ is surjective.
Observe that the trivial solutions of correspond precisely to the elements of $\units{\K}$.
Laurent series and valuation {#sec:org7391b36}
----------------------------
Define the field of Laurent series over $\K$ $$\laurinx \K = \left\{\left. \lseries{N}{t}{n} \;\right|\; N \in \Z, t_n \in \K \right\}.$$ It contains $\K[X]$ and its fraction field $\K(X)$. Note that $\laurinx \K$ is the completion of $\K(X)$ with respect to the discrete valuation $\ios = \ord_\infty$ (the zero-order at infinity), defined by $$\io{f} = \ord_\infty(f) = -N \text{ where } f = \lseries{N}{t}{n}, \; f_N \neq 0.$$
\[rem-poly-no-poles\] For example if $f \in \K[X]$, then $\io{f} = - \deg f$. Moreover, $$\label{poly-no-poles}
\io{f} > 0 \text{ and } \quad f \in \K[X] \text{ implies } f = 0.$$
There is a truncation operation which takes a Laurent series and returns a polynomial, essential for the continued fraction process:
\[define-laurent-truncation\] For $\alpha = \lseries{N}{t}{n} \in \laurinx \K$, we define the *truncation* (or *principal part*) $$\gauss{\alpha} = \begin{cases}
0 & \text{ if } \io{\alpha} >0, \text{ i.e. } N < 0 \\
\poly{N}{t} & \text{ if } \io{\alpha} \leq 0, \text{ i.e. } N \geq 0
\end{cases}$$ as the *polynomial part* of $\alpha$.
\[truncation-unique\] We could also define $\gauss{\alpha}$ as the *unique* $a \in \K[X]$ satisfying $\io{\alpha - a} > 0$ – unicity is a consequence of Remark \[rem-poly-no-poles\].
\[truncation-of-sum\] The preceding remark implies for $\alpha, \beta \in \laurinx{\K}$ that $\gauss{\alpha + \beta} = \gauss{\alpha} + \gauss{\beta}$.
\[truncation-of-rational\] Recall that $\K[X]$ is Euclidean with respect to $\deg$. So for $p, q \in \K[X]$ with $q \neq 0$ there exist $a, r \in \K[X]$ satisfying $p = a \, q + r$ and $\deg r < \deg q$. Then $$\frac{p}{q} - a = \frac{r}{q} \text{ with } \io{\ifrac{r}{q}} > 0$$ implies $\gauss{\ifrac{p}{q}} = a$, and moreover $a, r$ are uniquely determined, again by Remark \[truncation-unique\].
We now explain how to compute $\sqrt{D}$ as a Laurent series in $\inv X$:
\[laurent-sqrt-d\] Let $D \in \K[X]$ with $\deg D = 2d$ and $\LC(D) \in \K$ a square. Then $\sqrt{D} \in \laurinx \K$, so $D$ is a square in $\laurinx \K$.
Let $D = \poly{2d}{d}$, where $d_{2d}$ is a square in $\K$. Hence we may reduce to the case $d_{2d} = 1$, and write $$D = X^{2d} \, (1 + f(X)) \text{ where } f(X) = d_{2d-1} \, \inv X + \dots + d_0 \, X^{-2d}.$$ Of course $X^{2d}$ is a square in $\laurinx \K$, and because $\io{f(X)} > 0$, we find that $$\sqrt{1+f(X)} = \sum_{n=0}^\infty \binom{1/2}{n} \, f(X)^n$$ converges in $\laurinx{\K}$, so also $(1 + f(X))$ is a square.
\[choose-sqrt-d\] We choose once and for all one square root of $D$, and denote it by $\sqrt{D}$. We also set $A = \gauss{\sqrt{D}}$. For example, if $D$ is monic of degree $2d$, then we choose $\sqrt{D} = X^d + \dots$.
\[completion-of-square\] We have $\deg A = \frac{1}{2} \deg D$, and $\deg (D - A^2) < \deg A$.
As $\ios$ is a valuation, clearly $- \deg D = \ios{D} = 2 \, \ios{\sqrt{D}} < 0$, hence $-\deg A = \ios{A} = \ios{\gauss{\sqrt{D}}} = \ios{\sqrt{D}}$ which implies the first claim.
Moreover, we can write $$\label{sqrt-d-A-plus-eps}
\sqrt{D} = A + \varepsilon \text{ with } \varepsilon \in \laurinx \K \text{ and } \io{\varepsilon} > 0.$$ So $$D = A^2 + 2 \, A \, \varepsilon + \varepsilon^2$$ where of course $$\io{2 \, A \, \varepsilon + \varepsilon^2} = \min(\io{A}, \io{\varepsilon}) + \io{\varepsilon} = \io{A} + \io{\varepsilon} > \io{A}$$ implies the second claim.
We can rephrase this as
\[completion-of-square-lemma\] There exist $A, \Omega \in \K[X]$ with $\deg \Omega < \deg A = \frac{1}{2} \, \deg D$ satisfying $$D = A^2 + \Omega$$ where $A$ is unique up to a factor $-1$.
Note that the lemma also holds if $D$ is a square.
\[rem-q=1-pell-solution\] If $\deg \Omega = 0$, then clearly $(A, 1)$ is a solution of the Pell equation .
Group structure of Pell solutions {#sec:org4f39766}
---------------------------------
We apply the definitions of the previous section directly to study the structure of the Pell solutions. The group of solutions of is essentially cyclic:
If $D$ is not Pellian, then $\sol{D} = \{ \pm 1 \}$ and $\solu{D} = \units{K}$. But if $D$ is Pellian, then $$\sol{D} \iso \{\pm 1\} \times \Z \quad \text{ and } \quad \solu{D} \iso \units K \times \Z.$$
We use that $\solu{D} \iso \units{\HER}$. By Proposition \[laurent-sqrt-d\], we can embed $\HER$ into $\laurinx \K$, and define $$o(p,q) = \io{p + \sqrt{D} \, q} \text{ for } (p, q) \in \solu{D}.$$ This defines a group homomorphism $o : \solu{D} \to \Z$. The kernel is made precisely of the trivial solutions: $$\io{p} = \io{p + \sqrt{D} \, q + p - \sqrt{D} \, q} \geq \min\left( \io{p + \sqrt{D} \, q}, \io{p - \sqrt{D} \,q}\right)$$ and $$\io{p + \sqrt{D} \, q} + \io{p - \sqrt{D} \, q} = 0$$ so $\io{p + \sqrt{D}} = 0$ implies $\deg p = - \io{p} \leq 0$, hence $q = 0$.
If $D$ is not Pellian, then the image of $o$ is $0$. But if $D$ is Pellian, then the image of $o$ is isomorphic to $\Z$.
We can of course restrict $o$ to $\sol{D}$, and then the kernel becomes $\{(\pm 1, 0) \} \iso \{\pm 1\}$.
The structure of $\sol{D}$ and $\solu{D}$ now follows from standard theorems about group homomorphisms.
We conclude our discussion of the polynomial Pell equation with the following observation:
\[deg-2-always-pellian\] If $\deg D = 2$ and the leading coefficient $\LC(D)$ is a square, then $D$ is *always* Pellian (unless it is square).
By Lemma \[completion-of-square-lemma\], in this case $\deg \Omega < \deg A = 1$ so forcefully $\deg \Omega = 0$, and Remark \[rem-q=1-pell-solution\] says that $(A, 1)$ is a Pell solution.
Rational approximations {#sec:orgbc894a6}
=======================
As mentioned before, the existence of non-trivial solutions is not guaranteed for the polynomial Pell equation. But one observes that the Pell solutions produce very good rational approximations for $\sqrt{D}$ (as in the numerical case). This chapter introduces two notions of rational approximation: convergents and best-approximations. We will study in this chapter how they are related to each other and to the non-trivial Pell solutions. Their complete classification is however best understood with the help of continued fractions, to be discussed later in Section \[sec:orga882f91\].
For our purposes, it is convenient to keep track of common factors in the numerator and denominator of the rational approximation. *Instead* of $\K(X)$, our candidate set for rational approximations is the set of tuples representing quotients $$\Batest{\K} = \{ (p,q) \in \K[X]^2 \mid q \neq 0 \}.$$ We loosely refer to $p$ as the *numerator* and to $q$ as the *denominator*, in spirit of the obvious map $\Batest{\K} \longto \K(X), \; (p,q) \mapsto p/q$.
For $r, p , q \in \K[X]$ with $r, q \neq 0$ we also write $$r \cdot (p, q) = (r \, p, r \, q).$$
With this terminology established, we can begin the study of different types of approximations. Of course, we are using the valuation $\ios = \ord_\infty$ (the zero-order at infinity) introduced in Section \[sec:org7391b36\] to measure how well we can approximate any Laurent series in $\laurinx \K$.
Convergents {#sec:org32d755d}
-----------
A classical type of rational approximation is given by the convergents. They arise very naturally from the continued fraction expansion – we will see details later in Chapter \[sec:org172eb73\]. For now, we give a different characterisation in the spirit of the famous Dirichlet Lemma. This definition also shows immediately that the convergents are a special case of Padé approximations.
Let $\alpha \in \laurinx \K$. A tuple $(p, q) \in \Batest{\K}$ is called a *convergent* of $\alpha$ over $\K[X]$ if it satisfies $$\label{convergent-condition}
% \tag{C}
\io{p - \alpha \, q} > \deg q.$$ We denote the set of all convergents by $\Coset{\alpha}{\K}$.
\[rem-convergents-existence\] It can easily be seen that convergents exist: The condition is a linear condition on the coefficients of $p$ and $q$. Clearly $p$ removes the coefficients of $X^n$ for $n \geq 0$ in $\alpha \, q$; then only the coefficients of $X^{-1}, \dots, X^{-\deg q}$ need to vanish, which can be accomplished by choosing the $1 + \deg q$ coefficients of $q$ appropriately. See Section \[sec:org542977d\] for more details.
\[convergent-cancel-factor\] Suppose $r, p, q \in \K[X]$. Then $$r \cdot (p, q) \in \Coset{\alpha}{\K} \implies (p, q) \in \Coset{\alpha}{\K}$$ because $0 \geq \io{r}$ implies $$\io{p - \alpha \, q} \geq \io{r \, p - \alpha \, r \, q} > \deg(r \, q) \geq \deg q.$$ Note that the implication in the converse direction does not hold in general because multiplication with $r$ decreases $\ord$ and increases $\deg$.
In principle, one could for any convergent $(p, q)$ assume that $p$ and $q$ are coprime, and identify it with the fraction. This might improve the approximation quality, however it turns out that the common factors help to understand the reduction of convergents modulo a prime better (to be discussed in Chapter \[sec:orgd5f1900\]).
Anyway the common factor usually has a small and controllable degree:
\[convergent-common-factor-degree\] Let $(p, q) \in \Batest{\K}$ and $r \in \mino{\K[X]}$. Suppose $$\io{p - \alpha \, q} = \xi + \deg q.$$ Then $r \cdot (p, q) \in \Coset{\alpha}{\K}$ is a convergent $\deg r < \xi/2$.
In particular, suppose $r' \in \mino{\K[X]}$ with $\deg r \leq \deg r'$. Then $r' \cdot (p, q) \in \Coset{\alpha}{\K}$ implies $r \cdot (p, q) \in \Coset{\alpha}{\K}$.
Note that the Proposition holds also when $\xi = \infty$ – but that happens only for $\alpha \in \K(X)$.
In order for $r \cdot (p, q)$ to be a convergent, the following expression must be positive: $$\label{convergent-factor-approx-quality}
\io{r\,p - \alpha\, r\,q} - \degb{r \, q} = \io{r} + \io{p - \alpha \, q} - \deg r - \deg q = \xi - 2 \, \deg r.$$ The second part of the Proposition follows immediately.
The above also suggests that for $(p, q)$ coprime we have the optimal relative approximation quality: higher is better.
Let $\alpha \in \laurinx \K$, set $a = \gauss{\alpha}$. Then $(a, 1) \in \Coset{\alpha}{\K}$ because $\io{a - \alpha} > 0 = \deg 1$.
\[convergent-q-determines-p\] If $(p, q) \in \Coset{\alpha}{\K}$ is a convergent, then $p$ is uniquely determined by $q$ via $p = \gauss{\alpha \, q}$.
This follows immediately from $\io{p - \alpha \, q} > \deg q \geq 0$, and Remark \[truncation-unique\] characterising $\gauss{\cdot}$.
Pell solutions are convergents {#sec:org0d42630}
------------------------------
Let us for a moment return to the polynomial Pell equation, and show that the non-trivial Pell solutions (up to conjugate) are convergents of $\sqrt{D}$. Obviously, not all convergents of $\sqrt{D}$ need to be Pell solutions.
\[weak-pell-solutions-are-convergents\] Let $(p, q) \in \Batest{\K}$ and $p^2 - D \, q^2 = \Omega$. Then the inequality $$\label{general-weak-pell-condition}
\deg \Omega < \tfrac{1}{2} \deg D$$ holds either $(p, q) \in \Coset{\sqrt{D}}{\K}$ or $(p, -q) \in \Coset{\sqrt{D}}{\K}$ is a convergent of $\sqrt{D}$.
In particular, if $(p, q) \in \solu{D}$ is a Pell solution with $q \neq 0$, then one of $(p, q), (p, -q)$ is a convergent of $\sqrt{D}$.
Let us begin with some observation useful to both directions of implication. Note that $$\label{omega-order-factors}
\io{\Omega} = \io{p^2 - D \, q^2} = \io{p + \sqrt{D} \, q} + \io{p - \sqrt{D} \, q}.$$ And if $\io{p - \sqrt{D} \, q} > 0$, the ultrametric inequality and $\io{\sqrt{D} \, q} \leq 0$ imply $$\label{cv-plus-bigger-minus}
\io{p + \sqrt{D} \, q} = \min\left(\io{2 \, \sqrt{D} \, q}, \io{p - \sqrt{D} \,q} \right) = \io{\sqrt{D} \, q} \leq 0.$$
Now assume that $(p, q) \in \Coset{\sqrt{D}}{\K}$ is a convergent, hence $\io{p - \sqrt{D} \,q} > \deg{q} \geq 0$. Then and yield $$\io{\Omega} > \deg q + \io{\sqrt{D} \, q} = \io{\sqrt{D}}$$ which implies $\deg \Omega < \tfrac{1}{2} \deg D$.
For the other direction, assume that $(p, q)$ satisfies , hence $\io{\Omega} > \io{\sqrt{D}} \geq \io{\sqrt{D} \, q}$. Without loss of generality, we may further assume $\io{p - \sqrt{D} \, q} \geq \io{p + \sqrt{D} \, q}$. It follows $$\io{\Omega} > \io{2 \, \sqrt{D} \,q} = \io{p + \sqrt{D} \, q - (p - \sqrt{D} \, q)} \geq \io{p + \sqrt{D} \, q}$$ so by $\io{p - \sqrt{D} \, q} > 0$, which in turn implies . Using again, we arrive at $$\begin{gathered}
\io{p - \sqrt{D} \, q} = \io{\Omega} - \io{p + \sqrt{D} \, q} \\
= \io{\Omega} - \io{\sqrt{D} \,q} > -\io{q} = \deg q\end{gathered}$$ as desired.
The universal property of best-approximation {#sec:orgd9280fe}
--------------------------------------------
The convergents have a useful universal property: they are in some sense the optimal approximations that we can find. For a discussion about where this particular universal property comes from, see [@khintchine-1956-kettenbruche]. See also [@cassels-1957-introduction-to-diophantine] where the continued fraction process for real numbers is defined using best-approximations.[^3]
As we did with the convergents, we modify our definition so that it allows common factors; and we prefer a category theoretic style of universal property.
Let $\alpha \in \laurinx \K$. A tuple $(p,q) \in \Batest{\K}$ is called a *best-approximation* (of second type) in $\K[X]$, if for every other tuple $(p',q') \in \Batest{\K}$ satisfying $$\label{bestapproxcondition}
% \tag{B}
\io{p' - \alpha \, q'} \geq \io{p - \alpha \, q} \text{ and } \deg{q'} \leq \deg{q}$$ we have $\ifrac{p'}{q'} = \ifrac{p}{q}$.
We denote by $\Baset{\alpha}{\K}$ the set of all best-approximations of $\alpha$.
\[best-approx-cancel-factor\] If $(p, q) \in \Batest{\K}$ and $r, r' \in \mino{\K[X]}$ with $\deg r' \leq \deg r$ (for example $r' = 1$), then $$r \cdot (p, q) \in \Baset{\alpha}{\K} \implies r' \cdot (p, q) \in \Baset{\alpha}{\K}.$$ because $$\io{r' \, p - \alpha \, r' \,q} \geq \io{r \, p - \alpha \, r \, q} \text{ and } \degb{r' \, q} \leq \degb{r \, q}.$$
So without loss of generality, one *could* assume that for a best-approximation $(p, q)$, we have $p$ and $q$ coprime. This could also be enforced by changing the phrasing of the definition slightly, as is in fact usually done in the literature. However, in that case, becomes harder to satisfy because removing a common (non-constant) factor decreases $\deg q$ and increases $\io{p - \alpha \, q}$.
As alluded to before, when studying the reduction of convergents modulo a prime, it is useful to allow common factors. The notion of best-approximation presented here gives even more freedom for such common factors than our notion of convergent. We can indeed find best-approximations $(p, q)$ for arbitrary $\deg q$, which may not be possible with convergents (see Section \[sec:orga882f91\]). This simplifies their classification, and hence the classification of convergents.
Before we investigate the relation between convergents and best-approximations, let us show that there are not so many best-approximations:
\[best-approx-for-given-degree\] Let $(p, q) \in \Batest{\K}$ coprime and $r \in \mino{\K[X]}$. Suppose $r \cdot (p, q) \in \Baset{\alpha}{\K}$ is a best-approximation.
Then any (other) best-approximation $(p', q') \in \Baset{\alpha}{\K}$ with $\deg{q'} = \degb{r \, q}$ has the shape $$(p', q') = r' \cdot (p, q) \text{ where } r' \in \K[X], \; \deg r = \deg r'.$$
Because $(p', q'), r \cdot (p, q) \in \Baset{\alpha}{\K}$ with $\deg q' = \degb{r \, q}$, at least one of $$\io{p' - \alpha \, q'} \geq \io{r\,p - \alpha \, r\,q} \text{ or } \io{p' - \alpha \, q'} \leq \io{r\,p - \alpha \, r\,q}$$ must be satisfied. Together with $\deg q' = \degb{r \, q}$ this implies $\frac{p'}{q'} = \frac{r\,p}{r\,q} = \frac{p}{q}$ by the best-approximation property of either $r \cdot (p, q)$ or $(p', q')$.
Finally because we assume $p, q$ are coprime, there exists $r' \in \K[X]$ with $q' = r' \, q$ and $p' = r' \, p$.
This proposition has two important consequences:
For any best-approximation $(p, q) \in \Baset{\alpha}{\K}$, the numerator $p$ is uniquely determined by the denominator $q$.
\[best-approx-coprime-unicity\] Given an integer $n \geq 0$, there exists up to a constant factor at most one best-approximation $(p, q)$ with $\deg {q} = n$ and $p, q$ coprime.
We proceed to show that best-approximations generalise the convergents.
\[best-approx-common-factor-degree\] Let $(p, q) \in \Batest{\K}$ and $r \in \mino{\K[X]}$. Suppose $$\io{p - \alpha \, q} = \xi + \deg q.$$ Then $\deg r < \xi$ implies $r \cdot (p, q) \in \Baset{\alpha}{\K}$ is a best-approximation.
Putting $r = 1$ with $\deg r = 0 < \xi$ by definition of convergents, we get:
\[convergents-are-best-approx\] Every convergent is a best-approximation: $\Coset{\alpha}{\K} \subset \Baset{\alpha}{\K}$.
With Corollary \[weak-pell-solutions-are-convergents\] this implies also:
\[pell-solutions-are-bestapprox\] For every non-trivial solution $(p, q)$ of the Pell equation , either $(p, q)$ or $(p, -q)$ is a best-approximation of $\sqrt{D}$.
Let $(p', q') \in \Batest{\K}$ satisfy $$\deg{q'} \leq \degb{r \, q} \text{ and } \io{p' - \alpha \, q'} \geq \io{r} + \io{p - \alpha \, q}.$$ Now $$\det \mfour{p}{p'}{q}{q'} = \det \mfour{1}{-\alpha}{0}{1} \mfour{p}{p'}{q}{q'} = \det \mfour{p-\alpha\,q}{p'-\alpha \, q'}{q}{q'}$$ and taking the valuation $\ios$ we get $$\begin{gathered}
\io{p \, q' - p' \, q} \geq \min\left(\io{q'} + \io{p-\alpha\,q}, \io{q} + \io{p'-\alpha \, q'} \right) \\
\geq \io{r} + \io{q} + \io{p - \alpha \, q} = \xi - \deg r > 0.\end{gathered}$$ But $p \, q' - p' \, q \in \K[X]$, so it must be $0$. This implies $\ifrac{p'}{q'} = \ifrac{p}{q} = \ifrac{r \, p}{r \, q}$ as desired.
Note that unlike Proposition \[convergent-common-factor-degree\], this is only a sufficient condition. It is not necessary: if we start with $(p, q)$ with $\xi > 1$ (for example $\xi = 2$), then multiplying with $r$ of maximal degree (for example $\deg r = 1$), we obtain a best-approximation $(p', q')= r \cdot (p, q)$ with $\xi' \leq 0$ (in the example $\xi' = 0$). Then $r' = 1$ does not satisfy $\deg r' < \xi'$, even though $(p', q')$ is a best-approximation.
We conclude our study of best-approximations by investigating their ordering. Indeed we expect that increasing the “height” of the convergent (i.e. $\deg q$) should also increase the approximation quality:
\[compare-best-approx-strict\] Let $(p, q), (p', q') \in \Baset{\alpha}{\K}$ different best-approximations, i.e. $\ifrac{p}{q} \neq \ifrac{p'}{q'}$. Then $$\deg{q} < \deg{q'} \iff \io{p - \alpha \, q} < \io{p' - \alpha \, q'}.$$
By the universal property, the statement $$\label{best-approx-condition-reverse}
\io{p - \alpha \, q} \geq \io{p' - \alpha \, q'} \text { and } \deg{q} \leq \deg{q'}$$ is false under the hypothesis of the fractions being different.
So if $\deg q < \deg q'$, necessarily the first inequality must not hold, giving the $\Rightarrow$ part.
Conversely, if $\io{p - \alpha \, q} > \io{p' - \alpha \, q'}$, then the second inequality is false, i.e. $\deg q > \deg q'$. But this is clearly the $\Leftarrow$ part, with the roles of $(p, q)$ and $(p', q')$ swapped.
If we restrict to coprime approximations, we don’t even need strict inequalities:
\[compare-best-approx-coprime\] Let $(p, q), (p', q') \in \Baset{\alpha}{\K}$ where $p, q$ and $p', q'$ respectively are coprime. Then $$\deg{q} \leq \deg{q'} \iff \io{p - \alpha \, q} \leq \io{p' - \alpha \, q'}.$$
This is also covered by Proposition \[compare-best-approx-strict\], unless $p/q = p'/q'$. But in this case, the best-approximations differ only by a constant factor, so both inequalities actually become equalities.
A linear system for computing convergents {#sec:org542977d}
-----------------------------------------
This thesis contains three different proofs for the existence of convergents of arbitrary approximation quality. There is a geometric argument to be explained in Chapter \[sec:org99a8e17\]. The most elegant approach uses the continued fraction expansion, and yields a complete classification of convergents and best-approximations at the same time; it is one of the main goals of Chapter \[sec:org172eb73\]. But here, we give an elementary proof which uses only some linear algebra and other results from this chapter.
We describe a linear system which allows to compute the convergents, alluded to already in Remark \[rem-convergents-existence\]. This already demonstrates the existence of convergents. We will also use these results in Chapter \[sec:org56e7cb7\] to produce estimates for the projective height of the convergents.
See also [@platonov-2014-number-theoretic-properties], where a version of this linear system with additional conditions/rows is used to determine the existence of Pell solutions.
From Proposition \[convergent-q-determines-p\] we know $p = \gauss{\alpha \, q}$ which gives a linear condition on the coefficients of $p$. Moreover, from the Cauchy product formula, it is clear that every coefficient of $\alpha \, q$ is a linear expression in the coefficients of $q$. And requires just finitely many coefficients of $p - \alpha \, q$ to vanish, so this produces a linear condition on the coefficients of $q$ as well.
We make this more precise now, and start by fixing notation:
Write $\alpha = \lseries{N}{A}{j}$ (with $A_N \neq 0$, so $\io{\alpha} = -N$), and $q = \poly{n}{Q}, \; p = \poly{n+N}{P}$. The $A_n$ are given, and we are solving for $P_{n+N}, \dots, P_0, Q_n, \dots, Q_0$, a total of $N + 2n + 2$ unknowns. For simplicity, we assume $N \geq 0$, but the argument works for negative $N$ as well. We get $$\begin{array}{lllllll}
\alpha \, q -p = & X^{n+N} & (-P_{n+N} &+ A_N \, Q_n) \\
&+ X^{n+N-1} & (-P_{n+N - 1} &+ A_{N-1} \, Q_n &+A_N \, Q_{n-1}) \\
& & \quad \vdots \\
&+ X^{n} & (-P_n &+ A_{0} \, Q_{n} & \dots &+ A_n \, Q_0) \\
&+ X^0 & (-P_0 &+ A_{-n} \, Q_n & \dots &+ A_{0} \, Q_0) \\
& & \quad \vdots \\
&+ X^{-n} & ( &+ A_{-2n} \, Q_n & \dots & + A_{-n} \, Q_0) \\
&+ \dots
\end{array}$$ and the condition $\io{\alpha \, q - p} > \deg q = n$ means that at the very least the coefficients of $X^{n+N}, \dots, X^{-n}$ vanish. We count a total of $N + 2n + 1$ conditions linear in the $P_i$ and $Q_i$.
So the matrix describing the linear system has $N + 2n + 2$ columns and $N + 2n + 1$ rows; the right part (and also the left) on its own has the shape of a Toeplitz matrix:[^4] $$\label{convergents-condition-matrix}
\pqmatrix_n = \left( \begin{array}{ccccc|ccc}
-1 & & & & 0 & A_N & & 0\\
&\ddots & & & & A_{N-1} & \ddots \\
& & \ddots & & & \vdots & \ddots & A_{N} \\
& & & \ddots & & \vdots & \ddots & \vdots \\
0 & & & & -1 & A_{-n} & \dots & A_{0} \\ \hline
& & & & & A_{-n-1} & \dots & A_{-1} \\
& & 0 & & & \vdots & \ddots & \vdots \\
& & & & & A_{-2n} & \dots & A_{-n}
\end{array} \right)
% \cdot
% \left( \begin{array}{c}
% P_{n+N} \\ \vdots \\ P_0 \\ Q_n \\ \vdots \\ Q_0
% \end{array} \right)
% = 0$$
Every non-zero element of $\ker \pqmatrix_n$ yields a convergent $(p, q)$. As always $\ker \pqmatrix_n \neq 0$, this implies that for any $\alpha \not \in \K(X)$ there exist convergents with arbitrarily high $\io{p - \alpha \, q}$.
From the discussion above, it is evident that an element of the kernel gives polynomials $(p, q)$ which are a convergent of $\alpha$ as soon as $q \neq 0$. But if an element of $\ker \pqmatrix_n$ has all $Q_i = 0$, then clearly it follows that also all $P_i = 0$. So we only need to avoid the zero element. And elementary linear algebra tells as that $\ker \pqmatrix_n \neq 0$ because there are more columns than rows.
If $\alpha \in \K(X)$, then of course at some point $\io{p - \alpha \, q} = \infty$, so the approximation quality can no longer be improved.
Note that for a single $\pqmatrix_n$, we do not get different convergents:
If $(p, q)$ and $(p', q')$ correspond to non-zero kernel elements, then $p/q = p'/q'$.
Let $(p_i, q_i)$ for $i=1,\dots,r$ correspond to a basis of $\ker \pqmatrix_n$. Then for any $(p, q)$ corresponding to a solution, we get $$(p, q) = \sum_{i=1}^r \eta_i \cdot (p_i, q_i) \text{ where } \eta_i \in \K$$ and hence $$\io{p - \alpha \, q} = \io{\sum_{i=1}^r \eta_i \, (p_i - \alpha \, q_i)} \geq \min_{i=1,\dots,r} \left(\io{p_i - \alpha \, q_i}\right)$$ so there exists $(p, q)$ in the kernel with $\io{p - \alpha \, q}$ minimal. Write $\io{p - \alpha \, q} = \xi + \deg q > n$. By Proposition \[best-approx-common-factor-degree\] also $X^{\xi - 1} \cdot (p, q)$ is a best-approximation.[^5] And by minimality of $\io{p - \alpha \, q}$, we have for every $(p', q')$ in the kernel $$\io{p' - \alpha \, q'} \geq \io{p - \alpha \, q} \geq \io{X^{\xi-1} \, (p - \alpha \, q)}$$ and moreover $\deg q' \leq n \leq \degb{X^{\xi -1} \, q}$ which implies $p'/q' = p/q$.
We can also compute the dimension of the kernel (i.e. the rank of $\pqmatrix_n$):
\[convergent-linear-matrix-full-rank\] There exists $(p, q)$ in the kernel with $p$ and $q$ coprime.
If $\io{p - \alpha\, q} = \xi + \deg q$, then $$\dim \ker \pqmatrix_n = \min(1 + \floor{(\xi-1)/2}_\Z, 1 + n - \deg q, \xi + \deg q - n)$$ where $\floor{\cdot}_\Z$ denotes the next lowest integer. So if $\xi \leq 2$ or $n = \deg q$, the matrix $\pqmatrix_n$ has full rank.
Removing a common factor decreases $\deg q$ and increases $\io{p - \alpha \, q}$, so the existence of any solutions implies the existence of a coprime solution. Of course, by the previous Proposition, we can produce all other solutions by adding back a common factor $r$, with has to satisfy $\deg r \leq n - \deg q$, $\deg r < \xi/2$, and also $$\io{r} + \io{p - \alpha \, q} = \io{r} + \xi + \deg q > n$$ which is equivalent to $\deg r < \xi + \deg q -n$.
These results hold for any $\alpha \in \laurinx \K$, even if $\alpha \in \K(X)$.
With Cramer’s rule we can compute an element of the kernel:
\[rem-cramers-rule\] Denote by $\det \pqmatrix_n(i)$ the $i$th minor obtained by striking the $i$th column. Then $$\begin{gathered}
\label{toeplitz-matrix-convergents}
\left(P_{n+N}, \dots, P_0, Q_n, \dots, Q_0\right) = \\
\left(\det \pqmatrix_n(1), - \det \pqmatrix_n(2), \det \pqmatrix_n(3), \dots \right. \\ \dots, (-1)^{N+2n} \det \pqmatrix_n(N+2n+1), \\ \left. (-1)^{N+2n+1} \det \pqmatrix_n(N+2n+2) \right)\end{gathered}$$ is an element of the kernel. If $\pqmatrix_n$ has full rank, then it is clearly non-zero.
These formulas present an alternative to computing convergents via the continued fraction, and we will later show that the convergents obtained in this way are actually optimally normalised (see Proposition \[prop-convergents-hankel-determinants-normalised\]).
A (hyper)elliptic curve {#sec:org99a8e17}
=======================
In this chapter, we describe the (hyper)elliptic curve corresponding to a given polynomial Pell equation. We additionally assume that $D$ is square-free, to avoid complications and so that we may work with the Jacobian of the curve.[^6]
We also explain how the convergents give rise to principal divisors of particular shape (Lemma \[convergent-divisor-lemma\]), and this gives rise to the torsion condition for $D$ being Pellian (Theorem \[thm-pellian-iff-torsion\]).
Most of the results of this chapter have long been known, probably already to Abel [@abel-1826-ueber-integ-differ] and Chebyshev [@chebyshev-1857-sur-integration], albeit not in our modern mathematical language. More recent publications are [@adams-razar-1980-multiples-points-on] for elliptic curves, or [@berry-1990-periodicity-continued-fractions] for arbitrary genus.
As in the previous chapters, we assume that $\K$ is a field of characteristic not $2$.
Defining the (hyper)elliptic curve {#sec:orgf4393ef}
----------------------------------
Let $D \in \K[X] \setminus \K$ *square-free* with even degree $2(g+1)$ and $\LC(D)$ a square in $\K$. Then $$\CCa : Y^2 = D(X)$$ defines an affine (plane) curve over $\K$ of genus $g$.
\[affine-curve-is-smooth-and-normal\] The curve $\CCa$ is smooth and normal in $\A^2_\K$.
The curve is defined by the equation $$F = Y^2 - D(X).$$ Applying the Jacobian criterion we calculate $$\pdiff{X}{F} = - \pdif_{X} D(X) = D'(X) \qquad
\pdiff{Y}{F} = 2 \, Y$$ which are never simultaneously $0$ because $D$ square-free implies that $D$ and $D'$ are coprime.
For normality, we need to show that if $p + Y \, q \in \Fr(\K[X,Y]/\spann{Y^2 - D(X)}) = \K(X)[Y]/\spann{Y^2 - D(X)}$ is integral, it is already contained in $\K[X,Y]/\spann{Y^2 - D(X)}$, i.e. $p, q \in \K[X]$ are polynomials. Recall that the integral closure is a subring of the fraction field, and $p + Y \, q$ integral implies that the conjugate $p - Y \, q$ is integral as well. It follows that $2 \, p$ and $p^2 - D \, q^2$ are integral. As we assume $\Char \K \neq 2$, this implies $p$ and also $D \, q^2$ are integral over $\K[X,Y]/\spann{Y^2 - D(X)}$, so in particular over the subring $\K[X]$. As $D$ is square-free, it follows that $p, q \in \K[X]$ as desired, and $\CCa$ is normal.
If $\deg D = 2$, then $\closure{\CCa} \subset \P^2_\K$ remains smooth at infinity, so it is isomorphic to $\P^1$ (see Proposition 7.4.1 in [@liu-2002-algebraic-geometry-arithmetic]).
But if $\deg D > 2$, then $\closure{\CCa} \subset \P^2_\K$ has a singularity at infinity (easily verified with the Jacobian criterion).
We build a smooth projective model for $\CCa$, as in Lemma III.1.7 of [@miranda-1995-algebraic-curves-riemann]:
Define the curve $$\CCinf : V^2 = D^\flat(U) = U^{2(g+1)} D(1/U)$$ where $D^\flat(U)$ is a polynomial of degree at most $2(g+1)$ – its coefficients are those of $D$ in reverse order. Note that $D^\flat(0) \neq 0$ because $\deg D = 2(g+1)$, and by Proposition \[affine-curve-is-smooth-and-normal\] the curve $\CCinf$ is smooth in $\A^2_\K$.
The relations $X \, U = 1$ and $U^{g+1} \, Y = V$ (respectively $X^{g+1} \, V = Y$) describe a birational map between $\CCa$ and $\CCinf$ which is an isomorphism outside of $U = 0$ and $X = 0$. So we may glue $\CCa$ and $\CCinf$ together to obtain a curve $\CC$. This simply adds two points $O_\pm$ with $U = 0$ to $\CCa$, the points at infinity.
\[projective-curve-is-smooth-and-normal\] The curve $\CC$, glued together from $\CCa$ and $\CCinf$ is a normal smooth projective curve over $\K$.
Normality and smoothness of $\CC$ are local conditions, hence they follow from Proposition \[affine-curve-is-smooth-and-normal\] applied to $\CCa$ and $\CCinf$.
We get a finite morphism $\CC \to \closure{\CCa} \subset \P^2_\K$, hence $\CC$ is proper over $\K$. As $\CC$ is an algebraic variety, this implies by Remark 3.3.33 (1) in [@liu-2002-algebraic-geometry-arithmetic] that it is projective.
There is an involution $\sigma$ defined by $X \mapsto X, \; Y \mapsto -Y$, or $U \mapsto Y, \; V \mapsto -V$. By abuse of notation, we also consider it as an automorphism of the function field $\K(X,Y)$. If we quotient $\CC$ by the group $\{1, \sigma\}$, we find that $\CC$ is (hyper)elliptic (we use Definition 7.4.7 from [@liu-2002-algebraic-geometry-arithmetic] which is essentially the content of the following proposition):
\[curve-is-hyper-elliptic\] There is finite morphism $\pi : \CC \to \P^1$ of degree $2$ defined by $(x,y) \mapsto (x:1)$ on $\CCa$ and $\pi(O_\pm) = (1:0)$. For $g = 1$, the curve $\CC$ is elliptic, and for $g \geq 2$ it is hyperelliptic.
The map $\pi$ is defined on $\CCa$ via $(x, y) \mapsto (x:1)$, and on $\CCinf$ via $(u, v) \mapsto (1:u)$. Clearly the definitions are compatible on the intersection (because there we have $x \, u = 1$). It is also clear that $\pi$ is a finite morphism of degree $2$ which means that $\CC$ is elliptic for $g = 1$ and hyperelliptic for $g \geq 2$.
Divisors and the Jacobian variety {#sec:org40007f2}
---------------------------------
We recall some basic notions about divisors and the Jacobian variety now. For more details, consult your favourite algebraic geometry book, for instance [@hartshorne-1977-algebraic-geometry], [@goertz-wedhorn-2010-algebraic-geometry-i] or [@liu-2002-algebraic-geometry-arithmetic]. For the rest of the chapter, we work over the algebraic closure $\closure \K$ to avoid complications.
### Divisors {#sec:orgeef8812}
For any $P \in \CC(\closure{\K})$, there is a discrete valuation $$\ord_P : \units{\closure{\K}(X,Y)} \to \Z,$$ the zero-order of $P$ of a function on $\CC$. In fact, all non-trivial discrete $\closure\K$-valuations (up to equivalence) on $\closure\K(X,Y)$ arise in this way.
By the *group of divisors* $\DIV(\CC)$ we understand the free abelian group generated by all points of $\CC(\closure\K)$ (we mark divisors in **bold**). For every divisor $$\di{D} = \divsum n_P \, \pd{P}, \text{ where } n_P \in \Z$$ we define the *degree* $$\deg \di{D} = \divsum n_P.$$ A divisor is called *effective* if $n_P \geq 0$ for all $P$.
For every element $f \in \units{\closure\K(X,Y)}$, only finitely many $\ord_P f$ are non-zero, so we can define the *divisor of $f$* as $$\Div f = \divsum (\ord_P f) \, \pd{P}.$$ The divisors arising in this way are called *principal divisors*, and they all have degree $0$. So there is group homomorphism $\Div : \units{\closure\K(X,Y)} \to \DIV^0(\CC)$ where $\DIV^0(\CC)$ denotes the divisors of degree $0$.
Recall that the Jacobian $\Jac$ of $\CC$ is an abelian variety of dimension $g$. If $g = 1$ (i.e. $\deg D = 4$), the curve $\CC$ is an elliptic curve (Corollary 7.4.5 in [@liu-2002-algebraic-geometry-arithmetic]), and it is isomorphic to its Jacobian.
The $\closure\K$-rational points of the *Jacobian* can be seen as the cokernel of the divisor map, more precisely the quotient $$\Jac = \Jac(\CC) = \DIV^0(\CC) / \im \Div,$$ with the projection $\DIV^0(\CC) \to \Jac$. By abuse of language, we call both the algebraic variety and its set of $\closure\K$-rational points “Jacobian”.
We write a divisor class in the Jacobian as $$\jdi{D} = \divsum n_P \, \j{P}.$$
### Order functions {#sec:org85f4b43}
If we restrict $\ord_{O_\pm}$ to $\closure\K(X)$, it becomes exactly $\ord_\infty = \ios$ from Section \[sec:org7391b36\]. As mentioned before, there are precisely two embeddings of $\closure\K(X,Y)$ into the completion $\laurinx{\closure\K}$. To distinguish them properly, we set for $p, q \in \closure\K(X)$ $$\ord_{O_+}(p + Y \, q) = \io{p + \sqrt{D} \, q}, \quad \ord_{O_-}(p + Y \, q) = \io{p - \sqrt{D} \, q}$$ for a fixed choice of $\sqrt{D}$ (see Definition \[choose-sqrt-d\]). Apart from this, the roles of $O_+$ and $O_-$ are essentially interchangeable (by the involution $\sigma$).
In a similar way, one may compute order functions for a finite point $P = (x, y) \in \CCa$ by choosing an uniformiser. Sending $X$ to $T + x$ gives a homomorphism $\closure\K(X) \to \laurent{\closure\K}{T}$, and if $y \neq 0$, one may compute $\sqrt{D(T+x)}$ in $\laurent{\closure\K}{T}$ with the constant coefficient $y$ determining the choice of square root. Sending $Y$ to $\sqrt{D(T+x)}$ then establishes a homomorphism $\HEF \to \laurent{\closure\K}{T}$ with $\ord_P$ corresponding to $\ord_{T=0}$.
If $y = 0$, one sends instead $X$ to $T^2 + x$, to ensure $\sqrt{D(T^2+x)} \in \laurent{\closure\K}{T}$. Because the latter has odd $\ord_{T=0}$, the choice of root does not matter, and one obtains as before the correspondence between $\ord_P$ and $\ord_{T=0}$. Note that in this case for $f \in \closure\K(X)$ the zero-order $\ord_P(f)$ is always *even*.
### Embedding the curve in the Jacobian {#sec:orgdaae3ac}
Choosing the base point $O_+$ (a natural choice here, but any other point on $\CC$ would do as well), define the map $j :\CC \to \Jac$ via $P \mapsto \j{P} - \j{O_+}$ which is an embedding for $g \geq 1$ (see Theorem A8.1.1 in [@hindry-silverman-2000-diophantine-geometry]). Actually, for $g = 1$, when $\CC$ is an elliptic curve, it is an isomorphism of curves, determined uniquely by the choice of the base point.
Of course, we can extend $j : \DIV(\CC) \to \Jac$ as a homomorphism of groups (using that $\DIV(\CC)$ is a free group on $\CC$).
For each $r \geq 0$, we may also define a subvariety of $\Jac$ $$W_r = j(\CC) + \dots + j(\CC) \quad (r \text{ copies})$$ remarking $W_g = \Jac$ (see again Theorem A8.1.1 in [@hindry-silverman-2000-diophantine-geometry]), while the Theta divisor $\Theta = W_{g-1}$ forms a proper subvariety which depends on the embedding $j$. We will use this divisor with the Weil height machine later, and likewise the canonical divisor.
The canonical divisor $\canondiv$ on $\CC$ is represented by $$\Div\left( \frac{\dif X}{Y} \right) = (g-1) \, ( \pd{O_+} + \pd{O_-} ).$$
From Riemann-Roch one deduces easily that $\deg \canondiv = 2(g-1)$. It is also clear that we obtain the canonical divisor class by computing the divisor of any differential on $\CC$.
Now outside of infinity, the sheaf of differentials is clearly generated by $\dif X$ and $\dif Y$ which enjoy the relation $$2 \, Y \, \dif Y = D'(X) \, \dif X$$ obtained by differentiating the equation of the curve. This tells us that outside of $D(X) = 0$, the sheaf of differentials is generated by $\dif X$, while outside of $D'(X) = 0$ it is generated by $\dif Y$. Moreover we see that $\dif X$ vanishes only on $Y = 0$ (i.e. $D(X) = 0$), while $\dif Y$ vanishes only on $D'(X) = 0$.
It follows that $\frac{\dif X}{Y}$ has poles and zeroes only at infinity. As the divisor of this differential is invariant under the involution $\sigma$ (which changes the differential only by a factor $-1$), and has to have degree $2(g-1)$, we obtain the above formula.
Divisors of convergents {#sec:org5685823}
-----------------------
Given a rational approximation $(p, q)$, it is very natural to build the function $p - Y\, q$ and study its divisor. For the convergents, we will see that this divisor describes how the multiples of the divisor at infinity $$\label{divisor-at-infinity}
% \tag{O}
\OO = \pd{O_+} - \pd{O_-} \in \DIV^0(\CC)$$ are represented as sums of $g$ points, i.e. as elements of $W_g = \Jac$. Note that $\OO$ is actually a $\K$-rational divisor, so $\j{\OO}$ is a $\K$-rational point of $\Jac$.
\[order-finpt-polynomial\] Let $p, q \in \closure\K(X)$ and $\phi_\pm = p \pm Y \, q \neq 0$.
1. $p, q \in \closure\K[X]$
2. For all $P \neq O_\pm$ holds $\ord_P \phi_+ \geq 0$
3. For all $P \neq O_\pm$ holds $\ord_P \phi_- \geq 0$
<!-- -->
1. and 3. are clearly equivalent because $\ord_P \phi_+ = \ord_{\sigma(P)} \phi_-$ for all $P$.
Together, they imply 1.: $$\begin{aligned}
\ord_P(p) = \ord_P(2\,p) &= \ord_P(\phi_+ + \phi_-) \geq \min( \ord_P(\phi_+), \ord_P(\phi_-)) \geq 0 \\
\ord_P(Y\,q) = \ord_P(2\,Y\,q) &= \ord_P(\phi_+ - \phi_-) \geq \min( \ord_P(\phi_+), \ord_P(\phi_-)) \geq 0\end{aligned}$$ so clearly $p$ has no poles outside infinity, hence it is a polynomial. If $\ord_P(Y) \neq 0$, then $\ord_P(Y) = 1$ because $D$ is square-free. But at the same time $\ord_P(q)$ must be even (see Section \[sec:org85f4b43\]). This shows that $\ord_P(q) \geq 0$, and that $q$ has no poles outside infinity which means it is a polynomial.
Conversely 1. implies 2.: if $p, q \in \closure\K[X]$, then $\ord_P(p) \geq 0$, $\ord_P(q) \geq 0$ and of course $\ord_P(Y) \geq 0$ for all $P \neq O_\pm$. Hence $$\ord_P(\phi_\pm) = \ord_P(p \pm Y \, q) \geq \min(\ord_P(p), \ord_P(Y) + \ord_P(q)) \geq 0$$ as desired.
\[convergent-divisor-lemma\] Let $p, q \in \closure\K(X)$, and $\phi_\pm = p \pm Y \, q \neq 0$. Set $m = \deg p$. Then $(p, q) \in \Coset{\sqrt{D}}{\closure\K}$ (it is a convergent of $\sqrt{D}$) $m > 0$ and there exists $0 \leq r \leq \min(g, m)$ and $P_1, \dots, P_r \in \CCa$ such that $$\label{convergent-divisor-equation}
\Div \phi_- = -m \, \pd{O_-} + (m-r) \, \pd{O_+} + \pd{P_1} + \dots + \pd{P_r}.$$ We call $\Div \phi_-$ a *convergent divisor*.
By Proposition \[order-finpt-polynomial\] we can clearly restrict to the case $p, q \in \closure\K[X]$ as the divisor in allows only poles at infinity, and convergents are always made of polynomials. The rest of the proof boils down to distinguishing the points at infinity and calculating $r$.
Obviously $\ord_{O_+} \phi_- = \io{p - \sqrt{D} \, q} \geq 0$ holds for both conditions and implies $$\begin{gathered}
\ord_{O_-} \phi_- = \ord_{O_+} \phi_+ = \io{p + \sqrt{D} \, q} \\= \min(\io{p}, \io{p- \sqrt{D}\, q}) = \io{p} = -m.\end{gathered}$$ Similarly, $m = \deg q + g + 1$ (see also the proof of Proposition \[weak-pell-solutions-are-convergents\]).
Now $P_1, \dots, P_r$ are the finite zeroes of $\phi_-$, accounted for with multiplicities. Of course $\Div \phi_-$ has degree $0$, hence $$\io{p - \sqrt{D} \, q} = \ord_{O_+} \phi_- = (m-r) = \deg q + g + 1 - r.$$ This is $> \deg q$ (i.e. $(p, q) \in \Coset{\sqrt{D}}{\closure\K}$) $r \leq g$, so we have the desired equivalence.
We will give a slight generalisation (extending to other elements of the function field) later in Section \[sec:org4e4485e\], to illustrate the connection with the continued fraction.
\[rem-convergent-lemma-jacobian\] In the Jacobian, we can write this divisor relation as $$m \cdot j(O_-) = j(P_1) + \dots + j(P_r).$$
\[convergent-omega-degree\] With the notation from Proposition \[weak-pell-solutions-are-convergents\], we get $\deg \Omega = r$ because $$\io{\Omega} = \ord_{O_+}(\phi_+ \cdot \phi_-) = \ord_{O_+}(\phi_+) + \ord_{O_+}(\phi_-) = -m + (m-r) = -r.$$ In the same proposition, the condition to obtain a convergent (up to sign of $q$) was $r = \deg \Omega < \frac{1}{2} \deg D = g+1$ which matches the above lemma.
For every $n \in \N$ there exists $\phi_n \in \mino{\closure\K(X,Y)}$ such that $$\Div \phi_n = -m \, \pd{O_-} + (m-r) \, \pd{O_+} + \pd{P_1} + \dots + \pd{P_r}$$ with $m \geq n$, $r \leq \min(g,m)$ and $P_1, \dots, P_r \in \CCa$.
For $n \in \N$ define the divisor $$\di{D}_n = (n+g) \, \pd{O_-} - n \, \pd{O_+}.$$ which has degree $\deg \di{D}_n = g$. Then the Riemann-Roch theorem (see Theorem IV.1.3 in [@hartshorne-1977-algebraic-geometry]) implies $$\dim \{ \phi \in \mino{\closure\K(X,Y)} \mid \Div \phi + \di{D}_n \geq 0 \} \geq \deg \di{D}_n - g + 1 = 1$$ so there exists $\phi_n$ with $\Div \phi_n \geq -\di{D}_n$. More precisely, we get $$\Div \phi_n = -(n+g) \, \pd{O_-} + n \pd{O_+} + \pd{P_1} + \dots + \pd{P_g}$$ where $P_i \in \CC$ (possibly $O_\pm$). We can write this as $$\Div \phi_n = -m \pd{O_-} + (m-r) \pd{O_+} + \pd{P_{i_1}} + \dots + \pd{P_{i_r}},$$ cancelling out any $O_-$ among the $P_i$ (hence $m \geq n + g - g = n$) and absorbing any $O_+$ from the $P_i$ (hence $m -r \geq n + g \geq 0$). And of course $r \leq g$.
Via the above lemma, we now have another proof for the existence of convergents:
$\sqrt{D}$ has convergents $(p, q)$ of arbitrarily high $\deg p$ (or $\deg q$).
\[thm-pellian-iff-torsion\] The Pell equation has a non-trivial solution $\j{\OO}$ is a torsion point in the Jacobian $\Jac$ of $\CC$.
From Proposition \[weak-pell-solutions-are-convergents\] we know that the Pell solutions (up to conjugation) form a subset of the convergents. By Remark \[convergent-omega-degree\] it is precisely the non-trivial Pell solutions for which we have $r = 0$ in Lemma \[convergent-divisor-lemma\].
By Remark \[rem-convergent-lemma-jacobian\], this implies $m \, \j{\OO} = m \, j(O_-) = 0$ with $m > 0$. In other words, $\j{\OO}$ is a torsion point in the Jacobian $\Jac$.
Conversely, if $\j{\OO}$ is torsion, then there exists some function $\phi$ with divisor $\Div \phi = m \, \left( \pd{O_+} - \pd{O_-}\right)$ and $m > 0$. By Lemma \[convergent-divisor-lemma\], we have $\phi = p - Y \, q$ where $(p, q)$ is a convergent (actually a Pell solution because $r = 0$).
Recall that the genus $g$ corresponds to the dimension of the Jacobian. For $g = 0$, the Jacobian is the trivial group, hence $\j{\OO}$ is trivially torsion. Hence $D$ is always Pellian as observed before in Corollary \[deg-2-always-pellian\].
\[finite-field-torsion-bound\] If the base field $\K$ is finite, i.e. $\K = \F_q$ with $q$ some prime power, the $\K$-rational points of the Jacobian form a finite group. The Hasse-Weil interval (conjectured by E. Artin in his thesis, then proved by Hasse for elliptic curves [@hasse-1936-zur-theorie-1; @hasse-1936-zur-theorie-2; @hasse-1936-zur-theorie-3], and generalised by Weil to higher genus curves in [@weil-1949-numbers-solutions-equations]) then provides the following bounds for the number of elements of the Jacobian: $$\ord( \j{\OO} ) \in [(\sqrt{q} - 1)^{2g}, (\sqrt{q} + 1)^{2g}].$$
Note that $\Jac(\F_q)$ can be cyclic (for elliptic curves, see for example [@gupta-murty-1990-cyclicity-generation-points]), so we cannot hope to improve this bound for the point $\j{\OO}$.
If $\K$ is not finite, then as mentioned in the introduction, this torsion condition allows to demonstrate the scarcity of Pellian polynomials. The polynomials of degree $2d$, after some normalisation, form an affine variety of dimension $2d-2$. The Pellian polynomials are then contained in a denumerable union of subvarieties of dimension at most $d-1$, corresponding to the possible torsion orders. See Section 12.2.2 in [@zannier-2014-pell-survey] for details.
Continued fractions {#sec:org172eb73}
===================
In this chapter, we develop the theory of polynomial continued fractions, to build a solid foundation for the specialization questions that form the main results of this thesis. Beginning with formal continued fractions, moving on to convergence questions in $\laurinx \K$ and a classification of the best-approximations, we conclude with a discussion of periodic continued fractions and reducedness which is relevant mostly for hyperelliptic continued fractions.
The first sections reiterate well-known facts about continued fractions in modern language. Already Abel [@abel-1826-ueber-integ-differ] and Chebyshev [@chebyshev-1857-sur-integration] worked with this type of polynomial continued fractions which they adapted from the numerical continued fraction expansion for square roots. Indeed there are not many differences with the theory of continued fractions for real numbers.
Most results in this chapter may already be found the literature, albeit presented differently. The formal definitions of continued fractions can be found in classical books on continued fractions, for example [@perron-1954-lehre-von-den], [@perron-1957-lehre-von-den], [@khintchine-1956-kettenbruche] and others. For polynomial continued fractions, see [@abel-1826-ueber-integ-differ], [@berry-1990-periodicity-continued-fractions], [@schmidt-2000-continued-fractions-diophantine] or the survey paper [@poorten-tran-2000-quasi-elliptic-integrals].
As before $\K$ is a field of characteristic not $2$.
Finite continued fractions {#sec:orgb81a872}
--------------------------
For our formal continued fractions, we begin by using a double index notation, as this should make some calculations much clearer and precise. We will drop the first index once we no longer need it.
\[def-finite-continued-fraction\] Let $m, n \in \Z, n \geq 0$. The expression $$\alpha_{m,n} = [a_m, a_{m+1}, \dots, a_{m+n}] = a_{m} + \dfrac{1}{a_{m+1} + \dfrac{1}{ \ddots + \dfrac{1}{a_{m+n}}}}$$ where we consider the $a_i$ as free variables is called a *finite continued fraction*. We define it recursively by $$\alpha_{m,0} = a_m \quad \text{ and } \quad \alpha_{m,n} = a_m + \frac{1}{\alpha_{m+1,n-1}} \text{ for } n \geq 1$$ respectively $$[a_m] = a_m \quad \text{ and } \quad [a_m, a_{m+1}, \dots, a_{m+n}] = a_m + \frac{1}{[a_{m+1}, \dots, a_{m+n}]} \text{ for } n \geq 1$$ in the square bracket notation.
\[cf-concatenation-rule\] By induction one obtains also for $l \geq 1$ $$\alpha_{m,n} = [a_m, a_{m+1}, \dots, a_{m+l-1}, \alpha_{m+l,n-l}],$$ so the concatenation of $[a_m, a_{m+1}, \dots, a_{m+l-1}]$ and $[a_{m+l}, \dots, a_{m+n}]$ is the same as inserting the second at the end of the first finite continued fraction: $$[a_m, a_{m+1}, \dots, a_{m+n}] = [a_m, a_{m+1}, \dots, a_{m+l-1}, [a_{m+l}, \dots, a_{m+n}]].$$
Continued fractions and matrix products {#sec:org932a99d}
---------------------------------------
Clearly a continued fraction $\alpha_{m,n}$ can be seen as an element of $\P^1(\Zamn)$, where the empty continued fraction corresponds to $[\,] = \frac{1}{0} \in \P^1$. This motivates the following viewpoint:
We can think of a finite continued fractions as a map on $\P^1$, via $$x \in \P^1 \mapsto [a_m, \dots, a_{m+n}, x] \in \P^1.$$
We can relate such a map to the natural (left) action of $\GL{2}{\Zai}$ on $\P^1$ via Moebius transformations: $$x \mapsto \frac{a \, x + b}{c \, x + d} \corresponds \mfour{a}{b}{c}{d}.$$ Then clearly $$x \mapsto [a_m, x] = a_m + \frac{1}{x} \corresponds \mcf{a_m}.$$ As concatenation is the same as composition, this extends to $$x \mapsto [a_m, \dots, a_{m+n}, x] \corresponds \mcf{a_m} \cdots \mcf{a_{m+n}}.$$
By multiplying out these matrices, we can canonically compute the numerator and denominator of the fraction represented by a finite continued fraction.
\[definition-convergents-matrix\] For every $m, n \in \Z, n \geq -1$, there exist polynomials $p_{m,n}, q_{m,n} \in \Zamn$ such that $$\label{convergents-matrix}
\mcf{a_m} \cdots \mcf{a_{m+n}} = \mfour{p_{m,n}}{p_{m,n-1}}{q_{m,n}}{q_{m,n-1}},$$ satisfying $\ifrac{p_{m,n}}{q_{m,n}} = \alpha_{m,n}$.
Take $x = [\,] = \frac{1}{0}$ the empty continued fraction, then we define $$\frac{p_{m,n}}{q_{m,n}} := \mcf{a_m} \cdots \mcf{a_{m+n}} \frac{1}{0} = [a_m, \dots, a_{m+n}, [\,]] = \alpha_{m,n}$$ where clearly $p_{m,n}, q_{m,n} \in \Zamn$ because the matrix entries are in that ring. Also note that $\mcf{a_{m+n}} \dfrac{0}{1} = \dfrac{1}{0}$, so $$\mcf{a_m} \cdots \mcf{a_{m+n}} \frac{0}{1} = \mcf{a_m} \cdots \mcf{a_{m+n-1}} \frac{1}{0} = \frac{p_{m,n-1}}{q_{m,n-1}}.$$
Transposing the matrix $\mfour{p_{m,n}}{p_{m,n-1}}{q_{m,n}}{q_{m,n-1}}$ corresponds to reversing the ordering of the variables $a_m, \dots, a_{m+n}$; and $p_{m,n-1}$ depends only on $a_m, \dots, a_{m+n-1}$, so one easily deduces that $q_{m,n}$ is independent of $a_m$: it follows $q_{m,n} \in \Z[a_{m+1}, \dots, a_{m+n}]$.
By taking the determinants of the matrix product, we get
\[canonical-convergent-coprime\] For fixed $m$ and $n$, we have the relation $$\label{canonical-convergent-determinant}
p_{m,n} \, q_{m,n-1} - q_{m,n} \, p_{m,n-1} = (-1)^{n+1}.$$ Consequently, the $p_{m,n}$ and $q_{m,n}$ are coprime.
This holds even if we assign values to the $a_i$. The sequences in $n$ of the $p_{m,n}$ and $q_{m,n}$ may also be computed independently:
The $p_{m,n}$ and $q_{m,n}$ satisfy the recursion relations $$\label{canonical-convergent-recursion}
\begin{aligned}
p_{m,n} &= a_{m+n} \, p_{m,n-1} + p_{m,n-2} \text{ for } n \geq 0, &p_{m,-1} &= 1, & p_{m,-2} &= 0, \\
q_{m,n} &= a_{m+n} \, q_{m,n-1} + q_{m,n-2} \text{ for } n \geq 1, &q_{m,0} &= 1, & q_{m,-1} &= 0.
\end{aligned}$$
Infinite continued fractions {#sec:org5c8f70a}
----------------------------
To give sense to infinite continued fraction, we need some topology. In our case, we use $\K[X]$ with the previously defined (non-archimedean) absolute valuation $\ios = \ord_\infty$ (see Section \[sec:org7391b36\]). We assume that all $a_n \in \K[X]$. Then the $\alpha_{m,n}$ are contained in $\K(X)$, and we can hope to find a limit in the completion $\laurinx \K$.
We define the *infinite continued fraction* $$\alpha_m = \alpha_{m,\infty} = [a_m, a_{m+1}, \dots] = \limn \alpha_{m,n}$$ if the limit exists.
From now on, we assume that all $a_n \in \K[X]$, and search for a sufficient condition for the convergence of $\folge{\alpha_{m,n}}{n}$.
\[canonical-convergents-degree-and-lc\] If $\deg a_n \geq 1$ holds for all $n \geq m+1$, then $$\begin{aligned}
\label{}
\label{convergent-deg-growth}
\deg{p_{m,n}} &= \sum_{j=0}^n \deg{a_{m+j}}, & \deg{q_{m,n}} &= \sum_{j=1}^n \deg{a_{m+j}}, \\
\label{convergents-leading-coeff}
\LC(p_{m,n}) &= \prod_{j=0}^n \LC(a_{m+j}), & \LC(q_{m,n}) &= \prod_{j=1}^n \LC(a_{m+j}).\end{aligned}$$
The proposition is a consequence of the following lemma:
Let $\folge{a_n}{n}$ a sequence in $\K[X]$, with $\deg{a_n} \geq 1$ for all $n \geq 1$. Define a sequence $\left(b_n\right)_{n \geq -1}$ via $$\label{canon-convergent-generalised-recursion-relation}
b_{-1} = 0, \quad b_0 = 1, \quad b_{n} = a_n \, b_{n-1} + b_{n-2} \text{ for } n \geq 1$$ Then $\deg{b_n}$ is strictly increasing and for $n \geq 0$ $$\begin{aligned}
\deg{b_n} &= \sum_{j=1}^n \deg{a_j}, & \LC(b_n) &= \prod_{j=1}^n \LC(a_j).\end{aligned}$$
We prove the statement by induction on $n$. For $n = 0$ we clearly have $\deg{b_{-1}} < \deg{b_0} = 0$ and $\LC(b_0) = 1$. For the induction step, note that by hypothesis $\deg{b_{n-2}} < \deg{b_{n-1}} < \degb{a_n \, b_{n-1}}$, so implies $$\deg{b_n} = \degb{a_n \, b_{n-1} + b_{n-2}} = \deg{a_n} + \deg{b_{n-1}} = \deg{a_n} + \sum_{j=1}^{n-1} \deg{a_j} %= \sum_{j=1}^n \deg{a_j},$$ as desired, and clearly $\deg{b_{n-1}} < \deg{b_{n}}$. It follows $$\LC(b_n) = \LC(a_{n} \, b_{n-1}) = \LC(a_n) \, \prod_{j=1}^{n-1} \LC(a_j).$$
We can now answer the question about the convergence of infinite continued fractions:
\[cf-cauchy-convergence\] Suppose $\deg a_n \geq 1$ holds for $n \geq m+1$. Then $\folge{\alpha_{m,n}}{n}$ is a Cauchy sequence and converges in $\laurinx{\K}$. We denote the limit by $\alpha_m = \alpha_{m,\infty} = [a_m, a_{m+1}, a_{m+2}, \dots]$.
Dividing by $q_{m,n-1} \cdot q_{m,n}$ implies $$\alpha_{m,n} - \alpha_{m,n-1} = \frac{p_{m,n}}{q_{m,n}} - \frac{p_{m,n-1}}{q_{m,n-1}} = \frac{(-1)^{n+1}}{q_{m,n-1} \cdot q_{m,n}},$$ hence $$\label{convergent-fraction-difference}
\io{\alpha_{m,n} - \alpha_{m,n-1}} = \deg q_{m,n} + \deg q_{m,n-1} \geq 2n-1$$ by Proposition \[canonical-convergents-degree-and-lc\]. This means the “distance” between $\alpha_{m,n}$ and $\alpha_{m,n-1}$ converges to $0$ as $n \to \infty$. Because we are working with a non-archimedean valuation, this already implies that $\folge{\alpha_{m,n}}{n}$ is a Cauchy sequence.
So a continued fraction (with non-constant coefficients $a_n \in \K[X]$) produces an element of $\laurinx \K$ (actually a sequence of elements of $\laurinx \K$). In the next section, we will reverse the process and produce a continued fraction for every element of $\laurinx \K$, thus establishing a bijection between $\laurinx \K$ and (a subset of) continued fractions over $\K[X]$.
Continued fraction process {#sec:org34409a6}
--------------------------
We now define a process which produces a continued fraction for elements of $\laurinx \K$, using the truncation $\gauss{\cdot}$ from Definition \[define-laurent-truncation\]. This is mostly analogous to classical continued fractions over $\Z$, but slightly nicer because here we have a unique truncation operation, and we avoid ambiguity as for example with $[2] = 2 = 1 + \frac{1}{1} = [1, 1]$ in the integer case.
Let $\alpha \in \laurinx \K$. We define the *complete quotients* of $\alpha$ as the (possibly finite) sequence $$\label{cf-complete-quotients}
\alpha_0 = \alpha, \quad \alpha_{n+1} = \frac{1}{\alpha_n - \gauss{\alpha_n}} \quad \text{ for } n \geq 0 \text{ and } \alpha_n \not\in \K[X].$$ One defines also the *partial quotients* $a_n = \gauss{\alpha_n}$ whenever the corresponding complete quotient is defined. As $\alpha_n = a_n + \inv{\alpha_{n+1}}$, this clearly gives rise to a (finite or infinite) continued fraction $$\CF(\alpha) = [a_0, a_1, \dots].$$
\[cf-absolute-values\] By definition of $\gauss{\cdot}$ we have always $\io{\alpha_n - \gauss{\alpha_n}} > 0$ which implies $\io{\alpha_{n+1}} < 0$ whenever $\alpha_{n+1}$ is defined. Then $\io{a_{n+1}} = \io{\alpha_{n+1}} < 0$ which means $\deg a_{n+1} \geq 1$. So if $\CF(\alpha)$ is an infinite continued fraction, it converges by Proposition \[cf-cauchy-convergence\].
The Euclidean algorithm works also in the ring $\K[X]$, establishing a complete correspondence between finite continued fraction and rational functions.
\[cf-euclidean-algorithm\] The continued fraction $\CF(\alpha)$ is finite $\alpha \in \K(X)$.
If $\CF(\alpha)$ is finite, it produces an element of $\K(X)$, and obviously $\alpha = \CF(\alpha)$.
Conversely, assume $\alpha \in \K(X)$. Write $\alpha = \frac{r_0}{r_1}$ with $r_0, r_1 \in \K[X]$ and $r_1 \neq 0$. In fact, we can write $\alpha_n = \frac{r_n}{r_{n+1}}$ whenever defined, with $r_n, r_{n+1} \in \K[X]$.
By Remark \[truncation-of-rational\] we write $r_n = a_n \, r_{n+1} + r_{n+2}$ where $\deg{r_{n+2}} < \deg{r_{n+1}}$ because $a_n = \gauss{\ifrac{r_n}{r_{n+1}}}$. Hence $$\alpha_n = \frac{r_n}{r_{n+1}} = a_n + \frac{r_{n+2}}{r_{n+1}} = a_n + \frac{1}{\alpha_{n+1}}.$$ So in this case, the continued fraction process corresponds to the Euclidean algorithm which is well known to terminate in a finite number of steps; so eventually $r_{n+1} = 0$ for some $n$ which means that $\alpha_n \in \K[X]$ and that consequently $\CF(\alpha)$ is finite.
Canonical convergents and classification of best-approximations {#sec:orga882f91}
---------------------------------------------------------------
The sequence of *canonical convergents* of $\alpha$ is defined by $$(p_n, q_n) = (p_{0,n}, q_{0,n}) \in \Batest{\K} \quad \text{ for } n \geq -1$$ where we plug the partial quotients into the formulas from Section \[sec:orgb81a872\]. If $\CF(\alpha)$ is finite, this sequence is also finite.
Note that Corollary \[canonical-convergent-coprime\] implies that $p_n$ and $q_n$ are coprime for a given $n$. And the canonical convergents are in fact convergents, and we have precise information about their approximation quality:
\[cf-expansion-yields-convergents\] Let $n \geq 0$. Then unless $\alpha = \ifrac{p_n}{q_n}$, $$\io{p_n - \alpha \, q_n} = \deg{q_{n+1}} = \deg a_{n+1} + \deg q_n > \deg q_n,$$ so $(p_n, q_n) \in \Coset{\alpha}{\K}$.
If $\alpha = \ifrac{p_n}{q_n}$, then clearly $\io{p_n - \alpha \, q_n} = \infty > \deg q_n$ and obviously $(p_n, q_n) \in \Coset{\alpha}{\K}$.
Unless $\CF(\alpha) = [a_0, \dots, a_n]$ is finite of length exactly $n+1$ which directly implies $p_n - \alpha \, q_n = 0$ by the Proposition \[cf-euclidean-algorithm\], we have $$\alpha = [a_0, \dots, a_{n}, \alpha_{n+1}]
\quad \text{ i.e. }
\alpha = \mfour{p_n}{p_{n-1}}{q_n}{q_{n-1}} \cdot \alpha_{n+1}.$$ Multiplying with the inverse matrix, we get the important formula $$\label{complete-quotient-convergent-quotient}
% \tag{Q}
\alpha_{n+1} = (-1)^{n+1} \, \mfour{q_{n-1}}{-p_{n-1}}{-q_n}{p_n} \cdot \alpha= -\frac{p_{n-1} - \alpha \, q_{n-1}}{p_n - \alpha \, q_n}.$$ Recall that $p_{-1} = 1, q_{-1} = 0$, so a telescoping product yields $$(-1)^{n+1} \prod_{j=0}^{n} \alpha_{j+1} = \prod_{j=0}^{n} \frac{p_{j-1} - \alpha \, q_{j-1}}{p_j - \alpha \, q_j} = \frac{1}{p_n - \alpha \, q_n}.$$ Taking valuations, note that $\io{\alpha_j} = \io{a_j} = - \deg a_j$ for $j \geq 1$, hence $$\io{p_n - \alpha \, q_n} = -\sum_{j=1}^{n+1} \io{\alpha_j} = \sum_{j=1}^{n+1} \deg a_j = \deg a_{n+1} + \deg q_n = \deg q_{n+1},$$ the last two equalities being a consequence of Proposition \[canonical-convergents-degree-and-lc\].
The continued fraction of $\alpha$ represents $\alpha$ as an element of $\laurinx \K$, i.e. $$\alpha = \CF(\alpha) \text{ in } \laurinx \K.$$
For $\alpha \in \K(X)$, this is was mentioned in the proof of Proposition \[cf-euclidean-algorithm\]. Otherwise, $\alpha \not \in \K(X)$, and from Proposition \[cf-expansion-yields-convergents\] we conclude $\alpha = \limn \frac{p_n}{q_n} = \CF(\alpha)$ as $\limn \deg{q_n} = \infty$.
With this information about the approximation quality of the canonical convergents, we can now give a complete classification of the best-approximations.
\[cf-best-approx-classification\] Let $\alpha \in \laurinx \K \setminus \K(X)$, and $(p, q) \in \Baset{\alpha}{\K}$ a best-approximation. Then there exist a unique $n \in \N_0$ and $r \in \mino{\K[X]}$ with $\deg r < \deg a_{n+1}$ such that $$(p, q) = r \cdot (p_n, q_n).$$ In particular, if $p$ and $q$ are coprime, then $r \in \units \K$.
Moreover, if $(p', q') = r' \cdot (p_{n'}, q_{n'}) \in \Baset{\alpha}{\K}$ is another best-approximation with $\deg q < \deg q'$, then $n \leq n'$.
With the sufficient condition for a best-approximation from Proposition \[best-approx-common-factor-degree\] applied to $(p_n, q_n)$ and $\xi = \deg a_{n+1}$, we see that for every possible $\deg q$ we can produce a best-approximation of the shape $r \cdot (p_n, q_n)$, with any $r \in \K[X]$ satisfying $0 \leq \deg r < \deg a_{n+1}$. Then by Proposition \[best-approx-for-given-degree\], all best-approximations have this shape. Because $p_n$ and $q_n$ are always coprime, and $\deg q_n$ is strictly increasing in $n$, no canonical convergent can be written as a multiple of another, so $n$ must be unique.
Finally, the monotony result is obvious from $\deg{q_n} \leq \degb{r \, q_n} < \deg q_{n+1}$.
If $\alpha \in \K(X)$, this argument works just as well, except for the last canonical convergent. However, if we put “$\deg a_{n+1} = \infty$”, the statement trivially holds even for the last canonical convergent.
For completeness, we also give the analogue for convergents (applying Proposition \[convergent-common-factor-degree\] instead of Proposition \[best-approx-common-factor-degree\]):
\[cf-convergent-classification\] Let $\alpha \in \laurinx \K \setminus \K(X)$, and $(p, q) \in \Coset{\alpha}{\K}$ a convergent. Then there exist $n \in \N_0$ and $r \in \mino{\K[X]}$ with $\deg r < \frac{1}{2} \deg a_{n+1}$ such that $$(p, q) = r \cdot (p_n, q_n),$$ and if $p$ and $q$ are coprime, then $r \in \units \K$.
Multiplication of a continued fraction by a constant {#sec:org976a8fd}
----------------------------------------------------
One nice feature of polynomial continued fractions is that it is possible to multiply them with a constant factor. In [@schmidt-2000-continued-fractions-diophantine], there is even a generalisation of this identity which holds also for non-constant factors. We limit ourselves to constants, however.
\[cf-scalar-multiplication\] Let $\mu \in \units \K$. Then $$\mu \, [a_0, a_1, a_2, a_3, \dots] = [\mu \, a_0, \inv \mu \, a_1, \mu \, a_2, \inv \mu \, a_3, \dots ].$$
Again, it is convenient to think of the continued fraction as a product of matrices: $$\begin{aligned}
\mdi{\mu}{1} \mcf{a} &= \mfour{\mu \, a}{\mu}{1}{} = \mcf{\mu \, a} \mdi{1}{\mu}, \\
\mdi{1}{\mu} \mcf{a} &= \mfour{a}{1}{\mu}{} = \mcf{\invfrac{a}{\mu}} \mdi{\mu}{1}.\end{aligned}$$ As multiplication by $\mu$ corresponds to $\mdi{\mu}{1}$ and division by $\mu$ corresponds to $\mdi{1}{\mu}$, we obtain for $n$ even $$\mdi{\mu}{1} \mcf{a_0} \cdots \mcf{a_n} = \mcf{\mu \, a_0} \mcf{\invfrac{a_1}{\mu}} \cdots \mcf{\mu \, a_n} \mdi{1}{\mu}$$ and for $n$ odd $$\mdi{\mu}{1} \mcf{a_0} \cdots \mcf{a_n} = \mcf{\mu \, a_0} \mcf{\invfrac{a_1}{\mu}} \cdots \mcf{\invfrac{a_n}{\mu}} \mdi{\mu}{1}$$ so the corresponding map would be for $n$ even $$x \mapsto [\mu \, a_0, \invfrac{a_1}{\mu}, \dots, \mu \, a_n, \invfrac{x}{\mu}]$$ and for $n$ odd $$x \mapsto [\mu \, a_0, \invfrac{a_1}{\mu}, \dots, \invfrac{a_n}{\mu}, \mu \, x]$$ as desired – because for the empty continued fraction, we have $\mu \cdot [\,] = \frac{\mu}{0} = \frac{1}{0} = [\,]$.
Periodic continued fractions {#sec:org3afbb4f}
----------------------------
For classical continued fractions, it is a well-known result that continued fractions of quadratics are always periodic. As in the real case, a periodic polynomial continued fraction must be quadratic. However, a continued fraction of a quadratic need not be periodic in the polynomial case. For $\sqrt{D}$ this in fact happens $D$ is Pellian which we will prove in Section \[sec:org8d34125\].
Indeed periodicity gives a solution of with $\omega = \pm 1$ (this follows from ). But if the base field $\K$ is very small, allowing arbitrary $\omega$ may give a solution with smaller $\deg q$. So one should not merely study periodicity, but periodicity up to a constant factor. We call this *quasi-periodicity* (sometimes it is also called pseudo-periodicity in the literature). For the continued fraction of $\sqrt{D}$, the period and the quasi-period are tightly linked, and one induces the other.
Later in Chapter \[sec:org5f9d2ce\], we will also see that quasi-periodicity is the more relevant notion for studying reductions of the continued fraction modulo a prime.
### Periods {#sec:org8b4d61a}
\[cf-periodicity-partial-quotients\] The (infinite) continued fraction $\alpha_m = \alpha_{m,\infty} = [a_m, a_{m+1}, \dots]$ is said to be *periodic* if for some $m' \geq m$ there exists $l \in \N$ (the minimal such $l$ is called the *period length*) such that $$\forall n \geq m': \; a_{n} = a_{n+l}$$ If $m' = m$, the continued fraction is called *pure periodic*, i.e. there is no preperiod. For compact notation, we usually write “$\CF(\alpha_m)$ is (pure) periodic”.
From a computational view, this definition is somewhat problematic because there is an infinite number of conditions to check. Fortunately, this can be reduced to a single condition on the complete quotients.
\[cf-periodicity-complete-quotients\]
1. The continued fraction $\CF(\alpha_m)$ is periodic.
2. There exist $m' \geq m$ and $l \in \N$ such that $\alpha_{m'} = \alpha_{m'+l}$.
3. There exist $m' \geq m$ and $l \in \N$ such that for all $n \geq m': \; \alpha_n = \alpha_{n+l}$.
Because $a_n = \gauss{\alpha_n}$, 3. directly implies 1.
On the other hand, $\alpha_{m'}$ is uniquely determined by $a_{m'}, a_{{m'}+1}, \dots$, and by periodicity of the $a_n$ one obtains $$\alpha_{{m'}+l} = [a_{{m'}+l}, a_{{m'}+l+1}, \dots] = [a_{m'}, a_{{m'}+1}, \dots] = \alpha_{m'}.$$ so 1. implies 2.
But through the continued fraction process, $\alpha_{n+1}$ is uniquely determined by $\alpha_n$ for every $n$, so $$\alpha_n = \alpha_{n+l} \implies \alpha_{n+1} = \alpha_{n+l+1}.$$ and by the induction principle, 2. implies 3.
### Quasi-periods {#sec:org7617d8f}
We now generalise periodicity to quasi-periodicity which is essentially periodicity up to a unit factor. For cleaner notation, we first define $$\parity{n} = (-1)^n = \begin{cases}
1 & \text{ if } n \text{ is even,}\\
-1 & \text{ if } n \text{ is odd}
\end{cases}.$$
\[cf-quasi-periodicity-partial-quotients\] The (infinite) continued fraction $\alpha_{m}$ is called *quasi-periodic*, if there exists $m' \geq m$, $\mu \in \units \K$ and $l > 0$ (if minimal, called the *quasi-period length*) such that $$\forall n \geq m': \; a_{n} = \mu^{\parity{n}} \, a_{n+l}.$$ If $m' = m$, then it is called *pure quasi-periodic*.
Any periodic continued fraction is also quasi-periodic, with $\mu = 1$. See below for a partial converse.
\[quasi-period-length-ideal\] It should be obvious that the $l \in \Z$ such that $\alpha_n / \alpha_{n+l} \in \units\K$ form an ideal, and the (quasi-)period length is the positive generator of it.
In particular, the period length must be a multiple of the quasi-period length.
We also have a complete analogue to Proposition \[cf-periodicity-complete-quotients\]:
\[cf-quasi-periodicity-complete-quotients\]
1. The continued fraction $\CF(\alpha_m)$ is quasi-periodic.
2. There exist $m' \geq m$, $\mu \in \units \K$ and $l > 0$ such that $\alpha_{m'} = \mu^{\parity{m'}} \, \alpha_{m'+l}$.
3. There exist $m' \geq m$, $\mu \in \units \K$ and $l > 0$ such that for all $n \geq m' : \; \alpha_{n} = \mu^{\parity{n}} \, \alpha_{n+l}$.
Using Proposition \[cf-scalar-multiplication\], and $\gauss{\mu \, \alpha} = \mu \, \gauss{\alpha}$ for $\mu \in \units \K$, and $$\alpha_n = \mu \, \alpha_{n+l} \implies \alpha_{n+1} = \inv \mu \,\alpha_{n+l+1},$$ the proof is completely analogous to the one of Proposition \[cf-periodicity-complete-quotients\].
\[odd-quasi-period-implies-periodic\] If $\CF(\alpha_m)$ is quasi-periodic with *odd* quasi-period length $l$ and $\mu \neq 1$, then $\CF(\alpha_m)$ is also periodic with period length $2 \, l$.
For all $n \geq m'$, we have $a_n = \mu^{\parity{n}} \, a_{n+l}$ and $a_{n+l} = \mu^{\parity{n+l}} \, a_{n+2 l}$. As $l$ is odd, we have $\parity{n+l} = - \parity{n}$ so $a_n = \mu^{\parity{n}+\parity{n+l}} \; a_{n+2 l} = a_{n+2 l}$.
\[quasi-period-length-shift-invariant\] The (quasi-)period length was above defined as the minimal $l$, and does not depend on where the (quasi-)period starts, so two complete quotients $\alpha_{m_1}$ and $\alpha_{m_2}$ have the same (quasi-)period length.
If $\CF(\alpha)$ is quasi-periodic, then $\alpha \in \laurinx \K$ is quadratic over $\K(X)$ (it cannot be in $\K(X)$ because it has an infinite continued fraction).
From Section \[sec:org932a99d\] we know $$\alpha_{m} = \mfour{p_{m,n}}{p_{m,n-1}}{q_{m,n}}{q_{m,n-1}} \cdot \alpha_{m+n+1} = \frac{p_{m,n} \, \alpha_{m+n+1} + p_{m,n-1}}{q_{m,n} \, \alpha_{m+n+1} + q_{m,n-1}}$$ so it suffices to treat the case where $\CF(\alpha)$ is pure quasi-periodic, i.e. $\alpha_l = \mu \, \alpha$. Then putting $m = 0$ and $n = l-1$ the above becomes $$\alpha = \mfour{p_{l-1}}{p_{l-2}}{q_{l-1}}{q_{l-2}} \cdot \mu \, \alpha = \frac{p_{l-1} \, \mu \, \alpha + p_{l-2}}{q_{l-1} \, \mu \, \alpha + q_{l-2}}.$$ Multiplying with the denominator, we then obtain $$q_{l-1} \, \mu \, \alpha^2 + (q_{l-2} - p_{l-1} \, \mu) \, \alpha - p_{l-2} = 0$$ and of course $q_{l-1} \neq 0$ so $\alpha$ is quadratic over $\K(X)$.
Reduced complete quotients {#sec:org6896ed2}
--------------------------
Our next goal is to understand the continued fraction expansion of $\sqrt{D}$ better. We will explain how we can usually go backwards in this continued fraction. This means we can only have very short preperiods (here just $a_0$ belongs to the preperiod), and allows to show that for $\CF(\sqrt{D})$, quasi-periodicity is equivalent to periodicity, also in the case of *even* quasi-period length.
In this case, the complete quotients are contained in the quadratic extension $\HEF$ of $\K(X)$ contained in $\laurinx{\K}$. It has precisely one non-trivial $\K(X)$-automorphism $\sigma$ which sends $\sqrt{D}$ to $-\sqrt{D}$. As we have chosen $\sqrt{D} \in \laurinx \K$, we have an embedding of $\HEF$ into $\laurinx \K$.
$\alpha \in \HEF$ is said to be *$\sigma$-reduced* (with $\sigma$ as above), if $$% \tag{$\sigma$}
\io{\sigma(\alpha)} > 0 > \io{\alpha}.$$
All elements of $\K(X)$ are invariant under $\sigma$, so none of them is $\sigma$-reduced.
\[sigma-reduced-no-translations\] Let $\alpha \in \HEF \setminus \K(X)$. Then there exists at most one $a \in \K[X]$ such that $a+\alpha$ is $\sigma$-reduced.
Clearly $\sigma(a+\alpha) = a+ \sigma(\alpha)$. Assume $\io{\sigma(a + \alpha)} = \io{a + \sigma(\alpha)} > 0$, then by Remark \[truncation-unique\] $a = - \gauss{\sigma(\alpha)}$ so there is at most one choice for $a$.
Note that this choice of $a$ does not yet guarantee $\io{a + \alpha} < 0$.
\[sigma-reduced-all-complete-quotients\] If the complete quotient $\alpha_m$ is $\sigma$-reduced, then so is $\alpha_{m'}$ for all $m' \geq m$.
Using the induction principle, it suffices to treat the case $m' = m+1$. By Remark \[cf-absolute-values\], we automatically have $\io{\alpha_{m+1}} < 0$. Moreover, $$\sigma(\alpha_{m+1}) = \frac{1}{\sigma(\alpha_m) - a_m}$$ and $\io{a_m} = \io{\alpha_m} < \io{\sigma(\alpha_m)}$ implies $\io{\sigma(\alpha_{m+1})} = -\io{\alpha_m} > 0$ as desired.
\[sigma-reduced-cfsb\] $\alpha$ is $\sigma$-reduced $\cfsb{\alpha}$ is $\sigma$-reduced.
This is an immediate consequence of $$\begin{aligned}
\io{\alpha} &= - \io{\sigma\left(\cfsb{\alpha}\right)}, &
\io{\sigma(\alpha)} &= - \io{\cfsb{\alpha}}.\end{aligned}$$
The most useful property of $\sigma$-reduced complete quotients is however that we may go backwards in the continued fraction expansion in a unique way:
\[cf-reduced-backwards\] Suppose $\alpha_1 \in \HEF$ is $\sigma$-reduced. Then there exists a unique $\alpha_0 \in \HEF$ which is $\sigma$-reduced and satisfies $$\alpha_1 = \frac{1}{\alpha_0 - \gauss{\alpha_0}}.$$
By Proposition \[sigma-reduced-no-translations\], there exists at most one $a_0 \in \K[X]$ such that $\alpha_0 = a_0 + \frac{1}{\alpha_1}$ is $\sigma$-reduced, namely $a_0 = \gauss{\cfsb{\alpha_1}}$. Rewriting this to $$\cfsb{\alpha_0} = \frac{1}{\cfsb{\alpha_1} - a_0},$$ we see that $\alpha_0$ is $\sigma$-reduced by applying twice Lemma \[sigma-reduced-cfsb\] and once Proposition \[sigma-reduced-all-complete-quotients\].
Finally, as $\io{\alpha_1} < 0$ it is also clear that $a_0 = \gauss{\alpha_0}$.
\[cf-reduced-backwards-n\] Generally, for any $n$ and $\alpha_n$ $\sigma$-reduced, we have $$\cfsb{\alpha_n} = \frac{1}{\cfsb{\alpha_{n+1}} - \gauss{\cfsb{\alpha_{n+1}}}}$$ so also the $\cfsb{\alpha_n}$ are the complete quotients of some continued fraction expansion, albeit with $n$ decreasing.
\[sigma-reduced-pure-period\] Suppose $\alpha_m$ is $\sigma$-reduced and $\CF(\alpha_m)$ is (quasi-)periodic, then $\CF(\alpha_m)$ is pure (quasi-)periodic.
We use Propositions \[cf-periodicity-complete-quotients\] and \[cf-quasi-periodicity-complete-quotients\] here.
Suppose $n > m, l \in \N$ and $\mu \in \units{\K}$ (where $\mu = 1$ in the case of periodicity) with $\alpha_{n} = \mu^{\parity{n}} \, \alpha_{n+l}$. By Proposition \[sigma-reduced-all-complete-quotients\], $\alpha_{n-1}, \alpha_n, \alpha_{n+l-1}, \alpha_{n+l}$ are all $\sigma$-reduced, and we have $$\begin{gathered}
\alpha_n = \frac{1}{\alpha_{n-1} - a_{n-1}} = \mu^{\parity{n}} \, \alpha_{n+l} \\ = \mu^{\parity{n}} \, \frac{1}{\alpha_{n+l-1} - a_{n+l-1}} = \frac{1}{\mu^{\parity{n-1}} \, \alpha_{n+l-1} - \mu^{\parity{n-1}} \, a_{n+l-1}}.\end{gathered}$$ With $\gauss{\mu^{\parity{n-1}} \, \alpha_{n+l-1}} = \mu^{\parity{n-1}} \, a_{n+l-1}$, Proposition \[cf-reduced-backwards\] implies $\alpha_{n-1} = \mu^{\parity{n-1}} \, \alpha_{n+l-1}$ as desired, and we may repeat this argument until we arrive at $\alpha_{m} = \mu^{\parity{m}} \, \alpha_{m+l}$.
\[cf-berrys-thm\] Suppose $\alpha \in \HEF$ is $\sigma$-reduced and has polynomial trace $\alpha + \sigma(\alpha) \in \K[X]$. If $\CF(\alpha)$ is quasi-periodic, it is even pure (quasi-)periodic.
Lemma \[sigma-reduced-pure-period\] already implies that $\CF(\alpha)$ is pure quasi-periodic, and once we prove it is periodic, it is automatically pure periodic. For odd quasi-period length, the general Proposition \[odd-quasi-period-implies-periodic\] already yields periodicity. For even quasi-period length, a bit more work is required.
From $\gauss{f} = f$ for $f \in \K[X]$ and $\io{\sigma(\alpha)} > 0$ we obtain $$\alpha + \sigma(\alpha) = \gauss{\alpha + \sigma(\alpha)} = \gauss{\alpha} = a_0$$ so $\alpha - a_0 = - \sigma(\alpha)$ which implies $$\alpha_1 = \cfsb{\alpha_0} \text{ and thus } \alpha_0 = \cfsb{\alpha_1}.$$ In the light of Remark \[cf-reduced-backwards-n\], the $\cfsb{\alpha_n}$, going backwards, are complete quotients of some continued fraction expansion and actually extend $\CF(\alpha)$ for negative $n$: $$\begin{array}{cccccccc}
\dots & \cfsb{\alpha_3} & \cfsb{\alpha_2} & \cfsb{\alpha_1} & \cfsb{\alpha_0} \\
& & & \alpha_0 & \alpha_1 & \alpha_2 & \alpha_3 & \dots
\end{array}$$ So we can define $\alpha_{n} = \cfsb{\alpha_{1-n}}$ for $n \leq 1$, with all $\alpha_n$ $\sigma$-reduced, and by Lemma \[sigma-reduced-pure-period\] the quasi-periodicity extends towards $-\infty$ as well.
Denote by $\QPL$ the quasi-period length of $\CF(\alpha)$, so we may write $$\begin{aligned}
\alpha_0 &= \mu \, \alpha_\QPL, &
\alpha_\QPL &= \mu^{\parity{\QPL}} \, \alpha_{2\QPL}, &
\alpha_{1-\QPL} &= \mu^{\parity{1-\QPL}} \, \alpha_1.\end{aligned}$$ It follows $$\alpha_\QPL = \cfsb{\alpha_{1-\QPL}} = \frac{1}{\mu^{\parity{1-\QPL}}} \, \cfsb{\alpha_1} = \mu^{\parity{\QPL}} \, \alpha_0$$ and further $\alpha_0 = \mu \, \mu^{\parity{\QPL}} \, \alpha_0$. Hence $\mu \, \mu^{\parity{\QPL}} = 1$ (if $\QPL$ is even, this means $\mu = \pm 1$), and then $\alpha_0 = \mu \, \alpha_\QPL = \mu \, \mu^{\parity{\QPL}} \, \alpha_{2\QPL} = \alpha_{2\QPL}$, so $\CF(\alpha)$ is periodic (with period length $\QPL$ or $2\QPL$).
This shows that the involution $x \mapsto \cfsb{x}$ acts as a reflection with centre $1/2$ on the $\Z$-series of $\alpha_n$ ($n \mapsto 1-n$ on the indices).
\[period-of-sqrt-d\] Obviously $\sqrt{D}$ is not $\sigma$-reduced. However $\alpha = A + \sqrt{D}$ (recall that $A = \gauss{\sqrt{D}}$) is $\sigma$-reduced, and $\sqrt{D} - \gauss{\sqrt{D}} = \alpha - \gauss{\alpha}$, so $$\CF(\sqrt{D}) = [A, a_1, a_2, \dots]$$ differs from $\CF(\alpha)$ only in the first complete (and partial) quotient. This means that if $\CF(\sqrt{D})$ is quasi-periodic, it is almost pure periodic, and the preperiod has length $1$ and consists just of $A$.
This reversibility of the continued fraction process also implies that the period must be a palindrome:
\[palindromic-period\] Let $\alpha \in \HEF$ $\sigma$-reduced with $\alpha + \sigma(\alpha) \in \K[X]$. Let $\QPL$ the quasi-period length.
- If $\QPL$ is even, then $\CF(\alpha)$ has actually period length $\QPL$, and the period is palindromic, i.e. $$\CF(\alpha) = \left[\overline{a_0, a_1, \dots, a_{\QPL/2}, \dots, a_1}\right]$$
- If $\QPL$ is odd, then $\CF(\alpha)$ has a “quasi-palindromic” quasi-period, i.e. $$\CF(\alpha) = \left[\overline{a_0, a_1, \dots, a_{(\QPL-1)/2}, \mu^{\pm 1} \, a_{(\QPL-1)/2}, \mu^{\mp 1} \, a_{(\QPL-3)/2}, \dots, \mu \, a_1}\right]$$
In the second case, either $\mu = 1$, or the period length $2 \QPL$ is even. Then we can apply the first case for the period instead of the quasi-period to get a palindromic period.
Recall how we defined the negative complete quotients, hence for any $n \in \Z$ $$\alpha_{n} = \cfsb{\alpha_{1-n}} = \sigma\left(- \frac{1}{\alpha_{1-n}}\right) = \sigma\left(\alpha_{-n} - a_{-n}\right)
= a_{-n} + \frac{1}{\cfsb{\alpha_{-n}}} = a_{-n} + \frac{1}{\alpha_{n+1}},$$ the crux of which is $a_{n} = \gauss{\alpha_{n}} = a_{-n}$.
Using quasi-periodicity, we then obtain $$a_n = a_{-n} = \mu^{\parity{-n}} \, a_{\QPL-n} = \mu^{\parity{n}} \, a_{\QPL-n}$$ and developing this for $n \leq \QPL/2$ we obtain $$a_0 = \mu \, a_\QPL, \quad
a_1 = \inv\mu \, a_{\QPL-1}, \quad
a_2 = \mu \, a_{\QPL-2}, \quad \dots$$ until for $\QPL$ odd we arrive at $a_{(\QPL-1)/2} = \mu^{\parity{(\QPL-1)/2}} \, a_{(\QPL+1)/2}$ and for $\QPL$ even we arrive at $a_{\QPL/2} = \mu^{\parity{\QPL/2}} \, a_{\QPL/2}$ which also implies $\mu = 1$.
Computation of hyperelliptic continued fractions {#sec:orge1a60a6}
================================================
We now give formulas for computing the continued fraction expansion for quadratic Laurent series. Optimising these formulas is not only useful for computing and studying examples, but it also serves to illustrate the connection between the Pell equation and periodicity of the continued fraction. Of particular interest is that everything can be expressed as operations on polynomials.
We assume as usual that $D$ is non-square of degree $2d$ and that $\LC(D)$ is a square in $\K$, a field of characteristic not $2$. Recall that we defined the polynomial part $A = \gauss{\sqrt{D}}$.
It is well-known that the complete quotients of $\sqrt{D}$ can be written as $\alpha_n = (r_n + \sqrt{D})/s_n$ with $r_n, s_n \in \K[X]$ of bounded degree. We can slightly improve upon this representation by writing $r_n = A + \text{terms of lower degree}$. This seems to be a new result:
\[thm-optimised-sqrt-cq-representation\] Let $\alpha = \sqrt{D}$. The complete quotients of $\alpha$ can be written as $$\label{quadratic-cq-representation}
\alpha_n = \frac{A + t_n + \sqrt{D}}{s_n} \quad \text{ for } n \geq 1$$ where $t_n, s_n \in \K[X]$ with $$\label{sigma-reduced-degree-condition-in-prop}
\deg t_n < \deg s_n < \deg A$$ for $n \geq 1$. Moreover, there are the following recursion formulas for $t_n$ and $s_n$: $$\label{quadratic-cq-recursion-formulas}
t_{n} + t_{n+1} = a_n \, s_n - 2 \, A, \quad s_{n} \, s_{n+1} = D - (A+t_{n+1})^2,$$ initialised with $t_0 = -A$ and $s_0 = 1$. Finally $\deg s_n = 0$ for $n \geq 1$ $\CF(\alpha)$ is periodic and the quasi-period length $\QPL$ divides $n$.
Note that $\alpha_n$ being $\sigma$-reduced is equivalent to by Proposition \[sigma-reduced-degree-prop\].
\[cor-pq-degree-periodicity\] The complete quotients satisfy $\io{\alpha_n} \geq \io{\sqrt{D}}$, so for the partial quotients we have $$1 \leq \deg a_n \leq \deg A$$ with equality $\deg a_n = \deg A = d$ for $n \geq 1$ the continued fraction $\CF(\sqrt{D})$ is periodic, and the quasi-period length $\QPL$ divides $n$.
In fact, we show more generally:
\[thm-quadratic-laurent-series-cf-representation\] Let $\alpha \in \laurinx \K$ any Laurent series quadratic over $\K(X)$. Then for a suitable $D$ depending only on $\alpha$, the complete quotients $\alpha_n$ may also be written as in , where $t_n$ and $s_n$ follow the recursion formulas .
Moreover, there exists $N \geq 0$, such that $t_n$ and $s_n$ satisfy for all $n \geq N$.
The theorem also gives a more elementary proof of periodicity over finite fields:
\[cor-finite-field-always-periodic\] If the base field $\K$ is finite, any Laurent series quadratic over $\K(X)$ has a periodic continued fraction expansion.
Using this representation of the complete quotients of $\sqrt{D}$, and our accumulated knowledge about the convergents, we also recover Abel’s result from [@abel-1826-ueber-integ-differ]:
\[thm-pellian-iff-cf-periodic\] $D$ is Pellian $\CF(\sqrt{D})$ is periodic.
We shall prove these results in the first part of this chapter. The second part then explores some further consequences.
Representing complete quotients with polynomials {#sec:org95cb74d}
------------------------------------------------
We begin by reiterating the formulas for hyperelliptic continued fraction expansions which can (with varying level of detail) be already found in [@abel-1826-ueber-integ-differ], [@berry-1990-periodicity-continued-fractions] and [@poorten-tran-2000-quasi-elliptic-integrals].
Let $\alpha \in \laurinx \K$ be quadratic over $\K(X)$, satisfying $s \, \alpha^2 - 2\, r \, \alpha + w = 0$ where $r, s, w \in \K[X]$. The discriminant $4 \, D = 4 \, (r^2 - s\,w)$ yields $D$, for which we choose a square root $\sqrt{D}$. Then we write $$\label{cf-quadratic-normalised-representation}
\alpha = \frac{r + \sqrt{D}}{s}$$ after possibly multiplying $r, s, w$ with $-1$ to accommodate our choice of $\sqrt{D}$. Note that here holds $s \div D - r^2$ which is crucial for the following computations. This allows a common factor in $r$ and $s$ which then must divide $D$ as well.
Clearly $\alpha$ is determined by the polynomials $r, s, D$ and our choice of $\sqrt{D}$. For example for $\alpha = \sqrt{D}$ we just put $r = 0$, $s = 1$ and $w = - D$.
All complete quotients of a given $\alpha$ can be written in this way; all of them with the same discriminant $D$:
\[prop-quadr-repr-rs\] The complete quotients of $\alpha$ as in have for all $n \geq 0$ the form $$\label{quadr-repr-rs}
\alpha_n = \frac{r_n + \sqrt{D}}{s_n}, \quad \text{ where } s_n \div (D - r_n^2) \text{ and } r_n, s_n \in \K[X].$$
We prove this using complete induction. For $n = 0$ we may take $r_0 = r$ and $s_0 = s$ which satisfy the desired conditions by hypothesis.
Suppose holds for $n$. Then write $$\frac{1}{\alpha_{n+1}} = \alpha_n - a_n
= \left( \frac{r_n + \sqrt{D}}{s_n} - a_n \right) \left( \fracsame{-r_n + \sqrt{D} + a_n \, s_n} \right)
= \frac{ D - (r_n - a_n \, s_n)^2}{s_n \, \left(a_n \, s_n - r_n + \sqrt{D}\right)}$$ and note that $$D - r_{n+1}^2 = D - (a_n \, s_n - r_n)^2 = D - a_n^2 \, s_n^2 + 2\, a_n \, s_n \, r_n - r_n^2$$ so by induction hypothesis $s_n \div D - r_n^2$, this is divisible by $s_n$ and we can set $$\begin{aligned}
\label{cf-rs-recursion}
r_{n+1} &= a_n \, s_n - r_n, &
s_{n+1} &= \frac{D - r_{n+1}^2}{s_n}.\end{aligned}$$ with $r_{n+1}, s_{n+1} \in \K[X]$ and moreover $s_{n+1} \div D - r_{n+1}^2$. This concludes the induction step.
It should be quite obvious that the discriminant does not change for the complete quotients. After all, the discriminant is invariant under the natural action of $\GL{2}{\K(X)}$ by linear change of variables on bilinear forms in two variables over $\K(X)$. Such a bilinear form gives of course a minimal polynomial for a quadratic $\alpha$. But advancing in the continued fraction expansion can exactly be expressed in terms of this action, as seen in Section \[sec:org932a99d\].
Berry (and Abel for $\deg D = 4$) give further simplifications of these formulas, see [@berry-1990-periodicity-continued-fractions] and [@abel-1826-ueber-integ-differ]. We prefer to perform simplifications of a different kind. And we still need to explain how to compute the $a_n$ from our representation.
We may rewrite as $$\label{quadr-repr-ats}
\alpha_n = \frac{A + t_n + \sqrt{D}}{s_n}$$ by setting $t_n = r_n - A$. The recursion formulas then obviously change to $$\label{cf-ats-recursion}
\begin{aligned}
t_0 &= r - A, & t_{n+1} &= a_n \, s_n - 2 \, A - t_n, \\
s_0 &= s, & s_{n+1} &= \frac{D - A^2 - 2 \, A \, t_{n+1} - t_{n+1}^2}{s_n}.
\end{aligned}$$
This already proves the first half of Theorem \[thm-quadratic-laurent-series-cf-representation\].
\[cf-ats-bound-t1\] We can compute $t_{n+1}$ and $a_n$ with a single polynomial division, i.e. $$(2 \, A + t_n)= a_n \, s_n - t_{n+1} \text{ with } \deg t_{n+1} < \deg s_n.$$
Recall from that $\sqrt{D} = A + \varepsilon$ with $\io{\varepsilon} > 0$. The equality follows directly from the formula for $t_{n+1}$ in , it remains to check $\deg t_{n+1} < \deg s_n$. Using $\gauss{\varepsilon} = 0$ and Remark \[truncation-of-sum\] ($\gauss{\cdot}$ is a homomorphism with respect to $+$) we find $$a_n = \gauss{\alpha_n} = \gauss{ \frac{A + t_n + \sqrt{D}}{s_n} } = \gauss{\frac{2 \, A + t_n + \varepsilon}{s_n}} = \gauss{\frac{2 \, A + t_n}{s_n}}.$$ So by Remark \[truncation-of-rational\] (taking $\gauss{\cdot}$ of rational functions corresponds to polynomial division) $-t_{n+1}$ must the remainder of the polynomial division of $2 \, A + t_n$ by $s_n$.
Complete quotients are eventually $\sigma$-reduced {#sec:org780327e}
--------------------------------------------------
The representation also gives a very simple way to check if some complete quotient is $\sigma$-reduced:
\[sigma-reduced-degree-prop\] $\alpha = \frac{A + t + \sqrt{D}}{s}$ is $\sigma$-reduced $$\label{sigma-reduced-degree}
\deg t < \deg s < \deg A,$$ and in this case $\io{\alpha} = \deg s - \deg A$.
With $A - \sqrt{D} = - \varepsilon$ we note that $$\io{\sigma(\alpha)} = \io{\ifracBb{A+t - \sqrt{D}}{s}} = \io{t - \varepsilon} - \io{s} = \io{t - \varepsilon} + \deg s.$$ Hence $0 < \io{\sigma(\alpha)}$ is equivalent to $\deg t < \deg s$: In the case $t = 0$, using $\io{\varepsilon} > 0$ we have $\io{\sigma(\alpha)} = \io{\varepsilon} + \deg s > 0$ we have $s \neq 0$, i.e. $\deg s > - \infty = \deg 0$. If on the other hand $t \neq 0$, then $\io{t - \varepsilon} = \io{t} = - \deg t$, hence $\io{\sigma(\alpha)} = \deg s - \deg t$.
So for the rest of the proof, we can assume $\io{\sigma(\alpha)} > 0$.
We may write $$\begin{gathered}
\io{\alpha} = \io{\ifracBb{A+t+\sqrt{D}}{s}} = \io{2 \sqrt{D} + t - \varepsilon} - \io{s} \\ \geq \min\left(\io{2 \sqrt{D}}, \io{t-\varepsilon}\right) + \deg s.\end{gathered}$$ If $\alpha$ is $\sigma$-reduced, then $0 > \io{\alpha} = \io{2 \sqrt{D}} + \deg s$ because $\io{t -\varepsilon} + \deg s > 0$. Hence $\deg s < \deg A = -\io{\sqrt{D}}$.
Conversely, if $\deg s < \deg A$, then $\io{\alpha} = \io{2 \sqrt{D}} + \deg s < 0$ by the ultrametric “equality”.
With $\io{2 \sqrt{D}} = \io{A} = -\deg A$, we also showed $\io{\alpha} = \deg s - \deg A$.
An immediate and important consequence is that the degrees of the partial quotients of a $\sigma$-reduced $\alpha$ are always bounded uniformly – once we show that every continued fraction of a quadratic $\alpha$ eventually becomes $\sigma$-reduced, this means all partial quotients have bounded degree.
\[maximal-degree-implies-quasi-period\] Suppose $\alpha$ as above is $\sigma$-reduced, and $a = \gauss{\alpha}$. Then $0 < \deg a \leq \deg A$.
Moreover, if $\deg{a} = \deg A$, then there exists $\mu \in \units \K$ such that $\alpha = \mu \, (A + \sqrt{D})$.
From $\alpha$ being $\sigma$-reduced, the preceding proposition yields $$0 > \io{\alpha} = \deg s - \deg A \geq - \deg A.$$ But $\io{\alpha} = \io{a} = - \deg a$, hence $0 < \deg a \leq \deg A$.
Additionally, if $\deg a = \deg A$ this means $\deg s = 0$ and thus $t = 0$. So we get $\mu = \inv s \in \units \K$.
The second half of Theorem \[thm-quadratic-laurent-series-cf-representation\] follows from
\[cf-compute-eventually-sigma-reduced\] Let $\alpha \in \laurinx \K$ quadratic over $\K(X)$. Then there exist $N \in \N$ such that for all $n \geq N$, the complete quotient $\alpha_n$ is $\sigma$-reduced.
Using Proposition \[sigma-reduced-degree-prop\], this boils down to an analysis of the degrees of $t_n$ and $s_n$.
From Proposition \[cf-ats-bound-t1\] follows $\deg t_{n+1} < \deg s_n$. Recall from Remark \[cf-absolute-values\] that we have $\deg a_n \geq 1$ for $n \geq 1$, hence $$\begin{gathered}
\deg t_{n+1} < \deg s_n < \degb{a_n \, s_n} = \degb{2 \, A + t_n + t_{n+1}} \\
\leq \max(\deg A, \deg t_n, \deg t_{n+1}) = \max(\deg A, \deg t_n).\end{gathered}$$ So if $\deg t_n \geq \deg A$ then $\deg t_{n+1} + 2 \leq \deg t_n$. Then after a finite number of steps we must have $\deg t_{n+j} < \deg A$ (actually $\deg t_{n+j} + 2 \leq \deg A$). And if $\deg t_n < \deg A$, then clearly also $\deg t_{n+1} < \deg A$ (actually $\deg t_{n+1} + 2 \leq \deg A$).
So we may now assume $\deg t_n < \deg A$ for all $n$ large enough.
Next, if $t_{n+1} = 0$, then $s_n \, s_{n+1} = D - A^2$ and $\degb{D - A^2} < \deg A$ (see Proposition \[completion-of-square\]). This implies $\deg s_n + \deg s_{n+1} < \deg A$, so clearly $\deg s_{n+1} < \deg A$, and trivially $-\infty = \deg t_{n+1} < \deg s_{n+1}$, hence $\alpha_{n+1}$ is $\sigma$-reduced.
If on the other hand $t_{n+1} \neq 0$, then $s_n \, s_{n+1} = D - A^2 - 2 \, A \, t_{n+1} - t_{n+1}^2$ and thus $$\begin{gathered}
\deg s_n + \deg s_{n+1} = \max( \degb{D - A^2}, \deg A + \deg t_{n+1}, 2 \, \deg t_{n+1}) \\ = \deg A + \deg t_{n+1} < \deg A + \deg s_n\end{gathered}$$ implies $\deg s_{n+1} < \deg A$. If moreover $\deg s_n < \deg A$ (if not, consider $s_{n+2}$ and $s_{n+1}$ instead), we also get $\deg t_{n+1} < \deg s_{n+1}$ and so $\alpha_{n+1}$ is $\sigma$-reduced.
All subsequent complete quotients then remain $\sigma$-reduced by Proposition \[sigma-reduced-all-complete-quotients\].
From the proof, we easily deduce an effective bound for $N$. The degree of $t_n$ decreases by at least $2$ in every step from $t_1$, so at most $\ifracBb{\deg t_1 - \deg A}{2}$ steps are required to arrive at $\deg t_n < \deg A$. From there, we need only one or two additional steps to arrive at a $\sigma$-reduced complete quotient. So $N \leq 3 + \ifracBb{\deg t_1 - \deg A}{2}$. This demonstrates the effectivity in Theorem \[thm-quadratic-laurent-series-cf-representation\].
The $\sigma$-reduced case allows even simpler computation of the partial quotient:
If $\alpha_n$ is $\sigma$-reduced, then we may use polynomial division of $2 \, A$ by $s_n$ to compute $t_{n+1}$ (improving minimally upon \[cf-ats-bound-t1\]): $$2 \, A = a_n \, s_n - (t_n + t_{n+1}),$$ as both $\deg t_n, \deg t_{n+1} < \deg s_n$.
Periodicity and Pell equation {#sec:org8d34125}
-----------------------------
Let us now check the theorems given at the beginning of this chapter.
We expand upon Remark \[period-of-sqrt-d\], and work with $A + \sqrt{D}$ instead of $\sqrt{D}$. This changes only $a_0$ and $\alpha_0$. Of course $A + \sqrt{D}$ has $t_0 = 0$ and $s_0 = 1$ which shows again (now using Proposition \[sigma-reduced-degree-prop\]) that it is $\sigma$-reduced, hence also all complete quotients $\alpha_n$ with $n \geq 1$ are $\sigma$-reduced.
Then Theorem \[thm-optimised-sqrt-cq-representation\] simply combines , (which follow from Proposition \[prop-quadr-repr-rs\]) and Proposition \[sigma-reduced-degree-prop\].
Additionally, Theorem \[cf-berrys-thm\] implies that $\CF(\sqrt{D})$ is periodic $\CF(A + \sqrt{D})$ is pure quasi-periodic, and both continued fraction have the same quasi-period length $\QPL$. With Proposition \[cf-quasi-periodicity-complete-quotients\] and Corollary \[maximal-degree-implies-quasi-period\] it follows that $\alpha_n = \frac{A + \sqrt{D}}{s_n}$ with $s_n \in \units \K$ (i.e. $\deg s_n = 0$) holds $\QPL \div n$ from minimality of the quasi-period length $\QPL$.
We give a few more details for
The degree inequalities were stated already in Corollary \[maximal-degree-implies-quasi-period\] and follow from $\deg a_n = \deg A - \deg s_n$. The corollary also says that $\deg a_n = \deg A$ implies pure quasi-periodicity of $\CF(A + \sqrt{D})$.
Set $\alpha = \sqrt{D}$, and recall from Section \[sec:org932a99d\] that (for $n
\geq 1$) $$\sqrt{D} = \mfour{p_{n-1}}{p_{n-2}}{q_{n-1}}{q_{n-2}} \, \alpha_n \iff \alpha_n = (-1)^n \, \mfour{q_{n-2}}{-p_{n-2}}{-q_{n-1}}{p_{n-1}} \, \sqrt{D}$$ which we rewrite as $$\begin{gathered}
\label{cf-moebius-pell-denom}
\alpha_n
= \frac{q_{n-2} \, \sqrt{D} - p_{n-2}}{p_{n-1} - q_{n-1} \, \sqrt{D}}
= \frac{q_{n-2} \, \sqrt{D} - p_{n-2}}{p_{n-1} - q_{n-1} \, \sqrt{D}} \cdot \fracsame{p_{n-1} + q_{n-1} \, \sqrt{D}} \\
= \frac{D \, q_{n-1} \, q_{n-2} - p_{n-1} \, p_{n-2} + \sqrt{D} \left(p_{n-1} \, q_{n-2} - p_{n-2} \, q_{n-1}\right)}{p_{n-1}^2 - D \, q_{n-1}^2} \\
= \frac{(-1)^n \, (\dots) + \sqrt{D}}{(-1)^n \left(p_{n-1}^2 - D \, q_{n-1}^2\right)}\end{gathered}$$ so $$\label{cf-sn-pell-eq}
s_n = {(-1)^n \left(p_{n-1}^2 - D \, q_{n-1}^2\right)}.$$ Recall Theorem \[cf-berrys-thm\] which states that periodicity and quasi-periodicity are equivalent in the current situation. So by Corollary \[maximal-degree-implies-quasi-period\] (proved just above), it follows that $\CF(\sqrt{D})$ is periodic for some $n \geq 1$ we have $\deg s_n = 0$ which means $(p_{n-1}, q_{n-1})$ solves the Pell equation .
On the other hand, we know that Pell solutions are convergents (Proposition \[weak-pell-solutions-are-convergents\]) and from the classification of convergents (Proposition \[cf-convergent-classification\]) follows that every non-trivial solution of has the shape $(p, q) = \mu \cdot (p_m, q_m)$ for some $m \geq 0$ with $\mu \in \units \K$ (because for a Pell solution $p, q$ are coprime). This implies that $(p_m, q_m)$ likewise solves , and then $\deg s_{m+1} = 0$.
Torsion order and period length {#sec:org61dfa9f}
-------------------------------
Recall the notation from Chapter \[sec:org99a8e17\], and assume again that $D$ is square-free. With $2(g+1) = \deg D$, we get the following inequalities between the torsion order and the quasi-period length:
\[prop-bounds-torsion-period-length\] Suppose $\j{\OO} \in \Jac$ is torsion of order precisely $m$, and let $\QPL$ the quasi-period length of $\CF(\sqrt{D})$. Then for $g \geq 1$ we have the inequality[^7] $$g + \QPL \leq m \leq 1 + g \, \QPL$$ which for $g = 1$ becomes the equality $m = \QPL + 1$.
Combining the knowledge from the proofs of Theorems \[thm-pellian-iff-torsion\] and \[thm-pellian-iff-cf-periodic\], we know that the minimal $n$ such that is satisfied with $r = 0$ by $(p_{n-1}, q_{n-1})$ is exactly $n = \QPL$, with $m = \deg p_{n-1}$. So this $m$ must be the torsion order of $\j{\OO}$.
We then calculate, using $1 \leq \deg a_i \leq g$ for $i=1, \dots, l-1$ which holds by Corollary \[cor-pq-degree-periodicity\], $$\begin{aligned}
m = \deg p_{l-1} = \deg a_0 + \deg q_{l-1} = g+1 + \deg q_{l-1} &\leq g+1 + (l-1) g = 1 + l \, g \\
& \geq g+1 + l-1 = l + g\end{aligned}$$ which yields the desired inequality. Clearly it collapses to an equality for $g = 1$.
So bounding the period length is as hard as bounding torsion.
Period lengths over finite fields {#sec:org4b81157}
---------------------------------
We now give an (elementary) proof of Corollary \[cor-finite-field-always-periodic\], by showing that over a finite base field $\K$ there are only finitely many possibilities for the $\sigma$-reduced complete quotients. As these form the tail of every continued fraction of a quadratic Laurent series, this means any repetition immediately implies periodicity. Of course we have to avoid characteristic $2$ again.
\[naive-period-length-bound\] Let $\K = \F_q$ a finite field of odd characteristic, and recall that $\deg D = 2d$. Then for a fixed $D$, there are precisely $$\label{sr-naive-period-bound}
\frac{q^{2 d }-1}{q+1}$$ $\sigma$-reduced expressions of type $\ifracBb{A + t + \sqrt{D}}{s}$.
Note that the above counting does not yet take into account that we usually have the additional condition $s \div D- r^2$. This further limits the number of possible complete quotients.
For fixed $e = \deg s$, there are $(q-1) \, q^e$ possibilities for $s$, and as $\deg t < \deg s$, there are $q^e$ possibilities for $t$. Summing over $e$, we compute $$\sum_{e=0}^{d-1} (q-1) q^e \; q^e = (q-1) \, \frac{q^{2 d}- 1}{q^2 - 1} = \frac{q^{2 d }-1}{q+1}$$ using the formula for geometric sums.
The above gives an elementary bound for the period length. Using our knowledge about quasi-periods, we could improve it further dividing by $2/(q-1)$.
But anyway we already have a far better bound for for the torsion order in the Jacobian (under the assumption that $D$ is square-free), see Remark \[finite-field-torsion-bound\].
Then we can do much better:
If $D$ is square-free, the quasi-period length is bounded by $$\QPL \leq m - g \leq (\sqrt{q} + 1)^{2g} - g.$$
Divisors of complete quotients {#sec:org4e4485e}
------------------------------
We now wish to expand upon the results of Section \[sec:org5685823\], and make the connection between the convergent divisors and the continued fraction more explicit. This will be useful later to give an additional viewpoint on the reduction of continued fractions. See also [@berry-1990-periodicity-continued-fractions], where it is shown that quasi-periodicity of arbitrary elements of $\HEF \setminus \K(X)$ is equivalent to $D$ being Pellian.
Recall the notation from Chapter \[sec:org99a8e17\], and the additional assumption that $D$ is square-free. Let $\alpha = \frac{r + w \, Y}{s} \in \K(X,Y)$ an arbitrary element of the function field of the (hyper)elliptic curve $\CC$ with $r, s, w \in \K[X]$. Put $\alpha_0 = \frac{r + w \, \sqrt{D}}{s} \in \laurinx \K$. We may assume $\io{\alpha_0} \leq 0$, otherwise we simply pass to the inverse of $\alpha$. We also require $w, s \neq 0$ and may of course assume $\gcd(r, s, w) = 1$.
Then the finite poles of $\alpha$ are zeroes of $s$. So the divisor has the shape $$\Div \alpha = -\pd{Q_1} - \dots - \pd{Q_h} + \dots, \quad \text{} Q_i \in \CCa$$ with $h \leq 2 \, \deg s$ and other poles only at infinity (the points $O_\pm$).
We now generalise Lemma \[convergent-divisor-lemma\] about the divisors of convergents of $\sqrt{D}$ to rational functions on $\CC$:
\[general-convergent-divisor-lemma\] Let $(p, q) \in \Coset{\alpha_0}{\K}$ a convergent, then $$\label{general-convergent-divisor-equation}
\Div(p - \alpha \, q) = -m \pd{O_-} - \pd{Q_1} - \dots - \pd{Q_h} + (m+h-e) \, \pd{O_+} + \pd{P_1} + \dots + \pd{P_e}$$ where $P_i \in \CCa$, $m \geq 0$ and $0 \leq e < h - \ord(\alpha_0) \leq h + m$.
Set $\phi = p - \alpha \, q$. Any finite poles (i.e. in $\CCa$) must be among the $Q_i$ because $\ord_P(p) \geq 0$ and $\ord_P(q) \geq 0$ imply $$\ord_P(\phi) \geq \min\left(\ord_P(p), \ord_P(\alpha) + \ord_P(q)\right) \geq \min(0, \ord_P(\alpha)).$$ From $(p, q)$ being a convergent, we know that $\ord_{O_+}(\phi) = \io{\phi} > \deg q \geq 0$. As in , this implies with $\io{\alpha_0 \, q} \leq 0$ that $$\ord_{O_-}(\phi) = \io{p + \alpha_0 \, q} = \io{p} = -\deg p = \io{\alpha_0} + \io{q}.$$ Hence $m = - \ord_{O_-} \geq 0$.
With all possible poles determined, we can write $\Div(\phi)$ as in , where possibly some of the $P_i \in \CCa$ coincide with some $Q_j$. The divisor must have degree $0$, so $\ord_{O_+}(\phi) = m + h -e$, and $$\deg q < m + h - e = \deg q - \io{\alpha_0} + h - e$$ implies $e < h - \io{\alpha_0}$.
We can make this even more precise for the canonical convergents $(p_n, q_n)$:
\[cor-canonical-convergent-general-divisor\] Let $\phi_n = p_n - \alpha \, q_n$, then $$\Div \phi_n = -(\deg p_n) \pd{O_-} - \pd{Q_1} - \dots - \pd{Q_h} + (\deg q_{n+1}) \pd{O_+} + \pd{P_1^n} + \dots + \pd{P_{e_n}^n}$$ where $P^n_{i} \in \CCa$ (perhaps some coincide with a $Q_j$) and $$e_n = \deg a_0 - \deg a_{n+1} + h \leq \deg a_0 + h - 1.$$
We obtain the formula for $e_n$ from $$% \ord{O_+}(\phi_n) =
\deg q_{n+1} = \deg q_n + \deg a_{n+1} = \deg p_n + h - e_n = \deg q_n + \deg a_0 + h - e_n,$$ because the principal divisor $\phi_n$ has degree $0$.
Via , we can now calculate the divisors of the complete quotients (thinking $Y = \sqrt{D}$):
Write $\mathbf P^n = \pd{P^n_1} + \dots + \pd{P^n_{e_n}}$, then $$\begin{gathered}
\Div \alpha_n = \Div\left( - \ifrac{\phi_{n-2}}{\phi_{n-1}} \right) \\
= (\deg p_{n-1}-\deg p_{n-2}) \, \pd{O_-} + (\deg q_{n-1} - \deg q_{n}) \, \pd{O_+} + \mathbf P^{n-2} - \mathbf P^{n-1} \\ % \sum_{i=1}^{e_{n-2}} \pd{P^{n-2}_{i}} - \sum_{i=1}^{e_{n-1}} \pd{P^{n-1}_{i}} \\
= (\deg a_{n-1}) \, \pd{O_-} + (-\deg a_n) \, \pd{O_+} + \mathbf P^{n-2} - \mathbf P^{n-1} % \sum_{i=1}^{e_{n-2}} \pd{P^{n-2}_{i}} - \sum_{i=1}^{e_{n-1}} \pd{P^{n-1}_{i}}.\end{gathered}$$
Note how $$\begin{aligned}
\ord_{O_+}(\alpha_n) &= \io{\alpha_n} = \io{a_n} = - \deg a_n, \\
\ord_{O_-}(\alpha_n) &= -\io{\cfsb{\alpha_n}} = -\io{a_{n-1}} = \deg a_{n-1}.\end{aligned}$$ This aligns with the observations in Section \[sec:org6896ed2\], in particular Remark \[cf-reduced-backwards-n\] about the “conjugate” continued fraction expansion.
So the $Q_i$ can no longer be seen directly in this divisor, but of course they could appear hidden among the $P^{n-1}_i, P^{n-2}_i$.
Let us now restrict to the case $w = 1$ and $s \div D - r^2$. This implies $h \leq \deg s$ because now it is impossible for both a point and its conjugate to appear as a pole, and a self-conjugate point can appear at most as a pole of order $1$ (assuming that $D$ is square-free).
If $s \in \units\K$, then there are no finite poles, and we are essentially in the situation of Lemma \[convergent-divisor-lemma\].
\[rem-general-convergent-divisor-translated-multiples\] Using $\io{\alpha_0} \leq 0$, we may also assume that $\deg r \leq d = \frac{1}{2} \deg D$ (otherwise we could subtract some multiple of $s$ from $r$ which does not change the subsequent complete quotients). This implies $\io{\alpha_0} \geq \io{\sqrt{D}} - \io{s}$, so $e < - \io{\alpha_0} \leq d + h - \deg s \leq d$ and hence $e \leq g$, so we get $$\label{jacobian-translate-point-multiples}
j(Q_1) + \dots + j(Q_h) + m \, j(O_-) = j(P_1) + \dots + j(P_e).$$ We are thus representing a translate of the multiples of $\OO$ as a sum of at most $g$ points in the Jacobian.
\[convergent-divisor-rationality\] The divisor $\pd{P_1} + \dots + \pd{P_e}$ is usually going to be a $\K$-rational divisor. Be aware that this does not mean that the $P_i$ are defined over $\K$. However they are defined over a field extension of degree at most $e$ over $\K$. So if $e = 1$, the single point $P_1$ is going to be defined over $\K$. We will make use of this later in Sections \[sec:org9851fc8\] and \[sec:org53fcc2b\].
Specialization of continued fractions {#sec:orgd5f1900}
=====================================
The first goal of this chapter is to explain and recover a theorem of van der Poorten (see Theorem 1 in [@poorten-1998-formal-power-series], Theorem 2.1 in [@poorten-1999-reduction-continued-fractions] and Theorem 6 in [@poorten-2001-non-periodic-continued]) stating that the convergents of some $\alpha$ modulo a prime number $\pp$ all arise by normalising and reducing the original convergents of $\alpha$ (which is a Laurent series with rational coefficients).
Here we actually prove this theorem (as Theorem \[convergent-reduction-surjective\]) in the general setting of Laurent series defined over a discrete valuation ring (or its fraction field), once some natural conditions are satisfied.
Before we look at the convergents, we however need to understand what we mean by reducing convergents, and likewise continued fractions. For the latter, this immediately leads to a notion of good or bad reduction of polynomial continued fractions. In the case of good reduction of a continued fraction, van der Poorten’s theorem becomes trivial, using the classification of convergents described in Chapter \[sec:org172eb73\]. This suggests that the bad reduction case is more interesting.
Understanding the reduction of the convergents also helps to understand reduction of the continued fraction better, and we will look at some simple cases at the end of the chapter. This goes already toward the calculation of the Gauss norms of the partial quotients and convergents. These will be further analysed for square roots of polynomials in the next chapter.
Specialization of Laurent series {#sec:org46a3df3}
--------------------------------
### Discrete valuation rings {#sec:orgb3fd29f}
We fix a discrete valuation ring $\O$ with its unique (principal) maximal ideal $\mm$. It produces two fields: the *fraction field* $K = \Fr(\O)$ and the *residue field* $k = \O/\mm$. In order to apply the theory from the preceding chapters, we require that $\Char k \neq 2$, which implies $\Char K \neq 2$ as well.
We denote the (non-archimedean) valuation of $\O$ by $\nu_0 : K \surject \Z \cup \{ \infty \}$. Recall that it satisfies
- $\nu_0(x) = \infty \iff x = 0$,
- $\nu_0(x \, y) = \nu_0(x) + \nu_0(y)$ for all $x, y \in \units K$,
- $\nu_0(x + y) \geq \min(\nu_0(x), \nu_0(y))$ for all $x, y \in K$.
In the last point, we can replace “$\geq$” with “$=$” if $\nu_0(x) \neq \nu_0(y)$.
Moreover we choose an uniformising parameter $\uni$ (a generator of the maximal ideal $\mm$ in $\O$), with satisfies $\nu_0(\uni) = 1$. Recall $$\label{dvr-valuation-defi}
\begin{aligned}
\O &= \{ x \in K \mid \nu_0(x) \geq 0 \},\\
\mm = \spann{\uni} &= \{x \in K \mid \nu_0(x) > 0 \},\\
\units \O &= \{ x \in K \mid \nu_0(x) = 0 \}.
\end{aligned}$$
We get the reduction map $\Redm : \O \to \O/\mm = k$; we usually write $\Red{x} = \RedM{x}$ for more compact notation.
Note that by choosing a discrete non-archimedean valuation $\nu_0$ on a given field $K$, we get a discrete valuation ring $\O$ through .
For example, starting with $K = \Q$ and some odd integer prime $\pp$ with its corresponding $\pp$-adic valuation $\nu_\pp$, one gets the localisation $\O = \Z_{\spann{\pp}}$ of $\Z$ at $\pp$. In this case, the residue field $k = \F_\pp$ is finite.
Another example would be $K = \C(t)$ with a zero-order $\ord_{t=t_0}$ (for some $t_0 \in \C$). Then $\O = \C[t]_{\spann{t-t_0}}$ is a localisation of $\C[t]$ at the prime ideal $\spann{t-t_0}$, and $t-t_0$ is a uniformising parameter. The residue field is now $k = \C$, hence infinite. In this example we could actually replace $\C$ by any field (of characteristic not $2$), even a finite field. The latter would make the residue field finite again.
### Gauss norms {#sec:orge83be0b}
It is natural to extend such a valuation to polynomials; for absolute values this is called a *Gauss norm*. In fact, we can extend the valuation even to a subset of Laurent series.
Define $\nu : \laurinx K \to \Z \cup \{+\infty, -\infty\}$ by setting for $u \in \laurinx K$, with $u_n \in K$: $$\nub{u} = \nub{\lseries{N}{u}{n}} = \inf \{ \nu_0(u_n) \mid n \in \Z, n \leq N \}.$$ To avoid $\nub{u} = -\infty$, we restrict to the subring $$\laurinx K_\nu = \{ u \in \laurinx K \mid \text{the } \nu_0(u_n) \text{ are bounded from below} \}.$$
If $x \in K$, note that because $\nu_0$ is non-archimedean, $u(x)$ converges $\nu_0(u_n \, x^n) = \nu_0(u_n) + n \, \nu_0(x) \to +\infty$ as $n \to \infty$. The boundedness condition ensures that $u(x)$ converges for every $x \in \mm$ (with $\nu_0(x) > 0$).
\[bounded-laurent-valuation\] $\laurinx K_\nu$ is a ring, and the extended $\nu$ is a discrete non-archimedean valuation on it.
It suffices to check that $\nu$ satisfies the usual properties of an ultrametric valuation on $\laurinx K_\nu$. Then $\laurinx K_\nu$ is automatically a ring (using the same arguments which show that $\O$ defined as in is a ring).
It is also obvious that $\nu$ is discrete because we take an infimum of a subset of $\Z$ bounded from below.
Clearly, we have $\nub{u} = \infty$ $u = 0$.
Take $u,v \in \laurinx K_\nu$ with $$u = \lseries{N}{u}{n}, \quad v = \lseries{M}{v}{m},$$
For the ultrametric inequality, let $$u + v = w = \lseries{\max(N,M)}{w}{l}.$$ Without loss of generality, one may assume $N=M$, and then $w_n = u_n + v_n$ for all $n \leq N$: $$\begin{gathered}
\nub{w} = \inf\{\nuOb{u_n + v_n} \mid n \leq N\}
\geq \inf\{\min(\nuOb{u_n}, \nuOb{v_n}) \mid n \leq N\} \\
= \min\left( \inf\{\nuOb{u_n} \mid n \leq N\}, \inf\{\nuOb{v_n} \mid n \leq N\} \right)
= \min(\nub{u}, \nub{v}).\end{gathered}$$
For multiplicativity, let $$u\, v = w = \lseries{(N+M)}{w}{l}.$$ As $\nu$ is invariant under multiplication with powers of $X$, we may assume $N=M=0$. From the definition of the Cauchy product $$\label{cauchy-product-u-v-eq-w}
w_l = \sum_{n+m=l} u_n \, v_m$$ it is obvious that $\nub{w} \geq \nub{u} + \nub{v}$ must hold: $$\begin{gathered}
\nu(w) = \inf\{\nu_0(w_l) \mid l \leq 0\} \geq \inf\left\{ \min(\nu_0(u_n) + \nu_0(v_m) \mid n + m = l ) \mid l \leq 0 \right\} \\
\geq \inf\left\{ \min(\nu(u) + \nu(v) \mid n + m = l ) \mid l \leq 0 \right\}
\geq \nu(u) + \nu(v).\end{gathered}$$
Because $\nu_0$ is discrete on $K$, there exist $n_0, m_0$ such that $$\nub{u} = \nuOb{u_{n_0}} \text{ and } \nub{v} = \nuOb{v_{m_0}}$$ and of course, we may choose $n_0$ and $m_0$ maximal. Then $$w_{n_0 + m_0} = \sum_{n+m = n_0 + m_0} u_n \, v_m = \sum_{n+m = n_0 + m_0, \atop n > n_0} u_n \, v_m + u_{n_0} \, v_{m_0} + \sum_{n+m = n_0 + m_0, \atop m > m_0} u_n \, v_m.$$ We have $\nuOb{u_n} > \nub{u}$ for all terms with $n > n_0$, hence the absolute value of the left sum is $> \nub{u} + \nub{v}$. And we have $\nuOb{v_m} > \nub{v}$ for all terms with $m > m_0$, hence the absolute value of the right sum is $> \nub{u} + \nub{v}$.
However, the middle term has absolute value $\nuOb{u_{n_0}} + \nuOb{v_{m_0}} = \nub{u} + \nub{v}$, implying $\nuOb{w_{n_0 + m_0}} = \nub{u} + \nub{v}$. It follows $\nub{u\,v} \leq \nub{u} + \nub{v}$, and hence $\nub{u\,v} = \nub{u} + \nub{v}$.
Clearly, $K \subset K[X] \subset \laurinx K_\nu$. For all $x \in K$ we have $\nu(x) = \nu_0(x)$, so henceforth we refer to $\nu_0$ also as $\nu$.
Of course also $\laurinx \O \subset \laurinx K_\nu$. Applying the reduction map on each coefficient, it extends naturally to $$\Redm : \O[X] \surject k[X], \qquad \Redm : \laurinx \O \surject \laurinx k.$$ For convenience, we use the same notation, including $\Red{x} = \RedM{x}$ for $x \in \laurinx \O$, and say that we *reduce mod $\nu$* or *specialize at $\nu$*.
In analogue to , we obviously get $$\begin{aligned}
% \O &= \{ u \in K \mid \nub{u} \geq 0 \} &
% \mm &= \{ u \in K \mid \nub{u} > 0 \} \\
\O[X] &= \{ u \in K[X] \mid \nub{u} \geq 0 \}, &
\mm[X] &= \{ u \in K[X] \mid \nub{u} > 0 \}, \\
\laurinx \O &= \{ u \in \laurinx K \mid \nub{u} \geq 0 \}, &
\laurinx \mm &= \{ u \in \laurinx K \mid \nub{u} > 0 \}.\end{aligned}$$ It is straightforward to check that $\mm[X]$ respectively $\laurinx \mm$ are the kernels of the (surjective) reduction map on $\O[X]$ respectively $\laurinx \O$ (consider the valuations of the coefficients of $u$). As $k[X]$ is an integral domain, this implies that $\mm[X]$ is a prime ideal of $\O[X]$. And as $\laurinx k$ is even a field, the ideal $\laurinx \mm$ is a maximal ideal of $\laurinx \O$.
Both $\mm[X]$ and $\laurinx \mm$ are obviously principal ideals in their respective ring, with generator $\uni$ (the uniformising parameter of $\nu_0$).
The fraction field of $\laurinx \O$ is $\laurinx K$. However $\laurinx \O$ is *not* a discrete valuation ring. It is not even a local ring, because $\nu(u) = 0$ is not a sufficient condition for having $u \in \units{\laurinx \O}$ (see Corollary \[laurent-inverse-bounded-corollary\] below).
For example $u = \pi + \inv X$ is not in $\laurinx \mm$, but neither is it a unit of $\laurinx \O$.
We say for $u \in \laurinx K$ that
- $u$ is *unbounded* if $\nub{u} = -\infty$ i.e. $u \not \in \laurinx K_\nu$,
- $u$ is *bounded* if $\nub{u} \neq -\infty$ i.e. $u \in \laurinx K_\nu$,
- $u$ *has* if $\nub{u} < 0$, in particular if it is unbounded,
- $u$ *has* if $\nub{u} > 0$.
For example, if $u \in K[X]$ is a polynomial, it has at least one of its coefficients has ; and it has all its coefficients are either $0$ or have . Note the different logical operations: For , we have **or**, for we have **and**.
Let us now investigate how far away $\laurinx K_\nu$ is from being a field (and $\laurinx \O$ from being a discrete valuation ring). For example, the Laurent polynomial $1 + u_{-1} \, \inv X$ with $\nub{u_{-1}} < 0$ does not have a bounded inverse:
\[laurent-inverse-bounded\] Let $u \in \laurinx K_\nu$ with $u_0 = \LC(u) \neq 0$ (so $u \neq 0$). Then $\inv u \in \laurinx K_\nu$ if and only if $\nub{u} = \nub{u_0}$.
If $u$ has a bounded inverse, we have $\LC(\inv u) = \ifrac{1}{u_0}$ with $\nub{\ifrac{1}{u_0}} \geq \nub{\inv u} = -\nub{u}$, hence $\nub{u} \geq \nub{u_0}$. But by definition $\nub{u_0} \geq \nub{u}$, so it follows $\nub{u_0} = \nub{u}$.
Conversely, assume $\nub{u_0} = \nub{u}$; dividing $u$ by $u_0$ and $X^{-\ord u}$ (both are bounded), we may without loss of generality write $u = 1 - v$ for $v \in \laurinx \O$ with $\nub{v} \geq 0$ and $\io{v} > 0$ (so actually $v \in \powerseriesinvx \O$ is a power series in $\inv X$ without constant coefficient). Then $$\inv{u} = \frac{1}{1-v} = \sum_{j=0}^\infty {v}^j = \lseries{0}{w}{m}$$ converges in $\laurinx K$. Only finitely many ${v}^j$ (always with $\nub{v^j} \geq 0$) contribute to each $w_m$, so clearly $\nub{w_m} \geq 0$ for all $m$, and $\inv u$ is bounded.
\[laurent-inverse-bounded-corollary\] Let $u \in \laurinx \O \setminus \{0\}$. Then $\inv u \in \laurinx \O$ (i.e. $u \in \units{\laurinx \O}$) $\LC(u) \in \units \O$.
If $\inv u \in \laurinx \O$, then both $\nu(u) \geq 0$ and $-\nu(u) = \nub{\inv u} \geq 0$ hence $\nu(u) = 0$. By Proposition \[laurent-inverse-bounded\] follows $\nu(\LC(u)) = 0$, i.e. $\LC(u) \in \units \O$.
Conversely, if $\LC(u) \in \units \O$, then $\nu(\LC(u)) = 0$ and so we clearly have $\nu(u) = 0$. Then Proposition \[laurent-inverse-bounded\] implies $\inv u \in \laurinx K_\nu$. With $\nub{\inv u} = 0$ we obtain $\inv u \in \laurinx \O$ as desired.
### Criterion for bounded square roots {#sec:org22171ff}
In the next chapter, we will be particularly interested in the specialization of Laurent series which are square roots of polynomials. Proposition \[laurent-sqrt-d\] already describes how to construct square roots that lie in $\laurinx K$, we now give additional conditions which are sufficient to have the square root lie in $\laurinx K_\nu$.
For a counterexample, take $u = 1 + u_{-1} \, \inv X$ where $u_{-1} \in K, \; \nub{u_{-1}} < 0$: then it is easy to see that $\nub{u} = -\infty$.
\[laurent-sqrt-bounded\] Let $u \in \laurinx K_\nu$ such that $\sqrt{u} \in \laurinx K$ and $\nub{u} = \nub{\LC(u)}$. Then $\sqrt{u} \in \laurinx K_\nu$, i.e. $\sqrt{u}$ is bounded.
Recall that $u_0 = \LC(u)$ must be a square, and $\io{u}$ must be even. We may thus divide $u$ by $u_0$ and an appropriate even power of $X$ (because both are squares and bounded), and assume $u = 1+v$ where $v \in \powerseriesinvx \O$, i.e. $\nub{v} \geq 0$, and $\io{v} > 0$.
Hence $$\sqrt{u} = \sqrt{1+v} = \sum_{j=0}^{\infty} \binom{1/2}{j} \, {v}^j = \lseries{0}{w}{m}$$ converges in $\laurinx K$. By the hypothesis $\Char k \neq 2$, we have $\nub{2} = 0$, so $\nub{\binom{1/2}{j}} \geq 0$ (see also Lemma \[lemma-binomial-half\]). As $\lim_{j\to\infty} \io{{v}^j} = \lim_{j\to\infty} j \, \io{v} = -\infty$, only a finite number of $\binom{1/2}{j} \, {v}^j$, each having $\nub{\cdot} \geq 0$, influence each $w_m$. Hence $\nub{w_m} \geq 0$ for all $m$, and $\sqrt{u}$ is bounded.
Specialization of polynomial continued fractions {#sec:org890b4d5}
------------------------------------------------
Given $\alpha \in \laurinx \O$, we can on the one hand see it as element of $\laurinx K$, or reduce it to $\Red{\alpha} \in \laurinx k$. For each, one gets a continued fraction over $K[X]$ respectively $k[X]$. If one is *lucky*, then $\CF(\alpha)$ has all “data” defined over $\O$, so one can apply $\Redm$, and ask: do $\CF$ and $\Redm$ commute?
The answer is yes, so the obstacle lies in $\CF(\alpha)$ not having all data defined over $\O$.
Let us fix notations for the rest of the chapter: Let $\alpha \in \laurinx \O$ with $\LC(\alpha) \in \units \O$ and $\io{\alpha} \leq 0$, so that $\alpha$ has a non-zero polynomial part. It has a continued fraction expansion $\CF(\alpha)$ over $K[X]$, with complete quotients $\alpha_n \in \laurinx K$, partial quotients $a_n \in K[X]$ and canonical convergents $(p_n, q_n) \in K[X]^2$, satisfying $$\begin{aligned}
\alpha &= [a_0, a_1, \dots],&
\alpha_n &= [a_n, a_{n+1}, \dots],&
p_n/q_n &= [a_0, \dots, a_n].\end{aligned}$$
For the *specialization*, we set $\gamma = \Red{\alpha} \in \laurinx k$. The condition $\LC(\alpha) \in \units \O$ ensures $\io{\gamma} = \io{\alpha} \leq 0$. Of course $\gamma$ has a continued fraction expansion $\CF(\gamma)$ with complete quotients denoted $\gamma_n \in \laurinx k$ and partial quotients denoted $c_n \in k[X]$. The canonical convergents of $\gamma$ are written as $(u_n, v_n) \in k[X]^2$ to distinguish them easily, and they satisfy $$\begin{aligned}
\gamma &= [c_0, c_1, \dots],&
\gamma_m &= [c_m, c_{m+1}, \dots],&
u_m/v_m &= [c_0, \dots, c_m].\end{aligned}$$
### Good reduction {#sec:org3baa196}
To answer the question about “commuting”, we want to apply the reduction map on the complete quotients, motivating the following definition:
\[def-cf-good-reduction\] We say that $\CF(\alpha)$ has *good reduction* at $\nu$ if for all $n \geq 0$ $$\alpha_n \in \laurinx \O \text{ and } \Red{\alpha_n} = \gamma_n.$$
It turns out the second condition is a consequence of the first, and that it is also possible to describe good reduction in terms of the partial quotients:
\[cf-good-red-partial-quotients\]
1. $\CF(\alpha)$ has good reduction.
2. $\alpha_n \in \laurinx \O$ for all $n \geq 0$.
3. $a_n \in \O[X]$ and $\LC(a_n) = \LC(\alpha_n) \in \units \O$ for all $n \geq 0$.
4. $\deg a_n = \deg c_n$ for all $n \geq 0$.
For $n=0$ we had $\LC(\alpha_0) \in \units \O$ as a hypothesis.
We begin to prove the theorem with the following observation:
If $\alpha_n \in \laurinx \O$, then clearly $a_n = \gauss{\alpha_n} \in \O[X]$.
Next, let us show that $a_n \in \O[X]$ cannot be a sufficient condition for good reduction:
\[cf-good-red-leading-coeffs\] Let $n \geq 0$. If $\alpha_n \in \laurinx \O$, then $\alpha_{n+1} \in \laurinx \O$ $\LC(a_{n+1}) = \LC(\alpha_{n+1}) \in \units \O$.
By Definition \[cf-complete-quotients\], we have $\inv{\alpha_{n+1}} = \alpha_n - a_n \in \laurinx \O$, and clearly $\LC(\inv{\alpha_{n+1}}) = \inv{ \LC(\alpha_{n+1})} \in \units \O$ $\LC(\alpha_{n+1}) \in \units \O$.
Then the statement follows from Corollary \[laurent-inverse-bounded-corollary\] applied to $u = \inv{\alpha_{n+1}}$.
This allows to show that the second condition in Definition \[def-cf-good-reduction\] is an automatic consequence of the first condition:
\[cf-good-red-second-condition-redundant\] Let $n \geq 0$. If $\alpha_n, \alpha_{n+1} \in \laurinx \O$ and $\Red{\alpha_n} = \gamma_n$, then $\Red{\alpha_{n+1}} = \gamma_{n+1}$.
Clearly $\Red{\alpha_n} = \gamma_n$ implies $\Red{a_n} = c_n$, and by Propositions \[laurent-inverse-bounded\] and \[cf-good-red-leading-coeffs\] we have $\alpha_{n+1} \in \units{\laurinx \O}$. Hence $$\inv{\gamma_{n+1}} = \gamma_n - c_n = \Red{\alpha_n} - \Red{a_n} = \Red{\inv{\alpha_{n+1}}} = \inv{\Red{\alpha_{n+1}}}$$ which implies $\gamma_{n+1} = \Red{\alpha_{n+1}}$ as desired.
\[good-reduction-preserve-degrees\] If $\LC(\alpha_n) \in \units \O$ and $\Red{\alpha_n} = \gamma_n$, we have $\io{\alpha_n} = \io{\gamma_n} \leq 0$ ($< 0$ for $n \geq 1$), and hence $\deg a_n = \deg c_n$.
Let us now describe good reduction in terms of the partial quotients; for this we first have a look at the convergents:
\[cf-good-red-convergents-bounded\] Let $n \geq 0$ and suppose $a_j \in \O[X]$ for $j = 0, \dots, n$ and $\LC(a_j) \in \units \O$ for $j = 1,\dots, n$. Then $p_n, q_n \in \O[X]$ and moreover $\ifrac{p_n}{q_n} \in \laurinx \O$.
The statement $p_n, q_n \in \O[X]$ follows directly from the recursion formulas for the canonical convergents . And the product formula for the leading coefficients implies $$\nub{\LC(q_n)} = \sum_{j=1}^n \nub{\LC(a_j)}.$$ But then $\nub{\LC(a_j)} = 0$ for $j = 1, \dots, n$ implies $\nub{\LC(q_n)} = \nub{q_n} = 0$. So by Corollary \[laurent-inverse-bounded-corollary\] we have $q_n \in \units{\laurinx{\O}}$, hence $\ifrac{p_n}{q_n} \in \laurinx{\O}$.
We conclude this section by proving the equivalence of the alternative characterisations of good reduction.
Equivalence of 1. and 2. is a consequence of Proposition \[cf-good-red-second-condition-redundant\] above.
Next, 2. implies 3. by Proposition \[cf-good-red-leading-coeffs\].
Conversely, 3. implies 2.: Let $m \geq 0$ and recall that $\io{\ifrac{p_{m,n}}{q_{m,n}} - \alpha_m} > 2 \, \deg{q_{m,n}}$ from Proposition \[cf-expansion-yields-convergents\]. Moreover, we have $\ifrac{p_{m,n}}{q_{m,n}} \in \laurinx \O$ by Proposition \[cf-good-red-convergents-bounded\], so the *first* coefficients of $\alpha_m$ are also in $\O$. As $\limn \deg{q_{m,n}} = \infty$, we cover all coefficients, and thus $\alpha_m \in \laurinx \O$.
Next, 1. and 3. imply $\Red{\alpha_n} = \gamma_n$, hence $\Red{a_n} = c_n$ and $\LC(a_n) \in \units \O$. The latter is equivalent to $\deg a_n = \deg \Red{a_n}$, so we get $\deg a_n = \deg c_n$.
Finally 4. implies 2.: by Proposition \[bad-reduction-minimal-pole\] (below, but independent of this theorem) there exists $n$ with $\deg a_n < \deg c_n$ if 2. is violated.
So continued fraction expansion and specialization commute as soon as the partial quotients are defined over $\O$ and do not “drop degree” on reduction, or even simpler, the degrees of the partial quotients match.
Theorem \[thm-vdp-intro\] of van der Poorten becomes almost trivial in this case:
\[cf-good-red-convergents-reduction\] If $\CF(\alpha)$ has good reduction, then for all $n \geq 0$ we have $u_n = \Red{p_n}$ and $v_n = \Red{q_n}$ which by the classification of convergents (Proposition \[cf-convergent-classification\]) implies that all convergents of $\gamma$ are obtained by reducing convergents of $\alpha$.
We can think of $p_n$ and $q_n$ as polynomials in $\Z[a_0, \dots, a_n]$ (see Proposition \[definition-convergents-matrix\]). Of course $u_n$ and $v_n$ are obtained by replacing $a_j$ with $c_j$ in those polynomials. But $c_j = \Red{a_j}$ for all $j \geq 0$, so $(u_n, v_n) = (\Red{p_n}, \Red{q_n})$.
An arbitrary convergent of $\gamma$ has perhaps an additional polynomial factor in $k[X]$ which we can however lift to a polynomial of same degree in $K[X]$. Because we have $\deg a_{n+1} = \deg c_{n+1}$, multiplying $(p_n, q_n)$ with this polynomial still produces a convergent of $\alpha$.
### Bad reduction {#sec:orgee082e8}
\[def-cf-bad-reduction\] The opposite of good reduction of $\CF(\alpha)$ is obviously *bad reduction* of $\CF(\alpha)$, by which we mean that there exists $n \geq 1$ such that $\alpha_n \not\in \laurinx \O$ (i.e. $\nub{\alpha_n} < 0$, so there is a coefficient with ).
The results for good reduction are still useful in this case, for example Propositions \[cf-good-red-leading-coeffs\] and \[cf-good-red-second-condition-redundant\] can be applied until we arrive at the complete quotient with bad reduction. They should also give an initial idea of what could go wrong in the case of bad reduction.
\[bad-reduction-minimal-pole\] Suppose $\CF(\alpha)$ has bad reduction and let $n$ minimal with $\alpha_n \not\in \laurinx \O$. Then in fact $\nub{\LC(\alpha_n)} < 0$, i.e. $\alpha_n$ has in the leading coefficient.
If $\gamma_n$ is defined, then $\deg c_n > \deg a_n$ and $\alpha_n$ is unbounded.
The first statement is an immediate consequence of Proposition \[cf-good-red-leading-coeffs\]: by minimality $\alpha_{n-1} \in \laurinx \O$, so $\nub{\LC(\alpha_n)} \neq 0$. But $\nub{\LC(\alpha_n)} > 0$ is impossible because $\inv{\LC(\alpha_n)} = \LC(\inv{\alpha_n}) = \LC(\alpha_{n-1} - a_{n-1}) \in \O$.
Now assume $\gamma_n$ is defined: As we have $\alpha_0, \dots, \alpha_{n-1} \in \laurinx \O$ (we could say we have “good reduction up to $\alpha_{n-1}$”), we certainly have $\Red{\alpha_{n-1}} = \gamma_{n-1}$ using Proposition \[cf-good-red-second-condition-redundant\] inductively. But by Proposition \[laurent-inverse-bounded\] we have $\LC(\alpha_{n-1} - a_{n-1}) \in \mm$, hence $$\deg a_n = - \io{\alpha_{n}} = \io{\alpha_{n-1} - a_{n-1}} < \io{\gamma_{n-1} - c_{n-1}} = -\io{\gamma_n} = \deg c_n.$$
In particular $\gamma_{n-1} - c_{n-1} \neq 0$ which implies $\nub{\alpha_{n-1} - a_{n-1}} = 0$. But as the leading coefficient is in $\mm$, Proposition \[laurent-inverse-bounded\] implies that the inverse $\alpha_n$ is unbounded.
If we are using the computation scheme with $t_n$ and $s_n$ from Chapter \[sec:orge1a60a6\] and we are already in the $\sigma$-reduced case, the in the leading coefficient of $\alpha_n$ corresponds to in the leading coefficient of $s_n$.
Unless $\gamma$ is rational,[^8] $\gamma_n$ is of course always defined.
### Reduction and normalisation of continued fractions {#sec:orgea2cbbe}
We can extend the reasoning of this section also to an arbitrary Laurent series $\alpha \in \laurinx K_\nu$, as long as $\alpha$ is bounded and satisfies $\nu(\alpha) = \nu(\LC(\alpha))$ and $\io{\alpha} \leq 0$. If these requirements are met, we can just divide $\alpha$ by $\LC(\alpha)$, or some $g \in \units K$ with $\nu(g) = \nu(\alpha)$. For the new series, we can apply the above results.
Of course reduction here must always be preceded by normalisation. But for example the existence of unbounded complete quotients is invariant under normalisation (see Proposition \[cf-scalar-multiplication\] about multiplying a continued fraction with a constant), and is characteristic for bad reduction.
We will revisit these issues later, first we need to study the reduction of the convergents in more detail.
Normalisation and reduction of convergents {#sec:orgd1cf6c9}
------------------------------------------
In the case of good reduction of the continued fraction, we were able to simply reduce the canonical convergents. In the case of bad reduction of the continued fraction, we cannot expect the canonical convergents to be polynomials defined over $\O$, so we need to normalise them first.
In other words, we wish to extend the reduction map in a useful way to all of $K[X]$ (or even $\laurinx K$) by normalising to valuation $0$ before reducing. Of course, extending the reduction map $\O \to k$ in this way from $\O$ to $K$ is not so useful. But for polynomials and Laurent series, there are usually several coefficients, so thinking projectively makes sense. For obvious reasons, this works only for bounded Laurent series.
Let $u \in \mino{\laurinx{K}_\nu}$, and recall that $\uni$ is a uniformising parameter of $\O$ satisfying $\nu(\uni) = 1$. Define the *normalisation* $\normal{u}$ for $u$ as $$\normal{u} = \uni^{-\nu(u)} \, u \in \laurinx \O.$$ Clearly, $\nu(\normal{u}) = 0$. For completeness, we also set $\normal{0} = 0$.
If $u \in K$, then $\normal{u} \in \O$, and if $u \in K[X]$, then $\normal{u} \in \O[X]$.
We denote the composition of reduction and normalisation by $$\Redn{u} = \RedM{\normal{u}}.$$
Before we start normalising convergents, we need to check that the normalisation factor is the same for the numerator and the denominator – otherwise we are unable to normalise the convergent as a whole:
\[normalise-convergents-preparation\] Suppose $\io{\RedM{\alpha}} \leq 0$, and let $(p, q) \in \Batest{K}$ a rational approximation with $\io{p - \alpha \, q} > 0$. Set $g = \uni^{\nu(q)} \in K$.
Then $(p, q) = g \cdot(\normal{p}, \normal{q})$ and in particular $\nub{p} = \nub{q} = \nub{g}$.
By definition, we have $q = g \, \normal{q}$, and $\nub{q} = \nub{g}$. The condition $\io{p - \alpha \, q} > 0$ implies $p = -\gauss{\alpha \, q} = - g \, \gauss{\alpha \, \normal{q}}$. Let $p' = - \gauss{\alpha \, \normal{q}} \in \O[X]$ with $p = g \, p'$.
It remains to show $p' = \normal{p}$: Indeed $\io{p' - \alpha \, \normal{q}} > 0$ implies $\io{\RedM{p'} - \RedM{\alpha} \, \RedM{\normal{q}}} > 0$. But $\io{\RedM{\alpha}\, \RedM{\normal{q}}} \leq 0$ by hypothesis, so also $\io{\RedM{p'}} \leq 0$. This means $\RedM{p'} \neq 0$, or $\nub{p'} = 0$, hence $p' = \normal{p}$ as desired.
\[normalise-convergents\] Every convergent and best-approximation $(p, q) \in \Baset{\alpha}{\K}$ (in particular the canonical convergents $(p_n, q_n)$) satisfies $\nub{p} = \nub{q}$.
Setting $g_n = \uni^{\nu(q_n)}$ we get $(p_n, q_n) = g_n \cdot (\normal{p_n}, \normal{q_n})$.
For $n = -1$ we have $q_{-1} = 0$ and $p_{-1} = 1$. We just set $g_{-1} = 1$, as no normalisation is required.
We finally state and prove the generalised version of Theorem \[thm-vdp-intro\] on the reduction of convergents by van der Poorten. First we check that convergents remain convergents after reduction.
\[convergent-reduction\] Let $(p, q) \in \Coset{\alpha}{K}$ a convergent. Then $\io{\Redn{p} - \gamma \, \Redn{q}} > \deg q \geq \deg \Redn{q}$, so $(\Redn{p}, \Redn{q}) \in \Coset{\gamma}{k}$ is also a convergent.
The important observation is that for $\beta \in \laurinx \O$ one has $\io{\Red{\beta}} \geq \io{\beta}$, and for $b \in \O[X]$ one has $\deg \Red{b} \leq \deg b$, hence $$\io{\Redn{p} - \gamma \, \Redn{q}} \geq \io{p - \alpha \, q} > \deg{q} \geq \deg {\Redn{q}}.$$
We restrict now to the conveniently enumerated canonical convergents. We find:
\[definition-convergent-reduction-map-lambda\] The reduction of a (normalised) convergent remains a convergent. In particular, there exists a (unique) map $\lambda : \N_0 \to \N_0$ defined by $$\ifrac{\Redn{p_n}}{\Redn{q_n}} = \ifrac{u_{\lambda(n)}}{v_{\lambda(n)}}.$$ More precisely, for each $n$ there exists $h_n \in \mino{k[X]}$ such that $$\Redn{p_n} = h_n \, u_{\lambda(n)}, \qquad \Redn{q_n} = h_n \, v_{\lambda(n)}.$$
The map $\lambda$ is well defined: every convergent of $\gamma$ is a multiple of a unique canonical convergent of $\gamma$ by Corollary \[cf-convergent-classification\].
Here one has to be careful, though: the factor $h_n$ need *not be constant*! We will investigate this closer for some special cases later. See also Example \[ex-cfp2-zero-pattern-deg6\] in Section \[sec:org0b837d5\], where non-constant $h_n$ in fact occur.
This possibility of non-constant factors make the following less obvious because $\deg \Redn{q_n}$ may not be non-decreasing:
\[reduced-convergents-increasing\] The map $\lambda$ is non-decreasing (it need not be increasing).
Let $n < n'$ and set $m = \lambda(n), m' = \lambda(n')$, hence $\deg{q_n} < \deg{q_{n'}}$.
If $\deg \Redn{q_n} \leq \deg \Redn{q_{n'}}$, Proposition \[cf-best-approx-classification\] (Classification of best-approximations) for $\gamma$ implies directly $m \leq m'$.
If however $\deg{\Redn{q_n}} \geq \deg{\Redn{q_{n'}}}$, then $$\begin{aligned}
\io{\Redn{p_n} - \gamma \, \Redn{q_n}} &> \deg{\Redn{q_n}} \geq \deg{\Redn{q_{n'}}}, \\
\io{\Redn{p_{n'}} - \gamma \, \Redn{q_{n'}}} &> \deg{q_{n'}} > \deg{q_n} \geq \deg{\Redn{q_n}}.\end{aligned}$$ Eliminating $\gamma$, one obtains $$\begin{gathered}
\io{\Redn{p_n} \, \Redn{q_{n'}} - \Redn{p_{n'}} \, \Redn{q_n}} =
\io{(\Redn{p_{n}} - \gamma \, \Redn{q_{n}}) \, \Redn{q_{n'}} - (\Redn{p_{n'}} - \gamma \, \Redn{q_{n'}}) \, \Redn{q_{n}}} \\
\geq \min\left(\io{\Redn{p_n} - \gamma \, \Redn{q_n}} + \io{\Redn{q_{n'}}},
\io{\Redn{p_{n'}} - \gamma \, \Redn{q_{n'}}} + \io{\Redn{q_n}} \right) > 0\end{gathered}$$ which implies $\ifrac{\Redn{p_n}}{\Redn{q_n}} = \ifrac{\Redn{p_{n'}}}{\Redn{q_{n'}}}$, hence $m = m'$.
\[best-approx-reduction-strictly-increasing-q\] If $m < m'$, then Proposition \[cf-best-approx-classification\] immediately implies $\deg{\Redn{q_n}} < \deg{\Redn{q_{n'}}}$.
Now we are ready to prove that the map $\lambda$ is in fact surjective, a result which appeared first [@poorten-1999-reduction-continued-fractions], and with a slightly different proof in [@poorten-1999-reduction-continued-fractions] and [@poorten-2001-non-periodic-continued]. [^9] Unfortunately, both proofs are somewhat confusing, perhaps because van der Poorten does not include an argument why the map $\lambda$ should be non-decreasing. He already seems to assume that property in his implicit definition of $\lambda$, where he uses an elaborate enumeration scheme.[^10]
\[convergent-reduction-surjective\] All the (coprime) convergents of $\gamma$ arise as reductions of convergents of $\alpha$. In other words, the map $\lambda : \N_0 \to \N_0$ is surjective. Moreover, if $n = \min \inv\lambda(m)$, then $\deg {v_m} = \deg{q_n}$.
First, we show that $\lambda$ has finite fibres. Indeed, for $n \geq 0$ and $m = \lambda(n)$ we have by definition of $\lambda$ $$\Redn{p_n} = h_n \, u_m, \quad \Redn{q_n} = h_n \, v_m \text{ where } h_n \in \mino{k[X]},$$ hence $\deg{q_{n+1}} \leq \deg{v_{m+1}}$: $$\begin{gathered}
\label{convergent-reduction-quality-improvement}
\deg{v_{m+1}} \geq \io{h_n} + \deg{v_{m+1}} = \io{h_n} + \io{u_{m} - \gamma \, v_{m}} \\
= \io{\Redn{p_{n}} - \gamma \, \Redn{q_{n}}} \geq \io{p_{n} - \alpha \, q_{n}} = \deg{q_{n+1}}\end{gathered}$$ Here we use Proposition \[cf-expansion-yields-convergents\] about the approximation quality of the canonical convergents $(u_m, v_m)$ and $(p_n, q_n)$ (first and last equality).
Now we know that $\limn \deg{q_{n+1}} = \infty$ so for fixed $m$ there can only by finitely many $n$ which satisfy the inequality.
Because we know that $\lambda$ is monotonous, we can prove its surjectivity by checking that there are no gaps in the image.
There is no gap at the start because $v_0 = 1$ and $q_0 = 1$ imply $\lambda(0) = 0$.
For $n \geq 0$, we either have $\lambda(n) = \lambda(n+1)$ in which case there is no gap.
Otherwise $m = \lambda(n) < \lambda(n+1) = m'$, and we need to show $m' = m+1$. Again, by definition of $\lambda$ $$\Redn{p_{n+1}} = h_{n+1} \, u_{m'}, \quad \Redn{q_{n+1}} = h_{n+1} \, v_{m'} \text{ where } h_{n+1} \in \mino{k[X]}.$$ and in particular $$\deg{v_{m'}} \leq \deg{h_{n+1}} + \deg{v_{m'}} = \deg{\Redn{q_{n+1}}} \leq \deg{q_{n+1}}.$$ But from $m+1 \leq m'$ and follows also $$\deg{q_{n+1}} \leq \deg{v_{m+1}} \leq \deg{v_{m'}},$$ so these are actually equalities, and as desired $m' = \lambda(n+1) = m+1 = \lambda(n) + 1$, so there is no gap. Note that $n+1$ is the minimal element of the fibre $\inv \lambda(m')$, and we have shown $\deg q_{n+1} = \deg{v_{\lambda(n+1)}}$.
\[convergent-reduction-minimal-maximal-coprime\] Observe that $\deg q_{n+1} = \deg v_{m+1}$ implies $\deg h_{n+1} = 0$, and from also $\deg h_n = 0$. Hence both for the minimal and maximal fibre element, the reduced convergent remains coprime.
\[cor-lambda-degree-sum\] Suppose that $\inv\lambda(m) = \{ n, \dots, n+l \}$. Then $$\label{eq-lambda-degree-sum}
\deg c_{m+1} = \sum_{i=1}^{l+1} \deg a_{n+i} = \deg a_{n+1} + \dots + \deg a_{n+l+1}.$$
Both $n$ and $n+l+1$ are the minimal elements of their respective fibres, hence $\deg q_n = \deg v_m$ and $\deg q_{n+l+1} = \deg v_{m+1}$. The degree formula for the convergents then gives the desired relation between the degrees of the partial quotients.
If the reduction is not rational, we also get an additional criterion for good reduction:
\[cf-good-reduction-lambda-bijective\] If $\gamma \not\in k(X)$, the map $\lambda$ is bijective $\CF(\alpha)$ has good reduction.
First observe that by Proposition \[reduced-convergents-increasing\], the map $\lambda$ is bijective it is the identity.
If $\CF(\alpha)$ has good reduction, Corollary \[cf-good-red-convergents-reduction\] implies that $\lambda$ is the identity.
Conversely, if $\lambda$ is the identity, then from Theorem \[convergent-reduction-surjective\] we obtain $\deg q_n = \deg v_n$ for all $n$, which in turn implies $\deg a_n = \deg c_n$ for all $n$. Then by Theorem \[cf-good-red-partial-quotients\] $\CF(\alpha)$ has good reduction.
We conclude this section by pointing out that while the canonical convergents are usually not normalised, the convergents we get as solutions of the linear system in Section \[sec:org542977d\] are in fact optimally normalised (even independently of the valuation):
\[prop-convergents-hankel-determinants-normalised\] Let $\alpha \in \laurinx \O$ and suppose that $\gamma = \Red{\alpha} \neq 0$. Let $n$ such that $\pqmatrix_n$ has full rank, and let $(p, q)$ correspond to an element of the kernel computed from the minors of $\pqmatrix_n$ as in Remark \[rem-cramers-rule\].
Then $p, q \in \O[X]$. Moreover, if $\deg q = \deg \Redn{q}$, we have $\nu(q) = 0$.
By hypothesis, the coefficients of the Laurent series $\alpha$ are in $\O$. The minors of $\pqmatrix_n$ are polynomials in these coefficients, so clearly the coefficients of $p$ and $q$ are in $\O$ too (recall that we need full rank so they do not all vanish).
The coefficients of $\gamma$ are obtained by reducing those of $\alpha$, hence the kernel elements of $\Red{\pqmatrix_n}$ correspond to convergents of $\gamma$. For example there is $(\Redn{p}, \Redn{q})$, and then $\deg q = \deg \Redn{q}$ implies that $\Red{\pqmatrix_n}$ has full rank as well, so we may compute a convergent using the minors. But of course the reduction map $\Redm$ is a ring homomorphism, so this convergent is exactly $(\Red{p}, \Red{q})$, with $\Red{q} \neq 0$. Then clearly $\nu(q) = 0$.
Calculating valuations {#sec:org7a10a6e}
----------------------
Once we understand the structure of $\lambda$ and the reduction of convergents thanks to Theorem \[convergent-reduction-surjective\], we can go further and attempt to compute the valuations (Gauss norms) for the partial quotients $a_n$, the canonical convergents $q_n$ and often even for the complete quotients $\alpha_n$. In the next chapter, we will see how there arise rather simple patterns in the case $\alpha = \sqrt{D}$ with $\deg D = 4$. For now, we remain in the general case which makes things a bit more complicated. However we will thus understand better the obstacles for generalising the degree $4$ case.
### Relating complete quotients with convergents {#sec:org826607c}
In the following, we always assume $\gamma = \Red{\alpha} \not\in k(X)$.
\[introduce-theta-n\] Define for $n \geq -1$ $$\label{define-theta-n}
\vartheta_n = \normal{p_n} - \alpha \, \normal{q_n}.$$ Then $\vartheta_n \in \laurinx \O$ with $\nub{\vartheta_n} = 0$, and $\io{\vartheta_n} = \deg q_{n+1}$.
With $g_n = \uni^{-\nu(q_n)}$, we may then write $$\label{cf-alphan-moebius-relation-normalised}
% \tag{Q}
\alpha_{n} = - \frac{g_{n-2} \, \vartheta_{n-2}}{g_{n-1} \, \vartheta_{n-1}}$$ as a quotient of elements of $\laurinx \O$ up to a normalisation factor.
Note that $\vartheta_{-1} = 1$ and $\vartheta_0 = a_0 - \alpha$.
By definition of normalisation, we have $\normal{p_n}, \normal{q_n} \in \O[X]$, and $\Redn{q_n} \neq 0$. Of course $p_n - \alpha \, q_n = g_n \, \vartheta_n$, so $\io{\vartheta_n} = \deg q_{n+1}$ is an immediate consequence of Proposition \[cf-expansion-yields-convergents\] and $\uni \in K$.
As we assume $\alpha \in \laurinx \O$, this implies $\vartheta_n \in \laurinx \O$. Moreover, $\gamma = \Red{\alpha} \not\in k(X)$ implies $\Red{\vartheta_n} \neq 0$, hence $\nub{\vartheta_n} = 0$.
Finally, from Proposition \[definition-convergents-matrix\] we obtain (see also ) $$\alpha_n = \frac{q_{n-2} \, \alpha - p_{n-2}}{-q_{n-1} \, \alpha + p_{n-1}} = -\frac{g_{n-2} \, (\normal{p_{n-2}} - \alpha \, \normal{q_{n-2}})}{g_{n-1} \, (\normal{p_{n-1}} - \alpha \, \normal{q_{n-1}})} = - \frac{g_{n-2} \, \vartheta_{n-2}}{g_{n-1} \, \vartheta_{n-1}}.$$
So in order to understand whether $\alpha_n$ is bounded, we need a criterion for when the $\vartheta_n$ have a bounded inverse:
\[spec-theta-lc-valuation\]
- $\inv{\vartheta_n} \in \laurinx K_\nu$,
- $\vartheta_n \in \units{\laurinx \O}$,
- $\io{\vartheta_n} = \io{\Red{\vartheta_n}}$,
- $\nub{\LC(\vartheta_n)} = 0$,
By the previous Proposition, we have $\vartheta_n \in \laurinx \O$ and $\nub{\vartheta_n} = 0$. So by Proposition \[laurent-inverse-bounded\] the inverse is bounded $$\nub{\LC(\vartheta_n)} = 0 \iff \Red{\LC(\vartheta_n)} \neq 0 \iff \io{\vartheta_n} = \io{\Red{\vartheta_n}}.$$ Finally, it is clear that if the inverse is bounded, then $\nu(\inv{\vartheta_n}) = 0$, so it is in $\laurinx \O$.
Of course $\inv{\vartheta_n} \in \laurinx K_\nu$ implies via that also $\alpha_{n+1} \in \laurinx K_\nu$.
We use this to show that there are always infinitely many bounded complete quotients:
\[cf-infinitely-bounded-complete-quotients\] Let $m \in \N$, and set $n = \min \inv\lambda(m)$. Then $\inv{\vartheta_{n-1}} \in \laurinx K_\nu$, hence $\alpha_n \in \laurinx K_\nu$.
With Theorem \[convergent-reduction-surjective\] follows from $n$ being minimal in the fibre $\inv\lambda(m)$ that $\deg q_n = \deg v_m$, and $\lambda(n-1) = m-1$. By Remark \[convergent-reduction-minimal-maximal-coprime\], we moreover know $\Redn{q_{n-1}} = h_{n-1} \, v_{m-1}$ with $h_{n-1} \in k$, hence $$\io{\Red{\vartheta_{n-1}}} = \io{\Redn{p_{n-1}} - \gamma \, \Redn{q_{n-1}}} = \io{u_{m-1} - \gamma \, v_{m-1}} = \deg v_m = \deg q_n = \ios{\vartheta_{n-1}},$$ so Proposition \[spec-theta-lc-valuation\] implies that $\vartheta_{n-1}$ has bounded inverse. Then implies that $\alpha_n$ is bounded.
Note that the condition for $\alpha_n$ bounded we give here is only sufficient, but not necessary.
### Fibre analysis of $\lambda$ {#sec:org84e8497}
Using the Lemmata for estimating valuations in quotients of Laurent/power series from Section \[sec:org5f19ed2\] in the appendix, we now attack the problem of computing valuations by doing case analysis for the different sizes of the fibres of $\lambda$, and the degrees of the partial quotients. This is successful mostly when we can read off the valuations (Gauss norms) from the leading coefficients.
The simplest case is the following, we get information on everything (recall that $\nu(g_n) = \nu(q_n)$ for all $n \geq 0$):
\[prop-single-element-fibre-analysis\] Let $m \in \N$ such that $\inv \lambda(m) = \{n\}$ has a single element. Then $\alpha_{n+1}$ is bounded and $$\label{eq-single-fibre-alpha-val}
\nu(\alpha_{n+1}) = \nu(\LC(\alpha_{n+1})) = \nu(a_{n+1}) = \nu(g_{n-1}) - \nu(g_{n}).$$ The normalised complete quotient reduces to $$\label{eq-1elem-red-cq}
\Redn{\alpha_{n+1}} = \frac{h_{n-1}}{h_n} \, \gamma_{m+1} \quad \text{ with } h_{n-1}, h_n \in \units k,$$ hence $\deg a_{n+1} = \deg c_{m+1}$.
For the corresponding convergent we have $$\nu(g_{n+1}) = \nu(q_{n+1}) = \nu(\LC(q_{n+1})) = \nu(g_{n-1}).$$
Both $n$ and $n+1$ are the minimal elements of their fibres, so Proposition \[cf-infinitely-bounded-complete-quotients\] implies that both $\vartheta_{n-1}, \vartheta_{n} \in \units{\laurinx \O}$. Hence $\alpha_{n+1}$ is bounded, and follows from and $\nu(\vartheta_{n-1}) = \nu(\vartheta_{n}) = 0$.
Normalising and reducing $\alpha_{n+1}$, we get $$\Redn{\alpha_{n+1}} = \RedM{\frac{g_{n}}{g_{n-1}} \, \alpha_{n+1}} = -\frac{\Red{\vartheta_{n-1}}}{\Red{\vartheta_{n}}} = - \frac{h_{n-1} \, (u_{m-1} - \gamma \, v_{m-1})}{h_{n} \, (u_{m} - \gamma \, v_{m})} = \frac{h_{n-1}}{h_n} \, \gamma_{m+1}.$$ Here $h_{n-1}, h_n \in \units k$ by Remark \[convergent-reduction-minimal-maximal-coprime\].
Again using that $n$ and $n+1$ are minimal in their fibres, Theorem \[convergent-reduction-surjective\] implies $\deg \Redn{q_n} = \deg q_n$ and $\deg \Redn{q_{n+1}} = \deg q_{n+1}$. This means $\nu(g_n) = \nu(q_n) = \nu(\LC(q_n))$ and $$\nu(g_{n+1}) = \nu(q_{n+1}) = \nu(\LC(q_{n+1})) = \nu(\LC(a_{n+1})) + \nu(\LC(q_n)) = \nu(g_{n-1}).$$ For $\deg a_{n+1} = \deg c_{m+1}$ see also Corollary \[cor-lambda-degree-sum\].
If there is more than one element in the fibre, we can say a few things in general. However boundedness of the complete quotients cannot be determined a priori, except for the first and last complete quotient. But even if the complete quotients are bounded, the reduction of the normalisation is *never* a complete quotient of $\gamma$ as in the single element case of Proposition \[prop-single-element-fibre-analysis\] above.
\[prop-multiple-element-fibre-analysis\] Let $m \in \N$ such that $\inv \lambda(m) = \{n, n+1, \dots, n+l \}$ has $l \geq 2$ elements. Then $\alpha_{n+1}$ is unbounded and $\alpha_{n+l+1}$ is bounded. The $\alpha_{n+i+1}$ for $1 \leq i < l$ can be bounded *or* unbounded.
If some $\alpha_{n+i+1}$ (for $1 \leq i \leq l$) is bounded, the reduction of the normalised complete quotient is a rational function (and a polynomial for $i = l$, as $h_{n+l} \in \units k$): $$\Redn{\alpha_{n+i+1}} = - \frac{h_{n+i-1}}{h_{n+i}}.$$ In particular $\io{\Redn{\alpha_{n+l+1}}} = - \deg h_{n+l-1}$. In this case, we also get $$\label{eq-multiple-fibre-alpha-val}
\nu(\LC(\alpha_{n+i+1})) \geq \nu(a_{n+i+1}) \geq \nu(\alpha_{n+i+1}) = \nu(g_{n+i-1}) - \nu(g_{n+i}),$$ and thus $$\label{eq-multiple-fibre-qn-val}
\nu(\LC(q_{n+i+1})) \geq \nu(q_{n+i+1}) \geq \nu(g_{n+i-1}).$$
Here $n$ and $n+l+1$ are minimal in their fibre, so $\vartheta_{n-1}, \vartheta_{n+l} \in \units{\laurinx \O}$ by Proposition \[cf-infinitely-bounded-complete-quotients\]; and $h_{n-1}, h_n, h_{n+l}$ are constant by Remark \[convergent-reduction-minimal-maximal-coprime\]. Moreover, Theorem \[convergent-reduction-surjective\] tells us that $\deg q_n = \deg v_m$ and $\deg q_{n+l+1} = \deg v_{m+1}$, from which we deduce $$\deg a_{n+1} + \dots + \deg a_{n+l+1} = \deg c_{m+1}$$ as in Corollary \[cor-lambda-degree-sum\].
Observe that $\vartheta_n$ has an unbounded inverse because $$\io{\vartheta_n} = \deg q_{n+1} = \deg q_n + \deg a_{n+1} < \deg q_{n} + \deg c_{m+1} = \deg v_{m+1} = \io{\Red{\vartheta_n}}.$$ Hence $\alpha_{n+1}$ is unbounded. But $\alpha_{n+l+1}$ is of course bounded by Proposition \[cf-infinitely-bounded-complete-quotients\], even if it need not have a bounded inverse. For the complete quotients in between, we cannot a priori say anything.
But assume that $\alpha_{n+i+1}$ (where $1 \leq i \leq l$) *is bounded*. Then it follows $$\Redn{\alpha_{n+i+1}} = \RedM{\frac{g_{n+i}}{g_{n+i-1}} \, \alpha_{n+i+1}} = -\frac{\Red{\vartheta_{n+i-1}}}{\Red{\vartheta_{n+i}}} = - \frac{h_{n+i-1} \, (u_{m} - \gamma \, v_{m})}{h_{n+i} \, (u_{m} - \gamma \, v_{m})} = -\frac{h_{n+i-1}}{h_{n+i}}.$$ As always $\nu(\vartheta_i) = 0$, we may deduce directly from , with the inequalities obvious from the definition of $\nu$ on polynomials and Laurent series as infimum over the coefficients. With the recurrence relation , we then get (again only in the bounded case) $$\nu(\LC(q_{n+i+1})) \geq \nu(q_{n+i+1}) \geq \min\left(\nu(a_{n+i+1}) + \nu(q_{n+i}), \nu(q_{n+i-1})\right) \geq \nu(g_{n+i-1}).$$
Observe that $\io{\Redn{\alpha_{n+l+1}}} = - \deg h_{n+l-1}$, while $\io{\alpha_{n+l+1}} = -\deg a_{n+l+1} \neq 0$. Its inverse, and hence $\alpha_{n+l}$, can be bounded only if $h_{n+l-1}$ is non-constant.
For a fibre with just two elements, we can under the simplest conditions precisely calculate the valuations:
\[prop-two-element-fibre-analysis\] Let $m \in \N$ such that $\inv \lambda(m) = \{ n, n+1 \}$ has two elements. Then $\alpha_{n+1}$ is unbounded, but $\alpha_{n+2}$ is bounded, with $$\Redn{\alpha_{n+2}} = - \frac{h_{n+1}}{h_{n}}, \quad \text{ where } h_{n}, h_{n+1} \in \units k.$$ If moreover $\deg a_{n+2} = 1$, then for the partial quotients we have $$\begin{aligned}
\label{eq-2elem-pq1}
\nu(a_{n+1}) &= \nu(g_{n-1}) - \nu(g_n) - (1+\deg a_{n+1}) \, \nu(\LC(\vartheta_n)), \\
\label{eq-2elem-pq1-lc}
\nu(\LC(a_{n+1})) &= \nu(g_{n-1}) - \nu(g_n) - \nu(\LC(\vartheta_n)), \\
\label{eq-2elem-pq2}
\nu(\alpha_{n+2}) = \nu(a_{n+2}) &= \nu(g_n) - \nu(g_{n+1}), \\
\label{eq-2elem-pq2-lc}
\nu(\LC(a_{n+2})) &= \nu(g_n) - \nu(g_{n+1}) + \nu(\LC(\vartheta_n)), \end{aligned}$$ and for the convergents we have $$\begin{aligned}
\label{eq-2elem-conv1}
\nu(g_{n+1}) = \nu(q_{n+1}) &= \nu(g_{n-1}) - (1+\deg a_{n+1}) \, \nu(\LC(\vartheta_n)), \\
\nu(\LC(q_{n+1})) &= \nu(g_{n-1}) - \nu(\LC(\vartheta_n)), \\
\label{eq-2elem-conv2}
\nu(g_{n+2}) = \nu(q_{n+2}) = \nu(\LC(q_{n+2})) &= \nu(g_n) + (1+\deg a_{n+1}) \, \nu(\LC(\vartheta_n)).\end{aligned}$$
The first part follows from Proposition \[prop-multiple-element-fibre-analysis\]. Here $n$ and $n+2$ are minimal in their fibre, so $\vartheta_{n-1}, \vartheta_{n+1} \in \units{\laurinx \O}$, and $h_{n-1}, h_n, h_{n+1}$ are all constant.
Now $\io{\vartheta_n} = \deg q_{n+1}$, but $\io{\Red{\vartheta_n}} = \deg v_{m+1} = \io{\vartheta_n} + \deg a_{n+2}$ because $\deg c_{m+1} = \deg a_{n+1} + \deg a_{n+2}$. So the first $\deg a_{n+2}$ coefficients of $\vartheta_n$ vanish after reduction, and when assuming $\deg a_{n+2} = 1$ we can apply the results of section \[sec:org5f19ed2\] to compute the valuations. In particular note that $\nu(\LC(\vartheta_n)) > 0$, while the next coefficient of $\vartheta_n$ is in $\units \O$.
With Proposition \[inverse-drop1-valuation-lemma\] on the valuations of a quotient of Laurent series, we easily compute and from the quotient presentation of $\alpha_{n+1}$. Of course $a_{n+1}$ contains precisely the first $1 + \deg a_{n+1}$ coefficients of $\alpha_{n+1}$.
Then allows to compute $$\nu(a_{n+1} \, q_{n}) = \nu(g_{n-1}) - (1 + \deg a_{n+1}) \, \nu(\LC(\vartheta_{n})) < \nu(q_{n-1}).$$ This implies via $q_{n+1} = a_{n+1} \, q_n + q_{n-1}$ and the ultrametric “equality”. As $n$ is minimal in the fibre, we have $\deg q_n = \deg \Redn{q_n}$ and hence $\nu(g_n) = \nu(q_n) = \nu(\LC(q_n))$, so $$\nu(\LC(q_{n+1})) = \nu(\LC(q_n)) + \nu(\LC(a_{n+1})) = \nu(g_{n-1}) - \nu(\LC(\vartheta_n)).$$
On the other hand, the first part of Lemma \[cauchy-valuation-lemma\] applied to gives and – there are just two coefficients in $a_{n+2}$. By Theorem \[convergent-reduction-surjective\], we also know that $\deg \Redn{q_{n+2}} = \deg q_{n+2}$, so we can compute the valuation of the convergent via the leading coefficient: $$\begin{gathered}
\nu(g_{n+2}) = \nu(q_{n+2}) = \nu(\LC(q_{n+2})) = \nu(\LC(q_{n+1})) + \nu(\LC(a_{n+2})) \\
= \nu(g_{n-1}) - \nu(\LC(\vartheta_n)) + \nu(g_n) - \nu(g_{n+1}) + \nu(\LC(\vartheta_n)) \\
= \nu(g_n) + (1 + \deg a_{n+1}) \, \nu(\LC(\vartheta_{n})).\end{gathered}$$
The $h_n$ and also the quotients $h_{n-1}/h_n$ do not seem to follow any larger (obvious) patterns. If they are all constants, we locally – in “areas” with only single element fibres – observe patterns as in Proposition \[cf-scalar-multiplication\]. But that is an unsurprising consequence of .
For $\deg a_{n+2} > 1$, there is more than one coefficient of $\vartheta_n$ that vanishes, and our reasoning which essentially boils down to geometric series arguments, breaks down. If we wanted to treat for example fibres $\inv\lambda(m) = \{n, n+1, n+2\}$ with three elements, we get additional complications, as $h_{n+1}$ can now be non-constant.
We have seen that the reduction of the normalisation of a bounded complete quotient of $\alpha$ yields a complete quotient of $\gamma$ we are at a single element fibre of $\lambda$. Otherwise, it becomes a rational (or even polynomial) function.
We have also seen that the $g_n$ do not change at the single element fibres. We will later investigate this closer for $\CF(\sqrt{D})$ with $\deg D = 4$ (see Theorem \[thm-genus1-zero-patterns\]).
Specialization of hyperelliptic continued fractions {#sec:org5f9d2ce}
===================================================
We now apply and extend the reduction theory for continued fractions from the previous chapter to square roots. After briefly treating reduction of periodic continued fractions, we finally prove Theorem \[thm-intro-infinite-poles-rationals\] from the introduction, after rephrasing it to include number fields. We go on to study the valuations more closely for $\deg D = 4$ which leads to Theorem \[thm-intro-genus1-unbounded-gauss-norm\] about unbounded valuations, also from the introduction.
We also explain how reduction of abelian varieties is related with the reduction of continued fractions via the reduction of the divisors of the convergents. This leads to a well-known effective method for testing if $D$ is Pellian by reducing modulo two primes. We conclude with a discussion of specialization of continued fractions, i.e. when the base field is $\C(t)$.
We continue using the notation from the previous chapter. From now on, let $D \in \O[X]$ non-square with even degree $2d$ and $\LC(D) \in \units \O$ a square. Then of course $\deg \Red{D} = 2d$ and Proposition \[laurent-sqrt-bounded\] implies $\alpha = \sqrt{D} \in \laurinx{\O}$ and $\gamma = \Red{\alpha} = \sqrt{\Red{D}}$. Recall we also defined $A = \gauss{\sqrt{D}}$ and note that $\Red{A} = \gauss{\sqrt{\Red{D}}}$ under the preceding hypotheses.
For example for $D \in \Z[X]$ we can ask that $D$ is monic to ensure that $\sqrt{D} \in \laurinx \Q_{\nu_\pp}$ for every prime number $\pp \neq 2$.
Reduction of periodic quadratic continued fractions {#sec:orgf6381b8}
---------------------------------------------------
In this section, we discuss reduction of periodic $\CF(\sqrt{D})$. Then all necessary information is contained in finitely many partial quotients, and we can study reduction by looking at this finite data.
First, we check that nothing strange can happen – we should not be able to reduce to a non-periodic continued fraction. Recall that periodicity of $\CF(\sqrt{D})$ is equivalent to $D$ being Pellian (see Theorem \[thm-pellian-iff-torsion\]).
If $D$ is Pellian, then either $\Red{D}$ is a square, or it is also Pellian.
Let $(p, q) \in \solu{D}$. By normalising it, we have also $(\normal{p}, \normal{q}) \in \solu{D}$, with reduction $\Redn{q} \neq 0$ in $k[X]$. Of course ${\normal{p}}^2 - D \, {\normal{q}}^2 = \omega \in \O$. If $\omega \in \mm$, then ${\Redn{p}}^2 - \Red{D} \, {\Redn{q}}^2 = 0$ which implies $\Red{D}$ is a square.
Otherwise we have $\omega \in \units \O$, hence $\Red{\omega} \in \units k$. Then clearly $(\Redn{p}, \Redn{q}) \in \solu{\Red{D}}$ and $\Red{D}$ is Pellian.
This proof shows that the degree $\deg q$ of the minimal solution can only decrease under reduction. This has been exploited by Platonov [@platonov-2014-number-theoretic-properties] to produce Jacobians of hyperelliptic curves over $\Q$ with torsion points of various order. In a previous article together with Petrunin [@platonov-petrunin-2012-the-torsion-problem], he gives $\Q$-rational torsion points of orders $36$ and $48$. It seems they employ a refined brute force approach for searching Pellian polynomials by checking that $D$ is Pellian only modulo several primes which speeds up the necessary calculations sufficiently (see also Example \[ex-periodic-good-red-2\] in Section \[sec:orgc648887\]).
The following does not even require that $D$ is Pellian:
\[prop-red-square-bad-reduction\] If $\Red{D}$ is a square, then $\CF(\sqrt{D})$ has bad reduction, with $\alpha_1 \not\in \laurinx \O$.
This is rather obvious because now $\gamma = \sqrt{\Red{D}} \in k[X]$, so $c_0 = \gamma_0$ and already $\gamma_1$ does not exist. So we must have bad reduction of $\CF(\sqrt{D})$ by Proposition \[bad-reduction-minimal-pole\].
\[rem-reduction-to-square-no-val-info\] If $\Red{D}$ is square, the map $\lambda$ has image $\{0\}$, so there is a single infinite fibre. We neglected to treat this case in Section \[sec:org84e8497\]. As already $\Red{a_0 - \alpha_0} = 0$, we do not get much information about the valuations. So we do not know whether $\alpha_1$ should be bounded or not.
\[bad-reduction-quasi-period-factor\] Suppose that $\CF(\sqrt{D})$ is quasi-periodic, with $\mu \in \units K$ such that $\alpha_\QPL = \mu \, (A + \sqrt{D})$. Then $\deg a_{\QPL} = d$ being maximal implies by Proposition \[bad-reduction-minimal-pole\] that bad reduction of the continued fraction cannot start at $\QPL$. As $\nu(A) = \nu(\sqrt{D}) = 0$, we have $\nu(\mu) = 0$ unless bad reduction of $\CF(\sqrt{D})$ occurred already before $\alpha_\QPL$, i.e. somewhere inside the quasi-period.
In particular, this means that $\mu \in K$ cannot have too many different factors.
\[rem- reduction-to-square-find-bad-reduction\] If $k$ has positive characteristic, it is possible that the (quasi-)period length shortens. This is best understood using the geometric viewpoint from Chapter \[sec:org99a8e17\] and will be analysed later in Section \[sec:orgaebd15f\].
Anyway, we can easily determine whether we have good or bad reduction of periodic $\CF(\alpha)$ by checking whether any of $\nu(\LC(a_1)), \dots, \nu(\LC(a_{\QPL}))$ is negative (with $\QPL$ the quasi-period length).
The quasi-period being palindromic (see Proposition \[palindromic-period\]) also implies that the bad reduction of the continued fraction must start at the latest at $\frac{\QPL}{2}$ for $\QPL$ even, or $\frac{\QPL-1}{2}+2$ for $\QPL$ odd (in the latter case, we have to account for $\nu(\mu) \neq 0$).
### Reduction in the $\deg D = 2$ case {#sec:org46de0ba}
Let us briefly describe what happens in the case $\deg D = 2$.
Suppose for simplicity that $D$ is monic, then we can write $D = (X+b)^2 + \omega$ with $b, \omega \in \O$ and $\omega \neq 0$ so that $D$ is not a square. Of course $A = X + b$, and one easily computes $$\alpha_0 = \sqrt{D}, \quad \alpha_{2i+1} = \frac{A+\sqrt{D}}{\omega}, \quad \alpha_{2i} = A + \sqrt{D}.$$ So bad reduction occurs $\Red{\omega} = 0$, in which case $\Red{D}$ is a square. Obviously we can reduce the $\alpha_{2i}$ directly, but the $\alpha_{2i+1}$ only after normalising. If $\Red{\omega} = 0$, then the map $\lambda$ has a single infinite fibre and clearly the $\Redn{\alpha_n}$ are all polynomials.
The partial quotients are $$a_0 = A, \quad a_{2i+1} = \frac{2 \, A}{\omega}, \quad a_{2i} = 2 \, A$$ with Gauss norms $$\nu(a_0) = 0, \quad \nu(a_{2i+1}) = - \nu(\omega), \quad \nu(a_{2i}) = 0$$ which of course remain bounded.
As to the convergents, it is easy to see that ($\ceil{\cdot}_\Z$ is the ceiling function) $$\nu(p_n) = \nu(q_n) \geq - \ceil{\frac{n}{2}}_\Z \, \nu(\omega) \text{ and } \nu(\LC(q_n)) = - \ceil{\frac{n}{2}}_\Z \, \nu(\omega)$$ hence $\nu(q_n) = - \ceil{\frac{n}{2}}_\Z \, \nu(\omega)$ and $\deg q_n = \deg \Redn{q_n}$. Indeed we expect this in the case of good reduction of $\CF(\sqrt{D})$.
Otherwise we have bad reduction of $\CF(\sqrt{D})$, hence $\Red{A}^2 = \Red{D}$ and $\nu(\omega) > 0$. Then we also know that $(\Redn{p_n}, \Redn{q_n}) = h_n \, (\Red{A}, 1)$ for all $n \geq 0$. We can even calculate this: set $\eta = \pi^{-\nu(\omega)}$ (so that $\eta/\omega \in \units \O$). Then for even $n$ we get $\inv{g_n} = \eta^{n/2}$ and for odd $n$ we get $\inv{g_n} = \eta^{(n+1)/2}$. We calculate for even $n$: $$\begin{aligned}
\normal{p_n} &= \eta^{n/2} p_n = 2A \, \eta^{n/2} \, p_{n-1} + \eta^{n/2} \, p_{n-2} = 2A \, \normal{p_{n-1}} + \eta \, \normal{p_{n-2}}, \text{ and similarly } \\
\normal{q_n} &= 2A \, \normal{q_{n-1}} + \eta \, \normal{q_{n-2}}\end{aligned}$$ which yields $$\Redn{p_n} = 2 \Red{A} \, \Redn{p_{n-1}}, \quad \Redn{q_n} = 2 \Red{A} \, \Redn{q_{n-1}}.$$ On the other hand, we get for $n$ odd $$\begin{aligned}
\normal{p_n} &= \eta^{(n+1)/2} p_n = \frac{2A}{\omega} \, \eta^{(n+1)/2} p_{n-1} + \eta^{(n+1)/2} p_{n-2} = 2A \, \frac{\eta}{\omega} \, \normal{p_{n-1}} + \eta \, \normal{p_{n-2}}, \text{ and } \\
\normal{q_n} &= 2A \, \frac{\eta}{\omega} \, \normal{q_{n-1}} + \eta \, \normal{q_{n-2}}\end{aligned}$$ so $$\Redn{p_n} = 2 \Red{A}\, \Red{\eta/\omega} \, \Redn{p_{n-1}}, \quad \Redn{q_n} = 2 \Red{A} \, \Red{\eta/\omega} \, \Redn{q_{n-1}}.$$ It follows that $h_n$ is $\Red{A}^n$ times some constant factor depending on $n$.
Reduction of non-periodic quadratic continued fractions {#sec:org326a39b}
-------------------------------------------------------
As before, let $D \in \O[X]$ non-square with even degree, and $\LC(D) \in \units \O$ a square, so that $\alpha = \sqrt{D} \in \laurinx{\O}$, and $\gamma = \Red{\alpha} = \sqrt{\Red{D}}$. But now, we assume that $\CF(\sqrt{D})$ is non-periodic. Recall that this requires $\deg D \geq 4$ (Corollary \[deg-2-always-pellian\] and Theorem \[thm-pellian-iff-cf-periodic\]).
### Reduction to square {#sec:org1ce08e7}
If $\Red{D}$ is a square, this implies bad reduction of $\CF(\sqrt{D})$ by Proposition \[prop-red-square-bad-reduction\]. Then $\lambda : \N_0 \to \N_0$ has image $\{0\}$, so we do not get a lot of information from it.
Anyway, for a fixed $D$, this can happen only for finitely many valuations $\nu$. From Proposition \[completion-of-square\] about completion of the square and Proposition \[laurent-sqrt-bounded\] about boundedness of the square root, it follows that $\Red{D}$ is a square $\nu(D-A^2) > 0$. So this can be checked easily, and concerns only finitely many valuations.
See Example \[ex-nonperiodic-to-square-2\] in Section \[sec:orge6e4f78\] for a non-periodic $\CF(\sqrt{D})$ where $\Red{D}$ is a square.
### Reduction to periodic and denominators {#sec:orge7a9aff}
We now study the case where $\CF(\gamma)$ becomes periodic. This happens automatically if $k$ is finite, for example with $D \in \Z[X]$ and reduction modulo some odd prime (see e.g. Corollary \[cor-finite-field-always-periodic\]). Instead of talking about denominators which is rather vague, we are looking for .
\[bad-reduction-to-periodic\] If $\CF(\gamma)$ is periodic, then infinitely many fibres of $\lambda$ have at least $2$ elements. Hence $\CF(\alpha)$ has bad reduction at $\nu$, and there exists $n > 0$ where $\alpha_n$ has in the leading coefficient.
Corollary \[cor-pq-degree-periodicity\] implies that for all $n \geq 1$ holds $\deg a_n < \frac{1}{2} \deg D$, and that there are infinitely many (because of pure periodicity of $\CF(\Red A + \sqrt{\Red D})$) $m \geq 1$ such that $\deg c_m = \frac{1}{2} \deg \Red{D}$.
However, $\deg D = \deg \Red{D} = 2 d$, and good reduction of $\CF(\alpha)$ would by Remark \[good-reduction-preserve-degrees\] imply that $\deg a_n = \deg c_n$ for all $n$. In fact, by Corollary \[cor-lambda-degree-sum\], for every $m \geq 1$ with $\deg c_m = d$, the fibre $\inv\lambda(m-1)$ has more than a single element, so $\lambda$ is certainly not bijective.
So $\CF(\alpha)$ must have bad reduction. The statement about then follows from Proposition \[bad-reduction-minimal-pole\].
\[bad-reduction-to-periodic-infinite-poles\] Proposition \[prop-multiple-element-fibre-analysis\] implies that in the case of bad reduction of $\CF(\alpha)$ each fibre with more than one element yields an unbounded complete quotient. It follows that there are infinitely many unbounded complete quotients. Compare also Proposition \[bad-reduction-minimal-pole\].
If $\deg D = 4$, the statement of the Lemma becomes an equivalence:
\[bad-reduction-is-periodic-deg4\] Suppose $\deg D = 4$ and $\Red{D}$ non-square. Then $\CF(\alpha)$ has bad reduction $\CF(\gamma)$ is periodic.
This extends Lemma \[bad-reduction-to-periodic\], it only remains to prove that bad reduction of $\CF(\alpha)$ implies periodicity. As $\deg D = 4$, we have $\deg a_n = 1$ for all $n \geq 1$. By Proposition \[bad-reduction-minimal-pole\], there is a minimal complete quotient $\alpha_n$ with $\nu(\alpha_n) < 0$. Because $\Red{D}$ is non-square, $\CF(\gamma)$ is infinite and hence the proposition also implies $\deg c_n > 1$.
Then from Corollary \[cor-pq-degree-periodicity\] follows $\deg c_n \leq \frac{1}{2} \deg \Red{D} = 2$, so $\deg c_n = 2$ and thus $\CF(\gamma)$ must be periodic.
If the residue field $k$ is finite (and $K$ is obviously infinite), then unless $\Red{D}$ is square, one always has periodic $\CF(\gamma)$ (see Corollary \[cor-finite-field-always-periodic\]). So it is impossible to avoid bad reduction of $\CF(\sqrt{D})$ in that case, for example if the base field $K$ is a number field.
Lemma \[bad-reduction-to-periodic\] works only if $\alpha$ is a square root of a polynomial (or shares a complete quotient with some $\sqrt{D}$). As we are interested in periodicity, we may assume that $\alpha$ is $\sigma$-reduced (the complete quotients eventually have this property, see Proposition \[cf-compute-eventually-sigma-reduced\]). But then $\deg a_n = \deg A$ is equivalent to $\alpha_n = \mu \,(A + \sqrt{D})$, so we would necessarily end up in the continued fraction expansion of $\sqrt{\mu^2 \, D}$.
### Primes occurring in infinitely many denominators {#sec:org1fe25a8}
We are now ready to attack the proof of Theorem \[thm-intro-infinite-poles-rationals\] from the introduction, about a prime occurring in infinitely many denominators of the $a_n$. We first give a more technical version for a fixed prime, which holds in full generality.
\[fibre-conditions-infinite-poles\] Suppose that there are infinitely many fibres of $\lambda$ with one element, and infinitely many fibres with at least two elements. Then there exist infinitely many $n$ with $\nu(\LC(\alpha_n)) < 0$, i.e. infinitely many complete (and partial) quotients have the “prime” as a factor in the denominator of the leading coefficient.
Let $N \geq 0$. By assumption, there exists $n \geq N$ such that $\{n \} = \inv\lambda(m)$ for some $m$. Then Proposition \[prop-single-element-fibre-analysis\] implies $$\label{eq-fibre-conditions-valuation}
\nu(\LC(\alpha_{n+1})) = \nu(\alpha_{n+1}) = \nu(g_{n-1}) - \nu(g_{n}) = \nu(g), \quad \nu(g_{n+1}) = \nu(g_{n-1}),$$ where we set $g = \ifrac{g_{n-1}}{g_{n}}$. Hence $\inv g \, \alpha_{n+1} \in \laurinx \O$ with leading coefficient in $\units \O$. As then also $\inv g \, \alpha_{n+1} - \inv g \, a_{n+1} \in \laurinx \O$, its leading coefficient is in $\O$, i.e. does not have . We deduce that the first complete quotient $g \, \alpha_{n+2}$ cannot have in the leading coefficient (compare the proof of Proposition \[bad-reduction-minimal-pole\]): $$\nu(\LC(g \, \alpha_{n+2})) \leq 0 \implies \nu(\LC(\alpha_{n+2})) \leq - \nu(g) = - \nu(\LC(\alpha_{n+1})).$$ So if $\nu(\LC(\alpha_{n+1})) \neq 0$, either $\alpha_{n+1}$ or $\alpha_{n+2}$ has the desired in the leading coefficient.
Otherwise $\nu(\LC(\alpha_{n+1})) = 0$, so $\alpha_{n+1} \in \laurinx \O$ and we can reproduce the argument from Proposition \[bad-reduction-minimal-pole\]: Let $n' > n$ minimal such that $\inv\lambda(\lambda(n'))$ has multiple elements. In this case we know $\nu(\LC(\alpha_{n'+1})) < 0$ by the minimality of $n'$. So the desired pole is in the leading coefficient of $\alpha_{n'+1}$.
\[infinite-poles-after-renormalising\] The proof also illustrates that there are infinitely many partial quotients with , even if we multiply $\alpha$ with $\pi^e$ for some $e \in \Z$ (or some other constant). This merely changes $g$ in in ; and recall Proposition \[cf-scalar-multiplication\] about multiplying a continued fraction with a constant factor. We do not even need to assume $\alpha \in \laurinx \O$ here, if we define $\lambda$ appropriately – it is determined by the sequences $\deg a_n$ and $\deg c_m$ which do not change under this multiplication.
Our results from Section \[sec:org84e8497\] tell us that multiple element fibres correspond to unbounded complete quotients, and hence bad reduction of the continued fraction. If we want this to occur repeatedly, it is very natural to ask for infinitely many such fibres.
On the other hand, asking also for infinitely many fibres with just a single element is a more technical condition. Right now we cannot avoid this because we do not understand how the valuations behave for multiple element fibres (except for $\deg D = 4$, to be treated in Theorem \[thm-genus1-zero-patterns\]). Recall that the complete quotients belonging to multiple element fibres, if at all, reduce to rational functions, about which we have hardly any information (see Proposition \[prop-multiple-element-fibre-analysis\]).
At least we have a simple criterion that guarantees the existence of infinitely many fibres of $\lambda$ with just a single element:
\[prop-criterion-infinite-single-element-fibres\] With $\alpha = \sqrt{D}$, suppose that $\CF(\alpha)$ is non-periodic, but $\CF(\gamma)$ is periodic. Let $\delta = \min\{\deg a_n \mid n \geq 0 \}$. If there exists $m$ such that $\deg c_m = \delta$, then $\lambda : \N_0 \to \N_0$ has infinitely many fibres with a single element.
Recall that $\deg D = 2d$. From Corollary \[cor-pq-degree-periodicity\] we know that $\deg a_n < d = \deg a_0 = \deg c_0$ for $n \geq 1$, so certainly $m \geq 1$. But the quasi-period of $\CF(\gamma)$ begins at $c_1$; this means there are actually infinitely many $m$ with $\deg c_m = \delta$.
Then Corollary \[cor-lambda-degree-sum\] implies for $\inv\lambda(m-1) = \{n-1,\dots,n-1+l\}$ that $\deg a_n + \dots + \deg a_{n+l} = \deg c_m = \delta$. Minimality of $\delta$ forces $l = 0$, hence the fibre has a single element. It follows that there are infinitely many fibres of $\lambda$ with a single element.
Note that along the way, we have also proved that there are infinitely many partial quotients with $\deg a_n = \delta$, so the minimal degree must be assumed infinitely often (however, we used periodicity of $\CF(\gamma)$ which is in general a rather strong hypothesis).
Now we restrict ourselves to $K$ being a number field. The ring of integers $\O_K$ of a number field, while it need not be a unique factorisation domain, has unique factorisations of ideals into prime ideals. Every $x \in K$ can be written as $x = \frac{a}{b}$ with $a \in \O_K$ and $b \in \N$. In the theorem below, we refer to $b$ as the denominator.
Each prime ideal $\PP$ of $\O_K$ corresponds to a non-archimedean valuation $\nu_\PP$ on $K$. By localising $\O_K$ at $\PP$, one obtains a discrete valuation ring with a finite residue field. The latter is a finite extension of $\F_\pp$, where $\pp$ is the unique prime number (of $\Z$) contained in $\PP$.[^11]
The following generalises Theorem \[thm-intro-infinite-poles-rationals\] from the introduction:
\[thm-infinite-poles-number-field\] Let $K$ a number field, suppose that $D \in K[X]$ is monic, non-square and has even degree, but is not Pellian.
Then for all but finitely prime numbers (in $\Z$), the prime $\pp$ appears in infinitely many $a_n$ (actually $\LC(a_n)$) in a (the) denominator.
The primes excluded are $2$ (because the residue field would have characteristic $2$), those which already appear in a denominator in $D$, and those which make $D \mod\PP$ a square polynomial. Additionally, we may need to exclude a finite number of primes, depending on where the first partial quotient of minimal degree $\delta$ occurs. This can be made effective, as discussed below in Remark \[minimal-an-degree-effective\].
The Theorem relies on $\alpha$ being a square root. For other elements of $K(X, \sqrt{D})$, we may in fact have good reduction of the continued fraction at infinitely many primes, see Section \[sec:org53fcc2b\] for an example.
Removing the finitely many primes with $\nu_\PP(D) < 0$ (i.e. $\pp$ is in the denominator of $D$), and ignoring the primes $\PP$ above $2$, the conditions on $D$ ensure that $\sqrt{D} \in \laurinx \O$ for $\nu = \nu_\PP$. Let us also ignore the $\PP$ for which $D - A^2 \in \PP[X]$, i.e. where the reduction $\Red{D}$ is a square (there are only finitely many, as $D - A^2 \in K[X]$ is a polynomial).
Of course $D$ not Pellian means that $\CF(\sqrt{D})$ is non-periodic. However, the residue field $k$ is finite for every prime, so $\CF(\sqrt{\Red{D}})$ must necessarily be periodic. Then Lemma \[bad-reduction-to-periodic\] implies that we are always in the case of bad reduction (of the continued fraction), and there are infinitely many fibres of $\lambda$ which have at least two elements.
In order to apply Proposition \[fibre-conditions-infinite-poles\], we use Proposition \[prop-criterion-infinite-single-element-fibres\], so we need to check that there exists $m$ with $\deg c_m = \delta = \min\{\deg a_n \mid n \geq 0 \}$.
Let $n_0$ the minimal $n$ with $\deg a_n = \delta$ (obviously $n_0 \geq 1$). We restrict to primes $\PP$ for which we have good reduction of $\CF(\sqrt{D})$ up to $n_0$, i.e. where $\alpha_0, \alpha_1, \dots, \alpha_{n_0} \in \laurinx \O$. This excludes only finitely many $\PP$: we can factor each $\LC(a_n) \in K$ for $n=0,\dots,n_0$ into a product (with possibly negative exponents) of prime ideals of $\O_K$. Of course $\nu_\PP(\LC(a_n)) < 0$ happens $\PP$ appears with a negative exponent in the factorisation. Of these there are obviously just finitely many, and by Proposition \[bad-reduction-minimal-pole\] the bad reduction of $\CF(\sqrt{D})$ starts only later for all other primes.
For the remaining primes $\PP$, the complete quotients up to $\alpha_{n_0}$ are thus contained in $\laurinx \O$. By Proposition \[cf-good-red-leading-coeffs\] and Remark \[good-reduction-preserve-degrees\] this implies $\deg c_{n_0} = \deg a_{n_0} = \delta$.
Hence $\nu_\PP(\LC(a_n)) < 0$ for infinitely many $n$. If we write $\LC(a_n) = \ifrac{a}{b}$ with $a \in \O_K$ and $b \in \N$, then naturally $\nu_\PP(a) \geq 0$, hence $\nu_\PP(b) > 0$. Applying the Norm, $b \in \N$ must have $\pp \div b$ as desired.
Let us briefly discuss effectivity of $\delta = \min\{ \deg a_n \mid n \geq 0 \}$. In [@zannier-2016-hyper-contin-fract], it is shown how a Skolem-Mahler-Lech theorem for algebraic groups implies that the sequence of the $\deg a_n$ is eventually periodic (even if $\CF(\sqrt{D})$ is not periodic!). While in certain cases it seems possible to obtain a bound for the period length (of the degrees) from this result, there is unfortunately no information on the pre-period. Summing upper bounds for the pre-period and the period would of course produce an upper bound for $\delta$.
However, the issues with effectivity are actually related to finding $\max\{ \deg a_n \mid n \geq 1 \}$. For finding the minimal degree, we do not need to know the entire period (of degrees):
\[minimal-an-degree-effective\] Let $\OOmultszar$ the Zariski closure of $\{ n \, \OO \mid n \in \Z \}$ in the Jacobian of $\CC$; we need to find the maximal $r$ such that $\OOmultszar \subset W_r$ but $\OOmultszar \not\subset W_{r-1}$. Using the divisor relations coming from the convergents, explained in Sections \[sec:org5685823\] and \[sec:org4e4485e\], this implies $\delta = g - r + 1$ for said maximal $r$ (recall from Section \[sec:orgdaae3ac\] that $W_r$ is the $r$ fold symmetric sum of $\CC$ embedded in its Jacobian variety).
We can effectively compute $\OOmultszar$ from a factorisation of the Jacobian as in Theorem 1.2 of [@gaudron-remond-2014-polarisations-isogenies] (which extends a deep result of Masser and Wüstholz, [@masser-wuestholz-2014-polarization-estimates-abelian]). As the $W_r$ can also be effectively represented, we can determine in which of the $W_r, \; (r=1, \dots, g-1)$ our subvariety $\OOmultszar$ is not contained. Certainly $\OOmultszar$ is contained in $W_g = \Jac(\CC)$.
In practice finding $\delta$ is not a big issue because we usually immediately find a partial quotient with $\deg a_n = 1$ (which in fact *must* occur for $\deg D = 4$ or $6$ if $D$ is non-Pellian and non-square; see Theorem 1.3 of [@zannier-2016-hyper-contin-fract], stated below as Theorem \[thm-zannier-an-deg-bound\]), and then we know that $\delta = 1$.
Theorem \[thm-genus1-zero-patterns\] below gives another (similar) proof for the occurrence of a prime in the denominators of infinitely many $a_n$ in the case $\deg D = 4$. This relies on being able to control cancellation issues sufficiently, so we do not need the single element fibres to estimate the Gauss norms.
Genus 1 valuation patterns {#sec:orgbb63dc1}
--------------------------
We analyse the case $\deg D = 4$ much closer now, and will describe how the valuations of the complete quotients, partial quotients and convergents behave in the case of bad reduction at $\nu$. When studying examples (see Tables \[cfr-mod5-valuations-table\] and \[cfr-mod19-valuations-table\] in Section \[sec:orgec32865\]), one notes that the valuations (Gauss norms) of the partial quotients $a_n$ are often divisible by $4$, with alternating signs, while the valuations of the convergents $q_n$ are always divisible by $2$, again with alternating signs. Both also exhibit an almost pseudo-periodic behaviour. The theorem below aims to explain these patterns:
\[thm-genus1-zero-patterns\] Suppose $\deg D = 4$, and that $\CF(\sqrt{D})$ is non-periodic, while $\CF(\sqrt{\Red{D}})$ is periodic with quasi-period $\QPL$, so that we have bad reduction as shown in Proposition \[bad-reduction-is-periodic-deg4\]. Then we observe the following:
- The unbounded complete quotients $\alpha_n$ are exactly those with $$\label{deg4-unboundedset-eq}
n \in \unboundedset = \{ j \, (\QPL+1) - 1 \mid j \geq 1 \}.$$
- Defining $$f_n = \nu(\LC(\vartheta_{n-1})) \geq 0$$ we have $f_n > 0$ $n \in \unboundedset$ (so $f_n = 0$ otherwise).
- Recursively defining $F_0 = 0$ and $F_n = -(F_{n-1} + f_n)$, we get formulas for the valuations $$\begin{aligned}
\nu(a_n) &= 2 \, (F_{n-2} + F_n), & \nu(\LC(a_n)) &= \nu(a_n) + f_{n-1} + f_n, \\
\nu(q_n) &= 2 \, F_n, & \nu(\LC(q_n)) &= \nu(q_n) + f_n.\end{aligned}$$
Note that if $n-1, n \not \in \unboundedset$, we have $F_{n-2} = F_n$ and thus $\nu(a_n) = 4 \, F_n$, which explains the divisibility by $4$.
For higher genus, one probably has to consider other coefficients besides $f_n$. But it is not at all clear how this generalises.
The following is the general version of Theorem \[thm-intro-genus1-unbounded-gauss-norm\] (recall that $\QPL + 1$ is the torsion order of $\j{\OOred}$, see Proposition \[prop-bounds-torsion-period-length\]):
\[cor-genus1-unbounded-gauss-norm\] Under the same hypotheses as Theorem \[thm-genus1-zero-patterns\], and additionally assuming the quasi-period $\QPL$ of $\CF(\sqrt{\Red{D}})$ is odd, the Gauss norms grow at least linearly (in particular they are unbounded): $$(-1)^n \nu(a_{n}) \geq 2 \left( \floor{\frac{n-1}{\QPL+1}}_\Z + \floor{\frac{n+1}{\QPL+1}}_\Z \right), \qquad
(-1)^n \nu(q_{n}) \geq 2 \floor{\frac{n+1}{\QPL+1}}_\Z.$$
We can easily write $F_n$ as an alternating sum of the $f_n$: $$F_n = \sum_{j=0}^n (-1)^{n-j+1} \, f_j = \sum_{j \in \unboundedset, \atop j \leq n} (-1)^{n-j+1} \, f_j = (-1)^n \sum_{j \in \unboundedset, \atop j \leq n} (-1)^{j+1} \, f_j.$$ In case $\QPL$ is odd, for every $j \in \unboundedset$ we have $j+1 = i \, (\QPL + 1)$ even. For $j + 1 \leq n + 1$, we have $1 \leq i \leq \frac{n+1}{\QPL+1}$. As every $f_j \geq 1$, this implies $(-1)^n \, F_n \geq \floor{\frac{n+1}{\QPL+1}}_\Z$. With the formulas from the theorem, we get the desired estimates for the Gauss norm.
For even $\QPL$, it is completely unclear if the $F_n$ could be bounded. In example calculations, we sometimes observe cancellation, but not always (see Table \[cfr-mod19-valuations-table\]). As we currently have almost no control over the $f_n$ for $n \in \unboundedset$, any result in this direction would be quite surprising.
In some examples, we see almost periodic patterns in the values of the $f_n$. Usually though, there comes a disturbance in these patterns at some point. We will revisit this issue briefly in Section \[sec:org9851fc8\].
However, we can check that the valuations are negative for infinitely many $n$ (giving another proof of Theorem \[thm-infinite-poles-number-field\] for $\deg D = 4$):
\[cor-infinite-poles-deg4\] Under the hypotheses of the Theorem \[thm-genus1-zero-patterns\], there are infinitely many $n$ with $\nu(a_n) < 0$, and infinitely many $n$ with $\nu(q_n) < 0$.
The previous Corollary \[cor-genus1-unbounded-gauss-norm\] gives a stronger statement when the quasi-period length $\QPL$ of $\CF(\gamma)$ is odd, so we only need to check the case where $\QPL$ is even. In particular, this means $\QPL \geq 2$.
So if we take $n \in \unboundedset$, this implies (from the structure of $\unboundedset$) that $f_n > 0$ but $f_{n+1} = f_{n+2} = 0$, hence $$F_n = -(F_{n-1} + f_n), \quad F_{n+1} = - F_n = F_{n-1} - f_n, \quad F_{n+2} = - F_{n+1} = F_n.$$ If both $F_{n-1} \geq 0$ and $F_{n} \geq 0$, then also $F_{n-1} + F_{n} = - f_n \geq 0$; but that contradicts our choice of $n$. So one of $\nu(q_{n-1}) < 0$ or $\nu(q_{n}) < 0$ must be satisfied.
Similarly, if both $F_{n-1} + F_{n+1} = 2 \, F_{n-1} - f_n \geq 0$ and $F_{n} + F_{n+2} = 2 \, F_n = -2 \, F_{n-1} - 2 \, f_n\geq 0$, then also $F_{n-1} + F_{n+1} + F_{n} + F_{n+2} = -3 \, f_n \geq 0$; this is again a contradiction. Hence at least one of $\nu(a_{n+1}) < 0$ or $\nu(a_{n+2}) < 0$ is satisfied.
As $\unboundedset$ is infinite, we find infinitely many of these partial quotients and convergents.
\[red-deg4-coprime-convergents\] For all $n$, we have $\Redn{p_n}$ and $\Redn{q_n}$ coprime because for single and two element fibres we observed that all the $h_n$ must be constant (see Propositions \[prop-single-element-fibre-analysis\] and \[prop-two-element-fibre-analysis\]).
We begin the proof of Theorem \[thm-genus1-zero-patterns\] by analysing the fibres of $\lambda$. For the rest of this section, assume the hypotheses on $D$ from the Theorem are satisfied.
\[bad-reduction-periodic-deg4-fibres\] The fibres of $\lambda$ have at most $2$ elements. The fibres with $2$ elements are given by $$\inv\lambda(j \, \QPL - 1) = \{ j (\QPL + 1) - 2, j (\QPL + 1) - 1 \}, \quad j \geq 1,$$ all other fibres have just one element.
The degrees of the $a_n$ are given by the sequence $2,1,1,1,\dots$, while the degrees of the $c_n$ are given by the sequence $2, 1, \dots, 1, 2, 1, \dots, 1,2,1, \dots$ with precisely $\QPL-1$ “1” between the “2” (the quasi-period is determined by the degrees of the partial quotients, see Corollary \[maximal-degree-implies-quasi-period\]). Recall that for a fibre $\{n, n+1, \dots, n+l \} = \inv\lambda(m)$ we always have $\deg a_{n+1} + \dots + \deg a_{n+l+1} = \deg c_{m+1}$ (see Corollary \[cor-lambda-degree-sum\]). So clearly, we can have at most two elements in a fibre.
The first fibre with two elements, due to $\deg c_{\QPL} = 2$ by the properties of the quasi-period, is $$\inv\lambda(\QPL - 1) = \{ \QPL - 1, \QPL \}.$$ In fact, we generally have $\deg c_{j \, \QPL} = 2$. In between, there are always $\QPL - 1$ fibres with a single element, so the minimal element increases by $\QPL + 1$ each time: $$\inv\lambda(j \, \QPL - 1) = \{ \QPL - 1 + (j-1) \, (\QPL + 1), \QPL + (j-1) \, (\QPL + 1) \} = \{ j \, (\QPL +1) - 2, j \, (\QPL + 1) -1 \}.$$
With our analysis of fibres with one or two elements (Proposition \[prop-single-element-fibre-analysis\] and \[prop-two-element-fibre-analysis\]), Proposition \[bad-reduction-periodic-deg4-fibres\] above implies directly : the unbounded complete quotients come only from the minimal element of the two element fibres (index of course shifted by $1$). These results also show that $\io{\Red{\vartheta_n}} > \io{\vartheta_n}$, i.e. $f_{n+1} = \nu(\LC(\vartheta_n)) > 0$, happens just for the minimal element of the two element fibres. As the definition of $f_n$ corrects for the index shift, it is clear that $f_n > 0$ happens precisely if $\alpha_n$ is unbounded.
It remains to check the valuation formulas, for which we use a complete induction. Recall that $\nu(g_n) = \nu(q_n)$.
For $n = 0$, by our assumption on $D$ we have $\nu(\LC(a_0)) = \nu(a_0) = 0$. As $q_0 = 1$, the valuation formulas are clearly satisfied.
Actually, we should also check $n = 1$. But the careful reader will find that we use the induction hypothesis for “$n-2$” only for $\nu(q_{n-2})$. So we can check $n = -1$ instead of $n = 1$.
By convention, we have $p_{-1} =1, q_{-1} = 0$, so $\vartheta_{-1} = 1$. So $\nu(q_{-1}) = \infty$ looks like a problem, but in fact we only need $\nu(g_{-1}) = 0$. Recall that $g_{-1} = 1$ is the normalisation factor of the “canonical convergent” $(p_{-1}, q_{-1}) = (1, 0)$.
For the induction step, we first check the single element fibre case:
Suppose $\{ n \} = \inv\lambda(m)$, so we refer to Proposition \[prop-single-element-fibre-analysis\]. In this case, $f_n = f_{n+1} = 0$. Hence $$\nu(a_{n+1}) = \nu(\LC(a_{n+1})) = \nu(\alpha_{n+1}) \\= \nu(g_{n-1}) - \nu(g_{n}) = 2 (F_{n-1} - F_n) = 2 (F_{n-1} + F_{n+1})$$ which covers $a_n$ and its leading coefficient.
We also get $$\nu(\LC(q_{n+1})) = \nu(q_{n+1}) = \nu(g_{n-1}) = 2 \, F_{n-1} = 2 \, F_{n+1}.$$
Next we verify the valuation formulas for the two element fibre: for $\{ n, n+1 \} = \inv \lambda(m)$, we use Proposition \[prop-two-element-fibre-analysis\]. As observed above, $f_n = f_{n+2} = 0$, but $f_{n+1} > 0$. We already computed in $$\nu(a_{n+1}) = \nu(g_{n-1}) - \nu(g_n) -2 \, f_{n+1} = 2 (F_{n-1} - F_n - f_{n+1}) = 2 \, (F_{n-1} + F_{n+1}),$$ and also $\nu(\LC(a_{n+1})) = \nu(a_{n+1}) + f_{n+1}$ as desired. For the convergent, we had $$\nu(q_{n+1}) = \nu(g_{n-1}) - 2 f_{n+1} = 2 (F_{n-1} - f_{n+1}) = 2 \, F_{n+1}.$$ Moreover, using $F_n = - F_{n-1}$, we find $$\begin{gathered}
\nu(\LC(q_{n+1})) = \nu(\LC(a_{n+1})) + \nu(\LC(q_{n})) \\ = 2 \, (F_{n-1} + F_{n+1}) + f_{n+1} + 2 \, F_n + f_n
= 2 \, F_{n+1} + f_{n+1}.\end{gathered}$$
For the second element of the fibre, we have (analogous to the calculation for the single element fibre, but using only $f_{n+2} = 0$) $$\nu(a_{n+2}) = \nu(g_n) - \nu(g_{n+1}) = 2 (F_{n} + F_{n+2})$$ and $\nu(\LC(a_{n+2})) = \nu(a_{n+2}) + f_{n+1}$, as desired. For the convergent, we have $$\nu(\LC(q_{n+2})) = \nu(q_{n+2}) = \nu(q_n) + 2 \, f_{n+1} = 2 \, (F_n + f_{n+1}) = -2 \, F_{n+1} = 2 \, F_{n+2}.$$
This concludes the proof of Theorem \[thm-genus1-zero-patterns\].
To visualise these formulas better, have a look at the three following tables, with horizontal lines directly before the unbounded complete quotients:
$n$ $\deg a_n$ $\deg q_n$ $\io{\vartheta_n}$ $m$ $\deg c_m$ $\deg v_m$ $\io{\Red{\vartheta_n}}$
---------- ------------ ------------ -------------------- -------- ------------ ------------ --------------------------
0 2 0 1 0 2 0 1
1 1 1 2 1 1 1 2
$\vdots$
$l-1$ 1 $l-1$ $l$ $l-1$ 1 $l-1$ $l+1$
$l$ 1 $l$ $l+1$ $l-1$ $l-1$ $l+1$
$l+1$ 1 $l+1$ $l+2$ $l$ 2 $l+1$ $l+2$
$l+2$ 1 $l+2$ $l+3$ $l+1$ 1 $l+2$ $l+3$
$\vdots$
$2l$ 1 $2l$ $2l+1$ $2l-1$ 1 $2l$ $2l+2$
$2l+1$ 1 $2l+1$ $2l+2$ $2l-1$ $2l$ $2l+2$
$2l+2$ 1 $2l+2$ $2l+3$ $2l$ 2 $2l+2$ $2l+3$
$2l+3$ 1 $2l+3$ $2l+4$ $2l+1$ 1 $2l+3$ $2l+4$
$\vdots$
For simplicity, we assume here that all $f_n \leq 1$. For $l$ odd, we get
$n$ $f_n$ $\nu(a_n)$ $\nu(\alpha_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
---------- ------- ------------ ----------------- ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 0 0 0 0 0 0
$\vdots$ 0 0
$l-1$ 0 0 0 0 0 0
$l$ 1 -2 $-\infty$ -1 -2 -1
$l+1$ 0 2 3 2 2 2
$l+2$ 0 -4 -4 -4 -2 -2
$l+3$ 0 4 4 4 2 2
$\vdots$
$2l-1$ 0 -4 -4 -4 -2 -2
$2l$ 0 4 4 4 2 2
$2l+1$ 1 -6 $-\infty$ -5 -4 -3
$2l+2$ 0 6 6 7 4 5
$2l+3$ 0 -8 -8 -8 -4 -4
$2l+4$ 0 8 8 8 4 4
$\vdots$
But for $l$ even, we get
$n$ $f_n$ $\nu(a_n)$ $\nu(\alpha_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
---------- ------- ------------ ----------------- ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 0 0 0 0 0 0
$\vdots$ 0 0
$l-1$ 0 0 0 0 0 0
$l$ 1 -2 $-\infty$ -1 -2 -1
$l+1$ 0 2 3 2 2 2
$l+2$ 0 -4 -4 -4 -2 -2
$l+3$ 0 4 4 4 2 2
$\vdots$
$2l-1$ 0 4 4 4 2 2
$2l$ 0 -4 -4 -4 -2 -2
$2l+1$ 1 2 $-\infty$ 3 0 1
$2l+2$ 0 -2 -2 -1 0 0
$2l+3$ 0 -8 -8 -8 0 0
$2l+4$ 0 8 8 8 0 0
$\vdots$
See also the tables for Example \[ex-cfp1-zero-pattern-deg4\] in Section \[sec:orgec32865\]
Reduction of abelian varieties {#sec:org376166a}
------------------------------
To understand how the quasi-period length may change under reduction, and hence to understand bad reduction to a periodic continued fraction, it serves to study reduction of torsion points on the Jacobian of the (hyper)elliptic curve.
### Reduction of curve and its Jacobian {#sec:orgd3ba0b6}
Our first step is to define a model of $\CC$ over $\O$. Here we can mostly retrace the steps from Section \[sec:orgf4393ef\]: instead of over the field $\K$, we are working over the discrete valuation ring $\O$. Note that $\Spec \O$ is a local affine Dedekind scheme of dimension $1$ with just two points: the generic point and a closed point corresponding to $\mm$.[^12]
For this section, we assume that both $D$ and $\Red{D}$ are square-free to ensure that the curve $\CC$ (and its Jacobian) has good reduction at $\nu$.
Gluing together $$\Spec \O[X,Y]/\spann{Y^2 - D(X)} \text{ and } \Spec \O[U,V]/\spann{V^2 - D^\flat(U)}$$ via the morphisms given by $X \, U = 1$ and $U^{g+1} \, Y = V$ (respectively $X^{g+1} \, V = Y$) we get a scheme $\XX$ of dimension $2$ which is our model of $\CC$ over $\O$. Note that the coefficients of $D^\flat$ are those of $D$ in reverse order, so $D^\flat \in \O[U]$.
Think of the surface $\XX$ as containing two curves: the fibre $\XX_0$ over the generic point of $\Spec \O$ which is essentially our curve $\CC$, and the fibre $\XX_\mm$ over the closed point of $\Spec \O$ which is the curve $\CCred$ defined over $k$, with $D$ replaced by $\Red{D}$.
The fibered surface $\XX \to \Spec \O$ is normal, regular, projective and flat, in other words it is a normal arithmetic surface.
Normal and regular are local conditions and may be checked at the stalks. Hence this follows from $\CC$ and $\CCred$ being normal and smooth (and thus regular), see Propositions \[affine-curve-is-smooth-and-normal\] and \[projective-curve-is-smooth-and-normal\].
Flatness follows from surjectivity via Proposition 4.3.9 of [@liu-2002-algebraic-geometry-arithmetic]: clearly the generic point of $\XX$ maps to the generic point of $\Spec \O$.
As the fibre $\XX_0$ is proper, and both fibres are geometrically connected, Remark 3.3.28 of [@liu-2002-algebraic-geometry-arithmetic] with surjectivity implies that $\XX \to \Spec \O$ is proper.
Then we can apply the second part of Remark 9.3.5 in [@liu-2002-algebraic-geometry-arithmetic] to obtain that $\XX \to \Spec \O$ is projective.
In order to properly define the reduction map, we need our field $K$ to be Henselian, i.e. complete with respect to the valuation $\nu$. See also Section 10.1.3 in [@liu-2002-algebraic-geometry-arithmetic] for further details.
Let $\hat K$ the completion of $K$, and $\hat \O = \{ x \in \hat K \mid \nu(x) \geq 0 \}$. This remains a discrete valuation ring with residue field still $k$. We now consider $\XX$ as a scheme over $\hat \O$.
For a closed point $P \in \XX_0 = \CC$, the Zariski closure $\closure{\{P\}}$ in $\XX$ is irreducible and has a unique closed point, the point of $\closure{\{P\}} \cap \XX_\mm$. This defines a reduction map $\Redm : \CC(\hat K) \to \CC(k)$ which extends linearly to Weil divisors.
For a point $P = (x, y) \in \CCa$ we easily see that $\nu(x) \geq 0$ implies $\nu(y) \geq 0$, so in this case we set $\Red{P} = (\Red{x}, \Red{y})$. Otherwise $\nu(x) < 0$, but then write $P = (u, v) \in \CCinf$ where now $\nu(u) \geq 0$, so we may set $\Red{P} = (\Red{u}, \Red{v})$. This also covers $O_\pm$, clearly $\Red{O_+} = O_+$ and $\Red{O_-} = O_-$.
Actually, for rational points $P \in \CC(K)$ we do not have to worry about $K$ being Henselian because the minimal polynomial of $x$ remains irreducible after reduction.
The reduction map extends to the algebraic closure $\Kbb$ of $\hat K$. Namely, for $x$ algebraic over $\hat K$, we define the valuation $$\nu(x) = \ifrac{\nu\left(\Nm_{\hat K(x)/\hat K}(x)\right)}{[\hat K(x):\hat K]}.$$ In fact the integral closure $\Obb$ of $\hat \O$ in $\K$ is still a valuation ring (but no longer discrete).
We also get a reduction map for the Jacobian: it is defined as a quotient of divisors of degree $0$ modulo principal divisors, and the latter are preserved by the reduction map:
\[reduction-principal-divisor\] Let $\di{D}$ a principal divisor over $\CC(\Kbb)$. Then also the divisor $\Red{\di{D}}$ over $\CCred(\closure k)$ is principal.
Let $$\di{D} = \Div f = \divsum n_P \cdot \pd{P},$$ a principal divisor over $\CC(\Kbb)$ with $f \in \Kbb(X, Y)$. As only finitely many $n_P \neq 0$, we may assume all $P \in \CC(\hat K)$, and $f \in \hat K(X, Y)$ by passing to a finite extension of $\hat K$.
Of course we may write $f = g/h$ with $g, h \in \hat \O[X, Y]$, and multiply with a suitable power of the uniformiser $\pi$ such that $\nu(g) = \nu(h) = 0$ (recall that $\hat \O[X, Y] \subset \laurinx{\hat \O}$). Thus $f$ is also a rational function on $\XX$ which does not vanish nor has a pole on all of $\XX_\mm$ (so there is no vertical component), because both $\Red{g} \neq 0$ and $\Red{h} \neq 0$.
Moreover, note that for every $P \in \CC(\hat K)$ we have that $\closure{\{ P \}}$ is a zero or pole (with multiplicity $n_P$) of $f$ on $\XX$. This follows from zeroes and poles being Zariski-closed, or the stalks being isomorphic: $\O_{\XX_0,P} \iso \O_{\XX,\closure{\{P\}}}$ (see the proof of Lemma 8.3.3 and Definition 7.1.27 of multiplicities in [@liu-2002-algebraic-geometry-arithmetic]). So $$\di{D}_\XX = \Div f_\XX = \divsum n_P \cdot \pd{\closure{\{P\}}},$$ which we intersect with $\XX_\mm$ to get the divisor $$\di{D}_{\XX_\mm} = \divsum n_P \cdot \pd{\Red{P}}.$$ We wish to show that this is the divisor of the function $f_\mm = \Red{g} / \Red{h} \in k(X, Y)$. Let $P \in \XX$ a closed point (i.e. a closed point in $\XX_\mm$), and consider the intersection number $i_P(\cdot, \cdot)$. Without loss of generality, we may assume that $n_P \geq 0$ (otherwise pass to $-\di{D}$ and $1/f$). Now by Corollary 9.1.32 in [@liu-2002-algebraic-geometry-arithmetic], we have $i_P(\pd{\closure{\{P\}}}, \XX_\mm) = 1$. This implies (using Definitions 7.1.27 and 9.1.1 in [@liu-2002-algebraic-geometry-arithmetic]) that $$n_P = i_P(\Div f_\XX, \XX_\mm) = \mathrm{length} \, \O_{\XX,P} /\! \left( \spann{f_\XX} + \mm \, \O_{\XX,P} \right) = \mathrm{length} \, \O_{\XX_\mm,P} /\! \spann{f_\mm} = \ord_P \left( f_\mm \right)$$ and hence $\di{D}_{\XX_\mm} = \Div f_\mm$ is principal as desired.
### Reduction of torsion points and periodicity test {#sec:org36b2a71}
While it is not so clear how the period length changes when reducing a periodic continued fraction, it is quite well understood how the torsion order of $\j{\OO}$ can change:
\[thm-serre-tate-torsion-reduction\] Suppose $D$ and $\Red{D}$ are square-free. Let $P \in \Jac$ be torsion of order $n$, and suppose that $\RedM{P} \in \Jacred$ has order $m$. If $\Char k = 0$, then $n = m$, otherwise there exists $e \in \N$ such that $n = \pp^e \, m$ with $\pp = \Char k$.
As $0 = \RedM{n \, P} = n \, \RedM{P}$, we see that $\Redm : \Jac \to \Jacred$ restricts to a homomorphism of groups $\Redm : \Jac[n] \to \Jacred[n]$ (the subgroups of points with torsion order dividing $n$). But the conditions we pose on $D$ and $\nu$ ensure that $\Jac$ has good reduction at $\nu$. So by Theorem 1 and Lemma 2 of [@serre-tate-1968-good-reduction-abelian], for $\Char k \notdiv n$, this map $\Jac[n] \iso \Jacred[n]$ is actually an isomorphism of groups; this is always the case in zero characteristic.
For positive characteristic $\Char k = \pp$, we may write $n = \pp^e \, n'$ with $\pp \notdiv n'$, and assume that $\pp \notdiv m$: because $m \div n$, we can remove any common power of $\pp$ and go to a multiple of $P$.
Now $\pp^e \, P$ has order precisely $n'$ not divisible by $\pp$, so $\RedM{\pp^e \, P}$ has the same order $n'$. However, $\pp^e$ is coprime with the order of $\RedM{P}$, implying that $\pp^e \, \RedM{P} = \RedM{\pp^e \, P}$ has likewise order $m$. So $n' = m$, and we are done.
The above theorem enables an old trick to effectively test if a point is torsion, mentioned already in [@davenport-1981-the-integration-algebraic], and described in [@yu-1999-arith] for hyperelliptic continued fractions.
\[rem-periodicity-test-reduction-two-primes\] Given a square-free $D \in K[X]$ with $K$ some number field, $\deg D$ even and $\LC(D)$ a square as usual, we can always find two prime ideals $\PP_1$ and $\PP_2$ such that $D \in \O_{\PP_i}[X]$, $\nu_{\PP_i}(\LC(D)) = 0$ and $D$ is square-free modulo $\PP_i$, for $i = 1,2$. Of course the residue fields are finite, and we may assume they are of different characteristics $\pp_1$ and $\pp_2$. Then $\j{\OO_{\PP_i}}$ is torsion of order $m_i$. Assuming that also $\j{\OO}$ is torsion of order $m$, we can write $$m = {\pp_1}^{e_1} \, m_1 = {\pp_2}^{e_2} \, m_2, \quad e_i \geq 0.$$ This implies $e_1 \leq e'_1 = \nu_{\pp_1}(m_2)$ and $e_2 \leq e'_2 = \nu_{\pp_2}(m_1)$, and moreover $$m \div \gcd({\pp_1}^{e'_1} \, m_1, {\pp_2}^{e'_2} \, m_2) \div \lcm(m_1, m_2).$$ This already gives a bound for the torsion order $m$ which translates into a bound for the period length via Proposition \[prop-bounds-torsion-period-length\]. So we can test for periodicity effectively.
Indeed as $m_1, m_2 \div m$, often it is even possible to immediately find a contradiction if $m_1$ and $m_2$ have too many different prime factors.
Also if $K$ is finitely generated over a number field, we can specialize $D$ to be defined over a number field. If $\j{\OO}$ is already torsion, this should not alter the torsion order, so we can lift the torsion bound as obtained above, and still determine effectively if $\CF(\sqrt{D})$ is periodic.
However, we need to be careful to avoid bad reduction of the continued fraction: it might happen that specializing a non-Pellian $D$, we end up with a Pellian $D$. Usually, it should however not be a problem to find a specialization where this does not happen. See for example Proposition \[specialization-only-countably-many-bad-reduction\] below.
### Shortening of quasi-period {#sec:orgaebd15f}
Theorem \[thm-serre-tate-torsion-reduction\] also gives a little bit of information on how the quasi-period may change in the case of bad reduction of the continued fraction.
Suppose that $\CF(\sqrt{D})$ has quasi-period length $\QPL$. Set $d_i = \deg a_i < d = \deg D, \; i = 1, \dots, \QPL-1$. The torsion order of $\j{\OO}$ is $$\label{shortening-degrees-sum-above}
m = \deg p_{\QPL - 1} = d + d_1 + \dots + d_{\QPL-1}.$$ Because the quasi-period is palindromic (see Proposition \[palindromic-period\]) we have $d_i = d_{\QPL - i}$.
Assuming that $\Red{D}$ is not a square, with $\CF(\sqrt{\Red{D}})$ having quasi-period length $\QPL'$, we set $d'_i = \deg c_i < d, \; i = 1, \dots, \QPL'-1$, with $d'_i = d'_{\QPL-i}$. The torsion order of $\j{\OOred}$ is then $$\label{shortening-degrees-sum-below}
m' = \deg u_{\QPL' - 1} = d + d'_1 + \dots + d'_{\QPL'-1},$$ where $m = \pp^e \, m'$ for some non-negative integer $e$ if $\Char k = \pp$, and $m = m'$ if $\Char k = 0$.
If $m' \neq m$, then has to be repeated $\pp^e$ times to make up . Recall from Corollary \[cor-lambda-degree-sum\] that each $d'_i = d_{i_1} + \dots + d_{i_j}$.
For example $$\begin{aligned}
d'_1 &= d_1 + \dots + d_{j_1}, & d'_{\QPL-1} &= d_{\QPL-1} + \dots + d_{\QPL - j_1} \\
d'_2 &= d_{j_1 + 1} + \dots + d_{j_2}, \\
d'_3 &= d_{j_2 + 1} + \dots + d_{j_3},\end{aligned}$$ and so on. Of course $m' = m$ does not prevent bad reduction of $\CF(\sqrt{D})$, as the $d'_i$ might just be larger than the $d_i$ – a better criterion is to check if $\QPL' = \QPL$.
Unfortunately, we do not get a lot more information about these degrees in general. But in some special cases, we can at a glance exclude the possibility of bad reduction of the continued fraction:
- If the sequence of $\deg a_n$ starts with $2, 1, 1, 1, 2, 1, 1, 1, 2, 1, \dots$, then bad reduction of $\CF(\sqrt{D})$ is impossible because $\deg c_n$ cannot follow the sequences $2, 2, 1, 2, 2, 1, \dots$ or $2, 1, 2, 2, 1, 2, \dots$. Then some complete quotients $\gamma_n$ would have quasi-period length $1$, but others would have quasi-period length $2$, which is impossible.
- Similarly, if the $\deg a_n$ start with $3, 1, 2, 1, 3, 1, 2, 1, 3, 1, \dots$, then bad reduction of $\CF(\sqrt{D})$ would make $\deg c_n$ start with $3, 3, 1, 3, 3, 1, \dots$ or $3, 1, 3, 3, 1, 3, \dots$. As above, this is not possible.
### Reduction of convergent divisors {#sec:org9851fc8}
We now attempt to give a geometric description for the reduction of a hyperelliptic continued fraction, in terms of the divisors associated to convergents.
Recall from Section \[sec:org4e4485e\] that we can write the divisors of the canonical convergents of $\alpha \in K(X, Y)$ as $$\begin{gathered}
\Div (p_n - \alpha \, q_n) = \Div (\vartheta_n) = \\ -(\deg p_n) \pd{O_-} - \pd{Q_1} - \dots - \pd{Q_h} + (\deg q_{n+1}) \pd{O_+} + \pd{P_1^n} + \dots + \pd{P_{e_n}^n}\end{gathered}$$ where $e_n = \deg a_0 - \deg a_{n+1} + h$. If $\alpha = Y$ ($= \sqrt{D}$), then this divisor satisfies $P_i^n \neq \sigma(P_j^n)$ if $i \neq j$ because $p_n$ and $q_n$ are coprime.
What happens when we reduce this divisor, and pass to $$\begin{gathered}
\Div (\Redn{p_n} - \gamma \, \Redn{q_n}) = \Div (\Red{\vartheta_n}) = \\ -(\deg p_n) \pd{\Red{O_-}} - \pd{\Red{Q_1}} - \dots - \pd{\Red{Q_h}} + (\deg q_{n+1}) \pd{\Red{O_+}} + \pd{\Red{P_1^n}} + \dots + \pd{\Red{P_{e_n}^n}}\end{gathered}$$ as in the proof of Proposition \[reduction-principal-divisor\]?
1. Of course $\Red{O_\pm} = O_\pm$.
2. The $\Red{Q_i}$ are always the same, and we can control them from the start.
3. It is possible that $\Red{P^n_i} = O_+$ which means $\io{\Red{\vartheta_n}} > \io{\vartheta_n}$.
4. Or $\Red{P^n_i} = O_-$ which means $\degb{\Redn{q_n}} < \deg q_n$.
5. Or if $\alpha = Y$, then possibly $\Red{P^n_i} = \sigma(\Red{P^n_j})$ for some $i \neq j$. This corresponds to $\Redn{p_n}$ and $\Redn{q_n}$ sharing a common factor.
6. Otherwise, $\Red{P^n_i}$ is just a finite point.
In the case $\alpha = Y$ with $g = 1$, we have $e_n \leq 1$, so there is at most $P^n_1$ and we do not have to worry about case 5. We also know that $P^n_1$ must be $K$-rational. But for higher genus, we may need to work over an algebraic extension of $K$ (not necessarily the algebraic closure because the degree of the equations defining the $P^n_i$ is uniformly bounded in terms of $\deg D$).
Let us have a closer look at the genus $1$ case, and study how it is related to the valuation analysis from Theorem \[thm-genus1-zero-patterns\].
Under the same hypotheses as for Theorem \[thm-genus1-zero-patterns\] and additionally $D, \Red{D}$ square-free, we have for $n \geq 0$ $$-(n+2) \, \j{\OO} = j(P_n) \quad \text{ where } P_n = (x_n, y_n) \in \CCa.$$ Let $\j{\OOred}$ have torsion order $m = \QPL +1$, then
1. If $m \div n+2$, then $\Red{P_n} = O_+$ and $\nu(x_n) = - f_{n+1} < 0$.
2. If $m \div n+1$, then $\Red{P_n} = O_-$ and $\nu(x_n) = - f_n < 0$.
3. Otherwise $\Red{P_n}$ is a finite point and $\nu(x_n) \geq 0$.
Notice that $P_n$ reduces to infinity precisely when $n$ is in a two element fibre (compare Proposition \[bad-reduction-periodic-deg4-fibres\]).
We have $\alpha = Y$, $g = 1$, and $\CF(\sqrt{D})$ non-periodic. This implies $\deg p_n = n+2$ and $$\Div(\vartheta_n) = -(n+2) \, \pd{O_-} + (n + 1) \, \pd{O_+} + \pd{P_n}$$ where $P_n = (x_n, y_n) \in \CCa(K)$ for all $n \geq 0$ because $\deg a_n = 1$ for $n \geq 1$.
The reduction of this divisor is $$\Div(u_{\lambda(n)} - Y \, v_{\lambda(n)}) = -(n+2) \, \pd{O_-} + (n + 1) \, \pd{O_+} + \pd{\Red{P_n}}.$$ With $\CF(\gamma)$ periodic, the point $\OOred$ over $k$ has torsion order $m = \QPL + 1$.
1. If $m \div n+2$, this forces $\Red{P_n} = O_+$.
2. If $m \div n+1$, this forces $\Red{P_n} = O_-$.
3. Otherwise $\Red{P_n} \in \CCa(k)$.
Recall from Proposition \[introduce-theta-n\] and that $$\vartheta_n \, \sigma(\vartheta_n) = {g_n}^{-2} \, (-1)^{n+1} \, s_{n+1} = b_n (X - x_n)$$ for some $b_n \in \units K$. From the normalisation of $\vartheta_n$ with $\nub{\vartheta_n} = 0$ (and analogously $\nub{\sigma(\vartheta_n)} = 0$), we get $\nub{b_n (X - x_n)} = 0$.
1. In the case $m \div n+2$, we have $$\io{\Red{\vartheta_n}} = 1 + \io{\vartheta_n}, \quad \io{\Red{\sigma(\vartheta_n)}} = \io{\sigma(\vartheta_n)}$$ which means $f_{n+1} > 0$. In fact, $$f_{n+1} = \nub{\LC(\vartheta_n)} + \nub{\LC(\sigma(\vartheta_n))} = \nub{b_n}.$$ This forces $\nub{x_n} = - \nub{b_n} = - f_{n+1} < 0$ because $\nub{b_n (X - x_n)} = 0$, so $x_n$ has as expected.
2. In the case $m \div n+1$, we have $$\io{\Red{\vartheta_n}} = \io{\vartheta_n}, \quad \io{\Red{\sigma(\vartheta_n)}} = 1 + \io{\sigma(\vartheta_n)}.$$ so $f_{n+1} = 0$. But $\nub{\LC(\sigma(\vartheta_n))} = \nu(\LC(\normal{q_n})) = f_n > 0$, and similarly as above $\nub{x_n} = - \nub{\LC(\sigma(\vartheta_n))} = - f_n < 0$. We find that $x_n$ has again .
3. Otherwise, we have $$\io{\Red{\vartheta_n}} = \io{\vartheta_n}, \quad \io{\Red{\sigma(\vartheta_n)}} = \io{\sigma(\vartheta_n)}$$ and hence $f_{n+1} = 0$. This implies $\nub{b_n} = 0$ and thus $\nub{x_n} \geq 0$, i.e. $x_n \in \O$.
Observe how this matches Theorem \[thm-genus1-zero-patterns\] and that $-f_{n+1}$ is the valuation of both $x_n$ and $x_{n+1}$ at the two element fibre.
So we have found a second description of the $f_n$ from Theorem \[thm-genus1-zero-patterns\]. Unfortunately, this still does not suggest what type of patterns they might follow, or whether they might be bounded. Generally, we should not expect the $f_n$ to be bounded, see for example the proposition on page 55 of [@silverman-tate-2015-rational-points-on]. It suggests that for an elliptic curve defined over $\Q$, there are rational points $P = (x,y)$ with arbitrarily low $\nu_\pp(x)$ for any prime $\pp$.
Good reduction at infinitely many primes {#sec:org53fcc2b}
----------------------------------------
We mentioned before that Theorem \[thm-infinite-poles-number-field\] holds only for $\CF(\sqrt{D})$, but not for other elements of $K(X, \sqrt{D})$.
\[thm-good-reduction-infinite-primes\] Let $D = X^4 + 16 \, X^2 + 24 \, X + 9$ which is not Pellian (the torsion orders of $\OO_3$ and $\OO_{17}$ differ just by $1$). Set $\alpha = \frac{\sqrt{D} - 3}{X}$.
There are infinitely many primes $\pp$ for which $\CF(\alpha)$ has good reduction (hence $\pp$ never divides a denominator of a partial quotient $a_n$).
This is related to questions treated in [@corrales-schoof-1997-support-problem-its], and earlier in [@schinzel-1960-the-congruence-a]. Here we present an explicit proof for our particular example, to illustrate these arguments more concretely. In fact, the given problem boils down to an analogue for elliptic curves (more generally abelian varieties) of
There exist infinitely many prime numbers $\pp$ such that for all $n \in \Z$ we have $2^n \not\equiv 5 \mod \pp$.[^13]
Note that $D$ non Pellian implies that $\CF(\alpha)$ is not quasi-periodic by Theorem A in [@berry-1990-periodicity-continued-fractions].
Recall from Proposition \[general-convergent-divisor-lemma\] that the divisors induced by the convergents have the shape $$\Div(p_n - \alpha \, q_n) = (\deg p_n) \, \OO - \pd{Q} + \pd{P_n}$$ because $\alpha$ has a single pole at $Q = (0, ?)$. Note that $X \div D - 3^2$, so we are in the situation of Theorem \[thm-quadratic-laurent-series-cf-representation\]. In principle, we could have $P_n = O_+$ for a single $n$, but the reduction arguments below imply that $P_n$ must always be a finite point.
Recall from Section \[sec:org9851fc8\] that here bad reduction of $\CF(\alpha)$ is equivalent to $P_n$ reducing to a point at infinity $O_\pm$ which means $$j(\Red{Q}) + (\deg p_n) \, j(O_-) = j(O_\pm) = 0 \text{ or } j(O_-) \quad \mod \pp.$$ So if we ensure that for all $m \in \Z$ $$\label{eq-good-red-inf-reduced-translate}
j(\Red{Q}) + m \, j(O_-) \neq 0 \quad \mod \pp$$ then we know that we must have good reduction of $\CF(\alpha)$ at this prime $\pp$ (and additionally we confirm that $P_n \neq O_\pm$ for all $n$).
We deduce from the Čebotarev density theorem (see Theorem 13.4 and Lemma 13.5 in [@neukirch-1999-algebraic-number-theory]) that this holds for infinitely many primes $\pp$. For reasons of space, we will assume the reader is already familiar with this famous theorem, and also ramification of prime ideals, Galois theory and the Frobenius automorphism.
Our curve $\CC$ is an elliptic curve, isomorphic to its Jacobian. We write it in Weierstrass form $$\EC : V^2 = U^3 + 16 \, U^2 - 36 \, U = U (U - 2) (U + 18)$$ using the transformation $$U = 2 X^2 + 2 Y, \qquad V = 2 X (U + 16) + 24$$ which sends $j(O_-)$ to $R_1 = (-16, -24)$ and $j(Q)$ to $R_2 = (6, 24)$ (over $\Q$). These are non-torsion rational points. Note that the $2$ torsion points of $\EC$ are by design rational. This implies that for any of the four choices for a point $R_i' \in \CC$ with $2 \, R_i' = R_i$, the point $R_i'$ is defined over the same number field $K_i$ (for $i=1,2$). Here the fields are $$K_1 = \Q(\zeta), \text{ where } \zeta^4 + 1 = 0, \qquad K_2 = \Q(\sqrt{6}).$$ The composite field $K_1 K_2$ has degree $8$ and is Galois, with abelian Galois group $G = \Gal(K_1 K_2 / \Q)$. We denote by $H_i$ the subgroup of $G$ whose fixed field is $K_i$.
Ignoring the finitely many primes where the curve $\EC$ has bad reduction (just $2, 3, 5$, the discriminant of $\EC$ is $2^{12} \cdot 3^{4} \cdot 5^{2}$), we now wish to find infinitely many primes $\pp$ such that $\Red{R_1'}$ is a $\F_\pp$-rational point, but $\Red{R_2'}$ is not. In that case, the point $m \, \Red{R_1'} + \Red{R_2'}$ cannot be $\F_\pp$-rational for any $m \in \Z$. In particular it is not $0$ or torsion of order $2$. This implies that for all $m \in \Z$ $$m \, R_1 + R_2 \neq 0 \mod\pp$$ which is equivalent to .
If we restrict to the (infinitely many) primes $\pp$ which are unramified over $K_1 K_2$ (and hence over $K_1$ and $K_2$), the condition on rationality of $\Red{R_1'}$ and $\Red{R_2'}$ amounts to saying that $\pp$ has a prime divisor $\PP_1$ of degree $1$ over $K_1$ (i.e. the residue field $k(\PP_1)$ has degree $1$ over $\F_\pp$), but over $K_2$ all the prime divisors of $\pp$ have degree $> 1$ (i.e. $[k(\PP_2):\F_\pp] > 1$ for any prime divisor $\PP_2$).
As described in Lemma 13.5 of [@neukirch-1999-algebraic-number-theory], this happens the conjugacy class of the Frobenius automorphisms of prime ideals of $K_1 K_2$ lying over $\pp$ intersects $H_1$, but not $H_2$. Here $H_1 = \{\id, \sigma_1\}$ where $\sigma_1$ is defined by $\sigma_1(\zeta) = \zeta$ and $\sigma_1(\sqrt{6}) = - \sqrt{6}$. The conjugacy classes are trivial because $G$ is abelian, so we are looking precisely for the primes $\pp$ for which $\sigma_1$ is a Frobenius automorphism of some prime $\PP$ of $K_1 K_2$ over $\pp$.
But the set of these primes has positive density $\geq \frac{1}{8}$ by the Čebotarev density theorem (Theorem 13.4 in [@neukirch-1999-algebraic-number-theory]), so in particular there are infinitely many of them.
In our computations, we observed that among the first 100 odd prime numbers, there are 62 prime numbers $\pp$ for which $\CF(\alpha)$ has good reduction at $\pp$. This is a much higher density than predicted by our Čebotarev density estimate, but of course the latter gives only a sufficient condition.
Moreover, we argued with $R_i'$ satisfying $2 \, R_i' = R_i$. Instead we could argue with $m \, R_i' = R_i$ for any $m$; then we probably get additional primes with the desired property.
Complex functions case {#sec:org6b10a31}
----------------------
Let us now discuss the situation of specialization, i.e. where $K = \C(t)$ and the reduction map works by assigning a special value $t_0 \in \C$ to $t$. The corresponding valuation is $\nu = \ord_{t_0}$ measuring the zero-order at $t_0$, with uniformising parameter $t - t_0$, and the discrete valuation ring $\O = \C[t]_{\spann{t - t_0}}$, the localisation of the maximal ideal $\spann{t - t_0}$ in $\C[t]$. We also write $\Red{\alpha} = \alpha_{t=t_0}$ to distinguish different specializations.
### Results of Masser and Zannier {#sec:org087aad4}
This is precisely the situation found in the article of Masser and Zannier on the connection between the Pell equation and Unlikely intersections [@masser-zannier-2015-torsion-points-on]. Let me restate their results in our language of specialization of continued fractions.
For genus $1$, they mention the following result:
\[mz-thm-spec-periodic-deg4\] Let $D = X^4 + X + t$, then $\CF(\sqrt{D})$ is non-periodic. The set of $t_0$ such that $\CF(\sqrt{D_{t=t_0}})$ is periodic (i.e. $\CF(\sqrt{D})$ has bad reduction at $t - t_0$ by Proposition \[bad-reduction-is-periodic-deg4\]), is infinite and denumerable.
For genus $2$ however, their Theorem P1 says:
\[mz-thm-spec-periodic-deg6\] Let $D = X^6 + X + t$, then $\CF(\sqrt{D})$ is non-periodic. The set of $t_0$ such that $\CF(\sqrt{D_{t=t_0}})$ is periodic is *finite*.
For example $\CF(\sqrt{D_{t=0}})$ is periodic. But because $\deg D = 6$, it is now possible that $\CF(\alpha)$ has bad reduction at $t-t_0$, even if $\CF(\sqrt{D_{t=t_0}})$ is non-periodic.
[|l|l|]{} $D = X^{6} + X + t$ & basefield $\C(t)$\
Discriminant of $D$: $(-46656) \cdot (t^{5} - \frac{3125}{46656})$ & $D$ never reduces to a square.\
$D$ is not Pellian &\
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To describe the $t_0$ in the theorem, we need to search for an increase in the degree of the partial quotients when specialising, as seen in Lemma \[bad-reduction-minimal-pole\]. Clearly $\deg c_0 = 3, \deg c_1 = 2$ for the partial quotients of any specialization. Their Theorem P2 says:
\[mz-thm-spec-weak-pell-deg6\] Let $D = X^6 + X + t$, with $\CF(\sqrt{D})$ non-periodic. The set of $t_0$ such that for $\gamma = \sqrt{D_{t=t_0}}$ there exists $n \geq 2$ with $\deg c_n = 2$, is an infinite and denumerable subset of $\closure \Q$.
This set also includes the $t_0$ with $\CF(\sqrt{D_{t=t_0}})$ periodic: because the period always begins at $c_1$, there are then infinitely many $n$ with $\deg c_n = 2$ for each of these $t_0$. By Theorem \[cf-good-red-partial-quotients\] the increase of degrees is necessary for bad reduction of $\CF(\sqrt{D})$, so this infinite set is actually the set of all $t_0$ with bad reduction of $\CF(\alpha)$ at $t-t_0$.
The hard part in the proof of this theorem is showing that this set of $t_0$ is *infinite* which is done in Section 11 of [@masser-zannier-2015-torsion-points-on].
With the theory of Chapter \[sec:orgd5f1900\], it is however not so hard to show:
\[specialization-only-countably-many-bad-reduction\] Let $\alpha \in \laurinx{\C(t)}$ with $\LC(\alpha) = 1$. Then there exist at most countably many $t_0 \in \C$ such that $\CF(\alpha)$ has bad reduction at $t-t_0$ (the valuation being $\nu = \ord_{t=t_0}$).
For every $n \geq 0$, let $d_n$ the denominator of $\LC(\alpha_{n}) \in \C(t)$ which is a monic polynomial in $\C[t]$. Clearly $\ord_{t=t_0}(\LC(\alpha_n)) < 0$ holds $d_n(t_0) = 0$.
Then if $\CF(\alpha)$ has bad reduction at $t-t_0$, there exists by Proposition \[bad-reduction-minimal-pole\] at least one $n$ such that $d_n(t_0)$.
Of course there are only countably many polynomials $d_n$, each with finitely many zeroes (even though $\deg d_n$ might increase with $n$). Hence there are only countably many possibilities for bad reduction of $\CF(\alpha)$.
Masser and Zannier also give another example in degree $6$, with different behaviour (see Section 3.4.5 in [@zannier-2012-some-problems-unlikely]):
\[mz-spec-infinite-periodic-deg6\] Let $D = X^6 + X^2 + t$, then $\CF(\sqrt{D})$ is non-periodic. The set of $t_0$ such that $\CF(\sqrt{D_{t=t_0}})$ is periodic, is infinite and denumerable.
### Repeated occurrences of $t-t_0$ {#sec:orgf9f9759}
Also for the specialisation case we can say something about the occurrence of “primes” in infinitely many denominators of partial quotients $a_n$. In degree $4$, we can simply use Corollary \[cor-infinite-poles-deg4\] to deduce this from Proposition \[mz-thm-spec-periodic-deg4\] above:
Let $D = X^4 + X + t$ as in Proposition \[mz-thm-spec-periodic-deg4\]. For each of the infinitely many $t_0$ where $\CF(\sqrt{D})$ has bad reduction, there exist infinitely many $n$ such that $t-t_0$ appears in a denominator of $a_n$.
In degree $6$, the situation becomes more subtle, and we need to use Proposition \[fibre-conditions-infinite-poles\]. If $D = X^6 + X + t$, then for $t_0 = 0$, there is the problem that the only fibre of $\lambda : \N_0 \to \N_0$ (the map describing the reduction of convergents, introduced in Section \[sec:orgea2cbbe\]) with a single element is $\inv\lambda(0) = \{0 \}$, so we cannot apply the proposition. This happens because for this specialization we have $\deg c_n \neq 1$ for all $n$.
[|l|l|]{} $D = X^{6} + X$ & basefield $\Q$\
Discriminant of $D$: $5^{5}$ & $D$ never reduces to a square.\
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$p_{1} = 2 X^{5} + 1$ & $q_{1} = 2 X^{2}$\
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For all other $t_0$, we can use Proposition \[prop-criterion-infinite-single-element-fibres\] because clearly $\deg c_2 = 1$ for any specialization to $t_0 \neq 0$, and the minimal degree $\delta$ is of course $1$. This implies infinitely many fibres with a single element, and from Lemma \[bad-reduction-to-periodic\] and Proposition \[fibre-conditions-infinite-poles\] then follows:
Let $D = X^6 + X + t$ as in Theorem \[mz-thm-spec-periodic-deg6\]. For each of the finitely many $t_0 \neq 0$ where $\CF(\sqrt{D_{t=t_0}})$ is periodic, there exist infinitely many $n$ such that $t-t_0$ appears in a denominator of $a_n$.
It is likely this property also holds for $t_0 = 0$, but this would require a different argument, for example an analogue to Theorem \[thm-genus1-zero-patterns\] for degree $6$ which is unfortunately not in sight.
For $D = X^6 + X^2 + t$, we have however $a_1 = 2 \, X$, so Proposition \[prop-criterion-infinite-single-element-fibres\] implies infinitely many fibres of $\lambda$ with a single element for every $t_0$, so again by Lemma \[bad-reduction-to-periodic\] and Proposition \[fibre-conditions-infinite-poles\] follows:
Let $D = X^6 + X^2 + t$ as in Proposition \[mz-spec-infinite-periodic-deg6\]. For each of the finitely many $t_0 \neq 0$ where $\CF(\sqrt{D_{t=t_0}})$ is periodic, there exist infinitely many $n$ such that $t-t_0$ appears in a denominator of $a_n$.
Now let us return to $D = X^6 + X + t$ and consider the remaining $t_0$ with non-periodic bad reduction of the continued fraction. To understand this better, we need Theorem 1.3 from [@zannier-2016-hyper-contin-fract] (which we state for arbitrary base field $\K$):
\[thm-zannier-an-deg-bound\] Let $D \in \K[X]$ of degree $2d$, non-square, but $\LC(D)$ square. Suppose further that if $D = E^2 \, D'$ with $D'$ Pellian, then $\deg D' \leq \frac{3}{2} d$ (for example assume $D$ is square-free). Then there are only finitely many $n$ with $\deg a_n > \frac{d}{2}$.
This is a consequence of a Skolem-Mahler-Lech theorem for algebraic groups (for example the Jacobian of $\CC$) explained in the same article.
For $\deg D = 6$, in particular for $D = X^6 + X + t$ and its specializations, this means that if $\CF(\sqrt{D})$ is not periodic, only finitely many partial quotients have degree $\deg a_n > 1$; and the same property holds for $\CF(\sqrt{D_{t=t_0}})$. This means that for the $t_0$ with non-periodic $\CF(\sqrt{D_{t=t_0}})$, the corresponding map $\lambda$ has only finitely many fibres with more than one element. Again, we cannot apply Proposition \[fibre-conditions-infinite-poles\]. This does not mean there might not be infinitely many $a_n$ with $t-t_0$ in the denominator, but for every $n$ large enough, we can normalise the complete quotient $\alpha_n$ to $$\begin{gathered}
\normal{\alpha_n} = (t-t_0)^{-\ord_{t_0}(\alpha_n)} \, \alpha_n = \mu \, \alpha_n \\ = [\mu \, a_n, \inv{\mu} \, a_{n+1}, \mu \, a_{n+2}, \inv \mu \, a_{n+3}, \dots] = [b_0, b_1, b_2, b_3, b_4, \dots]\end{gathered}$$ such that none of the $b_i$ has $t-t_0$ in a denominator. So $\CF(\normal{\alpha_n})$ has good reduction at $t-t_0$.
In fact, we cannot even exclude the possibility that $\mu = 1$, in which case only finitely many $a_n$ have $t- t_0$ in the denominator, despite there being bad reduction of $\CF(\alpha)$ at $t -t_0$ (compare Remark \[infinite-poles-after-renormalising\]).
Heights {#sec:org56e7cb7}
=======
While the valuations used in the previous chapters give a local estimate for the complexity of the partial quotients, affine and projective heights provide a global measure of complexity. For the convergents, and more generally Padé approximations, the projective logarithmic height of the convergents has known lower bounds (in the non-Pellian case), see [@bombieri-cohen-1997-siegels-lemma-pade]: they should increase at least quadratically. A lower bound for the height of the partial quotients is more delicate, and has been found only recently: [@zannier-2016-hyper-contin-fract] gives a lower bound for affine logarithmic height, with at least quadratic growth for a frame of fixed length of partial quotients.
Upper bounds for the projective heights of the partial quotients follow from those of the convergents.
Heights {#sec:org743079c}
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For the convenience of the reader, I will list some definitions and properties of heights to be used in this chapter, following mainly [@bombieri-gubler-2006-heights-diophantine-geometry] in notation and normalisation. For the Weil height machine, I follow [@hindry-silverman-2000-diophantine-geometry].
### Places and Product formula {#sec:orgfdc4da2}
For a number field $K$, one defines the set of *places* $M_K$ as the equivalence classes of non-trivial absolute values on $K$, where two absolute values are equivalent if they induce the same topology. A place contains either only archimedean absolute values, and then is called *infinite*, or only non-archimedean absolute values, in which case we call it *finite*. The infinite places correspond to embeddings of $K$ into $\C$ up to complex conjugation, hence there is only a finite number. The finite places correspond to prime ideals of the ring of integers of $K$, so there are infinitely many.
In order to define heights, one carefully chooses and fixes an absolute value to represent a place. For $\Q$, there is one infinite place, the restriction of the standard complex absolute value, and countably many finite places corresponding to the prime numbers. We represent these places by $$\begin{aligned}
M_\Q &= \{ \pp \in \N \text{ prime } \} \cup \{ \infty \}, \\
\abs{x}_\infty &= \max(x,-x), \\
\abs{x}_\pp &= \pp^{-n} \text{ for } x \neq 0 \text{ where } x = \pp^n \, \frac{a}{b} \text{ with } n \in \Z, a, b \in \Z \setminus \pp \Z\end{aligned}$$
This ensures $K = \Q$ satisfies the *product formula* $$\prod_{\nu \in M_\Q} \abs{x}_\nu = 1 \text{ for } x \in \units \Q.$$ The product on the left is well defined because for a fixed $x$, only finitely many $\nu \in M_\Q$ have $\abs{x}_\nu \neq 1$.
On any number field $K$, a place $\omega$ on $K$ restricts to a unique place $\nu$ on $\Q$, written $\omega \div \nu$, and we can choose a cleverly normalised representative $\omega$ such that $$\prod_{\omega \div \nu} \abs{x}_\omega = \abs{x}_\nu \text{ for all } x \in \units \Q.$$ In consequence, on any number field $K$, there is a *product formula* $$\prod_{\omega \in M_K} \abs{x}_\omega = 1 \text{ for } x \in \units K,$$ where on the left side only finitely $\omega$ have $\abs{x}_\omega \neq 1$.
We also remark that $\omega$ normalised in this way satisfies an *improved triangle equality*: Let $x_1, \dots, x_r \in K$, then $$\label{improved-triangle-equality}
\abs{x_1 + \dots + x_r}_\omega \leq \max(1, \abs{r}_\omega) \, \max\left(\abs{x_1}_\omega, \dots, \abs{x_r}_\omega\right).$$
### Height on projective and affine space {#sec:org5c9a4ee}
The product formula allows defining an *exponential absolute projective height* on $\P^n(K)$ for a number field $K$ by setting $$\Hproj(x_0 : \dots : x_n) = \prod_{\nu \in M_K} \max\left( \abs{x_0}_\nu, \dots, \abs{x_n}_\nu \right)$$ By considering $\A^n(K) \subset \P^n(K)$, this also defines an *affine height* on $\A^n(K)$, $$\Haff(x_1, \dots, x_n) = \Hproj(1 : x_1 : \dots : x_n)$$ and in particular we get a height $H$ on $K = \A^1(K)$. Note that the affine height is always larger than the projective height, i.e. $$\Hproj(x_1, \dots, x_n) \leq \Haff(x_1, \dots, x_n).$$
It can be shown that this definition does not depend on the number field $K$, and thus extends uniquely to $\closure \Q$.
Often it is more convenient to work with the *logarithmic heights*, $\hproj = \log \circ \Hproj$, $\haff = \log \circ \Haff$ and $h = \log \circ H$.
### Height of polynomials {#sec:org3ad38dd}
A non-zero polynomial in $K[X]$ of degree $\leq n$ can be considered both as a point in $\P^n(K)$, or in $\A^{n+1}(K)$. So we define $$\begin{aligned}
\Hproj(a_n \, X^n + \dots + a_0) &= \Hproj(a_n : \dots : a_0) \\
\Haff(a_n \, X^n + \dots + a_0) &= \Haff(a_n, \dots, a_0)\end{aligned}$$ and likewise the logarithmic heights $\hproj$ and $\haff$. Note that this means the projective height of a polynomial depends only on its zeroes, while the affine height coincides with the height on $K$ for constant polynomials.
Similarly as in Section \[sec:orge83be0b\], we define the *Gauss norm* for $\nu$ on $K[X]$ by $$\abs{a_n \, X^n + \dots + a_0}_\nu = \max\left( \abs{a_n}_\nu, \dots, \abs{a_0}_\nu\right).$$ If $\nu$ is non-archimedean, this is even an non-archimedean absolute value (see Proposition \[bounded-laurent-valuation\]). Nevertheless, the notation is useful also for archimedean absolute values.
In the following, $K$ is always a number field which has at most $[K:\Q]$ archimedean places.
\[proj-height-poly-mult\] Let $f_1, \dots, f_r \in K[X]$ and $f = f_1 \cdots f_r$. Then $$-\deg f \, \log 2 + \sum_{i=1}^r \hproj(f_i) \leq \hproj(f) \leq \deg f \, \log 2 + \sum_{i=1}^r \hproj(f_i)$$
For a proof see [@bombieri-gubler-2006-heights-diophantine-geometry], Theorem 1.6.13.
\[aff-height-poly-add\] Let $f_1, \dots, f_r \in K[X]$ and $f = f_1 + \cdots + f_r$. Then $$\haff(f_1 + \dots + f_r) \leq \haff(f_1) + \dots + \haff(f_r) + \log r.$$
This follows from Proposition 1.5.15 in [@bombieri-gubler-2006-heights-diophantine-geometry].
\[proj-height-poly-division\] Let $a, b, q, r \in \mino{K[X]}$ with $a = q \, b + r$ and $\deg r < \deg b$. Set $N = \deg q = \deg a - \deg b$. Then $$\begin{aligned}
\label{height-poly-div-q}
\hproj(q) &\leq \hproj(a) + N \left(\log 2 + \hproj(b)\right), \\
\label{height-poly-div-r}
\hproj(r) &\leq \hproj(a) + (N+1) \left( \log 2 + \hproj(b) \right).\end{aligned}$$
The bound for $\hproj(q)$ holds also if $r = 0$.
We can assume $\LC(b) = 1$, as the projective height for polynomials is invariant under multiplication with a constant factor. This conveniently implies $\abs{b}_\nu = \max(1, \abs{b}_\nu)$ for every $\nu \in M_K$.
Using the standard algorithm for division, we define a sequence of polynomials, beginning with $a_0 = a$ and continuing for $i \geq 0$ via $$a_{i} = \LC(a_i) \, X^{N_i} \, b + a_{i+1} \text{ where } \deg a_{i+1} < \deg a_i \text{ and } N_i = \deg a_i - \deg b.$$ Using on the coefficients, we estimate $$\begin{gathered}
\abs{a_{i+1}}_\nu = \abs{ \LC(a_i) b - a_i }_\nu \\ \leq \max(1,\abs{2}_\nu) \, \max\left(\abs{\LC(a_i)}_\nu \abs{b}_\nu, \abs{a_i}_\nu\right) \leq \max(1,\abs{2}_\nu) \, \abs{a_i}_\nu \, \abs{b}_\nu\end{gathered}$$ and obtain $$\abs{a_i}_\nu \leq \abs{a}_\nu \, \left(\max(1, \abs{2}_\nu) \abs{b}_\nu\right)^i.$$
There are at most $N+1$ steps necessary in the algorithm to reach $\deg a_i < \deg b$ (because in every step, the degree decreases by at least $1$, hence $\deg a_i < \deg a - i$, so $i \leq N+1$), at which point $a_i = r$. Consequently $$\abs{r}_\nu \leq \abs{a}_\nu \, \left(\max(1, \abs{2}_\nu) \, \abs{b}_\nu\right)^i \leq \abs{a}_\nu \, \left(\max(1, \abs{2}_\nu) \, \abs{b}_\nu\right)^{N+1}$$ which implies . The coefficients of $q$ are precisely $\LC(a_0), \dots, \LC(a_{i-1})$, so $$\abs{q}_\nu \leq \max\left( \abs{a_0}_\nu, \dots, \abs{a_{i-1}}_\nu\right) \leq \abs{a}_\nu \, \left(\max(1, \abs{2}_\nu) \, \abs{b}_\nu\right)^{i-1} \leq \abs{a}_\nu \, \left(\max(1, \abs{2}_\nu) \, \abs{b}_\nu\right)^{N}$$ whence .
\[proj-height-poly-zeroes\] Let $f \in K[X]$ a polynomial of degree $r$ with roots $\alpha_1, \dots, \alpha_r \in \closure \Q$ (accounted for multiplicities). Then $$-r \, \log 2 + \hproj(f) \leq \haff(\alpha_1) + \dots + \haff(\alpha_r) \leq r \, \log 2 + \hproj(f).$$
This follows directly from Proposition \[proj-height-poly-mult\], noting that $f$ factors as $$f = \mu \, (X - \alpha_1) \cdots (X - \alpha_r)$$ with $\mu \in K$ having $\hproj(\mu) = 0$, while $\hproj(X - \alpha_i) = \hproj(1: \alpha_i) = \haff(\alpha_i)$.
### Weil’s Height Machine and Néron-Tate height {#sec:org721d5ba}
On varieties defined over a number field $K$, there is a plethora of different height functions. However, many of them differ only by a bounded function, so they produce essentially the same height. We capture this notion of *quasi-equivalence of heights* in the following notation:
Let $V$ a variety defined over a number field $K$, and let $f_1, f_2 : V(\closure{\Q}) \to \R$ two functions. We write $f_1 \qeq f_2$ if there exists $C \in \R$ such that $$\abs{f_1(P) - f_2(P)} \leq C \text{ for all } P \in V(\closure{\Q}).$$
We reproduce the following from Theorem B.3.2 in [@hindry-silverman-2000-diophantine-geometry]
\[weil-height-machine\] Let $K$ a number field. For every smooth projective variety $V/ K$ there exists a map $$h_V : \DIV(V) \longto \{ \text{functions } V(\closure{\Q}) \to \R \}$$ satisfying the following:
1. (Normalisation) For $\di{H} \subset \P^n$ a hyperplane holds $h_{\P^n, \di{H}} \qeq \hproj$.
2. (Functoriality) For $\phi : V \to W$ a morphism and $\di{D} \in \DIV(W)$ a divisor holds $h_{V, \phi^*(\di{D})} \qeq h_{W,\di{D}} \circ \phi$.
3. (Additivity) For $\di{D}, \di{E} \in \DIV(V)$ holds $h_{V,\di{D}+\di{E}} \qeq h_{V,\di{D}} + h_{V,\di{E}}$.
4. (Linear Equivalence) For $\di{D}, \di{E} \in \DIV(V)$ with $\di{D} \sim \di{E}$ holds $h_{V,\di{D}} \qeq h_{V,\di{E}}$.
There are further properties which we omit because we will not use them directly.
\[neron-tate-height\] Let $K$ a number field, and $\mathcal A/K$ an abelian variety. Let $\di{D} \in \DIV(\mathcal A)$ have symmetric divisor class (i.e. $[-1]^*\di{D} \sim \di{D}$). Then there exists the (unique) *canonical height on $\mathcal A$ relative to $\di{D}$*, a height function $\Th_{\mathcal A,\di{D}} : \mathcal A(\closure{\Q}) \longto \R$ satisfying the following:
1. It is equivalent to the height from the height machine: $\Th_{\mathcal A,\di{D}} \qeq h_{\mathcal A,\di{D}}$.
2. For all integers $m$ and $P \in \mathcal A(\closure{\Q})$ we have $\Th_{\mathcal A,\di{D}}(m \, P) = m^2 \, \Th_{\mathcal A,\di{D}}(P)$.
3. It is a quadratic form.
\[height-zero-iff-torsion\] Take $\mathcal A$ and $\di{D}$ as in Theorem \[neron-tate-height\], and $\di{D}$ moreover ample, then for all $P \in \mathcal A(\closure{\Q})$ we have $\Th_{\mathcal A,\di{D}}(P) \geq 0$, and $\Th_{\mathcal A,\di{D}}(P) = 0$ $P$ is torsion on $\mathcal A$.
The preceding Theorem and Proposition are adapted from Theorem B.5.1 and Proposition B.5.3 in [@hindry-silverman-2000-diophantine-geometry], respectively.
### Heights on the Jacobian {#sec:orgbf62da0}
With these tools, we are finally able to setup our heights on our (hyper)elliptic curve $\CC$ and its Jacobian $\Jac$. On the Jacobian, we use the height corresponding to the Theta divisor. The Theta divisor induced by the map $j: \CC \to \Jac$ defined in Section \[sec:orgdaae3ac\] is unfortunately not symmetric for $g > 1$. So we use a different embedding, which differs only by a translation on the Jacobian.
Indeed let $P_0 \in \CC$ one of the Weierstrass points, i.e. $P_0 = (\xi, 0)$ where $\xi$ is one of the roots of $D$. This implies $2 \pd{P_0} \sim \pd{O_+} + \pd{O_-}$ (via the function $X - \xi$), hence the canonical divisor is a multiple of $P_0$: $\canondiv \sim 2(g-1) \, \pd{P_0}$.
Now embed the curve into the Jacobian via $$j_0 : \CC \longto \Jac, \quad P \mapsto \j{P} - \j{P_0}$$ and note that $j_0(\canondiv) = 0$ (recall that $j_0$ extends naturally to divisors).
Now Theorem A.8.2.1. in [@hindry-silverman-2000-diophantine-geometry] implies that $$\Theta_0 = j_0(\CC) + \dots + j_0(\CC) \quad (g-1 \text{ copies})$$ is symmetric, i.e. $[-1]^* \Theta_0 = \Theta_0$. Then Theorem \[neron-tate-height\] implies that the Néron-Tate height $\Th = \Th_{\Jac, \Theta_0}$ associated to the height $h_{\Jac, \Theta_0}$ is a quadratic form.
Theorem A.8.2.1. also says that $j_0^*\Theta_0 \sim g \, \pd{P_0}$, so $\Th \circ j_0 \qeq g\, h_{\CC,P_0}$.
Recall that the hyperelliptic curve comes with a degree 2 map $\pi : \CC \to \P^1$, with the hyperplane $H = \{(\xi : 1) \}$ in $\P^1$ having $\pi^*(H) = 2 \, \pd{P_0}$, hence $\hproj(\pi(P)) \qeq 2 \, h_{\CC,P_0}$. So we get $2 \, \Th \circ j_0 \approx g \, \hproj(\pi(P))$, or more precisely for $P = (x, y) \in \CCa$: $$\Th(j_0(P)) \qeq \frac{g}{2} \, \haff(x).
%, \quad \Th(j_0(O_\pm)) \qeq 0$$
Height of convergents {#sec:orgcf28bd0}
---------------------
We are now ready to study the height of the convergents. We begin by analysing the coefficients of the Laurent series $\sqrt{D}$, and comparing the heights of the numerator and denominator of a convergent.
### Height bounds for series coefficients of $\sqrt{D}$ {#sec:org584c9f9}
We need some bounds for the absolute values (and height) of the coefficients of the power series $\sqrt{D}$.
\[sqrt-d-coefficient-height\] Recall that $\deg D = 2\,d$ and write $$\label{sqrt-d-coefficients-for-height}
\sqrt{D} = X^d \sum_{n=0}^\infty w_n \, X^{-n}.$$ For any place $\nu$ on a number field, we have $$\abs{w_n}_\nu \leq \abs{\sqrt{\LC(D)}}_\nu \cdot \left( \max(1, \abs{1/4}_\nu) \cdot \max(1, \abs{(2d)^2}_\nu) \cdot \abs{D}_\nu / \abs{\LC(D)}_\nu \right)^n,$$ which implies $$h(w_n) \leq \frac{1}{2} h(\LC(D)) + n \, \left( \log 4 + 2 \, \log(2d) + \hproj(D) \right).$$
Before we can prove this, we need an estimate for the growth of the binomial coefficient:
\[lemma-binomial-half\] For $n \geq 1$, there exists an integer $b_n \in \Z$ with $\abs{b_n}_\R \leq 2^{2n-3}$ such that the binomial coefficient $\binom{1/2}{n} = \ifrac{b_n}{2^{2n-1}}$.
For $\nu$ a place on a number field, not over $2$, this implies $\abs{\binom{1/2}{n}}_\nu \leq 1$ for all $n \geq 0$.
But if $\nu$ represents a place over $2$ (with the normalisations introduced at the beginning of the chapter), we find $\abs{\binom{1/2}{n}}_\nu \leq 2^{2n}$ for all $n \geq 0$.
This is an easy exercise. Note that the $b_n$ are closely related to the Catalan numbers (see for example [@aigner-2007-course-enumeration], pages 101, 102). See also Theorem 5 in [@siegel-2014-some-appl-diophantine] for a generalisation of this lemma.
We write $$D = d_{2d} \, X^{2d} + d_{2d-1} \, X^{2d-1} + \dots + d_0 = d_{2d} \, X^{2d} (1 + f(X)) \text{ with } f(X) \in K[\inv X].$$ We may then compute $$\sqrt{D} = \sqrt{d_{2d}} \, X^d \; \sum_{n=0}^\infty \binom{1/2}{n} \, f(X)^n$$ which converges in $\laurinx K$ because $\io{f(X)} > 0$. Now let $\nu$ any place on $K$, and write $f(X) = f_1 X^{-1} + \dots + f_{2d} X^{-2d}$, to define $$C_\nu = \max(1, \abs{f_1}_\nu, \dots, \abs{f_{2d}}_\nu) = \abs{1 + f}_\nu = \abs{D}_\nu / \abs{\LC(D)}_\nu.$$
Studying for $i \geq 0$ the power $$\binom{1/2}{i} \, f(X)^i = \sum_{j_1 + \dots + j_{2d}=i} \binom{1/2}{i} \, \binom{i}{j_1, \dots, j_{2d}} \left( \prod_{l=1}^{2d} {f_l}^{j_l} \right) \; X^{-( j_1 + 2 \, j_2 + \dots + (2d) \, j_{2d})},$$ we note that $\binom{i}{j_1, \dots, j_{2d}} \leq {2d}^i$ is an integer, so the coefficient of every summand is bounded in $\abs{\cdot}_\nu$ by $(\max(1, \abs{1/4}_\nu \, \max(1, \abs{m}_\nu) \, C_\nu)^i$.
Now observe that every $w_i/w_0$ (clearly $w_0 = \sqrt{\LC(D)}$) is a sum of at most $(2d)^i$ of these, so with the improved triangle inequality we obtain the desired result $$\abs{w_n}_\nu \leq \abs{w_0}_\nu \, \max(1, \abs{(2d)}_\nu)^n \, (\max(1, \abs{1/4}_\nu \, \max(1, \abs{m}_\nu) \, C_\nu)^n$$
With $\hproj(1 + f) = \hproj(D)$ and $C_\nu \geq 1$ the inequality for the height follows after we replace $\abs{w_i}_\nu$ with $\max(1, \abs{w_i}_\nu)$ (also for $i = 0$).
We can now bound the projective height of the numerator of a convergent in terms of the height of the denominator.[^14]
\[convergent-numerator-denominator-height\] For every convergent $(p, q) \in \Coset{\sqrt{D}}{K}$ holds $$\hproj(p) \leq \hproj(q) + (\deg p) \left(\log 2 + \log 4 + \log(2d) + \hproj(D) \right).$$
Recall from Proposition \[convergent-q-determines-p\] that $p = \gauss{\sqrt{D} \, q}$. However we did not define a height for $\sqrt{D}$. To workaround this problem, let $m = \deg q$ and write $D = A_m + \varepsilon_m$ with $A_m$ a Laurent polynomial and $\io{\varepsilon} > m$. This ensures $p = \gauss{A_m \, q}$, so $\hproj(p) \leq \hproj(A_m \, q)$. With Proposition \[sqrt-d-coefficient-height\], we bound the projective height $$\hproj(A_m) \leq (d + m) \, \left( \log 4 + 2 \, \log(2d) + \hproj(D)\right).$$ Note that $A_m$ has precisely $d + m = d + \deg q = \deg p$ coefficients. The overall bound then follows from the bound for the product (Proposition \[proj-height-poly-mult\]).
### Lower bound {#sec:org1835e85}
In [@bombieri-cohen-1997-siegels-lemma-pade], a general result about the height of Padé approximations predicts that the projective height of the convergents of a square root should grow quadratically in the degree of the convergent.
If we just want to prove this for square root, a shorter and simpler proof by Zannier suffices. It is explained briefly in [@zannier-2016-hyper-contin-fract], here we give a bit more detailed version.
\[convergent-height-lower-bound\] If $D$ is not Pellian, there exists a constant $C = C(D) > 0$ such that for every convergent $(p, q) \in \Coset{\sqrt{D}}{\closure Q}$ we have for $\deg q$ large enough: $$C \cdot (\deg q)^2 \leq \hproj(q).$$
Recall that by Lemma \[convergent-divisor-lemma\] and subsequent remarks, the convergents produce an equality on the Jacobian (recall $\j{\OO} = -j(O_-) = \j{O_+} - \j{O_-}$) $$-m \, \j{\OO} = j(P_1) + \dots + j(P_r)$$ with $r \leq g$ and $P_i = (x_i, y_i) \in \CCa$. The $x_i$ are precisely the zeroes of $\Omega = p^2 - D \, q^2$ (accounted for multiplicities). And we have $m = \deg p = \deg q + g + 1 \geq \deg q$.
Applying the Néron-Tate height, and using Lemma \[quadratic-form-sum-bound\] from the Appendix, we obtain $$m^2 \Th(\j{\OO}) = \Th(-m \, \j{\OO}) = \Th(j(P_1) + \dots + j(P_r)) \leq g \left( \Th(j(P_1)) + \dots + \Th(j(P_r)) \right).$$ Next, there is a constant $C_1 \geq 0$ depending only on $D$ such that $\Th(j(P_i)) \leq C_1 + \haff(x_i)$ (see Section \[sec:orgbf62da0\]). Moreover, we know that $$h(x_1) + \dots + h(x_r) \leq g \, \log 2 + \hproj(\Omega).$$ We combine these estimates to $$m^2 \Th(\j{\OO}) \leq g \, C_1 + g^2 \, \log 2 + g \, \hproj(\Omega).$$ Because $D$ is not Pellian, the point $\j{\OO}$ is not torsion in the Jacobian, so $\Th(\j{\OO}) > 0$. We get a constant $C_2 > 0$ such that $$C_2 \, (\deg q)^2 \leq \hproj(\Omega).$$
As we are only interested in the projective height of $q$, we may without restriction normalise the convergent $(p, q)$ such that $p$ is monic, so that both $p^2$ and $D \, q^2$ have to be monic. Of course, for a monic polynomial, affine and projective height coincide, and using Proposition \[aff-height-poly-add\] we can estimate $$\hproj(\Omega) \leq \haff(\Omega) \leq \log 2 + \haff(p^2) + \haff(D\,q^2) = \log 2 + \hproj(p^2) + \hproj(D\,q^2).$$
By Proposition \[convergent-numerator-denominator-height\], we have $\hproj(p) \leq \hproj(q) + \deg p \, C_3$ for some $C_3 \geq 0$ depending only on $D$. With Proposition \[proj-height-poly-mult\] we get $$\begin{aligned}
\hproj(D \, q^2) &\leq 2 \, \deg p \, \log 2 + \hproj(D) + 2 \, \hproj(q), \\
\hproj(p^2) &\leq 2 \, \deg p \, \log 2 + 2 \, \hproj(p) \leq 2 \, \deg p \, \log 2 + 2 \deg p \, C_3 + 2 \, \hproj(q).\end{aligned}$$ Combining these, and noting $\deg p = \deg q + d$, we find $$C_2 \, (\deg q)^2 \leq C_4 + C_5 \, \deg q + 4 \, \hproj(q)$$ with $C_4, C_5 \geq 0$, so for example $C = C_2 / 8$ yields the desired constant.
This does not yet give a lower bound for the height of partial quotients. This seems more challenging, especially for the projective height – see also Example \[ex-dbh1-bounded-height\] in Section \[sec:orge27ae2f\]. But there are new results for the affine height if we take some type of average, see Theorem 1.4 in [@zannier-2016-hyper-contin-fract]:
There exist $M \in \N$ and $C > 0$ such that for $n$ large enough $$C \, n^2 \leq \max( \haff(a_{n-i}) \mid i=0,\dots,M ).$$
### Upper bound {#sec:orgdaa4c63}
An upper bound for the height of the convergents can be deduced with more elementary tools, using the Toeplitz determinants from Section \[sec:org542977d\]. It is then straightforward to deduce an upper bound also for the height of the partial quotients.
\[convergents-upper-proj-height-bound\] For the canonical convergents $(p_m, q_m)$ of $\sqrt{D}$ we obtain the height bounds $$\begin{aligned}
\label{eq-height-pm-upper-bound}
\hproj(p_m) &\leq ((n+1) \,d + \tfrac{3}{2} (n^2 + n)) \, (\log 4 + 2 \, \log(2d) + \hproj(D)), \\
\label{eq-height-qm-upper-bound}
\hproj(q_m) &\leq (n \,d + \tfrac{1}{2} (3 \, n^2 + n)) \, (\log 4 + 2 \, \log(2d) + \hproj(D)),\end{aligned}$$ where $\deg D = 2d$ and $n = \deg q_m$.
We wish to apply the results of Section \[sec:org542977d\] for $\alpha = \sqrt{D}$. Connecting the notations of and , we have $N = d$ and $A_{j} = w_{d-j}$. As we chose $n = \deg q_m$ and the canonical convergents are coprime, Proposition \[convergent-linear-matrix-full-rank\] tells us that the matrix $\pqmatrix_n$ has full rank. So the kernel has dimension $1$, and we can compute a solution $(p, q)$ using which differs from $(p_m, q_m)$ only by a constant factor, and thus has the same projective height. The coefficients of $p$ and $q$ are (up to signs) the minors of $$\pqmatrix_n = \left( \begin{array}{ccccc|ccc}
-1 & & & & & w_0 \\
&\ddots & & & & w_{1} & \ddots \\
& & \ddots & & & \vdots & \ddots & w_{0} \\
& & & \ddots & & \vdots & \ddots & \vdots \\
& & & & -1 & w_{d+n} & \dots & w_{d} \\ \hline
& & & & & w_{d+n+1} & \dots & w_{d+1} \\
& & 0 & & & \vdots & \ddots & \vdots \\
& & & & & w_{d+2n} & \dots & w_{d+n}
\end{array} \right).$$ If we strike any column (to get the minor), we obtain a $(d+2n+1)\times(d+2n+1)$ matrix. Now set $$C_\nu = \max(1, \abs{1/4}_\nu) \cdot \max(1, \abs{(2d)^2}_\nu) \cdot \abs{D}_\nu / \abs{\LC(D)}_\nu$$ so that $\abs{w_j}_\nu \leq \abs{w_0}_\nu \, {C_\nu}^j$.
If we strike a column in the right block (to compute the coefficients of $q$), by using Laplace development we get (up to sign) a minor $\mathcal M'$ of the lower right block of dimensions $n\times n$, with determinant $$\det \mathcal M' = \sum_{\sigma \in S_n} \sign(\sigma) \, \mathcal M'_{1 \;\sigma(1)} \dots \mathcal M'_{n \; \sigma(n)}$$ with $\abs{\mathcal M'_{ij}}_\nu \leq \abs{w_0}_\nu \, {C_\nu}^{d+n+i}$, hence $$\abs{q}_\nu \leq \prod_{i=1}^n \abs{w_0}_\nu \, {C_\nu}^{d+n+i} \leq {\abs{w_0}_\nu}^n {C_\nu}^{n\,d + n^2 + n(n+1)/2}.$$ When taking the product over all $\nu$, the first term with $w_0$ vanishes by the product formula, and likewise the term with $\LC(D)$. We arrive at by a straightforward calculation.
If we strike a column in the left block (to compute the coefficients of $p$), we can use similar arguments, with Laplace development we only get a $(n+1) \times (n+1)$ matrix $\mathcal M''$, with $\abs{\mathcal M''_{ij}}_\nu \leq \abs{w_0}_\nu \, {C_\nu}^{d+n+i-1}$, so $$\abs{p}_\nu \leq \prod_{i=1}^{n+1} \abs{w_0}_\nu \, {C_\nu}^{d+n+i-1} \leq {\abs{w_0}_\nu}^{n+1} {C_\nu}^{(n+1) \, d + (n+1) n + n(n+1)/2}.$$ Again, follows by a straightforward calculation.
The projective height of the convergents grows at most quadratically: $$\hproj(p_m) = O(m^2), \qquad \hproj(q_m) = O(m^2).$$
The partial quotients have bounded degree $1\leq \deg a_i \leq d$, hence $m \leq \deg q_m = n \leq d \, m$, and the above theorem gives $\hproj(p_m) = O(n^2)$ and $\hproj(q_m) = O(n^2)$.
\[partial-quotients-upper-proj-height-bound\] The projective height of the partial quotients also grows at most quadratically: $$\hproj(a_m) = O(m^2)$$
We can compute the partial quotients from subsequent convergents as in $$a_m = \gauss{\frac{p_m}{p_{m-1}}} \qquad a_m = \gauss{\frac{q_m}{q_{m-1}}}$$ and then Proposition \[proj-height-poly-division\] yields $$\hproj(a_m) \leq \hproj(q_m) + (\deg a_m) ( \log 2 + \hproj(q_{m-1}) ) = O(m^2).$$
An explicit bound is $$\begin{gathered}
\hproj(a_m) \leq \left( ((a+1) n + a) \, d + a \, \log 2 + \tfrac{1}{2} \left( 3 (a+1) n^2 + (7 a+ 1) n + 3 a^2 +a \right) \right)
\\ \cdot \left(\log 4 + 2 \, \log(2d) + \hproj(D)\right)\end{gathered}$$ where $a = \deg a_m$, $n = \deg q_{m-1}$.
Connecting heights and valuations {#sec:orgba963e8}
---------------------------------
From the definitions of the height of polynomials in Section \[sec:org3ad38dd\] and the Gauss norms in Chapter \[sec:orgd5f1900\] it is clear that there is a direct connection between the height and the valuations computations, as for example in Theorem \[thm-genus1-zero-patterns\]. Note that for a polynomial $f \in K[X]$, we have $$\begin{aligned}
\hproj(f) & = \sum_{\nu \in M_K} \, \log \abs{f}_\nu, \\
\haff(f) & = \sum_{\nu \in M_K} \, \max\left(0, \log \abs{f}_\nu\right).\end{aligned}$$ For $\nu$ non-archimedean, $\log \abs{f}_\nu$ is essentially $c \cdot \nu(f)$ for some $c < 0$.
However, in Chapters \[sec:orgd5f1900\] and \[sec:org5f9d2ce\] we do not treat the places over $2$, and certainly not the archimedean places. So this connection must remain incomplete. Still, we try to point out some phenomena relating the global picture of heights and the local picture of Gauss norms.
We restrict to the genus $1$ case with $\deg D = 4$ and $D$ non-Pellian, so that we may use Theorem \[thm-genus1-zero-patterns\].
Corollary \[cor-genus1-unbounded-gauss-norm\] says that for places with $\j{\OOred}$ having even torsion order, the Gauss norms of $q_n$ grow at least linearly in $n$. But $\hproj(q_n)$ should grow quadratically, which suggests that either the Gauss norms grow faster than linearly, or the number of places with bad reduction of the continued fraction before $n$ also grows linearly.
Computational evidence suggests that the latter is the case. Also, if we work over $\Q$, the Hasse-Weil interval (see Remark \[finite-field-torsion-bound\]) predicts that $\Jacred(\F_\pp)$ grows about linearly in $\pp$, so the quasi-period length of $\CF(D_\pp)$ and hence the first occurrence of $\pp$ in a denominator of $a_n$ grows linearly in $\pp$. However, the number of primes $\pp \leq n$ grows only as $n/\log n$.
The valuations $\nu(q_n)$ mostly alternate between positive and negative signs, so for the projective height they might cancel each other out. But for the affine height, there is no such cancellation, and in fact computations for examples suggest that $\haff(q_n)$ grows more or less cubically. This is in line with Remark 4.8 (ii) in [@zannier-2016-hyper-contin-fract], which says that $\haff(q_n)$ should at most grow cubically in $n$.
Similar observations can be made for the partial quotients.
Examples {#sec:orgda48525}
========
We now apply the theory developed in this thesis to some examples. Hopefully, this illustrates our theorems and their limitations. To this end, we include examples also for the corner cases that have been somewhat neglected in the theoretical part.
For $D$ defined over the rationals (or perhaps a number field), and $\pp$ some prime (in the ring of integers), we use the notations $D_\pp$ for $\Red{D}$ in $\F_\pp[X]$, $\nu_\pp$ for the $\pp$-adic valuation, we denote by $\OO_\pp$ the torsion divisor $\pd{O_+} - \pd{O_-}$ on $\CCred$ over $\F_\pp$.
For $D$ defined over $\C(t)$, we use analogous notation, with $t - t_0$ or $t = t_0$ instead of $\pp$.
Reduction to a square {#sec:orge6e4f78}
---------------------
We begin with some examples where $D$ reduces (or specializes) to a square.
[|l|l|]{} $D = (X - 1) \cdot X \cdot (X - t) \cdot (X + t - 1)$ & basefield $\C(t)$\
Discriminant of $D$: $(4) \cdot (t - \frac{1}{2})^{2} \cdot (t - 1)^{4} \cdot t^{4}$ & Primes with $\Red{D}$ square: $t - 1, t$\
period length $2$ for $\CF(\sqrt{D})$ & quasi-period length $1$ for $\CF(\sqrt{D})$\
\
$p_{0} = X^{2} - X - \frac{1}{2} t^{2} + \frac{1}{2} t$ & $q_{0} = 1$\
\
\
\
\
Example \[ex-periodic-to-square-1\] is very simple, but it illustrates already that bad reduction of the continued fraction is not the same as bad reduction of the elliptic curve. Only $D_{t=0}$ and $D_{t=1}$ are square, and $t, t-1$ are the only irreducible/prime factors appearing in the coefficient denominators (of $a_1$).
By the way, this means we can specialise $t$ to say an integer $t_0 \in \Z \setminus\{0, 1\}$ to get a periodic continued fraction over $\Q$. This continued fraction has bad reduction precisely at the prime numbers dividing $t_0 \, (t_0 - 1)$. In this way we obtain an example also for the reduction modulo $\pp$ case.
[|l|l|]{} $D = X^{4} + 2 X^{2} + t X + 1$ & basefield $\C(t)$\
Discriminant of $D$: $(-27) \cdot t^{2} \cdot (t^{2} - \frac{256}{27})$ & Primes with $\Red{D}$ square: $t$\
$D$ is not Pellian &\
\
\
\
\
\
Example \[ex-nonperiodic-to-square-2\] clearly reduces to a square at $t = 0$. We check that $\CF(\sqrt{D})$ is non-periodic by specializing to $t = 3$. Then reduction of the continued fraction $\CF(D_{t=3})$ modulo $5$ and $7$ yields torsion orders $5$ respectively $10$ which implies non-periodicity for both $\CF(\sqrt{D_{t=3}})$ and $\CF(\sqrt{D})$. As we are in the degree $4$ case, this implies good reduction of $\CF(\sqrt{D})$ at $t-3$ (by Proposition \[bad-reduction-is-periodic-deg4\]). Of course $D_{t=3}$ reduces then to a square modulo $3$, so the example works for the reduction modulo $\pp$ case too.
So both in the periodic and non-periodic case, it is possible that $D$ reduces to a square.
Reduction periodic to periodic {#sec:orgc648887}
------------------------------
For $\deg D = 4$, we have seen that torsion order $m$ and quasi-period length $\QPL$ satisfy $m = \QPL +1$ (see Proposition \[prop-bounds-torsion-period-length\]). Together with the discussion of Section \[sec:orgaebd15f\] on how the quasi-period may shorten, and rational torsion on elliptic curves being bounded by $12$, there cannot be many examples of bad reduction of a periodic continued fraction for $D \in \Q[X]$.
[|l|l|]{} $D = X^{4} - 8 X^{3} - 42 X^{2} + 424 X - 119$ & basefield $\Q$\
Discriminant of $D$: $-1 \cdot 2^{29} \cdot 3^{5}$ & Primes with $\Red{D}$ square: $3$\
\
\
$\deg p_{7} = 9$ & $\deg q_{7} = 7$\
\
$a_{0} = X^{2} - 4 X - 29$ & $a_{4} = \frac{4}{3} X - \frac{44}{3}$\
$a_{1} = \frac{1}{96} X + \frac{1}{96}$ & $a_{5} = \frac{1}{32} X + \frac{5}{32}$\
$a_{2} = -4 X + 12$ & $a_{6} = -4 X + 12$\
$a_{3} = \frac{1}{32} X + \frac{5}{32}$ & $a_{7} = \frac{1}{96} X + \frac{1}{96}$\
Indeed this is the case in Example \[ex-periodic-good-red-1\], where we see only $2$ and $3$ in the denominators. As $\OO$ has order $9$, the only candidate for bad reduction of $\CF(D)$ is $3$, but $D_3$ is already a square. Everywhere else we have good reduction of $\CF(\sqrt{D})$.
But if we are working over number fields, and can increase the torsion orders, then in principle one should be able to construct example with bad reduction of the continued fraction (by applying Theorem \[thm-serre-tate-torsion-reduction\]).
We also analysed an example with $\deg D = 6$ given in [@platonov-2014-number-theoretic-properties] ($f_{33}$ in Section 6). In Example \[ex-periodic-good-red-2\], we have torsion order $33$, so both $3$ and $11$ are good (but a priori not the only) candidates for bad reduction of $\CF(\sqrt{D})$. But again $D_3$ is already square. And we do not see $11$ in the denominators (it suffices to check the first half of the palindromic quasi-period as mentioned in Section \[sec:orgaebd15f\]). We have good reduction of $\sqrt{D}$ at all other primes which is not surprising given that the example seems to have been constructed by testing for this. Also observe that the factor for the quasi-period is $\mu = 3$, as predicted by Remark \[bad-reduction-quasi-period-factor\].
[|p[9cm]{}|p[5cm]{}|]{} $D = 4 X^{6} + 28 X^{5} + 37 X^{4} - 30 X^{3} + 87 X^{2} - 54 X + 9$ & basefield $\Q$\
Discriminant of $D$: $-1 \cdot 2^{22} \cdot 3^{14} \cdot 127$ & Primes with $\Red{D}$ square: $3$\
period length $54$ for $\CF(\sqrt{D})$ & quasi-period length $27$ for $\CF(\sqrt{D})$\
\
$\deg p_{26} = 33$ & $\deg q_{26} = 30$\
\
$a_{0} = 2 X^{3} + 7 X^{2} - 3 X + 3$ & $a_{9} = 6 X$\
$a_{1} = \frac{1}{9} X + \frac{1}{2}$ & $a_{10} = \frac{1}{9} X + \frac{7}{18}$\
$a_{2} = 3 X - \frac{9}{2}$ & $a_{11} = -3 X - \frac{3}{2}$\
$a_{3} = \frac{2}{27} X^{2} + \frac{1}{3} X + \frac{2}{9}$ & $a_{12} = -\frac{1}{3} X - \frac{5}{6}$\
$a_{4} = 6 X^{2} + 21 X - 9$ & $a_{13} = \frac{2}{3} X + 1$\
$a_{5} = \frac{1}{9} X - \frac{1}{6}$ & $a_{14} = 2 X + 3$\
$a_{6} = X + \frac{11}{2}$ & $a_{15} = -\frac{1}{9} X - \frac{5}{18}$\
$a_{7} = -2 X + 2$ & $a_{16} = -9 X - \frac{9}{2}$\
$a_{8} = -\frac{1}{9} X - \frac{1}{2}$ & $a_{17} = \frac{1}{27} X + \frac{7}{54}$\
Reduction non-periodic to periodic {#sec:org3de9d88}
----------------------------------
### Genus 1 {#sec:orgec32865}
For a polynomial of degree $4$, our Theorem \[thm-genus1-zero-patterns\] describes the behaviour of the valuations. We now give an example to illustrate this, both for odd and even quasi-period length of $\CF(\sqrt{D_\pp})$.
[|l|l|]{} $D = X^{4} + 5 X^{2} - 3 X + 19$ & basefield $\Q$\
Discriminant of $D$: $3^{2} \cdot 7^{2} \cdot 11^{2} \cdot 17$ & Primes with $\Red{D}$ square: $3$\
$D$ is not Pellian because of incompatible torsion orders & torsion order $8$ modulo $5$\
& torsion order $3$ modulo $7$\
\
\
\
\
\
Example \[ex-cfp1-zero-pattern-deg4\] is chosen randomly. Note again that while the discriminant has a finite number of prime divisors, only $D_3$ is a square polynomial, and of course $\CF(\sqrt{D})$ has bad reduction for every odd prime number $\pp$ by Lemma \[bad-reduction-to-periodic\] and Corollary \[cor-finite-field-always-periodic\].
#### modulo 5 {#sec:org421d2d5}
$n$ $\lambda(n)$ $\nu(\alpha_n)$ $\nu(a_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
----- -------------- ----------------- ------------ ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 1 0 0 0 0 0
2 2 0 0 0 0 0
3 3 0 0 0 0 0
4 4 0 0 0 0 0
5 5 0 0 0 0 0
6 6 0 0 0 0 0
7 6 $-\infty$ -2 -1 -2 -1
8 7 2 2 3 2 2
9 8 -4 -4 -4 -2 -2
10 9 4 4 4 2 2
11 10 -4 -4 -4 -2 -2
12 11 4 4 4 2 2
13 12 -4 -4 -4 -2 -2
14 13 4 4 4 2 2
15 13 $-\infty$ -6 -5 -4 -3
16 14 6 6 7 4 4
17 15 -8 -8 -8 -4 -4
18 16 8 8 8 4 4
19 17 -8 -8 -8 -4 -4
20 18 8 8 8 4 4
21 19 -8 -8 -8 -4 -4
22 20 8 8 8 4 4
23 20 $-\infty$ -10 -9 -6 -5
24 21 10 10 11 6 6
25 22 -12 -12 -12 -6 -6
26 23 12 12 12 6 6
27 24 -12 -12 -12 -6 -6
28 25 12 12 12 6 6
29 26 -12 -12 -12 -6 -6
30 27 12 12 12 6 6
31 27 $-\infty$ -14 -13 -8 -7
32 28 14 14 15 8 8
33 29 -16 -16 -16 -8 -8
34 30 16 16 16 8 8
: \[cfr-mod5-valuations-table\] 5-adic valuations for Example \[ex-cfp1-zero-pattern-deg4\]
Table \[cfr-mod5-valuations-table\] lists the 5-adic valuations (Gauss norms). Note how the changes in the patterns, and the unbounded $\alpha_n$ are aligned with the 2-element fibres of $\lambda$. We can also read off the quasi-period length of $\CF(\sqrt{D_{5}})$ from the first occurrence of non-zero valuations, and determine it to be $7$ (this works only in degree $4$).
As the quasi-period length is odd, we can observe (as predicted by Corollary \[cor-genus1-unbounded-gauss-norm\]) that the valuations increase in absolute value. We also see that the sign of the exponent is alternating, and that almost all the $\nu(a_n)$ are divisible by $4$ (as predicted by Theorem \[thm-genus1-zero-patterns\]). Pay attention to the valuations of the leading coefficients being larger. For $q_n$ this indicates that $\Redn{q_n}$ has a lower degree.
#### modulo 19 {#sec:org46eabc7}
$n$ $\lambda(n)$ $\nu(\alpha_n)$ $\nu(a_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
----- -------------- ----------------- ------------ ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 1 0 0 0 0 0
2 2 0 0 0 0 0
3 3 0 0 0 0 0
4 4 0 0 0 0 0
5 5 0 0 0 0 0
6 5 $-\infty$ -2 -1 -2 -1
7 6 2 2 3 2 2
8 7 -4 -4 -4 -2 -2
9 8 4 4 4 2 2
10 9 -4 -4 -4 -2 -2
11 10 4 4 4 2 2
12 11 -4 -4 -4 -2 -2
13 11 $-\infty$ 2 3 0 1
14 12 -2 -2 -1 0 0
15 13 0 0 0 0 0
16 14 0 0 0 0 0
17 15 0 0 0 0 0
18 16 0 0 0 0 0
19 17 0 0 0 0 0
20 17 $-\infty$ -2 -1 -2 -1
21 18 2 2 3 2 2
22 19 -4 -4 -4 -2 -2
23 20 4 4 4 2 2
24 21 -4 -4 -4 -2 -2
25 22 4 4 4 2 2
26 23 -4 -4 -4 -2 -2
27 23 $-\infty$ 2 3 0 1
28 24 -2 -2 -1 0 0
29 25 0 0 0 0 0
30 26 0 0 0 0 0
31 27 0 0 0 0 0
32 28 0 0 0 0 0
33 29 0 0 0 0 0
34 29 $-\infty$ -2 -1 -2 -1
35 30 2 2 3 2 2
36 31 -4 -4 -4 -2 -2
37 32 4 4 4 2 2
: \[cfr-mod19-valuations-table\] 19-adic valuations for Example \[ex-cfp1-zero-pattern-deg4\]
Table \[cfr-mod19-valuations-table\] is for the 19-adic valuations. The patterns are very similar to the table for $5$. We can also read off the quasi-period of $\CF(\sqrt{D_{19}})$: it is $6$, hence even. The alternating signs of the valuations then lead to cancellation of exponents at the 2-element fibres. However it remains an open question whether these valuations are eventually periodic. If so, our computations suggest that their period length must be significantly larger than the quasi-period length of $\CF(\sqrt{D_\pp})$.
Note that the torsion order of $\OO_{19}$ is 7, while the torsion order of $\OO_5$ is 8 (from Proposition \[prop-bounds-torsion-period-length\]). This implies that $\CF(\sqrt{D})$ cannot be periodic, using the arguments from Remark \[rem-periodicity-test-reduction-two-primes\] in Section \[sec:org36b2a71\] with reduction modulo two primes.
### Genus 2 {#sec:org0b837d5}
We also give an example of degree $6$, to illustrate the difficulties arising in higher genus, and the more complicated patterns of the valuations in this case. Moreover, we will find convergents where $\Redn{p_n}$ and $\Redn{q_n}$ share a common linear factor.
[|l|l|]{} $D = X^{6} + 7 X^{4} + 8 X^{3} + 9 X^{2} + 5$ & basefield $\Q$\
Discriminant of $D$: $-1 \cdot 2^{10} \cdot 5 \cdot 7^{2} \cdot 353^{2}$ & $D$ never reduces to a square.\
$D$ is not Pellian &\
\
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\
\
\
Example \[ex-cfp2-zero-pattern-deg6\] is again a random non-periodic example. See how the coefficient size explodes worse than in the genus 1 example. And observe that the prime $13$ appears already in $a_1$. However, it turns out that $\CF(\sqrt{D_{13}})$ has a quite long quasi-period length: it is $126$.
So unlike in genus $1$, the first occurrence of a prime $\pp$ in a denominator of the $a_n$ does not give so much information on the quasi-period length of $\CF(\sqrt{D_\pp})$.
#### modulo 3 {#sec:org2ff323f}
$n$ $m$ $\deg a_n$ $\deg c_m$ $\deg q_n$ $\deg \Redn{q_n}$ $\deg v_m$
----- ----- ------------ ------------ ------------ ------------------- ------------
0 0 3 3 0 0 0
1 1 1 1 1 1 1
2 2 1 1 2 2 2
3 2 1 1 3 2 2
4 3 1 2 4 4 4
5 4 1 1 5 5 5
6 5 1 1 6 6 6
7 5 1 1 7 7 6
8 5 1 1 8 6 6
9 6 1 3 9 9 9
10 7 1 1 10 10 10
11 8 1 1 11 11 11
12 8 1 1 12 11 11
13 9 1 2 13 13 13
14 10 1 1 14 14 14
15 11 1 1 15 15 15
16 11 1 1 16 16 15
17 11 1 1 17 15 15
18 12 1 3 18 18 18
19 13 1 1 19 19 19
20 14 1 1 20 20 20
21 14 1 1 21 20 20
22 15 1 2 22 22 22
23 16 1 1 23 23 23
24 17 1 1 24 24 24
25 17 1 1 25 25 24
26 17 1 1 26 24 24
27 18 1 3 27 27 27
28 19 1 1 28 28 28
: \[cf2-mod3-degrees-table\] Degrees for reduction mod 3 in Example \[ex-cfp2-zero-pattern-deg6\]
In Table \[cf2-mod3-degrees-table\], we compare the degrees of partial quotients between $\CF(\sqrt{D})$ and its reduction $\CF(\sqrt{D_3})$. We put $m = \lambda(n)$, and be aware that the columns depending on $m$ contain *repeated entries*.
Note particularly that the sequence of the $\deg \Redn{q_n}$ is also decreasing, and sometimes is larger than the corresponding $\deg v_m$. This means that $\Redn{p_n}, \Redn{q_n}$ have a common linear factor. This of course happens here only in the 3-element fibres of $\lambda$ (in the table $m = \lambda(n)$). For example $$\begin{aligned}
\Redn{p_7} &= (X + 1)\cdot(2 X^{9} + 2 X^{8} + X^{7} + 2 X^{6} + 2 X^{5} + 2 X^{4} + 2),\\ \Redn{q_7} &= (X + 1)\cdot(2 X^{6} + 2 X^{5} + 2 X^{3} + X^{2}).\end{aligned}$$
$n$ $\lambda(n)$ $\nu(\alpha_n)$ $\nu(a_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
----- -------------- ----------------- ------------ ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 1 0 0 0 0 0
2 2 0 0 0 0 0
3 2 $-\infty$ -2 -1 -2 -1
4 3 2 2 3 2 2
5 4 -4 -4 -4 -2 -2
6 5 4 4 4 2 2
7 5 $-\infty$ -5 -5 -3 -3
8 5 5 6 6 2 3
9 6 -5 -5 -5 -2 -2
10 7 4 4 4 2 2
11 8 -4 -4 -4 -2 -2
12 8 $-\infty$ 0 2 -2 0
13 9 0 0 2 2 2
14 10 -4 -4 -4 -2 -2
15 11 4 4 4 2 2
16 11 $-\infty$ -5 -5 -3 -3
17 11 5 6 6 2 3
18 12 -5 -5 -5 -2 -2
19 13 4 4 4 2 2
20 14 -4 -4 -4 -2 -2
21 14 $-\infty$ 2 3 0 1
22 15 -2 -2 -1 0 0
23 16 0 0 0 0 0
24 17 0 0 0 0 0
25 17 $-\infty$ -2 -2 -2 -2
26 17 2 4 4 0 2
27 18 -2 -2 -2 0 0
28 19 0 0 0 0 0
: \[cf2-mod3-valuations-table\] 3-adic valuations for Example \[ex-cfp2-zero-pattern-deg6\]
The patterns for the valuations in Table \[cf2-mod3-valuations-table\] are now more interesting, as there are fibres of $\lambda$ with $2$ or $3$ elements. But at least these are still isolated. Observe the differences between the valuation of the entire polynomial (the Gauss norm) and of the leading coefficient between 2-element fibres and 3-element fibres. Note that now odd valuations are occurring.
Also, we cannot read off the quasi-period length of $\CF(\sqrt{D_3})$ just by counting the rows with only zero valuations. From Table \[cf2-mod3-degrees-table\], we know that it is actually $6$, not $3$ (by looking for $c_m$ of degree $3$ which first occurs for $m = 6$). This corresponds to torsion order $9 = \deg p_{5}$ of $\OO_{3}$ (via Theorem \[thm-pellian-iff-torsion\] and Remark \[convergent-omega-degree\]).
#### modulo 19 {#sec:org5ccc0d4}
Another interesting prime would be $19$. There $\CF(\sqrt{D_{19}})$ has (quasi-)period length $6$, with degrees of the $a_n$ having the periodic pattern $\deg a_n = \overline{3, 1, 1, 1, 1, 1}.$
This degree pattern implies that $\lambda$ has only fibres with $1$ or $3$ elements.
$n$ $\lambda(n)$ $\nu(\alpha_n)$ $\nu(a_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
----- -------------- ----------------- ------------ ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 1 0 0 0 0 0
2 2 0 0 0 0 0
3 3 0 0 0 0 0
4 4 0 0 0 0 0
5 5 0 0 0 0 0
6 5 $-\infty$ -1 -1 -1 -1
7 5 1 2 2 0 1
8 6 -1 -1 -1 0 0
9 7 0 0 0 0 0
10 8 0 0 0 0 0
11 9 0 0 0 0 0
12 10 0 0 0 0 0
13 11 0 0 0 0 0
14 11 $-\infty$ -1 -1 -1 -1
15 11 1 2 2 0 1
16 12 -1 -1 -1 0 0
17 13 0 0 0 0 0
18 14 0 0 0 0 0
19 15 0 0 0 0 0
20 16 0 0 0 0 0
21 17 0 0 0 0 0
22 17 $-\infty$ -1 -1 -1 -1
23 17 1 2 2 0 1
24 18 -1 -1 -1 0 0
25 19 0 0 0 0 0
26 20 0 0 0 0 0
27 21 0 0 0 0 0
28 22 0 0 0 0 0
29 23 0 0 0 0 0
30 23 $-\infty$ -1 -1 -1 -1
31 23 1 2 2 0 1
32 24 -1 -1 -1 0 0
33 25 0 0 0 0 0
34 26 0 0 0 0 0
35 27 0 0 0 0 0
: \[cf2-mod19-valuations-table\] 19-adic valuations for Example \[ex-cfp2-zero-pattern-deg6\]
In Table \[cf2-mod19-valuations-table\], note how the valuations are reset to $0$ after the 3-element fibres. This illustrates nicely why we require infinitely many fibres of $\lambda$ with multiple elements in Proposition \[fibre-conditions-infinite-poles\].
#### modulo 5 {#sec:org0144a9b}
So far, the regularity of these valuation patterns for $\deg D = 6$ has been deceiving, so let us look at the 5-adic valuations too. The quasi-period length of $\CF(\sqrt{D_5})$ is just $6$. The partial quotients period is $\deg a_n = \overline{3, 1, 1, 2, 1, 1}.$
So compared to $\pp = 3$, there are also 2-element fibres. This makes the patterns much more complicated, as seen in Table \[cf2-mod5-valuations-table\] (and in other examples, this might be even worse).
$n$ $\lambda(n)$ $\nu(\alpha_n)$ $\nu(a_n)$ $\nu(\LC(a_n))$ $\nu(q_n)$ $\nu(\LC(q_n))$
----- -------------- ----------------- ------------ ----------------- ------------ -----------------
0 0 0 0 0 0 0
1 1 0 0 0 0 0
2 2 0 0 0 0 0
3 2 $-\infty$ -6 -3 -6 -3
4 3 6 6 9 6 6
5 4 -12 -12 -12 -6 -6
6 5 12 12 12 6 6
7 5 $-\infty$ -13 -13 -7 -7
8 5 13 14 14 6 7
9 6 -13 -13 -13 -6 -6
10 7 12 12 12 6 6
11 8 -12 -12 -12 -6 -6
12 8 $-\infty$ 6 9 0 3
13 9 -6 -6 -3 0 0
14 10 0 0 0 0 0
15 11 0 0 0 0 0
16 11 $-\infty$ -1 -1 -1 -1
17 11 1 2 2 0 1
18 12 -1 -1 -1 0 0
19 13 0 0 0 0 0
20 14 0 0 0 0 0
21 14 $-\infty$ -8 -4 -8 -4
22 15 8 8 12 8 8
23 16 -16 -16 -16 -8 -8
24 17 16 16 16 8 8
25 17 $-\infty$ -17 -17 -9 -9
26 17 17 18 18 8 9
27 18 -17 -17 -17 -8 -8
28 19 16 16 16 8 8
29 20 -16 -16 -16 -8 -8
30 20 $-\infty$ 10 13 2 5
31 21 -10 -10 -7 -2 -2
32 22 4 4 4 2 2
33 23 -4 -4 -4 -2 -2
34 23 $-\infty$ 3 3 1 1
35 23 -3 -2 -2 -2 -1
36 24 3 3 3 2 2
37 25 -4 -4 -4 -2 -2
: \[cf2-mod5-valuations-table\] 5-adic valuations for Example \[ex-cfp2-zero-pattern-deg6\]
Non-constant degrees of partial quotients {#sec:orge7e06d0}
-----------------------------------------
The following example was constructed in collaboration with Prof. Zannier and Francesca Malagoli, to answer a question raised during preparation of [@zannier-2016-hyper-contin-fract]: In the article, it is a consequence of the Skolem-Mahler-Lech Theorem for algebraic groups (mentioned before) that the sequence of the $\deg a_n$ (for $\alpha = \sqrt{D}$) becomes eventually periodic. However, in any non-periodic examples known previously, these degrees stabilised on a single value. Of course, in that case periodicity of the degrees is not very interesting.
So we searched for an non-periodic example where the degrees assume multiple values infinitely often.
We found Example \[ex-dnc1-non-constant-deg\] which has infinitely many partial quotients $a_n$ both of degree $1$ and of degree $2$ (we remark that this would be impossible for $\deg D = 4$ or $6$, so we cannot do better than $\deg D = 8$).
[|p[9cm]{}|p[5cm]{}|]{} $D = X^{8} - X^{7} - \frac{3}{4} X^{6} + \frac{7}{2} X^{5} - \frac{21}{4} X^{4} + \frac{7}{2} X^{3} - \frac{3}{4} X^{2} - X + 1$ & basefield $\Q$\
Discriminant of $D$: $-1 \cdot 2^{2} \cdot 3 \cdot 13 \cdot 173^{2}$ & Primes in denominators of $D$: $2$\
$D$ never reduces to a square. &\
$D$ is not Pellian because of incompatible torsion orders & torsion order $10$ modulo $3$\
& torsion order $40$ modulo $11$\
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This is related to the fact that the Jacobian in Example \[ex-dnc1-non-constant-deg\] is not simple. It contains an elliptic curve, and infinitely many multiples of the point $\OO$ lie on a certain translate of it. This causes the degrees of the $a_n$ to follow the pattern $4, 1, 1, \overline{2, 1, 1, 1, 1, 1, 1, 1, 1}$. For details, we refer to an article in preparation together with Malagoli and Zannier.
Here we remark only that if we reduce modulo 3, we actually get a square-free polynomial in $(X+1)^2$ (and divisible by $(X+1)^2$ too, hence the $3$ in the discriminant). So all the partial quotients have at least degree $2$ (see also Table \[cfnc1-mod3-degrees-table\]).
$n$ $m$ $\deg a_n$ $\deg c_m$ $\deg q_n$ $\deg \Redn{q_n}$ $\deg v_m$
----- ----- ------------ ------------ ------------ ------------------- ------------
0 0 4 4 0 0 0
1 0 1 4 1 0 0
2 1 1 2 2 2 2
3 2 2 2 4 4 4
4 2 1 2 5 4 4
5 3 1 2 6 6 6
6 3 1 2 7 7 6
7 3 1 2 8 7 6
8 3 1 2 9 6 6
9 4 1 4 10 10 10
10 4 1 4 11 10 10
11 5 1 2 12 12 12
12 6 2 2 14 14 14
13 6 1 2 15 14 14
14 7 1 2 16 16 16
15 7 1 2 17 17 16
16 7 1 2 18 17 16
17 7 1 2 19 16 16
18 8 1 4 20 20 20
19 8 1 4 21 20 20
20 9 1 2 22 22 22
21 10 2 2 24 24 24
22 10 1 2 25 24 24
23 11 1 2 26 26 26
24 11 1 2 27 27 26
25 11 1 2 28 27 26
26 11 1 2 29 26 26
: \[cfnc1-mod3-degrees-table\] Degrees modulo 3 for Example \[ex-dnc1-non-constant-deg\]
Note that $\lambda$ has still infinitely many fibres with a single element. Observe the sequence $\deg \Redn{q_n}$ is sometimes decreasing, so there are again convergents with a common factor between $\Redn{p_n}$ and $\Redn{q_n}$.
Recurring partial quotients {#sec:orge27ae2f}
---------------------------
Recall that [@zannier-2016-hyper-contin-fract] gave a lower bound for an average of the affine heights of partial quotients (see the end of Section \[sec:org1835e85\]). A strengthening of this would be a bound like $$C \, n^2 \leq \hproj(a_n)$$ for the non-Pellian case.
However, together with Prof. Zannier and Francesca Malagoli, and some assistance from Solomon Vishkautsan for the computations, we have found Example \[ex-dbh1-bounded-height\] below. There for $n = 7 + 17 \, j \pm 1, \; j \in \N_0$ the partial quotients have the shape $$a_n = C_n \, (X-2), \quad C_n \in \closure{\Q}$$ so in particular $\hproj(a_n)$ remains constant on this subsequence and the above lower bound is *impossible* in general.
[|p[9cm]{}|p[5cm]{}|]{} $D = X^{12} + (-8 \tau^{4} + 6 \tau^{3} - 28 \tau^{2} + 22 \tau + 22) X^{10} + (-8 \tau^{4} + 6 \tau^{3} - 28 \tau^{2} + 22 \tau + 22) X^{9} + (83 \tau^{4} - 62 \tau^{3} + 291 \tau^{2} - 225 \tau - 309) X^{8} + (166 \tau^{4} - 124 \tau^{3} + 582 \tau^{2} - 450 \tau - 618) X^{7} + (-127 \tau^{4} + 92 \tau^{3} - 447 \tau^{2} + 327 \tau + 529) X^{6} + (-630 \tau^{4} + 462 \tau^{3} - 2214 \tau^{2} + 1656 \tau + 2514) X^{5} + (-538 \tau^{4} + 398 \tau^{3} - 1893 \tau^{2} + 1434 \tau + 2115) X^{4} + (158 \tau^{4} - 102 \tau^{3} + 546 \tau^{2} - 336 \tau - 758) X^{3} + (552 \tau^{4} - 384 \tau^{3} + 1926 \tau^{2} - 1332 \tau - 2394) X^{2} + (368 \tau^{4} - 256 \tau^{3} + 1284 \tau^{2} - 888 \tau - 1596) X + 92 \tau^{4} - 64 \tau^{3} + 321 \tau^{2} - 222 \tau - 399$ & basefield $K = \Q(\tau)$, where $\tau$ has minimal polynomial $t^{5} + 3 t^{3} - 6 t - 3$\
& $D$ never reduces to a square.\
$D$ is not Pellian because of incompatible torsion orders & torsion order $42$ modulo $\tau$\
& torsion order $861$ modulo $3 \tau^{4} - 2 \tau^{3} + 11 \tau^{2} - 8 \tau - 11$\
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\
This also gives an example where $\deg a_n$ assumes three different values infinitely often, and again this is related to the Jacobian containing an elliptic curve. We hope to describe this example in much more detail in the article in preparation together with Malagoli and Zannier mentioned above.
Appendix {#sec:orgbd3afba}
========
Polynomial Pell equation in characteristic 2 {#sec:orgfe2a066}
--------------------------------------------
Let us quickly have a look at the polynomial Pell equation in characteristic $2$ and give a criterion which allows to easily test for and construct solutions in this case.
Let $\K$ a field of characteristic $2$ and $D \in \K[X]$. There exists a non-trivial solution (with $q \neq 0$) of $$p^2 - D \, q^2 = \eta, \qquad p, q \in \K[X], \eta \in \units \K$$ if and only if there exist $E \in \K[X], r \in \K$ such that $D = E^2 + r$.
Moreover, $r = 0$ is possible if and only if there exists a non-trivial solution with $\eta$ a *square*.
Let us first treat the second case $r = 0$. Suppose $D = E^2$, and choose $\mu \in \units\K$, $p = E-\mu, \; q = 1$. This yields $$p^2 - D \, q^2 = (E-\mu)^2 - E^2 = \mu^2 = \eta,$$ hence $\eta$ can be chosen a square.
On the other hand, suppose $(p,q) \in \K[X]^2$ with $q \neq 0$ is a solution with $\eta = \mu^2$ a square, then $$D \, q^2 = p^2 - \mu^2 = (p - \mu)^2$$ implies $D$ is a square because $\K[X]$ is a unique factorisation domain.
For the general case, note that if $D = E^2 + r$ with $r \neq 0$, then $$p = E, q = 1, \eta = r \implies p^2 - D \, q^2 = -r = \eta$$ gives the desired non-trivial solution.
Conversely, if there exists with a solution $(p,q) \in \K[X]^2$ with $q \neq 0$, set $K = \K(\sqrt{\eta})$ and reduce to the case with $r = 0$ – we now write $D = E^2$ with $E \in K[X]$, or rather $E = E_0 + \mu \, E_1$ with $E_0, E_1 \in \K[X]$ (here again $\mu = \sqrt{\eta}$). We obtain $$D = E^2 = E_0^2 + \mu^2 \, E_1^2 = E_0^2 + \eta \, E_1^2$$ and plugging it into the Pell equation we have $$0 = p^2 - q^2 \, \left(E_0^2 - \eta \, E_1^2\right) + \eta = (p - q \, E_0)^2 - \eta \, (q \, E_1 +1)^2$$ If $(q \, E_1 + 1) \neq 0$, then $\mu = \ifracBB{p-q\,E_0}{q \, E_1 + 1}\in K \cap \K(X) = \K$ and we are actually in the first case. Otherwise, $q \, E_1 = 1$, so $E_1 \in \units \K$ (because $q \in \K[X]$), hence $D = E_0^2 + \eta \, E_1^2 = E_0^2 + r$ with $r = \eta \, E_1^2 \in \K$.
So if we require $\eta = 1$, we see that in characteristic $2$ non-trivial solutions to the Pell equations only exist if $D$ is actually a square.
The proof also yields a classification of the Pell solutions:
For the first case with $D = E^2$, the solutions always have the shape $p = q \, E - \mu$. And obviously, we are free to choose $q$ here, so there are a lot of non-trivial solutions in this case.
In the second case with $D = E^2 + r$, we need to expand this observation. But note that we actually showed $q \in \units \K$ in the above proof, so essentially $q = 1$ after multiplying $\eta$ with a square factor. Hence there is *only one* non-trivial solution up to a constant factor.
Valuations in Laurent series quotients {#sec:org5f19ed2}
--------------------------------------
The problem that arises with bad reduction is that we can no longer read off $\nu(\alpha_n)$ from the leading coefficient. This also means that $\nu(a_n)$ could be different, so we need to compute the valuations of the coefficients of $\alpha_n$. As the latter can be written as a quotient of $\vartheta_i$’s, we naturally need to study quotients of Laurent series.
Indeed we may work with quotients of power series, as multiplying with powers of $X$ only shifts coefficient indices. For convenience, we work in $\powerseries{K}{Z}$ (think $Z = \inv X$) to avoid negative indices.
As in Chapters \[sec:orgd5f1900\] and \[sec:org5f9d2ce\], $K$ is the fraction field of a discrete valuation ring $\O$ with maximal ideal $\mm$ and valuation $\nu$.
Let $a_n, c_n \in \O, b_n \in K$, and consider the Cauchy product $$\left(\sum_{n=0}^\infty a_n \, Z^n\right) \left( \sum_{n=0}^\infty b_n \, Z^n\right) = \sum_{n=0}^\infty c_n \, Z^n.$$
For the coefficients, we get the relations $$c_n = \sum_{i+j=n} a_i \, b_j$$ which we can recursively solve to $b_n$ as $$\label{series-inverse-rec-formula}
b_n = \frac{1}{a_0} \left( c_n - \sum_{i+j=n,\atop i\neq 0} a_i \, b_j \right).$$ For the first couple of indices, we compute $$\begin{aligned}
b_0 &= \frac{c_0}{a_0} \\
b_1 &= \frac{1}{a_0^2} \, \left(a_0 \, c_1 - a_1 \, c_0 \right) \\
b_2 &= \frac{1}{a_0^3} \, \left(a_{1}^{2} c_{0} - a_{0} a_{2} c_{0} - a_{0} a_{1} c_{1} + a_{0}^{2} c_{2} \right) % \\
% &\vdots\end{aligned}$$ So we can try to calculate or estimate the valuations of the coefficient with these formulas.
The following Lemma addresses the simplest case (sufficient to treat $\deg D = 4$).
\[cauchy-valuation-lemma\]
- Suppose $\nu(c_0) > 0$, but $\nu(c_1) = \nu(a_0) = 0$. Then $\nu(b_0) = \nu(c_0) > 0$ and $\nu(b_1) = 0$.
- Suppose $\nu(a_0) > 0$ and $\nu(c_0) = \nu(a_1) = 0$. Then $\nu(b_0) = - \nu(a_0) < 0$ and $\nu(b_1) = -2 \, \nu(a_0) < 0$.
The valuation of $b_0$ is obvious. In the first situation, we deduce from $\nu(c_0) > 0$ and $\nu(a_1) \geq 0$ $$\nu(b_1) = \nu(c_1 \, a_0 - c_0 \, a_1) = \min(0, \nu(c_0) + \nu(a_1)) = 0.$$
In the second situation, $\nu(c_1) \geq 0, \nu(a_0) > 0$ implies $$\nu(b_1) = - 2\, \nu(a_0) + \nu(c_1 \, a_0 - c_0 \, a_1) = - 2 \, \nu(a_0) + \min(\nu(c_1) + \nu(a_0), 0) = -2 \, \nu(a_0).$$
We can actually generalise this somewhat, but first we need a better description of the formulas for the $b_n$:
Define $B_n = - (-a_0)^{n+1} \, b_n$. Then we find $$B_n = \sum_{i_0 + \dots + i_l = n\atop 0\leq i_0 \leq n, 1 \leq i_1, \dots, i_l \leq n} c_{i_0} \, a_{i_1} \cdots a_{i_l} \, (-a_0)^{n-l}.$$
Essentially, we are summing over integer partitions of $n$ with (at most) $n+1$ parts. However, except for the parts which are $0$, the ordering of the parts matters.
We prove this by induction. Clearly $B_0 = c_0$, precisely what the formula produces as no $a_i$ appears in the sum.
For the induction step, we use the recursion formula $$\begin{gathered}
B_n = (-a_0)^n \, c_n + \sum_{i+j=n,\atop i \neq 0} a_i \, (-a_0)^{i-1} \, B_j \\
= (-a_0)^n \, c_n + \sum_{i+j=n,\atop i \neq 0} a_i \, (-a_0)^{i-1} \, \sum_{i_0 + \dots + i_l = l\atop 0\leq i_0 \leq j, 1 \leq i_1, \dots, i_l \leq j} c_{i_0} \, a_{i_1} \cdots a_{i_l} \, (-a_0)^{j-l} \\
= \sum_{i_0 + \dots + i_l + i = n\atop 0\leq i_0 \leq n, 1 \leq i_1, \dots, i_l, i\leq n} c_{i_0} \, a_{i_1} \cdots a_{i_l} \, a_{i} \, (-a_0)^{n-l-1}.\end{gathered}$$ Essentially, we are recursing by fixing the last (or first) $a_i$.
We can now generalise the second part of Lemma \[cauchy-valuation-lemma\]:
\[inverse-drop1-valuation-lemma\] If $c_0, a_1 \in \units \O$ and $a_0 \in \mm$, then for all $n \geq 0$ we have $B_n \in \units \O$. This implies $\nu(b_n) = -(n+1) \, \nu(a_0)$.
It is clear that $B_n \in \O$, as all the summands are in $\O$ (recall that $a_i, c_i \in \O$). We show that precisely one summand lies in $\units \O$, while all others are in $\mm$.
Of course, with $i_0 = 0$ and $i_j = 1$ for the rest, we get $c_0 \, a_1^n \in \units \O$.
For all other summands, we show that $l < n$ which implies that $a_0$ appears in $c_{i_0} \, a_{i_1} \cdots a_{i_l} \, (-a_0)^{n-l}$, so the product is in $\mm$.
If still $i_0 = 0$, but one of the $i_j \neq 1$, i.e. $i_j \geq 2$, then clearly $l < i_1 + \dots + i_l = n$.
If on the other hand $i_0 > 0$, then immediately $l \leq i_1 + \dots + i_l < n$.
A lemma for a quadratic form {#sec:orgab72b2f}
----------------------------
Let $G$ a $\Z$-module (an abelian group) and $q: G \to \R$ a quadratic form. By abuse of notation, we also denote the corresponding $\Z$-bilinear form by $q : G \times G \to \R$.
\[quadratic-form-sum-bound\] Suppose that $q$ is positive (i.e. $q(g) \geq 0$ for all $g \in G$). Let $g_1, \dots, g_r \in G$. Then $$\label{eq-quadratic-form-sum-bound}
q(g_1 + \dots + g_r) \leq r \cdot \left( q(g_1) + \dots + q(g_r) \right) \leq r^2 \, \max\{q(g_i) \mid i=1,\dots,r \}.$$
Because $q$ is a quadratic form, we have $$q(g_1 + \dots + g_r) = \sum_{i=1}^r q(g_i) + 2 \, \sum_{1 \leq i < j \leq r} q(g_i, g_j).$$ Moreover $q$ positive implies that $$0 \leq q(g_i - g_j) = q(g_i) + q(g_j) - 2 \, q(g_i, g_j)$$ so we deduce $$q(g_1 + \dots + g_r) \leq \sum_{i=1}^r q(g_i) + 2 \, \sum_{1 \leq i < j \leq r} q(g_i) + q(g_j) = \sum_{1\leq i, j \leq r} q(g_i) = r \, \sum_{i=1}^r q(g_i).$$ The second inequality in is then obvious.
[^1]: The situation in characteristic $2$ is however completely different, see Section \[sec:orgfe2a066\] in the Appendix.
[^2]: This group is a twisted $\G_m$. We can see $D(X) \, Q^2 = P^2 - 1$ as a twist of $Q^2 = P^2 - 1$ by the (hyper)elliptic curve $Y^2 = D(X)$, via $(P, Q) \mapsto (P, Y \, Q)$. Of course $Q^2 = P^2 - 1$ written as $P^2 - Q^2 = 1$ is isomorphic to $\G_m$. See [@hazama-1997-pell-equations-polynomials] for more details.
[^3]: The polynomial case is even simpler than the integer case treated there: because the absolute value (corresponding to the valuation $\ios$) is non-archimedean, there are no intermediate fractions to worry about.
[^4]: Or the shape of a Hankel matrix if we reverse the ordering of the columns.
[^5]: Here we profit already from allowing common factors for best-approximations.
[^6]: If $D$ is not square-free, we have to use generalised Jacobians instead. See [@zannier-2016-hyper-contin-fract] on how generalised Jacobians relate to the Pell equation and continued fractions, and [@serre-1988-algebraic-groups-class] for an introduction to generalised Jacobians.
[^7]: Note that the case $g=0$ can easily be treated using Corollary \[deg-2-always-pellian\]. See also Section \[sec:org46de0ba\].
[^8]: In the case $\alpha = \sqrt{D}$ the reduction $\gamma = \sqrt{\Red{D}}$ clearly is rational $\Red{D}$ is a square.
[^9]: Note that van der Poorten speaks of good reduction only for the hyperelliptic curve, not for the continued fraction.
[^10]: Van der Poorten does not explicitly define the map $\lambda$ as we do it here.
[^11]: See for example [@neukirch-1999-algebraic-number-theory], Chapter I §8. Or any other decent textbook on algebraic number theory.
[^12]: Instead of $\Spec \O$, we could also work with any Dedekind scheme, for example $\Spec \Z$ if $D \in \Z[X]$. We stick to the local case for simplicity and consistency of notation.
[^13]: The reader might find it enjoyable to try and prove this exercise for himself.
[^14]: It is not quite clear if there is a similar bound in the other direction.
|
---
abstract: 'The ability to track a moving vehicle is of crucial importance in numerous applications. The task has often been approached by the importance sampling technique of particle filters due to its ability to model non-linear and non-Gaussian dynamics, of which a vehicle travelling on a road network is a good example. Particle filters perform poorly when observations are highly informative. In this paper, we address this problem by proposing particle filters that sample around the most recent observation. The proposal leads to an order of magnitude improvement in accuracy and efficiency over conventional particle filters, especially when observations are infrequent but low-noise.'
author:
- |
Kira Kempinska [email protected]\
Department of Security and Crime Science\
University College London\
London, WC1E 6BT John Shawe-Taylor [email protected]\
Department of Computer Science\
University College London\
London, WC1E 6BT\
bibliography:
- 'references.bib'
title: Improved Particle Filters for Vehicle Localisation
---
Introduction
============
Tracking a moving vehicle is a central and difficult problem arising in different contexts ranging from military applications to robotics [@Thrun2002; @Gordon2002]. It consists of computing the best estimate of the vehicle’s trajectory based on noisy sensor measurements. In this paper, we are interested in vehicle tracking when the road network is known.
Several strategies have been developed to track a vehicle on a road network [@Lou2009; @Chawathe2007; @Wenk2006; @Alt2003; @HuabeiYin; @Pink2008]. We focus on the particle filter method [@Gordon1993]. The method has had numerous successes in this area due to its flexibility to handle cases where the dynamic and observation models are non-linear and/or non-Gaussian. It is an importance sampling technique that approximates the target distribution by sampling from a series of intermediate proposal distributions.
Critically, in common with any important sampling method, the performance of particle filters is strongly dependent on the choice of the proposal distribution. If the proposal is not well matched to the target distribution, then the method produces samples that have low effective sample size and, as a result, it requires a prohibitively large number of particles to represent the target distribution accurately. The problem typically arises under highly informative observation regimes, in which the current observation provides significant information about the current state but the state dynamics are weak.
The particle filter community has developed various approaches to mitigate the deficiency. One approach attaches a post-sampling step that moves particles sampled from the proposal distribution towards the target distribution using Markov Chain Monte Carlo moves [@Andrieu2010; @Fox2001; @VanDerMerwe2000] or by solving partial differential equations [@Li2016; @Daum2010; @Daum2007; @Khan2014]. An alternative approach improves the proposal distribution by giving it additional information about the current [@Montemerlo2007; @Kong1994; @Liu1998] or even future observations [@Lin2013] or their approximations [@Pitt1999]. Several authors considered conditioning the proposal distribution on the current observation only [@Lin2005; @Fox2001]. The approaches successfully increased the effective sample size, but, often at the cost of high computational complexity or analytical intractability. The construction of good, but also computationally efficient proposal distributions is still an open research question.
In this paper, we propose an improved particle sampling scheme that is both computationally efficient and mathematically robust. The approach generates proposals based on the current sensor observation only, leading to good alignment between the proposal and the target distribution even with a small sample size. It converges to the desired target distribution at faster rates than standard particle filters, especially when observations are highly informative, e.g. infrequent but low-noise. It is easy to implement and avoids the computational and analytical complexity of the discussed alternatives with other proposal distributions or post-sampling moves. It also presents a simpler approach to sample weighing than those previously proposed with the same proposal distribution [@Lin2005; @Fox2001].
The paper is structured as follows. We present the problem statement in Section \[problem\_statement\], followed by a description of the standard particle filters in Section \[particle\_filters\]. We introduce the proposed particle filters method in Section \[improved\_particle\_filters\]. We outline the application of the method to vehicle tracking in Section \[application\] and present results in Section \[results\]. We conclude by summarising the paper’s contributions in Section \[conclusions\].
Problem statement {#problem_statement}
=================
The key idea of particle filters is to estimate the marginal posterior distribution $p(x_{t} \mid z_{0:t})$ where $x_{t}$ is the state of the system at time $t$ and $z_{0:t} = \{z_0,\ldots,z_t\}$ is a sequence of measurements collected up to time step $t$. We call the posterior the *belief* and use the following notation
$$Bel(x_t)=p(x_{t} \mid z_{0:t})
\label{eq:bel1}$$
In the context of vehicle tracking, the belief is our estimate of the vehicle position at time $t$ given all measurements collected until then. The measurements include *GPS readings* and *controls*, which carry information about vehicle motion between consecutive timestamps. Denoting a GPS reading at time $t$ by $y_t$ and a control in the time interval $(t-1;t]$ by $u_{t-1}$, we have
$$Bel(x_t) = p(x_{t} \mid y_{0:t},u_{0:t-1})
\label{eq:bel2}$$
Particle filters estimate $Bel(x_t)$ recursively. In order to arrive at a recursive equation, we note we can use Bayes rule to decompose Equation \[eq:bel2\] to
$$Bel(x_t)=\frac{p(y_t \mid x_t,y_{0:t-1},u_{0:t-1})p(x_t \mid y_{0:t-1},u_{0:t-1})}{p(y_t \mid y_{0:t-1},u_{0:t-1})}
\label{eq:bel3}$$
The underlying assumption of particle filters is that the system follows the *Markov assumption*, that is, measurements $y_t$ are conditionally independent of past measurements and controls given knowledge of the state $x_t$:
$$p(y_t \mid x_t,y_{0:t-1},u_{0:t-1}) = p(y_t \mid x_t)$$
This conveniently simplifies Equation \[eq:bel3\] to
$$Bel(x_t)=\frac{p(y_t \mid x_t)p(x_t \mid y_{0:t-1},u_{0:t-1})}{p(y_t \mid y_{0:t-1},u_{0:t-1})}$$
We integrate out the position at $x_{t-1}$ in order to arrive at the following recursive form
$$Bel(x_t)=\frac{p(y_t \mid x_t)}{p(y_t \mid y_{0:t-1},u_{0:t-1})}\int p(x_t \mid x_{t-1},y_{0:t-1},u_{0:t-1})p(x_{t-1} \mid y_{0:t-1},u_{0:t-1}) dx_{t-1}$$
which can be simplified again using *Markov assumption* by noting that:
$$p(x_t \mid x_{t-1},y_{0:t-1},u_{0:t-1})=p(x_t \mid x_{t-1},u_{t-1})$$
Finally, we arrive at a recursive estimator known as *Bayes filter*:
$$\begin{aligned}
\begin{split}
Bel(x_t)= & \frac{p(y_t \mid x_t)}{p(y_t \mid y_{0:t-1},u_{0:t-1})}\int p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1}) dx_{t-1} \\
= & \,\eta\,\,p(y_t \mid x_t)\int p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1}) dx_{t-1}
\end{split}
\label{eq:bf}\end{aligned}$$
where $\eta$ is a normalising constant. The *Bayes filter* equation is the basis for particle filters and the improved particle filters that we propose in this paper.
Particle Filters {#particle_filters}
================
Particle filters approximate the belief $Bel(x)$ by a set of $m$ weighted samples distributed according to $Bel(x)$:
$$Bel(x)=\{x^{(i)},w^{(i)}\}_{i=1,\ldots,m}$$
where each $x^{(i)}$ is a sample (a state) and $w^{(i)}$ are non-negative weights called *importance factors* that determine the importance of each sample.
The particle filters method operates recursively. It begins by generating $m$ samples $x_0^{(i)}$ from the initialisation distribution $Bel(x_0) = p(x_0)$ and annotates them by the uniform importance factor $1/m$. Subsequently, it estimates $Bel(x_t)$ at any future timestamp $t$ by performing a three-step recursive update, computing the expression in Equation \[eq:bf\] *from the right to the left*.
for $k=1,\ldots,m:$
1. Sample a state $x_{t-1}$ by drawing a random $x_{t-1}^{(i)}$ from the sample set representing $Bel(x_{t-1})$ according to the distribution defined through the importance factors $w_{t-1}^{(i)}$.
2. Use the sample $x_{t-1}^{(i)}$ and the control $u_{t-1}$ to generate a sample $x_{t}^{(j)}$ according to the so-called *transition probability* $p(x_t \mid x_{t-1},u_{t-1})$.
3. Finally, use the observation $y_t$ to weigh the sample $x_{t}^{(j)}$ by the non-normalized importance factor given by the so-called *observation probability* $p(y_t \mid x_{t}^{(j)})$, the likelihood of the sample $x_{t}^{(j)}$ given the observation $y_t$.
Further below, it will be important to notice that the particle filters method is, in fact, an importance sampling scheme. It approximates $Bel(x_t)$ using a proposal distribution given by
$$Q = p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})
\label{eq:q_pf}$$
The proposal approximates the desired posterior
$$P = \frac{p(y_t \mid x_t) p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})}{p(y_t \mid y_{0:t-1},u_{0:t-1})}$$
Consequently, the importance factors are given by the quotient
$$\begin{aligned}
\begin{split}
\frac{P}{Q}= & \,\,[p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})]^{-1}\frac{p(y_t \mid x_t) p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})}{p(y_t \mid y_{0:t-1},u_{0:t-1})} \\
\propto & \,\,p(y_t \mid x_t)
\end{split}
\label{eq:if_pf}\end{aligned}$$
Improved Particle Filters {#improved_particle_filters}
=========================
We propose an improved particle sampling scheme in which $x_t$ are sampled directly around the most recent observation $y_t$ according to the proposal distribution:
$$Q_{new}= \frac{p(y_t \mid x_t)}{\pi(y_t)} \qquad\text{with}\qquad \pi(y_t )=\int p(y_t \mid x_t)dx_t
\label{eq:q_new}$$
This new proposal distribution possesses orthogonal strengths to the one in Equation \[eq:q\_pf\], in that it generates samples that are highly consistent with the most recent sensor measurement but ignorant of past measurements and controls. As such, we expect it to outperform conventional particle filters in systems where the current observation provides more information about the current state than the underlying state dynamics.
The importance factors for these samples are again calculated by the quotient:
$$\begin{aligned}
\begin{split}
\frac{P}{Q_{new}}=&\left[\frac{p(y_t \mid x_t)}{\pi(y_t)}\right]^{-1}\frac{p(y_t \mid x_t) p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})}{p(y_t \mid y_{0:t-1},u_{0:t-1})} \\ \\
=& \,\,\frac{p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1}) \pi(y_t) }{p(y_t \mid y_{0:t-1},u_{0:t-1})} \\ \\
\propto & \,\,p(x_t \mid x_{t-1},u_{t-1})Bel(x_{t-1})
\end{split}\end{aligned}$$
Since $Bel(x_{t-1})$ is represented by a set of samples $x_{t-1}^{(i)}$ weighted by importance factors $w_{t-1}^{(i)}$, the (non-normalised) importance factor for any sample $x_t^{(j)}$ can be approximated by
$$\sum_{i=1}^m p(x_t^{(j)} \mid x_{t-1}^{(i)},u_{t-1})w_{t-1}^{(i)}
\label{eq:if_new}$$
The importance factor reflects the likelihood of the sample given *past* measurements and controls. This is orthogonal to the previous definition in Equation \[eq:if\_pf\], where it depends on the *current* measurement only.
Overall, the proposed sampling scheme changes how data and controls are used in belief estimation: the current measurement is now used for sampling (instead of weighing); past measurements and controls are used for calculating importance factors (instead of sampling).
The scheme is implemented recursively. It initialises $Bel(x_0)$ by generating $m$ samples around the first observation $y_0$ according to the observation probability $p(y_t \mid x_t)$. The samples are assigned the uniform importance factor of $1/m$. Subsequently, it estimates $Bel(x_t)$ at timestamps $t>0$ using a two-step recursive update:
for $k=1,\ldots,m:$
1. Generate a sample $x_t^{(i)}$ according to the observation probability $p(y_t \mid x_t)$.
2. Use the sample set representing $Bel(x_{t-1})$ to weight the sample $x_t^{(i)}$ by the importance factor in Equation \[eq:if\_new\] , the likelihood of the sample $x_t^{(i)}$ given past measurements and controls.
In the context of vehicle tracking, the estimates of $Bel(x_t)$ approximate the vehicle *position* at time $t$. If instead of the single-time approximation, you are interested in finding the most likely *trajectory* that the vehicle traversed until time $t$, it can be computed via the following dynamic programming routine. It corresponds to finding the sequence $x_{0:t}$ that maximises the posterior $p(x_{0:t} \mid y_{0:t}, u_{0:t-1})$.
1. Choose a sample $x_t^{(i)}$ from the sample set representing $Bel(x_t)$ that has the highest importance factor $w_t^{(i)}$.
2. Use the sample $x_t^{(i)}$ to find a preceding sample $x_{t-1}^{(j)}$ from $Bel(x_{t-1})$ that maximises $p(x_t^{(i)} \mid x_{t-1}^{(j)},u_{t-1})w_{t-1}^{(j)}$, i.e. is the most likely preceding state. Repeat this step until you reach $t=0$.
Application to Vehicle Tracking {#application}
===============================
Data
----
We tested the improved particle filters method on a GPS trajectory of a police patrol vehicle during its night shift (9am to 7am) in the London Borough of Camden on February $9^{th}$ 2015. The dataset contains 4,800 GPS points that were emitted roughly every second when moving. It was acquired for research purposes as part of the “Crime, Policing and Citizenship” project[^1].
Implementation
--------------
In order to apply the improved particle filters to vehicle tracking, we need to specify the form of the observation probability $p(y_t \mid x_t)$ and the transition probability $p(x_t \mid x_{t-1},u_{t-1})$. Their forms depend on the vehicle’s dynamics and the type of sensor used for localisation (a GPS receiver in our case). The distributions are time-invariant; hence we will omit the time index $t$ in the following derivations.
### Observation probability
#### Definition
We model the conditional probability $p(y \mid x)$ of observing a GPS point $\boldsymbol{y}$, represented by its easting and northing coordinates:
$$\boldsymbol{y} = \left(\begin{array}{cc} y_{e} \\ y_{n} \end{array}\right)
\label{eq:y_t}$$
as a two-dimensional Gaussian distribution
$$\boldsymbol{y} \sim \mathcal{N} (\boldsymbol{\mu},\boldsymbol{\Sigma})$$
with the mean vector $\boldsymbol{\mu}$ representing the true vehicle position $\boldsymbol{x}$
$$\boldsymbol{\mu}=\boldsymbol{x}=\left(\begin{array}{cc} x_{e} \\ x_{n} \end{array}\right)
\label{eq:mu}$$
and the covariance $\boldsymbol{\Sigma}$ that is constant across space, i.e. *isotropic* covariance
$$\boldsymbol{\Sigma} = \left[\begin{array}{cc} \Sigma_{ee} & \Sigma_{en} \\ \Sigma_{ne} & \Sigma_{nn} \end{array}\right] =\left[\begin{array}{cc} \sigma^2 & 0 \\ 0 & \sigma^2 \end{array}\right]
\label{eq:cov}$$
This representation of $p(y \mid x)$ reflects our expectation that GPS observations are normally distributed around the true vehicle positions.
#### Proposal Generation
In the proposed method, we use the observation probability $p(y \mid x)$ to sample possible vehicle positions on the road network (see Equation \[eq:q\_new\]). In order to efficiently generate samples on the road network (as shown in Figure \[fig:pf\_init\_network\]), we want to analytically project the two-dimensional $p(y \mid x)$ onto individual road segments.
[0.45]{} ![Sampling proposal positions around a GPS point (red).[]{data-label="pf_init"}](figures/initialisation_space_cropped.png "fig:"){width="\textwidth"}
[0.45]{} ![Sampling proposal positions around a GPS point (red).[]{data-label="pf_init"}](figures/initialisation_network_cropped.png "fig:"){width="\textwidth"}
We begin with the general form of a two-dimensional Gaussian distribution for $p(y \mid x)$
$$p(y \mid x) = \mathcal{N} (\boldsymbol{y} \mid \boldsymbol{\mu},\boldsymbol{\Sigma})=\frac{1}{2\pi|\boldsymbol{\Sigma}|^{1/2}}exp\left \{-\frac{1}{2}(\boldsymbol{y}-\boldsymbol{\mu})^T\boldsymbol{\Sigma}^{-1}(\boldsymbol{y}-\boldsymbol{\mu})\right \}
\label{eq:p_y_given_x1}$$
We precompute the inverse of the covariance matrix
$$\boldsymbol{\Sigma^{-1}}=\frac{1}{\sigma^4}\left[\begin{array}{cc} \sigma^2 & 0 \\ 0 & \sigma^2 \end{array}\right]=\left[\begin{array}{cc} \sigma^{-2} & 0 \\ 0 & \sigma^{-2} \end{array}\right]$$
and use it together with the partitioning (\[eq:y\_t\]), (\[eq:mu\]), and (\[eq:cov\]) to rewrite (\[eq:p\_y\_given\_x1\]) as
$$\begin{aligned}
\begin{split}
p(y \mid x) =& \,\,\frac{1}{2\pi\sigma^2}exp\left \{-\frac{1}{2}\left[\frac{(y_e-\mu_e)^2}{\sigma^2}+\frac{(y_n-\mu_n)^2}{\sigma^2}\right]\right \} \\
=& \,\,\frac{1}{(2\pi\sigma^2)^{1/2}}exp\left \{-\frac{1}{2\sigma^2}(y_e-\mu_e)^2\right \} \times \frac{1}{(2\pi\sigma^2)^{1/2}}exp\left \{-\frac{1}{2\sigma^2}(y_n-\mu_n)^2\right \} \\ \\
=& \,\,\mathcal{N} (y_e \mid \mu_e,\sigma) \times \mathcal{N} (y_n \mid \mu_n,\sigma)
\end{split}\end{aligned}$$
We successfully factor $p(y \mid x)$ into a product of two Gaussian distributions along the *easting* and *northing* directions due to the isotropic properties of the covariance matrix in (\[eq:cov\]). In fact, the factorisation of $p(y \mid x)$ holds for any other orthogonal coordinate system. Therefore, we replace the easting-nothing coordinates with orthogonal distances from $x$ dictated by the road segment that $x$ is on: *$a$* (distance *to* the road segment), *$b$* (distance *along* the road segment).
Under the new coordinate system $x$ and $y$ are partitioned as $$x=\left(\begin{array}{cc} x_{a} \\ x_{b} \end{array}\right)=\left(\begin{array}{cc} 0 \\ 0 \end{array}\right) \qquad\qquad y = \left(\begin{array}{cc} y_{a} \\ y_{b} \end{array}\right)$$ and $p(y \mid x)$ becomes $$\begin{aligned}
\begin{split}
p(y \mid x) =& \mathcal{N} (y_{a} \mid \mu_{a},\sigma) \times \mathcal{N} (y_{b} \mid \mu_{b},\sigma) \\
=& \mathcal{N} (y_{a} \mid 0,\sigma) \times \mathcal{N} (y_{b} \mid 0,\sigma)
\end{split}
\label{eq:p_y_t_new}\end{aligned}$$ The above definition enables us to generate proposals $x$ in accordance with the observation model $p(y \mid x)$ (as specified in Equation \[eq:q\_new\]):
1. Firstly, sampling a road segment that $x$ in on such that $y_{a}$ $\sim \mathcal{N} (0,\sigma)$
2. Secondly, sampling the position of $x$ along the segment such that $y_{b}$ $\sim \mathcal{N} (0,\sigma)$
### Transition probability
We set the transition probability $p(x_t \mid x_{t-1},u_{t-1})$ to be a linear estimate equal to the Cartesian distance between GPS points $x_{t-1}$ and $x_t$ (the control $u_t$) plus an additive Gaussian noise. This is a simplistic assumptions that could be further explored, however, it is not the focus of this paper.
Validation
----------
In the absence of the ground truth about vehicle positions at any point in time, we propose a validation framework based on the well-established technique of cross-validation [@Barber2012]. We remove every 10th GPS points from the available GPS trajectory. We then infer the path taken by the vehicle given the incomplete trajectory and the road network. Finally, we measure the distance between each removed point and the inferred path. The distances across all removed points form the distribution of the prediction error.
Results
=======
A series of tests was conducted to elucidate the difference between the standard and the proposed particle filters. We found that the modified proposal distribution consistently outperforms conventional particle filters in terms of accuracy. As expected, largest gains in accuracy are observed on datasets with long sampling intervals as their observations are infrequent and hence become highly informative. Figure \[fig:pf\_pfla\_accuracy\] plots the prediction error (in meters) of both algorithms for different sampling intervals and levels of sensor noise, using $m=10$ samples only. It shows that the proposed method has lower *median* error across all examined sampling rates and sensor noise levels, as well as much lower error *variation*.
[0.48]{} ![Accuracy of the improved particle filters (red) and the standard particle filters (blue) on GPS data with varied sampling rate and sensor noise, represented as 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_pfla_accuracy"}](figures/pf_and_pfla_accuracy_with_bounds_per_sampling_rate.png "fig:"){width="\textwidth"}
[0.48]{} ![Accuracy of the improved particle filters (red) and the standard particle filters (blue) on GPS data with varied sampling rate and sensor noise, represented as 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_pfla_accuracy"}](figures/pf_and_pfla_accuracy_with_bounds_per_measurement_error.png "fig:"){width="\textwidth"}
We evaluated the ability of both methods to track a vehicle over time. When they fail to track a vehicle, it means that all positions that they propose are completely unlikely given sensor data, i.e. particle weights sum up to zero. The standard particle filter basically fails when sensor measurements are infrequent (with $m=10$ samples). Figure \[fig:pclass\_sr\] shows that it is unable to track the vehicle nearly 70% of the time when the sampling interval increases to one minute. In the same scenario, the proposed method gives excellent results that show little variation to changes to sampling intervals.
On the contrary, the proposed method fails to track when sensors are very noisy. Although it shows high accuracy (see Figure \[fig:accuracy\_me\]), it is prone to high failure rates as the level of sensor noise increases (Figure \[fig:pclass\_me\]). This weakness reflects the orthogonal limitations of the two approaches: our method generates samples that are highly consistent with the most recent measurement (which makes it sensitive to sensor noise), whereas the conventional approach samples in accordance with past measurements (inefficient when sampling rates are low).
[0.48]{} ![Percentage of time the improved particle filters (red) and the conventional particle filters (blue) lost track of the position of the vehicle as a function of the GPS sampling rate and the sensor noise.[]{data-label="fig:pf_st_pclass"}](figures/pf_and_pfla_unclassified_points_per_sampling_rate.png "fig:"){width="\textwidth"}
[0.48]{} ![Percentage of time the improved particle filters (red) and the conventional particle filters (blue) lost track of the position of the vehicle as a function of the GPS sampling rate and the sensor noise.[]{data-label="fig:pf_st_pclass"}](figures/pf_and_pfla_unclassified_points_per_measurement_error.png "fig:"){width="\textwidth"}
Finally, we tested the sensitivity of the proposed method to the number of samples used. Figure \[fig:pf\_st\_samples\] shows comparative results on GPS data with the sampling interval of 70 seconds. The proposed method yields significantly better results, both in terms of accuracy and robustness to failure. When only $m=10$ samples are used, it reduces the estimation error by almost 10 meters and the percentage of failure by as much as 68%. The performance is further improved when more samples are used, but the gain is small compared to the conventional particle filters. In fact, the proposed method with $m=100$ samples is more accurate and robust than the conventional particle filters with as many as $m=10000$ samples (see Figure \[fig:pf\_longterm\]). Therefore, it can be reliably used with a small number of samples, making it highly computationally efficient.
[0.48]{} ![Accuracy and robustness of the improved particle filters (red) and the conventional particle filters (blue) as a function of the number of samples used. Accuracy is shown as the 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_st_samples"}](figures/pf_and_pfla_accuracy_with_bounds_per_number_particles.png "fig:"){width="\textwidth"}
[0.48]{} ![Accuracy and robustness of the improved particle filters (red) and the conventional particle filters (blue) as a function of the number of samples used. Accuracy is shown as the 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_st_samples"}](figures/pf_and_pfla_unclassified_points_per_number_particles.png "fig:"){width="\textwidth"}
[0.48]{} ![Accuracy and robustness of the standard particle filters as the number of samples is increased to very large values. Accuracy is shown as the 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_longterm"}](figures/pf_accuracy_with_bounds_per_number_particles_longterm.png "fig:"){width="\textwidth"}
[0.48]{} ![Accuracy and robustness of the standard particle filters as the number of samples is increased to very large values. Accuracy is shown as the 25th, 50th and 75th percentiles of prediction errors.[]{data-label="fig:pf_longterm"}](figures/pf_unclassified_points_per_number_particles_longterm.png "fig:"){width="\textwidth"}
Conclusions
===========
This paper describes a modified particle filters method that shows uniformly superior accuracy to the conventional particle filters. The improved algorithm utilizes a different proposal distribution which uses only the most recent observation in the position prediction process. In doing so, it makes more efficient use of the particles, particularly in situations in which the transition noise is high in relation to the observation noise.
The main contribution of the paper is the proposal distribution itself and the derivation of the associated importance weights that guarantees convergence to the same posterior distribution as the standard particle filters. An important contribution is also the projection of a two-dimensional Gaussian onto a network of roads, which enables efficient sampling on the road network from a spatial Gaussian.
The theoretical contributions are complemented by experimental results of vehicle tracking using a police GPS dataset. The new algorithm is consistently more accurate than the standard particle filters, with largest gains in accuracy on sparse GPS data. It requires much fewer samples to yield good performance. In fact, as few as fifty samples are sufficient to outperform the standard method with 10,000 particles in terms of accuracy and proneness to failure. We believe that our results illustrate that particle filters can be radically improved if one carefully chooses a proposal distribution, such that it extracts the most information from the available data.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is part of the project - Crime, Policing and Citizenship (CPC): Space-Time Interactions of Dynamic Networks (www.ucl.ac.uk/cpc), supported by the UK Engineering and Physical Sciences Research Council (EP/J004197/1). The data provided by Metropolitan Police Service (London) is greatly appreciated.
We would also like to show our gratitude to Dr Simon Julier for very helpful discussions during the course of this research.
[^1]: UCL Crime Policing and Citizenship: <http://www.ucl.ac.uk/cpc/>
|
---
abstract: 'The interaction of whispering gallery modes (WGM) of optical microresonators with subwavelength imperfections has been studied both experimentally and theoretically. This interaction is responsible for the formation of spectral doublets in place of single resonance peaks, and for degrading of Q-factors of the resonances. Within the currently accepted framework the spectral doublets are explained as a result of degeneracy removal of clockwise and counterclockwise WGMs due to their coupling caused by defect-induced backscattering, while the degrading of the Q-factor is described phenomenologically as an additional contribution to the overall decay rate of WGM due to coupling between WGM and radiative modes. Here we show that the existing understanding of this phenomenon is conceptually wrong and develop an exact theory of WGM interaction with a single defect, which provides a unified treatment for both aspects of this interaction explaining existing experiments and predicting new phenomena.'
author:
- 'L. Deych'
- 'J. Rubin'
title: 'Rayleigh Scattering of Whispering Gallery Modes of Microspheres due to a Single Scatterer: Myths and Reality'
---
Elastic (with no change in frequency) scattering of light due to small (compared to wavelength) particles is one of the most fundamental and intensively studied optical phenomena. Its modern history began almost one hundred fifty years ago with the explanation of the blue color of sky in a series of papers by Lord Rayleigh[@Rayleigh], where the now famous $1/\lambda^4$ cross section law, where $\lambda$ is the wavelength of light in vacuum, was derived. Since then it has been customary to refer to processes of elastic interaction of light with subwavelength particles as Rayleigh scattering. Besides providing us with beautiful blue skies and red sunsets, Rayleigh scattering is important for a large number of fundamental optical phenomena as well as for numerous applications. Recent developments in optics and photonics have created new situations in which the manifestations of Rayleigh scattering are significantly modified. Particularly drastic modification of this process is expected when light is confined in all three dimensions inside optical microresonators in the form of whispering gallery modes (WGM) [@VahalaNature2003]. Given the fundamental nature of this process it is not surprising that it has attracted a significant amount of attention in recent years [@WeissOL95; @LittleOL1997; @Gorodetsky2000; @KippenbergOL2002; @BorselliOE2005].
While whispering gallery modes can occur in various types of geometries [@BoriskinaReview2006] we will focus on spherical microresonators. WGMs in this case correspond to Mie resonances [@Mie1908] with ultra narrow widths, $\gamma_{ls}\ll \omega_{ls}$, where $\omega_{ls}$ is the frequency of the mode, and respectively high (up to $10^9$ for silica microspheres [@BoriskinaReview2006]) Q-factors defined as $Q_{ls}=\omega_{ls}/\gamma_{ls}$. WGMs are characterized by polar and azimuthal indexes, $l$ and $m$, and a radial number $s$ determining, respectively, the angular and radial dependence of the fields in a spherical coordinate system centered at the sphere. The resonance frequency $\omega_{ls}$ does not depend on the azimuthal number, which reflects the degeneracy of the resonances due to full spherical symmetry of the problem. WGMs are also characterized by their the mode volume, which can be very different for modes with the same $l$ but different $m$. Modes with the smallest volume correspond to $|m|=l$, and $s=1$ in which case the field is concentrated mostly in the equatorial plane and at the surface of the sphere. Such modes are called fundamental (FM) and their interaction with defects is of the primary interest.
This interaction causes two observable effects: (i) formation of spectral doublets in place of a single peak, and (ii) reduction of Q-factors of WGMs below theoretically predicted limits [@BorselliOE2005; @GrudininOC2006]. In the existing approaches these two effects are considered separately, even though they are two manifestations of the same phenomenon. The accepted explanation of the spectral doublets is based on the hypothesis likely first suggested by D.S. Weiss et al. in the following form: “We have observed that very high-Q Mie resonances in silica microspheres are split into doublets. This splitting is attributed to internal backscattering that couples the two degenerate whispering-gallery modes propagating in opposite directions along the sphere equator” [@WeissOL95]. This idea of the defect-induced backscattering was further developed in subsequent publications [@LittleOL1997; @Gorodetsky2000; @BorselliOE2005] and was claimed to be experimentally confirmed in Ref. [@KippenbergOL2002]. In a recent paper by A. Mazzei, et al. [@MazzeiPRL2007] double peak features in the spectra of microspheres were studied under conditions of controlled scattering, where the role of the defect was played by a tip of a near-field optical microscope. This work directly confirmed a connection between the interaction of WGMs with a *single* defect and the formation of the spectral doublets.
The influence of defects on Q-factors of microresonators is usually prescribed to defect-induced coupling between WGMs and radiative modes and is taken into consideration phenomenologically by adding a “scattering” loss rate to the total losses of the resonator. In the case of disk resonators this rate was calculated in Ref. [@BorselliOE2005] under the assumption that the surface roughness couples WGMs with free space electromagnetic radiation.
Results {#Results}
=======
In this work we develop an *ab initio* theory of interaction between WGMs of microspheres and a *single* subwavelength defect, which treats both aspects of this interaction in a unified way. We show that the generally accepted picture of interaction between FM and defects is conceptually wrong: the interaction cannot be described in terms of “backscattering”, and it does not result in coupling between counterpropagating “clockwise” (cw) and “counterclockwise” (ccw) FMs. The theory predicts that the observed spectral doublets are actually a part of a triplet of peaks, the third component of which has not yet been found. It also provides an accurate “from first principles” description of the broadening of the resonances. Similar to the Rayleigh scattering of propagating waves, the solution of the single-defect problem presented here constitutes the first fundamental step toward a theory of interaction between WGMs and multiple defects. When this interaction is small, which is usually the case, the single-defect solution can be directly used to make conclusions about the role of multiple scatterers. Since the results of Ref. [@MazzeiPRL2007] directly confirm that the spectral features formed due to a single discrete scatterer are similar to those caused by distributed surface or volume disorder, the results presented here are relevant not only for interaction of WGMs with discrete non-interacting defects, but can also be used to understand effects due to continuously distributed nonuniformities.
The main assumption of the theory is that the defect is small enough to be treated as a dipole. In this case the shape of the defect is irrelevant, and can be taken to be spherical. In this way the problem is reduced to describing two electromagnetically coupled spheres of radii $R_0$ and $R_d\ll R_0$ with refractive indexes $n$ and $n_d$ respectively, whose centers are positioned at a distance $d$ from each other (see Fig. \[fig:coordinates\]). We will also assume that the defect lies in the plane of the FM in order to maximize the strength of the interaction, although the theory developed here can readily be generalized for arbitrarily positioned defects. The goal is to find electromagnetic field induced by this system in the presence of an incident wave (imitating a mode of a tapered fiber) which, in the absence of the defect, would have excited a FM with given polar number $L$. The incident, $\mathbf{E}_{inc}$, and induced, $\mathbf{E}_s$, fields can be presented as linear combinations of vector spherical harmonics (VSH) of the form: $$\begin{aligned}
\mathbf{E_{inc}}&=&\sum_{m=-L}^{L}\eta_{L,m}\mathbf{N}_{L,m}(\mathbf{r}-\mathbf{r}_1)\label{eq:inc_ext}\\
\mathbf{E_{s}}&=&\sum_{i=1}^2\sum_{l=1}^\infty\sum_{m=-l}^l\left[a_{l,m}^{(i)}\mathbf{N}_{l,m}(\mathbf{r}-\mathbf{r}_i)+b_{l,m}^{(i)}\mathbf{M}_{l,m}(\mathbf{r}-\mathbf{r}_i)\right]\label{eq:scat_ext}
%\mathbf{E_{in}}=\sum_{i=1}^N\sum_{l,m}\left[c_{l,m}^{(i)}\mathbf{N}_{m,l}(\mathbf{r}-\mathbf{r}_i)+d_{l,m}^{(i)}\mathbf{M}_{m,l}(\mathbf{r}-\mathbf{r}_i)\right]\label{eq:intern_ext}.\end{aligned}$$ where index $i$ enumerates the spheres ($i=2$ refers to the defect), $\mathbf{r_i}$ is a position vector of the center of $i-th$ sphere, $\mathbf{M}_{l,m}$ and $\mathbf{N}_{l,m}$ are the VSH of TE and TM polarization respectively as defined in Ref. [@stratton_book1941], and $\eta_{L,m}$ are coefficients describing the TM incident wave of frequency $\omega$, which in the coordinate system $XYZ$ defined in Fig. \[fig:coordinates\] and used in all subsequent calculations have the following form $$\label{eq:FM}
\eta_{L,m}=(-1)^{\epsilon(L+m)}\frac{(-i)^L}{2^L}\displaystyle{\sqrt{\frac{(2L)!}{(L+m)!(L-m)!}}};
\hskip 4pt \epsilon=\left\{
\begin{array}{cc}
1 & \textit{cw FM} \\
0 & \textit{ccw FM}
\end{array}\right.$$ (See details in Section \[Methods\].) In order to find the induced field one needs to determine expansion coefficients $a_{l,m}^{(i)}$ and $b_{l,m}^{(i)}$. For simplicity we disregard the defect-induced coupling between TE and TM modes, and since the incident field is of TM polarization we set the TE coefficients $b_{l,m}^{(i)}=0$ and solve for coefficients $a_{l,m}^{(1,2)}$. The dipole approximation for the defect is introduced by setting $a_{l,m}^{(2)}=0$ for all $l>1$. The resulting system of equations for the scattering coefficients is solved exactly to yield: $$\begin{aligned}
a_{l,m}^{(1)}&=&\left\{ \begin{array}{lcc}
\displaystyle{\frac{\eta_{L,m}}{[\alpha_L^{(1)}]^{-1}+(-1)^L\alpha_1^{(2)}A_{1,m}^{L,m}A_{L,m}^{1,m}+\alpha_1^{(2)}[\alpha_L^{(1)}]^{-1}\displaystyle{\sum_{\nu\ne L}(-1)^\nu\alpha_\nu^{(1)}A_{1,m}^{\nu ,m}A_{\nu ,m}^{1,m}}}} & l=L;|m|\le 1& (a) \\
-\alpha_l^{(1)}\eta_{L,m}\displaystyle{\frac{(-1)^{L}\alpha_1^{(2)}\alpha_L^{(1)}A_{1,m}^{L,m}A_{l,m}^{1,m}}{1+\alpha_1^{(2)}\displaystyle{\sum_{\nu} (-1)^\nu\alpha_\nu^{(1)}A_{1,m}^{\nu ,m}A_{\nu,m}^{1,m}}}} & l\ne L;|m|\le 1& (b) \\
\alpha_l^{(1)}\eta_{L,m}\delta_{l,L} & |m|>1 & (c)
\end{array}\right.\label{eq:a1_coeff}\\
a_{l,m}^{(2)}&=&-\alpha_1^{(2)}\sum_\nu a_{\nu ,m}^{(1)}(-1)^\nu A_{1,m}^{\nu ,m}\label{eq:a2_coeff}\end{aligned}$$ where $\alpha^{(1,2)}$ are the single sphere scattering parameters for the main sphere and the defect respectively, defined in Eq. \[eq:alpha\_exact\] and \[eq:alpha2\_approx\] of section \[Methods\]. The scattering parameter of the main sphere has poles at the complex-valued frequency of WGMs; we assume that the frequency $\omega$ of the incident field is in the vicinity of the pole $\omega_L^{(0)}-i\Gamma_L^{(0)}$ corresponding to the frequency of our FM. Since the defect is assumed to be small so that $n_d\omega R_d/c \ll 1$, its scattering parameter does not have any poles of its own.
Parameters $A_{l,m}^{\nu ,m}$ in Eq. \[eq:a1\_coeff\] and \[eq:a2\_coeff\] describe the electromagnetic interaction between the spheres and are called translation coefficients. They appear when a VSH defined in one coordinate system needs to be expressed in terms of VSH defined in a system with a shifted origin [@SteinApplMath1961; @CruzanApplMath1962; @Mishchenko_book2002]. These coefficients depend on the translation vector $\mathbf{r}_1-\mathbf{r}_2$ and the coordinate system used to define VSHs. In the coordinate system $XYZ$ they are diagonal in terms of azimuthal number $m$, which reflects the fact that the polar axis $Z$ runs along the line connecting the centers of the spheres, thus preserving the axial symmetry of the two-sphere structure.
Eqs. \[eq:a1\_coeff\] and \[eq:a2\_coeff\] contain all the information about the electromagnetic field of the sphere-defect system. First of all, Eq. \[eq:a1\_coeff\]c shows that components of the initial FM with $|m|>1$ are not effected by the defect. Formally, this result is a consequence of the translation coefficients $A_{l,m}^{\nu ,\mu}$ being equal to zero when either $m$ or $\mu$ exceeds either $\nu$ or $l$ [@Mishchenko_book2002]. Physically, this result reflects the simple fact that a dipole can only produce a field with $l=1$ and $m=0,\pm 1$. Since in the $XYZ$ coordinate system $m$ remains a conserving quantity even in the presence of a defect there will be no coupling between the field of defect and WGMs with $|m|>1$. Thus, the expansion coefficients $a_{l,m}^{(1)}$ with $l=L$ and $|m|>1$ will produce a resonance at the original single-sphere frequency, which we will characterize by size parameter $x_L^{(0)}=\omega_L^{(0)}R_0/c=k_L^{(0)}R_0$. The width of this resonance, which can be described by a dimensionless parameter $\gamma_L^{(0)}=\Gamma_L^{(0)}R_0/c$ is also not affected by the defect. Eq.\[eq:a1\_coeff\]a, on the other hand, shows that $|m|\le 1$ components of the FM do interact with the defect, and that this interaction results in appearance of new resonance frequencies determined by poles of this expression. From the sum over $\nu$ we singled out a term with $\nu=L$, which in the frequency range around $\omega_L^{(0)}$ gives the biggest contribution to the shift of the new poles from their single-sphere value. Discarding the rest of the sum (the resonance approximation) we can obtain an analytical expression for the positions of the new poles $$\label{eq:res_pos}
\begin{split}
&x_{L,m}=x_L^{(0)}(1+\delta x_{L,m}) -i\gamma_L^{(0)}(1+\delta\gamma_{L,m})\\
\delta x_{L,m} = &-\gamma_L^{(0)}\frac{f_{L,m}\left(k_L^{(0)}d\right)}{(2L+1)R_0^2d}p[x_L^{(0)}]; \hskip 4pt
\delta\gamma_{L,m} = \frac{2}{3}\frac{f_{L,m}\left(k_L^{(0)}d\right)}{(2L+1)R_0^5d}p^2[x_L^{(0)}]^5
\end{split}$$ where $$\label{eq:polar_unnorm}
p=\frac{n_d^2-1}{n_d^2+2}R_d^3$$ is the standard polarizability of a small dielectric sphere, and function $f_{L,m}\left(k_L^{(0)}d\right)$ for $m=0,\pm 1$ is defined as $$\label{eq:f_funct}
f_{L,m}(kd)=\left[{(-1)^m}\sqrt{\frac{(L+1)(L+m^2)}{1+m^2}}g_{L-1}(kd)+\sqrt{L(L+1)(1-m^2)+L^2\frac{m^2}{2}}g_{L+1}(kd)\right]^2$$ with $$g_L(kd) = {\frac{1}{\sqrt{\rho\xi}}}{e^{\xi(atanh{\rho}-\rho)}} \ \ ; \ \
\rho = \sqrt{1-\left(\frac{kd}{\xi}\right)^2} \ \ ; \ \
\xi = L + \frac{1}{2} \nonumber$$ where we used an asymptotic form of the Hankel function valid for $l\gg kd$. Eq. \[eq:res\_pos\] predicts two new resonances, in addition to the original single sphere resonance, one for $m=0$ and another for $m=\pm 1$, both red shifted with respect to the initial frequency. Function $f_{Lm}(kd)$ specifies the dependence of the resonance frequencies on $m$ and distance $d$. The later is determined by the exponential decay of the spherical Hankel functions outside of the main sphere, which reflects the evanescent nature of the interaction with the defect. Thus, our theory predicts the existence of a triplet of peaks rather than the doublet expected in the current cw-ccw coupling picture. To verify this result we carried out numerical calculations of the frequency dependence of the energy emitted by the main sphere given by the standard expression $\sum_l(2l+1)|a_{l,m}|^2$ [@stratton_book1941], using complete Eq. \[eq:a1\_coeff\]. For these calculations we choose $L=39$ and take into account enough coefficients $a_{l,m}$ and terms in the sum over $\nu$ to ensure convergence of the procedure, which was achieved with $1\le l\le 50$ and $\nu\le 50$. The results of these calculations are shown in Fig. \[fig:scatter\_energy\] for different distances between the defect and the sphere so that one can see how the peaks shift toward the single-sphere resonance with increasing $d$ and eventually merge with it. There are indeed three peaks, which are not seen at curves 1 and 2 because the third peak on these curves is out of the range of the figure. The $m=0$ resonance is shifted further from $x_L^{(0)}$ and is weaker than the $m=\pm 1$ resonance, making it more difficult for experimental identification. We suggest, therefore, that the experimentally observed spectral doublets correspond to the original single-sphere resonance and the $m=\pm 1$ resonance introduced by the defect.
![Relative broadening versus frequency shift of the $|m| = 1$ resonance revealed through varying distance $d$. The points are obtained from the spectra computed at different $d$, while the line represents a fit with a quadratic polynomial\[fig:shift\_broad\]](Fig3.eps){width=".4\linewidth"}
The validity of the resonance approximation depends on the convergence of the sum $\sum_{\nu\ne L}(-1)^\nu\alpha_\nu^{(1)}A_{1,m}^{\nu,m}A_{\nu,m}^{1,m}$ appearing in the denominator of Eq. \[eq:a1\_coeff\]a. In the limit $\nu\rightarrow\infty$ we find $$\label{eq:asympt}
(-1)^{\nu+1}\alpha_\nu^{(1)}A_{1,m}^{\nu,m}A_{\nu,m}^{1,m}\asymp \frac{p^2}{(kd)^3}\left(\frac{R_0}{d}\right)^{2\nu+1}$$ which, given that $R_0/d<1$, proves the convergency of the sum. However, in the case of small defects positioned close to the surface of the sphere, $R_0/d$ differs from unity by a small amount and the convergence of the sum is slow. In this case the terms with $p\gg L$ become important and should be taken into account. It can be shown that incorporating these terms does not change the form of Eq. \[eq:res\_pos\] but renormalizes the polarizability, which becomes: $$\label{eq:renorm_p}
\tilde{p}=p\left[1+\left(1-\frac{m^2}{2}\right)\frac{dR_d^3(R_0^2+d^2)}{(kd)^3(d^2-R_0^2)^3}\frac{n^2-1}{n^2+2}\right]^{-1}$$ The renormalized polarizability acquires dependence on the distance $d$ between the defect and the sphere, thereby affecting the relation between the frequency shift and the broadening of the defect-induced resonances which is revealed through variation of $d$. Indeed, in the absence of the renormalization both these quantities decrease with $d$ by the same factor determined by the function $f_{L,m}$, resulting in $\delta x_{L,m}\propto \delta\gamma_{L,m}$. The renormalized polarizability, however, also changes with distance. Since the frequency shift of the resonance is linear in $p$ while the broadening is $quadratic$, this effect must result in deviations from this linear dependence. In order to confirm this conclusion we used numerical spectra obtained for different distances in order to plot $\delta x_{L,m}$ versus $\delta\gamma_{L,m}$ for the $|m|=1$ resonance. The obtained data shown in Fig. \[fig:shift\_broad\] are found to be better fit by a quadratic rather than a linear function confirming this conclusion.
With scattering coefficients known we can also compute the internal field inside the main sphere. Fig. \[fig:field\] shows the variation of the field in the $YZ$ plane of the $XYZ$ system (which corresponds to the plane of the FM) obtained by varying radial and polar coordinates at the azimuthal angles $\phi=\pi/2$ and $\phi=3\pi/2$. The computed field profile for the defect-induced peak demonstrates $2L$ oscillations and a drastic increase in intensity in the vicinity of the defect. At the frequency of a single-sphere resonance the situation is reversed: $2L$ oscillations, which are phase shifted compared to the defect-induced resonance are accompanied by a significant decrease in the field’s intensity in the defect’s proximity.
Finally Eq. \[eq:a1\_coeff\]b describes coupling of the FM to other WGMs, most important of which are terms with $l<L$. There are two reasons for this. First, modes with lower $l$ and higher radial numbers can spectrally overlap with the $l=L$, $s=1$ mode [@DeychRoslyakPRE2006] and thus have a large effect on the field distribution. Second, these modes usually have lower $Q$-factors and contribute more significantly to the radiation losses of the system. Note that $m$-components with $m>1$, which are responsible for the resonance at the single-sphere frequency do not couple to any other modes, so that this resonance is not affected by the coupling to the low-Q WGMs. The effects of coupling to these modes at the frequency of the defect-induced peak is shown in Fig. \[fig:radiative\_coupling\], where we plot the spectrum of the radiated energy in its vicinity with and without contributions of terms with $l\ne L$.
![Internal field intensity of the microsphere in the YZ plane at the frequency of the standard Mie resonance (left) and the defect induced resonance (right).\[fig:field\]](Fig4.eps){width=".8\linewidth"}
Discussion {#Discussion}
==========
The theory presented in the paper gives a complete picture of the interaction between WGMs and a single defect based on fundamental principles with no *ad hoc* assumptions, and replaces the currently accepted paradigm, which is proved to be inadequate. The results obtained on the basis of this theory show that a number of experiments previously “explained” within the cw-ccw coupling picture such as the backscattering experiment of Ref. [@KippenbergOL2002] must be reinterpreted. They also provide a natural explanation to other experimental results that the current paradigm was not able to explain in addition to predicting new effects that await experimental confirmation.
To begin with, the developed theory gives a natural explanation of an asymmetry between the two peaks of a doublet, which was seen in all experimental observations of this effect, but most clearly in Ref. [@MazzeiPRL2007]. Since, according to our calculations, the higher frequency component of the doublet corresponds to a single sphere resonance unaffected by the defect, this peak is supposed to be narrower than its counterpart and not to shift with the change in the position of the defect. This behavior is in complete agreement with observations of Ref. [@MazzeiPRL2007]. In addition, our theory predicts the existence of the third peak, which is, however, weaker than the other two making its experimental observation more difficult. It should be noted, however, that since within the prevailing paradigm the presence of the third peak was not expected, it is possible that more careful experimental observations will reveal its presence. Eq. \[eq:res\_pos\] also predicts quite specific dependence of the peak’s position and its broadening on the position of the defect and the polar number $L$ and frequency $\omega_L^{(0)}$, which also can be verified experimentally.
We also found that while the main contribution to the width of the resonance comes from coupling to the dipole field of the defect, there is also an additional contribution from coupling to lower Q WGMs of the main sphere. This contribution is emphasized in Fig. \[fig:radiative\_coupling\], where we compare defect-induced resonance with and without terms with $l\ne L$. One can see that these terms make the resonance wider while increasing its height. This effect, which cannot be described by simply adding an additional loss term to the Q-factor, is relatively small for a single defect case, but can be expected to become more significant with multiple defects.
![Radiated energy with (the curve with the higher peak) and without contributions from $l\neq L$.\[fig:radiative\_coupling\]](Fig5.eps){width=".5\linewidth"}
Another important effect predicted in our theory is the position dependent renormalization of the polarizability of the defect. This effect results in a deviation from the linear dependence between the frequency shift of the defect-induced resonance and its broadening, which was recently observed in Ref. [@MazzeiPRL2007]. The authors of that paper suggested that the position dependence of the polarizability could account for this finding, but were unable to explain its origin. In our theory this dependence appears naturally as a result of coupling between the defect and WGMs with high polar numbers $l$.
An important characteristic of the interaction between the FM and the defect is the resulting distribution of the electromagnetic field along the surface of the sphere. The backscattering paradigm predicts the formation of standing waves with $2L$ oscillations of the field’s intensity along the circumference of the FM. These waves are assumed to be due to interference of cw and ccw modes and are described by either $\sin$- or $\cos$-like behavior, depending on which component of the doublet is considered. Using Eq. \[eq:FM\] we can see that in $XYZ$ coordinate system used in our calculations these standing waves should be described as $$\mathbf{E_{sw}}=\sum_{m=-L}^{L}\frac{(-i)^L}{2^L}\displaystyle{\sqrt{\frac{(2L)!}{(L+m)!(L-m)!}}}\mathbf{N}_{L,m}(\mathbf{r}-\mathbf{r}_1)\label{eq:stand_wave}$$ where for the symmetric combination of the cw and ccw modes $m$ takes on only even values for even $L$, and odd values for odd $L$; for antisymmetric combination the situation is reversed. Our results show that for both the defect-induced and single sphere resonance there are indeed $2L$ oscillations, which, however, do not have the form prescribed by Eq. \[eq:stand\_wave\]. The field distribution at the frequency of the defect-induced peak is explained by the fact that the field at this frequency is mainly comprised of the $m$-components with $|m|=1$. The field of these WGMs is characterized by $L-|m|+1=L$ oscillations for $\theta$ changing between $0,\pi$ giving their total number equal to $2L$. These modes are also characterized by the enhancement of the field in the vicinity of $\theta=0$, which explains a drastic rise in the intensity around the location of the defect. The field distribution at the single sphere resonance can be understood by noting that this field is comprised of modes with $|m|>1$, which when added to the remaining $|m|\le 1$ components, would have produced a flat distribution of the intensity. Therefore, removal of these components obviously results in the decrease of the field around the defect and phase shifted oscillations elsewhere. The presence of these oscillations of the field’s intensity demonstrate that one can explain experimental results of Ref. [@KippenbergOL2002] without reliance on the “backscattering” paradigm.
Finally, we should comment on the relation between our results and the multi-defect problem. In the approximation of non-interacting defects, the field in the presence of multiple defects can be found as simple sum of fields due to each defect separately. Therefore, the generalization of these results would include finding the field distribution for a generic position of the defect relative to the plane of the FM. Since interaction with each defect can be considered independently we can always analyze it in a coordinate system with polar axis passing through the centers of the defect and the main sphere. Then, we can repeat all our calculations with only one adjustment: Eq. \[eq:FM\] needs to be generalized to incorporate an arbitrary inclination of the plane of the FM with respect to the coordinate axes. It is clear, therefore, that while the resulting field distribution and the heights of the resonance peaks will be different from the ones obtained here, the real and imaginary parts of the resonance frequencies will remain the same as in the single-defect case as long as all defects are identical [^1]. It is also important to note that these quantities do not depend on the type of the initial FM, cw or ccw, which means that even in the presence of a defect or multiple defects there are still two modes originating from cw or ccw initial FMs, which remain degenerate. This conclusion has a number of far reaching implications and dispels another important myth of the cw-ccw coupling paradigm, which suggests that in the presence of the defects each resonant peak corresponds to a single non-degenerate mode.
Methods {#Methods}
=======
Results presented in this paper were obtained with the help of a number of qualitative and quantitative methods. We will start with the approach based on symmetry arguments, which provides a qualitative explanation of many of the findings.
Symmetry considerations and choice of the system of coordinates
---------------------------------------------------------------
The logic behind the ccw-cw coupling paradigm is based upon an implicit assumption that the degeneracy between cw and ccw modes, which persists even in the absence of the complete spherical symmetry, is due to the rotational symmetry with respect to the axis perpendicular to the plane of the FM. Indeed, if this were the case, then any defect would have violated this symmetry, lifting the cw-ccw degeneracy and resulting in the spectral doublet. This assumption, however, is not correct, which can be immediately seen if one recalls that the group of rotations about a single axis is Abelian and, therefore, can only have one-dimensional representations. This means that axial symmetry alone cannot explain the cw-ccw degeneracy and one needs to invoke an additional symmetry such as inversion with respect to the azimuthal angle $\phi$. In the case of a single sphere the inversion symmetry is “hidden” behind the more powerful spherical symmetry, but in the system with a defect it starts playing a significant role. Indeed, two interacting spheres, described in $X^\prime Y^\prime Z^\prime$ coordinates (Fig. \[fig:coordinates\]), which is not consistent even with the remaining axial symmetry of the system, still exhibits a symmetry with respect to replacement $\phi \rightarrow -\phi$ if the $X$-axis of that coordinate system is chosen along the line connecting the centers of the spheres. Since this is the symmetry ultimately responsible for the cw-ccw degeneracy and it is not destroyed even in the system with the defect, the alleged coupling between cw and ccw modes cannot take place.
The $X^\prime Y^\prime Z^\prime$ coordinate system is convenient to describe the FM excited in the main sphere shown in Fig. \[fig:coordinates\]. In this case the respective field can be presented as a single vector spherical harmonic of $TE$ or $TM$ polarization [@stratton_book1941], meaning that the expansion coefficients of Eq. \[eq:inc\_ext\] takes a simple form $\eta_{L,m}=\delta_{L,m}$ instead of those given in Eq. \[eq:FM\]. However, in the two-sphere problem this coordinate system is not consistent with the symmetry of the configuration, therefore, it is more convenient, following Ref. [@MiyazakiPRB2000; @deychPRA2008], to switch to a coordinate system with polar axis directed along the line connecting the centers of the spheres (designated as $XYZ$ in Fig.\[fig:coordinates\]). It is important to realize, however, [@deychPRA2008] that the field of this FM cannot be presented as a single VSH in the spherical coordinates based on the coordinate system $XYZ$. To obtain such representation we notice that this system is obtained from $X^\prime Y^\prime Z^\prime$ system by means of a rotation characterized by Euler angles $\alpha=\pi/2$, $\beta=\pi/2$, $\gamma=0$, where we are following notations from Ref. [@Mishchenko_book2002]. Now using the transformation properties of VSH [@Mishchenko_book2002] we obtain the representation of the FM in the $XYZ$-based spherical coordinates given in Eqs. \[eq:inc\_ext\] and \[eq:FM\].
The $XYZ$ coordinate system reflects the presence of the axial symmetry of our configuration with respect to rotation about the axis connecting the centers of the spheres. Because of this symmetry, even though the modes of the sphere with the defect can no longer be classified according to the polar number, $l$, they still can be characterized by azimuthal number, $m$. Respectively, each of the $m$-components comprising the FM interacts with the defect independently making the analysis of the interaction simpler.
Multi-sphere Mie theory
-----------------------
In order to find the expansion coefficients of the induced field introduced in Eq. \[eq:scat\_ext\] we use the standard multi-sphere Mie theory [@MiyazakiPRB2000; @Mishchenko_book2002; @FullerApplOpt1991]. In this approach, the field outside of the spheres is separated into incident field given by Eq. \[eq:inc\_ext\] and the induced field given by Eq.\[eq:scat\_ext\]. In addition the field inside the spheres is also presented as a linear combination of VSHs centered at each sphere $$%\mathbf{E_{inc}}=\sum_{i=1}^N\sum_{l,m}\left[\zeta_{l,m}^{(i)}\mathbf{N}_{m,l}(\mathbf{r}-\mathbf{r}_i)+\eta_{l,m}^{(i)}\mathbf{M}_{m,l}(\mathbf{r}-\mathbf{r}_i)\right]\label{eq:inc_ext}\\
%\mathbf{E_{s}}=\sum_{i=1}^N\sum_{l,m}\left[a_{l,m}^{(i)}\mathbf{N}_{m,l}(\mathbf{r}-\mathbf{r}_i)+b_{l,m}^{(i)}\mathbf{M}_{m,l}(\mathbf{r}-\mathbf{r}_i)\right]\label{eq:scat_ext}\\
\mathbf{E_{in}^{(i)}}=\sum_{l,m}\left[c_{l,m}^{(i)}\mathbf{N}_{l,m}(\mathbf{r}-\mathbf{r}_i)+d_{l,m}^{(i)}\mathbf{M}_{l,m}(\mathbf{r}-\mathbf{r}_i)\right]\label{eq:intern_ext}.$$ In order to apply Maxwell boundary conditions on the surface on a sphere $i$ the VSHs centered at different spheres must be rewritten in the coordinate system translated to the center of the $i$-th sphere. This is accomplished with the help of the addition theorem for the vector spherical harmonics [@CruzanApplMath1962; @SteinApplMath1961], which introduces translation coefficients $A_{l,m}
^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right)$ and $B_{l,m}
^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right)$, and which allows one to derive a system of equations relating expansion coefficients $a_{l,m}^{(i)}$ and $b_{l,m}^{(i)}$ to the coefficients of the incident field $\eta_{l,m}^{(i)}$: $$\begin{aligned}
a_{l,m}^{(i)}&=&\alpha_{l}^{(i)}\left\{\eta_{l,m}^{(i)}+\sum\limits_{j\neq i}\sum\limits_{l^\prime,m^\prime}\left[a_{l^\prime,m^\prime}^{(j)}A_{l,m}
^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right) +b_{l^\prime,m^\prime}^{(j)}B_{l,m}^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right)\right]\right\}\label{eq:a_coeff_expan} \\
b_{l,m}^{(i)}&=&\zeta_{l}^{(i)}\sum\limits_{j\neq
i}\sum\limits_{l^\prime,m^\prime}\left[b_{l^\prime,m^\prime}^{(j)}A_{l,m}
^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right)
+a_{l^\prime,m^\prime}^{(j)}B_{l,m}^{l^\prime,m^\prime}\left(x,\mathbf{r}_j-\mathbf{r}_i\right)\right]\label{eq:b_coeff_expan}\end{aligned}$$ where $\alpha_l^{(i)}$ and $\zeta_l^{(i)}$ are single sphere Mie scattering parameters for $TM$ and $TE$ polarizations respectively. For the TM polarization this parameter, defined in terms of dimensionless frequency parameter $x=R_0\omega/c$, is given by the well-known expression [@Mishchenko_book2002] $$\label{eq:alpha_exact}
\alpha^{(1)}_l= - \frac{{j_l(x){\frac{d}{dx}}[xj_l(nx)]-{{n}^2}j_l(nx){\frac{d}{dx}}[xj_l(x)]}}{h_l(x){\frac{d}{dx}}[xj_l(nx)]-{{n}^2}j_l(nx){\frac{d}{dx}}[xh_l(x)]}$$ where $j_l(x)$ and $h_l(x)$ are Bessel and Hankel functions respectively. For the defect we only need $l=1$ and if $n_dx_d\ll 1$, where $x_d=xR_d/R_0$ the scattering parameter $\alpha^{(2)}_1$ does not have any poles and can be approximated as $$\label{eq:alpha2_approx}
\alpha^{(2)}_1\approx -\left(1+i\frac{3}{2}\frac{1}{p(n_dx_d)^3}\right)^{-1}$$ Explicit expressions for translational coefficients $A_{l,m}^{l^\prime,m^\prime}\left(\mathbf{r}_j-\mathbf{r}_i\right)$ and $B_{l,m}^{l^\prime,m^\prime}\left(\mathbf{r}_j-\mathbf{r}_i\right)$, which describe optical coupling between the spheres via modes of the same or different polarizations respectively, can be found, for instance in Ref. [@Mishchenko_book2002; @FullerApplOpt1991; @MiyazakiPRB2000]. Important property of the translation coefficients is that they take a diagonal form in $m$ if the translation vector is parallel to the polar axis of the coordinate system used to define spherical coordinates. This significantly simplifies the equations for expansion coefficients eliminating summation over the azimuthal number and decoupling equations for coefficients with different $m$. We take advantage of this property by working in the $XYZ$ coordinate system of Fig. \[fig:coordinates\].
For WGM with $l\gg 1$ the cross-polarization translation coefficients are usually much smaller than their same-polarization counterparts. Since we assumed incident wave to be of TM polarization we can set $b_{l,m}=0$ in Eq. \[eq:a\_coeff\_expan\] obtaining as a result a closed system of equation for coefficients $a_{l,m}$.
Dipole approximation
--------------------
The field of the dipole is described by VSHs with $l=1$. Therefore, we introduce the dipole approximation by assuming that $a_{l,m}^{(2)}=0$ for $l>1$. This reduces the system of Eq. \[eq:a\_coeff\_expan\] to a simpler form $$\begin{aligned}
a_{l,m}^{(1)}&=&\alpha_{l}^{(1)}\left\{\eta_{l,m}^{(i)}+a_{1,m}^{(2)}A_{l,m}^{1,m}\left(\mathbf{r}_1-\mathbf{r}_2\right)\right\}\label{eq:a1_coeff_simpl} \\
a_{1,m}^{(2)}&=&\alpha_{l}^{(2)}\sum\limits_{\nu}(-1)^{1+\nu}a_{\nu,m}^{(1)}A_{1,m}^{\nu,m}\left(\mathbf{r}_1-\mathbf{r}_2\right)\label{eq:a2_coeff_simpl}\end{aligned}$$ which can be solved exactly by multiplying Eq. \[eq:a1\_coeff\_simpl\] by $(-1)^{1+l}A_{1,m}^{l,m}$ and summing over $l$. Substituting Eq. \[eq:a2\_coeff\_simpl\] into the resulting expression, we obtain a closed equation for the quantity $\sum_{\nu}(-1)^{1+\nu}a_{\nu,m}^{(1)}A_{1,m}^{\nu,m}$, which can be easily solved. As a result we arrive at Eq. \[eq:a1\_coeff\] and \[eq:a2\_coeff\] of Section \[Results\].
Financial support by AFOSR via grant F49620-02-1-0305, as well as support by PSC-CUNY grants is acknowledged.
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[^1]: A size dispersion of the defects will result in additional inhomogeneous broadening of the resonances, but provided its statistical distribution is uniform would not change the position of the peak
|
---
abstract: 'We give a simple combinatorial criterion allowing to recognize whether a string (or, more generally, a special biserial) algebra is a laura algebra or not. We also show that a special biserial algebra is laura if and only if it has a finite number of isomorphism classes of indecomposable modules which have projective dimension and injective dimension greater than or equal to two, solving a conjecture ok Skowro[ń]{}ski for special biserial algebras.'
address: 'Département de mathématiques, Université de Sherbrooke, 2500, boul. de l’Université, Sherbrooke, Québec, Canada, J1K 2R1'
author:
- Julie Dionne
bibliography:
- 'bibliolaura.bib'
title: Laura string algebras
---
,
laura algebras, string algebras, special biserial algebras
*MSC* : 16S35 ,18E30
Let $k$ be an algebraically closed field. A finite-dimensional $k$-algebra $R$ is biserial if the radical of every projective indecomposable $R$-module is the sum of two uniserial modules whose intersection is simple or zero [@F79]. In 1983, Skowro[ń]{}ski and Waschbüsch characterized biserial algebras of finite representation type by the fact that almost split sequences have at most two non-projective middle terms [@SW83]. The biserial algebras which have at most two middle terms in their almost split sequences are called string algebras (see [@BR87]). The following definition is equivalent:
\[stringAlgebra\] A $k$-algebra $R$ is a string algebra if it admits a presentation $R=kQ/I$ such that :
1. Each point has at most two arrows entering and two arrows exiting;
2. For each arrow $\alpha:x \rightarrow y$ there is at most one arrow $\beta:y \rightarrow z $ such that $\alpha\beta$ is not in $I$ and at most one arrow $\gamma: z \rightarrow x$ such that $\gamma\alpha$ is not in $I$;
3. The ideal $I$ is monomial.
The characterizations of string algebras which are of finite representation type [@BR87], tilted [@HL00], quasi-tilted [@HL99] and shod [@BT05] are known. For a string algebra $R=kQ/I$, we have the following definitions. A walk in the quiver of $R$ is *reduced* if it contains no subwalk of the form $\alpha\alpha^{-1}$ or of the form $\alpha^{-1}\alpha$. A *cycle* is a non-oriented cycle, that is, a reduced walk starting and ending in the same point of the quiver. A walk $\omega$ is a *string* if it is reduced and contains no relation. Given a string $\omega$, the representation admitting a copy of the field $k$ at $x$ for each passage of $\omega$ on $x$ and with the obvious morphisms is the *string module* over $\omega$, denoted by $M(\omega)$. A walk $\omega$ is a *band* if it is a cyclic string which is not the power of another cyclic string and if there is no $n$ such that $\omega^n$ is in $I$.
Laura algebras have been defined independently by Assem and Coelho [@AC03] and Skowro[ń]{}ski [@S03]. Let $A$ be an algebra, its left part is the full subcategory of ind$A$ defined as follows: $$\mathcal L_A=\{M \in \mbox{ind}A \textit{ }|\textit{ }\textit{ for all }\textit{ }
L \rightsquigarrow M,\textit{ }\mbox{dp}L \leq 1\}$$ Its right part is dually defined and is denoted by $\mathcal R_A$.
An algebra $A$ is laura if *ind*$A \setminus (\mathcal L_A \cup
\mathcal R_A)$ contains finitely many objects.
We say that $R$ is *strict laura* if it is laura but not quasi-tilted. Skowro[ń]{}ski conjectured that an algebra $R$ is laura if and only if the number of indecomposable modules with projective and injective dimension greater than or equal to two is finite.
Our aim is to characterize laura string algebras and to show Skowro[ń]{}ski’s conjecture for these algebras.
Let $R \cong kQ/I$ be a string algebra, and $\omega$ be a double-zero on $Q$ (see [@HL99] for instance). We say that $\omega$ is an interlaced double-zero, abbreviated by DOZE, if there exist a band $\omega_2$, two walks $\omega_1$ and $\omega_3$ and two relations $\rho_1$ and $\rho_2$ such that $\omega=\rho_1\omega_1{\omega_2}\omega_3\rho_2$.
Remark that if $\omega=\rho_1\omega_1{\omega_2}\omega_3\rho_2$ is a DOZE, then $\rho_1\omega_1{\omega_2}^n\omega_3\rho_2$ is a double-zero for all $n \geq 0$.
The first two sections will be useful for proving the following theorem. Recall that an algebra is quasi-tilted of canonical type if its bounded derived category is equivalent to the bounded derived category of a category of coherent sheaves coh$\mathbb X$ on a weighted projective line $\mathbb X$ in the sense of Geigle and Lenzing.
\[TheoA\] Let $R=kQ/I$ be a string algebra having no DOZE and such that $Q$ contains a band.
1. If $Q$ has at least one band with at least one arrow entering and at least one arrow exiting, then $R$ is quasi-tilted of canonical type.
2. If $Q$ contains at least one band and is such that each band has either only exiting arrows, or only entering arrows, then $R$ is strict laura or tilted.
In the third section, we prove the validity of Skowro[ń]{}ski’s conjecture for string and more generally for special biserial algebras. Our main results are the following two theorems.
\[TheoPresqueFinal\] Let $R=kQ/I$ be a string algebra. The following are equivalent:
1. There is no DOZE on $(Q,I)$;
2. $R=kQ/I$ is laura;
3. $R$ has only a finite number of isomorphism classes of indecomposable modules having projective and injective dimension greater than or equal to 2.
If $R$ is a special biserial algebra, let $J$ be the minimal ideal containing all paths lying in non-monomial relation. Then $R/J$ is a string algebra.
\[TheoFinal\] Let $R=kQ/I$ be a special biserial algebra. The following are equivalent:
1. There is no DOZE on $(Q,I)$;
2. $R=kQ/I$ is laura;
3. $R$ has only a finite number of indecomposable modules having a projective and an injective dimension greater or equal than 2;
4. $R/J$ is laura.
Preliminaries
=============
A quiver $Q$ is a tuple ($Q_0$, $Q_1$, $s:Q_1\rightarrow Q_0$, $t:Q_1\rightarrow Q_0$). We call $Q_0$ the set of vertices, $Q_1$ the set of arrows, $s(\alpha)$ the source of the arrow $\alpha$ and $t(\alpha)$ the target of the arrow $\alpha$. The algebra $kQ$ is the $k$-vector space generated by all the paths on $Q$ with the multiplication defined by the composition of paths. If $I$ is an ideal of $kQ$, consider the algebra $kQ/I$. We have a complete set of idempotents of $kQ/I$ given by the trivial paths on each vertex $x$, denoted by $\varepsilon_x$. For a finite dimensional, basic and connected $k$-algebra $R\cong kQ/I$, we denote by $\mod R$ the category of finite dimensional left $R$-modules. The indecomposable projective, injective and simple modules associated to $\varepsilon_x$ will be denoted respectively by $P_x$, $I_x$ and $S_x$. A path in mod$R$ from $M$ to $N$ is a sequence $$\xymatrix@R=10pt@C=10pt{
(*)M=M_0\ar[rr]^{f_1}&&M_1\ar[rr]&& \ldots&&\ar[rr]^{f_t}&&M_t=N}$$ of non-zero morphisms between indecomposable modules. The path $(*)$ is called sectional if, for any $i$, we have that $X_{i}\ncong\tau X_{i+2}$. A refinement of $(*)$ is a path $$\xymatrix@R=10pt@C=10pt{
(*)M=X_0\ar[rr]^{g_1}&&X_1\ar[rr]&& \ldots&&\ar[rr]^{g_s}&&X_s=N}$$ with $s\geq r$ such that there is an order-preserving function $\sigma: \{1,$ $2,$ $3,$ $...,$ $t-1\} \rightarrow \{1,$ $2,$ $3,$ $...,$ $s-1\}$ such that $M_i \cong X_{\sigma(i)}$ for all $i$. The path $(*)$ is called sectionally refinable if it has a sectional refinement. It is well known that an algebra is laura if and only if there is only a finite number of isomorphism classes of indecomposable modules on a path going from an injective module to a projective module. Let $R$ be a strict laura or a tilted algebra then $R$ has a unique faithful nonsemiregular component which is quasidirected. Let $(_l {\Gamma_\lambda})_{\lambda \in \Lambda}$ be the left stable parts of the faithful non semiregular components (since $R$ is laura, $\Lambda$ is a finite set). Let $(_l {\Sigma_\lambda})_{\lambda \in \Lambda}$ be complete slices of each one of those stable parts (since each component has a finite number of orbits, such slices exist). For all $\lambda \in \Lambda$, we define $_\infty R_\lambda$ to be the full subcategory generated by the support of $ _l \Sigma_\lambda$. The left end algebra $_\infty R$ of $R$ is by definition the product of the $_\infty R_\lambda$. We define dually the right end algebra $R_\infty$ of $R$.
String algebras without DOZE {#Structure}
============================
Throughout this section, let $R=kQ/I$ be a string algebra. The following lemma will be useful for the next two sections:
\[cyclesdisjoints\] If $R=kQ/I$ has no DOZE, then two bands of $R$ intersect in at most one point.
Proof: We verify that for all cases with more than one point in common, we obtain a DOZE. If $\omega_1$ and $\omega_2$ have a non trivial walk $u$ in common, we have : $$\xymatrix@R=10pt@C=10pt{
\cdot\ar[rdd]^{\beta_1} &&&& \cdot \ar@{~}[llll]_{\omega'_1}&&&&\cdot &&&& \cdot \ar@{~}[llll]_{\omega'_1}\\
&&&&&&&&&&&&\\
& \cdot\ar@{~}[rr]^u \ar[ldd]^{\beta_2}& &\cdot\ar[ruu]^{\alpha_1}&&&or&& & \cdot\ar@{~}[rr]^u \ar[luu]_{\beta_1}& &\cdot\ar[ruu]^{\alpha_1}& \\
&&&&&&&&&&&&\\
\cdot &&&& \cdot \ar@{~}[llll]_{\omega'_2} \ar[luu]^{\alpha_2}
&&&&\cdot\ar[ruu]_{\beta_2} &&&& \cdot \ar@{~}[llll]_{\omega'_2}\ar[luu]^{\alpha_2}
}$$ where $\omega_1= u \alpha_1\omega'_1 \beta_1$ and $\omega_2=u {\alpha_2}^{-1}\omega'_2 {\beta_2}^{-1}$ in the first case and $\omega_1= u \alpha_1\omega'_1 {\beta_1}^{-1}$ and $\omega_2=u {\alpha_2}^{-1}\omega'_2 {\beta_2}$ in the second case. In the first case, $\alpha_2\omega_1'\beta_2$ is a DOZE. In the second, $\alpha_2\omega_1' u \omega_2' \beta_1$ is a DOZE. If $\omega_1$ and $\omega_2$ have no non trivial walks in common, we have one of the following three cases: $$\xymatrix@R=8pt@C=8pt{
1)&\cdot\ar[rdd]_{\alpha_1} &&&&&& \cdot \ar@{~}[llllll]_{\omega'_1}&2)&\cdot\ar[rdd]_{\alpha_1} &&&&&& \cdot \ar@{~}[llllll]_{\omega'_1}&&3)&\cdot\ar[rdd]_{\alpha_1} &&&&&& \cdot \ar@{~}[llllll]_{\omega'_1}\ar[ldd]^{\delta_1}\\
&&&\cdot&&\cdot\ar@{~}[ll]_u\ar[rd]_{\gamma_1}&&&&&&\cdot\ar[dl]^{\beta_1}&&\cdot\ar@{~}[ll]\ar[rd]_{\gamma_1}&&
&&&&&\cdot\ar[dl]^{\beta_1}&&\cdot\ar@{~}[ll]\ar[rd]_{\gamma_1}&&\\
&& \cdot\ar[ru]_{\beta_1} \ar[rd]^{\beta_2}&&& &\cdot\ar[ruu]_{\delta_1}\ar[rdd]^{\delta_2} && && \cdot \ar[rd]^{\beta_2}\ar[ldd]_{\alpha_2}&&& &\cdot\ar[ruu]_{\delta_1}\ar[rdd]^{\delta_2} & &&&& \cdot \ar[rd]^{\beta_2}\ar[ldd]_{\alpha_2}&&& &\cdot\ar[rdd]^{\delta_2}\ar[dl]_{\gamma_2} & \\
&&&\cdot&&\cdot\ar@{~}[ll]\ar[ru]^{\gamma_2}&&&&&&\cdot&&\cdot\ar@{~}[ll]^v\ar[ru]^{\gamma_2}&&&&&&&\cdot&&\cdot\ar@{~}[ll]^v&&\\
&\cdot\ar[ruu]^{\alpha_2} &&&&&& \cdot \ar@{~}[llllll]_{\omega'_2}&&\cdot &&&&&& \cdot \ar@{~}[llllll]_{\omega'_2}& &&\cdot&&&&&& \cdot \ar@{~}[llllll]_{\omega'_2}
}$$ where $\omega_1= \beta_1 u \gamma_1 \delta_1\omega'_1 \alpha_1$ and $\omega_2=\beta_2 v \gamma_2 \delta_2\omega'_2 \alpha_2$ in the first case, $\omega_1= {\beta_1}^{-1} u \gamma_1 \delta_1\omega'_1 \alpha_1$ and $\omega_2=\beta_2 v \gamma_2 \delta_2\omega'_2 {\alpha_2}^{-1}$ in the second case and $\omega_1= {\beta_1}^{-1} u \gamma_1 {\delta_1}^{-1} \omega'_1 \alpha_1$ and $\omega_2=\beta_2 v {\gamma_2}^{-1} \delta_2\omega'_2 {\alpha_2}^{-1}$ in the last case.
Then, in the first case $\alpha_1\beta_1 u\gamma_1\delta_1$ is a DOZE, in the second $\beta_1\beta_2 v\gamma_2\delta_2$ is a DOZE, and in the third $\beta_1\beta_2 v\gamma_2^{-1}\delta_1^{-1}w\alpha_1\alpha_2$ is a DOZE.\
$\square$
Quasi-tilted algebras without DOZE
----------------------------------
Let $R=kQ/I$ be a string algebra without DOZE having a band $\Theta$ with entering arrow $\beta$ and exiting arrow $\alpha$. Let $\beta^+$ be an arrow of $\Theta$ such that $\beta \beta^+ \in I$. Then there is a string $\omega$ on $\Theta$, going from the source of $\beta^+$ to the source of $\alpha$, such that $\omega \alpha$ is a string.
Let $Q$ be a quiver with a band having at least one entering arrow $\beta$ and at least one exiting arrow $\alpha$. If $R=kQ/I$ is a string algebra having no DOZE, then $R$ has no double-zero.
Proof: We denote by $\beta^+$ the arrow such that $\beta\beta^+ \in I$ and by $\alpha^-$ the arrow such that $\alpha^- \alpha \in I$. Suppose that we have a double-zero. If $\beta\beta^+$ is the relation at the beginning of the double-zero then we have a DOZE. Thus, suppose that this is not the case. Then we can find a walk of minimal length between the end point of $\beta^+$ and a point on the double-zero. If the first relation on the walk of minimal length composed with a part of the double-zero points in the same direction than $\beta\beta^+$, we have a DOZE. Otherwise, let $\omega$ be the walk going from $t(\beta)$ to $s(\alpha^-)$ such that $\beta\omega$ is a string. Then we have a DOZE.\
$\square$
Cycles on a string algebra without DOZE
---------------------------------------
We now consider the case where each cycle has only entering arrows or only exiting arrows. We ignore the trivial case where $R$ has only one band with neither entering nor exiting arrow (in this case, $R$ is hereditary). We denote by $\Theta_1, \Theta_2, ..., \Theta_n$ the bands (up to a cyclic permutation) having only exiting arrows. Since $R$ is finite dimensional, a band $\Theta$ has a point $a$ such that all arrows starting at $a$ belong to $\Theta$. We arbitrarily choose such a point on each cycle $\Theta_i$ and denote it by $a_i$. Let $\alpha$ be an exiting arrow of $\Theta_i$. We denote by $\alpha^-$ the arrow of $\Theta_i$ such that $\alpha^-\alpha \in I$. We denote by $\rho'_{(i,\alpha)}$ the minimal reduced walk going from $a_i$ to $t(\alpha)$ and passing through $\alpha^-\alpha$.
We denote by $\Delta_1, \Delta_2, ..., \Delta_m$ the bands (up to a cyclic permutation) having only entering arrows.
Let $R=kQ/I$ and $\omega$ be a string of $(Q, I)$. We denote by $W(\omega)$ the set of strings $\omega'$ of $(Q, I)$ such that $\omega'
= \omega_1\omega\omega_2$. We denote by $D(\omega)$ the subcategory of $(Q, I)$ whose objects are the points $x$ such that there exists a string in $W(\omega)$ passing through $x$ and whose morphisms are the composition of arrows $\alpha$ for which there exists a string of $W(\omega)$ passing through $\alpha$.
Let $R=kQ/I$ be a string algebra such that each band on its quiver has only entering arrows or only exiting arrows. We define $A_i$ to be the subcategory of $R$ whose objects are the points of $D(\varepsilon_{a_i})$ and whose morphisms are given by the linear combinations of paths of arrows of $D(\varepsilon_{a_i})$. The algebra $A_i$ is the quotient of the path algebra of $D(\varepsilon_{a_i})$ by $I \cap D(\varepsilon_{a_i})$. Note that $A_i$ does not depend on the choice of $a_i$.
We define dually the categories $B_j$.
**Example**: Let Q be the quiver $$\xymatrix@R=10pt@C=10pt{
1 && 2 \ar@/^/[ll]^{\rho_1}\ar@/_/[ll]_{\rho_2}&&&&&& & & 10\ar[lld]^{\delta_1}\ar@{.}@/_0.5cm/[lllldd]&& 11\ar@/^/[ll]^{\rho_5}\ar@/_/[ll]_{\rho_6}\ar@{.}@/^0.5cm/[lllld]\\
&&&&5\ar@{.}@/^0.5cm/[llllu]\ar[llu]^{\alpha_1}&&&&8\ar[lld]^{\gamma_1}\ar@{.}@/_0.5cm/[llll]&&&& \\
&&&&&&7\ar@{.}@/_0.5cm/[lllluu]\ar[llu]^{\beta_1}\ar[lld]_{\beta_2}\ar@{.}@/^0.5cm/[lllldd]&&&&&& \\
&&&&6\ar[lld]_{\alpha_2}\ar@{.}@/_0.5cm/[lllld]&&&&9\ar[llu]_{\gamma_2}\ar@{.}@/^0.5cm/[llll]&&&& \\
3&& 4 \ar@/^/[ll]^{\rho_3}\ar@/_/[ll]_{\rho_4} && && &&& & 12\ar[llu]_{\delta_2}\ar@{.}@/^0.5cm/[lllluu]&& 13 \ar@/^/[ll]^{\rho_7}\ar@/_/[ll]_{\rho_8}\ar@{.}@/_0.5cm/[llllu]\\
}$$\
with $I$ the ideal generated by $\alpha_1\rho_1$, $\alpha_2\rho_4$, $\rho_5\delta_1$, $\rho_8\delta_2$, $\beta_i\alpha_i$ for $i$ such that $i\in\{1,2\}$, $\gamma_i\beta_i$ for $i$ such that $\in\{1,2\}$ and $\delta_j\gamma_j=0$ for $j$ such that $j\in\{1,2\}$. Then $R=Q/I$ contains no DOZE. We have $B=B_1\times B_2$ with $B_1$ the full subcategory generated by $\{1,2,5\}$ and $B_2$ the full subcategory generated by $\{3,4,6\}$. We also have $A=A_1\times A_2$ with $A_1$ the full subcategory generated by $\{8,10,11\}$ and $A_2$ the full subcategory generated by $\{9,12,13\}$.
The categories $D(\varepsilon_{a_i})$ are full in $R$.
Proof: By contradiction, suppose that $x$ and $y$ are objects of $A_i$ and let $\gamma: x \rightarrow y$ be an arrow which is not in $D(\varepsilon_{a_i})$. Then there exist strings $\omega: a_i \rightsquigarrow x$ and $\omega': a_i \rightsquigarrow y$, and since $\gamma$ is not in $A_i$, $\omega$ and $\omega'$ are such that $\omega \gamma$ and $\omega'\gamma^{-1}$ are reduced walks containing a zero-relation. But in this case, the zero-relation must contain $\gamma$ and thus $\omega= u\alpha_1...\alpha_n$ and $\omega'= u'\alpha_{m}^{-1}...\alpha_{n+2}^{-1}$, with $\alpha_1...\alpha_n\gamma$ and $\gamma\alpha_{n+2}...\alpha_m$ some relations. Thus, $\gamma\omega'^{-1}\Theta_i\rho'_{(i,\alpha)}$ is a DOZE, a contradiction. We can apply the same proof to every morphism in the category $R$.\
$\square$
\[aiaGauche\] Let $x \in (Q_{A_i})_0$ and $y \notin (Q_{A_i})_0$. Then there is no arrow $\alpha:y \rightarrow x$. As a consequence, if a relation $\rho$ does not start in $(Q_{A_i})_0$ then it does not end in $(Q_{A_i})_0$.
Proof: Suppose that such an arrow exists. Let $\omega$ be a string from $a_i$ to $x$. Then $\omega\alpha^{-1}$ is not a reduced walk or else contains a zero-relation. If it is not a reduced walk, then $\omega=\omega'\alpha$, and $y \in (Q_{A_i})_0$, a contradiction. If it contains a zero-relation, then $\omega=\omega'\beta_n^{-1}...\beta_1^{-1}$ with $\alpha\beta_1...\beta_n$ in $I$ and so $\alpha\beta_1...\beta_n\omega'^{-1}\rho'_{i,\gamma}$, with $\gamma$ an exiting arrow of $\Theta_i$, is a DOZE, another contradiction.\
$\square$
The categories $D(\varepsilon_{a_i})$ are convex in $R$.
Proof: This follows directly from lemma \[aiaGauche\].\
$\square$
\[aucunCycle\] Each of the categories $D(\varepsilon_{a_i})$ contains only one simple cycle. In particular, it has no oriented cycle. Therefore, no relation $\rho$ starts in $(Q_{A_i})_0$ and ends outside the cycle $\Theta_i$.
Proof: Suppose that there are two distinct cycles $\Theta$ and $\Theta_i$ in $D(\varepsilon_{a_i})$. If $\Theta$ contains a relation, we have a DOZE by gluing the relation with a walk relating the two cycles and which is chosen such that the two relations point in the same direction (this is possible because $\Theta$ and $\Theta_i$ are cycles). If not, we have a cycle with an exiting or an entering arrow and so we have a relation of the form $\alpha^-\alpha$ or of the form $\beta\beta^+$. In each case, we can construct a DOZE. The last statement follows from Lemma \[aiaGauche\].\
$\square$
Recall that an algebra $R$ is left glued if $\mathcal R_R$ is cofinite in ind$R$ (see [@AC94]).
\[AlgebreInclinee\] For all $i$, with $1
\leq i \leq n$, the algebra $A_i$ is a tilted algebra containing a complete slice in its postprojective component. In particular, it is left glued and all but a finite number of isomorphism classes of indecomposable $A_i$-modules are in $\mathcal R_{A_i}$.
Proof: Let $\alpha$ be an arrow from $x$ to $y$ in $A_i$. We want to show that the projective $A_i$-modules $P_x$ and $P_y$, with respective tops $S_x$ and $S_y$, are in the same component of the Auslander-Reiten quiver of $A_i$. The radical of $P_x$ is the direct sum of at most two terms, one of them is a uniserial module $L$ with top $S_y$. Moreover, there is an epimorphism $f$ from $P_y$ to $L$. Since the inclusion of the radical of a projective module into this projective module is irreducible, we only have to show that $f$ lies in a finite power of the radical of the module category.\
Suppose $f_1:M\rightarrow L$ and $f_2:P_y\rightarrow M$ is a factorisation of $f$, where $M$ may be decomposable. $$\xymatrix@R=10pt@C=10pt{
P_y\ar[rr]^{f_2} && M\ar[rr]^{f_1} && L\ar[rr] && P_x}.$$ Since $f$ is an epimorphism, $L$ is a quotient of $M$. Let $z_1$ belong to the support of $L$, $z_2$ belong to the support of $M$ but not to that of $L$ and $\beta$ be an arrow from $z_1$ to $z_2$. Since $L$ is a direct summand of rad$P_x$, there exists a walk $\omega'$ such that $\omega'\beta$ is in the support of $M$ and $\alpha\omega'\beta \in I$ ($\omega'$ may be trivial).\
We have a relation of $I$ starting at $x$, thus $x$ is on the cycle $\Theta_i$ by Lemma \[aucunCycle\]. Since there is no relation on a band, $\beta$ is not on a cycle (by Lemma \[aucunCycle\]), and every arrow on a reduced walk beginning with $\alpha\omega'\beta$ and being the successor of $\beta$ is not in $\Theta_i$.\
Thus, every arrow of the support of $M$ lies in the support of $L$ or is not in $\Theta_i$. Since $\Theta_i$ is not included in the support of $L$, it is not in the support of $M$.\
The $k$-dimensions of the direct summands of $M$ are bounded above since the support of $M$ contains no band.\
We showed that there exist only a finite number of isomorphism classes of indecomposable modules which are direct summands of a module through which $f$ factors.\
Now, we only have to see that $A_i$ contains no double-zero (otherwise we have a DOZE since we have a string between every point of $A_i$ and $\Theta_i$). So $A_i$ is tilted (see [@HL00]) and we have our result by Theorem 3.4 of [@HL00].\
$\square$
Let $E$ be the set of strings which are not in one of the subcategories $A_i$ or $B_j$. We define the middle part $C$ to be the subcategory which has as set of objects $\bigcup_{\omega \in E}(D(\omega))_0$ and as set of morphisms $\Sigma_{\omega \in E}(D(\omega)(x,y))$ from $x$ to $y$.
\[remarqueCfini\] The algebra $C$ is of finite representation type since it contains no band and it is a string algebra (see [@BR87]).
Module Category
---------------
We now begin the study of the category of indecomposable modules of $R$, with the aim to prove that it is a laura algebra under assumptions of this subsection. We denote by $A=\prod_{i=1}^n A_i$ the emph[right side algebra]{} of $R$ and by $B=\prod_{j=1}^m B_j$ the emph[left side algebra]{} of $R$.
$\emph{ind}R=\emph{ind}A\cup\emph{ind}B\cup\emph{ind}C$
Proof: Every $A_i$-module or $B_j$-module can be considered as an $R$-module since $A_i$ and $B_j$ are full and convex subcategories of $R$, by completing their representations by zeros. Every $C$-module is a string module and thus is a string $R$-module.\
Let $M$ be an indecomposable $R$-module and $\omega$ be a string lying in the support of $M$. Then each point and each arrow of the support of $M$ are in $D(\omega)$. Suppose that all points of $M$ are in one of the $A_i$, or in one of the $B_j$, but not all in the same. Let $x$ be a point of $A_i$ which is not in $B_j$ and $y$ be a point of $B_j$ which is not in $A_i$. Then the string $\omega: x \rightsquigarrow y$ cannot be in $A_i$, nor in $B_j$, and thus by definition lies in $C$. $\square$
\[fermeSucc\] Let $M$ be an indecomposable $A_i$-module, $N$ be an indecomposable $R$-module which is not an $A_i$-module and $f:M \rightarrow N$ be a non-zero morphism. Then $M$ is a $C$-module or $M$ is a $B$-module.
Proof: Suppose that $M$ is neither a $B$-module, nor a $C$-module. Then the support of $M$ contains at least a point $z'$ which is not an object of $B\cup C$. So it is an object of $A_i$ for some $i$.\
The image of $f$ is a submodule of $N$, so there exist a string $\omega$ going from $z'$ to $x$ and an arrow $\alpha:y\rightarrow x$, with $x$ in the support of the image of $f$ and $y$ in the support of $N$ but not in the support of the image of $f$.\
Since $N$ is not an $A$-module, the support of $N$ contains at least a point $z$ which is not an object of $A_i$. Since $z$ and $y$ are in the support of $N$, there exists a string $\nu$ going from $y$ to $z$ of the form. $$\xymatrix@R=10pt@C=10pt{
z'\ar@{~}[rrrr]^{\omega}&&&&x&&y\ar[ll]^{\alpha}\ar@{~}[rrrr]^{\nu}&&&&z}.$$ But $z'$ is not in $B\cup C$ and $z$ is not in $A_i$. Then there is no string between them, since such a string would be neither in $A$, nor in $B$, so it would be in $C$. If $\omega\alpha^{-1}\nu$ is a reduced walk, it contains a relation and this relation must contain $\alpha$. But in this case, we obtain a relation entering in $A_i$, a contradiction.\
Thus $\omega\alpha^{-1}\nu$ is not reduced and the relation between $z$ and $z'$ goes in the other direction. Suppose that $\alpha_1\alpha_2...\alpha_n$ is going from $z'^*$ to $z^*$, as illustrated bellow. $$\xymatrix@R=10pt@C=10pt{
z'^*\ar[rr]^{\alpha_1}\ar@{.}@/^0.5cm/[rrrrrrrr]&&...\ar[rr]^{\alpha_i}&&\cdot\ar@{~}[d]^{\eta}\ar[rr]^{\alpha_{i+1}}&&...\ar[rr]^{\alpha_n}&&z^*\\
&&&&x&&&&\\
&&&&y\ar[u]&&&&}.$$ Since $\omega$ and $\nu$ are strings, so are $\alpha_1...\alpha_i\eta$ and $\eta^{-1}\alpha_{i+1}...\alpha_n$. Since $R$ is a string algebra, this implies that $n=2$. $$\xymatrix@R=10pt@C=10pt{
z'^*\ar[rr]^{\alpha_1}\ar@{.}@/^0.5cm/[rrrr]&&\cdot\ar@{~}[d]^{\eta}\ar[rr]^{\alpha_{2}}&&z^*\\
&&x&&\\
&&y\ar[u]&&}.$$ The target of $\alpha_1$ is on $\omega$ and on $\nu$, so it is in the support of Im$f$. Since Im$f$ is a quotient of $M$, $z'^*$ is in Im$f$. And since Im$f$ is a submodule of $N$, $z^*$ is in Im$f$, a contradiction to the fact that $\alpha_1\alpha_2 \in I$ and to Lemma \[aucunCycle\].\
$\square$
**Proof of Theorem \[TheoA\]:** The first part of Theorem \[TheoA\] follows now directly from [@HL00] (Theorems 2.6 and 3.4) and [@H01] (Theorem 3.1).
For the second part, the last proposition with the fact that\
$|$ind$A_i\cap$ ind$B_j|\leq |$ ind$(A_i\cap B_j)|<\infty$ say that there exists only a finite number of isomorphism classes of indecomposable $A_i$-modules which admit successors from ind$R \setminus$ ind$A_i$. Let $\mathcal X_i$ be the subcategory of ind$A_i$ which contains modules without successors from ind$R \setminus$ ind$A_i$, that is $$\mathcal X_i=\{X\in \textit{ind}A_i\textit{ | for all }0\neq f:X\rightarrow Y\textit{ in ind}R\textit{, }Y \in \textit{ ind}A_i\}.$$ We have that $\mathcal X_i$ is such that if $M$ is an $A_i$-module in $\mathcal X_i$ and if there is a non-zero morphism $f$ from $M$ to $M'$, then $M'$ is in $\mathcal X_i$. We define dually $\mathcal Y_j$.
We have that if $f:M \rightarrow N$ is a morphism, where $M$ and $N$ are objects of $\mathcal X_i$, then $f$ is irreducible in $A_i$ if and only if $f$ is irreducible in $R$, since the morphisms are preserved (this follows from the fullness of the categories $A_i$ and $B_j$ and from the fact that there are no points lying in $A_i \cap A_j$ if $i \neq j$). By studying the Auslander-Reiten translation in $A_i$ and in $R$, we can show that it is preserved. Take $$\xymatrix@R=10pt@C=10pt{
(*): &&0\ar[rr]&&M\ar[rr]&&_RI_0\ar[rr]^{i_1}&&_RI_1}$$ a minimal injective resolution of $M$ in $R$. We know that $\tau_R^{-1}(M)=\mbox{Coker }\nu^{-1}(i_1)$, where $\nu$ denotes the Nakayama functor. $$\xymatrix@R=10pt@C=10pt{
\nu^{-1}(_RI_0)\ar[rr]^{\nu^{-1}(i_1)}&&\nu^{-1}(_RI_1)\ar[rr]&&\mbox{Coker }\nu^{-1}(i_1)}$$ We compute $\tau_{A_i}^{-1}(M)$ in the same way. In general, $\nu^{-1}(_{A_i}I_0) $ is a quotient of $\nu^{-1}(_RI_0)$ and $\nu^{-1}(_{A_i}I_1)$ is a quotient of $\nu^{-1}(_RI_1)$. The kernels $K_0$ and $K_1$ of those projections contain no points of $A_i$ in their supports. $$\xymatrix@R=15pt@C=15pt{
0\ar[d]&&0\ar[d]&&\\
K_0\ar[d]\ar[rr]&&K_1\ar[d]\ar[rr]&&L\ar[d]\ar[rr]&&0\\
\nu^{-1}(_RI_0)\ar[rr]^{\nu^{-1}(i_1)}\ar[d]&&\nu^{-1}(_RI_1)\ar[rr]\ar[d]&&\tau^{-1}_R(M)\ar[d]^g\ar[rr]&&0\\
\nu^{-1}(_{A_i}I_0)\ar[rr]^{\nu^{-1}(i_1)}\ar[d]&&\nu^{-1}(_{A_i}I_1)\ar[rr]\ar[d]&&\tau^{-1}_{A_i}(M)\ar[rr]&&0\\
0&&0&&}$$ Let $L$ be the cokernel of the induced morphism from $K_0$ to $K_1$. It contains no points of $A_i$. But $\tau^{-1}_R(M)$ is an $A_i$-module and so $L$ is zero and $g$ is a monomorphism. It is also an epimorphism. Thus a path in $\mathcal X_i$ is sectionally refinable in $\mathcal X_i$ if and only if it is also in ind$R$.
Moreover, let $P$ be an $A_i$-modules in $\mathcal X_i$, if $P$ is $R$-projective, then it is $A_i$-projective. If $P$ is a $R$-projective module, than for every $R$-epimorphism, and in particular for every $A_i$-epimorphism $f:M\rightarrow N$ with $g: P\rightarrow N$, there exists an $R$-morphism $h : P\rightarrow M$ such that $hf=g$. Since $h$ is a morphism between two $A_i$-modules, it is a morphism in mod$A_i$. On the other hand, an $A_i$-module $I$ in $\mathcal X_i$ is an injective $R$–module if and only if $I$ is an injective $A_i$-module (this follows from Lemma \[aiaGauche\] and from the construction of injective modules).
Now, let $(Q,I)$ be a string bound quiver having a band and such that each band has only exiting arrows or only entering arrows. Let $R=kQ/I$ have no DOZE. We show that if $M$ is not in $\mathcal L_R \cup \mathcal R_R$, then one of the following conditions is satisfied:
1. The module $M$ is in ind$C$;
2. There exist $i$ and $j$ such that $1 \leq i \leq n$ and $1 \leq j \leq m$, and such that the module $M$ is in ind$A_i \cap$ind$B_j$;
3. The module $M$ is in ind$A_i$, but not in $\mathcal R_{A_i}$;
4. The module $M$ is in ind$B_j$, but not in $\mathcal L_{B_j}$.
If $M$ does not satisfy the first two conditions, then by Proposition \[fermeSucc\] there exists $i$ such that $M$ is in $\mathcal X_i$ or $j$ such that $M$ is in $\mathcal Y_j$. In the first case, if $M$ is not in $\mathcal R_R$, then there exists a path of $R$-morphisms which is not sectionally refinable from $M$ to an $R$-projective indecomposable module. From the results above, we obtain that $M$ is not in $\mathcal R_{A_i}$. In the second, if $M$ is not in $\mathcal L_R$, then there exists a path of $R$-morphisms which is not sectionally refinable from an $R$-injective indecomposable module to $M$. We obtain that $M$ is not in $\mathcal L_{B_j}$. We have shown our statement.
In the four cases, $M$ belongs to a finite set and thus all but a finite number of isomorphism classes of modules of ind$R$ are in $\mathcal L_R \cup \mathcal R_R$. In particular, $R$ is laura. It cannot be quasi-tilted of canonical type since none of its bands have entering and exiting arrows (see Theorems 2.6 and 3.4 of [@HL00]).\
$\square$
\[Domestique\] Let $(Q, I)$ be a string bound quiver having bands and such that each band has only exiting arrows or only entering arrows. Then, if $R=kQ/I$ is a strict laura or a tilted string algebra without DOZE, its left and right end algebras are tilted of type $\widetilde{\mathbb{A}}_n$. Consequently, each string strict laura or tilted algebra without DOZE is domestic.
Proof: We have that $\mathcal X_i$ is such that if $M$ is in $\mathcal X_i$ and there exist a non-zero morphism from $M$ to $M'$, then $M'$ is in $\mathcal X_i$, and such that only a finite number of $A_i$-modules are not in $\mathcal X_i$. Therefore, if $R$ is not quasi-tilted, we have that $\mathcal X_i$ contains a complete slice of mod$A_i$ and this complete slice gives us a connected component of the right end algebra of $R$. Moreover, the right end algebra cannot contain another factor (in this case, by the arguments of the proof of Theorem \[TheoA\], we obtain an infinite number of $R$-modules which are not in $\mathcal L_R \cup \mathcal R_R$). The domesticity of $R$ follows from [@AC03].\
$\square$
DOZED string modules
====================
We now study string algebras whose bound quiver contains at least a DOZE.
Let $R=kQ/I$ be a string algebra with a DOZE $$\alpha_1...\alpha_l\omega_1{\omega_2}^n\omega_3\beta_m...\beta_1$$ for some walks $\alpha_1\ldots\alpha_l$, $\omega_1$, $\omega_3$, $\beta_m...\beta_1$ and a band $\omega_2$. The string module $M_n = M(\sigma_n)$ corresponding to the string $$\sigma_n=\alpha_3...\alpha_l\omega_1{\omega_2}^n\omega_3\beta_m...\beta_{3}$$ is called the DOZED module of power $n$. By convention, $\alpha_3...\alpha_l$ represents the trivial path if the relation $\alpha_1...\alpha_l$ is of length 2.
\[TheoC\] Let $R$ be a string algebra containing a DOZE. Then the DOZED string modules have projective and injective dimension greater than one. Thus, $R$ is not laura.
Proof: Let $\rho_1\omega_1\omega_2\omega_3\rho_2$ be a DOZE where $\omega_2$ is a non-trivial band, $\rho_1=\alpha_1...\alpha_l$ and $\rho_2=\beta_m...\beta_1$. Let $M_n=M(\sigma_n)$ be the DOZED module and $I$ the injective envelope of $M_n$.\
Let $\alpha_3...\alpha_l\alpha_{l+1}...\alpha_k$ be the longest path such that $\sigma_n=\alpha_3...\alpha_l\alpha_{l+1}...\alpha_k\sigma'_n$ and $x_3$, $x_4$, ... $x_{k+1}$ the vertices of this path. Then $x_{k+1}$ is a sink of $\sigma_n$ and thus $I_{x_{k+1}}$ is a direct summand of $I$. Let $\pi_1$ be the canonical projection on $I_{x_{k+1}}$, $M_n(x)$ and $M_n(\alpha)$ be the vector spaces and the linear maps corresponding to the vertex $x$ and the arrow $\alpha$ in the representation $M_n$, respectively and $f_{x_i}$ be the linear map associated to $x_i$ in the representation of the morphism $f$. Then we have the following identities:
- $\pi_1\circ M_n(\alpha_i)={\pi_1}_{x_i}$ for every $i$ such that $3 \leq i \leq k$;
- $\pi_1\circ M_n(\alpha_2)=0$ by definition of $M_n$;
- $\pi_1\circ I(\alpha_i)={\pi_1}_{x_i}$ for every $i$ such that $2 \leq i \leq k$, since $\alpha_i...\alpha_k$ is the first non-zero path going to $t(\alpha_k)$ which is a sink of $\sigma_n$;
- $\pi_1\circ I(\alpha_1)=0$ since $\alpha_1...\alpha_l$ is a relation ending in $x_{l+1}$ and of minimal length for this property.
Let $\iota$ be the inclusion of $M_n$ in $I$, $\iota_x$ the linear map induced by $\iota$ between $M_n(x)$ and $I(x)$, $C=$Coker$ \iota$ and $c=$coker$ \iota$. Then:
- $\pi_1\circ \iota_{x_i}={\pi_1}_{x_i}$ for every $i$ such that $3 \leq i \leq k$;
- $\pi_1\circ \iota_{x_2}=0$, by applying $\pi_1$ to the equation $\iota_{x_3}\circ M_n(\alpha_2)=I(\alpha_2)\circ \iota_{x_2}$;
- $\pi_1(C(x_2))\cong k$ since $\pi_1\circ \iota_{x_2}=0$ and since $\pi_1(I(x_2))\cong k$;
- $\pi_1 \circ c_{x_2}={\pi_1}_{x_i}$;
- $\pi_1 \circ c_{x_3}=0$;
$$\xymatrix@R=10pt@C=30pt{
&\cdot\ar[rr]\ar[ddd]&&\cdot\ar[rr]^{0}\ar[ddd]^{0}&&
\cdot\ar[rr]^{id_k}\ar[ddd]^{id_k}&&\cdot\ar[ddd]^{id_k}\\
M_n(x_1)\ar[rr]^{M_n(\alpha_1)}\ar[ddd]^{\iota_{x_1}}\ar[ru]^{\pi_1}&&
M_n(x_2)\ar[rr]^{M_n(\alpha_2)}\ar[ddd]^{\iota_{x_2}}\ar[ru]^{\pi_1}&&
M_n(x_3)\ar[rr]^{M_n(\alpha_3)}\ar[ddd]^{\iota_{x_3}}\ar[ru]^{\pi_1}&&
M_n(x_4)\ar[ddd]^{\iota_{x_4}}\ar[ru]^{\pi_1}&\\
&&&&&&&\\
&\cdot\ar[rr]^{0 \qquad}\ar[ddd]&&\cdot\ar[rr]^{id_k
\qquad}\ar[ddd]^{id_k}&&
\cdot\ar[rr]^{id_k \qquad}\ar[ddd]^{0}&&\cdot\ar[ddd]\\
I(x_1)\ar[rr]^{I(\alpha_1)}\ar[ddd]^{c_{x_1}}\ar[ru]^{\pi_1}&&
I(x_2)\ar[rr]^{I(\alpha_2)}\ar[ddd]^{c_{x_2}}\ar[ru]^{\pi_1}&&
I(x_3)\ar[rr]^{I(\alpha_3)}\ar[ddd]^{c_{x_3}}\ar[ru]^{\pi_1}&&
I(x_4)\ar[ddd]^{c_{x_4}}\ar[ru]^{\pi_1}&\\
&&&&&&&\\
&\cdot\ar[rr]^{0 \qquad}&&k \ar[rr]^{0 \qquad}&&\cdot\ar[rr]&&\cdot\\
C(x_1)\ar[rr]^{C(\alpha_1)}\ar[ru]^{\pi_1}&&C(x_2)\ar[rr]^{C(\alpha_2)}\ar[ru]^{\pi_1}&&
C(x_3)\ar[rr]^{C(\alpha_3)}\ar[ru]^{\pi_1}&&C(x_4)\ar[ru]^{\pi_1}&\\
}$$
By applying $\pi_1$ to the equation $C(\alpha_1)\circ c_{x_1}=c_{x_2}\circ I(\alpha_1)$, we obtain that $\pi_1\circ
C(\alpha_1)\circ c_{x_1}=\pi_1 \circ c_{x_2} \circ I(\alpha_1)=\pi_1 \circ I(\alpha_2)=0$. Since $c_{x_1}$ is an epimorphism, we have $\pi_1 \circ C(\alpha_1)=0$. We show in the same way that $\pi_1 \circ C(\alpha_2)=0$, and that $\pi_1 \circ C(\beta)=0$ for every arrow $\beta$ having $x_2$ for its source.\
Thus $C$ admits an indecomposable direct summand whose support does not contain $x_1$ and admits $x_2$ as a sink. So this direct summand (and thus $C$) is not an injective module.\
$\square$
**Proof of Theorem \[TheoPresqueFinal\]:** a) implies b):
1. If there is no band, the algebra is of finite representation type by [@BR87], and we are done;
2. Otherwise, Theorem \[TheoA\] gives us the statement.
The statement b) implies trivially c). The statement c) implies a) by Theorem \[TheoC\].\
$\square$
We now generalize Theorem \[TheoPresqueFinal\] to special biserial algebras.
Laura special biserial algebras and Skowro[ń]{}ski’s conjecture
===============================================================
\[biserielleSpeciale\] Let $R$ be an algebra. It is a special biserial algebra if it admits a presentation $R=kQ/I$ such that :
1. Each point has at most two arrows entering and two arrows exiting;
2. For each arrow $\alpha:x \rightarrow y$ there is at most one arrow $\beta:y \rightarrow z $ such that $\alpha\beta$ is not in $I$ and at most one arrow $\gamma: z \rightarrow x$ such that $\gamma\alpha$ is not in $I$.
For a special biserial algebra $R$, we denote $J$ the ideal generated by the paths appearing in the commutativity relations of $I$. Then, $R/J$ is a string algebra.
Let $R=kQ/I$ be a special biserial algebra and $(\rho_1,\rho_2)$ be two consecutive zero-relations on $Q$. We say that they form a DOZE if they do so on $R/J$.
By [@WW85], if $R$ is special biserial, any indecomposable $R$-module which is not projective-injective is also an $R/J$-module. This implies that any indecomposable $R$-module which is not projective-injective is a string module or a band module. Moreover, the restriction of scalars functor mod$R/J \rightarrow$ mod$R$ is full, faithful and sends an irreducible morphism in mod$R/J$ onto an irreducible morphism in mod$R$.
**Proof of Theorem \[TheoFinal\]:** The proof that (b) implies (c) follows from the definition of laura algebras.\
Suppose that there exists a DOZE on $(Q,I)$ and let $M_n$ be a DOZED module on $R/J$. Then $M_n$ is also an $R$-module. We have shown in the proof of Theorem \[TheoC\] that the injective dimension over $R/J$ of $M_n$ is greater than or equal to two by building $I_{x_{l+1}}$ and the cokernel $C$. Remark that a DOZE on $R$ contains no path involved in a binomial relation. By applying the same technique, we show that the injective dimension of $M_n$ over $R$ is greater than or equal to two and we get that c) implies a).\
The proof that a) implies d) follows from Theorem \[TheoC\].\
For the last implication, let $R$ be a special biserial algebra such that $R/J$ is laura. Let $M$ be a non projective-injective $R$-module which is not in $\mathcal L_R \cup \mathcal R_R$. Then there exist a non sectionally refinable path (\*) going from $M$ to a $R$-projective module $P$ and a non sectionally refinable path (\*\*) going from an $R$-injective module $I$ to $M$. $$\xymatrix@R=10pt@C=10pt{
(*)&M\ar[rd] &&\tau^{-1} M \ar@{~>}[rr]&& P &&(**)&I\ar@{~>}[rr] &&\tau M\ar[rd] &&M \\
&&\cdot \ar[ru]&&&&&&&&&\cdot \ar[ru]&}$$\
The fact that (\*) and (\*\*) are not sectionally refinable is preserved in $R/J$ since an almost split sequence admitting a projective-injective middle term has at least another middle term [@ASS97]. Moreover, if $P$ and $I$ are not projective-injective modules, then they are $R/J$-projective and $R/J$-injective modules respectively. If $P$ is projective-injective, then $P/$soc$P$ is an $R/J$-projective module. We only have to compose (\*) with the left minimal almost split morphism going from $P$ to $P/$soc$P$, and we get the desired path from $M$ to an $R/J$-projective module. $$\xymatrix@R=10pt@C=10pt{
M\ar@{~>}[rr]&& P\ar[r]&P/\emph{soc}P }$$ In all cases, if $M$ is a non projective-injective $R$-module which is not in $\mathcal L_R \cup \mathcal R_R$, then it is not in $\mathcal L_{R/J} \cup \mathcal R_{R/J}$, and so the set of indecomposable $R$-modules which are not in $\mathcal L_R \cup \mathcal R_R$ is finite.\
$\square$
We conclude with an example.
**Example:** Let $R=kQ/I$, where $Q$ is the following quiver: $$\xymatrix@R=10pt@C=10pt{
x_1\ar@{.}@/_0.5cm/[rrd]\ar[r]^{\alpha_1}& x_3\ar[rd]^{\beta_1}&& x_6\ar[r]^{\delta_1} &x_8 \\
&&x_5\ar[ru]^{\gamma_1}\ar[rd]^{\gamma_2}\ar@{.}@/^0.5cm/[rrd]\ar@{.}@/_0.5cm/[rru]& &&\\
x_2\ar@{.}@/^0.5cm/[rru]\ar[r]^{\alpha_2}& x_4\ar[ru]_{\beta_2}&& x_7\ar[r]^{\delta_2}&x_9
}$$\
and $I$ is generated by $\alpha_i\beta_i$ and $\gamma_i\delta_i$. This algebra is not a string algebra, but by applying the evident action of the group $\mathbb Z_2$, we obtain the skew group algebra $A[G]$ given by the following quiver: $$\xymatrix@R=10pt@C=10pt{
& & x_4\ar@{.}@/^0.5cm/[rrd]\ar[rd]^{\gamma_1} & & \\
x_1\ar@{.}@/^0.5cm/[rru]\ar@{.}@/_0.5cm/[rrd]\ar[r]^{\alpha} & x_2\ar[ru]^{\beta_1}\ar[rd]_{\beta_2} & & x_5\ar[r]^{\delta} & x_6\\
& & x_3\ar@{.}@/_0.5cm/[rru]\ar[ru]_{\gamma_2}&&
}$$\
where $I$ is generated by $\alpha\beta_1$, $\alpha\beta_2$, $\gamma_2\delta$ and $\gamma_1\delta$. It is a string algebra which contains the DOZE $\alpha\beta_1\gamma_1\delta$, this latter algebra is not laura by Theorem \[TheoPresqueFinal\] and so the initial algebra $R$ is not laura by [@ALR07].
[**ACKNOWLEDGEMENTS.**]{} The author warmly thanks Ibrahim Assem as well as Patrick Le Meur and Shiping Liu for their suggestions. She also thanks ISM and Sherbrooke University for their financial support.
|
---
abstract: '[The terrestrial fossil record shows that the exponential rise in biodiversity since the Precambrian period has been punctuated by large extinctions, at intervals of $40$ to $140$ Myr. These mass extinctions represent extremes over a background of smaller events and the natural process of species extinction. We point out that the non-terrestrial phenomena proposed to explain these events, such as boloidal impacts (a candidate for the end-Cretaceous extinction), and nearby supernovae, are collectively far more effective during the solar system’s traversal of spiral arms. Using the best available data on the location and kinematics of the Galactic spiral structure (including distance scale and kinematic uncertainties), we present evidence that arm crossings provide a viable explanation for the timing of the large extinctions.]{}'
author:
- 'Erik M. Leitch & Gautam Vasisht'
title: 'Mass Extinctions and The Sun’s Encounters with Spiral Arms'
---
The literature is replete with suggestions of non-terrestrial phenomena as the candidate causes for large scale extinctions. The most frequently invoked are supernovae and boloidal impacts (e.g. comets from the Oort Cloud), the latter a strong candidate for the K/T extinction ever since the discovery of the Iridium anomaly in the K/T boundary clay [@ALV90]. Certainly the most violent events in the solar neighborhood during geologic history would have been supernovae (barring the possibility of a nearby $\gamma$-ray burst, which is far less likely; Thorsett 1995); that the structure of the very local interstellar medium is considered to be the result of a supernova, possibly related to the Geminga pulsar (at 150 pc) [@BIG96], is an impressive reminder of their potential impact. Supernovae and young supernova remnants, especially those occurring at distances $\simlt 10$ pc [@RUD74], can result in biospheric imbalance through a variety of processes, including ozone depletion by enhanced ionizing radiation and cosmic rays ([@SCH95], [@KOY95]), the absorption of visible light by the formation of NO$_2$ [@CRU96], and in rare cases the direct deposition of supernova debris.
Tidal and collisional encounters with intermediate-sized gas or dust clouds might focus cometary activity ($\sim 10^9$ comets) to the inner solar system, by scattering of Oort Cloud member bodies, a mechanism proposed to explain the K/T boundary event. In addition, the passage of the Sun through a cloud of density $n \simgt
10^4$ cm$^{-3}$ could raise the solar luminosity significantly, through Bondi accretion, as well as raise the opacity of Earth’s atmosphere, directly affecting the insolation on Earth [@McC75]. While the recent association of the Chicxulub crater with the K/T boundary lends credence to the boloidal impact model, the large concentration of Ir deposited at the boundary may indicate that accretion also played an important role [@YABU].
The proposed mechanisms constitute a set of plausible external agents for any one extinction, yet do not of themselves suggest any explanation for the timing of the mass extinctions, or for the large variation in severity of the observed extinctions. Hatfield and Camp [@HAT81] were among the first to suggest that extinctions might be correlated with Galactic-plane crossing due to the solar orbit’s vertical oscillations. Rampino and Stothers (1984), as well as Schwartz and James [@SCH84], have invoked these z-oscillations in connection with the suggested $\sim 26 - 30$ Myr periodicity of minor extinctions, as virtually all of the postulated extinction mechanisms concentrate toward the Galactic plane. However, the fact that we are presently half-way between extinction cycles and that the Sun’s position is nearly midplane implies that Galactic plane passages are unlikely causes for the extinctions, unless the Sun has suffered a violent gravitational encounter in the last 15 Myr ([@CLU96]). Moreover, even if the correlation were exact, it does not explain the enormous difference in magnitude between the 6 largest extinctions and extinctions which occur on 30 Myr timescales.
While acknowledging that the apparent quasi-periodicity of mass extinctions may in fact be spurious, and that the extinction record may be one of chance encounters of varying magnitude, or indeed merely a record of terrestrial cataclysms, we suggest that the spiral arm environment of the Galaxy provides a natural framework in which all of the astrophysical mechanisms discussed thus far would operate most efficiently. Pre-supernova stars (the luminous O and B stars) are born primarily in the spiral arms, and spend much of their short lifetimes ($\simlt2\times10^7$ yr) in their vicinity. The Type II/Ib supernovae, which are a consequence of the core-collapse of OB stars, have a Galactic rate of roughly $R_{S\!N} \approx 1/30$ yr$^{-1}$ [@VAN91] and are distributed with a scale height $z \simeq 10^2$ pc in the disk. The longest lifetime of a pre-supernova star is $\tau
\simeq 2\times 10^7$ yr (for masses $M \simgt 8-9$ ). Defining the effective distance over which a supernova may have a profound impact on the biosphere as $l_k\simeq 10$ pc ([@SCH95]), the number of significant supernova encounters at the solar Galactic radius is $N_{S\!N} \simeq R_{S\!N}l_k^3\tau/N_{sp}A_{sp}z$, where $N_{sp} \simeq 4$ is the number of Galactic arms. The influence area $A_{sp}$ is assumed to be roughly the product of the arm-length $\sim
10$ kpc and the arm-width, i.e. $(\Omega_\odot-\Omega_p)R_0\tau$, where $\Omega_\odot$ is the angular speed at the solar galactocentric radius, $\Omega_p$ is the pattern speed of the spiral arms (see below) and $\tau \sim 10^7$ is the average lifetime of a supernova progenitor star. Then, $$N_{S\!N} \simeq 0.5 \left({R_{SN}\over
0.033~\hbox{yr$^{-1}$}}\right)\left({l\over
10~\hbox{pc}}\right)^3\left({\tau \over
10^{7}~\hbox{yr}}\right)\left({z\over
100~\hbox{pc}}\right)^{-1}\left({A_{sp} \over
6~\hbox{kpc$^2$}}\right)^{-1}\left({N_{sp} \over 4}\right)^{-1}$$ is the typical number of supernovae encountered within $l_k$ during one spiral arm passage. In addition, recent X-ray observations have shown that young supernova remnants (of radius $\simlt$ 10 pc) are active sites of acceleration of cosmic rays to energies $\simgt 10^2$ TeV ([@KOY95]). The above estimate shows that the chances of intercepting a supernova shock front are significant, leading to sustained exposure ($\sim 10^4$ yr) of the upper atmosphere to cosmic ray bombardment, by factors $10^2 - 10^3$ over the mean level.
Besides supernovae, gravitational perturbers such as large complexes of molecular gas and dust (the giant molecular clouds and the intermediate sized clouds), with typical sizes of a few hundred parsecs and masses of up to $10^6$ , are also concentrated along spiral arms. It is instructive, therefore, to trace the first order solar orbit through the best estimate of the structure of the Milky Way and the position of its spiral arms, back to the beginning of the Phanerozoic period (0 – 500 Myr-ago, or $\simlt 5$% of the age of the Galactic disk). Episodes during the solar motion may then be compared directly with episodes in the geologic timeline.
In its simplest approximation, the solar revolution is circular with an adopted galactocentric radius $R_{0}\simeq 8.5$ kpc. Radial and vertical oscillations may be considered small departures from an otherwise circular orbit (e.g. the vertical oscillation has a period $P \simeq 62$ Myr and amplitude $\sim 35$ pc [@BIN87], smaller than the scale heights of the perturber populations). Severe gravitational encounters are unlikely to have distorted this orbit over the past 0.5 Gyr (i.e. two dynamical times) making it reasonable to assume that the Sun has preserved its nearly circular motion, with angular speed $\Omega_{\odot} \simeq 27$ km s$^{-1}$ kpc$^{-1}$ ($v_{\odot} \simeq 230$ km s$^{-1}$).
The spiral density waves trail the disk rotation with a characteristic pattern speed $\Omega_p \simeq 19\pm5 $ k m s$^{-1}$ kpc$^{-1}$ [@WADA], implying that the solar system streams through the arms at a mean relative speed $v_r \simeq 68$ km s$^{-1}$. Due to the inherent difficulty of the measurement, the pattern speed is not an accurately determined quantity and contributes the largest uncertainties to any estimate of the past structure of the Galaxy. Methods for estimating $\Omega_p$ have included measurement of the age gradient of objects along the Sagittarius-Carina arm (Avedisova 1989) and the velocity field of Cepheids (Mishurov et al. 1979). Amaral and Lépine (1997) estimated $\Omega_p$ based on a study of open clusters, an ideal population for such a study since their ages are well determined from the HR diagram. Their analysis suggests that $\Omega_p \simeq 20\pm5$ km s$^{-1}$kpc$^{-1}$. The estimate derived by Wada et al. (1994) is for the pattern speed of a putative end-on Galactic bar based on modeling of the molecular cloud longitude-velocity diagram. Other arguments have been summarized by Amaral (1995) in favor of $\Omega_p \simeq 20~$km s$^{-1}$ kpc$^{-1}$.
The present day positions of the spiral arms have been outlined using optical and radio observations of large H II regions (Georgelin & Georgelin 1976), supplemented by data from the 21-cm line of neutral hydrogen, the H109$\alpha$ radio recombination line, and the 2.6-mm line of carbon monoxide (which traces molecular hydrogen), used to resolve distance ambiguities. The highly excited H II regions define two pairs of arms (four major arms altogether), which intersect the solar orbit at angles of $10^{\circ}$–$12^{\circ}$. The face-on morphology of the Galaxy is shown in Figure 1, along with the location of the Sun during each of the six Phanerozoic extinctions. The data for the major arm (the Sagittarius-Carina arm) and the intermediate arm (the Scutum-Crux arm) are complete out to the solar galactocentric radius of 8.5 kpc. However, data is unavailable for the internal arm (the Norma arm) beyond a galactocentric radius of $\simeq 6.0$ kpc, due to obscuration by the Galactic center. We extend this arm to the solar orbital radius using a logarithmic spiral model [@AVE96]; this function provides excellent corroborating fits to arms for which data do exist.
Figure 2 displays the times of solar spiral arm crossings in graphical form (assuming a relative speed of 68 km s$^{-1}$) through the Galactic free-electron distribution, as modeled by Taylor and Cordes [@TAY93], based on the radio and optical data of giant H II regions and corroborated by $\gamma$-ray observations of Al-26, a tracer of massive star nucleosynthesis [@CHE96]. The free electron density (or equivalently the ionized gas density) is a tracer of spiral structure; ionized gas in the Galaxy is concentrated in the H II regions surrounding hot OB stars, young star clusters, and in the near exteriors and interiors of expanding supernova remnants. Dotted lines indicate the range of uncertainty in the past positions of the spiral arms due to unwinding. (A simple way to estimate the unwinding is to notice that the phase winding between the innermost regions of the arms (the so-called inner Lindblad resonance where $\Omega_d
\approx 0$ km s$^{-1}$ kpc$^{-1}$ and $R \simeq 4.0$ kpc) to those at radius $R_0$ is $\simeq \pi$ rad (Figure 1), over a time roughly equal to the age of the Galactic disk $\simeq 12$ Gyr. The phase unwinding for individual crossings is then $\sim 1^\circ$, $\sim 4^\circ$ and $\sim 8^\circ$, respectively, for the first three spiral arms. A more detailed calculation gives $2.4^\circ$, $10^\circ$ and $20^\circ$; Binney & Tremaine 1987).
Along with the crossing times, Figure 2 illustrates the extinction timeline adopted from Sepkoski (1994), where individual extinctions are modeled as gaussians after the subtraction of a mean background extinction rate at each epoch. Notice a correlation between the two timeseries, which [*a priori*]{} represent two rather disparate temporal sequences, i.e., the solar spiral arm crossing times and the geological times of terrestrial extinction. Perhaps most interesting, as it involves the smallest extrapolation in time, is the close coincidence of the K/T (Maastrichtian) event with the Sagittarius-Carina arm crossing 60 Myr-ago (0.6 % the disk age) for the above-mentioned kinematic parameters. Further back in time, the end-Permian and upper Norian events coincide with the crossing of the Scutum-Crux arm. The upper Botomian and possibly the late Ordovician (Ashgillian) extinctions may be associated with the Norma arm, but the large uncertainty in the position of this arm makes any definite association specious at best. The late Devonian (Frasnian) extinction does not coincide with any major arm, although a detailed statistical examination [@HUB] of the extinction record suggests that the Devonian and Norian events should only be regarded as candidates for extinctions. It is well worth mentioning that independent of our assumed distance scale and kinematic model, the ratios of timescales in the geologic record are a good match to those of the three spiral crossings (assuming an unperturbed solar orbit).
The uncertainties involved in the above analysis are admittedly quite large, and any quantitative comparison should naturally be regarded with caution. Yet in the face of the terrestrial geologic record, and the near certainty that at least one of the mass extinctions is linked to non-terrestrial mechanisms, we find an idea which unifies these mechanisms appealing. If a fraction of mass extinctions are indeed due to non-terrestrial phenomena, then the concentration of supernovae and other perturbers ought to make spiral arms far more hazardous than other locations in the Galaxy. A comparison between the extinction record and the best available data on the spiral structure of the Galaxy suggests that this may indeed be the case; although the large uncertainties in the spiral arm pattern speed, as well as the locations of the arms themselves prevent a more definitive comparison, it is nonetheless intriguing that the observed spacing of the major extinctions is approximately reproduced. Moreover, evidence from the fossil record [@OFF] (and possibly the Ir evidence; Yabushita & Allen 1997) that the mass extinctions were far more gradual than previously thought, lends credence to the spiral arm hypothesis, as the Sun spends tens of Myr in the vicinity of each arm, during which any or all of the aforementioned processes are not only possible, but likely. If both boloidal impacts (the Iridium evidence) and supernovae are established as culprits-in-common (as per the suggestions of Ellis, Fields and Schramm 1996) from either geological or ice-layer records, then spiral arm crossing must play an important role in the repeated extinction of terrestrial (or extra-terrestrial) life.
[**Acknowledgments:**]{} We thank J. H. Taylor and J. M. Cordes for making their Galactic free-electron density model widely available, as this paper makes use of their software. We thank T. Padmanabhan, A. C. S. Readhead, M. R. Metzger and S. R. Kulkarni for several useful discussions.
[99]{}
Amaral, L. H. 1995, PhD Thesis, Univ. Saõ Paulo, Instituto Astronômico e Geofísico Amaral, L. H. & Lépine, J. R. D. 1997, MNRAS, 286, 885. Alvarez, W., Asaro, F., & Montanari, A., 1990, Science, 250, 1700-1702. Alvarez, W. & Muller, R. A., 1984, Nature, 308, 718-720. Avedisova, V. S. 1989, Astrophys., Vol 30, No. 1, 83. Avedisova, V. S., 1996, Astronomy Letters, 22, 443-454: Translated from Pisma v Astronomicheskii Zournal, 1996, 22, No., 7-8. Bignami, G. F. & Caraveo, P. A., 1996, ARAA, 34, 331-381. Binney, J. & Tremaine, S., 1987, Galactic Dynamics, Princeton Univ. Press, 350. Chen, W., Gehrels, N., Diehl, R., & Hartmann, D., A&AS, 1996, 120, 315-316,. Clube, S. V. M. & Napier, W. M., 1996, QJRAS, 37, 617-642. Crutzen, P. J. & Brühl, C., 1996, PNAS, 93, 1582-1584. Ellis, J. & Schramm, D. N., 1995, PNAS, 92, 235-238. Ellis, J., Fields, B. D., Schramm, D. N., 1996, ApJ, 470, 1227-1236. Georgelin, Y. M. & Georgelin, Y. P., 1976, A&A, 49, 57-59. Hatfield, C. B. & Camp, M. J., 1970, Bull. geol. Soc. Am.,81, 911-914. Hubbard, A. E. & Gilinsky, N. L. 1993, 1992, Paleobiology, 18, 148-160. Koyama, K., Petre, R., Gotthelf, E. V., Hwang, U., Matsuura, M., Ozaki, M. & Holt, S. S., 1995, Nature, 378, 255-258. McCrea, W. H., 1975, Nature, 255, 607-609. Mishurov, Y. N., Pavlovskaya, E. D., Suchkov, A. A. 1979, AZh, 56, 286. Officer, C. B., Hallam, A. D., Drake, C. L. & Devine, J. D., 1987, Nature, 326, 143-149. Rampino, M. R. & Stothers, R. B., 1984, Nature, 308, 709-711. Rampino, M. R. & Haggerty, B. M., 1996, Earth Moon & Planets, 72, 441-460. Raup, D. M. & Sepkoski, J. J. Jr, 1982, Science, 25,1501-1503. Raup, D. M., 1991, Paleobiology, 17, 37-48. Ruderman, M. A., 1974, Science, 184, 1079-1081. Sepkoski, J. J., 1994, Geotimes, March 1994, 15-1. Schwartz, R. D. & James, P. B., 1984, Nature, 308, 712-713. Taylor, J. H. & Cordes, J. M., 1993, ApJ, 411, 674-684. Thorsett, S. E., ApJ, 1995, 444, L53-L55. van den Bergh, S. & Tammann, G. 1991, 1991, ARAA, 29, 363-407. Wada, K., Taniguchi, Y., Habe, A. and Hasegawa, T., 1994, ApJ, 437, L123. Yabushita, S. & Allen, A., 1997, Astronomy & Geophysics, 38 (2), 15.
APPENDIX
========
In this appendix, we present a somewhat more detailed comparison of the two timeseries shown in Figure 2. Mass extinctions are identified as in Sepkoski (1994). As described in the text, the individual extinctions are modeled as gaussians after subtraction of a mean extinction rate at each epoch, estimated from the troughs in the extinction record. We find that the cross-correlation of the two time series peaks at $\Delta t = 0$ for a relative velocity of $v_s = 68.4~$km/s (the expected value for $\Omega_p \approx 19$ km s$^{-1}$ kpc$^{-1}$).
The statistical significance of the observed cross-correlation is assessed using Monte Carlo methods. The locations of the individual extinctions are uniformly randomized within the time bounds of the Phanerozoic period. Two preconditions are applied when generating the fake datasets: (i) the individual extinctions are not allowed to overlap each-other within $2\sigma$-bounds (since geological methods have to distinguish them as distinct events), and (ii) wherever model extinctions do partially overlap, their sum is truncated at the 100 percent level. In addition, we let the orbital speed of the Sun in the frame corotating with the arms (i.e. $v_s = (\Omega_p
- \Omega_\odot)R_0$), be a gaussian random variable; that is, we let $\Omega_p = 19\pm5$ km s$^{-1}$ kpc$^{-1}$ (1$\sigma$). We generate $10^5$ fake extinction datasets, and for each we compute the zero-lag cross-correlation with the spiral arm crossing curve. We find that 99 percent of the randomly generated correlations were smaller than the actual data correlation. The highest tail-end cross-correlations, when examined carefully, displayed rough positional interchange and strong clumping of the extinction gaussians around the spiral arms, as expected.
|
CERN-TH/2000-261\
-.1 cm NEIP-00-016\
-.1 cm hep–th/0010048\
.5in
[**Current correlators in the Coulomb branch of ${\cal N}=4$ SYM**]{}
0.4in
[**Andreas Brandhuber**]{}${}^{1,}$[^1] and${}^2$ 0.1in [*${}^1\!$Theory Division, CERN\
CH-1211 Geneva 23, Switzerland\
*]{}\
.2in [*${}^2\!$Institut de Physique, Université de Neuchâtel\
Breguet 1, CH-2000 Neuchâtel, Switzerland\
*]{}\
.3in
**Abstract**
We study correlators of ${\cal R}$-symmetry currents in the Coulomb branch of ${\cal N} = 4$ supersymmetric gauge theory in the large-$N$ limit, using the AdS/CFT correspondence. In particular, we consider gauge fields in the presence of gravity and scalar fields parameterizing the coset $SL(6,\IR)/SO(6)$ in the context of five-dimensional gauged supergravity. From a ten-dimensional point of view these backgrounds correspond to continuous D3-brane distributions. We find the surprising result that all 2-point functions of gauge currents fall into the same universality class, irrespectively of whether they correspond to broken or unbroken symmetries. We show that the problem of finding the spectrum can be mapped into an equivalent Schrödinger problem for supersymmetric quantum mechanics. The corresponding potential is the supersymmetric partner of the potential arising in studies of the spectrum for massless scalars and transverse graviton fluctuations in these backgrounds and the associated spectra are also identical. We discuss in detail two examples where these computations can be done explicitly as in the conformal case.
.4in CERN-TH/2000-261\
August 2000\
16 pt
Introduction
============
For several years the dynamics of branes in string theory have been a fruitful playground to test strong coupling physics of gauge theories. For instance, the AdS/CFT correspondence [@malda; @gkp; @witten] provides us with precise prescriptions to calculate correlation functions, spectra of gauge invariant operators, Wilson loops and $c$-functions in ${\cal N}=4$ supersymmetric Yang–Mills (SYM) theory in four dimensions at large $N$ and large ’t Hooft coupling. The data obtained this way from supergravity can sometimes be compared with field theory or provide non-trivial predictions for strongly coupled field theories. This correspondence can be extended also to theories with spontaneously or manifestly broken superconformal symmetry. Such theories arise either by giving vacuum expectation values to fields [@malda], [@kraus]-[@bbs] or by deforming the conformal theory with relevant operators [@gppz1]-[@Evans]. Many of these deformations can be treated efficiently in the context of five-dimensional gauged supergravity [@PPN; @GRW] and the resulting backgrounds have four-dimensional Poincaré invariance and approach $AdS_5$ in the ultraviolet (in a field theory terminology). Typically, towards the infrared, singularities appear which are not fully understood and seem to require a proper inclusion of the string theory dynamics or the use of other methods developed in gravity.
In this letter we study correlation functions of ${\cal R}$-symmetry currents using the holographic description of large-$N$ gauge theories. For the conformal case correlation functions for operators in various representations of the ${\cal R}$-symmetry group $SU(4)\simeq SO(6)$ have been worked out in great detail (see, for instance, [@fmmr; @dhoker]). Less is known about correlators in deformed gauge theories which are described by more general domain wall solutions of gauged supergravity. So far mainly scalars have been studied, namely the minimally coupled scalar [@fgpw2; @brand1; @Anselmi] (which has the same equation as the transverse traceless graviton modes [@brand2]), active and inert scalars which parameterize deformations of the $S^5$ [@DeWolfe; @AFT; @massimo], but also fermionic and abelian vector field fluctuations for the ${\cal N}=1$ flow of [@gppz2] and the ${\cal N}=4$ Coulomb branch background of [@fgpw2; @brand1] have been considered recently in [@massimo].
We will show that for a specific class of examples this analysis can be extended to include fluctuations of non-abelian gauge fields which are dual to ${\cal R}$-symmetry currents of the gauge theory. We make a general connection between the fluctuation equation and supersymmetric quantum mechanics and find that, the relevant Schrödinger potential, associated with the spectrum, is just the supersymmetric partner of the potential arising from the corresponding massless scalar and transverse graviton-fluctuations equations. We show also that the corresponding spectra are identical. It seems plausible to us that this can be extended to the full set of fields in the supergravity multiplet. Using the AdS/CFT correspondence we calculate two-point functions of the symmetry currents in ${\cal N} = 4$ SYM on the Coulomb branch in two particular cases.[^2] As expected, we find deviations from the conformal $1/r^6$ fall-off for large separations $r$. From the non-analytic part of the correlator in momentum space we get contributions that are suppressed exponentially for large separation.
The choice of a particular state on the Coulomb branch breaks the ${\cal R}$-symmetry to a subgroup and therefore one might expect that broken and unbroken currents behave differently and in particular one would expect Goldstone bosons corresponding to the broken symmetry. From the dual supergravity point of view this symmetry is a local gauge symmetry and the massless bosons simply get eaten by the gauge fields and make them massive via the Higgs mechanism. Although the equations for broken and unbroken currents look quite different — they correspond, respectively, to massless gauge fields in a curved background and massive gauge fields — the associated spectra are identical. This result is not too surprising since on the Coulomb branch only conformal symmetry is broken but the currents still reside in the same supersymmetry multiplet. However, a small puzzle remains since the correlator has also an analytic piece that depends on which of the broken or unbroken currents are considered. For the two-point function of scalars such analytic terms give rise to contact terms and are usually dropped, but in the case of gauge field correlators they give rise to terms of the form $x_\m x_\n/r^6$, which might be interpreted in field theory as arising from Goldstone bosons. However, we do not find a one to one relation between broken currents and the presence of these terms in the correlators. We believe that these analytic terms are unphysical, since the corresponding mode is non-normalizable, and should be dropped.
The organization of this paper is as follows: In section 2 we present some background material on gauged supergravity and calculation of correlators in AdS/CFT. We also make a general connection between the fluctuation equation and supersymmetric quantum mechanics. In section 3 we focus on our two main examples where calculations can be performed explicitly. We obtain the exact fluctuation spectrum of gauge fields, and the two-point functions in momentum and position space. In section 4 we give a summary of our results and give some final remarks.
Generalities
============
Our starting point is a specific truncation of the ${\cal N} = 8$ gauged supergravity action [@PPN; @GRW] including $SO(6)$ gauge fields $A_{\widehat \m}^{ij}$, antisymmetric in $i,j$, with field strength $F_{\widehat \m \widehat \n}^{ij}$, where $\widehat \m, \widehat \n = 1,2,3,4,z$; unhatted indices $\mu, \nu =1,2,3,4$ will be used later to denote Euclidean directions along the boundary at $z=0$. For notational convenience we will occasionally use the collective index $a=1,2,\dots, 15$ to denote the adjoint representation of $SO(6)$, instead of $i$ and $j$ or we will omit such an index all together. Furthermore, scalars in the ${\bf 20^\prime}$ are represented by a symmetric traceless matrix $M^{ij}$. The action of the supergravity truncated to these fields has been constructed in [@cvetic] and we follow closely their conventions.
The Lagrangian density for the relevant fields of five-dimensional gauged supergravity is = \_[scalar]{} + \_[gauge]{} , \[sugralagr\] where $\cL_{\rm scalar}$ refers to the pure gravity-scalar sector and $\cL_{\rm gauge}$ contains the gauge fields and their interaction with the scalars and gravity. We first recall some results for the pure gravity-scalar sector since we are interested to study fluctuations of the gauge fields in the background of specific solutions of the gravity-scalar sector. The explicit form of the Lagrangian is \_[scalar]{} = [14]{} [R]{} - ( \_ M M\^[-1]{} \^ M M\^[-1]{} ) - P , \[actionsc\] where the potential is \[potential\] P = -[g\^232]{} , with $g$ being a mass scale. Alternatively we may use the length scale $R$ via the relation $g=2/R$.
Supersymmetric solutions of [(\[actionsc\])]{} preserving 16 supercharges and Poincaré symmetry in four-dimensions have been studied extensively and they correspond to states on the Coulomb branch of $\cN=4$ SYM theory. Their interpretation in ten dimensions is simply in terms of a continuous distribution of D3-branes. For these backgrounds the matrix of scalar fields can be brought to a diagonal form using a gauge transformation. Thus we are left with six scalar fields that parameterize \[M\] M = [diag]{} (e\^[2 \_1]{}, …, e\^[2 \_6]{} ) , obeying the constraint $\sum_{i=1}^6 \beta_i = 0$. There are five independent scalar fields, denoted by $\a_I$, $I=1,2,\dots 5$, and the relation to the $\b_i$’s is given by $\b_i= \sum^5_{I=1} \l_{iI} \a_I$, where $\l_{iI}$ is a $6\times 5$ matrix, with rows corresponding to the fundamental representation of $SL(6,\IR)$; the normalization conventions can be found in eq. (2.4) of [@bbs]. The metric ansatz reads ds\^2 = e\^[2 A(z)]{} (dz\^2 + \_ dx\^dx\^) = dr\^2 + e\^[2A(r)]{} \_ dx\^dx\^ , \[metriki\] where the relation between the coordinates $z$ and $r$ is such that $dr=-e^A dz$. In addition, all scalar fields depend on the variable $r$ or equivalently $z$. The most general solution preserving 16 supercharges has been found in [@bakas1] and is conveniently presented in terms of an auxiliary function $F(g^2 z)$. Specifically, the conformal factor is given by e\^[2 A]{} = g\^2 (-F\^)\^[2/3]{} , \[pro1\] where the prime denotes the derivative with respect to the argument of $F(g^2z)$. In addition, the profiles of the scalar fields are e\^[2\_i]{} = [f\^[1/6]{}F-b\_i]{} ,f = \_[i=1]{}\^6 (F-b\_i) , i=1,2,…, 6 . \[proo\] The constants of integration are ordered as $b_1\geq b_2 \geq \dots \geq
b_6$ and the function $F$ is constrained to obey the differential equation (F\^)\^4 = f . \[hd3\] Equating $n$ of the integration constants $b_i$ (or equivalently the associated scalar fields $\b_i$) corresponds to preserving an $SO(n)$ subgroup of the original $SO(6)$ $\cR$-symmetry group. We note in passing, that there is a deep connection between solutions of the gravity-scalar sector of the five-dimensional gauged supergravity that we just reviewed, and the theory of algebraic curves and associated Riemann surfaces to which the differential equation [(\[hd3\])]{} is related [@bakas1; @bbs].
Let us now turn to the part of the Lagrangian containing the gauge fields. First, we have to replace the partial derivatives in [(\[actionsc\])]{} by gauge-covariant ones $\partial_{\widehat \m} M^{ij} \to
\partial_{\widehat \m} M^{ij} + g
(A_{\widehat \m}^{ik} M^{kj} + A_{\widehat \m}^{jk} M^{ik})$, and, second, we add the gauge kinetic term \[gaugeaction\] \_[gauge]{} = - (M\^[-1]{})\^[ij]{} (M\^[-1]{})\^[kl]{} F\^[ik]{}\_ F\^[jl ]{} . Since we are interested in two-point functions we only need to keep terms in [(\[sugralagr\])]{} and [(\[gaugeaction\])]{} which are quadratic in the gauge fields and the scalar fluctuations in the symmetric unimodular matrix $M$. Note that although for our solution the matrix $M$ is diagonal as in [(\[M\])]{}, we have to consider fluctuations along the diagonal as well as off-diagonal ones. Using the fact that $M$ is diagonal [(\[M\])]{} for our backgrounds, we collect all terms that can give quadratic terms in the fluctuations of the scalars and the gauge fields \_[quad.]{} & = & - [18]{} e\^[-2 (\_i+\_j)]{} F\^[ij]{}\_ F\^\_[ij]{} - [g\^24]{} \^2(\_i -\_j) A\^[ij]{}\_ A\^\_[ij]{}\
&& -[g8]{} [Tr]{}((\_ M M- M\_ M) A\^)|\_[quad.]{} \[actiongor\]\
&& -[116]{} [Tr]{}(\_ M M\^ M M) - P|\_[quad.]{} . The first line above is already quadratic in the gauge field fluctuations. We emphasize that $F_{\widehat \m \widehat \n}^{ij}=
\del_{\widehat \m} A^{ij}_{\widehat \n} -
\del_{\widehat \n} A^{ij}_{\widehat \m} $ is, for our purposes, the relevant part of the gauge field strength. The second line in the above expression is already linear in the gauge field fluctuaction. Hence, we are supposed to expand it to linear order in the scalar fluctuations. Finally, the third line has to be expanded to quadratic order in the scalar field fluctuations. In this paper we are only interested in the gauge field fluctuations which, however, couple to fluctuations of the scalars. Therefore, it is not a priori correct to simply keep the terms in the first line in [(\[actiongor\])]{} and drop the rest. Nevertheless, we will now explain that this procedure gives the correct result since there is a field redefinition that effectively decouples the gauge field fluctuations from those of the scalars.[^3] To see that let us expand the second line in [(\[actiongor\])]{} and keep the linear term in the scalar field fluctuations. We find that && - [g8]{} [Tr]{} ((\_ M M- M\_ M) A\^) |\_[quad.]{} =\
&& = [g8]{} ( (e\^[-2\_j]{} - e\^[-2 \_i]{}) \_ M\_[ij]{} +2 (e\^[-2\_i]{}\_ \_j - e\^[-2 \_j]{} \_\_i) M\_[ij]{} )A\^\_[ij]{} . \[ghe\] From this we immediately deduce that the diagonal fluctuations $\d M_{ii}$ do not couple to the gauge fields. A less trivial fact is that the scalar fluctuations in $\d M_{ij}$ that belong to any unbroken subgroup of $SO(6)$ do not couple to the gauge fields as well. The reason is that in this case $\b_i=\b_j$, since then the corresponding integration constants in [(\[proo\])]{} are equal, i.e. $b_i=b_j$. Hence, let us consider the remaining cases with $\b_i \neq \b_j$ which arise when the indices $i, j$ belong to the coset. If we make the field redefinition A\_\^[ij]{} A\_\^[ij]{} + [1g]{} \_ (M\_[ij]{}e\^[2\_i]{} - e\^[2\_j]{}) ,\_i\_j , \[frfe\] the mixed terms between scalar and gauge field fluctuations in [(\[actiongor\])]{} (with the substitution [(\[ghe\])]{} understood) disappear and the fluctuations decouple. Note that the field redefinition [(\[frfe\])]{} acts as an abelian gauge transformation and as such it leaves the gauge field strength $F_{\widehat \m\widehat \n}^{ij}$ invariant (to the quadratic order we are working). We emphasize that the field redefinition [(\[frfe\])]{} does not guarantee that there will be no mixing between scalar and gauge field fluctuations at the cubic or at some higher order in the fluctuating fields, but only that the quadratic fluctuations decouple. We also note that a similar decoupling mechanism for vector and scalar fluctuations was found to be at work for the flow of [@gppz2] in [@massimo]. There, it was observed that decoupling was achieved since the gauge field and a (charged) scalar appeared in a gauge invariant combination.
The field redefinition [(\[frfe\])]{} removes the scalar fluctuations of $\d M_{ij}$ since it removes terms quadratic in first derivatives of $\d M_{ij}$ from the Lagrangian. The remaining terms are at most linear in first derivatives and of the form $B_{ij} \d M_{ij} \d M_{ij} + B^{\widehat\m}_{ij}
\d M_{ij} \del_{\widehat\m} \d M_{ij} $ for some space-depended $B_{ij}$ and $ B^{\widehat\m}_{ij}$ which are symmetric in $i,j$. Clearly the derivative-term can be removed by adding an appropriate total derivative so that we are left with a non-dynamical field $\d M_{ij}$ corresponding to no physical degrees of freedom. What we have is nothing but a manifestation of the Higgs effect in a curved background. As in flat space-time, the Goldstone bosons corresponding to the broken gauge symmetries are eaten by the gauge bosons which then become massive.
Since we are only interested in the gauge field fluctuations we ignore the scalar fluctuations for the rest of the paper and concentrate on those for the gauge fields which, after the redefinition [(\[frfe\])]{}, are described by the first line in [(\[actiongor\])]{} (A)\_[quad.]{} =- [18]{} e\^[-2 (\_i+\_j)]{} F\^[ij]{}\_ F\^\_[ij]{} - [g\^24]{} \^2(\_i -\_j) A\^[ij]{}\_ A\^\_[ij]{} . \[actiong\] The second term corresponds to mass terms for the gauge fields, if the scalar fields $\b_i$ are not equal. This implies that for general states on the Coulomb branch the bulk gauge symmetry $SO(6)$ is spontaneously broken and, hence, that the ${\cal R}$-symmetry group of the field theory on the boundary is reduced accordingly. Notice also that the kinetic term for the gauge fields is not canonically normalized as it gets “dressed” by the scalar fields. This will have important consequences, as we will see.
The equation of motion following from this quadratic action [(\[actiong\])]{} is: A\^[ij]{}\_ :D\_ (e\^[-2(\_i+\_j)]{} F\^\_[ij]{}) - g\^2 \^2 (\_i - \_j) A\^\_[ij]{}=0 . \[tade\] In solving these equations we have to distinguish two cases: First, for the unbroken symmetry (currents), for which $\b_i=\b_j$, we can use the gauge symmetry to choose the gauge $A^{ij}_z=0$. This still allows for restricted gauge transformations with parameters that depend only on the $x^\m$’s, but not on $z$. Then, the $\widehat \m=z$ component of the eqs. [(\[tade\])]{} yields the constraint $\del_z \del_\m A^\m=0$ which allows to eliminate unphysical longitudinal modes via a restricted gauge transformation. The equation of motion for the remaining physical (transverse) modes $A_\m^\bot$ which obey $\partial^\m A_\m^\bot = 0$ is the same for all components and can be written as an equation for a scalar field, which we denote by $\Phi$: \[unbroken\] \_z (e\^B \_z ) + m\^2 e\^B = 0 , with the definition B = A-2 (\_i +\_j) . \[h93\] To arrive at this equation we have performed a Fourier transform in the $x^\m$-directions with $k_\m k^\m = -m^2$.[^4]
For the broken symmetry currents for which $\b_i\neq\b_j$ we cannot use a gauge symmetry to eliminate degrees of freedom. In order to calculate the two-point functions we couple the gauge field to an external source by adding $-\frac{1}{2} A_{\widehat \mu}^{ij} J^{\widehat \mu}_{ij}$ to the gauge field action [(\[actiong\])]{}. The source is required to be covariantly conserved, i.e., $D^{\widehat \mu} J_{\widehat \mu}^{ij}=0$. We choose to decompose the gauge field into transverse modes $A_\m^\bot$, longitudinal modes $\partial_\mu \xi = A_\m - A_\m^\bot$, and the component $A_z$. The equations of motion [(\[tade\])]{} give \_z(e\^B(\_z A\^\_-\_A\_z + \_z \_))+ e\^B A\^\_ - e\^C(A\^\_+\_) = e\^[3 A]{} J\_z \[gh1\] and e\^B(A\_z - \_z) - e\^CA\_z = e\^[3 A]{} J\_z , \[gh2\] where $\square = \eta^{\m\n} \del_\m \del_\n$. The above coupled system of equations can be further simplified. By taking the derivatives $\del_\m$ and $\del_z$ in [(\[gh1\])]{} and [(\[gh2\])]{} respectively, adding up the resulting expressions and then using the condition $D^{\widehat \m} J_{\widehat \m}=0$, we obtain a relation that determines $\xi$ in terms of the component $A_z$, namely e\^C + \_z ( e\^C A\_z )=0 , \[gh3\] where e\^C = g\^2 e\^[3 A]{} \^2 (\_i - \_j) = [14]{} g\^2 (b\_i-b\_j)\^2 e\^[-B]{} . \[hgde\] The first equality defines $C$, whereas the second one follows with the help of [(\[proo\])]{} and relates $C$ to $B$ which was defined in [(\[h93\])]{}. Using [(\[gh3\])]{} to solve for $\square \xi$ and then substituting back the result into [(\[gh2\])]{} we find the equation for the mode $A_z$, which decouples from the transverse modes: \[Azeom\] e\^B A\_z + e\^B \_z ( e\^[-C]{} \_z ( e\^C A\_z ) ) - e\^C A\_z = e\^[3 A]{} J\_z . With further manipulations using [(\[Azeom\])]{}, we may cast [(\[gh1\])]{} into an equation for the transverse modes \[Atrans\] e\^B A\_\^+ \_z ( e\^B \_z A\_\^) - e\^C A\_\^= e\^[3 A]{} J\_\^ , \[hgjh2\] where we have defined the transverse current-source as $J^\bot_\m = (\d_{\m\n}
-\del_\m \del_\n/\square) J_\n$. In order to compute the two point functions in momentum space we need solutions of the homogeneous equations [(\[Azeom\])]{} and [(\[Atrans\])]{}. Actually, after a Fourier transform in the $x^\m$ brane-directions, we can write both equations as an equation for a scalar field \[broken\] \_z (e\^B \_z ) +( m\^2 e\^B -[14]{} g\^2 (b\_i-b\_j)\^2 e\^[-B]{} ) = 0 , \[fiiin\] where we have dropped the source term. Its effect will be implemented by imposing appropriate boundary conditions to the solutions. For the case of [(\[Atrans\])]{} the scalar $\Phi$ denotes any component of $A^\bot_\m$. In order to cast [(\[Azeom\])]{} into the form [(\[fiiin\])]{}, we have used [(\[hgde\])]{} and defined $\Phi=e^C A_z$. For $\b_i=\b_j$ we recover from [(\[broken\])]{} eq. [(\[unbroken\])]{} that describes the cases with unbroken symmetry. Hence, for full generality, we may use [(\[broken\])]{} in order to calculate current-current correlators. We will follow the standard procedure of [@gkp; @witten] and we will work in Euclidean signature unless stated otherwise.
In order to proceed we need a complete set of eigenfunctions of [(\[broken\])]{}, which for the examples we will discuss in the next section can be found explicitly and is given in terms of hypergeometric functions. Furthermore, we keep the solutions that blow up at the AdS boundary since they correspond to current operator insertions [@gkp; @witten]. Finally, we have to evaluate the on shell-value of the action ${1 \ov \kappa^2} \int d^5x {\cal L}$ with ${1 \ov \kappa^2} = \frac{N^2}{16 \pi^2}$ for solutions $\Phi$ of [(\[broken\])]{}.[^5] We find the boundary term - \_[0]{} e\^B \_z |\_[z=]{}\^[z\_[max]{}]{} k\^2 H(k) . \[onshell\] In order to keep formulas short in later sections we have written out the overall factor $1/\kappa^2$ in the definition of $H(k)$. In order to obtain the correct result we have to normalize $\Phi |_{z=\epsilon} = 1$ and take the limit in [(\[onshell\])]{}. Re-introducing Lorentz and group theory indices properly, we can present the current-current correlators in momentum space schematically as \[momcorr\] J\_\^a(k) J\_\^b(-k) = \^[ab]{} (\_ - ) k\^4 (k) , where a group theory factor and the momentum space version of the projector, which guarantees that the amplitude is transverse, have been included. The factor $H(k) \equiv k^2 \tilde{G}(k)$ depends also on the adjoint indices $a,b$, but for reasons similar to those explained in footnote 2 we have not explicitly displayed them.
In the explicit calculations performed later in section 3 we will not use $H(k)$ directly, as defined in [(\[onshell\])]{}, because the correlator in $x$-space is too singular to be Fourier transformed to momentum space. However, by using differential regularization one can make sense of such expressions by writing singular functions as derivatives of less singular ones and then defining the Fourier transform by formal partial integrations [@diffreg]. In our case we have to take the correlator to be of the form $\sim \square \square G(x)$ which is just $k^4 \tilde{G}(k)$ in momentum space. Hence, the correlator in $x$-space becomes J\_\^a(x) J\_\^b(0)= \^[ab]{}(\_ - \_\_) G(x) , \[cor1\] where G(x) = [14\^2]{} d\^4 k e\^[i kx]{} [H(k)k\^2]{} = [1r]{} \^\_0 dk H(k) J\_1(k r) , \[cor2\] with $J_1(kr)$ being a Bessel function.
Supersymmetric quantum mechanics
--------------------------------
In this subsection we want to study general aspects of the fluctuation equation [(\[broken\])]{}, before we proceed in section 3 to describe two special cases where calculations can be performed exactly. Writing $\Phi = e^{-B/2} \Psi$ the field equation [(\[broken\])]{} turns into the one-dimensional Schrödinger equation -\^ + V = m\^2 , with potential V=[14]{} (B\^)\^2 +B\^ + g\^2 e\^[2(A+\_i+\_j)]{} \^2(\_i-\_j) . \[kkk1\] This potential, though not at all obvious, can be cast into a form that appears in supersymmetric quantum mechanics. First, we rewrite it differently using the properties of our solution [(\[pro1\])]{} and [(\[proo\])]{} and in particular [(\[hd3\])]{} which proves useful in turning derivatives with respect to the variable $z$ into functions of the auxiliary function $F$ only: V = [g\^4 f\^[1/2]{}64]{} . \[hf8s\] Comparing with eq. $(4.16)$ of [@bbs] (after setting in there the parameter $\Delta=4$) we find that this can be written solely in terms of the conformal factor in the metric ansatz [(\[metriki\])]{} \[smth\] V= [94]{} [A\^]{}\^2 - [32]{} A\^ . \[kkk2\] This potential[^6] has the same form as the potential appearing in supersymmetric quantum mechanics [@SQMwit; @SQMrev] with superpotential $W=-3/2 A^\prime$. In fact, it is the supersymmetric partner of the potential V\_s= [94]{} [A\^]{}\^2 + [32]{} A\^ , \[jdhf1\] that appeared in studies of 2-point functions for scalar fields or transverse graviton fluctuations [@fgpw2; @brand2; @bakas1; @cglp; @DeWolfe0; @bbs]; the relation of [(\[jdhf1\])]{} to supersymmetric quantum mechanics in the context of gauged supergravity was first hinted in [@bakas1] and explicitly noted in [@DeWolfe0]. Note that, the Schrödinger problem is universal and does not depend on the indices $i,j$ of the gauge currents. Consequently, the mass spectrum is the same irrespectively of whether it is associated to currents corresponding to broken or unbroken symmetries. Instead, the wavefunction $\Phi$ does depend on the indices $i,j$ through the explicit dependence on them of the conformal factor $B$ defined in [(\[h93\])]{} (cf. footnote 3).
It is well known from the general theory of supersymmetric quantum mechanics that the spectra of superpartner potentials, such as [(\[smth\])]{} and [(\[jdhf1\])]{}, are identical except for a zero mode. However, in our case such a mode is not normalizable due to the asymptotic behavior of the function $A(z)$ as $z\to 0$ and, therefore, is not included in the spectrum. Hence, the spectra of current fluctuations, corresponding to [(\[smth\])]{} and those for dilaton and transverse graviton fluctuations, corresponding to [(\[jdhf1\])]{}, exactly coincide, as advertised in the introduction. We note, that related observations concerning a $SO(3)$ invariant sector of 5d gauged supergravity and a particular Coulomb branch flow have been made in [@massimo]. [^7]
The analysis of the qualitative features of the spectrum can be done in a similar fashion as in the case of the superpartner potential arising in the case of scalar correlators [@bakas1; @cglp]. At the boundary $z=0$ the potential goes to $+\infty$ as $V\simeq {3\ov 4 z^2}$. The behavior in the interior depends on the number $n$ of constants of integration $b_i$ that equal the maximum constant among them, $b_1$. We follow closely the discussion of [@bakas1; @bbs] to which we refer for further details. For $n=4,5$ the range of $z$ necessarily extends to $+\infty$, i.e. $0\leq z < \infty$, corresponding to $F=b_1$. We find that, for $n=5$, the potential goes to zero as $z\to \infty$ and the spectrum is continuous. For $n=4$ the potential approaches a constant value, as $z\to \infty$, which is given by $V_{{\rm min}}={g^4\ov 4} f_0^{1/2} $. Therefore, although the spectrum is continuous, there is a mass gap whose squared value is given by the minimum of the potential. For $n=5$ the potential behaves as n=5: V\_5 , z . \[hjf24\] For $n=1,2,3$ the potential goes to $+ \infty$ as $F\to b_1$ and therefore the spectrum must be discrete. Therefore there should be a maximum value for $z$, denoted by $z_{\rm max}$, that is determined by solving the algebraic equation $F(z_{\rm max}g^2)=b_1$. We find the behaviour n=1,2,3: V\_n && [C\_[n]{}(z-z\_[max]{})\^2]{} , zz\^-\_[max]{} ,\
C\_[n]{}&=& [4(4-n)\^2]{} -[14]{} . \[hjf23\] For more details on the full structure of the potentials [(\[smth\])]{} and [(\[jdhf1\])]{}, which generically can be written using elliptic functions, the reader is referred to the original literature [@bakas1; @bbs]. In the two special cases, to which we turn now in section 3, all computations and results can be written in terms of elementary functions.
The 2-point function
====================
In the previous section we introduced all necessary ingredients for the calculation of correlators of symmetry currents and pointed out the relation between supersymmetric quantum mechanics and the fluctuation equations. In this section we want to use these results and apply them to two specific backgrounds worked out in [@fgpw2; @brand1; @bakas1]. These backgrounds correspond to distributions of D3-branes on a disc or a three-sphere [@kraus; @sfe1] and they both break the bulk gauge symmetry down to $SO(2) \times SO(4)$. The broken symmetries form the coset $\frac{SO(6)}{SO(2) \times SO(4)}$. On the dual field theory side these backgrounds correspond to states on the Coulomb branch of ${\cal N} = 4$ SYM theory with reduced ${\cal R}$-symmetry. In the following we will calculate the correlators in momentum and position spaces.
Distribution of D3-branes on a three-sphere
-------------------------------------------
We begin our exactly solvable examples with the case of a model representing D3-branes uniformly distributed on a three-sphere. The expressions for the metric and the scalar fields have been given in [@fgpw2; @bakas1]. The five-dimensional metric [(\[metriki\])]{} has the conformal factor e\^[2 A]{} = [r\_0\^2R\^2]{} [\^[2/3]{} u\^2 u]{} ,0u , \[mett1\] where we have defined for notational purposes the dimensionless variable $u=r_0 z/R^2$. The parameter $r_0$ actually plays the rôle of the radius of the three-sphere. The $AdS_5$ boundary corresponds to $u=0$, whereas at $u=\pi/2$ there is a naked curvature singularity. This is however naturally interpreted, from a string theoretical point of view, as the location of the distribution of the D3-branes on the three-sphere.
The profiles of the scalar fields are \[profiles\] e\^[2\_1]{}=e\^[2\_2]{} = \^[-4/3]{}u , e\^[2\_3]{}= …=e\^[2\_6]{} = \^[2/3]{} u . From a ten-dimensional view point, these scalars deform the five-sphere line element that appears in the D3-brane solution in such a way that the subgroup $SO(2)\times SO(4)$ of the isometry group $SO(6)$ is preserved. The Schrödinger potential [(\[smth\])]{} is found to be V=[r\_0\^2R\^4]{}( -1 + [3\^2 2u]{} ) . It is not difficult to show that a complete orthonormal set of solutions to the corresponding Schrödinger equation is given by \_n = P\_n\^[(-1,1)]{}(2 u) ,0u , n=1,2,… , \[sooll1\] where the $P^{(-1,1)}_n$’s are Jacobi polynomials, provided that the spectrum is given by m\^2\_n = [4r\_0\^2R\^4]{} n(n+1) ,n=1,2,… , \[llso2\] Note that the case with $n=0$, giving rise to a zero-mass eigenvalue, is not included in the spectrum since the corresponding Schrödinger norm diverges. The eigenvalues [(\[llso2\])]{} coincide with those found for dilaton fluctuations in [@fgpw2; @brand1] using the same background as here, in agreement with our general discussion in section 2. Also the $n$-dependent overall constant in [(\[sooll1\])]{} has been chosen such that the $\Psi_n$’s are normalized to one.
The conformal factor appearing in the equation of the fluctuations [(\[broken\])]{} is: e\^B = [r\_0R]{} {
[ll]{} [\^3 u u]{} , & i,j=1,2 ,\
[1 uu]{} , & i,j =3,4,5,6 ,\
[uu]{} , & i=1,2 ,j=3,4,5,6 .
. \[limi16\]
### The 2-point functions
Using [(\[broken\])]{}, [(\[profiles\])]{} and [(\[limi16\])]{} we find the wave equation for the transverse modes of the gauge field in the unbroken $SO(2)$ subgroup, the coset and the unbroken $SO(4)$: (1-x) (x\^2 \^)\^- & = & 0 ,\
(1-x) (x \^)\^- - & = & 0 ,x \^2 u ,\
x (1-x) \^ - & = & 0 , where the prime denotes derivatives with respect to $x$ and $\tilde{k}^2 = R^4/r_0^2 k_\m k^\m$, i.e. is the length-square of the four-vector $k^\m$ rescaled for notational convenience with the indicated factor.
The wave-functions that blow up at the boundary at $x=1$ and are regular at the singularity at $x=0$ are given in terms of a hypergeometric function as[^8] = ( (3+)/2) ( (3-)/2) x\^łF( , ,2,x ) . \[sooll\] where $\D = \sqrt{1-\tilde{k}^2}$, and where we have introduced the parameter $\l= 0,\ha$ and $1$ for the currents corresponding to the unbroken $SO(2)$, the broken coset and the unbroken $SO(4)$ symmetries, respectively. The proportionality constant in [(\[sooll\])]{} has been fixed such that $\Phi(1)=1$ and hence at the boundary the solution becomes proportional to a $\d$-function, i.e., fully localized operator insertion. It is interesting to note that the wavefunctions $\Phi$ in all three cases differ only by different powers of $x$. This is related, as we have seen, to the fact that the mass spectra for broken and unbroken currents are identical. From [(\[sooll\])]{} we extract H() &= & + 1/4 (((1+)/2) + ((1-)/2)+ 2 )\
& =& -[ł\^2]{} + [12]{} \_[n=1]{}\^ , \[add1\] which has a discrete spectrum of poles at $\tilde{k}^2 = - 4 n(n+1)$, $n=1,2,\ldots$, corresponding precisely to the mass eigenvalues [(\[llso2\])]{}. However, if $\l \neq 0$, there is an additional pole at $\tilde{k}^2=0$. We will comment on this in various places below.
The three correlators differ only in the coefficient of the $1/\tilde{k}^2$ term. In the case of scalar correlators this would just give a contact term and could be ignored, but in the case of the symmetry-current correlators this has important consequences as we will explain shortly. Using [(\[cor2\])]{} we obtain the following exact expression for the function $G(x)$ in the correlator [(\[cor1\])]{}: G(x) = ł r + [r\_0 2 R\^2 r]{} \_[n=1]{}\^ K\_1(2) , \[ggg1\] where $K_1$ denotes the modified Bessel function and in writing the term containing $\ln r$ we discarded an infinite constant. We have also dropped a $1/r^2$ term, which, since $\square 1/r^2 \sim \d^{(4)}(r)$, contributes only contact terms to the correlator which we consistently ignore. Hence, we find G(x) = ł[r\_0\^2 R\^4 r\^2]{} + [2 r\_0\^3 R\^6 r]{} \_[n=1]{}\^(2n+1) K\_1(2) .
Let us perform the consistency check that for small $r$, or equivalently, in the limit $r_0\to 0$, we should recover the conformal result. The dominant contribution in this limit comes from the infinite sum which can be approximated by an integral G(x) = [12 r\^2]{} \_[1]{}\^[1/r]{} [dnn]{}+…-[1 4 r\^2]{} r\^2 , r0 . \[cv1\] This gives rise to \[bfsphere\] G(x) ,r0 , which in turn, gives a $1/r^6$ fall off for the correlator [(\[cor1\])]{} at short distances. As expected, this coincides with the result in the conformal case (see, for instance, eq. (30) of [@fmmr]).
The behavior of $G(x)$ for large $r$ is easily found from the asymptotic expansion of the modified Bessel function. For large $r$ each separate term in the infinite sum behaves as $e^{-m_{n} r}/r^{3/2}$, where $m_n$ are the mass eigenvalues in [(\[llso2\])]{} and hence gives rise to an exponential fall off. Keeping the two most dominant contributions in the right hand side of [(\[ggg1\])]{} we obtain G(x) ł[r\_0\^22 R\^4]{} r + [3 8 ]{} (R\^2r\_0 r)\^[3/2]{} e\^[-2 r\_0 r/R\^2]{} , r . \[jke\] For the cases corresponding to the broken coset currents and the unbroken $SO(4)$ currents we have $\l\neq 0$ and therefore the dominant contribution for large $r$ comes from the first term in [(\[jke\])]{}. When substituted into the correlator in [(\[cor1\])]{} it produces a contact term, which we drop, and a term of the form \[goldy\] J\_\^a(x) J\_\^b(0)ł\^[ab]{} (r\^2 \_ - 4 x\_x\_) , r . This term decays only with the forth power of the distance and at first sight it might be tempting to interpret it as arising from the massless Goldstone boson associated with the broken symmetry.[^9] From a physical point of view there are several problems with such an interpretation: First, this term does not appear on equal footing for all three types of currents although they reside in the same supersymmetry multiplet. Its existence might seemingly be acceptable or even desirable for the broken symmetry, but this term also appears for the unbroken $SO(4)$-symmetry currents. We also know from section 2 that the gauge fields dual to the broken currents become massive via the Higgs mechanism and, therefore, are not expected to produce any massless states. Second, the pole of the massless state corresponds to a non-normalizable mode and it is not expected to show up in the two-point function. The most plausible solution seems to be that these poles are actually unphysical and should be dropped from the correlators. Note that a similar problem was found in [@DeWolfe] for the two-point function of active scalars in the same backgrounds we are discussing here. The mysterious massless poles in that paper were later shown to be absent if a different prescription for the correlators is used [@AFT]. It seems likely, although we have not checked, that an improved prescription would resolve the puzzle in our case as well.[^10]
Distribution of D3-branes on a disc
------------------------------------
Our second exactly solvable model represents D3-branes uniformly distributed on a disc of radius $r_0$. The expressions for the metric and the scalar fields have been given in [@fgpw2; @bakas1]. The five-dimensional metric [(\[metriki\])]{} has the conformal factor e\^[2 A]{} = [r\_0\^2R\^2]{} [\^[2/3]{}u\^2 u]{} , 0u < , \[mett2\] where as before $u= r_0 z/R^2$. The scalar fields are given by e\^[2\_1]{}= …=e\^[2\_4]{} = \^[2/3]{} u , e\^[2\_5]{}=e\^[2\_6]{} = \^[-4/3]{} u . As before, from a ten-dimensional type-IIB view point, these scalars deform the five-sphere line element that appears in the D3-brane solution in such a way that the subgroup $SO(2)\times SO(4)$ of the isometry group $SO(6)$ is preserved. The Schrödinger potential [(\[smth\])]{} becomes V=[r\_0\^2R\^4]{} ( 1 + [3\^2 2 u]{} ) . The energy spectrum for this potential is continuous and has a mass gap m\^2 . \[mmmg\] As before the zero mode corresponds to a non-normalizable wavefunction.
The conformal factor appearing in the equation of the fluctuations [(\[broken\])]{} is: e\^B = [r\_0R]{} {
[ll]{} [\^3 uu]{} , & i,j=1,2 ,\
[1uu]{} , & i,j =3,4,5,6 ,\
[uu]{} , & i=1,2 ,j=3,4,5,6 .
. \[li16\]
### The 2-point function
The wave equation [(\[fiiin\])]{} for the gauge fields of the unbroken $SO(2)$, the broken coset and the unbroken $SO(4)$ symmetries, respectively, are: x\^2 (1-x) \^ -[k\^24]{} & = & 0 ,\
x(1-x)(x \^)\^- - (1-x) & = & 0 ,x , \[gh9\]\
(1-x) (x\^2 \^)\^ - [k\^24]{} & = & 0 , where, as before, $\tilde k^2 = k^2 R^4/r_0^2$. The properly normalized solution that is also regular in the interior is = [((1+)/2) ( (3+)/2)(1+)]{} x\^[(1+)/2-ł]{} F( ,, 1+,x ) , where $\D = \sqrt{\tilde k^2+1}$ and similarly to before, the parameter $\l= 0,\ha$ and $1$ for the currents corresponding to $SO(2)$, to the coset and to $SO(4)$, respectively. From this we obtain H(k) & = & [ł-1 k\^2]{} + ( ((1+)/2) + )\
& = & [ł-1 k\^2]{} + \_0\^dt [e\^[-t]{}-e\^[-[+12]{} t]{}1-e\^[-t]{}]{}\[add2\] and then G(x) = (1-ł) [r\_0\^2R\^4]{}r + \_0\^dy [yy]{} [e\^[-]{}]{} . Using this result it can be easily seen that the short distance behavior of the propagator is the same as in the conformal case and in particular [(\[cv1\])]{} is recovered. At large distances one finds that the two most dominant terms are G(x) (1-ł) [r\_0\^2R\^4]{} r + [\^28]{} [R\^2r\_0 r\^3]{} e\^[-r\_0 r/R\^2]{} , r , where naturally the range of the Yukawa-term is set by the mass gap in [(\[mmmg\])]{}. Hence, for the case where $\l\neq 1$, corresponding to the cases of the broken coset and the unbroken $SO(2)$ symmetries, the first term dominates for large $r$ giving a contribution to the correlator similar to [(\[goldy\])]{}, but with $\l$ replaced by $1-\l$. For similar reasons to those that we outlined for the case of the sphere-distribution of D3-branes after [(\[goldy\])]{}, the interpretation of such a term as being related to the Goldstone bosons is problematic and we believe that they are unphysical. (However, see footnote 9.)
Discussion
==========
In this letter we studied ${\cal R}$-symmetry current correlators in certain states on the Coulomb branch of ${\cal N} = 4$ SYM using the standard description of the AdS/CFT correspondence. The surprising result is that the spectra derived from the analytic structure of the correlators agree with spectra of other operators corresponding to dilaton and to the transverse graviton fluctuations. Furthermore, it turned out that the spectra are identical and do not depend on whether they are in the unbroken part of the left over global symmetry or reside in the coset, except for certain zero-mass poles which do depend on the sector. These poles give rise to a $1/r^4$ fall off of the correlators at large distances, the behavior expected of massless scalars, but we did not find good physical reasons to identify them with Goldstone bosons of the broken symmetry currents. We rather think that these poles are unphysical since they correspond to non-normalizable states and are inconsistent with the fact that the currents are all in the same supersymmetry multiplet.[^11]
Rephrasing the fluctuation equations into a supersymmetric quantum mechanics problem we found that they all fall into the same universality class and, furthermore, the Schrödinger potential are the supersymmetric partner potentials arising from the dilaton or from the transverse graviton fluctuations, which are identical. This indicates that all fluctuations in such backgrounds fall into the same class of supersymmetric quantum mechanics problems.
To obtain a more complete picture including the Goldstone bosons one probably has to include additional modes that live on the D3-branes which create the singularity in the infrared. In our set up with a continuous distribution of branes this seems a formidable task, and as a starting point it seems more feasible to study simpler examples, e.g. two stacks of coinciding branes or a single test brane separated from a stack of branes, in which case one would readily know the additional modes and their respective couplings to the bulk fields. We leave these issues for future work.
It will also be interesting to investigate current-correlators using solutions of $D=7$ and $D=4$ gauged supergravity that are dual to the (2,0) theories in six dimensions and the three-dimensional theories with sixteen supercharges, respectively, on the Coulomb branch. For a class of such backgrounds corresponding to a scalar-gravity sector analogous the one used in the present paper the most general solution has been found and is very similar to that in [(\[metriki\])]{}-[(\[hd3\])]{} [@bbs] (see also [@cglp]). The spectrum, of fluctuations corresponding to a massless scalar has been also exhaustively studied and in some cases the computations can be performed explicitly [@bbs]. Similarly to the present paper, in these cases as well, it is quite plausible that the current-correlators and the associated spectra are related via supersymmetric quantum mechanics to those of the massless scalar.
Acknowledgements {#acknowledgements .unnumbered}
================
A.B. would like to thank Y. Oz, J. Sonnenschein and S. Yankielowicz for discussions, and the universities of Tel-Aviv, Neuchâtel and Ludwig-Maximilians in Munich for hospitality and financial support. K.S. would like to thank the Theory Division at CERN for hospitality and financial support during a considerable part of this research. The research of K.S. was supported by the European Union under contracts TMR-ERBFMRX-CT96-0045 and -0090, by the Swiss Office for Education and Science, by the Swiss National Foundation and by the contract HPRN-CT-2000-00122.
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Addendum {#addendum .unnumbered}
========
The purpose of this addendum is to investigate the structure of the massless poles that appear in 2-point functions of broken symmetry currents in ${\cal N}=4$ SYM theory using purely field theoretical techniques and to compare the results with those obtained in section 3 using supergravity and the AdS/CFT correspondence. We had completed the essential part of this work around March of 2001. Parts of it are based on ideas developed around that time in collaboration with D. Freedman and K. Skenderis.
General formulation {#general-formulation .unnumbered}
--------------------
We start with the case of unbroken $\cal R$-symmetry where the vev’s corresponding to the six scalars of the theory are turned off. The $\cal R$-symmetry currents $J^a_\m$ are represented as bilinears in the scalar fields $X^i$, $i=1,2,\dots , 6$ transforming in the adjoint of $SU(N)$ J\^a\_= [1g\_[YM]{}\^2]{} T\^a\_[ij]{} [Tr]{}( X\^i\_X\^j) + , \[hl1\] where $T^a$ are $6\times 6$ matrices of $SO(6)$. The scalars $X^i$, being free fields, obey the following two-point function [^12] X\^i\_[pq]{}(x) X\^j\_[rs]{}(0)= g\_[YM]{}\^2 \^[ij]{} (\_[qr]{} \_[ps]{} - [1N]{} \_[pq]{} \_[rs]{})[1r\^2]{} , p,q,r,s=1,2,…, N . \[twopoint\] After performing the Wick contractions we compute the two-point function for the currents J\_\^a(x) J\_\^b(0)\~N\^2 \^[ab]{}(\_ - \_\_) [1r\^4]{} , \[cor1ne\] where we have kept only the leading term in the $1/N$-expansion.[^13] This is indeed the correct result for the two point function which also agrees with the AdS/CFT result [@fmmr].
In the case that the symmetry is broken by turning on non-zero scalar vev’s, we replace $X^i $ by $X^i_{vev} + \delta X^i$, where the $\delta X^i$ have the same free field two-point function as in [(\[twopoint\])]{}. Besides the bilinear term [(\[hl1\])]{} the current contains now a term linear in fluctuating fields J\^a\_= [1g\_[YM]{}\^2]{} T\^a\_[ij]{} [Tr]{}( X\_[vev]{}\^i\_X\^j) , \[hl11\] where we have introduced the vev’s X\^i\_[vev]{}=X\^i= [diag]{}(X\^i\_1,X\^i\_2,…, X\^i\_N) , \_[p=1]{}\^N X\^i\_p=0 . \[vvee\] At this point it is convenient to replace the adjoint $SO(6)$ indices by $a=[ij]$ and $b=[kl]$. Then, the matrix elements of the $SO(6)$ generators become $T^{ij}_{mn}= \d_{im} \d_{jn} -\d_{jm} \d_{in}$. The leading order correction to the conformal result [(\[cor1ne\])]{} is J\_\^[ij]{} (x) J\_\^[kl]{}(0)\~[1g\^2\_[YM]{}]{} H\^[ij,kl]{} \_\_ , \[corrector\] where the group theoretical factor $H_{ij,kl}$ takes the form H\^[ij,kl]{} = \_[ik]{} A\_[jl]{} - \_[jk]{} A\_[il]{} - \_[il]{} A\_[jk]{} + \_[jl]{} A\_[ik]{} ,A\_[ij]{} = \_[p=1]{}\^N X\^i\_p X\^j\_p . \[ggrop\] It is clear that, in the UV where the vev’s can be neglected, the conformal result [(\[cor1ne\])]{} dominates, whereas in the IR the dominant term is [(\[corrector\])]{}. The symmetric tensor $A_{ij}$ is given in terms of the scalar vevs only and depends on their distribution. In the following we think of the vevs $X^i_{vev}$ as defining $N$ points in ${\bf R}^6$. In most examples we use the fact that in the large $N$ limit such a discrete distribution can be well approximated by a continuous one.[^14] Furthermore, we will consider situations where the distribution spans only a lower dimensional submanifold embedded in ${\bf R}^6$. The tensor $H^{ij,kl}$ contains all the important information about the zero mass poles. It is antisymmetric in the indices $ij$ and $kl$ separately and symmetric under pairwise exchange. Note that only if both indices $i,j$ are along the vev-distribution $A_{ij}$ is non-zero. That implies that if all indices correspond to directions which are perpendicular to the distribution, then $H^{ij,kl}=0$.
### Basic examples {#basic-examples .unnumbered}
We digress to present a toy example of a discrete distribution of vevs in an $N$-polygon enclosed by a ring of radius $r_0$ in the $1$-$2$ plane [@sfe1] X\^i\_[vev]{} = ( r\_0 \_p, r\_0 \_p,0,0,0,0 ) , \_p=2p/N ,p=1,2,…, N . \[lv\] Computing the matrix elements $A_{ij}$ using the definition [(\[ggrop\])]{} is straightforward. We find that the only non-zero components are $A_{11} = A_{22} = N r_0^2/2$.[^15] We note that in this case we obtain the same result even if we approximate the discrete distibution by a continous uniform distribution of vev’s on the circumference of the circle.
We now turn to the specific examples of the distribution of vev’s on a disc and on a three-sphere, which we considered in section 3 using AdS/CFT correspondence. In these cases a direct comparison with the free field calculation can be performed. In particular, in accordance with the convention in [(\[momcorr\])]{}-[(\[cor2\])]{}, the momentum space version of [(\[corrector\])]{} can be expressed in terms of a function $H(k)$ H(k) \~- . \[hhh\]
For the distribution on a three sphere it is obvious that $A_{ii}=N r_0^2/4$, for $i=1,2,3,4$ and zero otherwise. These results are most easily derived in the continous approximation of the distributions. Hence, using [(\[hhh\])]{} and the facts that $g_{\rm YM}^2=g_s$ and $R^4=4\pi g_s N$, we obtain H(k) \~- , where the parameter $\lambda = 0, \frac{1}{2}$ and $1$ corresponds to currents in the transverse direction (unbroken $SO(2)$), broken currents in the coset and directions along the distribution (unbroken $SO(4)$), respectively. This agrees nicely with the AdS/CFT result [(\[add1\])]{}.
For the distribution on a disc we have similarly that $A_{ii}=N r_0^2/4$, for $i=1,2$ and zero otherwise. Using [(\[hhh\])]{} we compute H(k) \~ , where the parameter $\lambda = 0, \frac{1}{2}$ and $1$ corresponds to currents along the distribution (unbroken $SO(2)$), broken currents in the coset and directions orthogonal to the distribution (unbroken $SO(4)$), respectively. Again we find precise agreement with the AdS/CFT result [(\[add2\])]{}.
Generalization to a class of models {#generalization-to-a-class-of-models .unnumbered}
-----------------------------------
A natural question is whether the agreement between field theoretical results and those obtained from supergravity goes beyond the two specific examples we considered in detail. In fact, we may systematize our approach and show that such an agreement persists for all models with vev distributions corresponding to the five-dimensional supergravity solution [(\[metriki\])]{}-[(\[hd3\])]{}.
On the supergravity side the distribution of vev’s is encoded in the harmonic function appearing in the ten-dimensional metric describing the gravitational field of D3-branes. In our cases the harmonic function is [@bakas1] H\^[-1]{} = [4 R\^4]{} f\^[1/2]{} \_[i=1]{}\^6 [y\_i\^2(F-b\_i)\^2]{} , \[dhj1\] where $F$ is determined in terms of the six transverse coordinates $y_i$ as a solution of the algebraic equation \_[i=1]{}\^6 [y\_i\^2F-b\_i]{} =4 . \[jk4\] The harmonic function is in general H=\_[p=1]{}\^N[4g\_s |y - X\_p|\^4]{} , where the vev values $\vec X_p$ in [(\[vvee\])]{} became the centers of the harmonic function. In the continous approximation this takes the form H=4g\_sd\^6x [(x)|y-x|\^4]{} , where the density $\r(x)$ is normalized as $\int d^6 x \r(x)=N$. We would like to compute $A_{ij}$ in [(\[ggrop\])]{}, which for a continuous distribution reads A\_[ij]{}= d\^6x (x) x\_i x\_j . In general this can be found from the large $r$ expansion H=[R\^4r\^4]{} - 4g\_s [2 r\^6]{} ( \_[ij]{} -[6 y\^i y\^jr\^2]{}) A\_[ij]{} +… .
Returning to our cases where the harmonic function has the specific form [(\[dhj1\])]{}, we may cast its large $r$ expansion into the above form with A\_[ij]{} = N b\_[1j]{} \_[ij]{} , \[jefd\] where we define in general $b_{ij}=b_i - b_j$. We see that our general distributions allow a diagonal matrix $A_{ij}$. Hence, the only non-zero independent components of the group theoretical factor $H^{ij,kl}$ are $H^{ij,ij}$. If all indices correspond to directions which are perpendicular to the distribution then $H^{ij,kl}=0$, whereas if all directions are along the distribution $H^{ij,ij}=N (b_{1j} + b_{1i})$. If we are in the coset one index is along the distribution (say $i$) and one is orthogonal to it (say $j$), then one of the above terms is missing and therefore $H^{ij,ij}=N b_{1i}$. This agrees perfectly with the two special cases of the disc and sphere distribution that we considered before.
### Correlators from supergravity {#correlators-from-supergravity .unnumbered}
Let us consider the equation [(\[broken\])]{} but in terms of the variable $F$ ( (F-b\_i)(F-b\_j) [ddF]{}) - k\^2 [ (F-b\_i)(F-b\_j)f\^[1/2]{}]{}- [b\_[ij]{}\^24 (F-b\_i)(F-b\_j)]{}=0 , \[hjg\] where $F$ was defined in equation [(\[pro1\])]{}. Equation [(\[hjg\])]{} was solved exactly for the cases of the disc sphere and the sphere distribution. For the purposes of this addendum it suffices to concentrate on the limit $k^2\to 0$, where [(\[hjg\])]{} can be solved exactly for any distribution. This will give the leading contribution to the two-point function of currents for large distances. At the AdS boundary $F\to \infty$ we impose the usual boundary condition $\Phi\to 1$ corresponding to a point-like source. Furthermore, we require $\Phi$ to be smooth at the singularity $F=b_1$ in the interior. In the following we use units where $g=2/R=1$.
:
In this case the indices of the current $i,j$ are such that $b_i=b_j=b_1$. Demanding regularity at the singularity $F=b_1$ and imposing the normalization condition at the boundary gives =1 . Therefore [(\[onshell\])]{} gives H(k)=0 . As expected this agrees with the field theoretical result.
:
In this case the indices of the current are such that $b_i, b_j\neq b_1$. As before, regularity at the singularity at $F=b_1$ and the normalization condition at the boundary give = [1b\_[ij]{}]{} ( b\_[1j]{} ([F-b\_iF-b\_j]{})\^[1/2]{} -b\_[1i]{} ([F-b\_jF-b\_i]{})\^[1/2]{} ) , from which we compute using [(\[onshell\])]{} that H(k) = - [b\_[1i]{}+ b\_[1j]{}4 k\^2]{} . This is in perfect agreement with field theory expectations as spelled out after [(\[jefd\])]{}. A particularly interesting case is when $b_i=b_j\neq b_1$. Then the above expressions reduce to = [F-b\_1F - b\_i]{}and H(k) = - [b\_[1i]{}2 k\^2]{} . The case of the sphere and disc distributions correspond precisely to that with $b_{1i}=r_0^2/4$ ($b_1$ can be put to zero by a shift of the coordinate $F$), for $i=1,2,3,4$ and $i=1,2$, respectively.
:
In this case the currents indices are such that $b_i=b_1$ and $b_j\neq b_1$. Proceeding as before we find that = ([F-b\_1F-b\_j]{})\^[1/2]{} and that H(k) = - [b\_[1i]{}4 k\^2]{} . Again, one easily sees that this agrees with field theoretical expectations.
Comments on the masses of gauge bosons {#comments-on-the-masses-of-gauge-bosons .unnumbered}
--------------------------------------
Finally, we mention some usefull facts about the masses of the W-bosons that arise on a generic point of the Coulomb branch of the ${\cal N}=4$ SYM theory. The general mass matrix is read off from eq. (63) of [@sfe2] (M\^2)\_[pq]{}=|X\_p-X\_q |\^2,p,q=1,2, …, N , up to a numerical constant of order 1. Hence, the masses have the geometrical interpretation as the distances between the various vev positions distributed in the ${\bf R}^6$ scalar space. Equivalently, they are given by the masses of the strings stretched between the D3-branes located at these points. Since we may shift uniformly all vectors $\vec X_p$’s without changing the Physics, the number of elements are in a generic case $N^2-1$ as it should be. It is clear that depending on the specific vev distributions some of these masses might be degenerate. In particular, in the case of the discrete distribution of vev’s in the $N$-polygon we find, using [(\[lv\])]{}, that (see eq. (66) of [@sfe2]) M\_n = r\_0 (n/N) ,n=1,2,…, N , \[hwe\] which is an exact result for any $N$. The degeneracy for the zero mode is $d_N=N-1$ and for the rest $d_n=2(N-n)$. It is easily seen that $\sum_{n=1}^N d_n=N^2-1$. Hence, for large $N$ there are W bosons with masses of order $r_0$ and light masses of order $r_0/N$. In the case of vev’s distributed on a disc a similar result can also be derived starting from a discrete distribution [@sfe1; @brand1] whose limit is the continuous one we have been using.
[^1]: Since $30^\mathrm{th}$ September 2000: Department of Physics, CalTech, Pasadena, CA 91125
[^2]: Other studies of the Coulomb branch of the ${\cal N} =4$ SYM theory using the AdS/CFT correspondence can be found in [@OthersCoulomb].
[^3]: We thank M. Bianchi for prompting us to explain in detail how the decoupling between scalar and gauge field fluctuations actually works as well as for other related comments.
[^4]: For notational simplicity we did not include indices $i,j$ in defining $B$ in [(\[h93\])]{}. Nevertheless it should be kept in mind that different choices for the scalar fields $\b_i$ and $\b_j$ lead to different values for $B$.
[^5]: The overall normalization is found by carefully keeping track of all the prefactors in the dimensional reduction in the $S^5$-directions of the ten-dimensional type-IIB action to five dimensions. In particular, ${1\ov \kappa^2} = {V_{S^5}\ov
4 \kappa_{10}^2} R^{8}$. Then using $2 \kappa_{10}^2 = (2\pi)^7 \a'^4 g_s^2$, $R^4 = 4\pi g_s \a'^2 N$ and $V_{S^5}=\pi^3$ we find the result mentioned above.
[^6]: An alternative way to prove the equivalence of the potentials [(\[kkk1\])]{} and [(\[kkk2\])]{} is to use the differential equation obeyed by the $\b_i$’s, namely $\b_i^\prime =A^\prime +{g\ov 2} e^{A+2\b_i}$ [@bakas1].
[^7]: The authors of [@massimo] informed us that their arguments concerning graviphotons are actually broader and include all massive cases where $U(1)_R$ is broken.
[^8]: Throughout the paper we will make use of special functions and their properties following the conventions of [@tipologio].
[^9]: Work on the AdS/CFT correspondence and the Goldstone bosons has been reported using a different model in [@dz2].
[^10]: Actually, we were able to explain the presence of these massless poles we found in the supergravity calculation by a field theory calculation in the free field approximation. These results are added as an addendum at the end of this paper, since they were found after publication of the original version of the paper.
[^11]: After this paper was published in JHEP we found convincing evidence that these poles are actually physical. See footnote 9 on page 13 and especially the addendum at the end of the paper for more details on the resolution of this puzzle.
[^12]: In our conventions the field theory action has an overall factor of $1/g_{YM}^2$.
[^13]: For finite $N$, the $1/N$-term in [(\[twopoint\])]{} induces a shift which replaces the coefficient $N^2$ by $N^2\!-\!1$ corresponding to the dimension of the $SU(N)$ group. We also note that the contribution of the fermions only affects the result by an overall $N$-independent numerical constant which is not important for our purposes.
[^14]: This is correct as long as we work with energies (distances in the gravity side) $U$ not too close to the vev values. Typically the condition to be fulfilled for the continous approximation to be valied is $U/X_{\rm vev}-1\gg {\cal O}(1/N)$, where $X_{\rm vev}$ is a typical scalar vev value [@sfe1].
[^15]: We have used the fact that \_[p=1]{}\^N \^2(2 p/N) = \_[p=1]{}\^N \^2(2p/N) = N/2 , \_[p=1]{}\^N (2p/N) (2p/N) =0 .
|
---
abstract: 'The behavior of the nucleon structure functions in lepton nuclei deep inelastic scattering, both polarized and unpolarized, due to nuclear structure effects is reanalyzed. The study is performed in two schemes: an [*[x]{}*]{}-rescaling approach, and one in which there is an increase of sea quark components in the in medium nucleon, related to the low energy [*[N-N]{}*]{} interaction. In view of a recent interesting experimental proposal to study the behavior of the proton spin structure functions in nuclei we proceed to compare these approaches in an effort to enlighten the possible phenomenological interest of such difficult experiment.'
author:
- 'H. Fanchiotti'
- 'C. A. García Canal'
- 'T. Tarutina'
- 'V. Vento'
title: '[**Medium Effects in DIS from Polarized Nuclear Targets**]{}'
---
Introduction
============
More than 30 years ago, the European Muon Collaboration (EMC) discovered that the unpolarized structure function of a bound nucleon in a nucleus is different from that of a free nucleon, and different from one nucleus to another [@Aubert:1983xm]. This experimental fact triggered an innumerable series of analyses and their corresponding explanations (see [@Rith:2014tma] for an extensive list of references) and very interesting recent theoretical developments relating the EMC effect to Short Range Correlations [@Higinbotham:2010tb; @Weinstein:2010rt; @Piasetzky:2011zz; @Arrington:2012ax; @Hen:2012fm] ,[@Frankfurt:2009vv; @Vanhalst:2012ur; @GarciaCanal:2013dma].
Not long ago there has been a proposal for an experimental study of the nuclear effect for the polarized structure function $g_1$, an experiment in which both projectile and the nuclear target are longitudinally polarized [@Joo]. This work has been motivated by some detailed model dependent calculations [@Cloet:2005rt; @Cloet:2006bq; @Smith:2005ra; @Ganesamurthy:2011zza], which have coined the development: [*polarized EMC effect*]{}. This problem was theoretically studied long ago for the Deuteron by Frankfurt and Strikman [@Frankfurt:1981mk] and shortly thereafter for arbitrary spin by Jaffe and Manohar [@Jaffe:1988up] with a non-relativistic convolution model. The recent analyses are detailed model calculations which take into account a realistic low energy nuclear description, a model for hadron structure and include QCD evolution in their schemes. Their result shows a large effect in the nuclear to proton ratio and moreover, the contribution of the quark could dramatically affect it at small $x$.
Despite the fact that the proposed phenomenon is very interesting and might help to understand the behavior of the nucleons in nuclei, the name, as pointed out by the PAC29 committee [@PAC29] might be misleading. In the experimental proposal the chosen nucleus has been $^7 $Li, were the spin of the system, $3/2$, is naively mostly associated with a valence proton lying in a $p$-shell, while the remaining nucleons are coupled mostly in pairs to total angular momentum $0$. Thus the averaged medium behavior implied by the unpolarized EMC effect is not present in this polarized case. However, it is clear that the active valence proton is subject to the effect of its companions and therefore a medium effect must be present.
Our aim in this note is, by avoiding in as much as possible model dependence, to center our attention into physical ideas which might be at the origin of the discussed phenomenon. For that purpose we revisit the unpolarized EMC effect in terms of an $x$-rescaling description [@GarciaCanal:1984eh] and a pion content approach [@Epele:1994aq; @deFlorian:1993hx] to fix the ideas and the parameters connected with the experimental data available at present. We then proceed to study the in medium polarized case. We have in mind in all our discussion the experimentally proposed $^7$ Li, which represents an ideal system to distinguish conceptually between the [ *polarized*]{} and unpolarized EMC effects. We end by comparing the polarized and unpolarized phenomena with the intention of motivating further experimental research.
Unpolarized DIS
===============
In this section we briefly summarize the analysis of DIS with nuclear targets in the two approaches mentioned before, the $x$-rescaling approach [@GarciaCanal:1984eh], and a qualitative approximation to the so called pion content approach [@Epele:1994aq; @deFlorian:1993hx].
This $x$-rescaling approach contains only one parameter, $\eta$, and the EMC effect was described by suggesting that the true scaling variable for deep-inelastic scattering off nuclei should be taken to be $x^* = \eta \,x$ [@GarciaCanal:1984eh]. The main idea behind this approach is that the quark distributions in nuclei are shifted towards lower $x$ values as compared to those corresponding to free nucleons. Thus the name for the mechanism, $x$-rescaling. This approach was shown to be connected [@GarciaCanal:1986xe] to the $Q^2$-rescaling approach [@Close:1984zn].
Consequently, the measured ratios of the nuclear to deuteron structure functions can be written, for fixed $Q^2$ as, $$R(A) = F_2^{A}(x^*)/F_2^{D}(x^*)
\label{f2A}$$ where $F_2^{A}(x^*)$ is the nuclear structure function calculated using a rescaled variable $x^*$ in the free proton and neutron structure functions, and $F_2^{D}(x^*)$ is the Deuteron structure function where the effects of rescaling are small.
The $x$-rescaling mechanism leads to fits of very good quality for the EMC effect in the region $0.3< x < 0.75$ for all the nuclei experimentally analyzed [@GarciaCanal:1984eh; @GarciaCanal:2013dma]. Recently, this approach was used to show an interplay between the quark-gluon and hadronic degrees of freedom in the unpolarized EMC [@GarciaCanal:2013dma].
The second approach is a simplification of the so called pion content model [@Epele:1994aq; @deFlorian:1993hx]. In this approach one introduces the pion presence into the nucleon structure function and makes a convolution model which takes this presence and the pion structure into account. The end result is twofold, on the one hand there is a $x$-rescaling effect associated to momentum conservation in the pion emission and on the other hand there is an explicit contribution of the sea associated with the pionic structure function. This approach also leads to fits of very good quality for the EMC effect in the whole $x$-region for all the nuclei experimentally analyzed. The direct contribution of the sea is important for low $x$ but not so much for intermediate $x$. However, and this is a peculiarity of this approach, it is necessary to consider also the pion sea effect in the deuteron when one deals with the EMC ratio, if one wants to avoid the blowing up of the ratio at low $x$ which would destroy the agreement in the EMC ratio [@Epele:1994aq; @deFlorian:1993hx].
We incorporate this twofold mechanism, in our phenomenological scheme, by means of two parameters. One, $\delta$, describes the associated scaling mechanism into the free nucleon structure functions as before $ x \rightarrow \delta x $; analogously to $\eta$, $\delta >1$. Another, $\sigma$, which magnifies the contribution from the sea associated to the $u$ and $d$ flavors in the intermediate $x$ region, concentrated initially for very small $x$. For one nucleon in the medium the contribution to $F_2$ is $$F_2^{N} (x) = F_2^{N} (\delta x) + F_2^{udsea_N} (\sigma x).$$ This modified nucleon structure functions have to be incorporated into Eq. (\[f2A\]) to perform the corresponding EMC average. We shall call this approach modified sea scheme (MSS).
Our study is concentrated on the experimentally wishful nucleus $^7$Li [@Joo]. We use the proton and neutron structure functions and sea distributions from the analysis of Ref. [@Martin:2009iq] for fixed $Q^2 = 10 $GeV$^2$. The value of $\eta$ for $^7$Li is extracted from a linear extrapolation of a fit to the data of several nuclei. The value obtained for $^7$ Li is $\eta = 1.011 \pm 0.002$. The values of the parameters for the MSS scheme are obtained by fitting them to reproduce the Ratio, $R$, of the $x$-scaling description in the EMC region and are shown in the caption of Fig. \[f2Li\].
In Fig.\[f2Li\] we show our prediction of both approaches and the size of the sea contribution in the MSS method. A good agreement between both approaches is obtained by simply doubling the sea contribution. Note, as it was already mentioned, that we had to incorporate an extra sea contribution for the Deuteron in order to avoid a dramatic increase at the origin, where it tends to dominate. Fig. \[f2noDsea\] shows that effect.
Polarized DIS
=============
Several studies of the nuclear effects for the $g_1$ polarized structure functions in the typical EMC $x$ region have recently appeared. They are based on detailed dynamical nucleon structure models and models for nuclear matter. The analysis of Ref. [@Cloet:2005rt; @Cloet:2006bq] predicts a strong increase of the ratio $R_{pol}= g_1^A/A g_1^p$ of the order of twice the size in the unpolarized case. This calculation is based upon a convolution where the nucleon is described as a bound state of a quark-diquark in the Nambu-Jona-Lasinio (NJL) model and the presence of the nuclear medium is taken into account through mean fields which act on the quarks in the nucleon. This is mainly a valence quark picture where the sea appears by evolution. Another proposal [@Smith:2005ra] includes sea quarks explicitly in order to be consistent with both DIS and Drell-Yan results. Consequently, the nuclear medium effects are present both in the valence and the sea quark distributions. They are computed using the chiral quark-soliton model and the nuclear effects related to the valence quarks are similar to those obtained in Ref. [@Cloet:2005rt; @Cloet:2006bq], while the inclusion of sea quarks gives rise to an important increase for $x < 0.3$. Both description contain many parameters that have to be fitted. A third analysis [@Ganesamurthy:2011zza] is based on the phenomenological Thermodynamical Bag Model, a modification of the MIT bag model, and shows not much difference between the polarized and the unpolarized ratios.
For our analysis we use the fits to all polarized data of Ref. [@deFlorian:2008mr; @deFlorian:2009vb] for $Q^2 = 10 $GeV$^2$. We start by the rescaling approach and as in the case of the unpolarized structure function we replace the scaling $x$ variable by a rescaled variable $x^*$ in the free nucleon structure functions.
In order to find reasonable values for this $\eta$ parameter, we follow two arguments. The first, which we already discussed previously, is the definition of the polarized in medium effect. Let us for clarity limit ourselves to $^7 Li$. In this case, from a naive nuclear shell model point of view we have one active proton in a $p$-shell and the remaining nucleons are coupled to total angular momentum zero in $s$ and $p$ shells. More sophisticated calculations with Green Function Monte Carlo [@Pieper:2004qw] and a Cluster Model [@Walliser:1985zz] give a mean proton polarization of $87\%$ and $3\% $ for the neutron. For simplicity we assume the shell model picture and we define the ratio
$$R_{pol}(x) = \frac{g_1^p(x^*)}{g_1^p (x)}.
\label{g1x}$$
It is interesting to compare now with Eq. \[f2A\] to realize the difference between the polarized and the unpolarized effects, namely the conventional EMC averages over nucleons, while not so the polarized in medium effect.
Accepting the previous naive definition, the second problem we face is the relation of the $x$-scaling parameter $\eta$ with the dynamics. This parameter is related to the efective nucleon mass in the medium [@GarciaCanal:1984eh] which is an average property and therefore we expect that $\eta$ should not change. However, the fact that the nucleon is polarized might incorporate some spin interactions, that average out in the unpolarized case, which might modify it. The results of Refs. [@Cloet:2005rt; @Cloet:2006bq; @Smith:2005ra] seem to indicate that $\eta$ should increase. We plot in Fig. \[g1f2x\] the value of the ratio Eq. (\[g1x\]) in form of a band, whose smallest value corresponds to the unpolarized $\eta$ value and the largest to $\eta = 1.02$. The full curve shows the ratio $R$. We note that the $R_{pol}$ ratio is extremely sensitive to $\eta$ and therefore to the detailed dynamics of the polarized case.
We next proceed to the MSS approach. As before the nucleus structure function will be defined in terms of modified nucleon structure functions, $$g_1^N (x) = g_1^N (\delta x) + g_1^{udsea_N} (\sigma x).
\label{g1sea}$$ For $^7 Li$, $N$ is a proton. In Fig.\[g1f2sea\] we plot $ R_{pol}$ with the same parameters as for $R$, and following the same discussion as before we construct a band with a $2\%$ increase both in $\delta$ and $\sigma$.
Finally in Fig.\[g1comparison\] we compare our results with the two most extreme published analyses. Our result is intermediate between both. We could obtain a higher sea contribution close to the origin by increasing $\sigma$ in line of Ref.[@Smith:2005ra], but we can never get a result as low as that of Ref.[@Cloet:2005rt; @Cloet:2006bq] for low $x$.
We have only analyzed in here the low flavor sea contributions not worrying about heavy flavors and gluons, whose contribution are small in the two approches.
Conclusion
==========
The schemes presented in this note parametrize the change of parton distributions in medium in terms of one or two parameters. These parametrizations are motivated by QCD based dynamical pictures, $x$-scaling associated to a Renormalization Group Analysis, and the MSS, a chiral scheme convoluted into a partonic description. It is really surprising that in the unpolarized case, with one or two parameters, one is able to fit the EMC effect for all studied nuclei. Thus our fits lead, based on QCD and chiral dynamics, to parametrizations of the data with a minimal number of parameters.
What can a polarization experiment contribute to our undertanding of proton structure? As has been widely discussed this is not an EMC [“]{}average" type experimental analysis. The proposed nucleus $^7$Li is, with percentage corrections, basically a polarized proton in a nuclear medium. In some sense the experiment reminds us of the the proton spin analysis [@Anselmino:1994gn], but here with its parton distributions modified by nuclear dynamics.
Our calculation has shown two important results. The most relevant one is that a dramatic change in the EMC region in $R_{pol}$ would imply a dramatic change in the value of the scaling parameters $\eta$ and $\delta$ and therefore an important influence of the spin-spin interactions in nuclei which lie dormant in the average procedure of the unpolarized EMC effect. The second result, which has been confirmed by the MSS approach is that the sea contribution is important for low $x$ if the ratio is performed against the free proton $g_1$ structure function. If one would perform the ratio with respect to the Deuteron, with its pionic sea included, the sea effect would be quantitatively diminished. Therefore, our study strongly supports the realization of the proposed experiment [@Joo] in the whole range of the variable $x$ as the means of better understanding the structure of the nucleon parton distributions in nuclei and their relation to nuclear dynamics.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge Elliot Leader, who proposed the idea of the present analysis and for his careful reading of the manuscript. We thank Rodolfo Sassot and Robert Thorne for invaluable assistance in the use of their codes, and Arcadi Santamaria for sharing his ample knowledge of Mathematica. We have all been partially supported ANPCyT Argentina. V.V. has been also supported by the Ministerio de Economía y Competitividad and EU FEDER under contract FPA2010-21750-C02-01, by Generalitat Valenciana: Prometeo/2009/129 and by the EPLANET network under contract PIRSES-2009-GA-246806.
[ABC]{}
References {#references .unnumbered}
==========
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---
address: 'Dept. of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India.'
author:
- 'The Quark Gluon Plasma: lattice computations put to experimental test'
- Sourendu Gupta
---
\#1[[\#1]{}]{}
Introduction
============
[ *The quagma engineers? That huge ugly brown thing we saw? That was one of them?\
[Gregory Benford]{}, in “Around the curve of a cosmos”*]{}
QCD has been tested at zero temperature by its predictions for “hard processes”, , processes in which all relevant scales are much larger than the intrinsic scale, $\lqcd$. This convenience is due to asymptotic freedom in QCD; at scales much larger than $\lqcd$ the coupling $\alphas$ is small. At finite temperature, $T$, the scale relevant to most thermodynamic variables is of order $T$. Since $T_c/\lqcd=0.5$ for QCD with two light flavours of quarks [@precise], at experimentally accessible temperatures $T/T_c\sim$ 1–3, the scales are comparable to $\lqcd$, $g\equiv\sqrt{4\pi\alphas}={\cal O}(1)$, and one deals with soft physics [@expg]. Perturbation theory may remains a rough guide to intuition. However, since it is sensitive to the infrared, , non-perturbative length/mass scales, its domain of applicability really is $g\ll 1$, , $T\ge10^9T_c$. As a result, lattice gauge theory is the only theoretical tool of direct relevance to experiments currently being performed at the Relativistic Heavy-Ion collider (RHIC) at the Brookhaven Lab.
Until recently, the agreement of the energy density at freeze-out in relativistic heavy-ion collisions with that predicted at $T_c$ by lattice computations, and the connection between Debye screening and $J/\psi$ suppression, have been the main points of contact between fundamental QCD computations and experiments. In this talk I will concentrate on other comparisons, all potentially precise confrontations of lattice QCD predictions against experiments. Many of these have emerged in the last few years and are therefore less well-known. Specifically, I will deal with predictions of strangeness yields, event to event fluctuations of conserved quantities, extraction of the speed of sound from the centrality dependence of elliptic flow and the first estimates of relaxation times and photon/dilepton production rates. A secondary motive for this talk is to identify the ways in which thermal perturbation theory may guide our thinking even in the domain where it is not expected to work.
For $T\ll T_c$ strongly interacting matter is in the confined phase. Chiral symmetry is spontaneously broken, with pions emerging as pseudo-Goldstone boson. Since the Dirac operator for quarks has nearly vanishing eigenvalues, accurate lattice computations are hard. In this range of temperatures it may be much easier to use effective theories such as chiral perturbation theory to extract quantities of interest to experiments. Interesting predictions exist for a lukewarm pion gas [@son] and for the phases of cold and dense QCD [@dense]. At this time it seems that the role of lattice computations is to validate and determine some of the crucial inputs into such models. A discussion of this lies outside the scope of this talk.
QCD matter undergoes a phase transition, or at least a rapid cross over at $T=T_c$. This was the region on which the earliest lattice computations concentrated— successfully extracting $T_c$ with high precision, and estimating the order of the phase transition [@oldlat]. The universality class of the phase transition in the chiral limit still remains to be reliably extracted— the main problem here is that extracting physics at small quark masses requires very large lattices, thus pushing up the time required to perform accurate numerical lattice computations. This region of temperature remains of great interest, since the transition from quarks to hadrons stamps the physics of this region onto many observables studied at RHIC. Since highly accurate lattice computations for this region are still underway, this talk will touch only briefly on this.
Most of the material in this talk is of relevance to the physics of the temperature range $1.5\le T/T_c\le3$, where $g={\cal O}(1)$, and the perturbative and non-perturbative scales cannot be separated. As a result, perturbation theory cannot be numerically accurate and lattice computations are essential to extract the physics of the plasma. This talk is divided into four main sections. We begin by an examination of the quasiparticle modes of the plasma, which allows us to test perturbative expansions in a theoretically clean setting. The next two sections concentrate on two thermodynamic quantities of direct relevance to experiments— the equation of state and quark number susceptibilities. The following section is devoted to off-equilibrium phenomena such as relaxation times, electrical conductivity and photon (and dilepton) production rates in the plasma.
Perturbation theory: is the QCD plasma a quark gluon plasma?
============================================================
Perturbation theory is an expansion of the free energy of QCD in a series in $g$, and is effectively an expansion in terms of gluon and quark fields. One of the most basic quantities in Euclidean high temperature perturbation theory is the Debye screening mass. At leading orders in the perturbative series this has contribution only from the electric polarisation of the gluon [@pols], however at higher orders magnetic polarisations also contribute [@nadkarni] and, as a result, the perturbation expansion breaks down at finite order [@arnold]. Perturbative predictions for the Debye screening mass do not exist close to $T_c$, and lattice studies of Debye screening can give no meaningful test of perturbation theory [@olaf]. A couple of more limited tests are possible.
The first is to check whether a “constituent” gluon picture works [@bernd]. Correlations of the operators A\_1\^[++]{} = [R]{}e[Tr]{}L A\_2\^[–]{} = [I]{}m[Tr]{}L \[ops\]($L$ is the Polyakov line operator, , the flux due to a static quark) are obtained by two and three electric gluon exchanges to leading order. If this continues to be true in some sense non-perturbatively, then the screening masses obtained in these two channels should be in the ratio 3/2. A recent lattice computation (see Figure \[fg.saumen\]) shows that this is actually true in the range $1.25\le T/T_c\le3$ [@saumen]. However, detailed studies of other screening masses on the lattice show that no “constituent” picture can be built up in the sector of magnetic gluons [@saumenold]. In fact, magnetic Wilson loops have been shown to confine [@bali]. This is consistent with a picture of an effective theory for finite temperature QCD in which electric gluons and magnetic glueballs are the degrees of freedom [@gpy]. A detailed model consistent with the lattice data is under investigation [@rob].
The second is to test a systematic reduction of the theory which goes by the name of dimensional reduction (DR) [@dr]. This attempts to integrate out the high frequency ($\omega\ge2\pi T$) components of the theory and produce a long distance effective theory. The couplings in this effective theory are computed at the scale $2\pi T$ and hence perturbation theory should be fine as long as $\alphas$ is small enough. However, the effective theory is fairly complicated (probably confining) [@kajantie] and its long distance properties have to be extracted by a lattice computation. For quenched QCD, the spectrum of screening masses obtained from DR [@owe] agrees with that from the full theory for $T\ge2T_c$ [@talk00]. One such test is shown in Figure \[fg.saumen\].
For physics in thermal equilibrium, it seems fruitful to think of the quenched QCD plasma above $1.25T_c$ as containing electric gluons. The magnetic sector seems confined, thus solving the infrared (Linde) problems of hot perturbative QCD through the non-perturbative mechanism of generating “thermal glueballs” [@magscr]. Closer to $T_c$ there is not even any evidence for electric gluons. QCD with dynamical quarks may have a quantitative description in terms of gluons only for $T>6T_c$ [@obstr; @precise].
Flow and the equation of state
==============================
A clear signal of collective effects in the final state of a relativistic heavy-ion collision would be hydrodynamical flow. If flow can be unambiguously identified in experiments, then the equation of state (EOS) of QCD matter becomes accessible to measurement, since it is an input to the hydrodynamical equations. The EOS, , the temperature dependence of pressure ($P$) , energy ($E$) and entropy ($S$) densities, have been extracted on the lattice in quenched QCD [@eosnf0] as well as in QCD with two [@eosnf2] or four [@eosnf4] flavours of dynamical quarks. It is a remarkable lacuna that these EOS has not yet been put through the machinery of hydrodynamical codes to confront experiments [@jane].
$P$, $S$ and $E$ deviate from the Stefan-Boltzmann limit strongly near $T_c$ and by about 20% even at the highest temperatures at which lattice computations exist (about $4T_c$). This seems to have no explanation within perturbation theory, since the perturbative series for $P$ fluctuates wildly as more terms are added; a Borel [@borel] or Padé [@pade] summation of the series does not help. Screened perturbation theory [@scrpt] applied to the hard thermal loop resummation does not produce agreement with the lattice results [@eospert]. On the other hand, there have been reasonably successful attempts to fit the pressure by a gas of quasiparticles whose masses are the fit parameters [@quasi]. A partially self-consistent resummation also gives good agreement with the lattice data [@eosbir]. More recently the pressure has been obtained in the DR theory [@eosdr].
Signatures of hydrodynamic flow have been sought in particle spectra and in HBT radii in the past. At present one of the most promising signals is elliptic flow [@flow]. If hydrodynamics can be trusted, then, in off-center collisions of two nuclei, the spatial anisotropy leads to pressure gradients. These drive momentum anisotropies, whose second Fourier coefficient, $v_2$, is called elliptic flow [@v2]. This has been observed in experiments over a wide range of collider energies [@elliptic].
At RHIC energies, the variation of $v_2$ with the impact parameter $b$ (which determines the charged multiplicity $n_{ch}$) is claimed to have a good explanation in terms of hydrodynamic flow [@v2hydro]. So does the variation of $v_2$ with the transverse momenta, $p_t$, of the particles used to measure it [@altv2]. If the initial temperature is determined independently, then the slope of $v_2$ against $b$ depends on the speed of sound, $c_s$, since the pressure drives the evolution of $v_2$. In principle, then, $c_s$ can be measured directly from RHIC experiments and compared to predictions from the lattice.
Lattice predictions for $c_s$ can be obtained as a byproduct of the extraction of the EOS. In Figure \[fg.v2sound\] we show our extraction of $c_s$ from the data in [@eosnf0]. This computation is preliminary (a more detailed computation is underway), and the main uncertainty is connected with the fact that the lattice data used have finite lattice spacing artifacts which need to be compensated for. However, a dip in $c_s$ near $T_c$ has been seen with two-flavour dynamical quarks [@milc], and argued to follow from thermodynamic considerations [@thermo]. The most interesting observation is that at the highest temperatures $c_s$ is close to its ideal gas value, although both $P$ and $E$ are far from ideal. This has also been seen with two flavours of dynamical quarks [@milc].
Fluctuations, strangeness yields, and quark number susceptibilities
===================================================================
Event by event fluctuations in conserved quantities such as the charge or baryon number [@fluct] are proportional to quark number susceptibilities \_[fg]{} =.-TV |\_[\_f=\_g=0]{}, \[qns\]where $Z$ is the partition function of QCD and $\mu_f$ is the chemical potential for flavour $f$ [@gott]. Further details, including those of the evaluation of these susceptibilities on the lattice can be found in several recent reviews [@qns]. It is interesting to note that recent lattice computations [@qnslat] for the diagonal susceptibilities ($\chi_{ff}$) can be reproduced in a skeleton graph resummation [@qnsbir], dimensional reduction [@qnsdr] and also in a quasiparticle picture [@qnsqp]. The off-diagonal susceptibilities are found to be zero in lattice computations; there seems to be no explanation for this in models.
Measured fluctuations [@fluce] are thought to be proportional to the ratio $\chi/S$. Lattice computations for these are under good control for $T>T_c$, but the region $T<T_c$ requires more work. Present day lattice data [@qnslat] indicate a hierarchy of fluctuations for baryon number ($\chi_B$), electric charge ($\chi_Q$) and strangeness ($\chi_s$)— \_B<\_Q<\_s&& (T>T\_c),\
\_B>\_Q>\_s&& (T<T\_c). \[hier\]The inversion of the hierarchy as one crosses $T_c$ may be a possible experimental signal of the phase transition.
One of the most interesting pieces of information that the lattice can supply is for the strangeness yield, which is measured very accurately in experiments, and hence has attracted much attention [@strange]. This yield is parametrised as the Wroblewski parameter, $\lambda_s$, which is the relative number of primary produced strange to light quarks [@wrob; @cleymans]. Clearly, $\lambda_s$ is the ratio of imaginary parts of the complex susceptibilities in these flavour channels. Under reasonable (and testable) assumptions [@ours] \_s=, \[wrob\]thus allowing us to compute this quantity on the lattice. Results obtained in quenched QCD [@ours] are exhibited in Figure \[fg.wrob\]. We expect this ratio to be fairly insensitive to quenching artifacts. A computation in dynamical QCD with two flavours at $T_c$ is now underway.
Relaxation times, photon emissivity and the electrical conductivity of a plasma
===============================================================================
We turn next to non-equilibrium phenomena in the QCD plasma. These are of very direct relevance to heavy ion experiments, since the matter formed in the fireball is fully out of equilibrium initially. Of interest are limits on how fast it equilibrates with respect to the strong interactions, how fast local thermal fluctuations diffuse away, how quickly a hard probe (such as a jet) loses energy, whether the system remains forever out of equilibrium in electroweak interactions, and if so, the rate at which it radiates leptons and photons. Over the last two years perturbation theory and lattice computations have reached a stage where we can begin to constrain the answers seriously.
The most crucial piece of information that is required is of the equilibration time. Hydrodynamic explanations for particle spectra, HBT radii and, especially, elliptic flow, all require relatively small equilibration times in the plasma (0.6–1 fm) [@ttime], implying that transport related cross sections are huge. Experimental evidence for jet quenching [@jetq], particularly the damping of away-side jets [@jetaway], are also indicative of small relaxation times or rapid energy flows. These time scales, or the corresponding transport coefficients are intimately related to large angle or multiple small angle (Landau-Pomeranchuk-Migdal, LPM) scattering and are of the order of $1/g^4\log(1/g)T$ when $g$ is small enough [@kinetic]. The Kubo formul[æ]{} relate these transport coefficients to the zero energy ($\omega=0$) limits of the imaginary parts of certain retarded correlators. When these correlators are evaluated in perturbation theory, the multiparticle states which contribute to it have momenta $(k^i_0,\vec
k^i)$ which sum up to zero ($i$ labels particles). However, when these intermediate states are massless, each of the $k^i_0\simeq\omega$ can be zero. Then while integrating over $k^i_0$, the contour is pinched between these poles. Interactions, specifically the transport cross sections, throw these poles slightly off-axis, but the pinch still gives a bump in the imaginary part of the correlators. The effect of such bumps, which are seen to persist beyond the pertubative regime, is to give rise to transport coefficients [@aarts].
The simplest of this class of problems deals with electromagnetic interactions. The transport coefficient one deals with is the ohmic conductivity, $\sigma$, , the response of the QCD plasma to an external static and spatially uniform electric field, $E$. The result of applying such a field is to set up a current $j=\sigma E$ in the direction of the field. A Kubo formula relates $\sigma$ to the imaginary part, $\rho$, of the retarded current-current correlator in equilibrium— (T) = 16. \_i\^i(,[p=0]{},T)|\_[=0]{}, \[cond\]where all spatial components $i$ are summed over. There is a finite and non-vanishing ohmic conductivity as long as $\rho_i^i$ is linear near zero energy. The photon emissivity is given by = 1[8\^3]{} (,T) \_\^(,p,T), \[rate\]where $\Omega$ is the number of photons produced per unit volume per unit time. This is equal to the observed photon rate if the reabsorption rate is very small— in which case the medium is out of equilibrium with respect to the EM coupling $\alpha$. In this work we shall take $\omega={\mathbf p}=0$, and hence obtain the soft photon production rate. Since $\rho_{00}=0$ for $\vec p=0$, the soft photon rate can be obtained once $\sigma$ is computed. Extracting $\rho_i^i$ from lattice computations needs the maximum entropy method [@mem] or other Bayesian techniques [@sigma].
The soft photon production rate from the plasma phase of hadronic matter has long been of importance to searches for the QCD phase transition, especially due to persistent observations of enhancements in heavy-ion collisions over proton-proton rates [@wa98]. Consequently, there has been a long history of attempts at perturbative computations of this rate [@history]. The first lattice computation in quenched QCD of dilepton (off-shell photon) rates [@dilepton] showed good agreement with perturbative results for $\omega>3T$. Recently the leading order computation of the photon production rate was completed [@amy]. For the transport coefficient one has $\sigma\propto \alpha T/g^4\log
g^{-1}$, to leading-log accuracy, with a known proportionality constant [@amy2]. The first computation of $\sigma$ and hence of the soft photon emissivity from a quenched lattice computation has now been performed for $1.5\le T/T_c\le3$ [@sigma]. It turns out that T7 C\_[EM]{},([for ]{}1.5T/T\_c3)C\_[EM]{}=4\_f e\_f\^2, \[value\]and $e_f$ is the charge of a quark of flavour $f$. The corresponding soft photon emissivity is shown in Figure \[fg.photon\]. Clearly, for fireball dimensions less than $1/\sigma=1/7C_{EM}T\approx3$ fm, the plasma is transparent to photons and this emissivity is also the detection rate of photons.
The diffusion coefficient of quarks can also be obtained in the same computation using the Einstein relation $\sigma=4\pi\alpha\sum_f
e_f^2\chi_{ff} D_f$ — T D\_f = ()(), \[diffu\]where $\chi_{ff}$ is the quark number susceptibility defined in eq.(\[qns\]) [@amy2]. A characteristic relaxation time, $\tau_R$, is the time for quarks for diffuse a distance equal to the screening length $1/T$. Then, we have \_R 1[DT\^2]{}1[7T]{}. \[relax\]For $1.5\le T/T_c\le3$ this is much smaller than a fermi. However, the relaxation time for charge carries an extra power of $\alpha$ in the denominator and hence is two orders of magnitude larger. This is the reason why charge fluctuations may be detectable.
The relaxation time required in jet quenching has to do with the gluon-dominated transport coefficient $\hat q$, which measures momentum transport transverse to the external force [@transq]. This transport coefficient remains to be measured on the lattice, but there is no reason to suspect that it leads to a significantly longer relaxation time. A complete theory of equilibration does not exist at this time [@thermal], but given such small relaxation times near equilibrium, it does not seem implausible that equilibration times are also small.
On purely phenomenological grounds it is clear that extremely fast thermalization and jet quenching is not compatible with a fireball that is very transparent to photons. The ratio of the relevant scales is just $C_{EM}\approx1/20$. If the former scale is about 0.1–0.15 fm, then the latter scale must be in the range 2–3 fm. Thus, the fireball produced at RHIC is marginally transparent to soft photons, whereas the larger expected size of a fireball at LHC would only allow photon detectors to look 2–3 fm inside the surface of the fireball.
(not the) Conclusion
====================
Let me introduce a dimensionless parameter which classifies several aspects of the physics that I have been talking about— the liquidity, defined by = S\^[1/3]{} E\^[1/4]{}, \[liquid\]where $\tau$ is the transport mean free time. The non-relativistic analogue of $S$ is the number density, so that $\ell$ is the mean free path in units of the interparticle spacing. For gases we expect this number to be large. A liquid would be characterised by values of $\ell$ close to unity.
In the perturbative expansion, when $g\ll1$, we have $S\simeq T^3$, $\tau\simeq1/Tg^4\log(1/g)$, and hence $\ell\simeq1/g^4\log(1/g)\gg1$. As a result, perturbation theory describes only the dilute, gaseous, phase of the QCD plasma. In experiments one finds $E\simeq 1$ Gev/fm${}^3$ and $\tau<1$ fm, giving $\ell<1.5$, and matter that is definitely liquid. We shall continue to call this phase a plasma, in view of the screening phenomena that occur (but remain to be rigorously demonstrated in experiments). However it is important to remember that transport coefficients are dominated by interactions, as in liquids, and not by long mean free paths, as in gases. The lattice studies now seem to indicate liquid-like behaviour for $T\le3T_c$, thus bringing us closer to an interpretation of heavy-ion collisions as quark matter.
The departure of $c_s^2$ from its gas value for $T<2T_c$ and the rapid fall in $S$, also indicate that the plasma changes character in the temperature region 2–3$T_c$. However, there is no evidence of a phase transition between the gaseous and liquid like extremes of the QCD plasma. This is likely to be the reason that perturbative expansions around some quasi-particle pictures give a qualitative description of static quantities such as $S$, $E$ or $\chi$, not far from $T_c$. However, the experimental numbers indicate that this is unlikely to be the case for dynamics.
Liquid-like behaviour means that dissipative effects are important to the fluid dynamics— in the relation between the HBT, single particle spectra and elliptic flow. In addition, the supersonic motion of jets through the liquid should give rise to many interesting colour-MHD effects apart from jet quenching. One near-term target for the lattice theory is to estimate the various transport coefficients and thereby determine the relative efficiency of various physical mechanisms for entropy production.
It is a pleasure to thank my collaborators, Saumen Datta, Rajiv Gavai, and Pushan Majumdar, for discussions. I would also like to thank Gert Aarts, Jean-Paul Blaizot, Francois Gelis, Jean-Yves Ollitrault, Toni Rebhan and Jose Resco for communications and discussions. Our lattice computations are largely performed on Compaq Alphas of the Department of Theoretical Physics, TIFR.
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|
---
abstract: |
We introduce a categorical framework for the study of representations of $G(\bF)$, where $G$ is a reductive group, and $\bF$ is a 2-dimensional local field, i.e., $\bF=\bK((t))$, where $\bK$ is a local field.
Our main result says that the space of functions on $G(\bF)$, which is an object of a suitable category of representations of $G(\bF)$ with the respect to the action of $G$ on itself by left translations, becomes a representation of a certain central extension of $G(\bF)$, when we consider the action by right translations.
address: 'D.K.: Einstein Institute of Mathematics, the Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel; D.G.: Department of Mathematics, The University of Chicago, 5734 University Ave., Chicago, IL, 60637, USA.'
author:
- David Kazhdan and Dennis Gaitsgory
title: 'Representations of algebraic groups over a 2-dimensional local field'
---
Introduction {#introduction .unnumbered}
============
Let $\bK$ be a local field, and let us consider the field $\bF=\bK((t))$. In his paper [@Kap], Kapranov studied a certain representation of the group $G(\bF)$, where $G$ is a reductive group over $\bK$. He introduced a pro-vector space (we will denote it by $\BV$), on which the group $G(\bF)$ acts in a continuous way, and which may be thought of as an analogue of a principal series representation of usual $\fp$-adic groups.
Namely, $\BV$ is the (pro)-vector space of locally constant functions with compact support on the set of $\bK$-points on the base affine space of the loop group $G((t))$. (We remind that this base affine space is a principal $T$-bundle over the affine flag scheme corresponding to $G$, where $T$ is the Cartan subgroup.)
Kapranov wrote down a certain algebra of endomorphisms of $\BV$ generated by explicit intertwining operators, and proved that this algebra is isomorphic to the (modified) double affine Hecke algebra. This double affine Hecke algebra, which was introduced and studied by Cherednik, is clearly an object of great importance, and Kapranov’s work explained that it is related to groups over a 2-dimensional field, such as $\bF$, in the same way as the usual affine Hecke algebra is related to $\fp$-adic groups.
The present paper grew out of an attempt to put Kapranov’s ideas and results into a categorical framework. Our goal is to find a category of smooth representations, let us denote it $\on{Rep}(\BG)$, which would contain Kapranov’s representation (and its close relatives) as objects. Moreover, we want $\on{Rep}(\BG)$ to be abelian, so that the usual representation-theoretic questions, such as irreducibility, would make sense in it. We also want $\on{Rep}(\BG)$ to be as “rigid” or “constrained” as possible, and finally we want the definition of $\on{Rep}(\BG)$ to resemble the definition of the category of smooth representations for usual $\fp$-adic groups.
After some categorical preliminaries in [Sect. \[ind-pro\]]{}, we propose a definition of $\on{Rep}(\BG)$ in [Sect. \[categories of representations\]]{}. A somewhat surprising feature of $\on{Rep}(\BG)$ is that, unlike most abelian categories that arise in representation theory, the natural forgetful functor defined on $\on{Rep}(\BG)$ does not map to the category of vector spaces, but rather to the category $\BVect$ of pro-vector spaces. We remark that for the purposes of this paper, one could restrict to the subcategory $\BVect^{\aleph_0}$ of projective systems indexed by countable sets.
Let us recall that $\BVect$ is an abelian category, but it is not semi-simple. In fact, the subcategory $\BVect^{\aleph_0}$ has cohomological dimension $\leq 1$, and it can be visualized as follows: An object of $\BVect^{\aleph_0}$ is called strict if it can be represented as a (filtering, countable) inverse system of vector spaces $\bV_i$, such that the arrows $\bV_i\to \bV_j$ are surjective. Strict objects of $\BVect^{\aleph_0}$ are the same as vector spaces endowed with a linear topology, with a countable fundamental system of neighbourhoods of zero, in which they are separated and complete. However, as is well-known, the category of such topological vector spaces is not abelian, which corresponds to the fact that strict objects of $\BVect^{\aleph_0}$ do not form an abelian subcategory.
We do have a (left-exact) functor $limProj:\BVect\to Vect$, but the point of view taken in this paper, and which is largely borrowed from [@Kap], is that we really have to work with the abelian category $\BVect$, and avoid taking projective limits.
We justify the appearance of $\BVect$ by showing that $G(\bF)$ does not have representations in any reasonable sense, unless we admit pro-vector spaces.
In [Sect. \[induction\]]{} we show, generalizing the basic construction of [@Kap], how to produce non-trivial objects of $\on{Rep}(\BG)$.
Namely, let $\BH$ be a subgroup of $G(\bF)$, contained in the group $G[[t]](\bK)$ of $\bK$-points of the group $G[[t]]$ and equal to the preimage of a closed subgroup of $(G[[t]]/G^i)(\bK)$ for some congruence subgroup $G^i$. Then, representations of $\BH$ on vector spaces, as well as on pro-vector spaces, are notions that are easy to recover from the usual representation theory of $\fp$-adic groups.
We define two functors $\wt{i}^\BG_\BH$ and $i^\BG_\BH$ from $\on{Rep}(\BH,\BVect)$ to $\on{Rep}(\BG)$, such that the former is the right adjoint to the tautological restriction functor $\on{Rep}(\BH,\BVect)\to \on{Rep}(\BG)$. (In other words, $\wt{i}^\BG_\BH$ should be thought of as an ordinary induction functor, whereas we think of $i^\BG_\BH$ as some sort of semi-infinite induction, by analogy with the theory of modules over vertex algebras, cf. [@AG].)
Kapranov’s representation $\BV$ is exactly of the form $i^\BG_\BH(\BC)$, where $\BH$ is the group of $\bK$-points of the unipotent radical of the Iwahori subgroup of $G((t))$, and $\BC$ is the trivial representation. In [Sect. \[examples\]]{} we give a slight improvement of Kapranov’s main result by showing that the (modified) Cherednik’s algebra maps isomorphically onto the ring $\on{End}_{\on{Rep}(\BG)}(\BV)$.
In addition, in [Sect. \[examples\]]{} we discuss another series of examples of objects of $\on{Rep}(\BG)$ by applying the functor $i^\BG_\BH$ for $\BH=G[[t]](\bK)$ and $G[[t]](\bK)$-representations, which are restrictions of irreducible cuspidal representations of the $\fp$-adic group $G(\bK)$. By analogy with the corresponding result in the theory of $\fp$-adic groups, we conjecture that these objects are actually irreducible in $\on{Rep}(\BG)$, and give some evidence in support of this conjecture.
Finally, in [Sect. \[Schwartz space\]]{} we formulate and prove the main result of this paper.
Suppose that the group $G$ acts on an algebraic variety $S$. In the theory of $\fp$-adic groups one introduces the Schwartz space $\on{Funct}^{lc}_c(S(\bK))$ of locally constant compactly supported functions on the set of $\bK$-points of $S$, which is a smooth representation of the group $G(\bK)$.
The question that we want to address is whether one can define an analogue of the Schwartz space, denoted in this paper by $M(\BS)$, which would be related to functions and/or distributions on the set of $\bF$-valued points of $S$. Of course, one expects that $M(\BS)$ is an object of $\BVect$, underlying a $G(\bF)$-representation.
It appears that the answer to this question is negative in the simplest example of $G=SL_2$ acting on the projective line, and the situation seems to be analogous to the problem of developing the theory of D-modules on loop spaces, cf. [@AG].
However, there are two important examples of $G$-varieties $S$, for which we can define $M(\BS)$:
First, we consider the case of $S$ being the affine space $A^n$, with the natural action of $GL_n$. We introduce a space $M(\BA^n)$ and show that it is naturally an object in the category of representations of the group $\wh{\BG L}_n$ (here $\wh{\BG L}_n$ is the group of $\bK$-points of the canonical (i.e., Tate) central extension $1\to G_m\to \wh{GL}_n\to GL_n((t))\to 1$).
Next, we consider the case when the variety $S$ is isomorphic to the group $G$ itself, with the action by left translations, and we construct an object $M(\BG)\in \on{Rep}(\BG)$. Now the natural question to ask is, whether the action of $G(\bF)$ on itself by [*right translations*]{} defines on $M(\BG)$ another, commuting, structure of an object of $\on{Rep}(\BG)$.
The answer to this question is that the right action of $G(\bF)$ on $M(\BG)$ develops an anomaly (compare it with the main theorem from [@AG]). Namely, $M(\BG)$ does carry a commuting action, but of the group of $\bK$-points of the central extension $1\to G_m\to \wh{G}\to G((t))\to 1$ corresponding to the adjoint action of $G$ on its Lie algebra.
We would like to thank S. Arkhipov, I. Cherednik, P. Etingof, V.Ginzburg, M. Kapranov for useful discussions and communications. We are grateful to the anonymous reviewer for valuable comments on the previous version of the paper, and to E. Hrushovski for reading the revised version.
The research of D.G. is supported by the long-term fellowship at the Clay Mathematics Institute. He also wants to thank the Mathematics Department of the Hebrew University of Jerusalem, where the main part of this work was written.
Preliminaries {#ind-pro}
=============
We will work with inductive and projective limits of objects of various categories. Let $I$ be a set. Recall that $I$ is said to be filtering if it is endowed with a partial order, such that for any two elements $i_1,i_2\in I$, there exists an element $i'\in I$ with $i'\geq i_1,i_2$.
Let $I$ be a filtering set, which we can regard as a category, and $\Phi:i\mapsto S_i$ be a functor $I\to Set$. We will denote by $\underset{\longrightarrow}{lim}\, S_i$ the inductive limit of $\Phi$. In other words, $$\on{Hom}_{Set}(\underset{\longrightarrow}{lim}\, S_i,S)\simeq
\on{Hom}_{\on{Functors}}(\Phi,\Phi_S),$$ where $\on{Functors}$ denotes the category of functors $I\to Set$, and $\Phi_S$ is the “constant” functor corresponding to the set $S$.
Let $\CC$ be an arbitrary category. Recall from [@SGA4] that the ind-completion of $\CC$, denoted $\on{Ind}(\CC)$, is the full subcategory in the category of contravariant functors $\CC\to
Set$, which consists of objects (isomorphic to ones) of the form $$X\mapsto \underset{\longrightarrow}{lim}\,\on{Hom}_{\CC}(X,X_i),$$ where $i\mapsto X_i$ is a functor $I\to \CC$ and $I$ is a filtering set; we will denote by $"\underset{\longrightarrow}{lim}"\,X_i$ the corresponding object of $\on{Ind}(\CC)$, which we will call “the direct limit of the system $X_i$”. By definition, $"\underset{\longrightarrow}{lim}"\,X_i(X)=\underset{\longrightarrow}{lim}\,\on{Hom}(X,X_i)$, where the inductive limit is taken in the category of sets.
For example, let $Vect$ (resp., $Vect_0$) be the category of vector spaces (resp., finite-dimensional vector spaces) over a given ground field. We have $Vect\simeq \on{Ind}(Vect_0)$. (It is a good exercise to show $\on{Ind}(Vect)$ is NOT equivalent to $Vect$.)
For a cardinal $\aleph$, we will denote by $\on{Ind}^\aleph(\CC)$ the full subcategory of $\on{Ind}(\CC)$ obtained by imposing the condition that the sets of indices that we are considering are of cardinality $\leq\aleph$.
We have a canonical fully faithful embedding $\CC\to \on{Ind}(\CC)$. The (partially defined) left adjoint $\on{Ind}(\CC)\to \CC$, called the inductive limit, which we will denote by $limInd$, is always right-exact. We will say that $\CC$ is closed under inductive limits (resp., inductive limits of cardinality $\leq\aleph$) if the functor $limInd$ is defined on the entire $\on{Ind}(\CC)$ (resp., $\on{Ind}^\aleph(\CC)$). For example, it is easy to show that any category of the form $\on{Ind}(\CC)$, (resp., $\on{Ind}^\aleph(\CC)$), where $\CC$ is another category, is always closed under inductive limits (resp., of cardinality $\leq\aleph$).
[**Note on the terminology**]{}: Let us emphasize that for a functor $I\to \CC:i\mapsto X_i$, we denote by $"\underset{\longrightarrow}{lim}"\,X_i$ the corresponding object of $\on{Ind}(\CC)$, and call it the direct limit of the $X_i$’s, following [@SGA4]. By contrast, if the object $limInd("\underset{\longrightarrow}{lim}"\,X_i)\in \CC$ exists, we will call it the inductive limit of the $X_i$’s, and denote it also by $\underset{\longrightarrow}{lim}\,X_i$.
The following simple assertion is useful:
\[ind vs representability\] Assume that $\CC$ is closed under inductive limits of cardinality $\leq\aleph$, and $X\in \on{Ind}^\aleph(\CC)$. Then $X$ belongs to $\CC$ if and only if for every $"\underset{\longrightarrow}{lim}"X'_i=:X'\in \on{Ind}^\aleph(\CC)$, the canonical arrow $X\left(\underset{\longrightarrow}{lim}\, X'_i\right)\to
\underset{\longleftarrow}{lim}\, \left(X(X'_i)\right)\simeq
\on{Hom}_{\on{Ind}(\CC)}(X',X)$ is an isomorphism.
The pro-completion $\on{Pro}(\CC)$ (resp., $\on{Pro}^\aleph(\CC)$) and the functor $limProj: \on{Pro}(\CC)\to \CC$ are defined in the same way by inverting the arrows, i.e., $\on{Pro}(\CC)=\left(\on{Ind}(\CC^o)\right)^o$, where the superscript “$o$” means the opposite category.
Suppose now that $\CC$ is an additive (resp., $\BC$-linear) category. Then every object $F$ of $\on{Ind}(\CC)$, which is a priori a contravariant functor $\CC\to Set$, lifts in a natural way to an additive functor $\CC\to\{\text{Abelian groups}\}$ (resp., $\BC$-linear functor $\CC\to Vect$).
Indeed if for some $X_i\in \CC$, $X="\underset{\longrightarrow}{lim}"\,X_i$, for the corresponding $\on{Hom}$ sets we have: $X(Y)=\underset{\longrightarrow}{lim}\on{Hom}(Y,X_i)$, and this inductive limit of sets has a natural structure of an abelian group (resp., $\BC$-vector space).
Suppose now that $\CC$ is abelian. We will now give a simple criterion that establishes ind-representability of functors in this case. Together with [Lemma \[ind vs representability\]]{} this provides a tool to prove representability of various functors in the framework of abelian categories.
Assume that $\CC$ is such that for a given object the class if its subobjects is a set. Let $F:\CC\to \{\text{Abelian groups}\}$ be a contravariant left exact functor. Suppose that there exists another functor $F'$ and a morphism of functors $F\to F'$ such that $\forall X\in \CC$ the map $F(X)\to F'(X)$ is injective.
\[prorep dom\] Assume that for $\CC$, $F$ and $F'$ as above, the functor $F'$ is ind-representable. Then the functor $F$ is also ind-representable.
(Compare [@AM], Corollary 2.8)
Let $Z="\underset{\longrightarrow}{lim}"\, Z_i$ be the object of $\on{Ind}(\CC)$ ind-representing $F'$. For each index $i$ consider the functor $F_i$ equal to $F\underset{F'}\times \on{Hom}(\cdot,Z_i)$. Each $F_i$ is also left exact and its map to $\on{Hom}(\cdot,Z_i)$ is an injection. Obviously, $F(Y)=\underset{\longrightarrow}{lim}\, F_i(Y)$, so it enough to show that each $F_i$ is ind-representable. In other words, we can assume that $F'$ is representable by an object $Z\in \CC$.
Consider the category of pairs $(X\in \CC,\alpha:X\to Z)$, where $\alpha$ is an injective morphism in $\CC$ such that the corresponding element in $F'(X)$ belongs to $F(X)$. (Morphisms between $(X\in \CC,\alpha:X\to Z)$ and $(X'\in \CC,\alpha':X'\to Z)$ are maps $X\to X'$ in $\CC$, which commute with the data of $\alpha$ and $\alpha'$.) This category is obviously discrete, and it is small due to our assumption on $\CC$. This resulting poset is filtering and is endowed with a functor to $\CC$, i.e., $(X\in \CC,\alpha:X\to Z)\mapsto X$.
Let $W\in \on{Ind}(\CC)$ be the direct limit of this system. We claim that $W$ ind-represents the functor $F$. Indeed, for $Y\in \CC$, given an element in $\on{Hom}(Y,W)$ we have for some $X$ an element in $F(X)$ and a map $Y\to X$, which gives rise to an element of $F(Y)$.
And vice versa, given an element in $a_Y\in F(Y)$ consider the corresponding element $a'_Y\in F'(Y)$ and the resulting map $Y\to Z$. Let $X$ be the image of this map: $$Y\twoheadrightarrow X\hookrightarrow Z.$$ It is enough to show that $a_Y$ belongs to the image of $F(X)$. Since $F$ is left exact, it is enough to show that the image of $a_Y$ vanishes in $F\left(\on{ker}(Y\to X)\right)$. However, by assumption, the image of $a'_Y$ in $F'\left(\on{ker}(Y\to X)\right)$ is zero, which implies our assertion, since $F\to F'$ is injective.
The following is also well-known (cf. [@AM], Proposition 4.5):
\[abelian\] If $\CC$ is abelian, then so is $\on{Ind}(\CC)$. The functor $limInd:\on{Ind}(\on{Ind}(\CC))\to
\on{Ind}(\CC)$ is exact.
Of course, assertions similar to the above ones hold when we replace $\on{Ind}$ by $\on{Pro}$.
\[set notation\]
The following category will play an essential role in this paper: $$\BVect:=\on{Pro}(Vect)\simeq \on{Pro}(\on{Ind}(Vect_0)).$$ According to [Lemma \[abelian\]]{}, this is an abelian category.
We will also consider the categories $\bSet:=\on{Ind}(\on{Pro}(Set_0))$, and $$\BSet:=\on{Ind}(\on{Pro}(\bSet))\simeq
\on{Ind}(\on{Pro}(\on{Ind}
(\on{Pro}(Set_0)))),$$ where $Set_0$ is the category of finite sets.
Note that the category $\on{Pro}(Set_0)$ is equivalent to the category of compact totally disconnected topological spaces; let us denote this equivalence by $\bY\mapsto \bY^{\on{top}}$. If $\bY="\underset{\longleftarrow}{lim}"\, Y_j$, then $\bY^{\on{top}}\simeq \bY=\underset{\longleftarrow}{lim}\, Y_j$, where the projective limit is taken in the category of topological spaces. For $\bX\in \bSet$ presented as a direct limit $"\underset{\longrightarrow}{lim}"\, \bX_i$ with $\bX_i\in \on{Pro}(Set_0)$, set $\bX^{\on{top}}$ to be the topological space $\underset{\longrightarrow}{lim}\, \bX^{\on{top}}_i$ (where the inductive limit is again taken in the category of topological spaces).
We will use the following terminology. We will call an object $\bX\in \bSet$ compact, if it belongs to $\on{Pro}(Set_0)$, and a morphism $\bX\to \bY$ in $\bSet$ proper if every base change by a compact object is compact.
We will call an object $\bX\in \bSet$ locally compact, if it can be represented as a direct limit $\bX="\underset{\longrightarrow}{lim}"\, \bX_i$, $\bX_i\in \on{Pro}(Set_0)$, where the maps $\bX_i\to \bX_j$ are such that the corresponding maps of topological spaces $\bX^{\on{top}}_i\to \bX^{\on{top}}_j$ are open embeddings.
The full subcategory of $\bSet$ consisting of locally compact objects is equivalent to the category $Top^{Hlctd}$ of Hausdorff locally compact totally disconnected topological spaces. All objects of $\bSet$ that are relevant for the purposes of this paper will be locally compact. Therefore, the reader may safely replace $\bSet$ by $Top^{Hlctd}$ and $\BSet$ by $\on{Ind}(\on{Pro}(Top^{Hlctd}))$.
Similarly, we will call an object $\BX\in \BSet$ bounded if it actually belongs to $\on{Pro}(\bSet)$.
Let $\CA$ be a monoidal category, i.e., we have a functor $\otimes:\CA\times \CA\to \CA$, a unit object $\one_\CA\in \CA$ and functorial isomorphisms $$X\otimes (Y\otimes Z)\simeq (X\otimes Y)\otimes Z;\,\,\,
X\otimes \one_\CA\simeq X\simeq \one_\CA\otimes X,$$ obeying the usual axioms. Note that in this case the categories $\on{Ind}(\CA)$ and $\on{Pro}(\CA)$ also possess natural monoidal structures.
If $\CC$ is another category, there is a standard notion of action of $\CA$ on $\CC$, in which case we say that $\CC$ is a module category over $\CA$. Namely, a module structure is a functor $\otimes: \CA\times \CC\to \CC$, and for $X,Y\in \CA$ and $V\in \CC$ functorial isomorphisms $$(X\otimes Y)\otimes V\to X\otimes (Y\otimes V);\,\,\,
\one_\CA\otimes V\simeq V,$$ satisfying the natural axioms. In particular, for $X\in \CA$, $V,W\in \CC$ we have a well-defined Hom set $\on{Hom}(X\otimes V,W)$.
By definition, a [*pseudo-action*]{} of $\CA$ on $\CC$ (or a structure on $\CC$ of a [*pseudo-module*]{} over $\CA$) is a functor $\CA^o\times \CC^o\times \CC\to Set$, denoted $\CHom(\cdot\otimes
\cdot,\cdot)$, and a morphism of functors: for $X,Y\in \CA$, $V,U,W\in \CC$ $$\CHom(X\otimes V,W)\times \CHom(Y\otimes U,V)\Rightarrow
\CHom((X\otimes Y)\otimes U,W),$$ and a functorial isomorphism $\CHom(\one_\CA\otimes V,W)\simeq
\on{Hom}_\CC(V,W)$, such that the following compatibility conditions hold:
For $X,Y,Z\in \CA$, $V,U,W,Q\in \CC$, the arrows $$\begin{aligned}
&\CHom(X\otimes W,Q)\times \CHom(Y\otimes V,W) \times \CHom(Z\otimes U,V)
\to \\
&\to \CHom(X\otimes W,Q)\times \CHom((Y\otimes Z)\otimes U,W)\to
\CHom((X\otimes (Y\otimes Z))\otimes U,Q) \,\text{ and} \\
&\CHom(X\otimes W,Q)\times \CHom(Y\otimes V,W)
\times \CHom(Z\otimes U,V)\to \\
&\to \CHom((X\otimes Y)\otimes V,Q)\otimes \CHom(Z\otimes U,V) \to
\CHom(((X\otimes Y)\otimes Z)\otimes U,Q)\end{aligned}$$ coincide under the associativity isomorphism $(X\otimes Y)\otimes Z\simeq X\otimes (Y\otimes Z)$, and for $U,V,W\in \CC$ and $X\in \CA$, the squares $$\CD
\CHom(X\otimes V,U)\times \on{Hom}_\CC(W,V) @>>>
\CHom(X\otimes V,U)\times \CHom(\one_\CA\otimes W,V) \\
@VVV @VVV \\
\CHom(X\otimes W,U) @>>>
\CHom((X\otimes\one_\CA)\otimes W,U)
\endCD$$ and $$\CD
\on{Hom}_\CC(V,W)\times \CHom(X\otimes U,V) @>>>
\CHom(\one_\CA\otimes V,W)\times \CHom(X\otimes U,V) \\
@VVV @VVV \\
\CHom(X\otimes U,W) @>>> \CHom((\one_\CA\otimes X)\otimes U,W)
\endCD$$ are commutative.
Note that if $\CC$ is a pseudo-module over $\CA$, then $\CC^o$ is a pseudo-module over $\CA^{op}$, where the latter is the category $\CA$ with the opposite monoidal structure: $${\mathcal Hom}(X\otimes V^o,W^o):={\mathcal Hom}(X\otimes W,V).$$
When $\CC$ is additive (resp., $\BC$-linear), we will rather use the variant of the above definition, when we require that the sets $\CHom(X\otimes U,V)$ have a structure of an abelian group (resp., $\BC$-vector space), such that the natural transformations $\CHom(X\otimes U,V)\times \CHom(Y\otimes V,W)\Rightarrow
\CHom((X\otimes Y)\otimes U,W)$ and $\CHom(\one_\CA\otimes V,W)\simeq
\on{Hom}_\CC(V,W)$ are bilinear (resp., linear).
For example, the category $Set$ is a monoidal via $X\otimes Y:=X\times Y$, and any category $\CC$ has a pseudo-module structure over $Set$ via $\CHom(X\otimes U,V):=Hom(U,V)^X$ for $X\in Set$, $U,V\in \CC$.
Of course, when $\CC$ is a module category over $\CA$, it acquires a pseudo-module structure by setting $$\CHom(X\otimes U,V):=\on{Hom}_\CC(X\otimes U,V).$$
In what follows we will say that an element $\phi\in
\CHom(X\otimes U,V)$ defines an action $X\times U\to V$.
Let us now analyze how pseudo-actions behave when we Ind- and Pro- complete our categories.
First, we claim that if $\CA$ pseudo-acts on $\CC$, then so do $\on{Ind}(\CA)$ and $\on{Pro}(\CA)$. Indeed, if $X\in \on{Ind}(\CA)$ (resp., $X\in \on{Pro}(\CA)$) is $"\underset{\longrightarrow}{lim}"\,X_i$ (resp., $"\underset{\longleftarrow}{lim}"\,X_i$), we set $\CHom(X\otimes V,W)=\underset{\longleftarrow}{lim} \,
\CHom(X_i\otimes V,W)$ (resp., $\CHom(X\otimes V,W)=\underset{\longrightarrow}{lim} \,
\CHom(X_i\otimes V,W)$). It is easy to see that this definition is independent of the way we represent $X$ as a direct (resp., inverse) limit.
Also, if $\CC$ has a pseudo-module structure over $\CA$, so do $\on{Ind}(\CC)$ and $\on{Pro}(\CC)$. Indeed, for $V,W\in \on{Ind}(\CC)$ equal to $"\underset{\longrightarrow}{lim}"\,V_i$ and $"\underset{\longrightarrow}{lim}"\,W_j$, (resp., $"\underset{\longleftarrow}{lim}"\,V_i$ and $"\underset{\longleftarrow}{lim}"\,W_j$), we set $\CHom(X\otimes V,W)$ to be $$(\underset{i}{\underset{\longleftarrow}{lim}})
(\underset{j}{\underset{\longrightarrow}{lim}})\,\,\CHom(X\otimes V_i,W_j)
\text{ and }
(\underset{j}{\underset{\longleftarrow}{lim}})
(\underset{i}{\underset{\longrightarrow}{lim}})\,\,\CHom(X\otimes V_i,W_j),$$ respectively. One can easily see that this definition is independent of the presentation of $V$ and $W$ as directs (resp., inverse) limits.
Now, we obtain that there are two pseudo-actions of $\on{Ind}(\CA)$ on $\on{Ind}(\CC)$. One is (which we will call “naive”) when we first consider the pseudo-action of $\on{Ind}(\CA)$ on $\CC$ and then produce from it the corresponding pseudo-action on $\on{Ind}(\CC)$. The other is when we first consider the pseudo-action of $\CA$ on $\on{Ind}(\CC)$ and then produce from it the corresponding pseudo-action of $\on{Ind}(\CA)$. Unless specified otherwise, in the sequel we will use the pseudo-action of the second kind. Note that we have a canonical map $\CHom(X\otimes V,W)_{naive}\to \CHom(X\otimes V,W)$. In concrete terms, if $X="\underset{\longrightarrow}{lim}"\,X_k$, $"\underset{\longrightarrow}{lim}"\,V_i$ and $"\underset{\longrightarrow}{lim}"\,W_j$, we have: $$\begin{aligned}
&\CHom(X\otimes V,W)_{naive}=\underset{i}{(\underset{\longleftarrow}{lim})}
\underset{j}{(\underset{\longrightarrow}{lim})}
\underset{k}{(\underset{\longleftarrow}{lim})}\,\,
\CHom(X_k\otimes V_i,W_j); \\
&\CHom(X\otimes V,W)=\underset{k}{(\underset{\longleftarrow}{lim})}
\underset{i}{(\underset{\longleftarrow}{lim})}
\underset{j}{(\underset{\longrightarrow}{lim})}\,\,
\CHom(X_k\otimes V_i,W_j).\end{aligned}$$
For example, by taking $\CC=\CA$, the canonical action of $\on{Ind}(\CA)$ on itself corresponding to the monoidal structure coincides with the pseudo-action described above coming from the action on $\CA$ on itself.
Similarly, we obtain the corresponding notions concerning the pseudo-action of $\on{Ind}(\CA)$ on $\on{Pro}(\CC)$.
The situation with the pseudo-actions of $\on{Pro}(\CA)$ is the opposite. The naive pseudo-module structure on $\on{Ind}(\CC)$ is obtained when we first consider the pseudo-action of $\CA$ on $\on{Ind}(\CC)$, and then produce from it a pseudo-action of $\on{Pro}(\CA)$. The pseudo-module structure that we will normally consider is is obtained by first considering the pseudo-action of $\on{Pro}(\CA)$ on $\CC$, and then producing from it the corresponding pseudo-action on $\on{Ind}(\CC)$. As before, we have a canonical map $\CHom(X\otimes V,W)_{naive}\to \CHom(X\otimes V,W)$, and for $V="\underset{\longrightarrow}{lim}"\,V_i$, $W="\underset{\longrightarrow}{lim}"\,W_j$ and $X="\underset{\longleftarrow}{lim}"\,X_k$ $$\begin{aligned}
&\CHom(X\otimes V,W)_{naive}=\underset{k}{(\underset{\longrightarrow}{lim})}
\underset{i}{(\underset{\longleftarrow}{lim})}
\underset{j}{(\underset{\longrightarrow}{lim})}\,\,
\CHom(X_k\otimes V_i,W_j); \\
&\CHom(X\otimes V,W)=\underset{i}{(\underset{\longleftarrow}{lim})}\underset{j}
{(\underset{\longrightarrow}{lim})}
\underset{k}{(\underset{\longrightarrow}{lim})}\,\,
\CHom(X_k\otimes V_i,W_j).\end{aligned}$$
In a similar way, we obtain the two pseudo-actions of $\on{Pro}(\CA)$ on $\on{Pro}(\CC)$. As above, for $\CC=\CA$ this canonical pseudo-action coincides with the action corresponding to the monoidal structure on $\on{Pro}(\CA)$.
Finally, we see that there are 3 possible pseudo-actions of $\on{Ind}\on{Pro}(\CA)$ on $\on{Ind}(\CC)$. The one that we will consider is “the biggest”: we will first consider the pseudo-action of $\on{Pro}(\CA)$ on $\CC$, then produce from it the pseudo-action of $\on{Pro}(\CA)$ on $\on{Ind}(\CC)$, and then the pseudo-action of $\on{Ind}\on{Pro}(\CA)$ on $\on{Ind}(\CC)$.
Explicitly, if $X\in \on{Ind}\on{Pro}(\CA)$ is $\underset{k}{"\underset{\longrightarrow}{lim}"}
(\underset{l}{"\underset{\longleftarrow}{lim}"\, X^k_l})$, $V="\underset{\longrightarrow}{lim}"\,V_i$ and $W="\underset{\longrightarrow}{lim}"\,W_j$, then $$\CHom(X\otimes V,W)=
\underset{k}{\underset{\longleftarrow}{lim}}\,
\underset{i}{\underset{\longleftarrow}{lim}}\,
\underset{l}{\underset{\longrightarrow}{lim}}\,
\underset{j}{\underset{\longrightarrow}{lim}}\, \CHom(X^k_l\otimes V_i,W_j).$$
By inverting the arrows in $\CC$ we obtain the corresponding pseudo-action of $\on{Ind}\on{Pro}(\CA)$ on $\on{Pro}(\CC)$.
\[ex\]
Let us consider our main examples. Let $\CA=Set_0$, and $\CC=Vect_0$. Then for $\bX\in \bSet=
\on{Ind}(\on{Pro}(Set_0))$, $\bV,\bW\in Vect=\on{Ind}(Vect_0)$, we obtain the notion of an action $\bX\times\bV\to \bW$. However, it is easy to see that such an action is the same as a continuous map $\bX^{\on{top}}\times \bV\to \bW$, linear in $\bV$ and $\bW$, where $\bV$ and $\bW$ are endowed with the discrete topology, and $\bX^{\on{top}}$ is as in [Sect. \[set notation\]]{}.
Now set $\CA=\bSet=\on{Ind}(\on{Pro}(Set_0))$ and $\CC=Vect$. We obtain a pseudo-module structure on $\BVect$ with respect to $\BSet$.
Let us write down the last notion in more concrete terms. First, let $\BX$ be an object of $\on{Pro}(\bSet)$, and $\BV,\BW$ be two objects of $\BVect$. An action $\phi:\BX\times \BV\to \BW$ is the following data. Let $\BX="\underset{\longleftarrow}{lim}"\,\bX_j$, $\BV="\underset{\longleftarrow}{lim}"\,\bV_i$, $\BW="\underset{\longleftarrow}{lim}"\,\bW_{i'}$, with $\bX_j\in \bSet$, $\bV_i,\bW_{i'}\in Vect$. Then for every $i'$ there must exist $i_0$, $j_0$ and a compatible system of action maps $\phi_{j,i,i'}:\bX_j\times \bV_i\to \bW_{i'}$ defined for $i\geq i_0$, $j\geq j_0$. Another compatibility condition is imposed: for $i'_1\geq i'_2$ the corresponding diagrams $$\CD
\bX_{j_1}\times \bV_{i_1} @>{\phi_{j_1,i_1,i'_1}}>> \bW_{i'_1} \\
@VVV @VVV \\
\bX_{j_2}\times \bV_{i_2} @>{\phi_{j_2,i_2,i'_2}}>> \bW_{i'_2}
\endCD$$ must commute for $i_1$ and $j_1$ large enough. Two action maps $\phi$ and $\psi$ coincide if for every $i'$ the corresponding maps $\phi_{j,i,i'}$ and $\psi_{j,i,i'}$ coincide for $i$ and $j$ large enough.
If now $\BX$ is an object of $\BSet$, equal to $"\underset{\longrightarrow}{lim}"\,\BX^j$ and $\BV,\BW\in \BVect$, an action $\phi:\BX\times \BV\to \BW$ is a compatible system of actions $\phi^j:\BX^j\times \BV\to \BW$.
\[weakly strict\]
The following definition will be needed in the sequel. First, note that we have an obvious functor from the category of sets (denoted $Set$) to $\bSet$ via $$Set\simeq \on{Ind}(Set_0)\to \on{Ind}(\on{Pro}(Set_0))\simeq \bSet.$$ Let $\bX_1\to \bX_2$ be a map of objects of $\bSet$. We will say that it is weakly surjective if for any $Y\in Set$, the map $$\on{Hom}_{\bSet}(\bX_2,Y)\to \on{Hom}_{\bSet}(\bX_1,Y)$$ is injective.
Note that if $\bX_1,\bX_2$ are locally compact, the above notion that a morphism $\bX_1\to \bX_2$ is weakly surjective is equivalent to the condition that the corresponding map $\bX_1^{\on{top}}\to
\bX_2^{\on{top}}$ has dense image.
A map $\bX_1\to \bX_2$ in $\bSet$ is weakly surjective if and only if for any $\bV,\bW\in Vect$, the map $\CHom(\bX_2\otimes \bV,\bW)\to \CHom(\bX_1\otimes \bV,\bW)$ is injective.
We will call an object $\BX\in \on{Pro}(\bSet)$ weakly strict if it can be represented as $"\underset{\longleftarrow}{lim}"\,\bX_i$, where the maps $\bX_j\to \bX_i$ are weakly surjective.
Note that if $\BX$ is weakly strict and $\bV,\bW\in Vect$, for any element $\phi\in \CHom(\BX\otimes \bV,\bW)$ we have well-defined kernel and image of $\phi$. By definition, $\on{ker}(\phi)\subset \bV$ (resp., $\on{Im}(\phi)\subset\bW$) is the maximal (resp., minimal) subspace $\bV'$ of $\bV$ (resp., $\bW'$ of $\bW$) having the property that $\phi$ factors through an element $\phi'\in \CHom(\BX\otimes \bV/\bV',\bW')$ (resp., $\phi'\in \CHom(\BX\otimes \bV,\bW')$).
Indeed, both $\on{ker}(\phi)$ and $\on{Im}(\phi)$ are clearly well-defined when $\bX\in \bSet$. If now $\BX="\underset{\longleftarrow}{lim}"\,\bX_i$, with weakly surjective maps, and $\phi$ comes from an element $\phi_i\in \CHom(\bX_i\otimes \bV,\bW)$, then it is easy to see that $\on{ker}(\phi_i)\subset \bV$ and $\on{Im}(\phi_i)\subset \bW$ are the sought-for subspaces.
Categories of representations {#categories of representations}
=============================
In the abstract set-up of the previous section, let us recall that an object $X\in \CA$ is called a monoid (in the sense of the monoidal structure on $\CA$) if we are given a (multiplication) map $X\otimes X\to X$ and a (unit) map $\one_\CA\to X$, which satisfy the usual associativity and unit axioms.
In our examples, the monoidal structure on $\CA$ will be such that $X\otimes Y$ is isomorphic to the categorical direct product $X\times Y$. Moreover, $\on{Hom}_{\CC}(X,\one_\CA)$ will be a one-element set $\forall X\in \CC$. Note that this property is inherited by both $\on{Ind}(\CA)$ and $\on{Pro}(\CA)$.
In this case, it makes sense to speak about group-objects in $\CA$: a monoid $X$ is called a group if there exists a map $\gamma:X\to X$ (automatically unique) such that the two compositions $$X\overset{\Delta}\to X\times X\overset{\on{id}\times \gamma}
\longrightarrow X\times X\overset{\on{mult}}\longrightarrow X \text{ and }$$ $$X\overset{\Delta}\to X\times X\overset{\gamma\times \on{id}}
\longrightarrow X\times X\overset{\on{mult}}\longrightarrow X$$ are both equal to $X\to \one_\CA\to X$.
In the sequel we will only consider monoids, which are groups.
If $\CC$ is another category with a pseudo-action of $\CA$ and $X\in \CA$ is a monoid, a representation of $X$ in $\CC$ is a pair $\Pi=(V,\rho)$, where $V\in \CC$ and $\rho\in \CHom(X\otimes V,V)$, such that the following two conditions hold:
Associativity: The image of $\rho\times\rho$ under the associativity constraint $$\CHom(X\otimes V,V)\otimes \CHom(X\otimes V,V)\to
\CHom((X\otimes X)\otimes V,V)$$ equals the image of $\rho$ under the map $\CHom(X\otimes V,V)\to \CHom(X\otimes X)\otimes V,V)$ given by the multiplication $X\otimes X\to X$.
Unit: The image of $\rho$ in $\CHom(\one_\CA\otimes V,V)$ under $\one_\CA\to X$ equals the identity element in $\CHom(\one_\CA\otimes V,V)\simeq \on{Hom}(V,V)$.
Representations of $X$ in $\CC$ form a category, which we will denote by $\on{Rep}(X,\CC)$. When $\CC$ is additive (resp., $\BC$-linear), the category $\on{Rep}(X,\CC)$ is additive (resp., $\BC$-linear) as well.
Assume now that $\CC$ is abelian and that for a fixed $X\in \CA$, the functor $\CC^o\times \CC\to Set$ given by $V,W\mapsto {\mathcal Hom}(X\otimes V,W)$ is left-exact in both arguments.
Under the above circumstances the category $\on{Rep}(X,\CC)$ is abelian and the natural forgetful functor $\on{Rep}(X,\CC)\to \CC$ is exact.
If $\CA$ is a monoidal category with a pseudo-action on an abelian category $\CC$, such that the above left-exactness condition is satisfied, then the same holds for $\on{Ind}(\CA)$ (resp., $\on{Pro}(\CA)$) pseudo-acting on $\on{Ind}(\CC)$ (resp., $\on{Pro}(\CC)$), due to the fact that the functor $limInd$ (resp., $limProj$) is exact (resp., left-exact) on the category of abelian groups.
In particular, we obtain that this condition is satisfied in our examples of $\CA=\bSet$, $\CC=Vect$ and $\CA=\BSet$, $\CC=\BVect$.
\[condition star\]
Set first $\CA=\bSet$, and $\CC=Vect$. Thus, for a group-object $\bH\in \bSet$ the category $\on{Rep}(\bH,Vect)$ is the usual category of representations of $\bH$ appearing in the theory of $\fp$-adic groups. In other words, if $\bH$ is locally compact (cf. [Sect. \[set notation\]]{}) and $\bH^{\on{top}}$ is the corresponding topological group, then an object of $\on{Rep}(\bH,Vect)$ is the same as a smooth representation of $\bH^{\on{top}}$.
If $\BH$ is a group-object of $\on{Pro}(\bSet)$, we can consider its representations on $Vect$ and $\BVect$, and the resulting categories will be denoted by $\on{Rep}(\BH,Vect)$ and $\on{Rep}(\BH,\BVect)$, respectively.
We will say that $\BH\in \on{Pro}(\bSet)$ satisfies condition ($*$) if it is weakly strict as an object of $\on{Pro}(\bSet)$, cf. [Sect. \[weakly strict\]]{}.
The following assertion will play an important role in the sequel: [^1]
\[representations of pro-groups\] For $\BH$ satisfying ($*$), the categories $\on{Rep}(\BH,\BVect)$ and $\on{Pro}(\on{Rep}(\BH,Vect))$ are naturally equivalent.
Note that the proof given below is valid when $\BH$ is a just a monoid (not necessarily a group), satisfying condition $(*)$.
The functor in one direction: $\sF:\on{Pro}\on{Rep}(\BH,Vect))\to \on{Rep}(\BH,\BVect)$ is evident; moreover, it is easy to see that it is fully faithful. Let us show that it admits a left adjoint.
For $(\BV,\rho)\in\on{Rep}(\BH,\BVect)$, let us write $\BV="\underset{\longleftarrow}{lim}"\, \bV_i$, where the index $i$ runs over some filtering set $I$. Consider the category of quadruples $(\bV',\rho',i,\alpha:\bV_i\to \bV')$, where $(\bV',\rho')\in\on{Rep}(\BH,Vect)$, $i\in I$ and $\alpha$ is a map, such that its image generates $\bV'$ as an $\BH$-representation and the composition $\BV\to \bV_i\to \bV'$ is compatible with the $\BH$-actions. A morphism between $(\bV_1,\rho_1',i_1,\alpha_1:\bV_{i_1}\to \bV'_1)$ and $(\bV_2,\rho_2',i_2,\alpha_2:\bV_{i_2}\to \bV'_2)$ is by definition a relation $i_2\geq i_1$ and a map of $\BH$-representations $\bV'_1\to\bV'_2$, such that the square $$\CD
\bV_{i_1} @>>> \bV_{i_2} \\
@V{\alpha_1}VV @V{\alpha_2}VV \\
\bV'_1 @>>> \bV'_2
\endCD$$ commutes. The resulting category is evidently discrete, filtering and small. By definition, we have a forgetful functor from this category to $\on{Rep}(\BH,Vect)$ that sends a quadruple $(\bV',\rho',i,\alpha)$ to $(\bV',\rho')$. Let us denote by ${\mathsf G}(\BV,\rho)\in \on{Pro}(\on{Rep}(\BH,Vect))$ the resulting inverse limit. It is easy too see that the assignment $(\BV,\rho)\mapsto \sG(\BV,\rho)$ defines a functor left adjoint to $\sF$.
The fact that $\sF$ was fully-faifull means that the composition $\sG\circ \sF$ is isomorphic to the identity functor. Thus, it remains to see that for $(\BV,\rho)\in\on{Rep}(\BH,\BVect)$, the adjunction map $(\BV,\rho)\to \sF\circ\sG(\BV,\rho)$ is an isomorphism. For that, it suffices to show that if $\BV="\underset{\longleftarrow}{lim}"\, \bV_i$, then for every $i$ there exists a vector space $\bV'_i$ underlying an object $\Pi=(\bV'_i,\rho_i)\in \on{Rep}(\BH,Vect)$, such that the map $\BV\to \bV_i$ factors as $\BV\to \bV'_i\to \bV_i$, with the first arrow preserving the $\BH$-action. Indeed, this would show that the map $(\BV,\rho)\to \sF\circ\sG(\BV,\rho)$ is always injective, and combined with the fact that $\sG$ is right-exact, this implies that this map is an isomorphism.
Let $j$ be an index such that the map $\BH\times \BV\overset{\on{act}}{\longrightarrow} \BV\to \bV_i$ factors as $\BH\times \BV\to \BH\times \bV_j\to \bV_i$. Let us denote by $p_{j,i}$ the projection $\bV_j\to \bV_i$ and by $\on{act}_{j,i}$ the map $\BH\times \bV_j\to \bV_i$.
By the definition of the action, there exists another index $k$ such that the map $\BH\times \BV\overset{\on{act}}{\longrightarrow} \BV\to \bV_j$ factors as $$\BH\times \BV\to
\BH\times \bV_k\overset{\on{act}_{k,j}}\longrightarrow \bV_j,$$ and such that the diagram $$\CD
\BH\times \BH\times \bV_k @>{\on{mult}\times p_{k,j}}>> \BH\times
\bV_j \\
@V{\on{id}\times \on{act}_{k,j}}VV @V{\on{act}_{j,i}}VV \\
\BH\times \bV_j @>{\on{act}_{j,i}}>> \bV_i
\endCD$$ is commutative, where $p_{k,j}$ denotes the projection $\bV_k\to \bV_j$.
Let $\bW'\subset \bV_j$ be the kernel of the map $\BH\times \bV_j
\overset{\on{act}_{j,i}}\longrightarrow \bV_i$, and let $\bW''\subset \bV_j$ be the image of $\BH\times \bV_k\overset{\on{act}_{k,j}}\longrightarrow \bV_j$. The above kernel and image are well-defined due to the ($*$) assumption on $\BH$, cf. [Sect. \[weakly strict\]]{}.
Set $\bV'_i$ to be the image of $\bW''$ in $\bV_j/\bW'$, and let $\on{act}'$ denote the map $\BH\times \bV_k\to \bV'_i$. We claim that there exists a unique map $\BH\times \bV'_i\to \bV'_i$, which makes the diagram $$\CD
\BH\times \BH\times \bV_k @>{\on{id}\times \on{act}'}>> \BH\times \bV'_i \\
@V{\on{mult}\times \on{id}}VV @VVV \\
\BH\times \bV_k @>{\on{act}'}>> \bV'_i
\endCD$$ commute. The commutativity of the diagram implies that the action $\BH\times \bV'_i\to \bV'_i$ is unital and associative.
To construct the sought-for map $\BH\times \bV'_i\to \bV'_i$, let us write $\BH="\underset{\longleftarrow}{lim}"\, \bX_n$ with weakly surjective maps. Let $n_0$ be an index such that the maps $\on{act}_{j,i}$ and $\on{act}_{j,k}$ are defined on the level of $\bX_{n_0}$ (we will denote them $\on{act}^{n_0}_{j,i}$ and $\on{act}^{n_0}_{k,j}$, respectively).
Let $n_1\geq n_0$ be an index such that the multiplication on $\BH$ gives rise to a map $\on{mult}^{n_0}_{n_1,n_1}:\bX_{n_1}\times \bX_{n_1}\to \bX_{n_0}$, and let $n_2\geq n_1$ be another index, such that we have a multiplication $\on{mult}^{n_1}_{n_2,n_2}:\bX_{n_2}\times \bX_{n_2}\to \bX_{n_1}$, satisfying an obvious associativity with respect to $\on{mult}^{n_0}_{n_1,n_1}$. For $m=1,2$ let us denote by $\on{act}^{n_m}_{j,i}$, $\on{act}^{n_m}_{k,j}$ the maps obtained by composing $\on{act}^{n_0}_{j,i}$ and $\on{act}^{n_0}_{k,j}$, respectively, with $\bX_m\to \bX_0$.
We will construct a map $\bX_{n_2}\times \bV'_i\to \bV'_i$, which amounts to a map $\bX_{n_2}^{\on{top}}\times \bV'_i\to \bV'_i$. Let $v_j$ be an element in $\bW''\subset \bV_j$, and $h_{n_2}\in \bX_{n_2}^{\on{top}}$. We claim that there exists an element, denoted $v'_j\in \bW''\subset \bV_j$, which is unique modulo $\bW'$, satisfying $$\label{cond}
\on{act}^{n_2}_{j,i}(h'_{n_2},v'_j)=
\on{act}^{n_1}_{j,i}(\on{mult}^{n_1}_{n_2,n_2}(h'_{n_2},h_{n_2}),v_j)\in
\bV_i,$$ for any $h'_{n_2}\in \bX_{n_2}^{\on{top}}$.
By assumption, every element $v_j\in \bW''$ can be written as $\underset{a}\Sigma\, \on{act}^{n_2}_{k,j}(h_{n_2}^a,v_k^a)$ for $h^a_{n_2}\in \bX_{n_2}^{\on{top}}$, $v_k^a\in \bV_k$. For $h_{n_2}\in \bX_{n_2}^{\on{top}}$ as above we set $$v'_j=\underset{a}\Sigma\,
\on{act}^{n_1}_{k,j}(\on{mult}^{n_1}_{n_2,n_2}(h_{n_2},h^a_{n_2}),v_k^a)\in
\bW''\subset \bV_j.$$ It is easy to see that $v'_j$ satisfies .
For $\BH$ as above we have a natural embedding $\on{triv}:\BVect\to
\on{Rep}(\BH,\BVect)$, corresponding to “trivial” representations.
\[simple inv and coinv\] For $\BH$ satisfying ($*$), the functor $\on{triv}$ admits both right and left adjoints.
Note that in [Proposition \[inv and coinv\]]{} a more general statement is established.
First, from [Sect. \[weakly strict\]]{} it follows the the functor $\on{triv}:Vect\to \on{Rep}(\BH,Vect)$ admits right and left adjoints, denoted $\Pi\mapsto \Pi^\BH$ and $\Pi\mapsto \Pi_\BH$, respectively.
Therefore, using [Proposition \[representations of pro-groups\]]{}, it is enough to show that the functor $\on{triv}:\on{Pro}(Vect)\to \on{Pro}(\on{Rep}(\BH,Vect))$ has left and right adjoints. But these are simply given by sending $\Pi="\underset{\longleftarrow}{lim}"\, \Pi_i$ to $\Pi_\BH\simeq "\underset{\longleftarrow}{lim}"\, (\Pi_i)_\BH$ and $\Pi^\BH\simeq "\underset{\longleftarrow}{lim}"\, (\Pi_i)^\BH$, respectively.
As every right adjoint, the functor $\Pi\mapsto \Pi^\BH$ is left-exact, and similarly, the functor $\Pi\mapsto \Pi_\BH$ is right-exact.
\[exactness of Jacquet\] Assume that $\BH$ is the inverse limit of a weakly surjective family of $\bH_i$, where each $\bH_i$ is a group-object in $\bSet$ isomorphic to a direct limit of $\bH_{i,j}$, with each $\bH_{i,j}$ being a group-object of $\on{Pro}(Set_0)$. Then the functor of coinvariants $\on{Rep}(\BH,\BVect)\to \BVect$ is exact.
According to [Proposition \[representations of pro-groups\]]{} and [Corollary \[simple inv and coinv\]]{}, each $\Pi\in \on{Rep}(\BH,\BVect)$ is an inverse limit of $\Pi_k\in \on{Rep}(\BH,Vect)$, and $\Pi_\BH\simeq "\underset{\longleftarrow}{lim}" (\Pi_k)_\BH$. Therefore, it suffices to show that the functor of coinvariants is exact on $\on{Rep}(\BH,Vect)$. By the definition of the latter, we can replace $\BH$ by one of its quotients $\bH_i$, which we will denote by $\bH$.
However, the fact that functor $\Pi\mapsto \Pi_\bH$ is exact on the category $\on{Rep}(\bH,Vect)$ is well-known. Indeed, if $\bH="\underset{\longrightarrow}{lim}" \bH_j$, $\bH_j\in \on{Pro}(Set_0)$, $$\Pi_\bH\simeq \underset{j}{\underset{\longrightarrow}{lim}}\, \Pi_{\bH_j},$$ but the functor $limInd$ is exact on $Vect$, and the functor $\Pi\to \Pi_{\bH_j}$ is exact on $\on{Rep}(\bH_j,Vect)$, since $\bH^{\on{top}}_j$ is a compact group.
\[\*\* condition\]
Consider now the category $\BSet$ with its pseudo-action on $\BVect$. The main object of study of this paper is the category of representations $\on{Rep}(\BG,\BVect)$ of a group-object $\BG\in \BSet$ in $\BVect$. For brevity, we will denote the category by $\on{Rep}(\BG)$, when no confusion is likely to occur.
\[pro-completeness\] The functor $limProj:\on{Pro}(\on{Rep}(\BG))\to \on{Rep}(\BG)$ is defined on the entire category and is exact.
Recall (cf. [Lemma \[abelian\]]{}, with Ind replaced by Pro) that the category $\BVect$ is closed under projective limits. If $\Pi_i=(\BV_i,\rho_i)$ is an inverse system of objects of $\on{Rep}(\BG)$, we define $\BV\in \BVect$ as $\underset{\longleftarrow}{lim}\, \BV_i$. It is easy to see from the definitions that there exists an action $\rho:\BG\times \BV\to \BV$, such that $(\BV,\rho)$ represents the projective limit $\underset{\longleftarrow}{lim}\, \Pi_i$. The exactness follows from the fact that the functor $limProj:\on{Pro}(\BVect)\to \BVect$ is exact.
We will say that $\BH\in \BSet$ satisfies condition ($**$) if, as an object of $\on{Ind}(\on{Pro}(\bSet))$, $\BH$ can be represented as $"\underset{\longrightarrow}{lim}"\, \BX_k$, with $\BX_k\in \on{Pro}(\bSet)$ being weakly strict.
As before, we have an obvious functor $\on{triv}:\BVect\to \on{Rep}(\BH)$ corresponding to “trivial” representations.
\[inv and coinv\] The functor $\on{triv}:\BVect\to \on{Rep}(\BH)$ admits a left adjoint, and when $\BH$ satisfies ($**$), also a right adjoint.
Let us first construct the left adjoint of $\on{triv}$. Consider the covariant functor on the category $Vect$ that sends a vector space $\bV$ to $\on{Hom}_{\on{Rep}(\BH)}(\Pi,\on{triv}(\bV))$. This functor is a subfunctor of $\bV\mapsto \on{Hom}_{\BVect}(\Pi,\bV)$. Hence, by [Proposition \[prorep dom\]]{}, it is pro-representable.
Let us denote the resulting object of $\BVect$ by $\Pi_\BH$. It is strightforward to check that for $\BV\in \BVect$, we have a functorial isomorphism $\on{Hom}_{\on{Rep}(\BH)}(\Pi,\on{triv}(\BV))\simeq
\on{Hom}_{\BVect}(\Pi_\BH,\BV)$.
Now let us construct the right adjoint to $\on{triv}$. Let us write $\BH\in \BSet$ as $"\underset{\longrightarrow}{lim}"\, \BX_k$, where $\BX_k\in \on{Pro}(\bSet)$ are weakly strict.
For a weakly strict object $\BX\in \on{Pro}(\bSet)$, $\BV,\BU\in \BVect$, and an action map $\phi:\BX\times \BV\to \BU$, consider the kernel of $\phi$ as a functor on $\BVect$: $$\on{ker}(\phi)(\BW)=\{\psi:\BW\to \BV\,|\, \phi\circ \psi=0\}.$$ We claim that this functor is representable. If this is so, it is easy to see that the sought-for right adjoint of $\on{triv}$ is representable by $$(\BV,\rho)^\BH=\underset{k}{\underset{\longleftarrow}{lim}}\,
\on{ker}\left(p-\on{act}:\BX_k\times \BV\to \BV\right),$$ where $\underset{\longleftarrow}{lim}$ is taken in the category $\BVect$, and $p$ is the obvious projection map $\BX_k\times \BV\to \BV$.
To show the representability, we can assume that $\BU=\bU\in Vect$. Indeed, if $\BU="\underset{\longleftarrow}{lim}"\, \bU_i$, then $\on{Ker}(\phi)=\underset{i}{\underset{\longleftarrow}{lim}}\,
\on{ker}\left(\BX\times \BV\to \bU_i\right)$. In the latter case, we can assume that $\BV="\underset{\longleftarrow}{lim}"\, \bV_j$, and we have a compatible system of maps $\phi_j:\BX\times \bV_j\to \bU$. By [Sect. \[weakly strict\]]{}, $\on{ker}(\phi_j)\subset \bV_j$ is well-defined, and it is easy to see that $"\underset{\longleftarrow}{lim}"\,
\on{ker}(\phi_j)\in \BVect$ represents $\on{ker}(\phi)$.
The main source of examples of such $\BG$, i.e., of group-objects in $\BSet$, is provided by considering sets of points of algebraic groups with values in a two-dimensional local field.
Let $\bK$ be a local field, with the corresponding local ring $\CO_\bK$. We will denote by $\pi$ a uniformizer of $\bK$. Set $\bF=\bK((t))$, $\CO_\bF=\bK[[t]]$.
Let $Sch^{ft}$ denote the category of separated schemes of finite type over $\bK$. If $S$ is an object of $Sch^{ft}$, we will denote by $S(\bK)$ the corresponding set of $\bK$-points. It is well-known that $S(\bK)$ carries a natural locally compact totally disconnected topology; therefore, as a topological space, $S(\bK)\simeq \bS^{\on{top}}$ for a canonically defined locally compact object $\bS\in \bSet$.
Hence, we obtain a functor $S\mapsto \bS:Sch^{ft}\to \bSet$, and also the functors $\on{Pro}(Sch^{ft})\to \on{Pro}(\bSet)$, and $\on{Ind}(\on{Pro}(Sch^{ft}))\to \BSet$.
In particular, any affine scheme (not necessarily of finite type) over $\bK$ defines an object of $\on{Pro}(Sch^{ft})$, and hence, an object of $\on{Pro}(\bSet)$. In addition, for any scheme of finite type $S$, the corresponding scheme of arcs $S[[t]]$ is naturally an object of $\on{Pro}(Sch^{ft})$: $$S[[t]]\simeq "\underset{\longleftarrow}{lim}"\,S[t]/t^i.$$ We will denote the corresponding object of $\on{Pro}(\bSet)$ by $\bS[[t]]$.
If $S$ is smooth, the maps in this family defining $S[[t]]$ are fibrations into affine spaces; therefore the corresponding maps $\bS[t]/t^j\to \bS[t]/t^i$ are weakly surjective. Hence, if $S$ is smooth, the object $\bS[[t]]\in \on{Pro}(\bSet)$ is weakly strict.
For a scheme $S'$ over $\bF$, we define its “restriction of scalars” from $\bF$ to $\bK$ as a functor on the category of schemes over $\bK$ by $S\mapsto \on{Hom}_{\bF}(S\underset{\bK}\otimes \bF,S')$. If $S'$ is of finite type and affine, then by embedding it into an affine space one shows that the above functor is ind-representable by an ind-scheme, which is a direct limit of affine schemes under closed embeddings. By taking $S'=S\underset{\bK}\otimes \bF$ for $S$ an affine scheme of finite type over $\bK$, we obtain an object of $\on{Ind}(\on{Pro}(Sch^{ft}))$ that will be denoted by $S((t))$. The resulting object of $\BSet$ will be denoted by $\bS((t))$ or $\BS$.
By applying the functor of iterated inductive and projective limits $\BSet\to Set$, we obtain from $\BS$ (resp., $\bS[[t]]$) the set, which is tautologically identified with the set $S(\bF)$ of $\bF$-points of $S$ (resp., $S(\CO_\bF)$–the set of $\CO_\bF$-points of $S$).
\[group notation\]
If $G$ is a smooth linear algebraic group over $\bK$, by applying the functor $G\mapsto \bG$ we obtain the corresponding group-object in $\bSet$. In particular, we can consider the category of representations $\on{Rep}(\bG,Vect)$, which is tautologically equivalent to the category of smooth representations of the locally compact group $G(\bK)$.
For a non-negative integer $i$, let us denote by $G^i$ the congruence subgroup of $G[[t]]$, i.e., the kernel of $G[[t]]\to G[[t]]/t^i$; in particular, $G^0=G[[t]]$. Let $\bG^i$ be the corresponding object of $\on{Pro}(\bSet)$. A subgroup $\BH$ of $\bG[[t]]$ will be called [*thick*]{} if it contains $\bG^i$ for some $i$ and equals the preimage of a closed subgroup of $\bG[[t]]/\bG^i$ (we are slightly abusing the terminology by identifying $\bG[[t]]/\bG^i$ with the corresponding locally compact group).
For a thick $\BH \subset \bG[[t]]$ we can consider the corresponding categories $\on{Rep}(\BH,Vect)$ and $\on{Rep}(\BH,\BVect)$. As was remarked above, $\bG[[t]]\in \on{Pro}(\bSet)$ is weakly strict, and so are the groups $\bG^i$. From this it is easy to see that any thick subgroup $\BH\subset \bG[[t]]$ satisfies condition ($*$) of [Sect. \[condition star\]]{}.
Finally, for an algebraic group $G$ as above, we can consider $\BG$ (sometimes also denoted $\bG((t))$), which is a group-object in $\BSet$ and the corresponding category $\on{Rep}(\BG,\BVect)$, which we will denote for brevity by $\on{Rep}(\BG)$.
It is well-known that the ind-scheme $G((t))$ can be represented as a direct limit under closed embeddings of subschemes, each of which is stable under (both left and right) multiplication by $G[[t]]$, and is a principal $G[[t]]$-bundle over a scheme of finite type. (In fact, the above family of subschemes is obtained by taking the preimages of finite-dimensional subschemes of the affine Grassmannian of $G$, i.e., $\Gr_G=G((t))/G[[t]]$.) This implies, in particular, that $\BG$ satisfies condition ($**$), cf. [Sect. \[\*\* condition\]]{}.
Let us denote by $\BVect^{G(\bF)}$ the category consisting of objects of $\BVect$ with an action of the abstract group $G(\bF)$.
\[abstract group\] The natural forgetful functor $\on{Rep}(\BG)\to\BVect^{G(\bF)}$ is fully faithful.
We have to show that if $(\BV_1,\rho_1)$ and $(\BV_2,\rho_2)$ are two objects of $\on{Rep}(\BG)$, and $\BV_1\to \BV_2$ is a map preserving the action of $G(\bF)$, then it is compatible with the $\BG$-action.
This can be shown in the following general set-up: Let $\bV_1,\bV_2,\bW_1,\bW_2$ be vector spaces, and let $\BX$ be an object of $\on{Pro}(\bSet)$ endowed with action maps $\BX\times \bV_k\to
\bW_k$, $k=1,2$. Let $\bV_1\to \bV_2$, $\bW_1\to \bW_2$ be maps, such that the square $$\CD
\BX^{\on{top}}\times \bV_1 @>>> \BX^{\on{top}}\times \bV_2 \\
@VVV @VVV \\
\bW_1 @>>> \bW_2
\endCD$$ commutes, where $\BX^{\on{top}}$ is the topological space obtained from the corresponding object of $\on{Pro}(Top^{Hlctd})$ by taking the projective limit.
Assume now that $\BX$ can be presented as $"\underset{\longleftarrow}{lim}"\,
\bX_i$, where the maps $\bX_j\to \bX_i$ are such that the corresponding maps $\bX_j^{\on{top}}\to \bX_i^{\on{top}}$ are surjective. Then it is easy to see that the square $$\CD
\BX\times \bV_1 @>>> \BX\times \bV_2 \\
@VVV @VVV \\
\bW_1 @>>> \bW_2
\endCD$$ commutes as well.
The above assumption is satisfied in our situation for $\BX$ being the object of $\on{Pro}(\bSet)$ corresponding to a subscheme of $G((t))$, obtained as a preimage of a closed subscheme in $G((t))/G[[t]]$. The required surjectivity follows from the fact that the groups $G^i$ for $i>0$ are pro-unipotent.
\[cent ext\]
Suppose now that $\wh{G}$ is a group-indscheme, which is a central extension of $G((t))$ by the multiplicative group $G_m$, i.e., $$1\to G_m\to \wh{G}\to G((t))\to 1.$$ In other words, $\wh{G}$ is a group-object in the category of ind-schemes, such that if $G((t))="\underset{\longrightarrow}{lim}"\, X_k$, and $X_k="\underset{\longleftarrow}{lim}"\, X_{k,l}$ with $X_{k,l}\in Sch^{ft}$, then each $\wh{G}\underset{G((t))}\times X_k$ is a total space of a $G_m$-torsor over $X_k$, and this torsor is pulled back from $X_{k,l}$ for some index $l$. In what follows we will assume that we have a splitting $G[[t]]\to \wh{G}$.
We will denote by $\widehat{\BG}$ the corresponding group-object in $\BSet$, which is an extension of $\BG$ by $\bG_m$.
Let $c$ be a character $G_m(\bK)\to \BC^*$. We will denote by $\on{Rep}_c(\wh{\BG})$ the category of representations of $\widehat{\BG}$ with central character $c$. In other words, the objects of this category are pairs $\Pi=(\BV,\rho)$, where $\BV\in \BVect$, and $\rho$ is an action map $\widehat{\BG}\times \BV\to \BV$, satisfying the associativity and the unit axioms as above, and such that the composite action $$\bG_m\times \BV\to \widehat{\BG}\times \BV\to \BV$$ (where $\bG_m$ is viewed as an object of $\bSet\subset \BSet$) corresponds to the above character.
We propose the category $\on{Rep}(\BG)=\on{Rep}(\BG,\BVect)$ as a framework for the study of representations of the group $G(\bF)$. Let us explain why introducing pro-objects of $Vect$ appears to be necessary. For the remainder of this section, let us assume that $G$ is semi-simple, simply-connected and split.
The first question to ask is whether the category $\on{Rep}(\BG)$ contains any objects $\Pi=(\bV,\rho)$, where $\bV$ belongs to $Vect$. The answer is that such representations are necessarily trivial (i.e., they lie in the image of the functor $Vect\to\BVect\overset{triv}\to \on{Rep}(\BG)$), for the same reason as why $\fp$-adic groups usually have no finite-dimensional representations.
Indeed, suppose that $(\bV,\rho)$ is such a representation. By [Lemma \[abstract group\]]{}, it is sufficient to prove that the corresponding representation of the abstract group $G(\bF)$ on $\bV$ is trivial.
Consider the kernel $K$ of the action $G(\bF)\times \bV\to \bV$. This is a normal subgroup, and by definition, there exists an $i$ such that $K\supset G^i(\bK)$. But then we claim that $K$ must coincide with $G(\bF)$. Let $N$ be the maximal unipotent subgroup of $G$, and let $N^i(\bK):=N(\bF)\cap G^i(\bK)$ be the corresponding congruence subgroup. Then $N^i(\bK)\subset K$, but using the torus action and the normality of $K$, we obtain that the entire $N(\bF)$ is contained in $K$. Again, by normality, we obtain that all unipotent elements in $G(\bF)$ are contained in $K$. However, it is known that for a split simply-connected group, its set of field-valued points is generated by the subset of unipotent elements.
Another sense in which one may seek an alternative definition of $G(\bF)$-representations is to consider the pseudo-action of $\BSet$ on $\on{Ind}(Vect)=\on{Ind}(\on{Ind}(Vect_0))$. We claim that (under the same assumption on $G$) all objects of $\on{Rep}(\BG,\on{Ind}(Vect))$ are again trivial.
As before, we have a fully faithful functor $\on{Rep}(\BG,\on{Ind}(Vect))\to \on{Ind}(Vect)^{G(\bF)}$, and it suffices to show that for any object $(\bV,\rho)$, $\bV\in\on{Ind}(Vect)$, the action of the maximal unipotent group $N(\bF)$ on $\bV$ is trivial. Obviously, we can replace $G$ by an $SL_2$ corresponding to some simple root; let $B\subset G$ be the corresponding Borel subgroup, i.e., $N\simeq G_a$, and $B:=G_a\ltimes G_m$, where $G_m$ acts on $G_a$ by the square of the standard character.
Our $\bV$ is a direct limit $"\underset{\longrightarrow}{lim}"\bV_l$, with $\bV_l\in Vect$. Fix an index $l$, and it suffices to show that the action map $B(\bF)\times \bV_l\to \bV$ is trivial.
For a (not necessarily positive) integer $i$, let us denote by $N^i(\bK)$ the subgroup of $N(\bF)\simeq \bK((t))$ equal to $t^i\cdot \bK[[t]]$. If the action of $N(\bF)$ on $\bV_l$ is non-trivial, let $i$ be the minimal integer such that the restriction of this action to $N^i(\bK)$ is trivial. By assumption we have a non-trivial action map $(N^{i-1}(\bK)/N^i(\bK)\simeq \bK)\times \bV_l\to \bV$.
Let now $j$ be a sufficiently large integer, so that the corresponding congruence subgroup $(G_m)^j(\bK)$ acts trivially on $\bV_l$. Take $i'=(i-1)-2j$ and consider now the action of $N^{i'}(\bK)$ on $\bV_l$. Let $l'$ be a sufficiently large index such that the iteration of actions $$N^{i'}(\bK)\times (G_m)^j(\bK)\times N^{i'}(\bK)\times (G_m)^j(\bK)
\times \bV_l\to \bV_{l'}$$ is well-defined. We will show that the action $N^{i-1}(\bK)/N^i(\bK)\times \bV_l\to \bV_{l'}$ is necessarily trivial, which would be a contradiction. For that, it suffices to show that it is trivial on every element $v\in \bV_l$. For every such $v$ there exists an integer $k$ such that the action of $t^{i'}\cdot \pi^k\cdot \CO_\bK[[t]]\subset N^{i'}(\bK)$ on $v$ is trivial. Hence, for $g\in (G_m)^j(\bK)$ and $n\in t^{i'}\cdot \pi^k\cdot \CO_\bK[[t]]$ $$(n\cdot g\cdot n^{-1}\cdot g^{-1})\cdot v=v\in \bV_{l'}.$$ However, since $(G_m)^j(\bK)=1+t^j\cdot \bK[[t]]$, the subset of $N(\bK)$ consisting of elements of the form $(n\cdot g\cdot n^{-1}\cdot g^{-1})$, with $g$ and $n$ as above, equals the entire $t^i\cdot\bK[[t]]$, in particular, it projects surjectively onto $N^{i-1}(\bK)/N^i(\bK)$. Therefore, for $n'\in N^{i-1}(\bK)$, we have $n'\cdot v=v\in \bV_{l'}$, which is what we had to show.
The induction functor {#induction}
=====================
Let $G$ be a split reductive group over $\bK$, and let $\BH$ be a thick subgroup of $\bG[[t]]$. We have an obvious restriction functor $r^{\BG}_{\BH}:\on{Rep}(\BG)\to \on{Rep}(\BH,\BVect)$.
Our goal in this section is define the functors $\wt{i}^{\BG}_{\BH},i^{\BG}_{\BH}: \on{Rep}(\BH,\BVect)\to \on{Rep}(\BG)$, such that $\wt{i}^{\BG}_{\BH}$ will be the right adjoint of $r^{\BG}_{\BH}$.
We will have an injective functorial map $i^{\BG}_{\BH}(\Pi)\to \wt{i}^{\BG}_{\BH}(\Pi)$, and there is a certain analogy between the functors $\wt{i}^{\BG}_{\BH}$ and $i^{\BG}_{\BH}$ and the functors of induction and compact induction in the theory of $\fp$-adic groups. When $\BH$ corresponds to a parahoric subgroup of $G[[t]]$, we will have an isomorphism $i^{\BG}_{\BH}\simeq \wt{i}^{\BG}_{\BH}$.
The construction of the functor $i^{\BG}_{\BH}$ makes sense for any algebraic group $G$, but the construction of the functor $\wt{i}^{\BG}_{\BH}$ given below uses the fact that $G$ is reductive. However, we expect that the right adjoint to $r^{\BG}_{\BH}$ exists for any $G$.
\[induction functor\]
To an object $\bX\in \on{Pro}(Set_0)$ we can attach the vector space of locally constant $\BC$-valued functions, denoted $\on{Funct}^{lc}(\bX)$. Namely, if $\bX="\underset{\longleftarrow}{lim}"\, X_i$, $$\on{Funct}^{lc}(\bX)=\underset{\longrightarrow}{lim}\,\on{Funct}(X_i),$$ where the direct system is taken with respect to the pull-back maps between the spaces of functions. Of course, $\on{Funct}^{lc}(\bX)$ identifies with the space of locally constant functions on the topological space $\bX^{\on{top}}$.
For any $\bX\in \bSet$ we define ${\mathbb Funct}^{lc}(\bX)\in \BVect$ by setting for $"\underset{\longrightarrow}{lim}"\, \bX_j$, $\bX_j\in \on{Pro}(Set_0)$ $${\mathbb Funct}^{lc}(\bX)="\underset{\longleftarrow}{lim}"\,
\on{Funct}^{lc}(\bX_j),$$ with respect to the restriction maps. We define the space $\on{Funct}^{lc}(\bX)\in Vect$ of locally constant functions on $\bX$ as $limProj({\mathbb Funct}^{lc}(\bX))$.
If $\bX\in \bSet$ is locally compact (cf. [Sect. \[set notation\]]{}), we can introduce the vector space $\on{Funct}^{lc}_c(\bX)$, which can be called the space of locally constant functions with compact support. One way to introduce it is as the space of locally-constant compactly supported functions on $\bX^{\on{top}}$. Equivalently, if $\bX$ is represented as a direct limit as in [Sect. \[set notation\]]{}, we have the natural “extension by zero” maps $\on{Funct}^{lc}(\bX_i)\to \on{Funct}^{lc}(\bX_j)$, and we set $\on{Funct}^{lc}_c(\bX)=\underset{\longrightarrow}{lim}\,
\on{Funct}^{lc}(\bX_i)$. Note that we always have an inclusion $\on{Funct}^{lc}_c(\bX)\hookrightarrow \on{Funct}^{lc}(\bX)$.
Let $\bX$ be again locally compact, presented as a direct limit as in [Sect. \[set notation\]]{}. If $\bX'\to \bX$ is map between objects of $\bSet$, we define the vector space $\on{Funct}^{lc}_{c,rel}(\bX')$ as the inductive limit $\underset{\longrightarrow}{lim}\,
\on{Funct}^{lc}(\bX'\underset{\bX}\times \bX_i)$.
If $\bX\to \bY$ is a map in $\bSet$, we have the pull-back morphism $\on{Funct}^{lc}(\bY)\to \on{Funct}^{lc}(\bX)$, and if this is a proper map between locally compact objects, we also have the morphism $\on{Funct}^{lc}_c(\bY)\to \on{Funct}^{lc}_c(\bX)$.
Suppose now that $\bY^1,\bY^2\in \bSet$ are locally compact, and we have an action $\bX\times \bY^1\to \bY^2$ (in the sense of the canonical tensor structure on $\bSet$), such that the map $\bX\times \bY^1\to \bX\times \bY^2$ is proper. Then we obtain an action map $\bX\times \on{Funct}^{lc}_c(\bY^2)\to \on{Funct}^{lc}_c(\bY^1)$ (in the sense of the pseudo-action of $\bSet$ on $Vect$).
For example, the above properness condition is always satisfied if $\bX$ is a group-object acting on $\bY^1=\bY^2$.
Note that the above action does not always extend onto $\on{Funct}^{lc}(\bY)$.
Let now $\bY$ be an object of $\on{Ind}(\bSet)$. We will say that $\bY$ is “tame” if it can be represented as $"\underset{\longrightarrow}{lim}"\, \bY_i$ such that $\bY_i\in \bSet$ are locally compact, and the corresponding maps $\bY_i\to \bY_j$ are proper. If $\bY$ is “tame”, we can attach to it the object $\on{Funct}^{lc}_c(\bY) \in \BVect$ as $\on{Funct}^{lc}_c(\bY)="\underset{\longleftarrow}{lim}"\,
\on{Funct}^{lc}_c(\bY_i)$, where the maps are again given by restriction.
Suppose now that $\bY^1,\bY^2\in \on{Ind}(\bSet)$ are both “tame”, $\bY^j_i="\underset{\longrightarrow}{lim}"\, \bY^j_i$ for $j=1,2$, and let $\BX\times \bY^1\to \bY^2$ be an action of $\BX\in \BSet$ in the sense of the pseudo-action of $\BSet$ on $\on{Ind}(\bSet)$. That is $\BX="\underset{\longrightarrow}{lim}"\, \BX_l$, $\BX_l="\underset{\longleftarrow}{lim}"\, \bX_{l,k}$ and the action is given by the maps $\bX_{l,k}\times \bY_i^1\to \bY_{i'}^2$. We say that this action is proper if the above presentations can be chosen so that the maps $\bX_{l,k}\times \bY_i^1\to
\bX_{l,k}\times \bY_{i'}^2$ are proper. (This condition is satisfied if $\BH$ is a group-object in $\BSet$ acting on $\bY=\bY^1=\bY^2$.)
If the action $\BX\times \bY^1\to \bY^2$ is proper we obtain an action map $\BX\times \on{Funct}^{lc}_c(\bY_2)\to
\on{Funct}^{lc}_c(\bY_1)$ in the sense of the canonical pseudo-action of $\BSet$ on $\BVect$.
A little more generally, if $\bV$ is a vector space, instead of complex-valued functions, we can consider spaces of functions with values in $\bV$, denoted $\on{Funct}^{lc}(\bX,\bV)$ and $\on{Funct}^{lc}_c(\bX,\bV)$, respectively.
Let $i\geq 0$ be such that $\bG^i\subset \BH$. Consider the full subcategory of $\on{Rep}(\BH/\bG^i,Vect)\subset \on{Rep}(\BH,\BVect)$; we will first define the restrictions of the functors $i^\BG_\BH,\wt{i}^\BG_\BH$ to this subcategory.
Recall that there exists a strict ind-scheme of ind-finite-type $G((t))/G^i$ (“strict” means that it can be presented as a direct limit of schemes with transition maps being closed embedding). Its existence, i.e., the ind-representability of the corresponding functor, follows easily from the corresponding fact for $\Gr_G=G((t))/G[[t]]$ (see, for example, the Appendix to [@Ga]). As an object of $\on{Ind}(Sch^{ft})$ it carries an action of $G((t))\in \on{Ind}(\on{Pro}(Sch^{ft}))$ “on the left” and a commuting action of $G([[t]]/t^i)\in Sch^{ft}$ “on the right”.
Therefore, by applying the functor $S\mapsto \bS:Sch^{ft}\to \bSet$, we obtain a “tame” object, denoted $\BG/\bG^i$ in $\on{Ind}(\bSet)$, which carries the actions of $\BG$ and $\bG[[t]]/\bG^i$.
For an object $\Pi=(\bV,\rho)\in \on{Rep}(\BH/\bG^i,Vect)$, we obtain that $\on{Funct}^{lc}_c(\BG((t))/\bG^i,\bV)\in \BVect$ carries a natural $\BG$-action and a commuting $\BH/\bG^i$-action.
The object of $\BVect$ underlying $i^\BG_\BH(\Pi)$ is set to be $$\label{definition of i}
\left(\on{Funct}^{lc}_c(\BG/\bG^i,\bV)\otimes\mu(\BH/\bG^i)
\right)_{\BH/\bG^i},$$ where $\mu(\BH/\bG^i)$ is the $1$-dimensional vector space of left-invariant measures on the locally compact group $(\BH/\bG^i)^{\on{top}}$ (acted on naturally by $\BH/\bG^i$, being a subspace of all measures on $(\BH/\bG^i)^{\on{top}}$). The $\BG$-action on $\on{Funct}^{lc}_c(\BG/\bG^i,\bV)$ defines on $i^\BG_\BH(\Pi)$ a structure of an object of $\on{Rep}(\BG)$.
This definition of $i^\BG_\BH(\Pi)$ can be rewritten as follows. First, let us introduce the object $\BG/\BH\in \on{Ind}(\bSet)$. Let us write $G((t))/G[[t]]$ as $"\underset{\longrightarrow}{lim}"\, S_k$, $S_k\in Sch^{ft}$, and let $S_k^i$ be the preimage of $S_k$ in $G((t))/G^i$. Let $\bS_k$, $\bS_k^i$ be the corresponding objects of $\bSet$. By construction $\bS_k^i$ carries an action of the groups $\BH\subset\bG[[t]]/\bG^i$, and we claim that the categorical quotient $\bS^\BH_k:=(\bS^i_k)/(\BH/\bG^i)\in \bSet$ is well-defined and is locally compact. This follows for example from the fact that $S_k^i\to S_k$ is a fibration locally trivial in the Zariski toplogy. Let $\BG/\BH="\underset{\longrightarrow}{lim}"\, \bS_k^\BH$ be the corresponding object of $\on{Ind}(\bSet)$; this object is “tame” and it evidently does not depend on the way we presented $G((t))/G[[t]]\in \on{Ind}(Sch^{ft})$ as a direct limit.
For $\Pi=(\bV,\rho)\in \on{Rep}(\BH/\bG^i,Vect)$, let $\on{Funct}^{lc}_{c,rel}(\bS^i_k,\bV)$ be the space of locally-constant $\bV$-valued functions on $\bS^i_k$, whose support is contained in the preimage of a compact subset of $\bS_k^\BH$ (see [Sect. \[induction functor\]]{}).
We have: $$\label{integration}
\left(\on{Funct}^{lc}_c(\bS^i_k,\bV)\otimes\mu(\BH/\bG^i)
\right)_{\BH/\bG^i}\simeq
\left(\on{Funct}^{lc}_{c,rel}(\bS^i_k,\bV)\right)^{\BH/\bG^i},$$ where the isomorphism is given by integration along the fibers of $\bS^i_k\to \bS^\BH_k$.
The above isomorphism makes it clear that $i^\BG_\BH(\Pi)$, as an object of $\on{Rep}(\BG)$, is independent of the choice of the congruence subgroup $\bG^i$ contained in $\BH$. In particular, we obtain a well-defined functor $i^\BG_\BH:\on{Rep}(\BH,Vect)\to \on{Rep}(\BG)$. From we infer that $i^\BG_\BH$ is right-exact, and from that it is also left-exact.
Using [Proposition \[representations of pro-groups\]]{} we extend the above functor $\on{Rep}(\BH,Vect)\to \on{Rep}(\BG)$ to a functor $\on{Rep}(\BH,\BVect)\simeq \on{Pro}(\on{Rep}(\BH,Vect))\to
\on{Pro}(\on{Rep}(\BG))$, which is also exact. We extend it further to an exact functor $i^\BG_\BH:\on{Rep}(\BH,\BVect)\to \on{Rep}(\BG)$, using [Lemma \[pro-completeness\]]{}.
It is easy to see that our functor $i^\BG_\BH$ is isomorphic to the composition of two functors: $i^\BG_{\bG[[t]]}:\on{Rep}(\bG[[t]],\BVect)\to \on{Rep}(\BG)$ and $i^{\bG[[t]]}_\BH:\on{Rep}(\BH,\BVect)\to \on{Rep}(\bG[[t]],\BVect)$, where the latter functor is defined by a similar induction procedure.
Let us now define the functor $\wt{i}^\BG_\BH:\on{Rep}(\BH,\BVect)\to \on{Rep}(\BG)$. First, let us assume that $\BH$ is such that $\BG/\BH\in \on{Ind}(\bSet)$ is ind-compact, i.e., is a direct limit of compact objects of $\bSet$. (E.g., this condition is verified for $\bG[[t]]$, or more generally for any $\BH$ containing $\bI$, where $I\subset G[[t]]$ is the Iwahori subgroup. This follows from the fact that the affine flag variety $G((t))/I$ is ind-proper, i.e., is a direct limit of proper schemes of finite type.)
In this case we set $\wt{i}^\BG_\BH=i^\BG_\BH$.
\[adjointness\] If $\BG/\BH$ is ind-compact, the functor $i^\BG_\BH$ is the right adjoint to the restriction functor $r^\BG_\BH$.
The proof mimics the proof of the usual adjunction property for $\fp$-adic groups.
Let us first construct the adjunction map $r^\BG_\BH\circ i^\BG_\BH\to \on{id}_{\on{Rep}(\BH,\BVect)}$. By the definition of $i^\BG_\BH$, it is enough to construct a morphism $r^\BG_\BH\circ i^\BG_\BH(\Pi)\to \Pi$ for an object of $\on{Rep}(\BH/\bG^i,Vect)$ for some $i$.
For $\Pi=(\bV,\rho)\in \on{Rep}(\BH/\bG^i,Vect)$, consider the canonical restriction map $$\on{Funct}^{lc}_c(\BG/\bG^i,\bV)\otimes \mu(\BH/\bG^i)
\to \on{Funct}^{lc}_c(\BH/\bG^i,\bV)\otimes \mu(\BH/\bG^i),$$ which is bi-$\BH/\bG^i$-equivariant by construction.
Since $\on{Funct}^{lc}_c(\BH/\bG^i)\otimes \mu(\BH/\bG^i)$ identifies as a bi-module over $(\BH/\bG^i)^{\on{top}}$ with the Hecke algebra of (compactly supported, locally constant) measures on this group, we obtain a bi-$\BH/\bG^i$-equivariant map $\on{Funct}^{lc}_c(\BH/\bG^i,\bV)\otimes \mu(\BH/\bG^i)\to \bV$. By the $\BH/\bG^i$-equivariance on the right, we thus obtain a map $\left(\on{Funct}^{lc}_c(\BG/\bG^i,\bV)\otimes
\mu(\BH/\bG^i)\right)_{\BH/\bG^i}\to \bV$, as required.
Let us now construct the second adjunction map $(\BW,\rho')\to i^\BG_\BH\circ r^\BG_\BH(\BW,\rho')$ for $(\BW,\rho')\in \on{Rep}(\BG)$. Using [Proposition \[representations of pro-groups\]]{}, we can represent $r^\BG_\BH(\BW)$ as an inverse limit of $\bW_i$, where each $\bW_i$ is a vector space underlying an object of $\on{Rep}(\BH/\bG^{n_i},Vect)$ for some $n_i$. Let $\BG/\BH="\underset{\longrightarrow}{lim}"\, \bS^\BH_k$ be as before, and let $\bS_k^{n_i}$ be the preimage of $\bS^\BH_k$ in $\BG/\bG^{n_i}$.
Then the object of $\BVect$ underlying $i^\BG_\BH\circ r^\BG_\BH(\BW,\rho')$ is $$\underset{\underset{k,i}\longleftarrow}{"lim"}\,
\left(\on{Funct}^{lc}_c(\bS_k^{n_i},\bW_i)\otimes \mu(\BH/\bG^{n_i})
\right)_{\BH/\bG^{n_i}}.$$
For every fixed $k$ and $i$, let an index $j$ be such that the $\BG$-action on $\BW$ gives a map $\on{act}_{j,i}:\bS_k^{n_j}\times \bW_j \to \bW_i$. By further enlarging $j$, we may assume that this map is compatible with the $\BH$-action.
We define a map $\bW_j\to \left(\on{Funct}^{lc}_c(\bS_k^{n_i},\bW_i)\otimes
\mu(\BH/\bG^{n_i})\right)_{\BH/\bG^{n_i}}$ as follows. First, the above action map gives rise to a map $$\bW_j\to
\left(\on{Funct}^{lc}(\bS_k^{n_j},\bW_i)\right)^{\BH/\bG^{n_j}}.$$
Now, from the fact that $\bS_k^\BH$ is compact and isomorphism , we obtain $$\left(\on{Funct}^{lc}(\bS_k^{n_j},\bW_i)\right)^{\BH/\bG^{n_j}}\simeq
\left(\on{Funct}^{lc}_c(\bS_k^{n_i},\bW_i)\otimes \mu(\BH/\bG^{n_i})
\right)_{\BH/\bG^{n_i}}.$$ By composing, we obtain the required morphism.
It is easy to check that the constructed map from $\BW$ to the object of $\BVect$ underlying $i^\BG_\BH\circ r^\BG_\BH(\BW)$ respects the $\BG$-action. It is equally straightforward to see that the two adjunction maps indeed give rise to the adjointness of functors.
Thus, to define the functor $\wt{i}^\BG_\BH$ in general, it suffices to define the functor $\wt{i}^{\bG[[t]]}_\BH$, which is the right adjoint to the restriction functor $r^{\bG[[t]]}_\BH: \on{Rep}(\bG[[t]],\BVect)\to \on{Rep}(\BH,\BVect)$.
Let $\bG^i$ be a congruence subgroup contained in $\BH$. We define the functor $$\wt{i}^{\bG[[t]]}_\BH:
\on{Rep}(\BH/\bG^i,Vect)\to \on{Rep}(\bG[[t]]/\bG^i,Vect)$$ to equal the corresponding functor defined for locally compact groups.
Explicitly, for $\Pi=(\bV,\rho)\in \on{Rep}(\BH/\bG^i,Vect)$, $$\wt{i}^{\bG[[t]]}_\BH\simeq
\left(\on{Funct}^{sm}(\bG[[t]]/\bG^i,\bV)\right)^{\BH/\bG^i},$$ where $\on{Funct}^{sm}(\bG[[t]]/\bG^i)$ is the space of functions on $\bG[[t]]/\bG^i$, smooth with respect to the action of this group by left translations.
Note that for $\Pi=(\bV,\rho)\in \on{Rep}(\BH/\bG^i,Vect)$ as above, the object of $\BVect$ underlying $\wt{i}^\BG_\BH\circ \wt{i}^{\bG[[t]]}_\BH(\Pi)$ is $\underset{\underset{k}\longleftarrow}{lim}\, (\bW_k)^{\BH/\bG^i}$, where each $\bW_k$ is a certain subspace of $\on{Funct}^{lc}(\bS^i_k,\bV)$, and $\bS^i_k$ is as in .
The above functor $\on{Rep}(\BH/\bG^i,Vect)\to \on{Rep}(\bG[[t]]/\bG^i,Vect)$ extends to a functor $\wt{i}^{\bG[[t]]}_\BH:\on{Rep}(\BH,Vect)\to \on{Rep}(\bG[[t]],Vect)$. Using [Proposition \[representations of pro-groups\]]{}, from it we obtain the functor $\wt{i}^{\bG[[t]]}_\BH:\on{Rep}(\BH,\BVect)\to \on{Rep}(\bG[[t]],\BVect)$, which is the right adjoint to $r^{\bG[[t]]}_\BH: \on{Rep}(\bG[[t]],\BVect)\to \on{Rep}(\BH,\BVect)$; and hence also the functor $\wt{i}^\BG_\BH:\on{Rep}(\BH,\BVect)\to \on{Rep}(\BG)$ with the desired adjointness property.
\[induction from Levi\]
Consider the functor $r^\BG_\bG:\on{Rep}(\BG)\to \on{Rep}(\bG,\BVect)$ equal to the composition of $r^\BG_{\bG[[t]]}:\on{Rep}(\BG)\to \on{Rep}(\bG[[t]],\BVect)$ and the functor $\BV\mapsto \BV_{\bG^1}:
\on{Rep}(\bG[[t]],\BVect)\to \on{Rep}(\bG,\BVect)$. Note that by [Lemma \[exactness of Jacquet\]]{}, $r^\BG_\bG$ is exact. Its right adjoint, which we will denote by $i^\BG_\bG$ is the composition of the “obvious” functor $\on{Rep}(\bG,\BVect)\to \on{Rep}(\bG[[t]],\BVect)$ coming from the homomorphism $\bG[[t]]\to \bG$ and the functor $i^\BG_{\bG[[t]]}$ studied above.
More generally, let $P\subset G$ be a parabolic, with the Levi quotient $M$, and let $I_P\subset \bG[[t]]$ be the corresponding parahoric subgroup. (For $P=B$ we will denote $I_B$ simply by $I$, and $M$ by $T$). Let $\bI_P$ (resp., $\bP$, $\bM$) be the corresponding group-objects of $\on{Pro}(\bSet)$ (resp., $\bSet$).
In a similar fashion we obtain a pair of mutually adjoint functors $r^\BG_\bM:\on{Rep}(\BG)\to \on{Rep}(\bM,\BVect)$ and $i^\BG_\bM:\on{Rep}(\bM,\BVect)\to \on{Rep}(\BG)$.
Let now $\wh{G}$ be a central extension of $G((t))$ by means of $G_m$ (cf. [Sect. \[cent ext\]]{}), and let $\on{Rep}_c(\wh{\BG})$ be the corresponding category of representations. Since we are given a splitting of $\wh{\BG}$ over $\bG[[t]]$, and hence, over $\BH$, we have an obvious restriction functor $r^{\wh{\BG}}_{\BH}:\on{Rep}_c(\wh{\BG})\to\on{Rep}(\BH,\BVect)$.
By repeating the construction of the previous subsections, we obtain the functors $i^{\wh{\BG}}_\BH:\on{Rep}(\BH,\BVect)\to
\on{Rep}_c(\wh{\BG})$, and $\wt{i}^{\wh{\BG}}_\BH:\on{Rep}(\BH,\BVect)\to
\on{Rep}_c(\wh{\BG})$, such that $\wt{i}^{\wh{\BG}}_\BH$ is the right adjoint of $r^{\wh{\BG}}_\BH$, and $i^{\wh{\BG}}_\BH\simeq \wt{i}^{\wh{\BG}}_\BH$ when $\BH$ contains $\bI$.
We will denote by $r^{\wh{\BG}}_\bG$, $i^{\wh{\BG}}_\bG$ (resp., $r^{\wh{\BG}}_\bT$, $i^{\wh{\BG}}_\bT$) the corresponding functors between $\on{Rep}_c(\wh{\BG})$ and $\on{Rep}(\bG,\BVect)$ (resp., $\on{Rep}(\bT,\BVect)$).
Next we will establish an analogue of Bernstein’s geometric lemma, which describes the composition of the functors $i^{\wh{\BG}}_\bT$ and $r^{\wh{\BG}}_\bT$, cf. [@Be].
Let $\Lambda$ be the lattice of co-weights of the maximal torus $T$ of $G$, and $W$–the Weyl group. When we restrict $\wh{G}$ to $T((t))\subset G((t))$, the commutator defines a map $$T((t))/T[[t]]\times T[[t]]\to G_m,$$ which factors through $T((t))/T[[t]]\twoheadrightarrow \Lambda$ and $T[[t]]\twoheadrightarrow T$. In other words, we have a map $$\Lambda\times T\to G_m,$$ which defines a pairing $Q:\Lambda\otimes \Lambda\to \BZ$. This pairing is $W$-invariant, since the central extension of $T((t))$ was induced from that of $G((t))$. For $\lambda\in \Lambda$ we will denote by $\phi_{cQ}(\lambda)$ the character $\bT\to \BC^*$ equal to $$\bT\overset{Q(\lambda,\cdot)}\longrightarrow
\bG_m\overset{c}\to \BC^*.$$
Recall now the affine flag scheme of $G$, which is a strict ind-scheme, equal by definition to $G((t))/I$ (where $I$ is the Iwahori subgroup), and denoted $\on{Fl}_G$. Its existence, i.e., the ind-representability of the corresponding functor, follows easily from the case of $\Gr_G$, cf. [@Ga].
Recall also that the set of $I$-orbits on $\on{Fl}_G$ identifies naturally with $W_{aff}\simeq \Lambda\ltimes W$–the extended affine Weyl group of $G$. For $w\in W_{aff}$, let us denote by $\on{Fl}^w_G$ the corresponding orbit and by $\ol{\on{Fl}}{}^w_G$ its closure. Note that $W_{aff}$ is naturally partially ordered and $\ol{\on{Fl}}{}^w_G=\underset{w'\leq w}\cup\, \on{Fl}^w_G$.
Let $\check T$ be the Langlands dual torus of $T$ (over $\BC$), which identifies with the set of unramified characters of $\bT$. For $w\in W_{aff}$ let us denote by $w(\rho_{aff})-\rho_{aff}$ the character of $T$ equal to the projection on $T$ of the sum of negative affine roots which are turned positive by the action of $w^{-1}$. Let $\mu_{w}$ denote the element in $\check T$ equal to the value of $w(\rho_{aff})-\rho_{aff}:G_m\to \check T$ on $q\in \BC^*$, where $q$ is the order of the residue field of our local field $\bK$.
For $w\in W_{aff}$ and $\Pi\in \on{Rep}(\bT,Vect)$ we define a new representation $w\cdot \Pi$ by setting $$w\cdot \Pi:=\Pi^{\ol{w}}\otimes \phi_{cQ}(\lambda)\otimes \mu_{w},$$ where $w=\lambda\cdot \ol{w}$, $\lambda\in \Lambda$, $\ol{w}\in W$, and $\Pi^w$ is obtained from $\Pi$ by twisting the $\bT$-action using $w$ viewed as an automorphism of $T$.
\[geometric lemma\] For a representation $\Pi=(\bV,\rho)\in \on{Rep}(\bT,Vect)$, the object $r^{\wh{\BG}}_\bT\circ i^{\wh{\BG}}_\bT(\Pi)$ can be canonically written as $\underset{\underset{w\in W_{aff}}\longleftarrow} {"lim"}\, \bV_w$, $\bV_w\in \on{Rep}(\bT,Vect)$ in such a way that for for $w'\leq w$ the map $\bV_w\to \bV_{w'}$ is a surjection, and the kernel $\bV^w:=\on{ker}\left(\bV_w\to \underset{w'<w}\oplus\, \bV_{w'}\right)$ is isomorphic to $w\cdot \Pi$.
Let $\bFl_G$ be the object of $\on{Ind}(\bSet)$ corresponding to the ind-scheme $\on{Fl}_G$. Let us denote by $\bFl^w_G$ and $\ol{\bFl}{}^w_G$ the corresponding objects of $\bSet$.
Let $\bI^0$ denote the kernel of the map $\bI\to \bT$. Let $\bS^w$ (resp., $\ol{\bS}^w$) be the preimage of $\bFl^w_G$ (resp., $\ol{\bFl}{}^w_G$) in $\wh{\BG}/\bI^0$. By construction, $r^{\wh{\BG}}_\bI\circ i^{\wh{\BG}}_\bT(\Pi)$ is the inverse limit of $$\bW_w:=\left(\on{Funct}^{lc}_c(\ol{\bS}^w,\bV)\right)_{\bT\times \bG_m}
\simeq \left(\on{Funct}^{lc}(\ol{\bS}^w,\bV)\right)^{\bT\times \bG_m}.$$
Set $\bV_w:=\left(\bW_w\right)_{\bI^0}$. Since for $w'\leq w$, the restriction map $\on{Funct}^{lc}_c(\ol{\bS}^w,\bV)\to
\on{Funct}^{lc}_c(\ol{\bS}^{w'},\bV)$ is surjective, we obtain that $\bV_w\to \bV_{w'}$ are indeed surjective, by the right-exactness of the functors $(\cdot )_{\bT\times \bG_m}$ and $(\cdot )_{\bI^0}$.
Using [Lemma \[exactness of Jacquet\]]{}, we obtain that $$\bV^w:=\on{ker}(\bV_w\to \underset{w'<w}\oplus\, \bV_{w'})
\simeq \left(\left(\on{Funct}^{lc}_c(\bS^w,\bV)\right)_{\bT\times \bG_m}
\right)_{\bI^0}.$$
Let us choose a splitting $\bT\to \bB$, by means of which $\bT$ becomes a subgroup of $\bI$; let $g\in \on{Fl}^w_G(\bK)$ be the $\bT$-stable point, and let $St(g)_{\bI}$ be the stabilizer of $g$ in $\bI$. We obtain a homomorphism $St(g)_{\bI}\to \bT\times \bG_m$ and we have: $$\left(\left(\on{Funct}^{lc}_c(\bS^w,\bV)\right)_{\bT\times \bG_m}
\right)_{\bI^0}\simeq
\left(i^{\bI}_{St(g)_{\bI}}\circ r^{St(g)_{\bI}}_{\bT\times \bG_m}(\Pi)
\right)_{\bI^0}.$$
Observe that the character of $\bT$, corresponding to measures on the homogeneous space $\on{Fl}^w_G(\bK)\simeq \bI/St(g)_{\bI}$, equals $\mu_w$.
Write $w=\lambda\cdot \ol{w}$, $\lambda\in \Lambda$, $\ol{w}\in W$. Observe now that the pull-back of $\Pi$ under the composition $$\bT\to St(g)_{\bI}\to \bI\times \bG_m\to \bI\to\bT$$ is naturally isomorphic to $\Pi^{\ol{w}}$, and the pull-back of the character $c:\bG_m\to \BC^*$ under $\bT\to St(g)_{\bI}\to \bI\times \bG_m\to \bG_m$ is $\phi_{cQ}(\lambda)$.
This implies that $\bV^w \simeq w\cdot \Pi$ as $\bT$-representations.
One can formulate an analog of [Proposition \[geometric lemma\]]{} describing the composition of the functors $r^{\wh{\BG}}_\bG\circ i^{\wh{\BG}}_\bG:\on{Rep}(\bG,\BVect)\to
\on{Rep}(\bG,\BVect)$:
Set $\Gr_G:=G((t))/G[[t]]$, and recall that $G[[t]]$-orbits on $\Gr_G$ are in a natural bijection with the partially ordered set $\Lambda^+$ of dominant weights.
For every $\lambda\in\Lambda^+$, let $g\in \Gr_G^\lambda$ be a $T$-stable point, and let $St(g)_{G[[t]]}$ be its stabilizer in $G[[t]]$, so that $\Gr_G^\lambda\simeq G[[t]]/St(g)_{G[[t]]}$. Note that since $G^1$ is normal in $G[[t]]$, the quotient $G^1\backslash \Gr_G^\lambda$ is a $G$-homogeneous space isomorphic to $G/P^\lambda$ for a parabolic $P^\lambda\subset G$. Let $M^\lambda$ be the Levi quotient of $P^\lambda$, and let $\mu_\lambda$ be the character of $M^\lambda$ corresponding to measures on the homogeneous space $I_P/St(g)_{G[[t]]}(\bK)$. Note also that for $\lambda$ as above, the character $\phi_{cQ}(\lambda)$ of $\bT$ is in fact well-defined as a character of $\bM^\lambda$.
Recall from the theory of $\fp$-adic groups that for a parabolic $P$ with a Levi quotient $M$ we have a pair of mutually adjoint functors $r^\bG_\bM:\on{Rep}(\bG,Vect)\to \on{Rep}(\bM,Vect)$ and $i^\bG_\bM:\on{Rep}(\bM,Vect)\to \on{Rep}(\bG,Vect)$.
\[geometric lemma for Grassmann\] For a representation $\Pi=(\bV,\rho)\in \on{Rep}(\bG,Vect)$, the object $r^{\wh{\BG}}_\bG\circ i^{\wh{\BG}}_\bG(\Pi)$ can be canonically written as $\underset{\underset{\lambda\in \Lambda^+}\longleftarrow} {"lim"}\, \bV_\lambda$, $\bV_\lambda\in \on{Rep}(\bG,Vect)$ in such a way that for $\lambda'\leq \lambda$ the map $\bV_\lambda\to \bV_{\lambda'}$ is a surjection, and the kernel $\bV^\lambda:=\on{ker}(\bV_\lambda\to \underset{\lambda'<\lambda}\oplus\,
\bV_{\lambda'})$ is canonically isomorphic to $i^\bG_{\bM^\lambda}\left(r^\bG_{\bM^\lambda}(\Pi)
\otimes \mu_\lambda\otimes \phi_{cQ}(\lambda)\right)$.
The proof of this proposition is parallel to that of [Proposition \[geometric lemma\]]{}.
Examples
========
Assume now that the group $G$ is split, simple and simply-connected. In this case, a data of an extension $\wh{G}$ is equivalent to that of a $W$-invariant even symmetric bilinear form $Q:\Lambda\otimes\Lambda\to \BZ$ (cf. Sect. 4 of [@BrDe]), and we fix this form to be the minimal one, i.e., $\frac{1}{2\check h}Q_0$, where $Q_0$ corresponds to the Killing form, and $\check h$ is the dual Coxeter number.
We have previously worked with a fixed character $\bG_m\to \BC^*$, but now we will consider all representations of the group $\wh{\BG}$. For a thick subgroup $\BH\subset \bG[[t]]$ we have the corresponding functors $i^{\wh{\BG}}_{\BH\times \bG_m}$, $r^{\wh{\BG}}_{\BH\times\bG_m}$ between $\on{Rep}(\BG)$ and $\on{Rep}(\BH\times \bG_m,\BVect)$.
Let $\Lambda_{aff}$ be the lattice $\Lambda\oplus\BZ$; which identifies with the quotient of $\bT\times \bG_m$ by its maximal compact subgroup, and let $\BC[\Lambda_{aff}]$ be its group-algebra, viewed as a representation of $\bT\times \bG_m$.
Consider the object $\BV:=i^{\wh{\BG}}_{\bT\times \bG_m}(\BC[\Lambda_{aff}])\in
\on{Rep}(\wh{\BG})$, studied by Kapranov in [@Kap]. Let $\overset{\cdot\cdot}\sH_q$ be the modified Cherednik algebra of [*loc.cit.*]{} 2.3.3. In [@Kap] it was shown that $\overset{\cdot\cdot}\sH_q$ injects into $\on{End}_{\on{Rep}(\wh{\BG})}(\BC[\Lambda_{aff}])$.
By combining the results of [@Kap] and [Proposition \[adjointness\]]{} we will establish the following result:
\[Kapranov\] The map $\overset{\cdot\cdot}\sH_q\to \on{End}_{\on{Rep}(\wh{\BG})}(\BV)$ is an isomorphism.
Let $\BV^{rat}$ be the object of $\on{Rep}(\wh{\BG})$ equal to $i^{\wh{\BG}}_{\bT\times \bG_m}(\BC(\check T\times G_m))$, where $\BC(\check T\times G_m)$ is the field of rational functions on the torus $\check T\times G_m$, viewed as a $\bT\times \bG_m$-representation. Note that by construction, both $\BV$ and $\BV^{rat}$ carry an action of the algebra $\BC[\Lambda_{aff}]$ by endomorphisms.
Using [Proposition \[geometric lemma\]]{} and [Proposition \[adjointness\]]{} we obtain that $$\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV,\BV^{rat})\simeq
\underset{\underset{w\in W_{aff}}\longrightarrow}{lim}\,
\on{Hom}_{\Lambda_{aff}-\on{mod}}(\bV_w,\BC(\check T\times G_m)),$$ where $\bV^w:=\on{ker}\left(\bV_w\to \underset{w'<w}\oplus\, \bV_{w'}\right)$ is isomorphic to $w\cdot \BC[\Lambda_{aff}]$. In particular, we see that the restriction map $\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV^{rat},\BV^{rat})\to
\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV,\BV^{rat})$ is an isomorphism.
The subquotients $\on{Hom}_{\Lambda_{aff}-\on{mod}}(\bV^w,\BC(\check
T\times G_m))$ are all identified with $\BC(\check T\times G_m)$ as left $\Lambda_{aff}$-modules, with the right $\Lambda_{aff}$-module structure twisted by $w\cdot$ (see [Proposition \[geometric lemma\]]{}). Hence, we obtain a canonical direct sum decomposition $$\label{big Hom}
\on{End}_{\on{Rep}(\wh{\BG})}(\BV^{rat})\simeq
\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV,\BV^{rat})\simeq \BC(\check T\times
G_m)\ltimes W_{aff}.$$
Therefore, using the main Theorem 3.3.8 of [@Kap], it suffices to check that the isomorphism coincides with the map $$\overset{\cdot\cdot}\sH_q{}^{rat}\simeq
\BC(\check T\times G_m)\ltimes W_{aff}\to
\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV,\BV^{rat})\to
\on{End}_{\on{Rep}(\wh{\BG})}(\BV^{rat})$$ of [@Kap], Equation 3.3.7.
Since both isomorphisms preserve the ring structure, it suffices to check that the generators of $\BC(\check T\times G_m)\ltimes W_{aff}$ over $\BC(\check T\times G_m)$, corresponding to the simple reflections under the two homomorphisms, act on $\BV^{rat}$ in the same way.
If $s$ is a simple reflection in $W_{aff}$, there exists a parahoric $I_s\subset G((t))$ such that the corresponding Levi quotient $M_s$ is a reductive group of semi-simple rank $1$. As in [Sect. \[induction from Levi\]]{} we have an induction functor $i^{\wh{\BG}}_{\bM_s}:\on{Rep}(\bM_s,Vect)\to \on{Rep}(\BG)$, and $\BV^{rat}\simeq i^{\wh{\BG}}_{\bM_s}\circ i^{\bM_s}_{\bT\times \bG_m}
(\BC(\check T\times G_m))$, so that the endomorphism of $\BV^{rat}$ corresponding to $s$ via both and the integral operator $\tau_s$ of [@Kap] come from the corresponding endomorphisms of $i^{\bM_s}_{\bT\times \bG_m}(\BC(\check T\times G_m))$.
Therefore, we have reduced the question about the equality of two endomorphisms of $\BV^{rat}$ to a similar question about endomorphisms of $i^{\bM_s}_{\bT\times \bG_m}(\BC(\check T\times G_m))$ in the theory of $\fp$-adic groups. This reduces to the following (well-known) calculation:
Let $G$ be a split reductive group of semi-simple rank $1$, and consider the $G(\bK)$-representation $\bV:=i^\bG_\bT(\BC[\Lambda])$, which identifies with the space of locally-constant compactly supported functions on the quotient $G(\bK)/N(\bK)$, where $N$ is the maximal unipotent subgroup of $G$. We can view $\bV$ as a $\check T$-family of principal series representations, denoted $\bV_t, t\in \check T$. Let $\bV^{rat}$ be the $\bG$-representation $i^\bG_\bT(\BC(\check T))$. As above, we have $$\BC(\check T)\ltimes W\simeq \on{End}_{G(\bK)}(\bV^{rat})\simeq
\on{Hom}_{G(\bK)}(\bV,\bV^{rat}).$$ Consider the element $\tau_s$ of $\on{Hom}_{\bG}(\bV,\bV^{rat})$ corresponding to the (unique) simple reflection in $W\subset \BC(\check T)\ltimes W$. Then $\tau_s$ gives rise to a map $\bV_t\to \bV_{s\cdot t}$ defined for $t$ belonging to an open subset of $\check T$, and the claim is that this map is given by the meromorphic integral operator $f\mapsto f^{\tau_s}$ with $$f^{\tau_s}(g)=\underset{n\in N(\bK)}\int\, f(g\cdot n\cdot s).$$
Let us go back to the situation, when the parameter $c$ is fixed and unramified. Let $\BV_{c}:=i^{\wh{\BG}}_\bT(\BC[\Lambda])\in \on{Rep}_c(\wh{\BG})$ and let $\overset{\cdot\cdot}\sH_{q,c}$ be the specialization of $\overset{\cdot\cdot}\sH_q$ at $c$.
We have an isomorphism $\overset{\cdot\cdot}\sH_{q,c}\simeq \on{End}(\BV_c)$.
By applying [Proposition \[adjointness\]]{} and [Proposition \[geometric lemma\]]{} to $\BV$ and $\BV_c$, we obtain that $\on{Hom}_{\on{Rep}(\wh{\BG})}(\BV,\BV)$ is a flat module over $\BC[\Lambda_{aff}]$, and hence over $\BC[\BZ]$, whose fiber at $c\in \on{Spec}(\BC[\BZ])$ identifies with $\on{Hom}_{\on{Rep}_c(\BG)}(\BV_c,\BV_c)$.
In other words, $\on{Hom}_{\on{Rep}_c(\BG)}(\BV_c,\BV_c)$ is isomorphic to the fiber of $\overset{\cdot\cdot}\sH_q$ at $c$, which is the same as $\overset{\cdot\cdot}\sH_{q,c}$.
The representation $\BV_c$ studied above is an analogue of principal series representations. We will now introduce an object of $\on{Rep}(\BG)$ which should be thought of as a cuspidal representation of $\BG$, although at the moment we do not have a definition of cuspidality.
Let $\Pi$ be an irreducible cuspidal representation of $\bG$; and consider $i^{\wh{\BG}}_\bG(\Pi)\in \on{Rep}_c(\wh{\BG})$.
$\on{End}_{\on{Rep}_c(\wh{\BG})}(i^{\wh{\BG}}_\bG(\Pi))\simeq \BC$.
Using [Proposition \[adjointness\]]{} we have $\on{End}_{\on{Rep}_c(\wh{\BG})}(i^{\wh{\BG}}_\bG(\Pi))\simeq
\on{Hom}_{\on{Rep}(\bG,\BVect)}(r^{\wh{\BG}}_\bG\circ i^{\wh{\BG}}_\bG(\Pi),\Pi)$.
We claim that the natural map $r^{\wh{\BG}}_\bG\circ i^{\wh{\BG}}_\bG(\Pi)\to \Pi$ is an isomorphism, which would imply the assertion of the lemma. In fact, we claim that all the subquotients $\bV^\lambda$ of [Proposition \[geometric lemma for Grassmann\]]{} vanish except for $\lambda=0$.
Indeed, by [Proposition \[geometric lemma for Grassmann\]]{} each such subquotient involves the functor $r^\bG_{\bM^\lambda}$ applied to $\Pi$, which vanishes, since $\Pi$ was assumed to be cuspidal.
The objects $i^{\wh{\BG}}_\bG(\Pi)$, for $\Pi$ being a cuspidal representation of $\bG$, are irreducible.
Recall that an object $\Pi\in \on{Rep}(\bG,Vect)$ is called admissible if for every open compact subgroup $\bH\subset \bG$, the vector space $\Pi^\bH\simeq \Pi_\bH$ is finite-dimensional, i.e., belongs to $Vect_0$.
We can give an analogous definition in the case of $\on{Rep}_c(\wh{\BG})$:
An object $\Pi\in \on{Rep}_c(\wh{\BG})$ is called admissible if for every thick subgroup $\BH\subset \bG[[t]]$ the object $(\Pi)_\BH\in \BVect$ belongs, in fact, to $Vect$.
It is easy to see that the principal series representations $\BV_c$ are not admissible. However, we have the following assertion:
The representation $i^{\wh{\BG}}_\bG(\Pi)$, for $\Pi$ being a cuspidal representation of $\bG$, is admissible.
First, we can replace $\BH$ by a congruence subgroup $\bG^i$: indeed, if $\BH\supset \bG^i$, then the statement for $\bG^i$ would imply that for $\BH$.
Let $\bGr_G$ (resp., $\bGr^\lambda_G$, $\ol{\bGr}{}^\lambda_G$) be the objects of $\on{Ind}(\bSet)$ and $\bSet$ corresponding to $\Gr_G$, $\Gr^\lambda_G$ and $\ol{\Gr}{}^\lambda_G$, respectively.
Let $\bS^\lambda$ (resp., $\ol{\bS}{}^\lambda$) be the preimage of $\bGr^\lambda_G$ (resp., $\ol{\bGr}{}^\lambda_G$) in $\wh{\BG}/\bG^1$. As in [Proposition \[geometric lemma\]]{}, the object $r^{\wh{\BG}}_{\bG[[t]]}\circ i^{\wh{\BG}}_\bG(\Pi)\in \on{Rep}(\bG[[t]],\BVect)$ for $\Pi=(\bV,\rho)\in \on{Rep}(\bG,Vect)$ is the inverse limit over $\lambda\in \Lambda^+$ of $\on{Funct}^{lc}_c\left(\ol{\bS}{}^\lambda,\bV\right)_{\bG\times \bG_m}$.
Set $\bW^\lambda:=
\on{Funct}^{lc}_c\left(\bS^\lambda,\bV\right)_{\bG\times \bG_m}$, and we have to show that $\left(\bW^\lambda\right)_{\bG^i}\simeq 0$ for all but finitely many $\lambda$’s.
Let $g\in \Gr^\lambda_G(\bK)$ be a $\bT$-stable point and $\bSt(g)_{G[[t]]}$ its stabilizer in $\bG[[t]]$. By definition, we have a homomorphism $\bSt(g)_{G[[t]]}\to \bG[[t]]\times \bG_m$ and $$\bW^\lambda\simeq i^{\bG[[t]]}_{\bSt(g)_{G[[t]]}}\circ
r^{\bSt(g)_{G[[t]]}}_{\bG\times \bG_m}(\Pi).$$ Therefore, as representations of $\bG[[t]]/\bG^i$, $$\left(\bW^\lambda\right)_{\bG^i}\simeq
i^{\bG[[t]]/\bG^i}_{\bSt(g)_{G[[t]]}/\bSt(g)_{G[[t]]}\cap \bG^i}
\left(\left(r^{\bSt(g)_{G[[t]]}}_{\bG\times\bG_m}
(\Pi)\right)_{\bSt(g)_{G[[t]]}\cap \bG^i}
\otimes \mu\right),$$ where $\mu$ is a character.
Note that as an object of $Vect$, $\left(r^{\bSt(g)_{G[[t]]}}_{\bG\times\bG_m}
(\Pi)\right)_{\bSt(g)_{G[[t]]}\cap \bG^i}$ is isomorphic to $(\Pi)_{\bH^i}$, where $\bH^i$ is the image of $\bSt(g)_{G[[t]]}\cap \bG^i$ under the homomorphism $\bSt(g)_{G[[t]]}\to \bG[[t]]\to \bG$. Therefore, the assertion of the proposition follows from the fact that for all but finitely many $\lambda$’s, the subgroup $\bH^i\subset \bG$ contains the unipotent radical of a non-trivial parabolic.
The Schwartz space on $\BG$ {#Schwartz space}
===========================
Suppose now that $\bS$ is an object of $\bSet$, corresponding to a smooth scheme of finite type $S$ over $\bK$. Then it makes sense to consider the space of locally constant compactly supported measures on $S(\bK)$, denoted $M(\bS)$. To define it, we choose locally a top degree nowhere vanishing differential form $\omega$ on $S$, which defines a measure $\mu(\omega)$ on $S(\bK)$. We say that a measure is locally constant if it can be obtained from $\mu(\omega)$ by multiplication by a locally constant function. One readily checks that this definition is independent of the choice of $\omega$.
Suppose now that $\phi:S_1\to S_2$ is a smooth map between smooth schemes. In this case, the operation of push-forward of constantly supported measures preserves the subspaces of locally constant ones, i.e., it defines a map $\phi_!:M(\bS_1)\to M(\bS_2)$.
In particular, if an algebraic group $G$ acts on $S$, we obtain that $M(\bS)$ is naturally on object of $\on{Rep}(\bG,Vect)$.
For $S$ as above, consider now the object $\BS\in \BSet$. It appears that there is no invariant way to assign to $\BS$ an object of $\BVect$, which would be a replacement of locally constant compactly supported measures, and this is similar to the absence of a notion of D-module on $S((t))$, cf. [@AG].
In this section we will study this phenomenon first when $S$ is the affine space $A^n$, and then when $S$ is an affine algebraic group $G$.
For any scheme $S$ which is isomorphic to a projective limit of smooth schemes of finite type $S_i$ with smooth transition maps $S_i\to S_j$, we have $\bS\in \on{Pro}(\bSet)$, and we define $M(\bS)\in \BVect$ as $M(\bS):="\underset{\longleftarrow}{lim}"\, M(\bS_i)$, where the maps $M(\bS_j)\to M(\bS_i)$ for $j\geq i$ are the push-forwards of measures.
Recall that a lattice $L\subset \bK((t))^n$ is a finitely generated $\bK[[t]]$-submodule, which contains $t^i\cdot \bK[[t]]$ for some $i$. The “standard” lattice is by definition $L_0=\bK[[t]]$. By abuse of notation, we will denote by the same character $L$ the group-subscheme of $A^n((t))$ corresponding to a lattice $L$, and by $\bL$ the corresponding object of $\on{Pro}(\bSet)$. Since $L=\underset{i}{\underset{\longleftarrow}{lim}}\, (L/t^i\cdot L)$, we have a well-defined object $M(\bL)\in \BVect$.
For a finite-dimensional vector space $H$ over $\bK$ let $\on{det}(H)=\Lambda^{\on{top}}(H)$ denote its determant line. Let $\bH$ and $\on{det}(\bH)$ be the corresponding objects of $\bSet$, and let $\mu(\on{det}(\bH))$ denote the $1$-dimensional $\BC$-vector space of Haar measures on $\on{det}(\bH)$. [^2] Of course, an element of $\mu(\on{det}(\bH))$ determines also a Haar measure on $\bH$.
Recall that for two lattices $L,L'\subset \bK((t))^n$ we can assign their relative determinant line $\on{det}(L,L')$ so that $\on{det}(L,L'')\simeq
\on{det}(L,L')\otimes \on{det}(L',L'')$ and for $L\subset L'$, $\on{det}(L,L')=\on{det}(L'/L)$, where the vector space $L'/L$ is, by definition, finite-dimensional. Let $\mu(\on{det}(\bL,\bL'))$ be the corresponding line of Haar measures.
For $L\subset L'$ we have a canonical morphism $M(\bL')\to M(\bL)\otimes \mu(\on{det}(L,L'))$.
Let $L''$ be a sublattice in $L$. By definition, for every such $L''$ we must construct a morphism $M(\bL'/\bL'')\to M(\bL/\bL'')\otimes
\mu(\on{det}(\bL,\bL'))$. The required map is defined as a composition: $$\begin{aligned}
&M(\bL'/\bL'')\simeq \on{Funct}^{lc}_c(\bL'/\bL'')\otimes \mu(\on{det}(\bL'',\bL'))\to
\on{Funct}^{lc}_c(\bL/\bL'')\otimes \mu(\on{det}(\bL'',\bL'))\simeq \\
&\on{Funct}^{lc}_c(\bL/\bL'')
\otimes \mu(\on{det}(\bL'',\bL))\otimes
\mu(\on{det}(\bL,\bL'))\simeq M(\bL/\bL'')\otimes \mu(\on{det}(\bL,\bL')),\end{aligned}$$ where the arrow corresponds to the ordinary restriction of functions.
Finally, we are ready to define the object $M(\BA^n)\in \BVect$, which we propose as a candidate for the Schwartz space of functions on $A^n(\bF)$: $$M(\BA^n):=\underset{L}{\underset{\longleftarrow}{lim}}\,
M(\bL)\otimes \mu(\on{det}(\bL,\bL_0)),$$ where $\underset{\longleftarrow}{lim}$ is taken in the category $\BVect$, and the arrows are given by the lemma above.
It is easy to see that the action of $A^n((t))$ on itself by translations makes $M(\BA^n)$ an object of $\on{Rep}(\BA^n)$.
Recall also that the group-indscheme $GL_n((t))$, which acts naturally on $A^n((t))$, has a canonical central extension $\wh{GL}_n$ by means of $G_m$, whose $S$-points for a test-scheme $S$ are pairs $g\in \on{Hom}(S,GL_n((t)))$ and a trivialization of the line bundle $\on{det}(g\cdot L_0,L_0)$ on $S$.
\[flat space\] The action of $GL_n((t))$ on $A^n((t))$ makes $M(\BA^n)$ an object of $\on{Rep}(\wh{\BG L}_n)$, where $\bG_m\subset \wh{\BG L}_n$ acts via the character $\bG_m\to \BZ\overset{1\mapsto q}\longrightarrow \BC^*$.
By construction, as an object of $\BVect$, $$M(\BA^n)\simeq "\underset{\longleftarrow}{lim}"\,
M(\bL/\bL') \otimes \mu(\on{det}(\bL,\bL_0)),$$ where the inverse limit is taken over the partially ordered set of pairs of lattices $L'\subset L$ with $(L'_1\subset L_1)\leq (L'_2\subset L_2)$ if and only if $L_1\subset L_2$ and $L_1'\supset L'_2$.
For clarity, let us first define an action of the abstract group $\wh{GL}_n(\bK)$ on $M(\BA^n)$. For a pair of lattices $L'\subset L$ and $g\in GL_n((t))(\bK)$, the action of $g$ defines an isomorphism $M(\bL/\bL')\simeq M(g\cdot \bL/g\cdot \bL')$ and an isomorphism $\on{det}(L,L_0)\simeq \on{det}(g\cdot L,g\cdot L_0)\simeq
\on{det}(g\cdot L,L_0)\otimes \on{det}(L_0,g\cdot L_0)$. Hence, if we lift $g$ to an element of $\wh{GL}_n(\bK)$, we obtain an isomorphism $\on{det}(L,L_0)\simeq \on{det}(g\cdot L,L_0)$, i.e., we obtain a desired action.
Let us now repeat this construction in order to obtain an action map $\wh{\BG L}_n\times M(\BA^n)\to M(\BA^n)$. Let us write $\wh{GL}_n$ as $"\underset{\longrightarrow}{lim}"\, S_k$, and $S_k="\underset{\longleftarrow}{lim}"\, S_{k,l}$ with $S_{k,l}\in Sch^{ft}$. Set $\BS_k$ (resp., $\bS_{k,l}$) to be the corresponding objects of $\on{Pro}(\bSet)$ (resp., $\bSet$).
Recall that if $S$ is a scheme, there is a notion of an $S$-family of lattices in $\bK((t))^n$, which is in fact the same as an $S$-point of the affine Grassmannian of $GL_n$. If $L$ and $L'$ are two $S$-families of lattices with $L'\subset L$, then the quotient $L/L'$ is a vector bundle on $S$.
For a pair of lattices $L'\subset L\subset \bK((t))^n$ and an index $k$, using the action of $GL_n((t))$ on $\Gr_{GL_n}$, we obtain the $S_k$-families of lattices that we will denote by $S_k\cdot L'\subset S_k\cdot L$. Moreover, there exists another pair of lattices $L'_1\subset L_1$, thought of as constant $S_k$-families, such that $L'_1\subset S_k\cdot L'$ and $S_k\cdot L\subset L_1$. Consider the quotients $$H_{S_k}:=S_k\cdot L'/L'_1\subset H'_{S_k}:=S_k\cdot L/L'_1\subset
H''_{S_k}:=L_1/L'_1$$ as vector bundles on $S_k$. Note that both $H'_{S_k}/H_{S_k}$ and $H''_{S_k}$ are trivial bundles with fibers $L/L'$ and $L_1/L'_1$, respectively. By the definition of $\wh{GL}_n$, the line bundle $\on{det}(H''_{S_k}/H'_{S_k})$ is identified with the trivial line bundle with fiber $\on{det}(L,L_1)$. Finally, there exists an index $l$, so that $H_{S_k}$ and $H'_{S_k}$, together with their embeddings into $H''_{S_k}$, come from vector bundles on $S_{k,l}$, which we will denote by $H_{S_{k,l}}$ and $H'_{S_{k,l}}$, respectively.
We need to construct an action map $\bS_{k,l}\times
\left(M(\bL_1/\bL'_1)\otimes \mu(\on{det}(\bL_1,\bL))\right)\to
M(\bL/\bL')$, which on the level of $\bK$-points amounts to the one constructed above. For that, by Zariski localizing $S_{l,k}$, we may assume that the vector bundle $H'_{S_{k,l}}$ on $S_{k,l}$ can be trivialized, i.e., $H'_{S_{k,l}}\simeq H'\times S_{k,l}$.
Thus, we have a map $\bS_{k,l}\times \bH'\to \bL_1/\bL'_1$, such that the corresponding map $\bS_{k,l}\times \bH'\to \bS_{k,l}\times \bL_1/\bL'_1$ is proper (cf. [Sect. \[induction functor\]]{}). Therefore, it defines an action map $$\bS_{k,l}\times \on{Funct}^{lc}_c(\bL_1/\bL'_1)\to
\on{Funct}^{lc}_c(\bH').$$ By tensoring with $\mu(\on{det}(\bH'))$ we obtain an action map $\bS_{k,l}\times (M(\bL_1/\bL'_1)\otimes
\mu(\on{det}(\bL_1,\bL)))\to
M(\bH')$.
Similarly, we have a map $\bS_{k,l}\times \bH'\to \bL/\bL'$, and by integration we obtain an action map $\bS_{k,l}\times M(\bH')\to M(\bL/\bL')$. The composition $$\begin{aligned}
&\on{Hom}(\bS_{k,l}\otimes M(\bH'),M(\bL/\bL'))\times
\on{Hom}(\bS_{k,l}\otimes M(\bL_1/\bL'_1),M(\bH'))\to \\
&\on{Hom}((\bS_{k,l}\times \bS_{k,l})\otimes M(\bL_1/\bL'_1),M(\bL/\bL'))\to
\on{Hom}(\bS_{k,l}\otimes M(\bL/\bL'),M(\bH))\end{aligned}$$ yields the desired action.
Let now $G$ be an algebraic group over $\bK$. Let $G((t))$ be the corresponding loop group and $\wh{G}$ its central extension $1\to G_m\to \wh{G}\to G((t))\to 1$, as in [Sect. \[cent ext\]]{}. Let us fix a character $c:\bG_m\to \BC^*$. We will now define an object $M_c(\BG)\in \BVect$, which will underly an object $(M_c(\BG),\rho)\in \on{Rep}_c(\wh{\BG})$.
For every integer $i$ consider the trivial representation $\BC$ of the corresponding congruence subgroup $\bG^i$ and consider $i^{\wh{\BG}}_{\bG^i}(\BC)\otimes \mu(\bG[[t]]/\bG^i)\in \on{Rep}_c(\wh{\BG})$, where $\mu(\bG[[t]]/\bG^i)$ is the $1$-dimensional space of left-invariant Haar measures on the group $(\bG[[t]]/\bG^i)^{\on{top}}$.
By the construction of the functor $i^{\wh{\BG}}_{\bG^i}$ via compactly supported functions on $\wh{\BG}/\bG^i$, for $j\geq i$ we have the morphisms $$i^{\wh{\BG}}_{\bG^j}(\BC)\otimes \mu(\bG[[t]]/\bG^j)\to
i^{\wh{\BG}}_{\bG^i}(\BC)\otimes \mu(\bG[[t]]/\bG^i)$$ given by fiber-wise integration. Set $M_c(\BG):=\underset{\longleftarrow}{lim}\,\, i^{\BG}_{\bG^j}(\BC)$, where $\underset{\longleftarrow}{lim}$ is taken in $\BVect$.
For example, it is easy to see that when $G\simeq A^n$ (and the central extension is trivial), the space $M(\BG)$ obtained in this way identifies canonically with $M(\BA^n)$ considered above.
Since each $\bG^i$ is a normal subgroup in $\bG[[t]]$, the terms of the inverse system defining $M_c(\BG)$ carry a commuting $\bG[[t]]$-action on the right, which is respected by the arrows. Hence, $M_c(\BG)$ carries an additional $\bG[[t]]$-action “on the right”, which commutes with the action of $\wh{\BG}$ “on the left”.
This $\wh{\BG}-\bG[[t]]$-module structure on $M_c(\BG)$ allows to reinterpret the functor $i^{\wh{\BG}}_\BH$ introduced earlier:
Recall that the functor of tensor product $Vect\times Vect\to Vect$ extends naturally to $\BVect$: $$("\underset{\longleftarrow}{lim}"\, \bV_i)\otimes
("\underset{\longleftarrow}{lim}"\, \bW_j):=
"\underset{\longleftarrow}{lim}"\, (\bV_i\otimes \bW_j).$$
Let $\BH\subset \bG[[t]]$ be a thick subgroup, and let $\Pi=(\BV,\rho)$ be an object of $\on{Rep}(\BH,\BVect)$. Consider the tensor product $M_c(\BG)\otimes \BV\in \BVect$. The diagonal action of $\BH$ makes it into an object of $\on{Rep}(\BH,\BVect)$, which carries a commuting $\wh{\BG}$-action. Hence, $\left(M_c(\BG)\otimes \BV\right)_\BH$ is naturally an object of $\on{Rep}(\BG)$.
The following is straightforward from the definitions:
We have a natural isomorphism in $\on{Rep}(\BG)$: $i^{\wh{\BG}}_\BH(\Pi)\simeq \left(M_c(\BG)\otimes \BV\right)_\BH$.
Since we think of $M_c(\BG)$ as the space of functions on the group $\BG$, it is natural to expect that the $\bG[[t]]$-action on $M_c(\BG)$ considered above extends to an action “on the right” of the entire group $\BG=\bG((t))$, corresponding to right translations. The existence of such an action is given by the theorem below.
Let $\fg$ be the Lie algebra of $G$, let $GL_{\fg}((t))$ be the corresponding loop group, and let $\wh{GL}_{\fg}$ be its canonical central extension as in [Theorem \[flat space\]]{}. Let $\wh{G}_0$ be the central extension of $G((t))$ induced from $\wh{GL}_{\fg}$ by means of the adjoint action. For example, if $G$ is simple and simply-connected, the extension $\wh{G}_0$ corresponds to the pairing $\Lambda\otimes \Lambda\to \BZ$ given by the Killing form.
Let $\wh{G}{}'$ be the central extension of $G((t))$ equal to the Baer sum of $\wh{G}_0$ and the original extension $\wh{G}$. Let $c'$ be the character of $\bG_m\subset \wh{\BG}{}'$ equal to the inverse of the product of $c$ and $c_0$, where $c_0$ is the character $\bG_m\to \BZ\overset{1\mapsto q}\longrightarrow\BC^*$.
\[main\] We have a canonical action of $\wh{\BG}{}'$ on $M_c(\BG)$, with $\bG_m\subset \wh{\BG}{}'$ acting by the character $c'$. This action extends the natural action of $\bG[[t]]$ on $M_c(\BG)$ “on the right” and commutes with the action of $\wh{\BG}$ “on the left”.
Let us first construct an action of the abstract group $\wh{G}{}'(\bK)$ on $M_c(\BG)$. For an integer $i$ and a point $g\in G((t))(\bK)$, there exists an integer $j$ such that $g^{-1}(\bG^j)g$ is contained in $\bG^i$; therefore, the right multiplication map $\bG((t))\times g\to \bG((t))$ descends to a well-defined map $\bG((t))/\bG^j\times g\to \bG((t))/\bG^i$.
In particular, if we lift $g$ to an element of $\wh{G}(\bK)$, we obtain a map $$i^{\wh{\BG}}_{\bG^j}(\BC)\otimes \mu(\bG^i/g^{-1}(\bG^j)g)\to
i^{\wh{\BG}}_{\bG^i}(\BC),$$ which commutes with the left $\wh{\BG}$-action.
We claim now that a lift of $g$ to an element of $\wh{G}_0(\bK)$ define an identification of the line $\mu(\bG^i/g^{-1}(\bG^j)g)$ with $\mu(\bG^i/\bG^j)$. Indeed, let $\fg^i$ be the Lie subalgebra in $\fg((t))$ corresponding to the congruence subgroup $\bG^i$, then $$\mu(\bG^i/\bG^j)\simeq \mu(\on{det}(\fg^i/\fg^j));\,\,
\mu(\bG^i/g^{-1}(\bG^j)g)\simeq \mu(\on{det}(\fg^i/g^{-1}\cdot\fg^j)),$$ and $$\on{det}(\fg^i/\fg^j)\simeq \on{det}(\fg^i/g^{-1}\cdot \fg^j)\otimes
\on{det}(g\cdot \fg^j,\fg^j)\simeq
\on{det}(\fg^i/g^{-1}\cdot \fg^j)\otimes \on{det}(g\cdot \fg^0,\fg^0).$$
Therefore, if we take the Baer product of the extensions $\wh{\BG}$ and $\wh{\BG}_0$ we obtain an action of the group of $\bK$-points of $\wh{\BG}{}'$ on $M_c(\BG)$. Therefore, by passing to inverses, we obtain on $M_c(\BG)$ an action of $\wh{G}{}'(\bK)$, commuting with the left action of $\wh{\BG}$ and the prescribed value of the central character.
The fact the constructed point-wise action gives rise to a well-defined action map $\wh{\BG}{}'\times M_c(\BG)\to M_c(\BG)$ follows by considering families, as in the proof of [Theorem \[flat space\]]{}.
Namely, if $\wh{G}\underset{G((t))}\times \wh{G}_0="\underset{\longrightarrow}{lim}"\, S_k$, $S_k="\underset{\longleftarrow}{lim}"\, S_{k,l}$ with $S_{k,l}\in Sch^{ft}$, for every pair of indices $i,k$ there exists a large enough index $j$, such that the group-subscheme $\on{Ad}_{S_k}(G^j)\subset G((t))\times S_k$ is contained in $G^i\times S_k$. Moreover, the relative determinant line $\on{det}(\fg^i/\on{Ad}_{S_k}(\fg^j))$ is identified with the constant line bundle with fiber $\on{det}(\fg^j,\fg^i)$.
We have the map $$(\wh{G}/G^j)\times S_k\simeq (\wh{G}\times S_k)/\on{Ad}_{S_k}(G^j)
\to \wh{G}/G^i,$$ which comes from a map $\wh{G}/G^j\times S_{k,l}\to \wh{G}/G^i$ defined for a sufficiently large index $l$. The resulting map $\wh{G}/G^j\times S_{k,l}\to \wh{G}/G^i\times S_{k,l}$ is smooth over every finite-dimensional subscheme of $\wh{G}/G^i$.
Integration along the fiber defines the desired map $$\bS_{k,l}\times \left(i^{\wh{G}}_{\bG^j}(\BC)\otimes \mu(\on{det}(\fg^j,\fg^i))\right)\to
i^{\wh{G}}_{\bG^i}(\BC).$$
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[^1]: We would like to thank E. Hrushovski for pointing out the mistake in the previous version of the paper, where [Proposition \[representations of pro-groups\]]{}, was stated without the ($*$) assumption on $H$; in fact, he constructed a counterexample.
[^2]: Properly speaking, $\on{det}(H)$ is a super-vector space; however, in this paper it will appear only via $\mu(\on{det}(H))$, so the difficulties associated with the sign are irrelevant.
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abstract: 'We report on the latest results in the search for positively charged strangelets from E864’s 96/97 run at the AGS with sensitivity of about $8\times 10^{-9}$ per central collision. This contribution also contains new results of a search for highly charged strangelets with $Z=+3$. Production of light nuclei, such as ${}^{6}He$ and ${}^{6}Li$, is presented as well. Measurements of yields of these rarely produced isotopes near midrapidity will help constrain the production levels of strangelets via coalescence. E864 also measures antiproton production which includes decays from antihyperons. Comparisons with antiproton yields measured by E878 as a function of centrality indicate a large antihyperon-to-antiproton ratio in central collisions.'
address:
- ' Physics Department, Yale University, New Haven, CT 06520, USA'
- ' Univ Bari-BNL-UCLA-UC Riverside-Iowa State-Univ Mass-MIT-Penn State-Purdue-Vanderbilt-Wayne State-Yale'
author:
- 'Zhangbu Xu$\dag$ for the E864 Collaboration$\ddag$'
title: Search for positively charged strangelets and other related results with E864 at the AGS
---
\#1\#2 epsf
Introduction
============
Strangelets are small color-singlet hadrons with baryon number $A>1$ which contain about equal numbers of u, d and s quarks. Many of the theoretical calculations[@schaffner; @chin; @madsen] based on the phenomological bag model suggest that Strange Quark Matter (SQM) might be metastable or even absolutely stable . Ultimately only experiments can prove their existence or nonexistence.
Relativistic Heavy Ion Collisions create hot and dense nuclear (or quark) matter, which offer a unique opportunity to search for strangelets in accelerator facilities[@sandweiss]. The experimental signature of strangelets used to date is their low charge-to-mass ratio. This is based on the fact that strangelets have one more quark flavor with negative charge $(-{1\over 3})$ than the normal 2-flavor nuclear matters. Recent investigations[@schaffner] suggest that strangelets produced in heavy ion collisions might even be highly negatively charged.
There are three classes of strangelet production models in Heavy Ion Collisions: distillation of Quark-Gluon Plasma (QGP) scenario[@liu; @greiner], coalescence model[@baltz], and thermal model[@braun]. The QGP distillation scenario assumes that the strangelet formation is a two-step process. It is creation of QGP followed by QGP decay into strangelet. These two processes are in the speculative stage[@greiner2]. Some estimations are available in reference [@liu; @crawford] in which the QGP is assumed to break up into small droplets before distillation. Coalescence models calculate the production of hyperfragments. The hyperfragments are formed from individual nucleons and they subsequently decay to strangelets provided that strangelets are more stable. A thermal model[@braun] need not discuss specific reaction mechanisms, since its major ingredients are thermal and chemical equilibrium.
To prove the existence of strangelet one needs to find the strangelet. But to prove the nonexistence of strangelet, we need to know the production mechanism well. Unfortunately, even the formation of QGP is yet to be seen. The measurement of antihyperon-to-antiproton ratio will help us understand the system better even within the framework of hot and dense nuclear matter while large antihyperon-to-antiproton ratio[@lajoie] is consistent with QGP formation. Measurements of production of light nuclei are keys to understanding the possible strangelet production via coalescence[@baltz] and they also provide other important information about the colliding system[@heinz].
This contribution presents recent results in the search for positively charged strangelet with E864 and discusses the significance of the results and their implications with the help of light nuclei results. The antihyperon-to-antiproton ratios or $({\bar{\Lambda}+\bar{\Sigma^{0}}+1.1\bar{\Sigma^{+}}})/\bar{p}$ vs. centrality implied from $\bar{p}$ measurements by E864 and E878 are presented as well.
E864 apparatus
==============
E864 is an open geometry, high data rate spectrometer designed to search for strangelets[@barish; @e864slet; @e864slet2] and measure the production of many particle species in high energy nucleus-nucleus collisions at the AGS. We can have two relatively independent mass measurements with the E864 apparatus(Fig \[fig:E864\]) : the tracking system and the hadronic calorimeter. They identify particles and reject background powerfully with the confirmation of each other. The tracking system has two dipole analyzing magnets (M1 and M2) followed by three hodoscope planes(H1, H2 and H3). There are two straw stations (S2 and S3), each with three close-packed double planes. Each scintillating hodoscope plane has 206 vertical slats, and there are more than 1000 straw tubes in each straw plane. The tracking system measures momentum, charge and velocity ($\beta$) of charged particles with a mass resolution of about 3% in the region of interest. Charge misidentification is less than 1 in 10 billion due to three redundant charge measuremnts. There is an additional mass measurement from hadronic calorimeter[@calo] which has good energy ($\Delta E/{\sqrt{E}}={0.344/{\sqrt{E}}}+0.035$) and time ($\sigma_{t} \simeq 400ps$) resolutions. It is made of 754 towers of scintillating fibre embedded lead. A vacuum tank is along the beam line to reduce the background from beam particles interacting with air. The total length of the apparatus is about 28 meters. The incident beam is 11.5GeV/c Au beam on Pt target with 60% interaction length. The spectrometer fields of M1 and M2 are set to their highest field +1.5T to reduce background (sweep out high $Z/A$ particles) and achieve best tracking resolution for positively charged strangelets.
The trigger consists of good beam definition, a multiplicity requirement[@haridas] and a level II high mass trigger[@jhill]. Only events of the 10% most central collisions are collected since strangelets are most likely to be produced in central collisions. High sensitivity and open geometry are keys of E864. We achieve the high sensitivity by a high-mass level II trigger – Late-Energy Trigger (LET). There is a two-dimensional (Energy and TOF) programmable lookup table for every calorimeter channel to setup for different topics. The LET rejects those events without any high mass candidate and achieve a rejection factor of about 70 while maintaining good efficiency (${}^{>}_{\sim}85\%$) for high masses. There are about 200 million LET events or 13 billion 10% most central events sampled in the whole data set for positively charged or neutral strangelet searches.
Offline cuts are used to further refine the candidate selection. For any particle, tracking mass and calorimeter mass have to be consistent with each other. These confirmations are performed by energy and TOF consistency cuts. A upper limit cut on particle velocity ($\beta$) is used to maintain good mass resolution and clean up the background since $\sigma_{m}/m$ scales with $\gamma^{2}$. Tracking $\chi^{2}$ and shower quality cuts are studied extensively.
Results
=======
Strangelet searches
-------------------
We have conducted the full analysis of 1996/97’s ’+1.5T’ data set for strangelets with charge=+1, +2, +3.
There are two classes of background which we have to deal with. The background in the tracking system is the result from multiple scattering and neutron-proton charge exchange interactions. Fluctuations of energy measurements and overlapping showers are background in the calorimeter measurements. These two background sources are relatively independent. Simulations showed that we are able to achieve a rejection level of $<10^{-10}$ per central interaction[@prop].
With all the cuts we used in analysis of data from the previous run[@e864slet; @scott; @nagle], we observe 3 candidates with mass between 5 and 10 $Gev/c^{2}$. But when we check for additional evidence of double interactions using the calorimeter, we conclude that the 3 candidates are probably normal particles scattered and their showers in calorimeter are contaminated by a second interaction[@xzb; @marcelo]. The efficiency of this additional cut is about 85%. In fig \[fig:q2a6\], the mass measured from the tracking system of charge=+2 is plotted. It is clear that consistency cuts between tracking and calorimeter measurements and tighter $\beta$ cut do clean up the spectrum. We clearly see a ${}^{6}He$ peak but we do not see any ${}^{8}He$ or exotic particles. We conclude that there is no candidate with $1.0<y<2.1$ and $m>7GeV/c^{2}$. We extend our analysis to include $|Z|\geq3$ particles and indeed we are able to see a clear ${}^{6}Li$ peak. With a loose $\beta<0.985$ cut, no particle with $m>8GeV/c^{2}$ is found. However, when the limits are calculated, a tighter cut ($\beta<0.972$) is used. An analysis with the looser cut is being carried out and should give slightly better limits.
Because of the finite acceptance, we choose the following production model to calculate the sensitivities and limits : $${{dN}\over{dydp_{t}}} \propto p_{t}\exp{[-{{2p_{t}}\over{<p_{t}>}}]}
\exp{[-{{(y-y_{cm})^{2}}\over{2\sigma_{y}^{2}}}]}$$ where $<p_{t}>=0.6\sqrt{A}GeV/c$ is the mean transverse momentum, $y_{cm}=1.6$ is the center-of-mass rapidity and $\sigma_{y}=0.5$ is the standard width of the rapidity distribution of the produced strangelet. It is worth pointing out that because of the large rapidity and transverse momentum coverage, the results depend weakly on the model chosen. E864 is capable of detecting weak-decay strangelets with lifetime of about 50ns or greater.
We compute the limits at 90% Confidence Level (C.L.) for strangelets of Z=+1, +2 and +3 with mass between about 10 to 50$Gev/c^{2}$. These results represent the best limits at AGS energies[@sandweiss]. Fig \[fig:limits\] shows the preliminary limits together with our previous results[@e864slet] and predictions from coalescence, QGP distillation and thermal model. Interpretations and implications of the results will be discussed in next section.
Light nuclei
------------
In addition to searching for positively charged strangelet, we measure the production of light nuclei near midrapidity up to $A=6$ in this data set. Fig \[fig:q2a6\] shows preliminary measurements of the yields of ${}^{6}He$ ($J^{P}=0^{+}$) and ${}^{6}Li$ ($J^{P}=1^{+}$). Together with the analyses of other E864 data sets[@pope; @nigel], we observe an exponential decrease of the yields of particle production as a function of nuclear number A near midrapidity (center of mass of the system). That is $${{dN_{A}}\over{dydp_{t}}}|_{{p_{t}\simeq0},y\simeq1.9} \propto (1/50)^{A}$$ This means that the penalty factor of adding a nucleon to a nuclear cluster is about 50 for all nuclei up to $A=6$ near midrapidity at $p_{t}\simeq 0$. From this observation, one can make a naive model of the particle production, namely: $$\label{eq:na}
N_{A} \simeq 157\times ({1\over 50})^{A-1}\times{\lambda_{s}}^{|S|}$$ where 157 is the total number of initial protons, $\lambda_{s}$ is the strangeness penalty factor [@baltz; @dover] and $|S|$ is the total strangeness in the ‘cluster’. This naive model assumes complete stopping and does not take into account the change of the spectrum with mass and the spin factor. But it describes the trend of the production level.
$\bar{p}$ production vs. centrality
-----------------------------------
E864 has previously measured $\bar{p}$ production in 10% most central collisions[@lajoie; @john; @nagle]. Additional mininum bias data were taken in the 1996/97 run with LET trigger. When the measured $\bar{p}$ production shown in Fig \[fig:pbar\] is compared with that from E878[@e878; @lajoie] in different centralities, we see a strong centrality dependence of the level of disagreement between the two experiment measurements. This can be explained by the enhancement of antihyperon production in central collisions, because E864 accepts all the $\bar{p}$’s from antihyperon (${\bar{\Lambda}, \bar{\Sigma^{0}}, \bar{\Sigma^{+}}}$) decays while E878 accepts only a small fraction of them . Detail comparisons can be found in [@lajoie; @john; @nagle]. From the difference of the acceptance and difference of the detected $\bar{p}$’s, we compute the antihyperon-to-antiproton ratios $({\bar{\Lambda}+\bar{\Sigma^{0}}+1.1\bar{\Sigma^{+}}})/\bar{p}$ in each centrality bin, where acceptance and decay branching ratios are properly taken into account. In Fig \[fig:pbar\], we plot the low edge of the ratios at 98% C.L. together with the most probable values of the ratios as a function of centrality. It reaches the most probable value of 3.5 and is higher than 2.3 at 98% C.L. in central collisions. The interpretation is quite consistent since the ratio in peripheral collisions is consistent with that of p+p collisions at similar energy. We are looking forward to the forthcoming direct $\bar{\Lambda}$ measurements [@creig] at the AGS.
Interpretation of Strangelet Limits
===================================
Coalescence and thermal models
------------------------------
Coalescence[@baltz] and thermal[@braun] models predict the production of hyperfragments. It is noticed that [@baltz] [@braun] overpredict the production of light nuclei [@pope; @nigel] and therefore we expect overpredictions of hyperfragment/strangelet production from these models.
From and our most probable sensitivity at low mass range ($1.4\times10^{-8}/2.3$), we can translate our limits per central collision to limits in baryon number and strangeness content[@e864slet2]: $$A+(0.41\times{|S|}) < 7.1
\label{eq:as}$$ where $\lambda_{s}=0.2$[@dover]. For any combination of A and S satisfying , we have the sensitivity for strangelets produced by coalescence if they exist.
Another important message from is that it is very unlikely to produce a large strangelet in a normal medium. For example, production of an $A=10$ and $|S|=0$ object will be at the level of $8\times10^{-14}$ per central collision. On the other hand, this implies that positively charged strangelets can be a possible clean signature of QGP formation[@greiner] which will not be confused by normal nuclei or strangelets produced via ’coalescence’.
Distillation of QGP
-------------------
Our limits largely rule out the predictions from [@liu] and are at the same order of or below predictions from [@crawford] for $m\sim10$. However, predictions listed from [@crawford] are from their early predictions in which a QGP is assumed to happen in every central collision at the AGS energy and an optimistic mass formula is used. Further improvement in the mass formula[@crawford] indicates that strangelets with low A (${}^{<}_{\sim}20$) are unstable. Our limits are not sensitive enough to challenge the region where one might expect stability according to this model.
Our limits do constrain the sequence of QGP production followed by QGP distillation into strangelet in a model independent way[@e864slet]. For large strangelets which are predicted to be more stable , our data restrict these processes at the 90% confidence level as follows: $${\rm B}({\rm Au+Pt} \to {\rm QGP}) \times {\rm B}({\rm QGP} \to
{\rm Strangelet})\; ^{<}_{\sim} \; 8 \times 10^{-9},$$
Conclusions and future prospects
================================
In summary, we have found no evidence of positively charged strangelets produced in 11.5GeV/c per nucleon Au+Pt collision and set a 90% confidence level upper limit of about $8\times10^{-9}$ per 10% most central collision for $|Z|=+1,+2,+3$ strangelets over a wide mass range and with proper lifetimes of ${}^{>}_{\sim}50ns$. This represents the best sensitivity at the AGS energies. We have measured production of $A=6$ nuclei near midrapidity and found that the penalty factor of adding a nucleon to a cluster stays at about 50. Comparisons of $\bar{p}$ measurements between E864 and E878 as a function of centrality suggest an enhanced antihyperon-to-antiproton ratio in central collisions.
In the near future, we will combine two data sets (’+1.5T’ and ’-.75T’) together to improve our sensitivities.
The strangeness penalty factor for adding hyperon to a nuclear cluster has never been measured in high energy heavy ion collisions. We have collected about 250 million LET events ( 12 billion central events sampled). The unique high sensitivity and open geometry give us opportunity to measuring the production of ${}^{3}_{\Lambda}H$ and ${}^{4}_{\Lambda}H$ through their two-body mesonic decay channel[@h4l]. There are also data available for studying nuclear resonant states, such as ${}^{5}Li$, ${}^{5}He$. With these new studies, we hope to better understand the dynamics of the high energy heavy ion collisions and provide more information for future experiments in searching for strangelets via coalescence. We gratefully acknowledge the excellent support of the AGS staff. This work was supported by grants from the U.S. Department of Energy’s Hight Energy and Nuclear Physics Divisions, the U.S. National Science Foundation and the Istituto di Fisica Nucleara of Italy.
Reference {#reference .unnumbered}
=========
[10]{} Schaffner-Bielich J 1997 [**]{}[**23**]{} p 2107 Chin S A and Kerman A K 1979 [**]{}[**43**]{} p 1292 Madsen J 1995 [*Strangeness in Hadronic Matter (S ’95) ed J Rafelski (New York: AIP)*]{} p 32 Sandweiss J [*These proceedings*]{} Barish K N 1997 [**]{}[**23**]{} p 2127 Armstrong T A 1997 [**]{}[**79**]{} p 3612 Armstrong T A 1997 [*A*]{}[**625**]{} p 494 Liu H and Shaw G L 1984 [*D*]{}[**30**]{} p1137 Greiner C 1987 [**]{}[**58**]{} p 1825 Baltz A 1994 [*B*]{}[**325**]{} p 7 Greiner C [*These proceedings*]{} Kapusta J 1995 [*C*]{}[**52**]{} p 2725 Crawford H 1992 [*D*]{}[**45**]{} p 857 1993 [*D*]{}[**48**]{} p 4474 Braun-Munzinger P and Stachel J 1995 [**]{} p L17 Armstrong T A 1997 [**]{}[**79**]{} p 3351 (references therein) Armstrong T A 1998 [*A*]{} [**406**]{} p 227 Haridas P 1997 [*A*]{} [**385**]{} p 412 Hill J (to be published) Xu Z [*PhD Thesis, Yale University*]{} , in preparation Sandweiss J 1991 [*E864 proposal: Measurements of Rare Composite Objects and High Sensitivity Searches for Novel Forms of Matter Produced in High Energy Heavy Ion Collisions*]{} Coe S D 1997 [*PhD Thesis, Yale University*]{} Nagle J L 1997 [*PhD Thesis, Yale University*]{} Munhoz M 1998 [*These proceedings*]{}; Van Buren G 1998 [*These proceedings*]{} Pope J K 1996 [*Proceedings of Heavy Ion Physics at the AGS (HIPAGS) ed C A Pruneau WSU-NP-96-16*]{} p 119 1997 [PhD Thesis, Yale University]{} George N 1998 [*PhD Thesis, Yale University, in preparation*]{} Dover C 1995 [*A*]{} [**590**]{} p 333 Heinz U 1998 [*These proceedings*]{} Lajoie J G 1997 [*PhD Thesis, Yale University*]{} Beavis D 1995 [**]{} [**75**]{} p 3078 Ogilvie C 1998 [*These proceedings*]{} Xu Z
|
---
abstract: 'We introduce a new shape-constrained class of distribution functions on ${\mathbb{R}}$, the [*bi-$s^*$-concave*]{} class. In parallel to results of [@DuembgenKW:2017] for what they called the class of bi-log-concave distribution functions, we show that every $s-$concave density $f$ has a bi-$s^*$-concave distribution function $F$ and that every bi-$s^*$-concave distribution function satisfies $\gamma (F) \le 1/(1+s)$ where finiteness of $$\gamma (F) \equiv \sup_{x} F(x) (1-F(x)) \frac{| f'' (x)|}{f^2 (x)},$$ the Csörgő - Révész constant of $F$, plays an important role in the theory of quantile processes on ${\mathbb{R}}$.'
address:
- ' Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322, USA. '
- ' Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322, USA. '
author:
- Nilanjana Laha
- 'Jon A. Wellner'
bibliography:
- 'BiS-concave.bib'
title: 'Bi-$s^*$-concave distributions'
---
[^1]
Introduction: the bi-log-concave class {#sec:intro}
======================================
[@DuembgenKW:2017] investigated a shape constraint they called “bi-log-concavity” for distribution functions $F$ on ${\mathbb{R}}$: a distribution function $F$ is [*bi-log-concave*]{} if both $x \mapsto \log F(x) $ and $x \mapsto \log (1-F(x))$ are concave functions of $x$. They noted that [@MR2213177] showed that any log-concave distribution with density $f$ has a bi-log-concave distribution function $F$, but that the inclusion is proper: there are many bi-log-concave distributions that are not log-concave, and in fact bi-log-concave distributions may not be unimodal. [@DuembgenKW:2017] proved the following interesting theorem characterizing the class of bi-log-concave distributions.
First a bit of notation: $$J(F) \equiv \{ x \in {\mathbb{R}}: \ 0 < F(x) < 1 \} .$$ A distribution function $F$ is non-degenerate if $J(F) \ne \emptyset $.
\[thm:DKWthm\] (DKW, 2017) For a non-degenerate distribution function $F$ the following four statements are equivalent:\
(i) $F$ is bi-log-concave.\
(ii) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with derivative $f=F^{\prime}$ such that $$\begin{aligned}
F(x+t) \left \{ \begin{array}{l} \le F(x) \exp \left ( \frac{f(x)}{F(x)} t \right ) \\ \ge 1 - (1-F(x)) \exp \left ( - \, \frac{f(x)}{(1-F(x))} t \right )
\end{array} \right .\end{aligned}$$ for all $x \in J(F)$ and $t \in {\mathbb{R}}$.\
(iii) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with derivative $f = F^{\prime}$ such that the hazard function $f/(1-F)$ is non-decreasing and reverse hazard function $f/F$ is non-increasing on $J(F)$.\
(iv) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with bounded and strictly positive derivative $f = F^{\prime}$. Furthermore, $f$ is locally Lipschitz-continuous on $J(F)$ with $L^1-$derivative $f^{\prime} = F^{\prime \prime} $ satisfying $$\begin{aligned}
\frac{-f^2}{1-F} \le f^{\prime} \le \frac{f^2}{F} .\end{aligned}$$
An important implication of (iv) of Theorem \[thm:DKWthm\] is that the inequalities can be rewritten as follows: $$\begin{aligned}
-1 \le -F(x) \le F(x) (1-F(x)) \frac{f^{\prime} (x)}{f^2 (x)} \le 1-F(x) \le 1 .\end{aligned}$$ This implies that the bi-log-concave family of distributions satisfies $$\begin{aligned}
\gamma (F) \equiv \sup_{x \in J(F)} F(x) (1-F(x)) \frac{ | f^{\prime} (x) |}{f^2 (x)} \le 1.
\label{GammaFboundedbyOneForLogConcaveF}\end{aligned}$$ The parameter $\gamma (F)$ arises in the study of quantile processes and transportation distances between empirical distributions and true distributions on ${\mathbb{R}}$: see e.g. [@MR0501290], [@MR838963; @MR3396731] Chapter 18, page 643, [@Bobkov-Ledoux:2017], and [@MR2121458].
Questions and extensions: the bi$-s^*$-concave class {#sec:QuestAndExt}
====================================================
This immediately raises several questions:
Question 1
: What about distributions in classes larger than the log-concave class? In particular what happens for the $s-$concave classes described by [@MR0404559]? See [@MR954608], and [@MR0450480].
Question 2
: Is there a class of bi-$s^*$-concave distributions with the property that if $f$ is $s-$concave, then $F$ is bi-$s$-concave (or perhaps bi-$s^*$-concave with $s^*$ related to $s$)?
Question 3
: Is there a class of bi-$s^*$-concave distributions with a theorem analogous to Theorem \[thm:DKWthm\] with an analogue of Theorem \[thm:DKWthm\](iv) implying that $\gamma(F) $ is bounded by some function of $s$ for all bi-$s^*$-concave distributions $F$?
We provide positive answers to Questions 1-3 when $s \in (-1,\infty)$, beginning with the following definition of [*bi-$s^*$-concavity*]{} of a distribution function $F$.
\[defn:BiSConcaveDefn\] For $s \in (-1,\infty]$ we let $s^* \equiv s/(1+s) \in (-\infty, 1]$. For $s \in (-1,0)$ we say that a distribution function $F$ on ${\mathbb{R}}$ is bi$-s^*-$concave if both $x\mapsto F^{s^*} (x) $ and $x \mapsto (1-F(x))^{s^*}$ are convex functions of $x \in J(F)$. For $s \in (0,\infty)$ we say that $F$ is bi$-s^*-$concave if $x\mapsto F^{s^*} (x) $ is concave for $x \in (\inf J(F), \infty)$ and $x \mapsto (1-F(x))^{s^*}$ is concave for $x \in (-\infty, \sup J(F))$. For $s = 0$ we say that $F$ is bi-$0$-concave (or bi-log-concave) if both $x \mapsto \log F(x)$ and $x \mapsto \log (1-F(x))$ are concave functions of $x \in J(F)$. Note that this definition of bi-log-concavity is equivalent to the definition of bi-log-concavity given by [@DuembgenKW:2017].
To briefly explain this definition, recall that a density function $f$ (or just a non-negative function $f$) on ${\mathbb{R}}$ (or even on ${\mathbb{R}}^d$) is $s-$concave for $s < 0$ if $f^s$ is convex, while $f$ is $s-$concave for $s>0$ if $f^s$ is concave on $J(F)$. Furthermore, from the theory of concave measures due to [@MR0404559], [@MR0450480], and [@MR0428540], if $f$ is $s-$concave, the probability measure $P$ on $({\mathbb{R}},\mathcal{B} )$ defined by $P(B) = \int_B f(x) dx$ for Borel sets $B$, is $t - $concave with $t = s/(1+s) \equiv s^*$ if $s>-1$; see [@MR954608] for an introduction, and [@MR1898210] for a comprehensive review. From the basic theory of Borell, Brascamp and Lieb, and Rinott, it follows easily that if $f$ is $s-$concave with $s \in (-1,\infty]$, then $F$ and $1-F$ are $s^*-$concave; i.e. the distribution function $F$ corresponding to $f$ is bi$-s^*-$concave. This proof, as well as a simpler calculus type proof assuming that derivatives exist, is given in Section \[sec:SconcavityImpliesBsConcavity\]. The same argument also establishes the corresponding implication in the log-concave case since, in the log-concave case, $s=0$ and $s^* = 0$ as well. In Section \[sec:Theorem1s\] we provide a complete characterization of the class of bi-$s^*$-concave distributions on ${\mathbb{R}}$, answering Question 3.
For the moment we illustrate the definition with several examples.
\[exmpl:ex1\] Suppose $f_r$ is the $t-$density with $r >0$ “degrees of freedom”: $$f_r (x) = \frac{C_r}{\left (1 + \frac{x^2}{r} \right )^{(r+1)/2}} \ \ \ \mbox{for} \ \ x \in {\mathbb{R}}.$$ Here $C_r = \Gamma ((r+1)/2)/(\sqrt{\pi} \Gamma (r/2))$. It is well-known (see e.g. [@MR0404559]) that $f_r \in \mathcal{P}_s$, the class of $s-$concave densities, if $s \le - 1/(1+r)$. Note that $s$ takes values in $(-1,0)$ since $r \in (0,\infty)$. From the Borell-Brascamp-Lieb inequality we guess that the “right” transformation $h$ of $F$ and $1-F$ to define the Bi$-s^*-$concave class is $h(u) = u^{s^*} = u^{s/(1+s)}$ where $s = -1/(1+r)$, the largest possible value of $s$. This leads directly to Definition \[defn:BiSConcaveDefn\]. Note that $s^*$ in the Borell-Brascamp-Lieb inequality is well-defined since $s>-1$. Since $s = -(1+r)^{-1}$ we see that we can take $s^* = s/(1+s) = -1/r$ for the $t_r$ family. Then we want to know if $F_r^{-1/r}$ and $(1-F_r)^{-1/r}$ are convex. Direct computation shows that these are convex functions of $x$. Plotting these for $r \in \{1/2, 1, 4 \}$ we see that they are indeed convex. Moreover we find that $\gamma (F_r) = 1 + 1/r = 1/(1+s)$; this agrees nicely with the log-concave and bi-log-concave picture when $r = \infty$ (so $\gamma (F_{\infty} ) = \gamma (N(0,1)) = 1$), and it yields distributions with arbitrarily large values of $\gamma (F)$ by considering $\gamma (F_r)$ with $r $ arbitrarily small. Note, in particular, that this yields $\gamma (F_1) = \gamma (Cauchy) = 2$. Also note that this suggests the conjecture $\gamma (F) \le 1/(1+s) $ for all bi-$s^*$-concave distribution functions $F$ where $1/(1+s)$ varies from $1$ to $\infty$ as $s$ varies from $0$ to $-1$.
\[exmpl:ex2\] Suppose that $f_{a,b} $ is the family of $F-$distributions with “degrees of freedom” $a>0$ and $b>0$. (In statistical practice, if $T$ has the density $f_{a,b}$, this would usually be denoted by $T\sim F_{b,a}$ where $b$ is the “numerator degrees of freedom” and $a$ is the “denominator degrees of freedom”. ) The density is given by $$\begin{aligned}
f_{a,b} (x) = C_{a,b} \frac{x^{(b/2) -1}}{\left ( a + b x \right )^{(a+b)/2} } \ \ \ \mbox{for} \ \ x \ge 0 .\end{aligned}$$ (In fact, $C(a,b) = a^{a/2} b^{b/2} / \mbox{Beta} (a/2,b/2)$, and $f_{a,b} (x) \rightarrow g_b (x)$ as $a\rightarrow \infty$ where $g_b$ is the Gamma density with parameters $b/2$ and $b/2$.) It is well-known (see e.g. [@MR0404559]) that $f_{a,b} \in \mathcal{P}_s$, the class of $s-$concave densities, if $s \le - 1/(1 + \frac{a}{2} )$ when $a \ge 2$ and $b\ge 2$. This implies that $s \in [-1/2, 0)$, and the resulting $s^* = s/(1+s) $ is in $[-1,0)$. By Proposition \[prop:1s\] it follows that $F^{s^*}$ and $(1-F)^{s^*}$ are convex; i.e. $F$ and $1-F$ are $s^*-$ concave. This is confirmed by numerical computation.
\[exmpl:ex3\] Suppose that $f_{a,b} (x) \equiv f(x; a,b) = (a/b)(x/b)^{-(a+1)} 1_{[b,\infty)} (x) $, the Pareto distribution with parameters $a$ and $b$. In this case $f_{a,b} $ is $s-$concave for each $s \le -1/(1+a)$. Thus we take $s = -1/(1+a) \in (-1,0) $ for $a \in (0, \infty)$. Note that $s^* = s/(1+s) = -1/a$. Note that $f^s_{a,b} (x) = (x/b)\cdot (b/a)^{1/(1+a)}$ is certainly convex. Furthermore, it is easily seen that $$CR_R (x) \equiv (1-F(x)) \frac{f'(x)}{f^2 (x) } = 1 - s^* = 1+1/a \ \ \ \mbox{for all} \ \ x > b.$$ Thus the Pareto distribution is analogous to the exponential distribution in the log-concave case in the sense that it is exactly on the convex [*and*]{} concave boundary.
\[exmpl:ex4\] Suppose that $f_r (x) = C_r (1 - x^2/r)^{r/2} 1_{[-\sqrt{r} , \sqrt{r}]} (x)$ where $r \in (0,\infty)$. Here $$C_r =\Gamma ((3+r)/2)/ (\sqrt{\pi r} \Gamma (1+r/2)) .$$ Note that $f_r$ is $s-$concave with $s = 2/r \in (0,\infty)$ since $f_r^{2/r} (x) = C_r^{2/r}(1-x^2/r) 1_{[-\sqrt{r},\sqrt{r}]} (x)$ is concave. As $r\rightarrow \infty$ it is easily seen that $f_r (x) \rightarrow (2\pi)^{-1/2} \exp (-x^2/2)$, the standard normal density. Thus $r=\infty$ corresponds to $s=0$. On the other hand, $$\begin{aligned}
g_r (x) & \equiv &\sqrt{r} f_r (\sqrt{r} x ) = \sqrt{r} C_r (1-x^2)^{r/2} 1_{[-1,1]} (x)\\
& \rightarrow & 2^{-1} 1_{[-1,1]} (x) \ \ \mbox{as} \ r \rightarrow 0.\end{aligned}$$ Thus $r=0$ corresponds to $s = + \infty$.
![The $s-$concave densities $g_r$ of Example \[exmpl:ex4\] with $s = 2/r \in (0,\infty)$: $s=1/8$, magenta; $s=1/2$, green; $s= 2$, black; $s=4$, blue; $s=8$, red; $s=16$, purple.[]{data-label="fig:fig0sConPs"}](Plots/s-concaveWsPositive.pdf){width="\linewidth" height="5.5cm"}
$s$-concavity of $f$ implies $s^*$-concavity of $F$ and $1-F$ {#sec:SconcavityImpliesBsConcavity}
=============================================================
Motivated by Examples \[exmpl:ex1\]-\[exmpl:ex4\], we first give an extension of the log-concave preservation result of [@MR2213177]; also see Lemma 3 of [@MR1637480].
\[prop:BagnoliBergstrom\] (Bagnoli and Bergstrom; An; Barlow and Proschan)\
If $f$ is log-concave then both $F$ and $1-F$ are log-concave; i.e. $\log F$ and $\log (1-F)$ are concave.
\[prop:1s\] If $f$ is $s-$concave with $s \in (-1,\infty)$, then both $F $ and $1-F$ are $ s^* = s/(1+s)$ concave; i.e. $F^{s^*} $ and $(1-F)^{s^*}$ are convex when $s<0$; and $\log F $ and $\log (1-F)$ are concave when $s=0$; and $F^{s^*} $ and $(1-F)^{s^*}$ are convex when $s>0$. Equivalently, $F$ is bi-$s^*$-concave.
\[rem:1\] Results related to Proposition \[prop:BagnoliBergstrom\] have a long history in reliability theory and econometrics. [@MR0438625] (Lemma 5.8, page 77) showed that if $f$ is log-concave (i.e. $PF_2$, or Polya frequency of order $2$), then $f/(1-F)$ is non-decreasing (or “Increasing Failure Rate” in their terminology); they also noted that the IFR property is equivalent to $1-F$ being log-concave. Their proof of the IFR property using the equivalence of log-concavity of $f$ and $f \in PF_2$ is delightfully short and does not rely on existence of $f^{\prime}$. [@MR1637480] also proves Proposition \[prop:BagnoliBergstrom\] using $PF_2$ equivalences to log-concavity without requiring existence of $f^{\prime}$. The simple “calculus based” proof given here and taken from [@MR2213177], which relies on the classical “second-order conditions” for convexity (see e.g. [@MR2061575], section 3.1.4), was apparently given by [@Dierker:1991], but is likely to have a much longer history.
In the modern theory of convexity, Proposition \[prop:BagnoliBergstrom\] is an immediate consequence of the results of [@MR0404557]. As we will see in the second proof, Proposition \[prop:1s\] is an immediate consequence of the results of [@MR0404559], [@MR0450480], and [@MR0428540].
[**Proof of Proposition \[prop:BagnoliBergstrom\]**]{}\
[**First Proof, assuming $f^{\prime}$ exists:**]{}\
Fact 1: First note that $f$ is log-concave if and only if $f'/f$ is non-increasing.\
Fact 2: Note that $F(x) = \int_a^x f(y) dy$ is log-concave if and only if $f'(x) F(x) - f^2(x) \le 0$. To see this, note that $$\begin{aligned}
&& (\log F)^{\prime} (x) = \frac{f}{F} (x), \ \ \mbox{and}\\
&& (\log F)^{\prime \prime} (x) = \frac{f^{\prime \prime}}{F} - \frac{f^2}{F^2} = \frac{f' F - f^2}{F^2} \le 0 .\end{aligned}$$ Now if $f$ is log-concave we can use fact 1 to write $$\begin{aligned}
\frac{f^{\prime}}{f} (x) F(x)
& = & \frac{f^{\prime}}{f} (x)\int_a^x f(y) dy \\
& \le & \int_a^x \frac{f^{\prime} (y)}{f(y)} f(y) dy = \int_a^x f^{\prime} (y) dy \\
& = & f(x) - f(a) = f(x) .\end{aligned}$$ Rearranging this inequality yields $f^{\prime} (x) F(x) - f^2(x) \le 0$, and by Fact 2 we conclude that $F$ is log-concave. Note that this inequality also can be rewritten as $\frac{f^{\prime} (x)}{f^2 (x)} F(x) \le 1$, and hence we conclude that $$\begin{aligned}
\frac{f^{\prime} (x)}{f^2 (x)} F(x)(1-F(x)) \le 1-F(x) \le 1\end{aligned}$$ The argument for $1-F$ is analogous and yields the inequality $\frac{f^{\prime} (x)}{f^2 (x)} (1-F(x)) \ge -1$, and hence we conclude that $$\begin{aligned}
\frac{f^{\prime} (x)}{f^2 (x)} F(x)(1-F(x)) \ge -F(x) \ge -1\end{aligned}$$ Thus both $F$ and $1-F$ are log-concave, and $\gamma (F) \le 1$. $\Box$
[**Second Proof, general (without assuming $f^{\prime}$ exists):**]{} See the second proof of Proposition \[prop:1s\] below.
[**Proof of Proposition \[prop:1s\]**]{}\
[**First Proof, assuming $f^{\prime}$ exists:**]{}\
Suppose $s\in(-1,0)$; the proof for $s>0$ is similar.\
Fact 1-s: First note that $f$ is $s-$concave for $s<0$ if and only if $\varphi \equiv f^{s}$ is convex on $J(F)$, which is equivalent to $\varphi^{\prime}$ being non-decreasing. But we find $$\varphi^{\prime} (x) = (f^s)^{\prime} (x) = s f^{s-1} (x) f^{\prime} (x) = s f^s (x) (f^{\prime} (x) / f(x) ) .$$ Fact 2-s: Note that $F(x) = \int_a^x f(y) dy$ is $s^*-$concave for $\st<0$ if and only if $$(s^*-1) f^2 + F f^{\prime} \le 0\ \text{on }J(F).$$ To see this, note that for $x\in J(F)$ $$\begin{aligned}
( F^{s^*})^{\prime} (x) & = & s^* F^{s^*-1}(x) f(x) , \ \ \mbox{and}\\
(F^{s^*})^{\prime \prime} (x)
& = & s^* (s^*-1) F^{s^*} (x) \left ( \frac{f}{F} (x) \right )^2 + s^* \frac{f^{\prime}(x)}{F(x)} F^{s^*} (x) \\
& = & s^* \frac{F^{s^*}(x)}{F^2 (x)} \left \{ (s^*-1) f^2 (x) + F(x) f^{\prime} (x) \right \} \\
& \ge & 0\end{aligned}$$ if and only if (since $s^* = s/(1+s) < 0$) $$(s^*-1) f^2 (x) + F(x) f^{\prime}(x) \le 0.$$ Now if $f$ is $s-$concave and $x\in J(F)$ we can use fact 1-s to write $$\begin{aligned}
s f^s (x) \frac{f^{\prime}}{f} (x) F(x)
& = & s f^s (x) \frac{f^{\prime}}{f} (x)\int_a^x f(y) dy \\
& \ge & \int_a^x s f^s (y) \frac{f^{\prime} (y)}{f(y)} f(y) dy = \int_a^x s f^s (y) f^{\prime} (y) dy \\
& = & \frac{s}{s+1} \left ( f^{s+1} (x) - f^{s+1}(a)\right ) = \frac{s}{s+1} f^{s+1} (x) .\end{aligned}$$ Rearranging this inequality (and noting that $s<0$) yields $(s^*-1) f^2 + F f^{\prime} \le 0$, and by Fact 2-s we conclude that $F$ is $s^*-$concave. Note that for $x \in J(F)$ this inequality can also be rewritten as $\frac{f^{\prime} (x)}{f^2 (x)} F(x) \le \frac{1}{1+s}$, and hence we conclude that $$\begin{aligned}
\frac{f^{\prime} (x)}{f^2 (x)} F(x)(1-F(x)) \le \frac{1}{1+s} (1-F(x)) \le \frac{1}{1+s} = 1 - s^* .\end{aligned}$$ The argument for $1-F$ is analogous and yields the inequality $\frac{f^{\prime} (x)}{f^2 (x)} (1-F(x)) \ge - \frac{1}{1+s} = - (1-s^*)$, and hence we conclude that $$\begin{aligned}
\frac{f^{\prime} (x)}{f^2 (x)} F(x)(1-F(x)) \ge - \frac{1}{1+s} F(x) \ge - \frac{1}{1+s}\end{aligned}$$ Thus both $F$ and $1-F$ are $s^*-$concave, and $\gamma (F) \le 1/(1+s)$. $\Box$
[**Proof of Proposition \[prop:1s\]**]{}\
[**Second Proof, general (without assuming $f^{\prime}$ exists):**]{} First some background and definitions:\
$\bullet$ Let $a,b \ge 0$ and $\theta \in (0,1)$. The generalized mean of order $s \in {\mathbb{R}}$ is defined by $$\begin{aligned}
M_s (a,b; \theta)
= \left \{ \begin{array}{l l} ((1-\theta)a^s + \theta b^s )^{1/s}, & \mbox{if} \ \pm s \in (0,\infty), \\
a^{1-\theta} b^{\theta} , & \mbox{if} \ s = 0,\\
\max\{ a,b \} , & \mbox{if} \ s = \infty, \\
\min\{ a,b \} , & \mbox{if} \ s = - \infty .
\end{array}
\right .\end{aligned}$$ $\bullet$ Let $(M,d)$ be a metric space with Borel $\sigma-$field $\mathcal{M}$. A measure $\mu$ on $\mathcal{M}$ is called [*$t-$concave*]{} if for nonempty sets $A,B \in \mathcal{M}$ and $0 < \theta < 1$ we have $$\mu_{*} ((1-\theta ) A + \theta B) \ge M_t ( \mu_{*} (A), \mu_{*} (B) ; \theta ) .$$ $\bullet $ A non-negative real-valued function $h$ on $(M,d)$ is called [*$s-$concave*]{} if for $x,y \in M$ and $0 < \theta <1$ we have $$h((1-\theta)x + \theta y) \ge M_s ( h(x), h(y) ; \theta ).$$ $\bullet$ Suppose $(M, d) = ( {\mathbb{R}}^k, | \cdot | )$, $k-$dimensional Euclidean space with the usual Euclidean metric and suppose that $f$ is an $s-$concave density function with respect to Lebesgue measure $\lambda $ on $\mathcal{B}_k$, and consider the probability measure $\mu$ on $\mathcal{B}_k$ defined by $$\mu (B) = \int_B f d \lambda \ \ \ \mbox{for all} \ \ B \in \mathcal{B}_k .$$ Then by a theorem of Borell (1975), Brascamp and Lieb (1976), and Rinott (1976), the measure $\mu$ is $s^*$ concave where $s^* = s/(1+ks)$ if $s \in (-1/k,\infty)$ and $s^* = 0$ if $s= 0$.\
$\bullet$ Here we are in the case $k=1$. Thus for $s \in (-1,\infty)$ the measure $\mu$ is $s^*$ concave: for $s \in (-1,\infty)$, $A, B \in \mathcal{B}_1$, and $0 < \theta < 1$, $$\begin{aligned}
\mu_{*} ( (1-\theta )A + \theta B) \ge M_{s^*} ( \mu_{*} (A) , \mu_{*} (B) ; \theta ) ;
\label{GeneralS-concaveMeasInequalityForR}\end{aligned}$$ here $\mu_{*}$ denotes inner measure (which is needed in general in view of examples noted by [@MR0260958]). With this preparation we can give our second proof of Proposition \[prop:1s\]: if $A = (-\infty, x]$ and $B = (-\infty, y]$ for $x,y \in J(F)$, it is easily seen that $$\begin{aligned}
(1-\theta) A + \theta B
& = & \{ (1-\theta ) x' + \theta y' : \ x' \le x , \ y' \le y \}\\
& \subset & \{ (1-\theta ) x' + \theta y' : \ (1-\theta )x' + \theta y' \le (1-\theta) x + \theta y \} \\
& = & (-\infty, (1-\theta)x + \theta y ].\end{aligned}$$ Therefore, with the second inequality following from (\[GeneralS-concaveMeasInequalityForR\]) $$\begin{aligned}
F((1-\theta) x + \theta y)
& = & \mu ((-\infty, (1-\theta)x + \theta y])\\
& \ge & \mu ( (1-\theta ) (-\infty,x] + \theta (-\infty, y]) \\
& \ge & M_{s^*} ( \mu ((-\infty,x]) , \mu ((-\infty, y]); \theta ) = M_{s^*} (F(x), F(y); \theta ) ;\end{aligned}$$ i.e. $F$ is $s^*-$concave. Similarly, taking $A = (x,\infty)$ and $B = (y,\infty) $ it follows that $1-F$ is $s^*-$concave.
Note that this argument contains a second proof of Proposition \[prop:BagnoliBergstrom\] when $s=0$. $\Box$
Bi-$s^*$-concave is (much!) bigger than $s-$concave {#sec:BiIsBigger}
===================================================
Here we note that just as the class of bi-log-concave distributions is considerably larger than the class of log-concave distributions (as shown by [@DuembgenKW:2017]), the class of bi$-s^*-$concave distributions is considerably larger than the class of $s-$concave distributions. In particular, multimodal distributions are allowed in both the bi-log-concave and the bi-s-concave classes.
\[exmpl:ex5\] ([@DuembgenKW:2017], pages 2-3) Suppose that $f$ is the mixture $(1/2)N(-\delta,1) +(1/2) N(\delta,1)$. [@DuembgenKW:2017] showed (numerically) that the corresponding distribution function $F$ is bi-log-concave for $\delta \le 1.34$ but not for $\delta \ge 1.35$. This distribution has a bi-modal density for $\delta = 1.34$.
\[exmpl:5s\] Now suppose that $f$ is the mixture $(1/2) t_1 (\cdot - \delta) + (1/2) t_1 (\cdot + \delta ) $ with $\delta >0$ where $t_r$ is the standard $t$ density with $r$ degrees of freedom as in Example \[exmpl:ex5\]. By numerical calculation, this density is bi$-s^*-$concave for $\delta = 1.4$, but fails to be bi$-s^*-$concave for $\delta = 1.5$. Again by numerical calculations the $t_1$ mixture density with $\delta =1.475 $ is bi-$(-1/2)^*$-concave, but with $\delta =1.48$ it is [*not*]{} bi-$(-1/2)^*$-concave; see Figure \[fig:fig4Mixed\].
The following plots illustrate the bounds in Section \[sec:Theorem1s\].
![The bi$-s^*-$concave $t_1$ mixture distribution function $F$ (black) for $\delta = 1.3$ with its convex upper bound $F_U$ (red) and concave lower bound $F_L$ (blue) defined by (\[FUpperSNeg\]) and (\[FLowerSNeg\]).[]{data-label="fig:fig1Mixed"}](Plots/Bounds-t1-mixed-DF-labelled.pdf){width="\linewidth" height="5.5cm"}
Upper and lower bounds for the density $f = F^{\prime} $ of $F$ follow from (iii) of Theorem \[thm:1s\]. These bounds are illustrated for the bi$-s^*-$concave distribution $t_1$ mixture with $\delta = 1.3$ in Figure \[fig:fig2Mixed\].
![The bi-$s^*$-concave $t_1$ mixture density function $f$ (black), $\delta = 1.3$, with its bi-$s^*$-concave upper bounds $F_U^{\prime}$ (red) and $F_L^{\prime}$ (blue) defined by (\[FprimeUpperSNeg\]) and (\[FprimeLowerSNeg\]).[]{data-label="fig:fig2Mixed"}](Plots/Bounds-t1-mixed-Density-Labelled.pdf){width="\linewidth" height="5.5cm"}
![The bi-$s^*$-concave $t_1$ mixture density function derivative $f^{\prime}$ (black) for $\delta = 1.3$ with its bi-$s^*$-concave upper (blue) and lower (red) bounds as given in (iv) of Theorem \[thm:1s\].[]{data-label="fig:fig3Mixed"}](Plots/Bounds-t1-mixed-DensityDeriv-Labelled.pdf){width="\linewidth" height="5.5cm"}
To get some feeling for what is happening with the Csörgő - Révész condition, Figure \[fig:fig4Mixed\] gives plots of the two functions $$\begin{aligned}
CR(x) & \equiv & F(x) (1-F(x)) \frac{f^{\prime} (x)}{f^2 (x)} ,\\
CR_{min} (x) & = & \min\{ F(x), 1-F(x) \} \frac{f^{\prime} (x)} {f^2 (x)} .
\end{aligned}$$
![The Csörgő-Révész functions $CR$ (blue) and $CR_{min}$ (red) for the mixed $t_1$ density with $\delta = 1.475$ .[]{data-label="fig:fig4Mixed"}](Plots/CR-fcns-t1-mixed-shift1-475-labelled.pdf){width="\linewidth" height="5.5cm"}
The bi-$s^*$-concave analogue of Theorem \[thm:DKWthm\] {#sec:Theorem1s}
=======================================================
Characterization theorem, bi-$s^*$-concave class {#subsec:CharThm}
------------------------------------------------
Now we can formulate the natural bi-$s^*$-concave analogue of Theorem \[thm:DKWthm\].
\[thm:1s\] Let $s \in (-1,\infty]$. For a non-degenerate distribution function $F$ the following four statements are equivalent:\
(i) $F$ is bi-$s^*$-concave.\
(ii) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with derivative $f=F^{\prime}.$ Moreover when $s\leq 0,$ $$\begin{aligned}
\label{th12}
F(x+t) \left \{ \begin{array}{l} \le F(x) \cdot \left ( 1+ s^* \frac{f(x)}{F(x)} t \right )_{+}^{1/s^*} \\
\ge 1 - (1-F(x)) \cdot \left (1 - s^* \frac{f(x)}{1-F(x)} t \right )_{+}^{1/s^*}
\end{array} \right. \end{aligned}$$ for all $x\in{\mathbb{R}}$ and $t \in {\mathbb{R}}$. When $s>0,$ $$\begin{aligned}
\label{th12n}
F(x+t)\begin{cases} \ \le F(x) \cdot \left ( 1+ s^* \frac{f(x)}{F(x)} t \right )_{+}^{1/s^*},& \text{ for}\ t\in(a-x,\infty)\\
\ \ge 1 - (1-F(x)) \cdot \left (1 - s^* \frac{f(x)}{1-F(x)} t \right )_{+}^{1/s^*},& \text{ for}\ t\in(-\infty,b-x)
\end{cases}\end{aligned}$$ for all $x \in J(F)$ .\
(iii) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with derivative $f = F^{\prime}$ such that the $s^*-$hazard function $ f/(1-F)^{1-s^*}$ is non-decreasing, and the reverse $s^*-$hazard function $ f/F^{1-s^*}$ is non-increasing on $J(F)$.\
(iv) $F$ is continuous on ${\mathbb{R}}$ and differentiable on $J(F)$ with bounded and strictly positive derivative $f = F^{\prime}$. Furthermore, $f$ is locally Lipschitz-continuous on $J(F)$ with $L^1-$derivative $f^{\prime} = F^{\prime \prime} $ satisfying $$\begin{aligned}
\label{4eq3}
-(1-s^*)\frac{f^2}{1-F} \le f^{\prime} \le (1-s^*)\frac{f^2}{F} .\end{aligned}$$ Recall that $s^* = s/(1+s) \in (-\infty, 1]$ and $(1-s^*) = 1/(1+s) \in [0,\infty)$. Alternatively, $$\begin{aligned}
-\frac{f^2}{1-F} \le (1+s) f^{\prime} \le \frac{f^2}{F} .\end{aligned}$$
This yields the following corollary extending (\[GammaFboundedbyOneForLogConcaveF\]) from $s=0$ to $s \in (-1,\infty]$.
\[cor:CR-gammaBound\] Suppose that $F$ is bi-$s^*$-concave for $s \in (-1, \infty]$. Then $$\begin{aligned}
\gamma (F) = \sup_{x \in J(F)} F(x) (1-F(x)) \frac{| f' (x)|}{f^2(x)} & \le & 1- s^* = \frac{1}{1+s} , \end{aligned}$$ and $$\begin{aligned}
\tilde{\gamma} (F) = \sup_{x \in J(F)} \min\{ F(x) , 1-F(x) \} \frac{| f' (x)|}{f^2(x)} & \le & 1- s^* = \frac{1}{1+s} .\end{aligned}$$
\[rem:SW-connection\] The three distribution functions $F$ considered by [@MR838963; @MR3396731] page 644 all involved log-concave densities with the resulting bound for $\gamma (F)$ being $1$. Theorem \[thm:1s\] and Corollary \[cor:CR-gammaBound\] give a rather complete description of how the values of $\gamma(F)$ and $\tilde{\gamma} (F)$ depend on the index $s^*$ of bi-$s^*$-concavity.
If $s=0$, the proof follows from Theorem \[thm:DKWthm\] of [@DuembgenKW:2017]. When $s=\infty$, $s^*=1$ and $1-s^* = 0$. In this case $f'=0$ almost everywhere (Lebesgue) and $f$ is a uniform density on $(a,b)$. When $s \in (0,\infty)$ the proof is essentially the same as for $s=0$ with only two minor modifications (in the proof of (i) implies (ii) and in the proof of (iii) implies (iv)); see the Appendix section \[sec:appendix\] for complete details. It remains to consider the case when $s\in(-1,0)$. Our proof closely parallels the proof for the case $s=0$ given by [@DuembgenKW:2017]. Throughout our proof we will denote $\inf J(F)$ and $\sup J(F)$ by $a$ and $b$ respectively. Notice that if $F$ is continuous, $J(F)=(a,b).$
Proof of (i) implies (ii): Since $F$ is bi-$s^*$-concave with $s^{*}<0$, $\psi=F^{1/\st}$ is convex on $J(F)$. Since $\psi(x)=1$ and $\infty$ for $x\geq\sup J(F)$ and $x\le\inf J(F)$ respectively, $\psi$ is convex on ${\mathbb{R}}.$ By the convex version of Lemma $6$ of [@DuembgenKW:2017] $\psi$ is continuous on the interior of $\{\psi<\infty\}.$ Therefore $\psi$ and hence $F$ is continuous on the interior of the set $\{F>0\}$ or $(a,\infty)$. Similarly, the $s^*$-concavity of $1-F$ implies continuity of $1-F$ on the interior of the set $\{1-F>0\}:=(-\infty,b)$ where $b:=\sup\{F<1\}$. However unless $a<b$, $F$ would be degenerate. Hence, $a<b$ and $F$ is continuous on ${\mathbb{R}}$. More precisely $J(F)=(a,b)$.
Let $x\in(a,b)$. Convexity of $\psi$ implies that $$\begin{aligned}
F'(x\pm)=\lim_{t\to 0,\pm t>0}\dfrac{\psi^{1/s^*}(x+t)-\psi^{1/s^*}(t)}{t}=\dfrac{1}{s^*}\psi(x)^{1/s^*-1}\psi'(x\pm)
\end{aligned}$$ exist and satisfy $$\begin{aligned}
F'(x-) \le F'(x+) .
\end{aligned}$$ Similarly, convexity of $(1-F)^{s^*}$ yields $$\begin{aligned}
(1-F)'(x+)\geq (1-F)'(x-)
\end{aligned}$$ which implies that $$\begin{aligned}
-F'(x+)\geq -F'(x-).
\end{aligned}$$ Therefore $F'(x-)=F'(x+)$ which proves the differentiability of $F$. It also shows that $\psi'(x+)=\psi'(x-)=\psi'(x)$ on $(a,b)$.
By Lemma 6 (convex version) of [@DuembgenKW:2017] for each $x\in(a,b)$ and $c\in[\psi'(x-),\psi'(x+)]$ one has $$\begin{aligned}
\psi(x+t) -\psi(x)\geq ct\ \text{ for all }t\in{\mathbb{R}}.
\end{aligned}$$ Therefore $$\begin{aligned}
\psi(x+t)-\psi(x)\geq t \psi'(x).
\end{aligned}$$ Hence, $$\begin{aligned}
F^{s^*}(x+t)-F^{s^*}(x)\geq ts^{*}f(x)F(x)^{s^*-1} ,
\end{aligned}$$ or, with $x_{+}=\max\{x,0\}$, $$\begin{aligned}
\dfrac{F^{s^*}(x+t)}{F^{s^*}(x)}\geq\bigg ( 1+s^*\dfrac{f(x)}{F(x)}t\bigg )_{+}.
\end{aligned}$$ Hence, $$\begin{aligned}
\dfrac{F(x+t)}{F(x)}\leq\bigg ( 1+s^*\dfrac{f(x)}{F(x)}t\bigg )_{+}^{1/s^*}.\end{aligned}$$ Analogously it follows that $$\begin{aligned}
(1-F(x+t))^{s^*}-(1-F(x))^{s^*} \geq -ts^*f(x)(1-F(x))^{s^*-1}\end{aligned}$$ which yields $$\begin{aligned}
\bigg(\dfrac{1-F(x+t)}{1-F(x)}\bigg )^{s^*}\geq \bigg (1-ts^*\dfrac{f(x)}{1-F(x)}\bigg)_{+}\end{aligned}$$ or $$\begin{aligned}
F(x+t)\geq 1-(1-F(x))\cdot \bigg (1-ts^*\dfrac{f(x)}{1-F(x)}\bigg)_{+}^{1/s^*}.\end{aligned}$$ Hence is proved.
Since (ii) holds, $F$ is continuous and differentiable on $J(F)$ with derivative $f=F'$ and satisfies . Now let $x,y\in J(F)$ with $x<y.$ Let $$\begin{aligned}
h=f/F^{1-s^*}.
\label{defh}\end{aligned}$$ Then applying we obtain that $$\begin{aligned}
\dfrac{F^{s^*}(x)}{F^{s^*}(y)}\geq 1+s^*\dfrac{f(y)}{F(y)}(x-y).\end{aligned}$$ Hence, $$\begin{aligned}
F^{s^*}(x)
& \geq & F^{s^*}(y)+s^*\dfrac{f(y)}{F(y)^{1-s^*}}(x-y)\\
& = & F^{s^*}(y)+s^*h(y)(x-y)\\
&\geq & \ F^{s^*}(x)+s^*h(x)(y-x)+s^*h(y)(x-y).\end{aligned}$$ Therefore $$\begin{aligned}
s^{*}(x-y)(h(y)-h(x))\leq 0\end{aligned}$$ where $s^{*}(x-y)>0$, implying that $h(y)\leq h(x)$. Therefore $h$ is non-increasing. Now let $$\begin{aligned}
\tilde{h}=f/(1-F)^{1-s^{*}}. \label{defth}\end{aligned}$$ From we also obtain that $$\begin{aligned}
(1-F(x))^{s^{*}}-(1-F(y))^{s^*}\geq -ts^{*}\dfrac{f(y)}{(1-F(y))^{1-s^*}}=-ts^{*}\tilde{h}(y)\end{aligned}$$ or $$\begin{aligned}
(1-F(x))^{s^{*}}& \geq & (1-F(y))^{s^*} -(x-y)s^{*}\tilde{h}(y)\\
& = & (1-F(x))^{s^*}-(y-x)s^{*}\tilde{h}(x) -(x-y)s^{*}\tilde{h}(y)\\
& = & (1-F(x))^{s^*}-s^{*}(y-x)(\tilde{h}(y)-\tilde{h}(x) ).\\\end{aligned}$$ Since $s^{*}(y-x)<0,$ the last inequality leads to $$\begin{aligned}
0\leq \tilde{h}(y)-\tilde{h}(x), \end{aligned}$$ implying that $\tilde{h}$ is non-decreasing.
Proof of (iii) implies (iv): If the conditions of (iii) hold, then it immediately follows that $f>0$ on $J(F)$. If not, suppose that $f(x_0) = 0$ for some $x_0\in J(F)$. Now $J(F)=(a,b)$ since $F$ is continuous. Since $f(x)/F(x)^{1-s^*}$ is non-increasing, $f(x)=0$ for $x\in [x_0,b).$ Similarly since $f(x)/(1-F(x))^{1-s^*}$ is non-decreasing we obtain $f(x)=0$ for $x\in(a, x_0].$ Therefore, $F'=0$ or $F$ is constant on $J(F)$. Then $F$ violates the continuity condition of (iii). Hence $f>0$ on $J(F)$.
Suppose $h$ and $\tilde{h}$ are as defined in and . Then the monotonicities of $h$ and $\tilde{h}$ imply that for any $x,x_0\in J(F),$ $$\begin{aligned}
f(x)=\begin{cases}
F^{1-s^*}(x)h(x)\leq h(x_0) & \ \mbox{if} \ \ x\geq x_0,\\
(1-F(x))^{1-s^*}\tilde{h}(x)\leq \tilde{h}(x_0) & \ \mbox{if} \ \ x\leq x_0.
\end{cases}
\end{aligned}$$ Next, let $c,d \in J(F)$ with $c<d$. We will bound $(f(y)-f(x))/(y-x)$ for $x,y\in J(F)$ such that $x,y\in(c,d)$ with $x \neq y$. This will yield local Lipschitz-continuity of $f$ on $J(F)$. To this end, note that $$\begin{aligned}
\dfrac{f(y)-f(x)}{y-x}
& = & \ \dfrac{F^{1-s^*}(y)h(y)-F^{1-s^*}(x)h(x)}{y-x}\\
& = & \ h(y)\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}+F^{1-s^*}(x)\dfrac{h(y)-h(x)}{y-x}\\
& \leq & \ h(c)\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
& \to & \ h(c) (1-s^*)f(x) F^{-s^*}(x)=(1-s^{*})h(c)h(x)F^{1-2s^{*}}(x)\end{aligned}$$ as $y\to x$. Here the inequality followed from the fact that $$\begin{aligned}
\dfrac{h(y)-h(x)}{y-x}\leq 0\end{aligned}$$ which holds since $h$ is non-increasing. Now since $h(x)\leq h(c)$, $1-2 s^{*}>0$, $1-s^{*} >0$, and $F(x)\leq F(d)$, we find that $$\begin{aligned}
\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\leq (1-s^*)h(c)^2F^{1-2s^*}(d)
\label{4eq1}\end{aligned}$$ for all $x\in(c,d)$. Analogously with $\bar{F}=1-F$ we obtain that $$\begin{aligned}
\dfrac{f(y)-f(x)}{y-x}
& = & \ \dfrac{\bar{F}^{1-s^*}(y)\tilde{h}(y)-\bar{F}^{1-s^*}(x)\tilde{h}(x)}{y-x}\\
& = & \tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}+\bar{F}^{1-s^*}(x)\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}\\
& \geq & \ \tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\end{aligned}$$ since, by the non-decreasing property of $\tilde{h}$, for any $x,y\in J(F)$, $$\begin{aligned}
\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}>0.\end{aligned}$$ Next observe that since $1-s^*=1/(1+s)>0$, and $\bar{F}$ is nonincreasing, $$\begin{aligned}
\tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\geq \tilde{h}(d)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}.\end{aligned}$$ Hence as $y\to x$ it follows that $$\begin{aligned}
\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x}
& \geq &-\tilde{h}(d)(1-s^*)f(x)\bar{F}^{-s^*}(x)\\
& = & -\tilde{h}(d)\tilde{h}(x)(1-s^*)\bar{F}^{1-2s^*}(x).\end{aligned}$$ Therefore using the fact that $\tilde{h}(x)\leq \tilde{h}(d)$ and $1-2s^*>0$ we conclude that $$\begin{aligned}
\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x}\geq -\tilde{h}(d)^2(1-s^*)\bar{F}^{1-2s^*}(c).
\label{4eq2}\end{aligned}$$ Combining the above with we find that $f$ is Lipschitz-continuous on $(c,d)$ with Lipschitz-constant $$\begin{aligned}
\max\{(1-s^*)h(c)^2F^{1-2s^*}(d),(1-s^*)\tilde{h}(d)^2\bar{F}^{1-2s^*}(c)\}.\end{aligned}$$ This proves that $f$ is locally Lipschitz continuous on $J(F)$. Hence, $f$ is also locally absolutely continuous with $L^1$-derivative $f'$ such that $$\begin{aligned}
f(y)-f(x)=\int_{x}^{y}f'(t)dt\ \text{ for all }x,y \in J(F);\end{aligned}$$ hence $f'(x)$ can be chosen so that $$\begin{aligned}
f'(x)\in\bigg[\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x},\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\bigg].\end{aligned}$$ However and imply that for $c<x<d$, $$\begin{aligned}
\lefteqn{
\bigg[\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x},\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\bigg]}\\
&& \subset \bigg[-(1-s^*)\tilde{h}(d)^2\bar{F}^{1-2s^*}(c),(1-s^*)h(c)^2F^{1-2s^*}(d)\bigg]\end{aligned}$$ Now since $f$ and $F$ are continuous and $F>0$ on $J(F),$ so are $h$ and $\tilde{h}$. Therefore, letting $c,d\to x$ it follows that $$\begin{aligned}
\dfrac{-(1-s^*)f(x)^2\bar{F}^{1-2s^*}(x)}{\bar{F}^{2-2s^*}(x)}\leq f'(x)\leq (1-s^*)\dfrac{f(x)^2F^{1-2s^*}(x)}{F^{2-2s^*}(x)};\end{aligned}$$ and this implies .
Proof of (iv) implies (i): The fact that (iii) implies (i) can be easily proved since $f/F^{1-s^*}$ non-increasing on $J(F)$ implies that $F^{s^*}$ is convex on $J(F).$ Also $1< F^{s^*}<\infty$ on $J(F)$. Now $F^{s^*}(x)=\infty$ for $x< \inf J(F)$ and $F^{s^*}(x)=1$ for $x> \sup J(F)$. Therefore $F^{s^*}$ is convex on ${\mathbb{R}}$. Similarly one can show that $(1-F)^{s^*}$ is convex on ${\mathbb{R}}$. Hence $F$ is bi-$s^*$-concave. Therefore it is enough to prove that (iv) implies (iii).
By Lemma $7$ of [@DuembgenKW:2017] $h$ is non-increasing on $J(F)$ if and only if for any $x\in J(F)$ the following holds: $$\begin{aligned}
\limsup_{y\to x}\dfrac{h(y)-h(x)}{y-x}\leq 0.
\end{aligned}$$ Suppose $x\neq y\in J(F)$ and $r:=\min(x,y)$ and $s:=\max(x,y).$ Then it follows that $$\begin{aligned}
\lefteqn{\dfrac{h(y)-h(x)}{y-x} = \dfrac{f(y)/F^{1-s^*}(y)-f(x)/F^{1-s^*}(x)}{y-x}}\\
&& =\ \dfrac{1}{F^{1-s^*}(y)}\dfrac{f(y)-f(x)}{y-x}-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
&& = \ \dfrac{1}{F^{1-s^*}(y)}\dfrac{\int_{r}^{s}f'(t)dt}{s-r}-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
&& \leq \ \dfrac{(1-s^*)}{F^{1-s^*}(y)(s-r)}\int_{r}^{s}\dfrac{f(t)^2}{F(t)}dt-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}.
\end{aligned}$$ by . Since $F$ is continuous by (iv), $J(F)$ must be an interval. Also since $x,y\in J(F),$ $[r,s]\subset J(F)$. Since $f$ and $F$ are continuous on $J(F)$ and $F>0$ on $J(F)$, $f^2/F$ is continuous and integrable on $J(F)$ and hence also on $[r,s]$. Letting $y\to x$ we obtain that $$\begin{aligned}
\limsup_{y\to x}\dfrac{h(y)-h(x)}{y-x}\leq \dfrac{(1-s^*)f(x)^2}{F^{2-s^*}(x)}-\dfrac{(1-s^*)f(x)^2}{F^{2-s^*}(x)}=0.
\end{aligned}$$ Analogously by Lemma $7$ of [@DuembgenKW:2017], to show $\tilde{h}$ is non-decreasing it is enough to show that $$\begin{aligned}
\liminf_{y\to x}\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}\geq 0.
\end{aligned}$$ To verify this suppose $x\neq y\in J(F)$ and $r:=\min(x,y)$ and $s:=\max(x,y)$. As before we calculate $$\begin{aligned}
\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}
& = &\dfrac{f(y)/\bar{F}^{1-s^*}(y)-f(x)/\bar{F}^{1-s^*}(x)}{y-x}\\
& = & \dfrac{1}{F^{1-s^*}(y)}\dfrac{f(y)-f(x)}{y-x}-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\\
& = & \dfrac{1}{\bar{F}^{1-s^*}(y)}\dfrac{\int_{r}^{s}f'(t)dt}{s-r}-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\\
& \geq & - \ \dfrac{(1-s^*)}{\bar{F}^{1-s^*}(y)(s-r)}\int_{r}^{s}\dfrac{f(t)^2}{\bar{F}(t)}dt-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}
\end{aligned}$$ by . Since $f$ and $\bar{F}$ are continuous on $J(F)$, letting $y\to x$ it follows that $$\begin{aligned}
\liminf_{y\to x}\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}
\geq -\dfrac{(1-s^*)f(x)^2}{\bar{F}^{2-s^*}(x)}+\dfrac{(1-s^*)f(x)^2}{\bar{F}^{2-s^*}(x)}=0.
\end{aligned}$$
Bounds for $F$ bi-$s^*$-concave when $s<0$. {#subsec:boundsSNeg}
-------------------------------------------
First, upper and lower bounds on $F$: Note that $(1+y)^r \ge 1+ ry$ for any $r < 0$ and $y \ge -1$. Taking $y = -F(x)$ and $r = s^*$ yields $$\begin{aligned}
(1-F(x))^{s^*} \ge 1- s^* F(x) \end{aligned}$$ or, by rearranging, $$\begin{aligned}
F(x) \le \frac{1}{-s^*} \left \{ (1-F(x))^{s^*} - 1 \right \} \equiv F_{U,s} (x) \equiv F_U (x)
\label{FUpperSNeg}\end{aligned}$$ where $F_U$ is a convex function if $F$ is bi$-s^*-$concave. Similarly, taking $y= - (1-F(x))$ and $r=s^*$ yields, by rearranging terms $$\begin{aligned}
F(x) \ge \frac{1}{-s^*} \left \{ (1-s^*) - F(x)^{s^*} \right \} \equiv F_{L,s} (x) \equiv F_L (x)
\label{FLowerSNeg}\end{aligned}$$ where $F_L$ is a concave function if $F$ is bi$-s^*-$concave. Note that $$\begin{aligned}
F_U^{\prime} (x) = \frac{f(x)}{(1-F(x))^{1-s^*}} = \frac{f(x)}{(1-F(x))^{1/(1+s)}}
\label{FprimeLowerSNeg}\end{aligned}$$ is monotone non-decreasing, while $$\begin{aligned}
F_L^{\prime} (x) = \frac{f(x)}{F^{1-s^*}(x)} = \frac{f(x)}{F^{1/(1+s)} (x)}
\label{FprimeUpperSNeg}\end{aligned}$$ is monotone non-increasing. Therefore $$\begin{aligned}
&& 0 \le F_U^{\prime \prime} (x) = (1-F(x))^{s^*-2} \left \{ (1-s^*) f^2 (x) + (1-F(x)) f^{\prime} (x) \right \}, \\
&& 0 \ge F_L^{\prime \prime} (x) = F(x)^{s^*-2} \left \{ (s^*-1) f^2 (x) + F(x) f^{\prime}(x) \right \} . \end{aligned}$$ The upper and lower bounds in (iv) of Theorem \[thm:1s\] follow by rearranging these inequalities.
Taking $F$ to be the distribution function of $t_1$ and plotting the bounds for $F$, $F^{\prime} = f$ and $F^{\prime \prime} = f^{\prime}$ yields the following three figures.
![The bi$-s^*-$concave $t_1$ distribution function $F$ (black) with its convex upper bound $F_U$ (red) and concave lower bound $F_L$ (blue), where $F_U$ and $F_L$ are given in (\[FUpperSNeg\]) and (\[FLowerSNeg\]).[]{data-label="fig:fig1"}](Plots/Bounds-t1-DF-labelled.pdf){width="\linewidth" height="5.5cm"}
Upper bounds for the density $f = F^{\prime} $ of $F$ follow from (iii) of Theorem \[thm:1s\]: These bounds are illustrated for the bi-$s^*$-concave distribution $t_1$ in Figure \[fig:fig2\].
![The bi-$s^*$-concave $t_1$ density function $f$ (black) with its bi-$s^*$-concave upper bounds $F_U^{\prime}$ (red) and $F_L^{\prime}$ (blue) as given by (\[FprimeLowerSNeg\]) and (\[FprimeUpperSNeg\]).[]{data-label="fig:fig2"}](Plots/Bounds-t1-Density-labelled.pdf){width="\linewidth" height="5.5cm"}
Upper and lower bounds for the derivative $f'$ of $f$ are given in (iv) of Theorem \[thm:1s\]: These bounds are illustrated for the bi$-s^*-$concave distribution $t_1$ in Figure \[fig:fig3\].
![The bi-$s^*$-concave $t_1$ density function derivative $f^{\prime}$ (black) with its bi-$s^*$-concave lower (red) and upper (blue) bounds as given in (iv) of Theorem \[thm:1s\].[]{data-label="fig:fig3"}](Plots/Bounds-t1-DensityDeriv-labelled.pdf){width="\linewidth" height="5.5cm"}
Bounds for $F$ bi-$s^*$-concave when $s>0$. {#subsec:BoundsForSpos}
-------------------------------------------
Upper and lower bounds on $F$: Note that now $(1+y)^r \le 1+ ry$ for any $r \in (0,1]$ and $y \ge -1$ by concavity of $(1+y)^r$. Taking $y = -F(x)$ and $r = s^*>0$ (since $s>0$) yields $$\begin{aligned}
(1-F(x))^{s^*} \le 1- s^* F(x) . \end{aligned}$$ By rearranging, $$\begin{aligned}
F(x) & \le & \frac{1}{-s^*} \left \{ (1-F(x))^{s*} - 1 \right \} \nonumber \\
& = & \frac{1}{s^*} \left \{ 1- (1-F(x))^{s^*} \right \} \equiv F_{U,s} (x) \equiv F_U (x)
\label{FUpperSPos}\end{aligned}$$ where $F_U$ is a convex function if $F$ is bi$-s^*-$concave. Similarly, taking $y= - (1-F(x))$ and $r=s^*$ yields, by rearranging terms $$\begin{aligned}
F(x) \ge \frac{1}{s^*} \left \{ F(x)^{s^*} - (1-s^*) \right \} \equiv F_{L,s} (x) \equiv F_L (x)
\label{FLowerSPos}\end{aligned}$$ where $F_L$ is a concave function if $F$ is bi$-s^*-$concave. Note that $$\begin{aligned}
F_U^{\prime} (x) = \frac{f(x)}{(1-F(x))^{1-s^*}} = \frac{f(x)}{(1-F(x))^{1/(1+s)}}
\label{FprimeUpperSPos}\end{aligned}$$ is monotone non-decreasing, while $$\begin{aligned}
F_L^{\prime} (x) = \frac{f(x)}{F^{1-s^*}(x)} = \frac{f(x)}{F^{1/(1+s)} (x)}
\label{FprimeLowerSPos}\end{aligned}$$ is monotone non-increasing. Therefore $$\begin{aligned}
&& 0 \le F_U^{\prime \prime} (x) = (1-F(x))^{s^*-2} \left \{ (1-s^*) f^2 (x) + (1-F(x)) f^{\prime} (x) \right \}, \\
&& 0\ge F_L^{\prime \prime} (x) = F(x)^{s^*-2} \left \{ (s^*-1) f^2 (x) + F(x) f^{\prime}(x) \right \} . \end{aligned}$$ Again note that the upper and lower bounds in (iv) of Theorem \[thm:1s\] follow by rearranging these inequalities.
Taking $F$ to be the distribution function of $g(\cdot, r)$ with $r=1$ as in Example \[exmpl:ex4\] and plotting the bounds for $F$, $F^{\prime} = f$ and $F^{\prime \prime} = f^{\prime}$ yields the following three figures.
![The bi-$s^*$-concave distribution function $F$ (black) corresponding to $g(\cdot; 1)$ of Example \[exmpl:ex4\] with its convex upper bound $F_U$ (red) and concave lower bound $F_L$ (blue) (where $F_U$ and $F_L$ are given in (\[FUpperSPos\]) and (\[FLowerSPos\])).[]{data-label="fig:fig5"}](Plots/Bounds-Ct2-DF-v2-labelled.pdf){width="\linewidth" height="5.5cm"}
Upper and lower bounds for the density $f = F^{\prime} $ of $F$ follow from (iii) of Theorem \[thm:1s\]. These bounds are illustrated for the bi-$s^*$-concave distribution $F$ corresponding to $g(\cdot; 1)$ of Example \[exmpl:ex4\] in Figure \[fig:fig6\].
![The bi-$s^*$-concave density function $g(\cdot; 1)$ of Example \[exmpl:ex4\] (black) with its bi-$s^*$-concave upper bounds $F_L^{\prime}$ and $F_U^{\prime}$ given in (\[FprimeLowerSPos\]) and (\[FprimeUpperSPos\]).[]{data-label="fig:fig6"}](Plots/Bounds-Ct2-Density-labelled.pdf){width="\linewidth" height="5.5cm"}
Upper and lower bounds for the derivative $f'$ of $f$ are given in (iv) of Theorem \[thm:1s\] These bounds are illustrated for the bi-$s^*$-concave distribution function $F$ with density $g(\cdot ; 1)$ as in Example \[exmpl:ex4\] in Figure \[fig:fig7\].
![$F^{\prime \prime} = f^{\prime}$ (black) for the bi-$s^*$-concave function $F$ corresponding to the density $g(\cdot; 1)$ as in Example \[exmpl:ex4\] with its bi-$s^*$-concave upper (blue) and lower (red) bounds as given in (iv) of Theorem \[thm:1s\].[]{data-label="fig:fig7"}](Plots/Bounds-Ct2-DensityDeriv-labelled.pdf){width="\linewidth" height="5.5cm"}
A consequence for Fisher information {#sec:FisherInformation}
====================================
In this section we suppose that $F$ is a bi-$s^*$-concave distribution function with absolutely continuous density $f$ with respect to Lebesgue measure. Then from (\[4eq3\]) of Theorem \[thm:1s\] it follows that
$$\begin{aligned}
\frac{| f' (x) |}{f(x)} \le \frac{1}{1+s} \frac{f(x)}{F(x) \wedge (1-F(x))} \ \ \mbox{for all} \ \ x \in J(F),\end{aligned}$$
and hence that $$\begin{aligned}
\lefteqn{I_f \equiv \int_{{\mathbb{R}}} \left ( \frac{|f' (x)|}{f(x)} \right )^2 f(x) dx } \nonumber \\
&& \le \frac{1}{(1+s)^2} \int_{{\mathbb{R}}} \frac{f^2 (x)}{(F(x) \wedge (1-F(x)))^2} d F(x) \nonumber \\
&& \le \frac{1}{(1+s)^2} \left \{ \int_{{\mathbb{R}}} \frac{f^2 (x)}{F^2(x)} dF(x) + \int_{{\mathbb{R}}} \frac{f^2 (x)}{(1-F(x))^2} d F(x) \right \} \nonumber\\
&& \le \frac{2}{(1+s)^2} \max \left \{ \int_{{\mathbb{R}}} \left ( \frac{f}{F} \right )^2 dF, \int_{{\mathbb{R}}} \left ( \frac{f}{1-F} \right )^2 dF \right \} .
\label{FisherInformationUpperbounded}\end{aligned}$$ But with $h = f'/f$, we find that $$\int_{-\infty}^x h dF = \int_{-\infty}^x (f'(y) / f(y) ) f(y) dy = f(x) \ \ \mbox{and} \ \ \frac{f(x)}{F(x)} = \frac{\int_{-\infty}^x h dF}{F(x)} ,$$ while $$\int_x^{\infty} h dF = \int_x^{\infty} (f'/f) f dy = - f(x), \ \ \mbox{and} \ \ \frac{-f(x)}{1-F(x)} = \frac{\int_x^\infty h dF }{1-F(x)} .$$ Thus by the $L_2$ version of Hardy’s inequality $$\begin{aligned}
\int_{{\mathbb{R}}} \left ( \frac{f(x)}{F(x)} \right )^2 d F(x)
& \le &4 \int_{{\mathbb{R}}} \left ( \frac{|f' (x)|}{f(x)} \right )^2 f(x) dx = 4 I_f, \ \ \mbox{and} \nonumber \\
\int_{{\mathbb{R}}} \left ( \frac{f(x)}{1-F(x)} \right )^2 d F(x)
& \le & 4 \int_{{\mathbb{R}}} \left ( \frac{|f' (x)|}{f(x)} \right )^2 f(x) dx = 4 I_f .
\label{FisherInformationLowerBounded}\end{aligned}$$ Combining the inequalities in (\[FisherInformationUpperbounded\]) and (\[FisherInformationLowerBounded\]) yields $$\begin{aligned}
I_f & \le & \frac{2}{(1+s)^2} \max \left \{ \int_{{\mathbb{R}}} \left ( \frac{f(x)}{F(x)} \right )^2 d F(x) ,
\ \int_{{\mathbb{R}}} \left ( \frac{f(x)}{1-F(x)} \right )^2 d F(x) \right \} \nonumber \\
& \le & \frac{8}{(1+s)^2} I_f .
\label{FisherInfoUpperAndLowerBounded}\end{aligned}$$ But we note that the densities $f_r$ in Example 4 have $$\begin{aligned}
I_{f_r} = \frac{r}{2} \cdot \frac{\Gamma \left (\frac{r}{2}-1\right)
\Gamma \left ( \frac{r+3}{2} \right )}{\Gamma \left ( \frac{r}{2} +1 \right )^2} \nearrow \infty\end{aligned}$$ as $r \searrow 2$, and $I_{f_r} = \infty$ for $0 < r \le 2$. In this latter case all the integrals in (\[FisherInfoUpperAndLowerBounded\]) are infinite.
Questions and further problems {#sec:QuestAndProb}
==============================
[Application of bi-$s^*$-concavity to construction of confidence bands for $F$?]{} [@DuembgenKW:2017] use their bi-log-concave bounds to construct new confidence bands for bi-log-concave distribution functions $F$. Alternative confidence bands based on the bi-$s^*$-concavity assumption may be of interest.
[What can be said when $s \le -1$?]{} The only result we know in the direction of preserving $s-$concavity in the spirit of Borell, Brascamp and Lieb, and Rinott is due to [@MR572660], but we do not have an interpretation of their result. We also do not know if there is an approximation of the general (standardized) quantile process ${\mathbb{Q}}_n$ in terms of the uniform quantile process ${\mathbb{V}}_n$ in this case.
[Bi-log-concavity or bi-$s^*$-concavity in higher dimensions?]{} Although log-concave (and $s-$concave) densities and measures on ${\mathbb{R}}^d$ (and a variety of non-Euclidean spaces) exist, we do not know of any analogue of bi$-s^*$concavity or bi-log-concavity in higher dimensions.
[Tranportation distances for empirical measures when $d\ge2$?]{} The Csörgő - Révész condition has proved very useful for studying empirical transportation distances for empirical measures in one dimension, largely because of the connection with quantile processes. We do not know of comparable theory for transportation distances for empirical measures in higher dimensional settings.
Appendix: proof of Theorem \[thm:1s\] when $s \in (0,\infty)$ {#sec:appendix}
=============================================================
Our proof of Theorem \[thm:1s\] for the case $s \in (0,\infty)$ closely parallels the proof for the case $s\in(-1,0]$. The main difference is the proof of (iii) implies (iv). When $0 < s < \infty$, $s^* = s/(1+s) \in (0,1)$, and hence $1-2s^* < 0$ for $s>1$. This requires a slightly different argument in this range and results in different constants in the Lipschitz bounds.
Let us denote $\inf J(F)$ and $\sup J(F)$ by $a$ and $b$ respectively. Notice that $J(F)=(a,b)$ if $F$ is continuous.
Proof of (i) implies (ii): Since $F$ is bi-$s^*$-concave with $\st>0$, $\psi=F^{\st}$ is concave on $(a,\infty)$. Consequently $\psi$, and hence $F$ also, is continuous on $(a,\infty)$ by Lemma $6$ of [@DuembgenKW:2017]. Similarly, the $s^*-$concavity of $1-F$ implies continuity of $1-F$ on $(-\infty,b).$ Now if $a=b$, $F$ would be degenerate. Hence, $a<b$ and $F$ is continuous on ${\mathbb{R}}$. Therefore we can also conclude that $J(F)=(a,b)$.
Let $x\in(a,b)$. Concavity of $\psi$ implies that $$\begin{aligned}
F'(x\pm)=\lim_{t\to 0,\pm t>0}\dfrac{\psi^{1/s^*}(x+t)-\psi^{1/s^*}(t)}{t}=\dfrac{1}{s^*}\psi(x)^{1/s^*-1}\psi'(x\pm)\end{aligned}$$ exist and satisfy $$\begin{aligned}
F'(x+) \le F'(x-) .\end{aligned}$$ Similarly, concavity of $(1-F)^{s^*}$ yields $$\begin{aligned}
(1-F)'(x-)\geq (1-F)'(x+)\end{aligned}$$ which implies that $$\begin{aligned}
-F'(x-)\geq -F'(x+).\end{aligned}$$ Therefore $F'(x-)=F'(x+)$ which proves the differentiability of $F$. It also shows that $\psi'(x+)=\psi'(x-)=\psi'(x)$ on $(a,b)$.
By Lemma 6 of [@DuembgenKW:2017] for each $x\in(a,b)$ and $c\in[\psi'(x+),\psi'(x-)]$ one has $$\begin{aligned}
\psi(x+t) -\psi(x)\leq ct\ \text{ for }t\in(a-x,\infty)
\end{aligned}$$ since $\psi$ is concave on $(a,\infty).$ Therefore for such $x$ and $t,$ $$\begin{aligned}
\psi(x+t)-\psi(x)\leq t \psi'(x).
\end{aligned}$$ Hence, $$\begin{aligned}
F^{s^*}(x+t)-F^{s^*}(x)\leq ts^{*}f(x)F(x)^{s^*-1}
\end{aligned}$$ or, $$\begin{aligned}
\dfrac{F^{s^*}(x+t)}{F^{s^*}(x)}\leq 1+s^*\dfrac{f(x)}{F(x)}t .
\end{aligned}$$ Hence, $$\begin{aligned}
\dfrac{F(x+t)}{F(x)}\leq\bigg ( 1+s^*\dfrac{f(x)}{F(x)}t\bigg )^{1/s^*}.\end{aligned}$$ Analogously it follows that for $t\in(-\infty,b-x)$, $$\begin{aligned}
(1-F(x+t))^{s^*}-(1-F(x))^{s^*} \leq -ts^*f(x)(1-F(x))^{s^*-1}\end{aligned}$$ which yields $$\begin{aligned}
\bigg(\dfrac{1-F(x+t)}{1-F(x)}\bigg )^{s^*}\leq 1-ts^*\dfrac{f(x)}{1-F(x)}\end{aligned}$$ or $$\begin{aligned}
F(x+t)\geq 1-(1-F(x))\cdot \bigg (1-ts^*\dfrac{f(x)}{1-F(x)}\bigg)^{1/s^*}.
\end{aligned}$$ Hence is proved. Notice that for $\st< 0$ the inequalities in hold for all $t$ because if $\st<0$, unlike the present case, $F^{\st}$ and $(1-F)^{\st}$ are convex on the entire real line.
Proof of (ii) implies (iii): Since (ii) holds, $F$ is continuous and differentiable on $J(F)$ with derivative $f=F'$ and satisfies . Now let $x,y\in J(F)$ with $x<y.$ Let $$\begin{aligned}
h=f/F^{1-s^*}.
\label{defhsp}\end{aligned}$$ Then applying we obtain that $$\begin{aligned}
\dfrac{F^{s^*}(x)}{F^{s^*}(y)}\leq 1+s^*\dfrac{f(y)}{F(y)}(x-y).\end{aligned}$$ Hence, $$\begin{aligned}
F^{s^*}(x)
& \leq & F^{s^*}(y)+s^*\dfrac{f(y)}{F(y)^{1-s^*}}(x-y)\\
& = & \ F^{s^*}(y)+s^*h(y)(x-y)\\
&\leq & \ F^{s^*}(x)+s^*h(x)(y-x)+s^*h(y)(x-y).\end{aligned}$$ Therefore $$\begin{aligned}
s^{*}(x-y)(h(y)-h(x))\geq 0\end{aligned}$$ where $s^{*}(x-y)<0$, implying that $h(y)\leq h(x)$. Therefore $h$ is non-increasing. Now let $$\begin{aligned}
\tilde{h}=f/(1-F)^{1-s^{*}}.
\label{defthsp}\end{aligned}$$ From we also obtain that $$\begin{aligned}
(1-F(x))^{s^{*}}-(1-F(y))^{s^*}\leq -ts^{*}\dfrac{f(y)}{(1-F(y))^{1-s^*}}=-ts^{*}\tilde{h}(y)\end{aligned}$$ or $$\begin{aligned}
(1-F(x))^{s^{*}}
& \leq & (1-F(y))^{s^*} -(x-y)s^{*}\tilde{h}(y)\\
& = & (1-F(x))^{s^*}-(y-x)s^{*}\tilde{h}(x) -(x-y)s^{*}\tilde{h}(y)\\
& = & (1-F(x))^{s^*}-s^{*}(y-x)(\tilde{h}(y)-\tilde{h}(x)).\\\end{aligned}$$ Since $s^{*}(y-x)>0,$ the last inequality leads to $$\begin{aligned}
0\leq \tilde{h}(y)-\tilde{h}(x), \end{aligned}$$ implying that $\tilde{h}$ is non-decreasing.
Proof of (iii) implies (iv): If the conditions of (iii) hold, then it immediately follows that $f>0$ on $J(F)$. If not, suppose that $f(x_0) = 0$ for some $x_0\in J(F)$ where $J(F)=(a,b)$ since $F$ is continuous. Then since $f(x)/F(x)^{1-s^*}$ is non-increasing, $f(x)=0$ for $x\in[x_0,b)$. Similarly since $f(x)/(1-F(x))^{1-s^*}$ is non-decreasing we obtain $f(x)=0$ for $x\in(a, x_0]$. Therefore, $F'=0$ or $F$ is constant on $(a,b)$ or $J(F)$. Then $F$ violates the continuity condition of (iii). Hence $f>0$ on $J(F)$.
Suppose $h$ and $\tilde{h}$ are as defined in and . Then the monotonicities of $h$ and $\tilde{h}$ imply that for any $x,x_0\in J(F),$ $$\begin{aligned}
f(x)= \left \{ \begin{array}{l l} F^{1-s^*}(x)h(x)\leq h(x_0) & \ \mbox{if} \ \ x\geq x_0,\\
(1-F(x))^{1-s^*}\tilde{h}(x)\leq \tilde{h}(x_0) & \ \mbox{if} \ \ x\leq x_0.
\end{array} \right .\end{aligned}$$ Next, let $c,d \in J(F)$ with $c<d$. We will bound $(f(y)-f(x))/(y-x)$ for $x,y\in J(F)$ such that $x,y\in(c,d)$ with $x \neq y$. This will yield local Lipschitz-continuity of $f$ on $J(F)$. To this end, note that $$\begin{aligned}
\dfrac{f(y)-f(x)}{y-x}
& = & \ \dfrac{F^{1-s^*}(y)h(y)-F^{1-s^*}(x)h(x)}{y-x}\\
& = & \ h(y)\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}+F^{1-s^*}(x)\dfrac{h(y)-h(x)}{y-x}\\
& \leq & \ h(c)\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
& \to & \ h(c) (1-s^*)f(x) F^{-s^*}(x)=(1-s^{*})h(c)h(x)F^{1-2s^{*}}(x)\end{aligned}$$ as $y\to x$. Here the inequality followed from the fact that $$\begin{aligned}
\dfrac{h(y)-h(x)}{y-x}\leq 0\end{aligned}$$ which holds since $h$ is non-increasing, Now since $h(x)\leq h(c)$, $s^*>0,$ $1- s^{*}>0$, and $F(c)\leq F(x)\leq F(d)$, we find that $$\begin{aligned}
\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\leq (1-s^*)h(c)^2F^{1-s^*}(d)F^{-s^*}(c)
\label{4eq1sp}\end{aligned}$$ for all $x\in(c,d)$. Analogously with $\bar{F}=1-F$ we obtain that $$\begin{aligned}
\dfrac{f(y)-f(x)}{y-x}
& = & \dfrac{\bar{F}^{1-s^*}(y)\tilde{h}(y)-\bar{F}^{1-s^*}(x)\tilde{h}(x)}{y-x}\\
& = & \tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}+\bar{F}^{1-s^*}(x)\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}\\
& \geq & \tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\end{aligned}$$ since, by the non-decreasing property of $\tilde{h}$, for any $x,y\in J(F)$, $$\begin{aligned}
\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}>0.\end{aligned}$$ Next observe that since $1-s^*=1/(1+s)>0$, and $\bar{F}(y)\leq \bar{F}(x)$ if $y\ge x$, $$\begin{aligned}
\tilde{h}(y)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\geq \tilde{h}(d)\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}.\end{aligned}$$ Hence as $y\to x$ it follows that $$\begin{aligned}
\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x}
& \geq & -\tilde{h}(d)(1-s^*)f(x)\bar{F}^{-s^*}(x)\\
& = & -\tilde{h}(d)\tilde{h}(x)(1-s^*)\bar{F}^{1-2s^*}(x).\end{aligned}$$ Therefore using the fact that $\tilde{h}(x)\leq \tilde{h}(d)$ and $1-s^*,s^*>0$ we conclude that $$\begin{aligned}
\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x}\geq -\tilde{h}(d)^2(1-s^*)\bar{F}^{1-s^*}(c)\bar{F}^{-s^*}(d).
\label{4eq2sp}\end{aligned}$$ Combining the above with we find that $f$ is Lipschitz-continuous on $(c,d)$ with Lipschitz-constant $$\begin{aligned}
\max\{(1-s^*)h(c)^2F^{1-s^*}(d)F^{-s^*}(c),(1-s^*)\tilde{h}(d)^2\bar{F}^{1-s^*}(c)\bar{F}^{-s^*}(d)\}.\end{aligned}$$ This proves that $f$ is locally Lipschitz continuous on $J(F)$. Hence, $f$ is also locally absolutely continuous with $L^1$-derivative $f'$ such that $$\begin{aligned}
f(y)-f(x)=\int_{x}^{y}f'(t)dt\ \text{ for all }x,y \in J(F);\end{aligned}$$ hence $f'(x)$ can be chosen so that $$\begin{aligned}
f'(x)\in\bigg[\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x},\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\bigg].\end{aligned}$$ However and imply that for $c<x<d$, $$\begin{aligned}
\lefteqn{
\bigg[\liminf_{y\to x}\dfrac{f(y)-f(x)}{y-x},\limsup_{y\to x}\dfrac{f(y)-f(x)}{y-x}\bigg]}\\
&& \subset \bigg[-(1-s^*)\tilde{h}(d)^2\bar{F}^{1-s^*}(c)F^{-s^*}(d),(1-s^*)h(c)^2F^{1-s^*}(d)F^{-s^*}(c)\bigg]\end{aligned}$$ Now since $f$ and $F$ are continuous and $F>0$ on $J(F),$ so are $h$ and $\tilde{h}$. Therefore, letting $c,d\to x$ it follows that $$\begin{aligned}
\dfrac{-(1-s^*)f(x)^2\bar{F}^{1-2s^*}(x)}{\bar{F}^{2-2s^*}(x)}\leq f'(x)\leq (1-s^*)\dfrac{f(x)^2F^{1-2s^*}(x)}{F^{2-2s^*}(x)};\end{aligned}$$ and this implies .
Proof of (iv) implies (i): Notice that the fact that (iii) implies (i) can be easily verified since $f/F^{1-s^*}$ non-increasing on $J(F)$ implies that $F^{s^*}$ is concave on $J(F).$ Since $F$ is continuous, $J(F)=(a,b)$. Now $F^{s^*}\in(0,1)$ on $J(F)$ and $F^{s^*}(x)=1$ for $x\geq b$. Therefore $F^{s^*}$ is concave on $(a,\infty)$. Similarly one can show that $(1-F)^{s^*}$ is concave on $(-\infty,b)$. Therefore $F$ is bi-$s^*$-concave. Therefore it is enough to prove that (iv) implies (iii).
By Lemma $7$ of [@DuembgenKW:2017] $h$ is non-increasing on $J(F)$ if and only if for any $x\in J(F)$ the following holds: $$\begin{aligned}
\limsup_{y\to x}\dfrac{h(y)-h(x)}{y-x}\leq 0.
\end{aligned}$$ Suppose $x\neq y\in J(F)$ and $r:=\min(x,y)$ and $s:=\max(x,y).$ Then it follows that $$\begin{aligned}
\lefteqn{\dfrac{h(y)-h(x)}{y-x} = \dfrac{f(y)/F^{1-s^*}(y)-f(x)/F^{1-s^*}(x)}{y-x}} \\
&& = \dfrac{1}{F^{1-s^*}(y)}\dfrac{f(y)-f(x)}{y-x}
-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
&& = \dfrac{1}{F^{1-s^*}(y)}\dfrac{\int_{r}^{s}f'(t)dt}{s-r}
-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}\\
&& \leq \dfrac{(1-s^*)}{F^{1-s^*}(y)(s-r)}\int_{r}^{s}\dfrac{f(t)^2}{F(t)}dt
-\dfrac{f(x)}{F^{1-s^*}(x)F^{1-s^*}(y)}\dfrac{F^{1-s^*}(y)-F^{1-s^*}(x)}{y-x}
\end{aligned}$$ by . Since $F$ is continuous by (iv), $J(F)=(a,b)$. Also since $x,y\in J(F),$ $[r,s]\subset J(F)$. Since $f$ and $F$ are continuous on $J(F)$ and $F>0$ on $J(F)$, $f^2/F$ is continuous and integrable on $J(F)$ and hence also on $[r,s]$. Letting $y\to x$ we obtain that $$\begin{aligned}
\limsup_{y\to x}\dfrac{h(y)-h(x)}{y-x}\leq \dfrac{(1-s^*)f(x)^2}{F^{2-s^*}(x)}-\dfrac{(1-s^*)f(x)^2}{F^{2-s^*}(x)}=0 .
\end{aligned}$$ Analogously, by Lemma $7$ of [@DuembgenKW:2017], to show $\tilde{h}$ is non-decreasing it is enough to show that $$\begin{aligned}
\liminf_{y\to x}\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}\geq 0.
\end{aligned}$$ To verify this suppose $x\neq y\in J(F)$ and $r:=\min(x,y)$ and $s:=\max(x,y)$. As before we calculate $$\begin{aligned}
\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}
& = &\ \dfrac{f(y)/\bar{F}^{1-s^*}(y)-f(x)/\bar{F}^{1-s^*}(x)}{y-x}\\
& = &\ \dfrac{1}{F^{1-s^*}(y)}\dfrac{f(y)-f(x)}{y-x}-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}
\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\\
& = &\ \dfrac{1}{\bar{F}^{1-s^*}(y)}\dfrac{\int_{r}^{s}f'(t)dt}{s-r}-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}
\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}\\
& \geq & \ - \dfrac{(1-s^*)}{\bar{F}^{1-s^*}(y)(s-r)}\int_{r}^{s}\dfrac{f(t)^2}{\bar{F}(t)}dt
-\dfrac{f(x)}{\bar{F}^{1-s^*}(x)\bar{F}^{1-s^*}(y)}\dfrac{\bar{F}^{1-s^*}(y)-\bar{F}^{1-s^*}(x)}{y-x}
\end{aligned}$$ by . Since $f$ and $\bar{F}$ are continuous on $J(F)$, letting $y\to x$ it follows that $$\begin{aligned}
\liminf_{y\to x}\dfrac{\tilde{h}(y)-\tilde{h}(x)}{y-x}
\geq -\dfrac{(1-s^*)f(x)^2}{\bar{F}^{2-s^*}(x)}+\dfrac{(1-s^*)f(x)^2}{\bar{F}^{2-s^*}(x)}=0 .\end{aligned}$$
Acknowledgement {#acknowledgement .unnumbered}
===============
We owe thanks to Lutz Dümbgen for pointing out a simpler proof of Proposition \[prop:BagnoliBergstrom\] (not given here) and for noting several typos.
[^1]: Supported in part by NSF Grant DMS-1566514.
|
---
author:
- 'Hongfu Liu, Jun Li, Yue Wu and Yun Fu, [^1]'
bibliography:
- 'egbib.bib'
title: Clustering with Outlier Removal
---
[Shell : Bare Demo of IEEEtran.cls for Computer Society Journals]{}
Introduction
============
Cluster analysis is a fundamental task in data mining and machine learning area, which aims to separate a bunch of data points into different groups so that similar points are assigned into the same cluster. Although cluster analysis has been studied for long time, it is still catching rising attention in industrial scenarios due to its wide applications, from customer segmentation [@tsai2015customer] to information retrieval [@campos2015survey], and from recommendation systems [@shepitsen2008personalized] to resource allocation [@grandl2015multi]. Accordingly, cluster analysis has also been extensively explored in the academia. K-means is one of the most representative clustering methods, which seeks $K$ prototypes as the centroids to present the data points with the nearest distance. Spectral clustering designed for graph partition, minimizes the weights of cut edges to obtain disconnected sub-graphs with roughly even sizes. Gaussian mixture model estimates $K$ Gaussian distribution with means and variances to fit the data.
Although tremendous efforts have been devoted in the cluster analysis, most of the existing methods assume that all the data points should be assigned a cluster label. In another word, there are no anomaly data points for during clustering process. Unfortunately, this is not always true, especially for the unsupervised task. The potential anomalies or outliers inevitably degrade the clustering performance. For example, few outliers easily destroy the cluster structure derived from K-means and generate bizarre distributions of Gaussian mixture model. To handle outliers or noisy data, some robust clustering methods have been proposed to recover the clean data. Metric learning aims to learn a robust distance function to resist the outliers [@Davis07ICML]; $L_{1}$ norm is employed to alleviate the negative impact of outliers on the cluster structure [@Ding06ICML]. Beyond these, some methods aim to learn the more effective representation with some constraints. Low-rank representation assumes that the intrinsic or clean data lie in low-dimensional manifold [@Liu10ICML]; subspace sparse clustering explores self-expression property with sparse coefficient for representation learning. Recently, consensus clustering generates basic partition first, and employs the basic partitions as the representation for robust partition. Note that these methods still assign the cluster labels for each data point, rather than explicitly removing anomaly points.
To tackle the negative impacts of outliers during the clustering processing, some unsupervised outlier detection methods have been put forward from different aspects. Usually each data point is calculated a score to identify the outlier degree, returning top $K$ outlier candidates. Local outlier factor is one of the popular density-based methods, where outliers are identified by comparing the local density of the data point and its neighbors[@Breunig00SIR]. Similarly, local distance-based outlier detection uses the relative location of an object to its neighbours to determine the degree to which the object deviates from its neighbourhood [@Zhang09PKDD]. Angle-based outlier detection focuses on variance in the angles between the difference vectors of a point to the other points, where the angles of the outliers and other two randomly selected points have some deviations[@Kriegel08KDD; @Pham12KDD]. Other representative methods include ensemble-based iForest [@Liu08ICDM], eigenvector-based OPCA [@Lee13TKDE], cluster-based TONMF [@Kannan17SDM], and so on.
Although outlier detection methods can be regarded as a pre-process for cluster analysis, outlier detection and cluster analysis are usually conducted as two separated tasks. In fact, they are strongly coupled. Cluster structure can be easily destroyed by few outliers [@Georgogiannis16NIPS]; on the contrary, outliers are defined by the concept of cluster, which are recognized as the points belonging to none of the [@Breunig00SIR]. However, few of the existing works treat the cluster analysis and outlier detection in a unified framework. DBSCAN is one of the pioneering works for density-based cluster analysis with the outlier set as an extra output [@ester1996density], where all the data points are divided in three categories, core points, border points and outliers according to the density, then the clusters are generated by connecting core points and their affiliated border points. Strictly DBSCAN does not belong to the joint cluster analysis and outlier detection, which identifies and removes the outliers first and then follows the cluster generation. To our best knowledge, K-means[-]{}[-]{} [@Chawla13SDM] is the first work along this direction. It aims to detect $o$ outliers and partition the rest points into $K$ clusters, where the instances far away from the nearest centroid are regarded as outliers during clustering process. Since this problem is a discrete optimization problem in essence, it is natural that Langrangian Relaxation (LP) [@Ott14NIPS] formulates the clustering with outliers as an integer programming problem with several constraints, which requires the cluster creation costs as the input parameter. Although these two pioneering works provide new directions for joint clustering and outlier detection, the spherical structure assumption of K-means[-]{}[-]{} and the original feature space limit its capacity for complex data analysis, and the setup of input parameters and high time complexity in LP make it infeasible for large-scale data.
In this paper, we focus on the joint cluster analysis and outlier detection problem, and propose the Clustering with Outlier Removal (COR) algorithm. Since the outliers are relied on the concept of clusters, we transform the original space into the partition space via running some clustering algorithms (e.g. K-means) with different parameters to generate a set of different basic partitions. By this means, the continuous data are mapped into a binary space via one hot encoding of basic partitions. In the partition space, an objective function is designed based on Holoentropy [@Wu13TKDE] to increase the compactness of each cluster after some outliers are removed. With further analyses, we transform the partial problem of the objective function into a K-means optimization. To provide a complete and neat solution, an auxiliary binary matrix derived from basic partitions is introduced. Then COR is conducted on the concatenated matrix, which completely and efficiently solves the challenging problem via a unified K-means[-]{}[-]{} with theoretical supports. To evaluate the performance of COR, we conduct extensive experiments on numerous data sets in various domains. Compared with K-means[-]{}[-]{} and numerous outlier detection methods, COR outperforms rivals over in terms of cluster validity and outlier detection by four metrics. Moreover, we demonstrate the high efficiency of COR, which indicates it is suitable for large-scale and high-dimensional data analysis. Some key factors in COR are further analyzed for practical use. Finally, an application on flight trajectory is provided to demonstrate the effectiveness of COR in the real-world scenario. Here we summarize our major contributions as follows.
- To our best knowledge, we are the first to conduct the clustering with outlier removal in the partition space, which achieves simultaneous consensus clustering and outlier detection.
- Based on Holoentropy, we design the objective function from the aspect of outlier detection, which is partially solved by K-means clustering. By introducing an auxiliary binary matrix, we completely transform the non K-means clustering problem into a K-means[-]{}[-]{} with theoretical supports.
- Extensive experimental results demonstrated the effectiveness and efficiency of our proposed COR over the state-of-the-art rivals in terms of cluster and outlier detection.
The rest of this paper is organized as follows. Section 2 introduces the related work on robust clustering, outlier detection and joint learning. Section 3 provides the preliminary knowledge and our problem formulation. In Section 4, we elaborate the equivalent relationship between our addressed problem and K-means[-]{}[-]{} with an augmented matrix. Section 5 delivers a thorough discussion on the relationship among COR and cluster analysis, outlier detection and consensus clustering. Extensive experiments are conducted in Section 6. Finally, we conclude this paper in Section 7.
Related Work
============
In this section, we present the related work in terms of robust clustering, consensus clustering, outlier detection, and highlight the difference between existing work and ours.
Robust Clustering
-----------------
To alleviate the impact of outliers, robust clustering[^2] has been proposed from different aspects. From the distance function aspect, metric learning is used to learn a robust metric to measure the similarity between two points by taking the outliers into account [@Davis07ICML; @Yi12NIPS]; $L_{1}$ norm models the outliers as the sparse constraint for cluster analysis [@Ding06ICML; @Elhamifar13TPAMI]. From the data aspect, the outliers are assigned few weights during clustering process [@Dotto16SC]; low-rank representation treats the data as the clean part and outliers, and constrains the clean part with the lowest rank [@Liu10ICML]. From the model fusion aspect, ensemble clustering integrates different partitions into a consensus one to deliver a robust result [@Strehl02JMLR; @Liu17TKDE]. Although these robust clustering methods reduce the negative impacts of outliers on the cluster structure, they fail to explicitly detect or remove outlier points for clustering. In another word, each data point is assigned with a cluster label, even for the outliers.
Consensus Clustering
--------------------
Consensus clustering, also known as ensemble clustering, targets to integrate several diverse partition results from traditional clustering methods into a consensus one [@Strehl02JMLR]. It has been widely recognized of robustness, consistency, novelty and stability over traditional clustering methods, especially in generating robust partitions, discovering novel structures, handling noisy features, and integrating solutions from multiple sources. The process of consensus clustering generally has two steps: basic partitions generation and consensus fusion. Given basic partitions as input, consensus clustering is in essence a fusion problem rather than a partitioning problem, which seeks for an optimal combinatorial result from basic partitions. Over the past years, many clustering ensemble techniques have been proposed, resulting in various of ways to face the problem together with new fields of application for these techniques. Generally speaking, consensus clustering can be divided into two categories, *i.e.*, those with or without an explicit global objective function. The methods that do not set objective functions make use of some heuristics or meta-heuristics to find approximate solutions. Representative methods include co-association matrix-based [@Fred05TPAMI; @Lourenco13ML], graph-based [@Strehl02JMLR; @Fern04ICML], relabeling and voting based [@Ayad08TPAMI] and locally adaptive cluster-based algorithms [@Domeniconi09TKDD]. On another hand, the methods with explicit objectives employ global objective functions to measure the similarity between basic partitions and the consensus one. Representative solutions include K-means-like algorithm [@Topchy03ICDM], NMF [@Li07ICDM], EM algorithm [@Topchy04SDM], simulated annealing [@Lu08AAAI] and combination regularization [@Xie14KDD]. More information on consensus clustering can be found in the recent survey [@liu2019consensus].
Outlier Detection
-----------------
Outlier detection, also known as anomaly detection, seeks the points deviation from others and identifies these points as outliers, where most of the existing studies focus on unsupervised outlier detection. Some criteria are designed to assign a score to each point, and the points with large scores are regarded as the outlier candidates. Some representative methods include density-based LOF[@Breunig00SIR], COF[@Tang02PKDD], distance-based LODF [@Zhang09PKDD], frequent pattern-based Fp-outlier [@He08CSIS], angle-based ABOD [@Kriegel08KDD] and its fast version FABOD [@Pham12KDD], ensemble-based iForest [@Liu08ICDM], BSOD [@Liu16ICBD], eigenvector-based OPCA [@Lee13TKDE], cluster-based TONMF [@Kannan17SDM]. Recently, there are deep learning based outlier detection methods such as deep one-class SVM [@ruff2018deep] and GAN-based methods [@schlegl2017unsupervised; @zenati2018efficient; @li2018anomaly], which learns a non-linear transformation to project the original data into hidden space for effective recognition. However, these methods train the model only with clear data, and predict new data whether they are outliers, which is different from the problem we address here.
Joint Clustering and Outlier Detection
--------------------------------------
Cluster analysis and outlier detection are consistently hot topics in data mining area; however, they are usually considered as two independent tasks. Although robust clustering resists to the impact of outliers, each point including outliers is assigned the cluster label. Few of the existing works treat the cluster analysis and outlier detection in a unified framework. Two-stage frameworks, such as DBSCAN conduct the outlier detection first, then apply the clustering method for partition, which becomes struggled to handle complex data. K-measn[-]{}[-]{} [@Chawla13SDM] detects $o$ outliers and partitions the rest points into $K$ clusters, where the instances with large distance to the nearest centroid are regarded as outliers during the clustering process. Langrangian Relaxation (LP) [@Ott14NIPS] formulates the clustering with outliers as an integer programming problem, which requires the cluster creation costs as the input parameter. This problem has also been theoretically studied in facility location. Charikar *et al.* proposed a bi-criteria approximation algorithm for the facility location with outliers problem [@Charikar01SODA]. Chen proposed a constant factor approximation algorithm for the K-median with outliers problem [@Chen08SODA].
In this paper, we consider the clustering with outlier removal problem, which partitions the entire data sets into several clusters and one outlier set. Although some pioneering works provide new directions for joint clustering and outlier detection, none of these algorithms expect K-means[-]{}[-]{} are amenable to a practical implementation on large data sets, while of theoretical interests. Moreover, the spherical structure assumption of K-means[-]{}[-]{} and the original feature space limit its capacity for complex data analysis. In light of this, we transform the original feature space into the partition space, where based on Holoentropy, the COR is designed to achieve simultaneous consensus clustering and outlier detection.
Problem Formulation
===================
In this section, we first illustrate some preliminary knowledge and elaborate our objective function for clustering and outlier removal.
Preliminaries
-------------
Here we introduce some basic knowledge on K-means[-]{}[-]{} and Holoentropy.
K-means[-]{}[-]{} [@Chawla13SDM] is a variant of K-means, which is particularly designed for handling the sensitivity of K-means on outliers. It is widely recognized that few outliers deviate the centroids from their intrinsic positions. To tackle with this, some data points with far distance to their centroids are regarded as the outlier candidates, which are not assigned with any cluster label and involved into the centroid updating, either. Similar to K-means, K-means[-]{}[-]{} also has two iterative stages, data point assignment and centroid updating. During the data point assignment, we calculate the distances between each data point and its nearest centroid, and sort the distances, where the data points with top $o$ largest distances are outlier candidates. For the centroid updating, it is the same with K-means since these outlier candidates are not assigned with cluster labels. It is worthy to note that the outlier candidates are changing during the iteration. Compared with K-means, K-means[-]{}[-]{} requires two input parameters, the numbers of clusters and outlier $K$ and $o$. It enjoys many properties as K-means in terms of neat mathematical formulation, model efficiency and convergence.
As pointed out by Ref [@Wu13TKDE], it is not suitable to only employ entropy or total correlation for outlier detection. They proposes a new measure Holoentropy as follows.
\[def:holo\] Holoentropy $HL(\mathcal{Y})$ is defined as the sum of the entropy and the total correlation of the random vector $\mathcal{Y}$, and can be expressed by the sum of the entropies on all attributes.
Holoentropy is an outlier detection metric based on information theory, which handles the categorical data and takes both entropy and total correlation into consideration. In the rest of this paper, we elaborate our proposed objective function based on Holoentropy, and derive its to K-means[-]{}[-]{} algorithm for a neat and efficient solution.
Objective Function
------------------
Cluster analysis and outlier detection are closely coupled tasks. Cluster structure can be easily destroyed by few outlier points; on the contrary, outliers are defined by the concept of cluster, which are recognized as the points belonging to none of the clusters. To cope with this challenge, we focus on the Clustering with Outlier Removal (COR). Specifically, the outlier detection and clustering tasks are jointly conducted, where $o$ points are detected as the outliers and the rest instances are partitioned into $K$ clusters. Table \[tab:notation\] shows the notations used in the following sections.
The cluster structure is vulnerable to few outliers, and outliers request to be identified with cluster boundary. The coupling relationship among cluster analysis and outlier detection makes it like a chicken-and-egg problem. To escape the chicken-and-egg problem in joint clustering and outlier detection, we are inspired by consensus clustering [@Strehl02JMLR; @Fred05TPAMI; @Domeniconi09TKDD], which incorporates several basic partitions generated from the data for a robust fusion to alleviate the negative effects from outliers. Moreover, the definition of outliers relies on the clusters. The above two points motivate us to transform the data from the original feature space into partition space via generating several basic partitions. This process is similar to generate basic partitions in consensus clustering [@Liu15KDD; @Liu16KDD]. Let $X$ denote the data matrix with $n$ points and $d$ features. A partition of $X$ into $K$ crisp clusters can be represented as a collection of $K$ subsets of objects with a label vector $\pi=(L_\pi(x_1),\cdots,L_\pi(x_n)),1\le l\le n$, where $L_\pi(x_l)$ maps $x_l$ to one of the $K$ labels in $\{1,2,\cdots,K\}$. Some basic partition generation strategy, such as K-means clustering with different cluster numbers can be applied to obtain $r$ basic partitions $\Pi=\{\pi_i\},1\le i \le r$. Let $K_i$ denote the cluster number for $\pi_i$ and $R = \sum_{i=1}^rK_i$. Then a binary matrix $B=\{b_l\},1\le l\le n$ can be derived from $\Pi$ as follows: $$\label{eq:binary}
\begin{split}
b_l&=(
b_{l,1},\cdots,b_{l,i},\cdots,b_{l,r}),~\textrm{with} \\
b_{l,i}&=( b_{l,i1},\cdots,b_{l,ij},\cdots,b_{l,iKi}),~\textrm{and} \\
b_{l,ij}&=\left\{
\begin{array}{ll}
1,&\textrm{if}~L_{\pi_i}(x_l)=j\\
0,&\textrm{otherwise}
\end{array}
\right..
\end{split}$$
It is worthy to note that we do not require a specific algorithm to generate basic partitions. For the sake of simplicity and efficiency, K-means with different cluster numbers are recommended to generated basic partitions. Although K-means is vulnerable to outliers, our COR still delivers promising results based on the basic partitions generated by K-means. The benefits to transform the original space into the partition space lie in (1) the binary value indicates the cluster-belonging information, which is particularly designed according to the definition of outliers, and (2) compared with the continuous space, the binary space is much easier to identify the outliers due to the categorical features. For example, Holoentropy is a widely used outlier detection metric for categorical data [@Wu13TKDE].
Notation Domain Description
---------- ------------------------------- ----------------------------------
$n$ $\mathcal{Z}$ Number of instances
$d$ $\mathcal{Z}$ Number of features
$K$ $\mathcal{Z}$ Number of clusters
$o$ $\mathcal{Z}$ Number of outliers
$r$ $\mathcal{Z}$ Number of basic partitions
$X$ $\mathcal{R}^{\{n\times d\}}$ Data set
$O$ $\mathcal{R}^{\{o\times d\}}$ Outlier set
$\Pi$ $\mathcal{Z}^{\{n\times r\}}$ Set of basic partitions
$B$ $\{0,1\}^{n\times R}$ Binary matrix derived from $\Pi$
: The Contingency Matrix
\[tab:notation\]
In Ref [@Wu13TKDE], the authors aimed to minimize the Holoentropy of the data set with $o$ outliers removed. Here we assume there exists the cluster structure within the whole data set. Therefore, it is more reasonable to minimize the Holoentropy of each cluster. In such a way, the clusters become compact after the outliers are removed, rather than the entire data set. Therefore, based on Holoentropy of each cluster, we give our objective function of COR as follows. $$\label{eq:obj-outlier}
\min_{\pi} \sum_{k=1}^K p_{k}HL(C_k),$$ where $HL(\cdot)$ is defined in Definition 1, $\pi$ is the cluster indicator, including $K$ clusters $C_1 \cup \cdots \cup C_{K} = X\backslash O$, with $C_k \cap C_{k'} = \emptyset$ if $k\neq k'$ and $p_{k+} = |C_{k}|/(n-o)$. Actually, the objective function in Eq. is the summation of weighted Holoentropy of each cluster, where the weight $p_{k}$ is proportional to the cluster size. Here the number of cluster $K$ and the number of outliers $o$ are two parameters of our proposed algorithm, which is the same setting with K-means[-]{}[-]{} [@Chawla13SDM], and we treat determining $K$ and $o$ as an orthogonal problem beyond this paper. In the next section, we provide an efficient solution for COR by introducing another auxiliary binary matrix.
Clustering with Outlier Removal
===============================
To solve the problem in Eq. , we provide a detailed objective function on the binary matrix $B$ as follows. $$\label{eq:obj-outlier2}
\begin{split}
&\sum_{k=1}^K p_{k}HL(C_k) \propto \sum_{k=1}^K \sum_{b_{l} \in C_k}\sum_{i=1}^r\sum_{j=1}^{K_i} H(C_{k,ij}),\textrm{and} \\
&H(C_{k,ij})=-(1-p_{k,ij})\log (1-p_{k,ij})-p_{k,ij}\log p_{k,ij},\\
\end{split}$$ where $H$ denotes the Shannon entropy and $p_{k,ij}$ denotes the probability of $b_{l,ij}=1$ in the $ij$-th column of $C_k$.
To better understand the meaning of $p_{k,ij}$ in Eq. , we provide the following lemma.
\[lem:mk\] For K-means clustering on the binary data set $B$, the $k$-th centroid satisfies $$\label{eq:mk}
\begin{split}
m_{k} &=(m_{k,1},\cdots,m_{k,i},\cdots,m_{k,r}), ~\text{with}\\
m_{k,i} &=(m_{k,i1}, \cdots m_{k,ij} \cdots m_{k,iK_i}),~\textrm{and}\\
m_{k,ij} &= \sum_{b_{l,ij} \in C_k} b_{l,ij}/|C_k| = p_{k,ij}, \forall~k,i,j.\\
\end{split}$$
The proof of Lemma \[lem:mk\] is self-evident according to the arithmetic mean of the centroid in K-means clustering. Based on Lemma \[lem:mk\], we uncover the bridge between the problem in Eq. and K-means clustering on the binary matrix $B$.
\[the:second\] If K-means is conducted on $n-o$ inliers of the binary matrix $B$, we have $$\label{eq:equal}
\begin{split}
&\max \sum_{k=1}^K p_k\sum_{i=1}^r\sum_{j=1}^{K_i} p_{k,ij}\log p_{k,ij} \Leftrightarrow \min \sum_{k=1}^K \sum_{b_l \in C_k} f(b_l,m_{k}),\\
\end{split}$$ where $m_{k}$ is the $k$-th centroid by Eq. and the distance function $f(b_l,m_{k})=\sum_{i=1}^r\sum_{j=1}^{K_i} D_{\textrm{KL}}(b_{l,ij}||m_{k,ij})$, here $D_{\textrm{KL}}(\cdot || \cdot)$ is the KL-divergence.
According to the Bregman divergence [@Banerjee05JMLR], we have $D_{\textrm{KL}}(s||t) = H(t) - H(s) + (s-t)^\top\nabla H(t)$, where $s$ and $t$ are two vectors with the same length. Then we start on the right side of Eq. . $$\begin{split}
&\sum_{k=1}^K \sum_{b_l \in C_k} f(b_l,m_{k})\\
=&\sum_{k=1}^K \sum_{b_l \in C_k}\sum_{i=1}^r\sum_{j=1}^{K_i} (H(m_{k,ij})-H(b_{l,ij})\\
&+(b_{l,ij}-m_{k,ij})^\top\nabla H(m_{k,ij}))\\
=&\sum_{k=1}^K |C_k|\sum_{i=1}^r\sum_{j=1}^{K_i}H(m_{k,ij}) -\sum_{k=1}^K \sum_{b_l \in C_k}\sum_{i=1}^r\sum_{j=1}^{K_i}H(b_{l,ij}).\\
\end{split}$$ The above equation holds due to $\sum_{b_l \in C_k}(b_{l,ij}-m_{k,ij})=0$, and the second term is a constant given the binary matrix $B$. According to Lemma \[lem:mk\], we finish the proof.
Theorem \[the:second\] uncovers the equivalent relationship between the second part in Eq. and K-means on the binary matrix $B$. By this means, some part of this complex problem can be efficiently solved by the simple K-means clustering with KL-divergence on each dimension.
Although Theorem \[the:second\] formulates the second part in Eq. into a K-means optimization problem on the binary matrix $B$, there still remains two challenges. (1) The first part in Eq. is difficult to formulate into a K-means objective function, and (2) Lemma \[lem:mk\] and Theorem \[the:second\] are conducted on $n-o$ inliers, rather than the whole matrix $B$. In the following, we focus on these two challenges, respectively.
The second part in Eq. can be solved by K-means clustering, which inspires us to make efforts in order to transform the complete problem into the K-means solution. Since $1-p_{k,ij}$ is difficult involved into the K-means clustering by Theorem \[the:second\], which means $1-p_{k,ij}$ cannot be modeled by the binary matrix $B$, here we aim to model it by introducing another binary matrix $\widetilde{B}=\{\widetilde{b}_l\}, 1\leq l\leq n$ as follows. $$\label{eq:binary2}
\begin{split}
\widetilde{b}_l&=(
\widetilde{b}_{l,1},\cdots,\widetilde{b}_{l,i},\cdots,\widetilde{b}_{l,r}),~\textrm{with} \\
\widetilde{b}_{l,i}&=(\widetilde{b}_{l,i1},\cdots,\widetilde{b}_{l,ij},\cdots,\widetilde{b}_{l,iKi}),~\textrm{and} \\
\widetilde{b}_{l,ij}&=\left\{
\begin{array}{ll}
0,&\textrm{if}~L_{\pi_i}(x_l)=j\\
1,&\textrm{otherwise}
\end{array}
\right..
\end{split}$$
From Eq. , $\widetilde{B}$ is also derived from $\Pi$. Compared with the binary matrix $B$ in Eq. , $\widetilde{B}$ can be regarded as the flip of $B$. In fact, $B$ and $\widetilde{B}$ are the 1-of-$K_i$ and ($K_i$-1)-of-$K_i$ codings of the original data, respectively. Based on $\widetilde{B}$, we can define $\widetilde{m}_{k}$ according to Eq. , then we have $\widetilde{m}_{k,ij}=1-{m}_{k,ij} = 1-p_{k,ij}$.
Based on the binary matrices $B$ and $\widetilde{B}$, we transform the problem in Eq. into a unified K-means optimization by the following theorem.
$X$: data matrix;\
$K, o, r$: number of clusters, outliers, basic partitions.\
$K$ clusters $C_1, \cdots C_K$ and outlier set $O$; Generate $r$ basic partitions from $X$; Build the binary matrices $B$ and $\widetilde{B}$ by Eq. &; Initialize $K$ centroids from $[B\ \widetilde{B}]$; Calculate the distance between each point in $[B\ \widetilde{B}]$ and its nearest centroid; Identify $o$ points with largest distance as outliers; Assign the rest $n-o$ points to their nearest centroids; Update the centroids by arithmetic mean; the objective value in Eq. remains unchanged.
\[the:one\] If K-means is conducted on $n-o$ inliers of the binary matrix $[B\ \widetilde{B}]$, we have $$\nonumber
\min_{\pi} \sum_{k=1}^K p_{k}HL(C_k)\Leftrightarrow \min \sum_{k=1}^K \sum_{b_l \in C_k} (f(b_l,m_{k}) + f(\widetilde{b}_l,\widetilde{m}_{k})),$$ where $m_{k}$, $\widetilde{m}_{k}$ are the $k$-th centroid by Eq. , and the distance function $f(b_{l},m_{k})=\sum_{i=1}^r\sum_{j=1}^{K_i} D_{\textrm{KL}}(b_{l,ij}||m_{k,ij})$, $f(\widetilde{b}_{l},\widetilde{m}_{k})=\sum_{i=1}^r\sum_{j=1}^{K_i} D_{\textrm{KL}}(\widetilde{b}_{l,ij}||\widetilde{m}_{k,ij})$, and $D_{\textrm{KL}}(\cdot || \cdot)$ is the KL-divergence.
The problem in Eq. cannot be solved via K-means on the binary matrix $B$. Nontrivially, we introduce the auxiliary binary matrix $\widetilde{B}$, a flip of $B$, in order to model $1-p_{k,ij}$. By this means, the complete problem can be formulated by K-means clustering on the concatenated binary matrix $[B\ \widetilde{B}]$ in Theorem \[the:one\]. The benefits not only lie in simplifying the problem with a neat mathematical formulation, but also inherit the efficiency from K-means, which is suitable for large-scale data clustering with outlier removal.
Theorem \[the:one\] completely solves the first challenge that the problem in Eq. with inliers with the auxiliary matrix $\widetilde{B}$. This makes a partial K-means solution into a complete K-means solution. In the following, we handle the second challenge, which conducts on the entire data points, rather than $n-o$ inliers.
In this paper, we consider the clustering with outlier removal, which simultaneously partitions the data and discovers outliers. That means the outlier detection and clustering are conducted in a unified framework. Since the centroids in K-means clustering are vulnerable to outliers, these outliers should not contribute to the centroids. Inspired by [@Chawla13SDM], the outliers are identified as the points with large distance to the nearest centroid. The major difference is that K-means[-]{}[-]{} is proposed on the original feature space, while our problem starts from the Holoentropy on the partition space, and we formulate the problem into a K-means optimization with the auxiliary matrix $\widetilde{B}$. After delicate transformation and derivation, is used as a tool to solve the problem in Eq. , which returns $K$ clusters $C_1, \cdots, C_K$ and outlier set $O$. The complete process of our proposed clustering with outlier removal is summarized in Algorithm \[alg\].
Next, we analyze the property of Algorithm \[alg\] in terms of time complexity and convergence. In Line-1, we first generate $r$ basic partitions, which are usually finished by K-means clustering with different cluster numbers. This step takes $\mathcal{O}(rt'\overline{K}nd)$, where $t'$ and $\overline{K}$ are the average iteration number and cluster number, respectively. Line 5-8 denotes the standard K-means[-]{}[-]{} algorithm, which has the similar time complexity $\mathcal{O}(tKnR)$, where $R = \sum_{i=1}^rK_i$ is the dimension of the binary matrix $B$ and $\widetilde{B}$. It is worthy to note that only $R$ elements are non-zero in $[B\ \widetilde{B}]$. In Line 6, we find $o$ points with largest distances, rather than sorting $n$ points so that it can be achieved with $\mathcal{O}(n)$. It is worthy to note that $r$ basic partitions can be generated via parallel computing, which dramatically decreases the execution time. Moreover, $t'$, $t$, $r$ and $R$ are relatively small compared with the number of points $n$. Therefore, the time complexity of our algorithm is roughly linear to the number of points, which easily scales up for big data clustering with outliers. Moreover, Algorithm \[alg\] is also guaranteed to converge to a local optimum by the following theorem.
\[the:coverge\] Algorithm \[alg\] converges to a local optimum.
The proof holds due to the good convergence of K-means[-]{}[-]{}.
Discussions
===========
In this section, we launch several discussions on clustering with outlier removal. Generally speaking, we elaborate it in terms of the traditional clustering, outlier detection and consensus clustering.
*Comparison with cluster analysis.* Traditional cluster analysis aims to separate a bunch of points into different groups that the points in the same cluster are similar to each other. Each point is assigned with a hard or soft label. Although robust clustering is put forward to alleviate the impact of outliers, each point including outliers are assigned the cluster label. Differently, the problem we address here, clustering with outlier removal only assigns the labels for inliers and discovers the outlier set. Technically speaking, our COR belongs to the non-exhaustive clustering, where not all data points are assigned labels and some data points might belong to multiple clusters. NEO-K-Means [@Whang15SDM] is one of the representative methods in this category. In fact, if we set the overlapping parameter to be zero in NEO-K-Means, it just degrades into K-means[-]{}[-]{}. Our COR is different from K-means[-]{}[-]{} in the feature space. The partition space not only naturally caters to the definition of outliers and Holoentropy, but also alleviates the spherical structure assumption of optimization.
*Comparison with outlier detection.* Outlier Detection is a hot research area, where tremendous efforts have been made to thrive this area from different aspects. Few of them simultaneously conduct cluster analysis and outlier detection. Except K-means[-]{}[-]{}, Langrangian Relaxation (LP) [@Ott14NIPS] formulates the clustering with outliers as an integer programming problem, which requires the cluster creation costs as the input parameter. LP not only suffers from huge algorithmic complexity, but also struggles to set this parameter in practical scenarios, which leads LP to return the infeasible solutions. That is the reason that we fail to report the performance of LP in the experimental part. To our best knowledge, we are the first to solve the outlier detection in the partition space, and simultaneously achieve clustering and outlier removal. Our algorithm COR starts from the objective function in terms of outlier detection, and solves the problem via clustering tool, where demonstrates the deep connection between outlier detection domain and cluster analysis area.
*Comparison with consensus clustering.* Consensus Clustering aims to fuse several basic partitions into an integrated one. In our previous work, we proposed K-means-based Consensus Clustering (KCC) [@Wu13IJCAI; @Wu15TKDE], which transforms the complex consensus clustering problem into a K-means solution with flexible KCC utility functions. Similarly, the input of our COR is also a set of basic partitions, and it delivers the partition with outliers via K-means[-]{}[-]{}. The partition space derived from basic partitions enables COR not only to identify outliers, but also to fuse basic partition to achieve consensus clustering. From this view, Holoentropy can be regarded as the utility function to measure the similarity between the basic partition in $B$ or $\widetilde{B}$ and the final one. For the centroid updating, the missing values in basic partitions within KCC framework provide no utility, further do not contribute the centroids. For COR, we can automatically identify the outliers, which do not participate into the centroid updating either.
Experimental Results
====================
In this section, we first introduce the experimental settings and data sets , then showcase the effectiveness of our proposed method compared with K-means and K-means[-]{}[-]{}. Moreover, a variety of outlier detection methods are involved as the competitive methods. Some key factors in COR are further analyzed for practical use. Finally, an application on flight trajectory is provided to demonstrate the effectiveness of COR in the real-world scenario.
Experimental Settings
---------------------
**Data sets.** To fully evaluate our COR algorithm, numerous data sets in different domains are employed. They include the gene expression data, image data, high-dimensional text data and other multivariate data. These data sets can be found from [@Liu15SDM; @Liu17DMKD] and UCI[^3]. Here we treat the class with smallest size as outliers. For *ecoil*, three smallest classes in the original datasets are regarded as the outliers. Table \[tab:dataset\] shows the numbers of instances, features, clusters and outliers of these data sets.
**Competitive Methods.** K-means and K-means[-]{}[-]{} are used for comparisons. For our COR algorithm, 100 basic partitions are generated via K-means by different cluster numbers from 2 to $2K$, then K-means[-]{}[-]{} is employed with the distance function in Eq. for the partition and outliers. Note that K-means[-]{}[-]{} and COR are fed with $K$ and $o$ for fair comparisons, which are true numbers of clusters and outliers, respectively. For K-means, we set the cluster number as the true number plus one, the cluster found by K-means with the smallest size is regarded as the outlier set. Codes of K-means, K-measn[-]{}[-]{} and COR are implemented by MATLAB. Each algorithm runs 20 times, and returns the average result and standard deviation. Moreover, several classical outlier detection methods including density-based LOF[@Breunig00SIR], COF[@Tang02PKDD], distance-based LODF [@Zhang09PKDD], angle-based FABOD [@Pham12KDD], ensemble-based iForest [@Liu08ICDM], eigenvector-based OPCA [@Lee13TKDE], cluster-based TONMF [@Kannan17SDM] are also involved as the competitive methods to evaluate the outlier detection performance[^4]. $o$ points with the largest scores by these methods are regarded as outliers. For the outlier detection methods, some default settings in the original papers are used for stable results. The number of nearest neighbors in LOF, COF, LODF and FABOD is set to 50; the sub-sampling size and the number of trees in iForest are 200 and 100; the forgetting number is set to 0.1 in OPCA; the rank and two parameters in TONMF are 10, 10 and 0.1, respectively.
Data set Type \#instance \#feature \#cluster \#outlier
----------- ------- ------------ ----------- ----------- -----------
*ecoli* Gene 336 7 5 9
*yeast* Gene 1484 8 4 185
*caltech* Image 1415 4096 4 67
*sun09* Image 3282 4096 3 50
*fbis* Text 2463 2000 10 332
*k1b* Text 2340 21839 5 60
*re0* Text 1504 2886 5 218
*re1* Text 1657 3758 6 527
*tr11* Text 414 6129 4 87
*tr23* Text 204 5832 3 32
*wap* Text 1560 8460 10 251
*glass* UCI 214 9 3 39
*shuttle* UCI 58000 9 3 244
*kddcup* UCI 494021 38 3 54499
: Characteristics of data sets
\[tab:dataset\]
\[tab:performance\]
**Validation metric.** Although the clustering with outlier removal is an unsupervised task, we can still apply the ground truth to evaluate the performance with label information. Since we focus on the jointly clustering and outlier detection, four metrics are employed to evaluate the performance in terms of cluster validity and outlier detection. The outlier set is regarded as a special cluster in the ground truth.
Normalized Mutual Information ($NMI$) and Normalized Rand Index ($R_n$) are two widely used external measurements for cluster validity [@Wu09KDD]. $NMI$ measures the mutual information between resulted cluster labels and ground truth labels, followed by a normalization operation, while $R_n$ measures the similarity between two partitions in a statistical way. They can be computed as follows. $$\nonumber
NMI = \frac{\sum_{i,j} n_{ij}\log \frac{n\cdot n_{ij}}{n_{i+}\cdot n_{+j}}}{\sqrt{(\sum_{i} n_{i+}\log \frac{n_{i+}}{n})(\sum_{j} n_{j+}\log \frac{n_{+j}}{n})}},$$ $$\nonumber
R_n = \frac{\sum_{i,j}\binom{n_{ij}}{2} -\sum_{i}\binom{n_{i+}}{2}\cdot \sum_{j} \binom{n_{+j}}{2}/\binom{n}{2}}{\sum_{i}\binom{n_{i+}}{2}/2 + \sum_{j} \binom{n_{+j}}{2}/2 -\sum_{i}\binom{n_{i+}}{2}\cdot \sum_{j} \binom{n_{+j}}{2}/\binom{n}{2}},$$ where $n_{ij}$, $n_{i+}$, $n_{+j}$ are the co-occurrence number and cluster size of $i$-th and $j$-th cluster in the obtained partition and ground truth, respectively.
Jaccard index and F-measure are designed for the binary classification, which are employed to evaluate the outlier detection. They can be computed as follows. $$\nonumber
Jaccard = \frac{|O\cap O^*|}{|O\cup O^*|},$$ $$\nonumber
F{-}measure = 2*\frac{\textrm{precition}\cdot \textrm{recall}}{\textrm{precition}+ \textrm{recall}},$$ where $O$ and $O^*$ are the outlier sets by the algorithm and ground truth, respectively, and F-measure is the harmonic average of the precision and recall for outlier class.
To evaluate the overall performance on all used data sets, we propose a score as follows. $$\nonumber
sorce(A_i)=\sum_{j}\frac{P(A_i,D_j)}{\max_{i}P(A_i,D_j)},$$ where $P(A_i,D_j)$ denotes the performance of algorithm $A_i$ on data set $D_j$ in terms of some metric.
Note that these four metrics and the score are positive measurements, i.e, a larger value means better performance. Although $R_n$ is normalized, it can still be negative, which means that the partition is even worse than random label assignment.
**Environment.** All experiments were run on a PC with an Intel Core [email protected] GHz and a 64 GB DDR3 RAM.
\[tab:detection\]
Algorithmic Performance
-----------------------
Here we evaluate the performance of COR by comparing with K-means[-]{}[-]{} and outlier detection methods. Table \[tab:performance\] shows the performance of clustering with outlier removal via K-means, K-means[-]{}[-]{} and COR. There are three obvious observations. (1) few outliers can easily destroy the whole cluster structure. This point can be verified from the fact that K-means delivers poor clustering results on *fbis*, *tr23* and *kddcup* in terms of NMI and Rn. Moreover, K-means fails to capture the outliers by simply increasing the cluster number. (2) K-means[-]{}[-]{} jointly learns the cluster structure and detects the outliers, which alleviates the negative impact of outliers on the clusters and achieves better performance over K-means on the average level. Although K-means[-]{}[-]{} slightly outperforms COR on *sun09* in terms of outlier detection, the cluster structure provided by K-means[-]{}[-]{} is much worse than COR, even K-means clustering. (3) COR exceeds K-means and K-means[-]{}[-]{} by a large margin in both cluster analysis and outlier detection. For example, COR gains more than 10%, 20% and 40% improvements by cluster validity over rivals on *caltech*, *fbis* and *tr11*, respectively. Moreover, COR also provides better outlier detection results. On *yeast* and *caltech*, there exists more than 30%, 50% gains over K-means[-]{}[-]{}; especially, on *k1b*, COR achieves 25.53 and 34.06 in terms of Jaccard and F-measure; however, K-means[-]{}[-]{} fails to detect any outliers. Recall that COR is in essence K-means[-]{}[-]{} on the binary matrix $[B\ \widetilde{B}]$. The huge improvements result from the partition space, where defines the concept of clusters and achieves the joint consensus clustering and outlier removal. From the score, COR significantly outperforms K-means and in terms of all four metrics. Since COR is conducted in the partition space, we also compare with K-means-based Consensus Clustering (KCC) [@Wu15TKDE] with the same basic partitions by adding one more cluster to capture the outliers. Due to the limited page, we report that on the average level, KCC delivers the competitive cluster results, where COR slightly outperforms KCC by 1.21% and 3.95% in terms of NMI and Rn. Unfortunately, KCC fails to detect any outliers on all the datasets.
[l|rrrrr]{} Method & *sun09* & *k1b* & *wap* & *shuttle* & *kddcup*\
K-means & 1.12 & 4.55 & 1.25 & 0.22 & 0.62\
LOF & 65.16 & 150.38 & 26.81 & 11.93 & N/A\
COF & 79.50 & 154.02 & 30.18 & 181.45 & N/A\
LDOF & 277.25 & 2638.97 & 903.43 & 246.87 & N/A\
FABOD & 567.47 & 5373.76 & 1811.28 & 495.43 & N/A\
iForest & 12.55 & 12.88 & 8.53 & 165.42 & 1455.41\
OPCA & 0.40 & 6.18 & 1.75 & 0.30 & 2.51\
TONMF&7.87 & 31.76 & 7.67 & 1.18 & 18.17\
K-means[-]{}[-]{} & 3.56 & 65.28 & 12.73 & 0.33 & 5.98\
BP & 52.86 & 121.95 & 36.58 & 5.09 & 5.55\
COR & 2.31 & 0.15 & 0.19 & 0.57 & 2.89\
\
\[tab:time\]
Beyond K-means and K-means[-]{}[-]{}, we also compare COR with several outlier detection methods. Table \[tab:detection\] shows the performance of outlier detection in terms of Jaccard and F-measure. These algorithms are based on different assumptions including density, distance, angle, ensemble, eigenvector and clusters, and sometimes effective on certain data set. For example, COF and iForest get the best performance on *shuttle* and *kddcup*, respectively. However, in the most cases, these competitors show the obvious disadvantages in terms of performance. The reasons are complicated, but the original space and unsupervised parameter setting might be two of them. For TONMF, there are three parameters as the inputs, which are difficult to set without any knowledge from domain experts. Differently, COR requires two straightforward parameters, and benefits from the partition space and joint clustering with outlier removal, which brings the extra gains on several data sets. On *shuttle* and *kddcup*, COR does not deliver the results as good as the outlier detection methods. In the next subsection, we further improve the performance of COR via different basic partition generation strategy.
Next we continue to evaluate these algorithms in terms of efficiency. Table \[tab:time\] shows the execution time of these methods on five large-scale or high-dimensional data sets. Generally speaking, the density-based, distance-based and angle-based methods become struggled on high-dimensional data sets, especially FABOD is the most time consuming method, while the cluster-based methods including TONMF, K-means[-]{}[-]{} are relatively fast. It is worthy to note that the density-based, distance-based and angle-based methods require to calculate the nearest neighbor matrix, which takes huge space complexity and fails to deliver results on large-scale data sets due to out-of-memory on a PC machine with 64G RAM. For COR, the time complexity is roughly linear to the number of instances; moreover, COR is conducted on the binary matrix, rather than the original feature space. Thus, COR is also suitable for high-dimensional data. On *k1b*, COR only takes 0.15 seconds, over 400 times faster than K-means[-]{}[-]{}. Admittedly, COR requires a set of basic partitions as the input, which takes the extra execution time. In Table \[tab:time\], we report the execution time of generating 100 basic partitions as well. This process can be further accelerated by parallel computing. Even taking the time of generating basic partition, COR is still much faster than the density-based, distance-based and angle-based outlier detection methods.
Factor Exploration
------------------
In this subsection, we provide further analyses on the factors inside COR, the number of basic partitions and the basic partition generation strategy.
In consensus clustering, the performance of clustering goes up with the increase of basic partitions [@Wu15TKDE; @Liu16KDD; @Liu17DMKD]. Similarly, we test COR with different numbers of basic partitions. Figure \[fig:bp\] shows the boxplot of the performance of COR with 10, 30, 50, 70 and 90 basic partitions on *caltech* and *fbis* in terms of NMI and Jaccard. For a certain number of basic partitions, we generate 100 sets of basic partitions and run COR for the boxplot. From Figure \[fig:bp\], we have that COR delivers high quality partitions even with 10 basic partitions, and that for outlier detection, the performance slightly increases with more basic partitions and stabilizes in a small region. Generally speaking, 30 basic partitions are enough for COR to deliver a good result.
So far, we employ the Random Parameter Selection (RPS) strategy to generate basic partitions, which employs K-means clustering with different cluster numbers. In fact, Random Feature Selection (RFS) is another widely strategy to generation basic partitions, which randomly selects partial features for K-means clustering. In the following, we evaluate the performance of COR with RFS. Here we set the random feature selection ratio to be 50% for 100 basic partitions. Figure \[fig:rfs\] shows the performance of COR with different basic partition generation strategies on *shuttle* and *kddcup*. RFS achieves some improvements over RPS on these two data sets with different metrics, except on *shuttle* in terms of Rn. This indicates that RFS is helpful to alleviate the negative impact of noisy features, and further produces high quality basic partitions for COR. It is worthy to note that COR with RFS on *kddcup* achieves 21.18 and 34.95 in terms of Jaccard and F-measure, which exceeds the one with RPS over 5% and 7%, and competes with iForest. This means that COR with RFS gets the competitive performance with the best rival on *kddcup*, and it is over 170 times faster than iForest.
Application on Trajectory Detection
-----------------------------------
Finally, we evaluate our COR in the real-world application on outlier trajectory detection. The data come from Flight Tracker[^5], including flightID, flightNum, timestamp, latitude, longitude, height, departure airport, arrival airport and other information. We employ the API to request the flight trajectory every 5 minutes, and collect one-year data from October, 2016 to September, 2017 all over the world. After the data processing, we organize the data with each row representing one flight with evolutional latitude and longitude. Since these flights have different lengths of records, we uniformly sample 10 records for each flight, where only the latitude and longitude are used as features. Therefore, each flight is processed in a 20-length vector for further analysis. Here we select the Chinese flights between Beijing (PEK), Shanghai (PVG), Chengdu (CTU) and Guangzhou (CAN), and US flights between Seattle (SEA), San Francisco (SFO) and Atlanta (ATL) for further analysis. Figure \[fig:f-chinese\] & \[fig:f-us\] show the trajectories of Chinese and US flights. By this means, we have the Chinese and US flight trajectory data sets with 85,990 and 33,648 flights, respectively.
Then COR is applied on these two data sets for outlier trajectory detection. Here we set the cluster numbers to be 6 and 3 for these two data sets, and the outlier numbers are both 200. Figure \[fig:o-chinese\] & \[fig:o-us\] show the outlier trajectories in these two data sets. There are two kinds of outliers. The first category includes the outliers with extra ranges. Although we focus on 7 airports in China and US, there are some trajectories out of the scope of these airport locations in terms of latitude and longitude. The transmission error and loss lead to that the trajectories of different flights are mixed together. In such cases, the system stores a non-existence trajectory. The second category has the partial trajectories. The flight location is not captured due to the failure of the sensors. These two kinds of outliers detected by COR are advantageous to further analyze the problems in trajectory system, which demonstrates the effectiveness of COR in the real-world application.
Conclusion
==========
In this paper, we considered the joint clustering and outlier detection problem and proposed the algorithm COR. Different from the existing K-means[-]{}[-]{}, we first transformed the original feature space into the partition space according to the relationship between outliers and clusters. Then we provided the objective function based on the Holoentropy, which was partially solved by K-means optimization. Nontrivally, an auxiliary binary matrix was designed so that COR completely solved the challenging problem via K-means[-]{}[-]{} on the concatenated binary matrices. Extensive experimental results demonstrated the effectiveness and efficiency of COR significantly over the rivals including K-means[-]{}[-]{} and other state-of-the-art outlier detection methods in terms of cluster validity and outlier detection.
Acknowledgment {#acknowledgment .unnumbered}
==============
This research is supported in part by the NSF IIS Award 1651902 and U.S. Army Research Office Award W911NF-17-1-0367.
[Hongfu Liu]{} received his bachelor and master degree in Management Information Systems from the School of Economics and Management, Beihang University, in 2011 and 2014 respectively. He received the Ph.D. degree in computer engineering from Northeastern University, Boston MA, 2018. Currently he is a tenure-track Assistant Professor affiliated with Michtom School of Computer Science at Brandeis University. His research interests generally focus on data mining and machine learning, with special interests in ensemble learning. He has served as the reviewers for many IEEE Transactions journals including TKDE, TNNLS, TIP, and TBD. He has also served on the program committee for the conferences including AAAI, IJCAI, and NIPS. He is the Associate Editor of IEEE Computational Intelligence Magazine.
[Jun Li]{} (M’16) received the B.A. in Applied Mathematics from Pan Zhi Hua University in 2006. He received the M.S. in Computer Application from China West Normal University in 2009 and the PhD degree in pattern recognition and intelligence systems from the Nanjing University of Science and Technology in 2015. From Oct. 2012 to July 2013, he was a visiting student at Department of Statistics, Rutgers University, Piscataway, NJ, USA. From Dec. 2015 to Oct. 2018, he was a postdoctoral associate with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA. He is currently a postdoctoral associate with the Institute of Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA, USA. He is the Associate Editor of IEEE Access. His current research interests include deep learning, reinforcement learning, sparse representations, subspace clustering and recurrent neural networks.
[Yue Wu]{} received the BS and MS degree in Beijing University of Posts and Telecommunications at 2013 and 2016. He is currently a PhD student at Northeastern University. His current research interests are face recognition, object detection and deep learning.
[Yun Fu]{} (S’07-M’08-SM’11-F’19) received the B.Eng. degree in information engineering and the M.Eng. degree in pattern recognition and intelli- gence systems from Xian Jiaotong University, China, respectively, and the M.S. degree in statistics and the Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana- Champaign, respectively. He is an interdisciplinary faculty member affiliated with College of Engineering and the College of Computer and Information Science at Northeastern University since 2012. His research interests are Machine Learning, Computational Intelligence, Big Data Mining, Computer Vision, Pattern Recognition, and Cyber-Physical Systems. He has extensive publications in leading journals, books/book chapters and international conferences/workshops. He serves as associate editor, chairs, PC member and reviewer of many top journals and international conferences/workshops. He received seven Prestigious Young Investigator Awards from NAE, ONR, ARO, IEEE, INNS, UIUC, Grainger Foundation; nine Best Paper Awards from IEEE, IAPR, SPIE, SIAM; many major Industrial Research Awards from Google, Samsung, and Adobe, etc. He is currently an Associate Editor of the IEEE Transactions on Neural Networks and Leaning Systems (TNNLS). He is fellow of IEEE, IAPR, OSA and SPIE, a Lifetime Distinguished Member of ACM, Lifetime Member of AAAI, and Institute of Mathematical Statistics, member of AAAS, ACM Future of Computing Academy, Global Young Academy (GYA), INNS and Beckman Graduate Fellow during 2007-2008.
[^1]: Manuscript received XXX; revised XXX.
[^2]: The concept of robust clustering means that the partition is robust to outliers, rather than noisy features.
[^3]: <https://archive.ics.uci.edu/ml/datasets.html>
[^4]: The codes of outlier detection methods can be found at <https://github.com/dsmi-lab-ntust/AnomalyDetectionToolbox> and <https://github.com/ramkikannan/outliernmf>.
[^5]: <https://www.flightradar24.com>.
|
---
abstract: |
Cloud computing is becoming an almost ubiquitous part of the computing landscape. For many companies today, moving their entire infrastructure and workloads to the cloud reduces complexity, time to deployment, and saves money. Spot Instances, a subset of Amazon’s cloud computing infrastructure (EC2), expands on this. They allow a user to bid on spare compute capacity in Amazon’s data centres at heavily discounted prices. If demand was ever to increase such that the user’s maximum bid is exceeded, their instance is terminated.
In this paper, we conduct one of the first detailed analyses of how location affects the overall cost of deployment of a spot instance. We analyse pricing data across all available Amazon Web Services regions for 60 days for a variety of spot instance types. We relate the data we find to the overall AWS region as well as to the Availability Zone within that region.
We conclude that location does play a critical role in spot instance pricing and also that pricing differs depending on the granularity of that location - from a more coarse-grained AWS region to a more fine-grained Availability Zone within a region. We relate the pricing differences we find to the price’s reliability, confirming whether one can be confident in the prices reported and subsequently, in the ensuing bids one makes.
We conclude by showing that it is possible to run workloads on Spot Instances achieving both a very low risk of termination as well as paying very low amounts per hour.
author:
- |
Nnamdi Ekwe-Ekwe and Adam Barker\
School of Computer Science, University of St Andrews, UK\
Email: `{nnee, adam.barker}@st-andrews.ac.uk`
bibliography:
- 'location.bib'
title: 'Location, Location, Location: Exploring Amazon EC2 Spot Instance Pricing Across Geographical Regions - Extended Version '
---
INTRODUCTION
============
Amazon EC2 allows developers to provision compute resources, configure them to their needs, as well as scale resources up or down depending on application requirements. EC2 is widely used by organisations and developers as it makes it easy for them to provision compute resources on-demand without having to invest considerable time and money into building the underlying infrastructure needed in a data center.
With EC2, developers have three major ways to pay what they provision. Developers can pay “On-Demand" which allows the developer to pay for what they use by the hour with no “long-term commitments" [@amazonec2pricingwebsite]. Developers can also pay via a “Reserved Instance" model allowing them to commit to using a set of instances for either a “1 or 3 year term" [@amazonec2pricingwebsite]. Amazon claims that this model provides a “significant discount (up to 75%) as compared to On-Demand pricing" [@amazonec2pricingwebsite]. Finally, developers can pay for resources using a “Spot Instance" model [@amazonec2pricingwebsite]. The “Spot Instance" model allows developers to bid on spare compute capacity giving them savings of up to “90% off the On-Demand price" [@amazonec2pricingwebsite].
Spot Instances are becoming increasingly useful for certain use cases. For example, users that suddenly have “urgent computing needs for large amounts of additional capacity" [@amazonec2pricingwebsite] would find Spot Instances very useful. The user has access to both a sizeable amount of compute resources *and* at very low prices. The only caveat with this approach, however, is that the application that the user is running must be fault-tolerant due to the potential of an instance being terminated at any time.
The majority of research into Spot Instances has focused on modelling their pricing strategy as well as determining the best bid prices to make on an instance [@artur-andrzejak; @ben-yehuda; @liang-zheng]. Papers such as [@artur-andrzejak; @liang-zheng] have gone further - looking at how to find the lowest possible price for a user to spend, whilst maintaining the highest availability possible for the instance.
There is a distinct lack of research, however, which considers *where* (in terms of region and availability zone) a user can *reliably* deploy a spot instance in order to *minimise cost*. In order to address this gap, we make a number of core research contributions:
- We conduct one of the first detailed analyses of how location affects the overall cost of deployment of a spot instance.
- We analyse pricing data across all available Amazon Web Services regions for a variety of spot instance types. We relate the data we find to the overall AWS region as well as to the Availability Zone (AZ) within that region.
- For any pricing differences we find, we check whether those differences are *reliable* and as a result whether we can be confident in the ensuing bids we make.
We conclude that, 1) granularity of location has a significant impact on the overall price/price reliability of that instance; 2) the power of the instance type does not have as direct of an impact on price; some regions are universally substantially cheaper for certain instance types than others, while others are substantially more expensive.
The rest of this paper is structured as follows. Section II of this paper will examine what data we obtained from Amazon and its underlying structure. Section III will focus on the types of analyses we ran in order to explore the data. Section IV will give the results of our analyses. Section V will perform cross comparisons on our results to glean any interesting insights. Section VI will discuss related work. Finally Section VII will summarise and give our overall conclusions as well as discuss our future direction and work going forward.
EC2 SPOT PRICING DATA
=====================
Amazon provides spot price data to any Amazon user for a period of up to 90 days from when a request is made. This is done via its API endpoint [@amazonspotinstanceapiwebsite]. This price data comes in JSON format in the form of a price point per time period.
For example:
{
"Timestamp": "2017-06-25T00:20:56.000Z",
"ProductDescription": "Linux/UNIX",
"InstanceType": "d2.2xlarge",
"SpotPrice": "0.177800",
"AvailabilityZone": "eu-west-2a"
}
In the above example, the instance type (*d2.2xlarge*) was \$0.177800 (per hour) at 00:20:56 on the 25th of June 2017. The instance type was a Linux/UNIX instance - shown in the “ProductDescription" key. Finally, the Availability Zone in the EC2 region is *eu-west-2a*. We do not conduct any research related to the “ProductDescription" and so omit this column before we begin our analysis.
We used a data timeline of 60 days for our analysis. At time of carrying out this research, there were issues with retrieving the full 90 days worth of data from some zones. As a result, we worked with only 60 days worth of information. Additionally, not all Amazon regions provide a Spot Instance capability so our dataset comprised of data from only 4 AWS Regions - the EU, US, Asia Pacific and Canada Regions. These Amazon regions comprised of what, for the purposes of this paper, we’ll call *sub-regions*: EU - (*eu-central-1, eu-west-1, eu-west-2*), US (*us-east-1, us-east-2, us-west-1, us-west-2*), AP (*ap-southeast-1, ap-southeast-2*) and finally, Canada (*ca-central-1*). At time of running the analysis, there was a problem retrieving data from the *ap-northeast-1* *sub-region* and so those results have been omitted from the dataset.
Instance Type Mean$\pm$Standard Deviation(\$p/h)
--------------- ------------------------------------ -- --
*c3.large* 0.077$\pm$0.077
*c4.large* 0.068$\pm$0.104
*i3.large* 0.181$\pm$0.496
*m3.large* 0.078$\pm$0.088
*m3.medium* 0.063$\pm$0.051
*m4.large* 0.052$\pm$0.054
*r3.large* 0.081$\pm$0.082
*r4.large* 0.080$\pm$0.055
: Mean and Standard Deviation results of all instance types in CA over 60 days (to 3dp)[]{data-label="table:4"}
Instance Type Mean$\pm$Standard Deviation(\$p/h)
--------------- ------------------------------------ -- --
*c3.large* 0.074$\pm$0.104
*c4.large* 0.079$\pm$0.107
*i3.large* 0.081$\pm$0.115
*m3.large* 0.088$\pm$0.102
*m3.medium* 0.063$\pm$0.055
*m4.large* 0.084$\pm$0.118
*r3.large* 0.081$\pm$0.066
*r4.large* 0.090$\pm$0.127
: Mean and Standard Deviation results of all instance types in CA over 60 days (to 3dp)[]{data-label="table:4"}
Instance Type Mean$\pm$Standard Deviation(\$p/h)
--------------- ------------------------------------ -- --
*c3.large* 0.076$\pm$0.050
*c4.large* 0.096$\pm$0.139
*i3.large* 0.087$\pm$0.167
*m3.large* 0.085$\pm$0.056
*m3.medium* 0.063$\pm$0.047
*m4.large* 0.089$\pm$0.067
*r3.large* 0.088$\pm$0.058
*r4.large* 0.076$\pm$0.049
: Mean and Standard Deviation results of all instance types in CA over 60 days (to 3dp)[]{data-label="table:4"}
Instance Type Mean$\pm$Standard Deviation(\$p/h)
--------------- ------------------------------------ -- --
*c4.large* 0.043$\pm$0.040
*i3.large* 0.072$\pm$0.045
*m4.large* 0.027$\pm$0.032
*r4.large* 0.029$\pm$0.033
: Mean and Standard Deviation results of all instance types in CA over 60 days (to 3dp)[]{data-label="table:4"}
There are 68 different instance types that can be launched on EC2. If we had obtained the data for all 68, this would have led to an extremely large dataset. We therefore decided to restrict the dataset based on the most popular instance types available: small, medium and large instance types [@channelfutureswebsite]. Small instance types are not possible to deploy as a Spot Instance request so our final dataset consisted of medium and large instances. We then randomly picked a variety of instances (of varying compute power) from each of the AWS EC2 instance type categories - General Purpose, Compute Optimised, Memory Optimised and Storage Optimised [@amazonawsinstancetypes].
Our final selection of data included *m3.medium*, *m4.large*, *c4.large*, *c3.large*, *r3.large*, *r4.large* and *i3.large* instance types. The total pricing data over the 60 days for all the above instances was 4,112,000 data points.
ANALYSES
========
We focused on exploring the pricing data in relation to i) the *instance type* and ii) the *availability zone*.
We ran three main analyses.
- First, we ran an average price analysis of all instance types within an AWS region over the full 60 days.
- Second, we ran an average price analysis for each instance in our list of instance types for every AWS region.
- Finally, we ran a histogram analysis plotting the frequency of all price points in a particular AZ for each of the instances deployed in that zone.
We broke down the first two analyses by day of the week and by hour in the day - showing the mean price across both these time metrics. We also calculated the standard deviation and related the average price analysis to these in order to examine the price volatility. In this context, volatility means the propensity for the price to change across a time metric.
The lower the standard deviation, the lower the volatility. This meant that the data was closer to the mean and that we could be more confident in the mean price metrics reported.
Conversely, the higher the standard deviation, the less confident we could be in the price’s reliability.
In the hour in the day analysis, we divided the day into 5 portions (00:00 to 05:00, 05:00-10:00, 10:00-15:00, 15:00-20:00 and 20:00-23:59). On the weekly analysis, we divided the week into 7 portions with 0 standing for Monday and 6 for Sunday.
RESULTS
=======
Average Price Analysis of All Instance Types within an AWS region
-----------------------------------------------------------------
### EU
At first glance, the average pricing for the instances seem to be mostly concentrated around the \$0.075 to \$0.080 mark. We can clearly see that *i3.large* instances are very expensive in comparison to the other instances with *m4.large* instances looking like the cheapest overall across the day.
When we related these prices to their standard deviations (Table \[table:1\]), we noticed some interesting metrics. In general across the EU, prices were very volatile (high levels of standard deviation). This was evident in some instance types more than in others - across the EU, *i3.large*, **c4.large** and *m3.large* instances had the three highest levels of standard deviation in the dataset. The three instances that showed the most reliable and stable pricing were the *r4.large*, *m4.large* and *m3.medium* instances (lowest levels of standard deviation). *m3.medium* instances in fact, showed the most consistent and reliable pricing throughout the day, followed by *r4.large* and *m4.large* instances.
Already, we can begin to see the impact location has on spot instance pricing - *r4.large* instances are some of the most powerful within the AWS offering - however it displays pricing similar to two of the least powerful instances in the dataset. This hourly breakdown offers us a very helpful view of pricing - we can see which hours of the day are cheaper to deploy an instance than in others. For example, we can see that at the beginning of the day (around midnight to 5am and at the tail end of the day - 8pm to midnight) we get some of the cheapest pricing available for *m4.large* instances. *m3.medium* instances show relatively consistent pricing throughout the day, a trend also similarly seen in *r4.large* instances types.
We also performed average pricing analysis over the week - giving us a higher level overview of price differences that we might not necessarily see on a daily basis. As seen in Figure \[fig:eu-all-instances-dow\], for *r4.large* and *m4.large* instances, the end of the week seems to be the best time to use them - offering the user the cheapest possible pricing. This, coupled with a low standard deviation allows us to conclude that these are reliable price metrics. *m3.medium* instances’ pricing were also very consistent throughout the week. *i3.large* instances were clearly more expensive than all the other instances throughout the week. This, coupled with a very high standard deviation (also seen in **c4.large** and *m3.large* instance types) shows the user that the EU is not the best place to deploy such instance types.
### US
In the United States, we see significant and pronounced price volatility across nearly all the instance types. We see both very high levels of standard deviations (Table \[table:2\]) as well as high average price metrics throughout the day and throughout the week. Only two instance types (as seen in Figure \[fig:us-all-instances-per-hour\]) display significantly low levels of standard deviations throughout the entire time period - *m3.medium* and *r3.large* instances. *m4.large* instance types display the most volatile pricing in the region, closely followed by *i3.large* and *m3.large* instance types. We see a few interesting things here. First, the type of instance clearly does not have as large an impact on price as location does. *m3* instance types are the least powerful in our dataset, yet we’re seeing both very high and volatile pricing across the board. Conversely, we see reliable and low price metrics reported for *r3.large* instances, some of the most powerful in our dataset. We also see this replicated again in the week.
### Asia-Pacific
The Asia-Pacific region, in comparison to the EU and US regions show both low price points coupled with low standard deviations (Table \[table:3\]). We see very low and reliable pricing in the *m3.medium*, *r4.large*, **c3.large** instances (Figure \[fig:ap-all-instances-per-hour\]). We again see the same pattern we saw earlier - more powerful instance types appearing at very low and reliable price points. In general, the more powerful the instance type, the more favourable the pricing in the Asia-Pacific region. *i3.large* instances once again display the most volatile pricing across the board, but for the most part, our previous conclusion holds. When we further examined some of the price metrics, we saw a sizeable difference in pricing with the Asia-Pacific region and the EU and US: *r3.large* and *r4.large* instances have max prices of \$0.1934 and \$0.14 per hour. In the EU, we see max prices 17x and 18x that for *r3.large* and *r4.large* instance types respectively, and 15x that in the US for both instance types. Across the week (Figure \[fig:ap-all-instances-dow\]), we see also see these low prices reflected.
### Canada
The Canada region hosts the least amount of instance types from our original selection - only the more powerful **c4.large**, *m4.large*, *i3.large* and *r4.large* instance types are present.
From these graphs (Figure \[fig:ca-all-instances-per-hour\] and Figure \[fig:ca-all-instances-per-dow\]) the Canada region exhibits some of the cheapest pricing on average per hour and across the week for the different instance types. For the *i3.large* instance type, we see mean prices at around \$0.07 per hour for the instance with the lowest standard deviations recorded across all four regions at \$0.045 (Table \[table:4\]). In the Asia Pacific region, mean prices are at \$0.087 with standard deviations of \$0.167. In the US, \$0.081 with \$0.115 as the standard deviation and in the EU, \$0.181 with a standard deviation of \$0.496. From this we can clearly see that it would be much cheaper to deploy an *i3.large* instance in the Canada region than in any of the others. We can also apply the above analysis to all the other instance types.
We can therefore conclude, that the more powerful the instance type, the cheaper and more reliable it is to deploy in Canada.
Average Price Analysis of Each Instance for every AWS region
------------------------------------------------------------
This analysis plotted every single instance type’s pricing for every single AWS region. This allowed us to perform cross-regional comparisons and offered another view of the price data we had gathered. For some of the regions, some instance types were not deployed and as a result do not appear on some graphs.
We see some very clear patterns straight away. In any instance type that is deployed in Canada, we see both its lowest and most reliable price metrics. If we take a look at Figures \[fig:each-instance-every-region-per-hour-average-pricing\] and \[fig:each-instance-every-region-per-dow-average-pricing\], we see this very clearly. This also serves to reinforces the conclusions we came to earlier. The less powerful the instance, the more reliable it is to deploy in the EU or US - in the “m” instance types we generally see cheaper pricing in those two regions.
For example, if one was to deploy an *m4.large* instance and had the choice between the AP region and the EU, one would obtain much more reliable pricing in the EU with a lower mean price and standard deviation than in AP. There are however, some anomalies - for example *m4.large* instance types being amongst the cheapest to run in the EU but the most expensive in the US. What one can see however, is that with these different price metrics we can begin to make informed decisions about where to deploy an instance. If (for instance) there were certain constraints that meant we could only deploy in a specific region over another, we can use this analysis to make the best choice for a particular use case.
Spot Price Histogram Price Frequency Analysis
---------------------------------------------
### EU Region Analysis
We can see that *m3.medium* instances are generally the cheapest to use amongst the availability zones (AZs) in which they have been deployed. However, in some AZs we do not see as high a frequency of data for *m3.medium* instances as some of the others. Note that the *eu-west-2a* AZ did not record any data for the *m3.medium* instances.
In the *eu-central-1a* AZ (Figure \[fig:a1\]), it was generally cheaper to deploy on a *c4.large* instance than it was on an *m3.large* instance. This is particularly interesting as *c4* instances are “optimized for compute-intensive workloads" [@amazonawsinstancetypes]. c4 instances are also more powerful than both *m3* and *m4* instances. This is a trend that was only really noticeable in the *eu-central-1a* AZ due to the amount of pricing data generated for **c4.large** instances. Looking across the other AZs however, **c4.large** instances had in several cases much cheaper pricing per hour than the *m3* and *m4* instance types.
Towards the other end of the scale, *r3* and *r4* instance types displayed some unexpected pricing trends. *r3* and *r4* instance types are amongst the most powerful on AWS and are used to run “high performance databases...distributed web scale in-memory caches" [@amazonawsinstancetypes] etc. However, in the *eu-west-1b* AZ (Figure \[fig:a4\]) we see *r4.large* instances displaying modal prices of around \$0.02 p/hour with a frequency of over 4000 and then of around \$0.12 p/hour (with a frequency of just over 3000). *r3.large* instances also generally fall into these above two “price brackets" — (Figure \[fig:a3\] - *eu-west-1a* and Figure \[fig:a5\] - *eu-west-1c* AZs).
This is very interesting as it shows that 1) location of deployment is extremely important (specifically AZ) and 2) all these analyses must be taken together in context to find the best deployment area. A possible scenario - let’s say a developer wanted to run a workload on a “r” instance type but data protection rules limited them to running this workload in Europe. From this analysis, the developer could run this workload quite comfortably in the *eu-west-1b* region and get prices comparable to Canada or Asia Pacific regions.
### US Region Analysis
*r4.large* instance types appear regularly on the left most side of the chart (i.e. the least expensive). In the *us-west-2b* AZ (Figure \[fig:a15\]), we see that *r4.large* instances are cheaper than *m4.large* instances with prices at around \$0.02 per hour even though *r4.large* instances are a lot more powerful. We see this again in the *us-east-1c* AZ (Figure \[fig:a10\]) with the mode pricing being around \$0.02-\$0.03 per hour as well as \$0.015-\$0.125.
*m4.large* instances follow their own distinct pattern across all the different AZs. Their two highest (modal) pricing points are either \$0.025 or \$0.125 per hour. We can see this most clearly in the *us-west-2a*, *2b* and *2c* AZs. In the *us-west-2a* AZ (Figure \[fig:a14\]) the two modal price points occur 6000 times with the next closest price point occurring around 1800 times.
In all AZs where they’ve been deployed, *m3.medium* instance types are consistently the cheapest instance type to use.
### AP Region Analysis
*c3.large* instances show modal prices of either \$0.025 or \$0.125 an hour. The *c4.large* instance types generally fell within 3 modal pricing points in the *ap-southeast-2* regions - either \$0.03, \$0.10 or \$0.14 an hour.
The Asia Pacific region appears to be more popular with larger and more powerful instance types as that is where most of the data points are recorded for. Smaller and more General Purpose instance types have much smaller data frequencies – although where they do occur, the data points generally show cheap pricing. From this graph, we can clearly see advantages with deploying more powerful instance types (mainly c types) in this region than in the EU or US.
### Canada Region Analysis
m4.large instance types are the cheapest instances to use in both the *ca-central-1a* and *ca-central-1b* AZs. In general prices across the zone are very inexpensive with prices not going past \$0.11 an hour mark. They also occur with high frequency. If one was deploying one of these 4 *.large* instances, one would deploy them here than any of the other regions.
Also, this graph has very few data points meaning that the region is not widely used — a factor one could take advantage of to eliminate application “hotspots" as well as to get the lowest prices available.
CROSS-COMPARISONS
=================
Average Price Analysis
----------------------
The average price analysis over the day and week show us some interesting insights. We can see clear price differences in instance types as we go from region to region. In the Asia-Pacific and Canada regions we generally see much lower and more reliable pricing than in the EU and US regions. However, this is also instance type dependent. Overall, *m3.medium* instance types consistently show the most reliable pricing across all of the regions they are deployed in while *i3.large* instances show the most volatile. An interesting point to note is that *m3.medium* instances’ mean prices are around the same point across all regions in which they are deployed - \$0.063 per hour.
We can use this analysis to examine which regions offer best value for the instance types we require. If one was to deploy an *i3.large* instance type, one of the most powerful in our list of instance types, we’d would be looking at highly unstable pricing in the EU, US and Asia Pacific regions. However, in Canada, we’d would be obtaining very cheap and reliable pricing per hour.
If one wanted to use a quite powerful *c3.large* instance, it would be much cheaper to deploy in the Asia Pacific region with a mean price of \$0.0764 per hour, a low standard deviation of 0.05, and max price of \$0.364 than in the EU or US. If one wanted to deploy a less powerful *m4.large* instance, however, one would much rather deploy it in the EU where the prices are on average \$0.052 p/hour with a standard deviation of 0.054 in comparison to \$0.089 p/hour and a standard deviation of 0.067 in the Asia Pacific region.
Based on all these analyses, there is a clear pattern identified. Overall, the more powerful the instance, the cheaper and more reliable the pricing one would get in Canada than in any of the other three regions. This is closely followed by the Asia Pacific region. On the other hand, the less powerful the instance the cheaper and more reliable (in terms of pricing) it is to deploy in the EU and US regions.
Spot Price Histograms
---------------------
The Spot Price Histograms show us the frequency of prices for each instance type across the AZ. In the EU and US regions, we see a pattern - the “m" instances are generally the cheapest to run and appear in high frequencies towards the leftmost side of the graph.
However, in some AZs within these regions, we see that the more powerful instance types are among the cheapest to run - *r4.large* instances being an example. In the Asia Pacific and Canada regions, the *.large* instances present a double advantage of lower demand and lower prices.
These histograms can allow a developer to take a range of price points for a particular instance. A developer can then utilise the likelihood (modality) of those price points occurring and make a confident max bid price.
RELATED WORK
============
There has been a significant body of research done into Amazon EC2 Spot Instances. However, most of the research done has been mainly focused on trying to determine how the pricing for the instances is generated. Bodies of work such as [@artur-andrzejak; @ben-yehuda; @liang-zheng] look at modelling the pricing strategy for Spot Instances as well as determining the best bid prices to make on an instance. Papers [@artur-andrzejak; @liang-zheng] try to find the lowest price possible for a user to spend, whilst maintaining the highest availability possible for the instance.
Other bodies of research, more related to this paper, provide more meaningful insights. Wang et al., in [@pennsylvania-state-university-website] followed a more empirical approach to analysing Spot Instances, relating it to finding a more cost-effective procurement of compute resources for customers. They identified four key features a potential tenant of a Spot Instance should examine, “lifetime of an instance average spot price during lifetime simultaneous revocations, and startup delay", created a quantitative model, and then used this model to develop “computationally-efficient predictors" of pricing [@pennsylvania-state-university-website].
Their conclusions revealed that “the evolution of spot price depended on a spatially coarse (e.g., data center or availability zone wide) load metric" [@pennsylvania-state-university-website]. Their conclusions tie in fully with our own that the data center region or AZ had a significant impact on price. Approaching Spot Instances from an alternate angle, Li et al., recently in [@li-zheng] performed a comparative investigation between Spot Instances and fixed-price instances using a Systematic Literature Review. They highlighted the fact that cloud consumers were concerned at “unpredictably frequent interruptions when using spot services" [@li-zheng] - something that was putting them off utilising the spot market fully. The “academic community strongly advocate\[d\] the Cloud spot market" [@li-zheng] but that that was still not being wholly reflected by consumers.
In recent times, however, this has started to change. Spotinst [@spotinstwebsite], a cloud company, launched in 2015 to “Reduce 50-80%” of cloud computing costs for companies on EC2 via the use of Spot Instances. It does this by utilising predictive algorithms to “to predict Spot behaviour, capacity trends, pricing, and interruption rate” [@spotinstelastigroup], rebalancing whenever there is risk of an interruption, and falling back onto on-demand instances when Spot Instances are not available to use.
CONCLUSIONS
===========
In this paper, we have explored and gleaned fascinating insights from spot price data. We have seen the different dynamics that all play a role in affecting spot prices, including location (both on a coarse-grained regional level and on a finer grained availability zone level), type of instance, time of day, etc. We have analysed pricing volatility, determining when’s best to deploy an instance and whether we can be confident in the reliability of the average price point shown (via the standard deviation). We have also analysed modal price points for instance types in different AZs, giving us another viewpoint of spot price — and allowing us to make confident predictions on potential bid prices.
We have shown how important context is in interpreting these analyses and that taken together, they paint a revealing picture about where best to deploy instances of varying power for the lowest price.
The general conclusions we can therefore draw are:
I. The more powerful the instance, the cheaper it is in general to deploy in the Asia Pacific or Canada regions - the less powerful the instance, the EU and US regions.
II. The Canada and Asia Pacific regions have a lot less demand, making it more attractive for developers (both from a pricing point of view) but also as a means of eliminating potential application “hotspots".
III. AZs have a significant impact on price - it is possible to deploy very powerful instances in a cheaper AZ located in an overall more expensive region. Figure \[fig:a4\] revealed to us how in the *eu-west-1b* AZ, we could deploy a very powerful *r4.large* instance for a very low price (\$0.02 and \$0.12 an hour).
IV. There are some exceptions to these general conclusions — one example is in the US where *m4.large* instances are amongst the most expensive to deploy while not being that powerful an instance type.
V. Location plays a significant part in determining spot instance price, more than the instance type itself.
To conclude, spot Instances are very attractive as a means of using powerful compute power at very low prices. If a developer plans exactly where to deploy, examines all the pricing data available, and takes all the analyses performed together and in context, it is entirely possible to run workloads on Spot Instances at a very low risk of instance termination. Future work includes further analyses over longer periods of time, and building spot-price aware scheduling algorithms.
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